Differential Equations with Discontinuous Righthand Sides
Mathematics and Its Applications (Soviet Series)
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Differential Equations with Discontinuous Righthand Sides
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL Centre/or Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board: A. A. KIRILLOV, MGU, Moscow, U.SSR. Yu.1. MANIN. Steklov Institute o/Mathematics, Moscow, U.S.SR. N. N. MOISEEV. Computing Centre, Academy o/Sciences, Moscow, USS.R. S. P. NOVIKOV,LandauInstitute o/TheoreticaIPhysics,Moscow, US.S.R. M. C. POLYVANOV, Steklov Institute ofMathematics, Moscow, USSR. Yu. A. ROZANOV, Steklov Institute 0/ Mathematics, Moscow, U.S.SR.
A. F. Filippov Department of Mathematics. Moscow State University. U.S.S.R.
Differential Equations with Discontinuous Righthand Sides edited by
F. M. Arscott
KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON
Library of Congress Cataloging In Publication Data Fll ippov. A. F. (AlekSel Fedorovich) [Different'sia 1 'nye uravneni fa s razryvnnl pravol chast 'fu. Engllshl Diff~rential equations wlth discontlnuous righthand sides I A.F. Fi I lppov : edlted by F.M. Arscott. p. cm. <MathematIcs and its appl ications <Sovlet serles\) Translation of, D1fferentsial 'nye uravneni fa s razryvnol pravol chast 'fU. Bibl iography: p. Includes index. ISBN 902772S99X 1. DifferentIal equations. Partial. 1. Arscott, F. M. II. Title. III. SerIes: Mathematlcs and its applications . Soviet series. OA374.F4S13 1988 515.3·53dc19 884531 CIP
ISBN 902772699X
Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands, Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
All Rights Reserved © 1988 by Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner Printed in The Netherlands
SERIES EDITOR'S PREFACE
It isn't that they can't see the solution. It is that they can't see the prohlem.
Approach your prohlems from the right end and begin with the answers. Then one day, perhaps you will find the final question.
O.K. Chesterton. The Scandal of Father Brown 'The point of a Pin'.
'The Hermit Clad in Crane Feathers' in R. van Oulik's The Chinese Maze Murders.
Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (nontrivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD" , "completely integrable systems", "chaos, synergetics and largescale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This programme, Mathematics and Its Applications, is devoted to new emerging (sub)disciplines and to such (new) interrelations as exempla gratia:  a central concept which plays an important role in several different mathematical and/or scientific specialized areas;  new applications of the results and ideas from one area of scientific endeavour into another;  influences which the results, problems and concepts of one field of enquiry have and have had on the development of another. The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practitioners in diversified fields. Because of the wealth of scholarly research being undertaken in the Soviet Union, Eastern Europe, and Japan, it was decided to devote special attention to the work emanating from these particular regions. Thus it was decided to start three regional series under the umbrella of the main MIA programme. At first sight, a differential equation with a discontinuous righthand side (forcing term) is a difficult object to deal with. Indeed, the very concept of a solution needs to be reexamined and redefined in this setting. Yet they arise naturally and often, especially in engineering and physics. In engineering e.g. because (optimal) controls are often discontinuous (think, for example, about the bangbang optimality results; in physics because e.g. forcing terms of, say, electromagnetic field type are often discontinuous (in the presence of shielding materials). Thus from the applied point of
v
Series Editor's Preface
vi
view there is a definite need for a systematic treatise of equations with discontinuous righthand sides and that is precisely what is provided in this first book on the topic written by a foremost authority. The unreasonable effectiveness of mathematics in science ... Eugene Wigner Well, if you know of a better 'ole, go to it.
As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection.
Bruce Bairnsfather Joseph Louis Lagrange. What is now proved was once only imagined. William Blake
Russum, February 1988
Michiel Hazewinkel
CONTENTS v
Series Editor'S Preface
ix
Preface Introduction
1
Chapter 1 Equations with the RightHand Side Continuous in z and Discontinuous in t 1. Caratheodory Differential Equations 2. Equations with Distributions Involved as Summands 3. Differential Equations with Distributions in Coefficients
3 3 17 29
Chapter 2 Existence and General Properties of Solutions of Discontinuous Systems 4. Definitions of Solution 5. Convex Sets and SetValued Functions 6. Differential Inclusions 7. Existence and Properties of Solutions 8. Dependence of Solution on Initial Data and on the RightHand Side of the Equation 9. Change of Variables 10. Sufficient Conditions for Uniqueness 11. Variation of Solutions
48 48 59 67 75 87 99 106 117
Chapter 3 Basic Methods of Qualitative Theory 12. Trajectories of Autonomous Systems 13. Properties of Trajectories in a Plane 14. Bounded and Periodic Solutions 15. Stability
123 123 133 142 152
Chapter" Local Singularities of TwoDimensional Systems 16. Linear Singularities 17. Topological Classification of Singular Points vii
175 175 190
viii
Contents
18. Structurally Stable and Structurally Unstable Systems 19. Singular Points on a Line of Discontinuity 20. Singular Points on an Intersection of Lines of Discontinuity
205 217 250
Chapter 5 Local Singularities of ThreeDimensional and Multidimensional Systems 21. Basic Types of Singularities. TwoDimensional Singularities 22. Linear and Point Singularities on a Surface of Discontinuity 23. Singularities on an Intersection of Surfaces of Discontinuity
268 285
References
291
Subject Index
303
259 259
PREFACE The development of the theory of differential equations with discontinuous righthand sides has been to a great extent stimulated by its many applications. A large number of problems from mechanics, electrical engineering, and the theory of automatic control, described by these equations, is considered in the book [1]. The wide usage of switches (relays) in automatic control systems leads to the necessity of constructing an elaborate theory of such equations. Different aspects of this theory are elucidated in various sections and chapters of the books [2][5] and in a large number of journal papers. The theory of automatic control systems described in differential equations with discontinuous righthand sides (systems with variable structure and with sliding motions) is developed in the books [6], [7]. The main trends of the theory of differential equations with discontinuous righthand sides are presented in this book. The treatment of such equations requires from the very start a generalization of the concept of solution. Various definitions of the solutions of such equations are proposed, and the conditions of their applicability are indicated. Many results in the classical theory of differential equations are shown to be valid also for equations with discontinuous righthand sides. Applicability of the known methods of investigation, naturally under certain restrictions, is grounded. Properties of the solutions due to discontinuity of the righthand side are discussed. In particular, singularities on lines and surfaces of discontinuities are studied in detail and classified, and their bifurcations analyzed. Sufficient conditions are formulated for stability of equilibrium points on the lines and surfaces of discontinuity and on their intersection. The results obtained by various authors and the new results are presented, wherever possible, from a unified standpoint. In some cases this has allowed us to attain a wider generality or to simplify proofs. The book does not touch upon the following questions concerning equations with discontinuous righthand sides: the frequency method for studying stability, which is developed in [5], boundaryvalue problems, the analysis of equations of special form and of narrow classes. Chapter 1 and the subsequent chapters deal with different types of equations, so the reader may approach Chapters 25 without acquaintance with §2 and §3. Small print is used to present some cumbersome proofs as well as generalizations and special questions, which can be omitted without loss of understanding of the rest of the text. Theorems, lemmas, and formulae are numbered
ix
x
Preface
independently in each section. When referring the reader to material of some other section, we indicate its number, for instance: "By virtue of Lemma 4, §1." References are made to a paper (or a book), not necessarily a pioneering one, in which a given problem has been considered and do not always imply that the results mentioned are due to the author of that paper. In §4 and §15 the reader can find references to the papers containing a review of the history of the problems under discussion. When dealing with those aspects of the subject which are covered in the literature in sufficient detail (periodic solutions, stability)' we describe only the basic concepts and some methods of investigation, and give appropriate references.
INTRODUCTION As is known, a solution of the differential equation
dx dt
= J(t, x)
with a continuous righthand side is a function x(t), which has a derivative and satisfies this equation everywhere on a given interval. This definition is not, however, valid for differential equations with discontinuous righthand sides. As can be seen from the following examples (5; denotes the derivative dx/dt). EXAMPLE 1: 5; = sgn t. For t < 0 we have 5; = 1, the solution being given by x = t + CI; for t > 0 we have 5; = 1, the solution being x = t + C2 (Fig. 1). Proceeding from the requirement of solution continuity for t = 0, we obtain
Consequently the solution is expressed by the formula x(t) = It I + c. For t = 0 the derivative 5;(t) does not exist. EXAMPLE 2: x = 1  2 sgn x. For x < 0 we have 5; = 3, the solution being x(t) = 3t+CI; for x > 0 we have 5; = 1, the solution being given x(t) = t+c2 (Fig. 2). As t increases, each solution reaches the line x = O. The direction field prevents the solution from leaving this line either upwards or downwards. IT the solution is continued along this line, the function x(t) = 0 so obtained does not satisfy the equation in the usual sense since for it x(t) = 0, and for x = 0 the righthand side of the equation has the value 1  2 sgn 0 = 1 ::f: O. Thus, the consideration of differential equations with discontinuous righthand side requires a generalization of the concept of solution. In cases where the righthand side of the equation 5; = J(t, x) is continuous in x and discontinuous only in t, it usually proves possible to generalize the concept of solution using only a mathematical argument (in Example 1 this is the requirement of solution continuity). In cases where the righthand side of the equation is discontinuous in x, such simple mathematical arguments are often insufficient. Then the solution is defined by means of a limiting process taking into account the physical meaning of a given problem. The generalization of the concept of solution must necessarily meet the following requirements: 1) For differential equations with a continuous righthand side the definition of a solution must be equivalent to the usual one. 1
Introduction
2
Figure 1
Figure 2
2) For the equation x = f(t) the solutions must be the functions x(t) + conly. 3) Under any initial data x(to) = Xo in a given region of the solution must exist (at least for t > to) and continue to the boundary of this region or to infinity, i.e. (t, x) > 00. 4) The definition of a solution must serve as a description of a fairly wide class of processes in physical systems. In order that equations with discontinuous righthand sides be investigated by the wellknown methods, the following conditions should also be satisfied: 5) The limit of a uniformly convergent sequence of solutions must be a solution. 6) Under the commonlyused changes of variables a solution must be transformed into a solution. The bestknown definitions of a solution of a differential equation with a discontinuous righthand side are presented in §4. The applicability of one or another definition to different problems is considered in 3, §8. Many results from the theory of differential equations have already been extended (sometimes with necessary alterations) to differential equations with discontinuous righthand sides. Such equations are usually analyzed by the same methods as differential equations with continuous righthand sides.
f
f(t)dt
CHAPTER 1
EQUATIONS WITH THE RIGHTHAND SIDE CONTINUOUS IN 'x AND DISCONTINUOUS IN
t
In Chapter 1 Carath~odory differential equations and differential equations with distributions are considered. Existence theorems for solutions are established and the properties ofsolutions, especially the dependence of a solution on the righthand side, are investigated. Approximation of different types of equations by equations with continuous righthand side is studied.
§1 Caratheodory Differential Equations Existence, uniqueness and continuous dependence theorems for solutions under weakened assumptions and theorems on the properties of the set of solutions are proposed for Caratheodory differential equations. 1. The differential equation :i: = f(t,:c) with a continuous righthand side is known to be equivalent to the integral equation
(1)
:c(t)
= :c(to) +
r f (B, :C(B)) dB.
Jto
IT the function f(t,:c) is discontinuous in t and continuous in :c, then the functions satisfying the integral equation (1) can be called solutions of the equation :i: = f(t, :c). Using here the concept of the Lebesgue integral, one obtains the definition of a solution which is the basis of the theory of the Cara.theodory differential equations. Everywhere in §1 :c(t) and f(t,:c) are fl.dimensional vectorvalued functions. Integra.tion is to be understood in the sense of Lebesgue. The function f(t,:c) is assumed to satisfy the following conditions: The CaratModory conditions. In the domain D of the (t, :c)space, let 1) the function f(t,:c) be defined and continuous in :c for almost all tj 2) the function f(t,:c) be measurable in t for each :Cj 3) If(t,:c) I ~ m(t), the function m(t) being summable (on each finite interval if t is not bounded in the domain D). 3
Equations. .. Discontinuous only in t
4
Chapter 1
The equation :i; = I(t, x), where x is a scalar or a vector and the function
I satisfies conditions 1)3), is called the Carathiodory equation. A function x(t) defined on an open or closed interval l is called a solution of the Caratheodory equation if it is absolutely continuous on each closed interval [a,.B] eland satisfies almost everywhere this equation or, which under the conditions 1)3) is the same thing, if it satisfies the integral equation (1) for some to E l. 2. We now present the known existence and uniqueness theorems for solutions of Caratheodory differential equations (see, for instance, [8], [9], pp. 43, 97,
[10]). LEMMA 1 ([8]). Let the function I(t, x} satisfy the CaratModory conditions and let the function x(t)(a ~ t ~ b) be measurable. Then the composite function I(t, x(t)) is summable. THEOREM 1. For to ~ t ~ to + a, Ix  xol ~ b let the function I(t, x} satisfy the Caratheodory conditions. Then on a closed interval [to, to + d], where d > 0, there exists a solution of the problem
(2)
:i;
x(to) = Xo.
= I(t, x),
In this case one can take an arbitrary number d which satisfies the inequalities
(3)
0< d
~
a,
rp(to + d)
~ b,
rp(t) :;
it
m(s)ds.
to
PROOF: For any integer k ;
+ ih ~ t
~ to
+ (i + l)h,
= d/k. i
On the intervals
= 0, 1, ... , k 
1,
we construct iteratively an approximate solution by assuming Xk(t) t ~ to
(4)
(to < t ~ to
= Xo
for
+ d).
This integral has meaning, and by virtue of Lemma 1 and the estimate (3), we obtain IXk(t)  Xo I ~ b. Moreover, for arbitrary a,.B belonging to [to, to + d]
(5) The function rp(t) is continuous on the closed interval [to, to + d] and is therefore uniformly continuous. Hence, for each e > 0 there exists such a 6 > 0 that for I.B  al < 6 the righthand side of (5) is less than e. The functions Xk(t), k = 1,2, ... , are therefore equicontinuous and uniformly bounded. Using Arzela.'s theorem we choose from them a uniformly convergent subsequence; its limit will be denoted by x(t). Since
Co.ro.tModory Differentio.l Equo.tions
§1
5
and the first term on the righthand side is less than e for h = d/k < 6, it follows that X,.(8  h) tends to X(8), by the chosen subsequence. By virtue of continuity of the function J(t, x) in x, and the estimate IJ(t, x) I ~ met) one can pass to the limit under the integral sign in (4). We conclude that the limiting function x(t) satisfies equation (1) for x(to) = xo, i.e., it is a solution of the problem (2). REMARK: IT the Caratheodory conditions are satisfied for to  a ~ t ~ to, Ix  ';01 ~ band d ~ a, I~(to  d)1 < b, then a solution exists on the closed interval [to  d, tol. THEOREM 2. Let (to, xo) belong to D and let there exist a summable function let) such that for any points (t,,;) and (t,1/) of the domain D
IJ(t,,;) /(t, 1/)1 ~ let)
(6)
Ix  1/1·
Then in the domain D there exists at most one solution of the problem (2).
Here and in what follows, uniqueness of a solution implies that if there exist two solutions, the graphs of which lie in the domain D, these solutions coincide on the common part of their intervals of existence. REMARK [111: For uniqueness of a solution for t ~ to it is sufficient that instead of (6) there should hold the inequality (J(t,,;)  l(t,1/)) . (x  y) ~ let) Ix _ yl2
(7)
I, x, yare vectors, the product is understood as the scalar product). PROOF: Let x(t) and yet) be solutions of the problem (2), z(t) = x(t)y(t), to ~ (if
t ~ tl. Since
Izl2 =
z· z, we have
dlz(t)12 dz dt = 2z· dt = 2 (J(t, x)  J(t, y)) . (,;  y).
almost everywhere. Taking into account (7), we obtain Then
dlzl2 /dt
~ let)
Iz12.
almost everywhere. The absolutely continuous function Izl2 eL(t) does not increase, and it follows from z(to) 0 that z(t) 0 for t ~ to. Thus, uniqueness is proved for t ~ to in the case (7) and, therefore, under the condition (6). Under the condition (6) the case t ~ to is reduced to the case t ~ to by the substitution of t for t. Many other sufficient uniqueness conditions are known. Consider a linear system in the vector notation
=
(8)
=
z = A(t),; + bet).
all the elements of the matrix A(t) and the vectorvalued function bet) be summable on each segment contained in the interval (a,p).
THEOREM 3. Let
Equations. .. Discontinuous only in t
6
Chapter 1
Then for to E (a,,8) the solution of the system (8) with arbitrary initial data x(to) = Xo exists on the whole interval (a,,8) and is unique. PROOF: We pass over from (8) to an integral equation analogous to (1) and use successive approximations: xo(t) == :to,
(9)
XIo+1(t) = Xo
+
itor [A(s)xlo(s) + b(8)] ds,
k= 0,1, ....
From the assumptions of the theorem it follows that the functions 1(t) = IIA(t)1I and !pet) = Ib(t)1 are summable on each segment [al,,811 C (a,,8). Since for any vector z IA(s)zl ~ IIA(s)II'lzl = 1(S) Izi all approximations exist and are continuous on [al,,8lj, and k = 1,2, ....
Let
e= max Ixdt) 
xo(t)1 on the closed interval [all,81]
It can be proved by induction that on this closed interval
k
= 1,2, ....
The righthand side is the kth term of a series uniformly convergent on the closed interval [al,,81j. Hence, as k  00, limxlo(t) = x(t) exists, and in (9) a limit transition under the integral sign is possible. The function x(t) is therefore a solution of the integral equation and, accordingly, of equation (8) on the closed interval [aI, ,81]. Uniqueness of the solution follows from Theorem 2. Since [all ,811 is an arbitrary segment contained in the interval (a, ,8), the solution exists and is unique on this interval. Some other assertions can also be extended to linear Caratheodory systems (that is, systems satisfying the conditions of Theorem 3). These assertions concern the existence of a fundamental set of solutions, the representation of any solution by means of a fundamental set and the properties of the Wronskian. They are proved in the same way as for systems with continuous coefficients. 3. Solutions of Caratheodory equations possess many properties analogous to the properties of solutions of equations with continuous righthand sides (continuability of solutions, compactness of a set of solutions, the properties of integral funnels l considered in [12]) and are studied mainly by the same methods. 1 For
the meaning of this term, see p. 16
§1
CaratModory Differential Equations
LEMMA 2. 9n a finite interval c equation are equicontinuous.
~
t
~
7
d all the solutions of the CaratModory
PROOF: Let the function pet) be expressed by the integral (3), where to = c. For each closed interval [a,,8J C [c, dJ and for any solution ,; (t)
1,;(,8)  ,;(a) 1=
Iill I (t,,;(t)) dtl ~
!:
m(t)dt = p(,8)  pea).
From this and from the uniform continuity of the function pet) on the segment [c, d] there follows the equicontinuity of all the solutions. REMARK: The assertion of Lemma 2 is also valid for solutions satisfying different Caratheodory equations with the same summable majorant met). THEOREM 4. Let:i: = I(t,,;) be a CaratModory equation in a closed bounded domain D. Then each solution of this equation which lies within D can be continued on both sides up to the boundary r of the domain D. PROOF: Consider the solution ,;(t) which passes through the point Po(to,,;o) within D. Let £1 > 0 be not greater than half the distance p(po, r) from this point· to the boundary r. Let c ~ t ~ d inthe domain D. Since the function pet) in (3) is uniformly continuous on [c, dJ, there exists a 01 > 0, 01 ~ £1 such that for any a, f3 from [c, dJ satisfying the inequality 1,8  al < 01 we have Ip(,8)  pea) I < 1>'1. Then the cylinder is contained in D. By virtue of Theorem 1 and the remark following that theorem, the solution ,;(t) exists at least on the interval It  tol ~ 01 (or can be continued to this interval). If the distance from the point (to + 01, ,;(to + 01)) to r is not smaller than 2£1, the solution can be continued further to an interval of length 01, etc., until it reaches a point P1(tll';1) such that p(PlIr) < 2£1' Assuming £i  0, i = 1,2, ... , we continue the solution sequentially up to the points Pi(!;,';.) such that
t1 < t2 < ... ,
p(pi' r)  0
(i  00).
The bounded sequence t1, t2, . .. converges to some t*. Then by means of Lemma 2 and the Cauchy criterion we find that there exists a lim:r:(t) = x* for t  t·  O. Obviously, (t*, x*) belongs to r. Assuming x(t·) = ,;*, we obtain a solution which reaches the boundary r at the point (t*,:r:*). The solution is continued to the left in the same manner. The other known continuation theorems (for example, from [13], pp. 12 and 29 and from [11]) are also valid not only for differential equations with continuous righthand sides, but also for the Caratbeodory equations. Now consider compactness of sets of solutions.
8
Equations. .. Discontinuous only in t
Chapter 1
LEMMA 3. The limit of each sequence of solutions of a Caratheodory equation, which converges on a closed interval [a, 1'], is a solution of the same Caratheodory equation. PROOF: For a sequence of solutions x = x,,(t), k = 1,2, ... , the possibility of the limit transition in (1) is provided by the Caratheodory conditions. Hence, the limiting function also satisfies the equality (1) and is therefore a solution of the equation d; = f (t, x).
LEMMA 4. For a Caratheodory equation in a bounded closed domain D, the set M of all solutions, the graphs of which on the interval a ~ t ~ I' are contained in D, is a compactum in the metric C[a, 1'] (i.e., in a metric of uniform convergence on the segment [a, 1']). PROOF: All these solutions on the segment [a,l'] are uniformly bounded, and by Lemma 2 they are equicontinuous. By the ArzeIa theorem, out of any infinite set of such solutions one can choose a sequence which converges uniformly on [a,I']. By Lemma 3 the limit of this sequence is a solution of the same equation. The set M is therefore a compactum.
LEMMA 5. Let x" (t) (ale ~ t ~ 1'1e, k = 1,2, ... ) be solutions of a CaratModory equation, the graphs of which lie within a bounded closed domain D, and (10)
Then there exists a subsequence of these solutions which converges to a solution whose graph joins the points p(a, xo) and q(l', x*) and lies in D for a ~ t ~ 1'; for any 0> 0 the convergence is uniform on the segment [a+ 0, I'  0]. PROOF: Let O. + 0, i = 1,2, .... Using Lemma 2 and the Arzelli theorem we choose from x,,(t) a subsequence X1j(t), j = 1,2, ... , which converges uniformly for a + 01 ~ t ~ I'  01, and from it choose a subsequence X2j(t) which uniformly converges for a + 02 ~ t ~ I'  02, etc. The diagonal sequence Xjj(t) converges on the interval (a, 1') uniformly on each segment [a + 0., I'  0.]. By Lemma 3, the limiting function x(t) is a solution provided that a < t < 1'. By virtue of Lemma 2, it can be extended by continuity onto the segment [a, 1']. If e > 0 is arbitrarily small, we have for sufficiently small 0 and sufficiently large j
Ix(a)  :t(a + 0)1 < e/4, By Lemma 2, for a k such that Xjj(t)
l:t(a + 0)  3:1j(a + 0) I < e/4.
== :tle(t) we have (if j > j1(e))
13:1j(a + 0)  Xjj(ale) I < e/4, by virtue of (10). Hence, Ix(a)  :tol < e. Since e > 0 is arbitrary, x(a) = :to, that is, the solution x(t) passes through the point p. Similarly, it passes through the point q. The following theorem is proved in [10J on the assumption that the function f satisfies the Caratheodory equations in the whole domain G. This requirement is weakened in line with [12J.
§1
Caratklodory Differential Equations
9
THEOREM 5. Let the function f(t, x) satisfy the OaratM6dory conditions in each closed bounded subdomain of an open domain G. Let A be a point (to, xo) (or a closed bounded set), A c G. If all the solutions of the equation :i: = f(t, x) with the initial data x(to) = Xo (or with various initial data (to, x(to)) E A) exist for a ~ t ~ 13 and if their graphs for these t lie in G, then 1) the set of points lying on these graphs (i.e., a segment a ~ t ~ 13 of an integral funnel of the set A) is bounded and closed; 2) the set M of these solutions is a compactum in the metric O[a, 13]. PROOF: Let us take a sequence of closed bounded'domains DlJ D 2 ,. •• , such that A lies within D1 and D,. lies within D"+1, k = 1,2, ... , and that each bounded closed set KeG is contained in some of the domains D,.. We shall show that all the considered graphs lie in one of the domains D,.. Assume the contrary. Then for each of the domains D,. there exists a solution x,.(t), 13,. E [a,pl and a,. such that
(ll) From these solutions, we choose such a subsequece SoCk = k 1 ,k2 , ••• } for which PIc + pEA. Since A C D1 C D2 C ... , it follows from (11) that for E aDm(aDm is the each m ~ 1 and for each k ~ m there exists a point boundary of the domain Dm} on the graph of the solution x,.(t) such that the arc Pleqr of this graph is contained in Dm. From the subsequence So we choose a subsequence of solutions for which q~ + q1 E aD 1, and from it, by Lemma 5, we choose a subsequence Sl of solutions which converges to a solution whose graph joins the points pEA and q1 E aD1. By the same method, we choose from Sl a new subsequence S2 which converges to a solution whose graph joins the points pEA and q2 E aD2 . Proceeding with this process, we obtain a solution x(t), the graph of which passes through the points pEA, ql E D 1 , q2 E D 2 , ... . By assumption this solution exists on a closed interval containing a, 13 and the abscissa tp of the point p. The graph of the solution on this interval is a bounded closed set and, accordingly, it lies in some domain Dm, that is, within D m +1' This contradicts the fact that this graph passes through the point qm+1 E aDm +1' Thus, the assumption is incorrect, and all the considered graphs lie in one of the domains Die. Then the assertion 2} is proved as in Lemma 4, and from this assertion there follQwS the assertion 1). 4. The continuous dependence of solutions of the Caratheodory equations on initial data and on the righthand side of the equation, or on the parameter, has been considered in a number of papers, particularly in [10] and in [14J[27]. The differentiable dependence has been considered in [28][31]. Below we present two theorems on continuous dependence: the simplest (for the case where a sequence of functions f,.(t, x) on the righthand sides of differential equations converges) and a more general one (for the case where a sequence of integrals of these functions over t converges). In 4 we deal with solutions x(t, IL) of equations, the righthand sides of which depend on the parameter IL which changes on some set M (of a metric space)
qr
10
Equations. .. Discontinuous only in t
Chapter 1
with a limit point J.to EM and establish conditions for the convergence x(t, J.t) + x(t,J.to) for J.t + J.to, i.e., for p(J.t,J.to) + o. To this case one can reduce the case of the sequence Xk(t), k = 1,2, ... , if one puts x,.(t) = x(t, J.t), J.t = 11k + O. As is known, for differential equations with continuous righthand sides and for Caratheodory equations, uniqueness of a solution leads to its continuous dependence on initial data [32]. The following lemma generalizes this assertion. It is applicable not only to solutions of differential equations, but also to solutions of differential inclusions. It specifies the sense in which one can speak of convergence to a set of solutions in the absence of uniqueness. This lemma makes it possible to reduce the conditions for continuous dependence of a solution to a number of simpler conditions. LEMMA 6. Let there be given a point (to, ao), numbers tl > to, eo> 0, a finite open domain D in the (t, xlspace, a set M of values of the parameter J.t, and a family S of continuous functions e(t), each of which corresponds to its initial value a = e(to) and a certain value of the parameter J.t E M. Let 1) each function e(t) be defined on some interval, its graph lying in the domain D, the endpoints of this graph being two points of the boundary of the domain Dj 2) for any a, J.t(la  aol < 0, J.t E M) there exist at least one function of the family which corresponds to these a and J.tj 3) in each sequence of functions edt) E S, i = 1,2, ... , which correspond to the values ai + ao, J.ti + J.to, all functions be equicontinuousj 4) the limit of each uniformly convergent sequence of functions of the family, for which a = a. + ao, J.t = J.ti + J.to, be a function of the family for which J.t = J.to; 5) all the functions of the family for which J.t = J.to, a = ao, be defined at least on the segment [to, tIl; the set of these functions will be defined by Xo; 6) for each of the functions eo(t) of the set Xo the eotube
(12)
Ix 
eo(t) I < eo,
be contained in D. Then for any e > 0 there exist a 0 > 0 and an '1 > 0 such that for all a and J.t satisfying the conditions
(13)
la  aol < 0,
each of the functions e( t) of the family, which corresponds to these a, J.t exists on the segment [to, hJ, and differs from some function eo(t) E Xo less than bye:
(14)
le(t)  eo(t)1 < e
(For different e(t) the functions eo(t) may be different). PROOF: Suppose for some ai + ao, J.ti + J.to there exist functions e.(t) E S defined on less than the whole segment [to, tIl. According to 1) I each of them reaches the boundary of the domain D at some point qi(ti,X.), t. E (to,td. It
§1
CflrfltModory Dif/erentifll Equfltion!
11
follows from the conditions 3) and 6) that ti ~ ro > to for all i > i l • From the sequence {gi} we choose a subsequence which converges to some point q, and from a corresponding subsequence offunctions €i(t) we choose a new subsequence which converges to the function €o(t) whose graph joins the points (to, 0.0) and q as in Lemma 5. By virtue of 4), the function €o(t) E Xo. Then, by virtue of 6), the eotube (12) is contained in D, and q is the point (tl' €o(td). This contradicts the convergence of the subsequence of points qi(ti, Zi) of the boundary of the domain D to the point q because ti is les~ than tl' Thus, for some 6> 0 and '1 > 0 for all a and I' which satisfy (13), the graph of the function €(t) lies in D for to < t < tl' IT the lemma is not true, then for some e > 0 there exists a sequence of functions Zle (t) E 8, Ie = 2,3, ... , such that ZIe(to) = ale . 0.0, I'le . 1'0, and the graphs of these functions lie in D for to < t < tl, and for each Ie and each function €(t) E Xo
(15)
Iz,.(t,.)  €(t,.) I ~ e,
t,. E [to, tl]'
Ie = 2,3, ...
j
the points t,. may depend on the choice of the function €(t) E Xo. By virtue of 3) and 4), one can choose from the sequence {z,.(t)} a subsequence which converges uniformly to some function zo(t) E Xo. This is in contradiction with (15) for €(t) == zo(t). The lemma is proved. COROLLARY. Let the conditions 1)4) of Lemma 6 be fulfilled; for a = 0.0, I' = 1'0 in the family 8 let there exist only one function €o(t) and for this function let the eotube (12) be contained in D. Then the assertion of Lemma 6 is valid and each sequence €i(t) of functions from 8, for which ai . ao, I'i . 1'0, converges
uniformly to €o(t) on the segment Ito, tl]'
z
REMARK: For the family of solutions of the Caratbeodory equations = f(t, z) the conditions 1) and 2) of Lemma 6 are fulfilled by virtue of Theorems 1 and 4, and the conditions 3) and 4) are fulfilled by virtue of Lemmas 2 and 3. Hence, if a solution with the initial data z(to) = flO is unique, it depends continuously on the initial data. For the Caratheodory equation = f(t, 3:,1') with the parameter I' the conditions 1) and 2) are fulfilled, and one need only fulfillment of the conditions 3) and 4) and uniqueness of the solution for z(to) = 0.0, I' = 1'0.
z
THEOREM 6 ([10]; [9], p. 58). Let for (t,3:) E D, I' E M, 1° f(t, 3:, 1') be measurable in t for constant 3: and 1'; 2° If(t, 3:, 1') I ~ m(t), the function m(t) being summable; 3° for almost all t the function f(t, 3:, 1') be continuous in z, and for I' =
I'oin z,l'; 4° the solution z = €o(t) of the problem
(16)
;; = f(t, z, 1'),
3:(to) = a
for a = 0.0, I' = 1'0 be unique for t ~ to; let this solution exist for to ~ t ~ tl and let its graph have a neighbourhood ofthe type (12) which is contained in D. Then for any a and I' sulliciently near ao and 1'0 the solution of the problem (16) on the closed interval Ito, ttl exists (it is not necessarily unique) and converges uniformly to €o(t) as 0.'0.0, I' . p.o. REMARK: The condition 3° leads to the fact that for almost all t the function f(t, z, 1') tends to f(t, z, 1'0) uniformly in 3: (on any compactum) as I' . 1'0.
Equations. .. Discontinuous only in t
12
Chapter 1
PROOF: By virtue of 1°3°, equation (16) with the J.tindependent majorant met) is a Caratheodory equation, and the conditions 1)3) of Lemma 6 are therefore satisfied for its solutions; the conditions 5) and 6) are satisfied by virtue of 4°. The solution x = e(t, a, J.t) of the problem (16) satisfies the integral
equation
r 't
e(t; a, J.t) = a + I(s, e(s; a, J.t); J.t)ds. , ito
(17)
If the sequence of solutions e(t, ai, J.ti) converges uniformly as ai then, from 3° , for almost all s
I (s, e(s; ai, J.t.); J.ti)
+
+
ao, J.ti
+
J.to,
I (s, e(s; a, J.to); J.to) .
In this case, by virtue of 2° a limit transition is legitimate under the integral sign in an integral equation for e(t; ai, J.ti). Thus, the limiting function e(t; ao, J.to) satisfies the equation (17) for a = ao, J.t = J.to arid is a solution of the problem (16) with a = ao, I' = 1'0. That is, the condition 4) of Lemma 6 is fulfilled. The assertion of the theorem follows from this lemma. A further generalization of the continuous dependence theorem consists in the replacement of the requirement of the convergence I(t, X; 1') + j(t, x; 1'0) (i.e., continuity of the function I in I' for I' = 1'0) by the requirement of convergence of the integral of j(t, x; 1') over t to the integral of I(t, Xi 1'0) ([14], [17] and others). ' THEOREM 7 [17]. Let for J.t E M, to ~ t ~ tll X E B (B being a finite open region in R"') 1° the function l(t,x;l') be measurable in t for constant X,I'j 2° I/(t, x; J.t) I ~ met, 1'), the function met; 1') being summable in tj 3° there exist a summable function l(t) and a monotone function ,per) + 0 for r + 0 such that for each r > 0, if Ix  YI ~ r, and for almost all t
I/(t, Xj 1')  j(t, Yi 1') I ~ l(t),p{r);
(18) 4° for each x
E B for
rt
(19)
ito
I'
+ 1'0
I(s, x; I')ds
+
it to
I(s, x; I'o)ds
be uniform in t on the segment [to, tl]j 5° the solution x = eo(t) of the problem (16) for a = ao E B, I' = 1'0 be unique for t ~ to and lie in the domain B for to ~ t ~ tl. Then for any a and I' sufliciently near ao and 1'0 the solution of the problem (16) on the interval [to, tl] exists (it is not necessarily unique) and converges uniformly to eo(t) as a + 0,0, I' + 1'0. PROOF: First we will show that for each sequence I'i + 1'0 and each sequence of continuous functions Xi(t) E B, i = 1,2, ... , uniformly convergent to xo(t) we have, for all t E [to, tl],
(20)
r
ito
[/(s, Xi( s); I'i)  f(s, xo( s); 1'0)] ds
+
0
(i
+
00).
Carathlodory Differential Equations
§1
13
Since the function xp(t) (p = 1,2, ... ) is continuous, the sequence of piecewise constant functions
i = 1,2, ... ,2'1,
to + (i  l)hq ~ t < to + ihq,
where hq = 2'I(tl to), q = 1,2, ... , converges uniformly to xp(t). There exists such a q(p) that for the function zp(t) == !lP,'I(p)(t)
Since xp(t) . xo(t) as p . 00, so does zp(t). By virtue of 1° and 2°, (21)
r [/{8, Zp(8)j JLo)  1(8, Xo(8), JLo) Id8
(p . 00).
.0
ito
It follows from 4° that a relation similar to (19) holds also for integrals over any interval contained in [to, tll. In such a relation one can replace x by zp(t) on any interval where the function zp(t) is constant. Summing up over such intervals, we obtain for p = const
Ji,p(t) ==
r [/(8,zp{8)jJLd  l(s,zp(s);JLo)] ds .
ito
0
Thus, for each t E [to,tlJ and each p ~ 1 there exists anip{t} such that IJi,p(t)1 < 2 P for all i > ip(t). The number ip(t) can be increased and we assume therefore that ip+1(t) > ip(t). Suppose
v(i, t) = p for ip(t) < i Then if i . 00, we have for each t
(22)
~
ip+1(t),
p
= 1,2, ....
= const
v(i, t} . 00,
IJi,v(i,t) (t}1 < 2 v (i,t) . O.
By virtue of the condition 3°, for almost all
8
E (0, t)
The righthand side does not exceed the summable function 1(8}tjJ{d}, where d is the diameter of the domam B, and for almost all s it tends to zero as i . 00 since X'(8} . XO(8}, ZV(i,t) (8) . XO(8). Hence,
(23) as i . 00. From (21)(23) there follows (20). Now let x.(t) be the solution of the problem (16) with a = a. . ao, JL = JLi . JLo (i . 00) and with a function f which meets the requirements of Theorem 7. Then
(24)
Xi(t)
= at +
r1
ito
(8, X.(8)j JL.) d8.
Equations. .. Discontinuous only in t
14
Chapter 1
IT x.(t) tends to xo(t) uniformly on some interval [to, t*], then, by virtue of (20), one can pass to the limit in the equality (24), and the function xo(t) is a solution of the problem (16) with a = ao, I' = 1'0. The condition 4) of Lemma 6 is thus fulfilled for the family of solutions of the problem (16) with different a and 1'. Let us verify fulfillment of condition 3) of Lemma 6. Since the functions f(s, ao, 1'0) and l(s) are summable, for each e > 0 there exists a 0 > 0 such that for any a, P E [to, til it follows from IP  al < 0 that (25) where d is the diameter of the domain B. For a certain pde) the difference between the left and the righthand sides of the relation (19) is less than e for p(l', /Lo) < p1(e) and for all t E [to, t1l. From this and from (25) there follows (26)
liP f(s,aO;I')dsl < 3e Let Xi(t) (i = 1,2,.,.) be a sequence of solutions of the problem (16) with ~ ao, I' = I'i ~ 1'0, From (24) we have
a = ai (27)
By virtue of (18) the integrands in (26) for /L = /Lo and in (27) differ by not more than l(s),p(d) and therefore, on account of (25), the integrals differ by not more than e. Now for p(l', 1'0) < pde) we have from (26) and (27) (28) Since I'i ~ 1'0, the inequality P(I'i, 1'0) < pde) may fail to hold only for a finite number of i values. By virtue of continuity of the functions Xi(t), there exists a 01 for these i such that (28) holds for all a,p E [to,t1], IPal < 01. For IP  al < min {o; 01} the inequality (28) is thus satisfied for all i. Since e is arbitrary, the solutions under consideration are equicontinuous, and the condition 3) of Lemma 6 is fulfilled. The conditions 1) and 2) of this lemma are satisfied by virtue of 10_3 0 and Theorems 1 and 4. The assertion of the theorem is valid by virtue of the corollary of Lemma 6. REMARK: The assertion of Theorem 7 remains true if the condition 30 is replaced by the following: 3*, There exist functions l(t,r,l') and oo(e) > 0, 0 < e < eo, such that for each r > 0 for almost all t and Ix  YI ~ r
(29)
and for any a,p E
(30) (31)
let, r, 1') ~ 0
If(t,x,l')  f(t,y,I')1 ~ l(t,r,I'),
[to,td, IP  al < oo(e), for I[a,p] I[tO,t1]
'=
~ 0
f:
all r,l'
l(s,r,l')ds < e,
(r
~ 0,
I'
~ 1'0).
(r
~
0),
15
CaratModorll Differential Equations
§1
In this case only two changes must be introduced into the proof of Theorem 7. By virtue of (29), the integrand in (23) is not greater than
I (s, l.zi(8)  ZV(i,t)(s)l,iJi) and by virtue of (Sl) the integral (2S) therefore tends to zero as i + 00. IT IP  al < min{ojoo(e)} the integrands in (26) for iJ = iJi and in (27) differ by not more than l(8, d, iJi), and therefore, by virtue of (30), the integrals differ by not more than e. The rest of the argument in the proof of Theorem 7 remains the same. The following eXaJRple shows that neither the conditions 3° in TheoreJRS 6 and 7, nor the condition (Sl) of the remark can be omitted, even in the case where for each iJ the function f(t,.z, iJ) is continuous in t,.z and is bounded by a iJindependent constant. Let iJ = 11k + 0, k = 1,2, ... j.z e Rl,
f(t,.z,iJ) The function .z1o(t) limit lim,. ... oo .z,. (t)
1
= II.(t,.z) = [k 2 (.z _ t) _
= t + 11k is
kJ2 + 1
+
a solution of the equation :i:
= t does not satisfy the equation :i: = O.
O.
= II.(t, .z), but the
Let elements bij(t, iJ) of a matrix B(t, iJ) and a vectorvalued function get, iJ) for iJ E M be absolutely continuous on the segment [to, tIl and uniformly in t .
THEOREM 8.
get, iJ) + get, iJo)
(S2)
Let there exist a o(e) > 0 (0 < e < eo) such that for an i, i iJ E M and all a, P E [to, tlJ, IP  al < See) we have
=
1,2, ... , n, all
ifl!bii(t,iJ)! dt < e.
(SS)
Then on the segment [to, tIl the solution of the problem (S4)
for a with
+
ao, P
+
iJo converges uniformly to the solution of the same problem
a= ao, PROOF: Since equation (S4) is of CaratModory type, then, by Theorem 3, its solution .z(tj ao, Po) for a = ao, P = Po exists on the segment [to, tlJ and is unique. Then for the problem (34) in the region
l.zl
< 1 + max l.z(tj ao, iJo) 1 Ito,hl
the requirements of Theorem 7 and of the remark are met. Thus, the assertion is valid.
16
Equations. .. Discontinuous only in t
Chapter 1
COROLLARY. If, for a sequence of linear Caratheodory systems, coefficients and free terms converge in the metric L 1 , and a sequence of initial data converges, then the sequence of solutions converges uniformly on a given segment. Using the estimate obtained in [18] for the difference of solutions of two linear systems, one can evaluate the rate of this convergence in terms of the norms (in Ld of differences of their coefficients and of the difference of free terms. 5. The properties of integral funnels investigated in [12] for differential equations with continuous righthand sides remain the same also for Caratheodory differential equations [10]. For the differential equation x = !(t, x) (x ERn) an integral funnel of a point (to, xo) (or of a set A) is a set of the (t, x )space points lying on all solutions which pass through the point (to, xo) (or respectively through the points of the set A). A funnel segment is a part of the funnel lying in the interval a ~ t ~ f3. In the following theorems we assume that the equation x = !(t, x) satisfies the Caratheodory conditions in each finite part of the domain under consideration and that all the solutions with the initial data x(to} = Xo (or all the solutions which pass through the points of a given closed set A) exist for a ~ t ~ f3, and the point (to, xo) (correspondingly, the set A) is contained in the layer a ~ t ~ f3. Compactness of a funnel segment was proved in Theorem 5. THEOREM 9. If A is a point or a connected compactum, the crosssection of a funnel by any plane t = t1 E [a, f3] is a connected compactumj the set of solutions passing through points of the set A is a connected compactum in the metric C[a, f3]. The first assertion for a funnel of the point (to, xo) in the case of a sufficiently small segment [a, f3] is proved in [10]. The second assertion is proved similarly with the help of the metric C[a, f3]. In the case of a segment of any length and any connected compactum A the assertions are extended by the methods presented in [12] and [33]. THEOREM 10. An arbitrary point (t1' xd of a funnel boundary can be joined to a point (to, xo) by such an arc of the graph of a solution which passes along the funnel boundary. This assertion is proved in [10]. THEOREM 11. Let !(t, Xi 1'0) and A satisfy the conditions formulated before Theorem 9, and the functions
!(t, Xi 1'),
I' = I'k  1'0,
k
= 1,2, ... ,
meet the requirements of Theorem 6 or Theorem 7, except the requirement of solution uniqueness. Let Ak, k = 1,2, •.. , be a sequence of sets such that for each e > 0 all Ak, beginning with a certain one, are contained in the e neighbourhood of the set A. Then the same property is inherent in the segments a ~ t ~ f3 of funnels of the set Ak for the equations x = !(t, Xi I'k) with respect to the segment of the funnel of the set A for the equation x = !(t, Xi 1'0)'
§2
Equations with Distributions Involved as Summands
17
The assertion follows from Lemma 6 because if the requirements of Theorem 6 or of Theorem 7 (without the uniqueness requirement) are met, the requirements of Lemma 6 for the family of solutions are met also.
§2 Equations with Distributions Involved as Summands We deal here with different classes of differential equations with additively involved distributions, including differential equations with impulses, linear (and simple nonlinear) equations with distributions on the righthand sides, and linear systems not solved with respect to derivatives and possessing discontinuous solutions. We present the methods of reducing such equations and systems to Caratheodory systems, which enables us to prove the existence and to investigate the properties of solutions. 1. In [34] (pp. 169179) the equations
a: = f(t, z) + p(t),
(1)
are analyzed, where z eRn, the function f(t, z) satisfies the Caratheodory equations, and p(t) is a distribution or an ordinary, but not Lebesgue integrable, function. In 1 the function p(t) is assumed to be a distributional derivative of some measurable function q(t) bounded on each finite interval, that is
(2)
p(t) = q(t),
Iq(t) I ~ 'Y
(a < t < ,8).
In particular, p(t) can be a usual function integrable in one or another sense, and q(t) can be an integral of p(t) (Perron, Denjoy, DenjoyKhintchine integrals; for a more general formulation of the problem see [35]); p(t) can be a deltafunction (in this case equation (1) belongs to the class of equations with impulses, encountered in applications) or a distributional derivative of a continuous or a discontinuous function of bounded variation. In all these cases, one can reduce equation (1) to the Caratheodory equation
(3)
iJ = f(t, y + q(t)),
on making the substitution z = y + q(t). Measurability of the righthand side of (3) in t for any constant y follows from Lemma 1, §1. A solution of equation (1) is any function of the form z(t) = y(t) + q(t), where y{t) is a solution of equation (3). Such a function z{t) satisfies equation (1) if the derivative a: is understood in the sense of the theory of distributions (note that the derivative of the Perron and Denjoy integrals and the approximative derivative of the DenjoyKhintchine integral exists almost everywhere and is a derivative in the sense of the theory of distributions; this follows from [36], Ch. 8, §2). Since equation (3) has a solution with any initial data of the form y(to) = a, equation (1) has a solution for initial data of the form
(4)
z(t)  q(t)lt=to = a.
18
Equations. .. Discontinuous only in t
Chapter 1
If the function q is continuous at the point to, the condition (4) is equivalent to the initial condition
(5)
(b = a + q(to)) .
x(t o ) = b
If the function q is discontinuous at the point to, all solutions of equation (1) are also discontinuous at this point and the condition (5) has no uniquely defined meaning. If there exists lim t _ to  O q(t) = q(to  0) or lim t _ to + o q(t) = q(to + 0), the condition (5) can be replaced by the condition x(to  0) = a + q(to  0) or x(to + 0) = a + q(to + 0). In other cases one has to restrict oneself to setting the initial data in the form (4). Knowing the properties of the solutions ofthe Caratheodory equation (3), §1, and using the change x = 11 + q(t), we obtain corresponding properties of solutions of equation (1): existence of the solution, compactness of the set of solutions contained in a closed bounded domain, and uniqueness under the conditions (6) or (7), §1. The behaviour of the solution near the ends of its interval of existence is examined in [34] (pp. 176179). Let us make a more detailed analysis of equations with impulses. Consider the equation
z = I(t, x) + P6(t),
(6)
where I(t, x) is a known function; with regard to the function Pe(t), it is known only to be equal to zero outside a small interval (tl  e, tl + e), and its integral over this interval is known to be equal to tI. Such equations arise from problems of body motion in the presence of pushes and knocks if such a push or knock is known to occur at the moment t = h, to be of short duration, and if the total impulse, i.e., the impulsive force integral over the time interval during which the knock lasts, is known. To exclude from consideration the unknown values of the function P6(t) in the interval (tl  e, tl + e), one has to make a limit transition e + 0 with a retained constant value tI of the integral of P6(t). In the limit one obtains the equation
(7)
z=
I(t,x)
+ tlo(t 
tl),
where 0 is a deltafunction. In the theory of distributions oCt) = F1'(t) , where
F1(t) = 0 (t < 0),
F1(t) = 1 (t> 0).
Hence, by the change x = 11 + tlF1(t  td equation (7) is reduced to equation (3) with q(t) = tlF1(t  tl)' The solutions of equation (3) are absolutely continuous. Thus, the solutions of equation (7) are functions which for t < tl and t > tl are absolutely continuous and almost everywhere satisfy the equation z = I(t, x), and for t = tl have a jump x(t + 0)  x(t  0) = tI. Similarly, at points ti all solutions of the equation 00
(8)
z=
I(t, x)
+L ;=1
tliO(t  til
§2
Equations with. Distributions Involved as Summands
19
have jumps equal to tli (i = 1,2, ... ), and in the intervals between these jumps they are absolutely continuous and satisfy the equation a: = J(t, x). The following theorem motivates the transition from equation (6) to equation (7) and the corresponding step in more general cases. The case where vectors tli in (8) depend on z, i.e., the values of the jumps depend on z, will be discussed in 3, §3. THEOREM 1. In a bounded closed domain D consider the equations
(9) where the function (10)
k
= 1,2, ... ,
J satisfies the Caratheodory conditions, p",(t) = q",(t), (k=l,2, ... ),
Then each function x( t), being the limit of some sequence of solutions x", (t) of equations (9) is a solution of equation (1) with pet) = q(t).
PROOF: We pass over from equation (1) to (3), and using a similar change of variables z", = y", + q",(t), from equations (9) to the equations
(11)
k
= 1,2, ....
For these equations the Caratheodory condition 2) (I, §1) holds by virtue of Lemma I, §1, the condition 3) holds with one and the same function met) for all k. For almost all t the function J(t, x) is continuous in z. Taking into account (10) and making a limit transition k + 00 in the integral equation equivalent to (11), we find that the function yet) = limy",(t) satisfies a similar integral equation, but with the function q(t) instead of q",(t). Therefore, yet) is a solution of equation (3). Then z(t) = yet) + q(t) is a solution if equation (1). Differential equations with impulses have been examined in many papers, which cannot all be referenced here (below we refer only to some of them). Equations with impulses at given time instants, as in (8), or when solutions reach given surfaces in the (t, z)space have been investigated. The magnitudes of the jumps of solutions are either given in advance or depend on the point at which the jump occurs. Consideration has been given to existence and uniqueness of solutions, continuation of solutions, continuous dependence of solutions [37][40], the properties of integral funnels and absorbing sets from which the solutions do not go out [37], stability [41][47], existence and stability of periodic solutions [37], [48][52], application of the averaging method to equations with small parameter [39], [53][56], and the use of jumps for retaining solutions in the domain in case these solutions reach the boundary of the domain [57]. See also the book [206]. 2. The cases with distributions more complicated than in §1 have been studied in linear equations and systems and only in a few nonlinear cases. In applications the equation
20
Equations . .. Discontinuous only in t
Chapter 1
often appears, where m ~ n, the coefficients ai, bi are constants or sufficiently smooth functions of t,z = z(t) is a known function, y = y(t) is the unknown function. li z E em, then (12) is a familiar and thoroughly investigated linear equation. lithe function z is less smooth, the righthand side of (12) may involve distributions. It will be shown below that if z is an ordinary (locally summable) function, the solutions of equation (12) are ordinary functions and can be found without the use of the theory of distributions. To pick a unique solution, one may impose the usual initial data
y(to) = Yo,
y'(to) = y~, ... ,
y
(nl)
(t 0 ) _ Yo(nl) ,
only for such a to for which the functions z, z', . .. ,z(m) are continuous. One often considers the following problem. We"must find the solution y(t) for t ~ 0 if it is known that y(t) == z(t) == 0 for t < 0 (that is, we wish to find the reaction of the system to the external action which starts only at the moment
t = 0). Different methods can be applied to the solution of this problem. li all and bi are constant, then, for example, one can integrate the function z as many times as is needed for the derivative u(m) of the obtained function u to be continuous or at least Lebesgueintegrable on the interval 0 ~ t ~ t l , that is ai
i
=
1,2, ... , k.
The lower limit of integration a: < 0 is arbitrary since z == 0 for t < OJ the number k is such that the function u = Zk(t) E em. Let us seek the solution x(t) of the equation
(13)
x(n)
+ an_lX(nl) + ... + aox = bmu(m) + bm_lu(ml) + ... + bau,
vanishing for t < O. Having differentiated both sides of the equality (13) k times, we see that the function y = x(k) is a solution of equation (12) which vanishes for t < O. Another way [58] of solving equation (12) consists in reducing this equation to a system of equations containing no distributions. Let all the ai and bi be constantj if m < n, then bi = 0 for i > m, the function z(t)is continuous or Lebesgueintegrable on every finite interval. Let us introduce new unknown functions Xl, ••• ,Xn by the formulae
i
(14)
=
n  1, n  2, ... , 1.
Successively substituting Xn into the formula for Xnl, and then Xnl into the formula for X n 2, etc., and using equation (12), we obtain the first equation of the following system (the rest of the equations are presented in (14)): X~
(15)
x~
= boz  aoy, = blZ  alY + Xli
Equations with. Distributions Involved as Summands
§2
21
Substituting 11 = 3IA + bAz into the system (15), we derive a system of the normal form. H the function z is continuous, this is an ordinary linear system with constant coefficients, and if z is Lebesgueintegrable, this is a linear Caratheodory system. Having found the solution of this system, we obtain the solution of equation (12) by the formula 11 = 3IA + bAz. We shall prove this. Differentiating the equality y = x A + b,.z n times and replacing, after each differentiation, 3I~, 3l'A1,"" x~ by the righthand sides of equation (15), we obtain y' = b,.z' + b,._lZ  4,.lY + 31,.1, 11" = b,.z" + b,._lZ' + bn  2 z  4 n l11' 
(16)
a,.2Y
+ X,.2.
Therefore, y satisfies equation (12). H the function z is bounded on the interval a < t < P and if for t the functions z, z', . .. have jumps
z( l' + 0)  z( l'  0) = [z].
= ., E (a, P)
z'(1' + 0)  z'(1'  0) = [z'] •.•. ,
then for t = l' the jumps fyI, [11'],'" of the functions y. y' •... can be expressed [59] in terms of Iz]. [z'] •... and of the coefficients lli. b,. We should note in this connection that in the system (15) the functions Xlo" •• x,. are continuous. Hence, from the relations 11 = 3In + bAz and (16) we have
[y] = b,.[z]. [y'] = bA[z'] + bn 1 [z] a,.l[y], [1I"J = bn[z"J + bn 1[z'J + b"_2[Z] 4n 1[y'J  an2[1I),
(17)
The jumps [11(10)1 a.re therefore expressed, through the jumps [z), [z'I, . .• , [z("')J. by the formula
(18)
k
The coefficients Co
= bn • Ci
Co, Cl, • ..
C1
= 0, 1.2, ....
are determined successively:
= b,.1 
C04,.I,
C2
= bn 
2  COa,.2  ClanI, •.. ,
= b"i  COa,.i  Clani+1  ..•  Ci1a,.lJ
i = 1,2, ....
H the function z = 0 for t < 0, and z E em for t > 0, m is the same as in (12), then y(t) for t > 0 can be found as a. solution of equation (12) with the initial data
y(+O) = [1IJ,
11'(+0) = [11'],'" ,
Equations . .. Discontinuous only in t
22
Chapter 1
where [y], [y'], ... are given by formulae (17) or (18) in which [z] = z(+O), [z'] = z'(+O}, .... Now let the coefficients ai,bi in (12) depend on t,z(t) E L 1 (loc), that is, the function z(t) is Lebesgueintegrable on each finite interval contained in its domain of definition. Then z', z", ... are distributions. For the products b.(t)z(i) and ai(t)y(i) to have sense, we require that
bi E
(19)
0',
i = 0,1, ... ,no
Then the product b(t)z(k) (t) is given by the formula 160]
bz(k) == z(k)b == t(l)k i 0. (b(ki l z ) (iI, i=O
(20)
where Ck are binomial coefficients, and the derivatives are understood in the sense of theory of distributions. We shall prove formula (20) first for the case b E Coo. Using the definitions of the product of the distribution by the function from COO and the derivative of the distribution ([61], Chapter 1), we have for any test function IP E K
Expressing (bIP)(k) by means of the Leibniz formula, we obtain
(bZ(k),IP)
=
(Z,(l)ktCkb(ki)IP(i)) i=O
= (l)kt Ck (z,b(k'IIP(i l ). i=O
Using again the definitions of the product and of the derivative for the distributions, we get (21) k
(bz(kl,IP)
= (l)kE0.
k
(b(ki)z,IP(i l )
i=O
=E
((l)k iCk(b(ki)z)(i l ,IP).
i=O
From this there follows (20) for the functions bE Coo. Let now bECk. Approximating the function b by the sequence of functions bi to b (convergence in C k ), we find that the expressions derived from the righthand side of (21) by the replacement of b by bi converge to the righthand side of (21) for any function IP E K. For bECk the righthand side of (21) is a linear continuous functional on the functions IP E K, that is, a distribution. The product bz(kl can therefore be defined as a distribution which satisfies the equality (21) for any function IP E K. This is equivalent to (20). To find the solution of equation (12) under the condition (19), one may either reduce it to a linear Caratheodorytype system or use the representation of the solution in the integral form [62] (for the case y(t) = z(t) = 0 for t < 0)
y(t) = bn(t)z(t)
r ECI(S)tll(t,S)z(s)ds,
+ 10
o
n
1=1
t
~
0,
Equations with Distributions Involved as Summands
§2
23
where VI(t, 8) are solutions of a linear homogeneous equation with particular initial data, and CI (8) are expressed through the coefficients /Ii, bi and their derivatives (see [62]). To reduce equation (12) under the conditions (19) to a linear Caratheodory system, transform each product bkz(k) and aky(k) by formula (20). Combining terms with the same i value, reduce equation (12) to the form (22) Un
Uk
= ( ale: 
= Y  bnz,
k, C Ie:+l ale:+l
Unl
= anlY 
(b n 
l 
nb~)z,
... ,
(,.lk») Y
k a"le:+2  ... + (1)nlkck + C k+2 nl anl
 (b k  C:+1bk+1
+ C:+ 2bZ+2 
... +
(_1)nle:C!b~nk»)
z,
Mter the introduction of new unknowns :Z:nl
=
:z:~
equation (22) takes the form :z:~ + Uo system (23)
+ UnI, ••• ,
= O.
Thus, equation (12) is reduced to the
i
= 1, ... ,n 1,
where Uo, Ul, ... , Un l are expressed through Y and Z by means of the above formulae, and 1/ == Un + bnz should be replaced by :Z:n + bnz. Under the above assumptions concerning the functions z, Gi, bi the system so obtained is a linear Caratheodory system. REMARK: IT one adds an ordinary function (not a distribution) I(t, 1/, z) to the righthand side of equation (12), the first equation of the system (23) will have the form :z:~ = uo + I, and the other equations will remain unchanged. A linear equation with the coefficients ai E cmn+i with a righthand side which is a derivative of any order m > n of the integrable function (24) can be reduced to similar equations with smaller values of m. The change of variables 1/ = z + g(mn) gives
Each term of the righthand side can be transformed by formula (20). After this the righthand side will have the form h~ml)(t) + ... + hm(t), where hi(t) are integrable functions. By virtue of the linearity of the equation, its solution is a sum of solutions of equations of the form (24), but with smaller m values. By means of a finite number of such transformations, equation (24) is reduced to similar equations, but with m ~ n, that is, to equations of the form (12), but with variable coefficients.
24
Equations . . , Discontinuous only in t
Chapter 1
3. The linear equations
(25)
X'
=
A(t)x + I(t)
with distributions I(t); x E Rn, A is a matrix, are considered in [34], [66], and [63]. If A(t) belongs to oo , then I(t) can be any distribution vanishing for t < (see [63]). Let now I(t) = g(m+l) (t) be an (m + l)th derivative of the function g(t). Let us consider two cases: a) the function g(t) is measurable and locally bounded, A(t) belongs to Wi" (loc), that is, A (ml) (t) is locally absolutely continuous and A (m) (t) belongs to Ldloc), m ~ 0; b) the function g(t) is of locally bounded variation, and A(t) belongs to
°
e
emI.
After the change of variables x
(26)
= Y + g(m) (t)
we obtain from (25)
y' = A(t)y + A(t)g(m) (t).
Now consider the case a). If m = 0, A(t) E Lt{loc), then (26) is a linear Caratheodoryequation. If m ~ 1, then, by virtue of (20), we have
A solution of equation (26) can therefore be expressed in the form y where Yk (k = 0,1, ... ,m) is a solution of the equation
... + Ym,
= Yo + YI +
(28) For k = m, equation (28) is a Caratheodory equation. For k = 0, 1, ... , ml the function hk(t) is measurable and locally bounded. Equation (28) is therefore of the same type as (25), but the number m + 1 is replaced by a smaller number mk. Here one can again make the change Y = z+ (hk(t))(mkl), etc. After a finite number of changes of variables, equation (25) is reduced to Caratheodory equati6ns. Consider the case b). Since A E emI, m ~ 1, the product A(m)g in (27) is a distribution equal to
(29) The integral is understood in the sense of Stieltjes. Formula (27) remains valid. To prove this formula, we consider a subsequence Ai(t) + A(t) such that on every finite interval the derivatives A~k) (t), k = 0, 1, ... , m  1 are absolutely continuous and converge uniformly to A (k) (t) as i + 00. The possibility of a limit transition under the Stieltjes integral sign in (29) follows from [64] (pp. 250 and 254). In formula (27), with Ai instead of A and with the last term transformed according to (29), one can pass to the limit
§2
Equations with Distributions Involved as Summands
25
in the sense of the theory of distributions. We obtain that formula (27) holds also for A E cml, and for a function g of locally bounded variation. By virtue of (29), equation (28) for k = m has the form
where p(t) is a continuous function equal to the integral in (29). By the change Ym = Z + (l)mp(t) this equation is reduced to a Caratheodory equation. For k < m, equation (28) is reduced to a Carathedory equation in the same way as in the case a). Thus, in both cases equation (25) can be reduced to Caratheodory equations. By virtue of Theorem 3, §1, its general solution has the form
where Ul! ••• , u,. is a fundamental system of solutions of the homogeneous equation u ' = A(t)u, and :z:t(t) is a particular solution of equation (25); the function Xl (t) may be a distribution. Consider the system (30)
X'
= A(t):z: + B(t)y + C(t)y',
where y(t) is a known vectorvalued function, possibly a distribution, :z:(t) is an unknown, and the coefficients belong to the class Coo. SiIch a system can be reduced to a Caratheodory system in the same way as the system (25). In particular, the change :z: = z + C(t)y reduces the system (30) to the form
z'
= Az + (AC + B 
C/)y.
In [65], the wellknown Cauchy formula, which expresses the solution of a linear inhomogeneous equation through its righthand side and the reaction of a homogeneous equation to an impulse, is extended to linear equations of the general form with distributions. The general integrodifferential representation of solutions of such equations is given. Stability of solution of equation (25) in a special topology of the inputoutput operator, i.e., the operator transforming a given function !(t) into a solution x(t), is investigated in [66], [63], and [34] (pp. 18D187). Some nonlinear equations, for example those derived from (25) by adding to the righthand side the function Ip(t, x), and derived from (12) for z = Ip(t, y) + !(t), are analyzed in [63] and in [66][68]. In particular, sufficient existence and stability conditions for periodic solutions are obtained. 4. Solutions of a linear system with constant coefficients, which is not solvable for higher derivatives of each of the unknown functions, may appear to be discontinuous functions or distributions even if the righthand sides are ordinary continuous functions. The general method of solving such systems, using reduction of a bundle of matrices to a canonical form, is presented in [69] (Chapter 12, §7). Some other methods are proposed in [10]. [71] (elementary methods and application of the Laplace transform) and in [12][15] (application of generalization of the inverse matrix concept). Below we present an elementary method of solving such systems, which is applied in the general case and is a development of the method proposed in [10).
26
Equations . .. Discontinuous only in
Chapter 1
t
Consider a system of m equations with n unknown functions and with constant coefficients which is written in the vector form A:i: + Bz = Itt).
(31)
Here z and I are ndimensional vectors, A and Bare m A system with derivatives of any order AIoII 0, that is, the second set of equations in (34) is present, for a further simplification of the system we select q linearly independent columns from the matrix Co. For example, let them be columns from (s+ l)th to rth (ifthis is not so, we renumber the unknowns 11:1, ••• ,11:" in order that these columns might occupy the indicated positions). Let D be a square matrix composed of these columns of the matrix Co. Then det D ~ O. Multiplying the second row in (34) by the matrix Dl on the left, we obtain
=
=
(35)
Loz
= I(t).
The columns of the matrix Lo, from (8+ l)th to rth, form a unit matrix, that is, the system (35) is solved with respect to the unknowns Z'+l, ... , Zr. We express these unknowns in terms of the rest of the variables and I(t) and substitute them into the first row of (34) (that is, we multiply by some numbers the equations involved in the system (35) and the equations derived from them by differentiation with respect to t and subtract them from the equations· contained in the first row of (34». In that way, we eliminate the unknowns Z.+l, .•. , Zr from the first row of (34), and obtain
Aoz + Boz = lo{t),
(36)
Bo
where from (s + l)th to ,.th columns of the matrices AD and consist only of zeros. The rows of the matrix AD remain linearly independent. Hence the matrix AD contains s linearly independent columns, for instance, from 1st to sth (otherwise we change the numeration of the unknowns so that these columns occupy the leading positions). Let Do be a matrix of these columns. Then detDo ~ O. Multiplying (36) by Dolan the left, we have
Hz + KII: = k(t), where the first 8 columns of the matrix H form a unit matrix. Introducing the notation
(T denoting a transposition), we write the whole system in the form
=k(t) + Law =I(t)
u + Haw + KIU + Ksw (37)
L 1 u+
tI
0= h(t)
(8 equations),
(q
=r 
s equations),
(m  ,. equations).
Some groups of the equations are absent if at least one of the numbers s,q, m  r is equal to zero. The system (37) is equivalent to the initial system (31) because all the transformations are reversible. The system is resolvable only in two cases:
(38)
a) r
=
tn.j
b) r < m,
h(t) == O.
If one of these conditions is fulfilled, the vectorvalued function w can be chosen arbitrarily. Then from the first group of equations U is expressed in terms of wand It arbitrary constants
Chapter 1
Equations ... Discontinuous only in t
28
and from the second group of equations the vectorvalued function v is uniquely expressed in terms of the already found u and w. Thus, for the system to have a solution, it is necessary and sufficient that one of the conditions (38) be fulfilled. This gives m  ,. (scalar) compatibility conditions for the system. If these conditions are fulfilled, the whole set of solutions of the system depends on n  ,. arbitrary functions (components of the vector wet»~ and on s arbitrary constants. The numbers,. and s can be found directly through the coefficients of the system (31) without making the above transformations. The number,. is equal to the rank of the matrix pA + B, the elements and the minors of which are polynomials in p. In this case,. rank(pA + B) is the highest order of minors which are not identically zero. and s is the highest degree of the polynomials, which are equal to the minors of order,.. For the system (37) this is obvious, and in passing over the system (31) to (37) the numbers,. and 8 remain unchanged. The inequalities 0 ~ 8 ~ P ~ ,. ~ min{m,n} always hold; if p m, then 8 m. The case B ::: 0 is possible even for,. m n, p::: n  1. For example, for the system
=
(39) we have m
=
= = :i: + Ii  z = I(t),
= ,. :::
:i:+ti+II=O
= n =,. = 2, p = I, B =o. The solution is expressed by the formulae z =  i(t) /(t), II = i(t).
Thus, the system (39) has only one solution which is independent of arbitrary constants. As distinguished from the case I, the solution is less smooth. If in the system (39) I E om, then z, II E omI. If the function I has a jump, for example,
I(t)
=0
(t < 0),
I(t)
=1
(t > 0),
then z and II are distributions:
z
= c5(t) 
I(t),
II
= c5(t).
Even if the function I(t) is continuous, but not absolutely continuous (for an example of such a function see [64J, p. 232), z and II are distributions. III. Consider in more detail the case If < ,. ::: m n. Now the system is compatible and the general solution depends only on 8 arbitrary constants, 8 < n. Thus the solutions of the system occupy, not the entire space, but only a certain manifold of smaller dimension. Therefore, the solution does not exist under arbitrary initial conditions z(to) zoo It follows from (34) that if lo(t) and 90(t) are continuous, for the existence of a continuous solution it is necessary and sufficient that the initial conditions be such that
=
=
(40)
Ooz(to)
= 90(to).
= ,. 
The vector equality (40) contains q B compatibility conditions. For some systems describing real physical problems it is reasonable to consider discontinuous solutions. This happens, for instance, when in setting up a system of differential equations one disregards small "parasitic" parameters, taking account of which would have led to the appearance in the system of new terms with small parameters at derivatives (see [IJ, Chapter 10). If the initial conditions do not satisfy the relation (40). the solution jumps rapidly to a point where the relation holds. Let us find the coordinates of this point under the assumption that during such a jump the solution remains in a bounded domain of the zspace (this assumption is justified in those physical problems where boundedness of the solution is a priori clear, for example, from energy considerations) and that the function lo(t) is bounded and the function 90(t) in (34) is continuoue for t to. Now the third row in (34) is absent because,. m n. During a very small time interval T the solution with the initial data z( to) ::: Zo reaches the point ZI which satisfies the compatibility condition (40), that is, satisfies the condition OOZI 90(to + T). From (34) we have
=
= =
=
AO(ZI  zo)
=
[
IO+~
10
(fo(t)  Boz(t» dt.
§3
Differential Equations with Distributions in Coefficients
Since the functions fo(t) and z(t) are bounded, then as
'I'  
29
0, we have
(41) The rows of the matrices Ao and Co are linearly independent, and the coordinates of the point ZI are therefore uniquely determined from the system (41). If the system is reduced to the form (37), then in the case r m n the function 1U is absent, and the equalities (41) take the form
= =
In [70), [71) it is pointed out that the same result can be obtained by applying the Laplace transform to the solution of (31) for the case r = m = n.
§3 Differential Equations with Distributions in Coefficients Linear equations containing distributions in coefficients and some nonlinear equations are considered. Classes of such equations reducible by change of variables to systems of Caratheodory equations are pointed out. Various limit transitions from differential equations with continuous righthand sides to equations with distributions are analyzed. 1. The classification of distributions, which is an extension of the classification of summable functions (classes Lp and W:), is given in [60j. For 0 < "I ~ 1 the class Mb) coincides with Lp(loc) for p = 117, that is, consists of functions I(t), 00 < t < 00, such that I(t) and l/(t) IP are summable on each finite interval. Let, for any integer k ~ 1 and 0 < "I ~ 1, the class M( k + "I) offunctions be absolutely continuous on each finite interval, together with derivatives up to (k  1)th order and let these functions have a kth derivative which belongs to Mb), that is, M(k + "I) = W:(loc), p = 1/"1. For an integer k ~ 1 the class M(k +"1) is a class of distributions which are derivatives of order k of the functions of the class MbJ. The function I is regarded as belonging to M( 00) if and only if f == O. For functions which are defined for not all t E (00, +00), but only on a given finite interval (c,d), one can also consider the classes M(a). A function belongs to the class M(a) on the interval (c, d) if it is a restriction onto this interval of some function of the class M(a) defined for 00 < t < 00. The following properties of classes M(a) (on the entire real axis or on a finite interval) are obvious: 1° If a < b, then M(a) c M(b). 2° If I E M(a), then 1(1c) E M(a + k), and vice versa. 3° If IE M(a), c = const, then cl E M(a). 4° If I E M(a), 9 E M(b), then f + 9 E M(c), c = max{a; b}. 5° If I E M(a), 9 E M(b), a ~ 1, b ~ 0, then Ig E M(c), c = max{a;b}. 6° If I E M(a), 9 E M(b), a> 0, b > 0, 0,+ b ~ 1, then Ig E M(a + b). (Indeed, on any finite interval the function h = 1/1 1/ 4 + Igll/b E £1; since III ~ h a , Igl ~ h b , then I/gl ~ ha+b, i.e., Ig E L p , p = 1/(0, + b)). The product of the distribution I E M(a), a> 1 with the ordinary function 9 E M{b) in the case 0,+ b ~ 1 can be defined by means offormula (20), §2. Let (1)
a = k + a, k ~ 1 be an integer,
0 ~ a ~ 1;
b = k + {3,
a + {3 ~ 1
Chapter 1
Equations . .. Discontinuous only in t
30
Then f
= h(k), hE M(o:), g(k)
(2) gh(k)
= h(k) 9 = (hg)(k)

E M(f3). By formula (20), §2
C~ (hg') (kl)
+ C~ (hg,,)(k2) + ... + (_I)k hg(k).
According to 5° and 6°, hg, hg', ... , hg(kl) E M(o:), hg(k) E M(c), c = 0: + f3 or c = 0:. Hence, the righthand side of (2) is a distribution of class M(k + 0:) = M(a). Thus, under the condition (1) the product fg can be defined by formula (2), fg E M(a). Acceptance of this definition can be motivated by the limit transition, as for formula (21), §2, but in another, k and f3dependent metric. In the case 0: + f3 > 1 the product hg(k) may be not a locally integrable function. Then formula (2) cannot be applied. From what has been said we have the following result. LEMMA 1 [60J. If f E M(a), 9 E M(b), a
+b ~
1, then the product fg is
defined,
(3)
a 0 b = max{a;b;a + b}.
fgEM(aob),
If a =
00
or b =
00,
we assume a
0
b=
00.
Lemma 1 gives only sufficient conditions for the existence of the product f 9 and for it to belong to one or another class M(c). For instance, let 3
4'
< "I < 1,
f(t) = Itl 1 E
Mb + e),
g(t) =
It 
11'1 E M("!
e> 0 being any arbitrary number. Here a + b > 3/2 > 1, but fg E
+ e},
Mb + e).
If they are not absolutely continuous (locally), continuous functions, functions of bounded variation, and measurable functions do not belong to the class M(O), but belong to the class M(o:) for any 0 > O. If the function f is continuous and the function 9 is of bounded variation, the product f g' is meaningful (although a + b > 1) and is defined by means of the Stieltjes integral
(4)
d (t
f(t)g'(t) = dt}c f(s)dg(s),
where c is any point of continuity of the function g(t). We introduce the norm" IIa for the functions defined on the interval (c, d) and belonging to the class M(a), a ~ 1. If a = m + 0, m ~ 0 is an integer, 0< 0: ~ I, P = 1/0:, then
(m
~
I).
Using the norm introduced, we define convergence. For a ~ 1 the convergence Ii + f in M(a} implies that IIIi  flia + O. For a = k + 0: (k is an integer), 0 < 0: ~ 1, the convergence Ii .+ f in M(a) implies that for some gi E M(o:), 9 E M(o:) we have
f \.  g(k) i ,
g.
+
9 in M(o:).
Then for any a and integer m > 0, from the convergence there follows convergence f.(m) + f(m) in M(a + m).
Ii
+
f in M(a)
Differential Equations with Distributions in Coefficients
§3
~ 1 and
LEMMA 2. If 0 < max{a;b} M(b), a 0 b = "I. we have
a
+b
~ 1,
then for
31
f e M(a), g e
(5) PROOF: IT b ~ 0 < a ~ 1, then 7 = a, Ilglib ~ Ilgllo = max Igl and the inequality (5) holds. The case a ~ 0 < b ~ 1 is similar. IT a > 0, b > 0, a + b ~ I, then 7 = a + b. The inequality (5) takes the form
1/ (a + b)
Raising both the sides of the inequality to the power v =
Igl
V
a+b
= g1,
and writing
a+b b=q,
a=P'
we obtain the known Holder's inequality. REMARK: IT max{a; b} ~ 0, then there holds an inequality similar to (5), but with an b, and Id  cldependent numerical factor in the righthand side.
a,
LEMMA 3. Ifa+b~ 1 and fig,  fg in M(a 0 b).
It  f
in M(a}, g, 
gin M(b) as i 
00,
then
PROOF: IT max{a;b} ~ I, then, but virtue of Lemma 2 and the remark, the norm in M(a 0 b) of each summand in the righthand side of the equality "g,  fg = (t,  f)g,
+ f(gi 
g)
tends to zero as i tends to infinity. IT a = k + a, k ~ 1 is an integer, 0 < a then, by virtue of the above it follows from the relations
that for
g~i)
i
f
= h(le) ,
~
k
,,=
_ g 0), and ... is a convolution sign, then the function
(7)
f(t) ... w.(t) = f.(t) e 0 00 ,
f.(t)  f(t)
r w(t)dt = I, 11 1
32
Equations. .. Discontinuous only in t
Chapter 1
Convergence is understood in the same sense as in the theory of distributions. If f E Lp , 1 ~ p ~ 00, or fEW;", then f. + f in Lp , correspondingly, in W;". Therefore, if f E M(a), then f. + f in M(a). 2. Some equations and systems with distributions in coefficients are reduced to Caratheodory systems by a change of the unknown functions. Consider the equation [77]
y"
(8)
= a(t)y' + b'(t)y = g'(t).
Let the functions a, b, g, and (ba)b, (ba)g be summable and let the derivatives b'(t) and g'(t) be understood in the sense of distributions. One can write this equation in the form
+ by  g)' = (b  a)y' and using the change y' + by  9 = z reduce it to the Caratheodory system (y'
(9)
z' = (ba)(zby+g).
y' = z  by+ g,
To the system (9) one can set the usual initial data y(to) = Yo, z(to) To the equation (8) one can therefore add the following initial data
(10)
= zoo
y(to) = Yo, (y'
+ by  g) It=to
=
ZOo
Here to, Yo, Zo are arbitrary numbers. Initial data of the form y(to) = Yo, y'(to) = yb cannot be given for arbitrary to, but only for those for which the functions b(t) and g(t) are continuous. As is seen from the above, a reduction of differential equations with distributions to Caratheodory equations makes it possible to prove the existence of a solution and to establish the form of admissible initial data. The same method enables us to show that the solution of an equation with distributions in its coefficients depends continuously on the coefficients and is the limit of a sequence of solutions of equations with smooth coefficients ai (t), b.(t), ... , which tend to given distributions as i tends to infinity. This is proved below for equation (8), but the same method can also be applied to the more complicated equations considered in the following theorems. LEMMA 4.
Let a(t), b(t), g(t) E L2 on a closed interval [c,d]
b.(t)
(11)
+
b(t),
gi(t)
+
g(t) in L2
and let YOi + Yo, ZOi + zoo Then the sequence of the solutions of the problems
(12)
y~'
+ ai(t)y; + bHt)Yi = gHt) , = YOi, (y; + biYi 
Yi(tO)
i
= 1,2, ... ,
gi)lt=to = ZOi
§3
Differential Equations with Distributions in Coefficients
33
converges in WJ to the solution of the problem (8), (10). PROOF: Using the change
(13)
we reduce each equation under consideration to a Caratheodory system similar to (9). As i + 00, the coefficients of this system converge in L1 to coefficients of the system (9), the initial data 1/0. and zo., converging also. By virtue of the corollary to Theorem 8, §1, the sequence of solutions 1/., (i = 1,2, ... ) of these systems converges uniformly to the solution of the system (9). Then it follows from (11) and (13) that 1/; + 1/' in L 2 • Hence, 1/. + 1/ in WJ, so the lemma is proved. For any functions a, b, g E L2 we construct the sequence of smooth functions a.(t), b.(t), g.(t), like (7), taking s = 1/i, i = 1,2, .... Then by Lemma 4, the sequence of ordinary solutions 1/. of the problems (12) converges in WJ to the solution of the problem (8), (10) which contains distributions b'(t), g'(t) (the case aft) == get) == 0 is considered in [77)). This makes it possible to extend some known results from the qualitative theory of linear equations [78] to equations with distributions. Eigenvalue problems for the equation x" + Ap(t)X = 0, where pet) = q'(t), the function q(t) being nondecreasing, have also been discussed in
z.
[79], [80j. In the following theorem some of the components of the solution Xl, ••• , Xn are distributions of different classes. Although this property is not retained under linear transformation of coordinates, such systems should be considered, for instance, for the reason that the equation of order n is reducible to a system of this kind. THEOREM 1 [60]. Consider a linear system
(14)
~ dt' = L. a'i(t)xi + f.(t),
dx'
i = 1 1 ••• , n,
i=1
where
(15)
It E M(Ip.).
ati E M(a'i) ,
If there exist numbers A1,"" An, such that
,
(16)
max{Ip.;m~(a'i
(17)
ali + Ai
0
Ai)} ~ A. ~
1,
+ 1,
i
= 1, ... , n ,
i, i = 1" .. , n
(for the notation of a 0 A see (3»), then using a linear change of the unknown functions the system (14) is reduced to a linear Caratheodory system. The system (14) has an ndimensional linear manifold of solutions for which
x. E M(A.),
i= l, ... ,n.
Equations. .. Discontinuous only in t
34
Chapter 1
PROOF: Let the functions Zi be enumerated in such an order that >'1 ~ >'2 ~ ••• ~ >'n. If all >'i ~ 0, then it follows from (16) that all 'Pi ~ I, OI.ii ~ I, and the system (14) is a Caratheodory system. Hence, let
(18) O~m'1 > 0, then m 0 and the sums over indices less than or equal to m will be omitted from the following formulae). It follows from (16) and (17) that for all i and i
=
OI.ii ~ 1 + >'i,
(19) (20)
OI.ii
0
OI.ii ~ 1 + >'i  >'i'
>'i ~ 1 + >'i,
'P; ~ 1 + >.;;
if O;j ;'i ~ >.; + 1. We will show that by a change of variables the system (14) can be reduced to a system in which the number >'n is replaced by a smaller number (21)
=
(22) /Ion max{>.n  1; O}, and the rest of the >.; remain unchanged. Let bni and gn be functions such that b~i an;' g~ of the system (14) in the form
=In.
=
1t
=
We rewrite the last equation
t
t
f)niZi  gn) bnl;Zk + °njZj' i=1 1;=1 i=m+l In the righthand side we replace zk(/c 1, ... , m) by the corresponding righthand sides from (14). The products so obtained, bnl:ol:j and b"TelTe are meaningful since bnl; E M(OI."Te  I), (zn 
=
and by virtue of (19) and (20) 01.,,1;  1 ~ >"a OI.l:i ~ 1 + >'1;, 'PI; ~ 1 + >'1;. Sincek ~ m, >.1; ~ 0, and in the case OTei;..1: ~ 1>'i, 1 + >'1;  >'i' then by Lemma 1
OI..l:i ~
=
(23) b"kaTej E M(vi), Vi (OI."Te  1) 0 OI..l:j ~ 1  >'i' If we take into account the fact that anI:  1 ~ >'n, Ct.l:i ~ 1 + >'1: ~ I, then we have
=
(24) Vj ~ max{>'"i I} /Ion + 1. Similarly, bnTeh E M(lln + 1). In the system (14) with the last equation already transformed, we make the change m
(25)
ZI=II1,,,·,
Znl
= linI,
Z"
= lin + L
bnjlli + gn'
i=1 We derive the system
n
+
L
aiilli
+ aingn + ,;,
i
= 1,2, ... , n 
Ii'
i=m+l m
(26)
d;;
=L
[(ann  dnn)bni  dni] IIi
j=1
m
n
+
L
(anj  dni)lIi 
i=m+l
L
bnl;/I:
+ (an,.
 dnn)g,.,
10=1
m
(27)
dni
=L 1:=1
bnl:aTei E M(Vj),
Vi ~ 1 >'i,
Vi ~ /Ion
+ I,
§3
Differential Equations with Distributions in Coefficients
35
d,.i == 0 if Ai > 1 since in this case a1r.i == O. Ie ~ m (see (21». Using the inequalities (18)(21) and (27). we can prove. as we have done above. that the products occurring in the coefficients of the system (26) are meaningful and that ai,.bni.
(28) (29)
ainU,. E M(ai..
(ann  dnn ) Un EM «1 An)
Let us estimate of (19).
ai .. bni
0
in a different way. Since An
(30)
lI'i
=ain
0
+ A.). C M(I' .. + 1).
A.. ) C M(l
0
An)
>
i
O.
~ m. Ai ~ O. then. by virtue
(ani 1) ~ (1 An)
0
(Ai) < 1 Ai'
Similarly we derive
(31)
(ann  d.. n)bni E M(l  Ai)'
If we take into account that ani  1 ~ An. then instead of (30) and (31) we have
(32)
dnnbni E M(max{>.ni I})
=M(l'n + 1).
We write the system (26) in the form dll' ' = En C'i(t)lIi + h.(t). dt
(33)
i
= 1, ...• n.
i=1
In this case Cii E M("tii). h. E M(¢i). Let
(34)
1'.
= Ai.
i
< ni
I'n
=maX{An 
1; O}
< An.
We will show that the numbers 'l'i • .pi, I'i satisfy the inequalities analogous to (16) and (17). For i < n the inequality 'l'i + I'i ~ 1 follows from (17) and (30). and for i n from (17). (27) and (31) using the fact that 0 ~ 1' .. < A... For i < n the inequality.pi ~ 1'.+ 1 follows from (20) and (28). and for i n from (29) and from the obtained estimate of bn kl1r.. For i < n the inequality 'lii 0 I'i ~ I'i + 1 follows from (20) and (28). and fori = n from (17). (27). and (31). It follows from the inequalities obtained that
=
=
(35)
max{ ¢ii ml;\X('l'i 1
0
I'il} ~ 1'.
+ 1.
i
= 1..... n.
Thus. in the system (33), the coefficients possess the same properties as in the system (14). Consequently, the system (33) can be treated like the system (14). By virtue of (34), under each such transformation of the system the largest of the numbers Ai decreases either by 1, if it was not smaller than 1, or to 0, if it had values between 0 and 1. Hence, after a finite number of transformations we obtain the system n
(36)
dZi de
0() O( ) =~ ~ a'i t zi + Ii t,
i
= 1, ... ,n,
i=1
for which all the numbers A? obtained from Ai by successive decrease are nonpositive. Like the numbers Ai and I'i, they satisfy the inequalities similar to (16) and (35),
(37)
i
I?
= l, ... ,n,
where fP? and a?i are such that a?i E M(a~i)' E M(fP~). Since all A? ~ 0, it follows from (37) that a?i ~ 1, fP? ~ 1. In other words. a?i E M(l), If E M(l), and (36) is a
Equations . .. Discontinuous only in t
36
Chapter 1
Caratheodory system. It follows from Theorem 3, §1 that this system has an ndimensional linear manifold of solutions. We will show that for any solution of this system z, E M(A?), i 1, ... , n. The functions z, being absolutely continuous, Zi E M(O). Hence a?izj E M(a?i 00), a?i 00= max.{a?i; OJ. Thus the righthand side of (36) belongs to Mhf)
=
(38) Then Zi E Mh; 1). i = 1, ... , n. Using this more precise estimate of Zi, we deduce that the righthand side of (36) belongs to Mhll,
(39) Consequently, Zi E Mb; EMbn, k= 1,2, ... ,

1). This procedure of specifying estimates leads successively to
Zi
( 40)
=
We now show that for some k, ""I~  1 :::;;; A? For A :::;;; 0 we have a 0 A max{a; A}. For those i, for which the righthand side of (40) is nonnegative, it is equal to ""If in (38) and to the lefthand side of (37). Therefore, for these i, ""Ii  1 :::;;; A? For the remaining i the righthand side of (40) is negative and, accordingly, in (38) ""If = O. Thus, i 1, .. . 1 :::;;; max.{>.?i I},
""1;
=
,n.
Then, comparing (39) and (37), we have i
= 1, ... n.
Continuing, we have
""I~  1 :::;;; max{>.?; k},
=I, ... ,n; k = 1,2, ....
i
Hence, for a sufficiently large k (41)
Zi
E Mb~  1) C M(A?),
i
= 1, .. . ,n.
Returning from the system (36) by successive changes of variables to the initial system (14), we conclude that the system (14) has an ndimensional linear manifold of solutions. We will show that it follows from (41) that ZI E M(A,). It suffices to consider one of the singletype transitions that arise in going back from the system (36) to (14). Let it be proved for the system (33) that (42)
III E
M(Jl;) ,
i
= l, ... ,n..
The transition from the system (33) to (14) proceeds by formulae (25). From (42) and (34) for i : :; ; m we have IIi C M(Ai), and from (18) and (19) we have
Ai :::;;; 0,
bni E M(a n 1) C M(Ai)'
Differential Equations with Distributions in Coefficients
§3 The product
bnjllj
37
is therefore meaningful. Next,
an;  1 ~ An,
bn;lI; E M(>'n
0
>.;)
= M(>'n),
9n E M('Pn  1) C M(>'n). Taking into accoun.t (26) and (34), we obtain
z. =
tI.
e M(A.)
(i < n),
The theorem is proved.
z,. E M(>.,.).
x.
REMARK 1: The best estimate of the form E M(>..) is obtained if there is equality in all the relations (16). REMARK 2: The transition from the system (14) to a Caratheodory system makes it possible to indicate the initial data under which this system has a unique solution. Some other conditions ensuring uniqueness of a solution are mentioned in [601. REMARK 3: Some nonlinear systems can be reduced to a Caratheodory system in the same way as the system (14). For instance, those a.j(t) in the system (14), for which CIt,j = "'ij + 'Yij, "'ij ~ 0 is an integer, 0 < 'Yi; ~ 1, can be replaced by the bounded functions P'j(t, ... , Xlo,"') of class om;; (with respect to its arguments) dependent on t and on those for which >.,. ~ CIt,j' In the other terms of the equations ofthe system those x;. for which >'j = lj+6;. lj ~ 0 is an integer,O < 6; ~ 1, can be replaced by the bounded functions W'j(t, ... , x,., ... ) of class O'i (in its arguments) dependent on t and on those for which >.,. ~ >'j.
x,.
x,.
In the next theorem the assumption (44) is equivalent to the assumption of a similar theorem from [60], but is expressed in a much simpler form. THEOREM 2 [601. Oonsider the linear equation
(43) where
a. E M(CIt.), i = 1, ... ,
nj
CIt,  i
~
f E Mb). Let us denote max{Cltl,""
Clt n
, 'Y}
=
1'. If
(44)
1  1',
i
= l, ... ,n,
then equation (43) is reduced to a Oaratheodory system and has an ndinlensional linear manifold of solutions
(45) where Ydt) is a a partial solution of equation (43), Clt ••• , Cn are arbitrary constants, and Ul (t), ..• , un(t) are linearly independent solutions of a corresponding homogeneous equation. The solutions (45) belong to M(p.  n), and
(46)
U,(t)
E M(p.o 
n),
i = 1, ... ,n.
38 PROOF:
Equations... Discontinuous only in t
Chapter 1
By the usual change of variables
... ,
y(n1)
= Xn
we reduce equation (43) to the system i = 1, .. . ,n 1;
(47)
We will show that in the case when the conditions (44) are fulfilled, there exist numbers A1, ••• ,.An satisfying the conditions (16) (with equality signs) and (17) of Theorem 1 as applied to the system (47), that is, the conditions (48)
Ai+1 = Ai + 1, i = 1, ... , n; max {y; an 0 .Al; an1 0 A2; ••• ; al 0 An} = An
Expressing all Ai in terms of the number A = An (49)
max {y; an
0
(A  n); a n 1
0
+ 1.
+ 1 we obtain
(A  n + 1); ... ; a1
0
(A  I)}
= A.
We will show that the number .A = J.t satisfies this equation. For A = J.t each of the expressions ai 0 (J.t  i) is meaningful by virtue of (44) and is equal to max{ ai; J.t  i; ai + J.t  i}. It is clear that the sum ai + J.t  i should be included only in the case J.t  i > 0, that is, J.t > i ~ 1. But in this case, by virtue of (44), ai + J.t  i ::;;; 1 < J.t. Therefore, for A = J.t = max{ a1, ••• , an, 'Y} the lefthand side of (49) is equal to J.t, thus A = J.t satisfies equation (49). Thus, for .Ai = J.t  n + i  I the conditions (48), i.e., the conditions (16) for the system (47), are satisfied. The conditions (17) for i = n take the form a n H1
+ J.t  n + iI::;;;
1,
i=
1, ...
,n,
and hold by virtue of (44). For i < n the coefficients in equations (47) are constant and, therefore, ai; in (17) can be taken to be less than any negative number, and the conditions (17) are fulfilled. Thus, the system (47) meets the requirements of Theorem 1. It has, therefore, an ndimensional linear manifold of solutions, and in this case (50)
Y
= Xl
E M(Ad
= M(J.t 
n).
From this there follows the assertions of Theorem 2, except (46). To prove (46), one must apply the result (50) to equations of the form (43), but with f == O. REMARK 1: In practice it is, as a rule, more convenient to reduce equation (43) to a Caratheodory system without first using the system (47). With this objective, those products ai(t)y(ni) in which one of the factors is a distribution (for yeni) this question is settled by use of (50)) are transformed by means of formula (2). Next, as in §2, 2 one combines the terms represented in the form of one and the same order derivatives of some functions. From the derived equation of the form (22), §2 (but with other tli), one can pass over to the system (23), §2.
Differential Equations with Distributions in Coefficients
§3
39
In this way one can not only reduce formally equation (43) to a Caratheodory system, but also prove Theorem 2 by a method independent of Theorem 1. REMARK 2: Using the method proposed in Remark 1, one can also reduce to a system 80me nonlinear equations of order ", for instance, the equation
Carath~dory
n1
(51)
lien)
=L
"WPi ("",,,', .. . ,,,(2I:i »)
+ P1 ("",,,', ... ,,,(A:»)
i=1+1 A:1
+
L 6i(')lOd',,,,,,' ... ,,,(i») + 1('). i=o
Here 0 ~ k ~ n  1, Pj E ciA:, lOj E C" i (in its arguments), the functions IIj and be distributions, I E M(n  k + a), 0 < a ~ 1, (Ji ~ min{k 
for
i > 2k the functions Pi depend on ,
i +1
aj n  k
I
may
+ a},
only, p. EC.
In the case at > 1/2 the functions P' (i > k) are independent of II (2I:i) • Then equation (51) is reduc:;{ to a Carath40dory system and has an ndimensional set ofsolutions which belong to M(a  k). To reduce this equation to a ayltem, we assume that II E M(a  k). Then ,,(A:) E M(OI) and the composite functions
lOi (t,,,(,),. ~. ,,,W(t») E M(a  k + i), I'A:
(',II(t), ... , II (A:) (t» E M(l),
=
Pi E M(OI·
+k 
i),
i> k,
=
where 01· a (01 ~ 1/2), 01· a  1 (01 > 1/2). Therefore, in (51) the products ,,(i)Pi and lIilOj are meaningful and can be transformed by formula (2). Next, as in Remark 1, we derive the eqUation (n) + (ni) , 0 tl "
tin_A:
+ ... ,+ 1 + tlo =
,
and then a Carathfodory system. Estimating smoothness of its solution and going back to equation (51), we get" E M(OI  k). Note that in equation (51) each product 6ilOi can be replaced by the sum of a finite number of summands 6i m lOim, where the functions 6im and lOi". satisfy the same conditions as 6i and lOi' Nonlinear equations with distributions, but simpler than those in (51), were considered in (81]. EXAMPLE: Let us reduce the equation
,,(.) = ""'Pa(t,,,,,,') + 1'2(""''''''''')
(52)
+ h(')101(t,,,,,,') + 60(t)lOO(t,,,) + I(t). to a Carath~dory system. Let 1'2 E Cj 1'3 E C1, 101 E C1, 100 E C 2 , 111 E M(S/2), "0 E M(5/2), IE M(5/2), 11'21 ~ ,,"21'.(""'''')' p. E C. Then the conditions of Remark 2 are fulfilled, n 4, k 2, a 1/2. Hence,,,' E M(1/2) L2(loc). Using the identity (2), we write equation (52) In the form
=
=
=
=
Chapter 1
Equatwns ... Discontinuous only in t
40
=
=
=
where 11~ h. 11~ boo g" f. and the derivatives P~.w~. w&. w~ are total derivatives of the composite functions with respect to t. Assuming tJ2
= 110WO 
g.
we derive the equation 1/(4)
+ tJ~ + tJ~ + tJo = O.
Introducing new unknowns
we obtain the system
The functions formulae
tJi
depend on the variables t. II. II',I/"i here 1/.1/'.11" should be replaced by the
II'
= 2:3.
II"
= 2:2 
tJ2
= 2:2 + 110(t)wo(t. 2:4) + g(t).
This system satisfies the Caratheodory conditions.
3. In [821 the following linear system in vector notation
y' = B'(t)y + g'(t),
(53)
is considered. Here the matrix B(t) and the vectorvalued function g(t) are of bounded variation, B(t) is continuous, and the derivatives are understood in the sense of the theory of distributions. The integral equations (54)
y(t) =
y(~) + fat (dB(s)) y(s) + g(t) 
g(a)
(where the integral is understood in the Stieltjes sense) equivalent to (53) and several more general equations were considered earlier [83], [841. In [82] existence and uniqueness of a solution with initial data y(a) = Yo are proved, the fundamental matrix is shown to be continuous and of bounded variation. The solution is expressed in terms of a fundamental matrix and of the function g(t). The existence of a solution can be proved, for instance, by applying to (54) the successive approximation method and by using the known estimates of Stieltjes integrals ([641, p. 254). Another method of investigating the system (53) is to reduce it to a , Caratheodory system. Using the change y(t) = z(t) + g(t) from (53) and (54), we obtain
z' = B'(t)z + h'(t), (55)
z(t) = z(a) +
h(t) =
it
it
(dB(s)) z(s)
(dB(s)) g(s),
+ h(t).
§3
Differential Equations with Distributions in Coefficients
41
Using the estimates taken from [64], (p. 254) we find that the function h(t) is continuous. Denote the elements of the matrix B(t) by b'i(t) and the continuous function n
t+
L
var bii(S)
.. a~a~t ',J=l
by ret). For any t1, t2 > t1, we have (56) Let t(r) be inverse to ret). It follows from (56) that the functions t(r) and b'i(t(r)) are absolutely continuous. Therefore, the last integral in (55) is equal to ([64], p. 290)
l~(t) dB~~(r)) z(t(r))dr and equation (55) is equivalent to the Caratheodory system
(57)
dz _ dB(t(r)) dB(t(r)) dr dr z+ dr
(()) r .
g t
From this there follow the above assertions on solutions of the system (53). All this is not extended directly to the case where B(t) is a discontinuous function of bounded variation, even for get) == O. In this case the solutions, generally speaking, have discontinuities at the same points as B(t), and the Stieltjes integral in (54) and (55) may not exist (example in [64], p. 249). The theory of distributions does not work here either since, for instance, the product of the deltafunction and its indefinite integral is not defined. H for functions of bounded variation one distinguishes the values y( t 0), yet), y(t+O) and considers separately the left and the right jumps yet) y(t0) and y(t+O) yet), then one can define the integral in (54) where B(t) and y(t) are discontinuous functions of bounded variation. Under different assumptions (for instance, if yet) = yet  0) or yet) = [yet  0) + y(t+ 0)]/2, etc.) one obtains different conditions for the existence of the solution of equation (53) or (54). the solutions themselves being also different. Equations of this kind written in a differential or in an integral form were considered in [85][90]. Several nonlinear equations and systems of this type were analyzed in [90][92]. Equations with impulses belong to this type in the cases where the magnitude of the jump of a solution depends not only on t, but also on the value of the solution before the jump. We will show by a very simple example that in such cases different approaches to the definition of a solution yield different results. Consider the linear equation
(58)
y'
= k6(t)y
(k = const).
Since y' = 0 for t < 0 and for t > O. then
yet)
yet) = c = e(l + a)
(t < 0). (t > 0),
Equations. .. Discontinuous only in t
42 that is, y(t)
Chapter 1
= c(l + a71 (t)) , where 71(t) = 0
(59)
'1(t) = 1 (t> 0).
(t < 0),
To find a, we substitute yet) into (58); '1'(t) = o(t)
ao(t) = ko(t)
(60)
+ kao(t)'1(t).
In the theory of distributions, the product o(t)'1(t) is not defined. If one takes the sequences of smooth functions o,(t) + o(t), '1,(t) + '1(t), the limit of the product Oi(t)'7,(t) does not, generally speaking, exist. In the cases where it does exist, it depends on the choice of the sequences o,(t), '7i(t). Under natural assumptions this limit, if it exists, has the form yo(t), where y may be any number from the closed interval [0,1]. If we assume that o(t)'1(t) = yo(t), we obtain from (60)
k
a=.
(61)
1 k"l
Consider different approaches to the choice of the value of "I. 10 If "I is assumed to be zero, i.e., 0 (t)'7 (t) = 0 (this is equivalent to the statement that by definition the solutions of equation (58) must be continuous on the left), then a = k,
y(t) = c(l + k'1(t)) ,
(62)
c being an arbitrary constant. The same result is suggested by the limit transition
Yi(t) + y(t), where y,(t) is a solution of the delay equation
T. + +0
(63)
(i+oo);
the function o.(t) being different from zero only on the interval (a" (ii), not longer than T., which contracts to the point t = 0 as i + 00, and for some q = const
r
fi '
Jai' o.(t)dt + 1,
(64) In the case y
= 0,
k
= 1, from
y(t) = c (t < 0),
(62) we have [92]
y(t)
=0
(t> 0).
This implies that all the solutions which exist for t < 0 jump to the point y = 0 at the moment t = 0 and remain at that point. Under the initial data y(to) = Yo, to > 0, Yo =F 0, we have the solution y(t) = yo(t > 0) which is not continued to the region t < O. 20 If "I is assumed to be equal to 1/2, i.e., o(t)'1(tj = 1/2o(t) (for instance, from symmetry considerations or taking by the definition the solutions of equation (58) to be such that y(t) = [y(t  0) + y(t + 0)l/2), then from (61) we obtain a = 2k/(2  k). For k = 2, this result becomes meaningless. More precisely, for
Differential Equations with Distributions in Coefficients
§3
43
Ie = 2 a solution with any initial data of the form veto) = Yo =I 0, to < 0 cannot be continued to the region t > o. For Ie > 2 we obtain a solution yet) which changes sign. This is unnatural for equations of the form y' = !p(t)y (in the case of any continuous or summable functions !pet) solutions do not change sign). 3° We shall consider equation (58) as a limit of the equations y~
= kO.(t)y.,
where for each i the function S. is summable, vanishing outside the interval (a.,.8.), and meets the requirements (64); a.,p. + 0 as i + 00. Then
y.(t) For i
+
00 we obtain Yi(t) yet)
(65)
+
=c
= cexp (Ie
fa: 6. (s)dS) .
yet) (t =I 0)
(t < 0),
yet) = celc
(t> 0).
IT the functions (65) are assumed to be solutions of equation (58), then for all values of Ie all the solutions exist both for t < 0 and for t > o. The choice of such a definition of a solution corresponds to the case where in (61) elc
11e
'Y = Ie(elc  1) .
Note that 'Y + 1/2 as Ie + o. Using these arguments, one may come to the following conclusion. Let equation (58) be interpreted as an idealization of equation (63), where " ~ 0,
6.(t) = 0
(66)
(t
~
ai, t ~
.8.),
The integral of 6.(t) over the interval (a •• .8.) is equal to 1, and the numbers are small. Then in the case " ~ P. the solution is close to the function (62), and in the case " = Qto the function (65). Hence, in the case 1". ~ P. it is more convenient to write the limit equation not in the form (58), but as
'" a., P.
a.
a.
yl(t) = k6(t)y(t  O)i Solutions of such an equation are the functions (62). In [921 such limit transitions are considered for more complicated equations (67)
= 1,2, .... Let the functions J(t, z), g(z) and S.(t) be continuous (to tl, Z ERR), J,g ERR, and 6.(t) satisfy the conditions (64) and (66),
i
a.
+
0,
.8. + 0,
z.o
+
Zo
(i
+
00),
The solution of the problem (67) is not necessarily unique.
~ t ~
Equations. .. Discontinuous only in t
44
Chapter 1
THEOREM 3 [92]. Let the problems
u' = f(t, u) v' = g(v) w' = f(t, w)
(68)
(69) (70)
(to~t~O),
(0
~
t
~
u(to) = Xo v(O) = u(O), w(O) = v(l)
1),
(O~t~tl)'
have unique solutions u(t), v(t), w(t). Then for an arbitrary sequence of solutions of the problems (67) (i = 1, 2, ... ) we have
x.(t) x.(t) THEOREM 4. Let lems (68) and
7.
+
(to ~ t < 0), (0 < t ~ tIl.
u(t) + w(t) +
0, 0
i3(r7) and a. ~ t ~ f3i we have
Ix(t  Ti)  u(O) 1 ~ 2'7. Since the function g(x) is continuous, then for any e > 0, for a sufficiently small '7 and i > i 3 ('7) (76)
= go,
Introduce the notation Ig(u(O))/
Then, by virtue of (64) and (76), for ai ~ t ~ f3i, (77)
IJi(t)/
~ q(gO
+ e),
Ji (f3i)
+ g
(u(O))
(i
+
00).
We will show that for a sufficiently large i, and for all i > i4 the solution Xi(t) of the problem (72) for ai ~ t ~ f3i remains in the ball K{/x  u(O)/ ~ q(gO + e:) + 2}. For/tl ~ a, x E K, we have If(t,x)1 ~ mI. By virtue of (75), IXi(aJ)  u(O) I < 1 for large i. Next, (78) If for 0 < t  ai ~ f3i  ai < mIl the solution had at some point reached the boundary of the ball K, the lefthand side of (78) would have been greater than q(gO + e) + 1 at that point, and the righthand side (by (77)) would have been less than this number. This is impossible. According to (75), xi(ai) + u(O) as i + 00. Hence, for t = f3i and (77), we obtain from (78), as i + 00,
(79)
Xi(f3i)
+
u(O)
+ g(u(O).
For t ;;?; f3. the solution of the problem (72) coincides with the solution of equation (73) with the initial data Xi = Xi(f3.) for t = f3i. If f3i > 0, this solution, as the solution of equation (73), can be continued up to t = O. For such a solution of equation (73), we obtain from (75) and (79), both for f3i > 0 and for f3i ~ 0
X.(O)
+
u(O)
+ g(u(O)).
By virtue of the remark to Lemma 6, §1, the solution x.(t) converges uniformly to the solution of the problem (71) on the interval 0 ~ t ~ t l . This solution Xi(t) coincides with the solution of the problem (72) for f3i ~ t ~ tl, and the result follows.
Equatt"ons ... Discontinuous only in t
46
Chapter 1
It is obvious that the continuity condition for f(t, x) can be replaced by the CaratModory conditions; the function f(t, x) in (72) may also depend on the parameter /li, /li + /lo, as in Theorem 6, §1; the function g(x) can be replaced by the continuous function g(t, x). Very general theorems (but with complicated formulations) on continuous dependence on the parameter for differential equations with discontinuous solutions are presented in [92J. Summarizing, one can say that the concept of solution for equations of the form
x' = f(t, x) + ~(t)g(t, x),
where ~(t) is a deltafunction or a derivative of a discontinuous function of bounded variation, the same as for equations (53) with a discontinuous matrix B(t), is not uniquely defined. In the choice of the definition of a solution one must pay close attention to the character of the limit transition which has led to a given equation. 4. The generalized differential equations dz
dt = DF(z, t). are considered in the papers by J. Kurzweil ([15]. [16J, [92J, and others). It is pointed out that under certain assumptions such an equation can be written in the form
a dzdt = F(z t) at " the derivatives being understood in the sense of distributions. Under certain conditions (different in [15J and in [92]) the author proves theorems on existence and uniqueness of solutions, on continuous dependence of a solution on initial data and on the parameter. The solution of equation (80) is defined by means of a generalization of the Perron integral [15J. In [161, [92] it is assumed that
(81)
IF(z, t2)  F(z, tl) I ~ Ih(t2)  h(tl)I, IF(z, t2)  F(z, tl)  F(tI, t2)
+ F(tI, h)1 ~ w (Iz  tiD Ih(t2) 
h(ttll,
where the function h(t) is nondecreasing, continuous on the left, and w('7) is nondecreasing, continuous, and w(O) == O. Under these assumptions, the solution of equation (80) with the initial data z(to) Zo is proved to exist on some dosed interval [to, to + ul, u > O. The solution satisfies the condition
=
=
and is therefore continuous on the left. In the case w('7) k'7 the solution is unique for t ~ to. [921: In the region Izl < I, It I < I, the function
EXAMPLE
(82)
F(z, t) =
2:
~
(t
0),
satisfies both the conditions (81) with w('7)
h(t) = 0, With the initial condition z(to)
(t
= zo.
z(t) == Zo
(t
= 0 (t > 0)
='7,
~ 0)
to
F(z, t)
h(t) == 1
(t> 0).
< O. we have the solution
~
0).
z(t)
=0
(t > 0).
§3
Differential Equations with Distributions in Coefficients
47
With the initial condition z(to) = zo, to > 0, the solution z(t) = Zo (t > 0) cannot be continued to the region t ~ 0 if Zo ;i: O. In this example, equation (80) with the function (82) can also be written in the form
Zl(t)
= 6(t)z(t 
0).
We will show that under the con(}jtions (81) equation (80) can be reduced to an equation with impulses, similar to equation (72). As is known ([64), p. 290), the function h(t) of bounded variation on the closed interval [to, tIl can be represented as
h(t)
(83)
= tp(t) + ret) + ,(t),
where tp(t) is absolutely continuous, ret) is a continuous function of bounded variation which almost everywhere has r'(t) 0, ,(t) is the algebraic sum of the jumps of the function h(t) on the interval [to, f). If the function h(t) is nondecreasing, so are the functions tp, r,'. Let the function h(t) from (81) be represented in the form (83). Then the change of the independent variable t + ret) r, z(t) z(r) reduces equation (80) to the equation
=
=
(84)
d~~) = I
=
(r,z(r» +
E 6(t  tj)gj (z(r  0», j
a I(r,z) = a; F (z,t(r» ,
gj(z)
= F(z, rj + 0) 
F(z, rj),
the sum being taken ·over all the points rj of discontinuity of the function h(t(r». It can be shown that for almost all r
I/(r, z)1
~
tp' (t(r»
+ 1,
I/(r,z)  l(r,tI)1 ~ w (Iz  til) (tpl(t(r»
+ 1).
Hence, the function I satisfies the Caratheodory conditions, and if the Lipschitz condition (6), §1.
we,,) = Ie", it also satisfies
CHAPTER 2
EXISTENCE AND GENERAL PROPERTIES OF SOLUTIONS OF DISCONTINUOUS SYSTEMS Various definitions of solutions of differential equations and systems with discontinuous righthand sides are considered for the cases where, by contrast with Chapter 1, the righthand sides are not continuous in 2:. The range of applicability of different definitions is indicated. For differential equations with discontinuous righthand sides and for differential inclusions, existence ofsolutions is proved and the properties of these solutions are analyzedj in particular, the dependence of solutions on initial data and on righthand sides of equations, and the properties of integral funnels are examined.
§4 Definitions of Solution Several definitions of solutions of differential equations with discontinuous righthand sides are proposed. The connection between such equations and differential inclusions is established. The velocity of motion along a surface of discontinuity (the derivative dx/dt for a solution lying on a surface of discontinuity) is determined for different cases. The velocity of motion along an intersection of surfaces of discontinuity is found in two main cases. 1. The definition of a solution (given in §1) as an absolutely continuous function satisfying the equation almost everywhere is not always applicable for equations, the righthand sides of which are discontinuous on an arbitrary smooth line or on a surface S. This definition can be applied in the case where the solutions approach S on one side and leave S on the other side. Here the solution passes through S and satisfies the equation everywhere except at the intersection point at which the solution does not have a derivative (Example 1 of the Introduction) . In the other case where on both sides of a line or a surface of discontinuity S the solutions approach S, this definition is unsuitable because there is no indication of how a solution which has reached S may be continued (Example 2 of the Introduction). To provide the existence and the possibility for solutions to be continued in this case, it is necessary either to change the value of the righthand side of the equation in some way at its points of discontinuity or to define it at these points in case it has not already been defined there. 48
§4
Definitions of Solution
49
It is necessary to have a definition of the solution which will cover both these main cases and be formulated irrespective of the position of lines and surfaces of discontinuity. The solutions thus obtained must meet the requirements mentioned in the Introduction. The definitions of solutions presented below are usually applied to equations and systems with piecewise continuous righthand sides, but some of them are also suitable in more general cases. In what follows, :z; denotes a point of the ndimensional space Rn with coordinates :Z;1, .•• , :Z;nj 1:z;1 = (:z;~ + ... + :z;~)1/2. A function ,or a vector:'valued function f(t,:z;) is called piecewise continuous in a finite domain G of an (n+ 1)dimensional (t,:z;) space if the domain G consists of a finite number of domains G. (i = 1 ... , I), in each of which the function f is continuous up to the boundary, and of a set M of measure zero which consists of boundary points of these domains. The function is continuous in the domain up to the boundary if, when its argument approaches each point of the boundary, the function tends to a finite limit, possibly to different limits for different boundary points. IT the domain G is infinite, then in the definition of a piecewise continuous function each finite part of the domain G must have common points only with a finite number of domains G•. The most frequent case is the one where a set M of points of discontilluity of the function f consists of a finite number of hypersurfaces. In an mdimensional space a set S is called a kdimensional hypersurface if in the neighbourhood of each of its points a (or a cluster point a of the points of S) all the coordinates of the points of the set S are continuous functions of some k of these coordinates varying over a certain kdimensional domain Gk(a). For example,
i = 1, ..• ,mj IT in this case, a point a is the image of a boundary point of the domain Gk(a), the point a belongs to the boundary of the hypersurface S. IT all !Pi belong to CP, that is, if they have continuous derivatives up to order p inclusive, the hypersurface S belongs to class CP. Hypersurfaces of class Cl are called smooth. IT all the functions !Pi are linear and the domain Gk(a) is a kdimensional space, the hypersurface is called a hyperplane. A onedimensional hypersurface is a line. For brevity an (m  I)dimensional hypersurface (hyperplane) in an mdimensional space is called hereafter a surface (plane), and as far as other hypersurfaces are concerned, their dimension will be specially indicated each time. 2. Consider an equation or a system in vector notation
(1)
x = f(t,:z;)
with a piecewise continuous function f in a domain Gj :z; ERn, X = dx/dtj M is a set (of measure zero) of points of discontinuity of the function f. Most of the known definitions of solution may be presented as follows. For each point (t,:z;) of a domain G, a set F(t,:z;) in an ndimensional space is specified. IT at the point (t,:z;) the function f is continuous, the set F(t, x) consists of one point which coincides with the value of the function f at this point. IT (t,:z;)
Solutions of Discontinuous Systems
50
Chapter 2
is a point of discontinuity of the function I, the set F(t, x) is given in some other way. A solution of equation (1) is called a solution of the differential inclusion
X E F(t, x),
(2)
that is, an absolutely continuous vectorvalued function x(t) defined on an interval or on a segment I for which x(t) E F(t, x(t)) almost everywhere on I. Of major interest are the methods of definition of F(t, x) at the points of discontinuity of the function I, under which the differential inclusion (2) can be applied to an approximate description of processes in real physical systems. Let a physical system be described outside a certain sufficiently small 5neighbourhood of a set M by the differential equation (1). Then to construct a set F(t, x), one needs some information on the behaviour of the physical system in this 5neighbourhood. To justify the use of a mathematical description (2) of a physical system, one must show that for a sufficiently small 5 the motion of the physical system is arbitrarily close to a certain solution of the differential inclusion (2) (for instance, tends to it as 5 + 0). A rigorous justification of the possibility of using the mathematical description (2), under different assumptions concerning the behaviour of the system in the 5neighbourhood of the set M, is given in 3, §8. Here we present only the most frequently used methods of definition of the function F(t, x) on the set M. The definition a) given below is applicable, in particular, to delay systems of any kind (for more details see 10 _3 0 , 3, §8), as well as to some systems with dry friction. a) The simplest convex definition [93J. Let for each point (t, x) E G the set F(t, x) be the smallest convex closed set containing all the limit values of the vectorvalued function I(t, x*) for (t, x*) ¢ M, x* + x, t = const. A solution of equation (1) is a solution of the inclusion (2) with the F(t, x) so constructed. Since M is a set of measure zero, then for almost all tEl the measure of the crosssection of the set M by the plane t = const is equal to zero ([64], p. 371). For such t the set F(t, x) is defined for all (t, x) E G. At points of continuity of the function I the set F(t, x) consists of one point I(t, x), and the solution satisfies equation (1) in the usual sense. H the point (t, x) E M lies on the boundaries of crosssections of two or several domains G 1 , .•• , Gk intersected by the plane t = const, the set F(t, x) is a segment, a convex polygon, or a polyhedron with vertices t. (t, x), i ~ k, where
h(t, x) =
(3)
lim
(t,"'*)EG.,,,,*,,,
I(t, x*).
All the points I.(t, x) (i = 1, ... , k) are contained in F(t, x), but it is not necessary that all of them be vertices. Consider the case where the function I(t, x) is discontinuous on a smooth surface S given by the equation cp(x) = O. The surface S separates its neighbourhood in the x space into domains G and G+. For t = const and for the point x* approaching the point xES from the domains G and G+, let the function I(t, x*) have the limit values lim I(t,x*) = r(t,x),
"'OEG, :&:+::1:
lim "'*EG+, :z:·+z
I(t, x*) = j+ (t, x).
Definitions of Solution
§4
51
Then the set F(t,3;) is a linear segment joining the endpoints of the vectors f (t, 3;) and f+ (t, 3;). Here and below we assume these vectors to start from the point 3;. IT for tl < t < t2 this segment lies on one side of the plane P tangent to the surface S at the point 3;, the solutions for these t pass from one side of the surface S to the other (Fig. 3).
Figure 9
Figure 4.
If this segment intersects the plane P, the intersection point is the endpoint of the vector fO(t, 3;) which determines the velocity of motion .2: = fO(t, 3;) along the surface S in the 3; space (Fig. 4). This means that the function 3;(t) satisfying the equation
(4) is assumed to be a solution of equation (1) by virtue of the definition (2). If /0 ;j: f, f O;j: f+, such a solution is often called a sliding motion. A continuous function 3;(t), which on a part of the time interval under consideration lies in the domain G (or in G+) and there satisfies equation (1), and on the rest of this interval lies on the surface S and satisfies equation (4), is of course also a solution of equation (1) in the sense of the definition (2). Do all such motions occur in real physical systems? Here one should distinguish between two main cases. Let fii(t,x) and fit(t,3;) be projections of the vectors f (t, 3;) and f+ (t, 3;) onto the normal to the surface S at the point Xj the normal is directed towards the domain G+. If the vectors f(t,3;) are directed to the surface S on both sides, that is, Iii > 0, fit < 0, then near the surface S all the solutions are approaching it from both sides as t increases, and none of them can leave S. A solution which at some moment tl passes through a point of the surface S will therefore remain on S for t > tl' If the vectors f(t,3;) are directed away from the surface S on both sides, that is, fii < 0, fii > 0, then a solution which passes through a point of the surface S at t = tl may either go off the surface S into the domain G or G+ or remain on S for t > tl. In the latter case'the solution may go off S at any
52
Solutions of Discontinuous Systems
Chapter 2
moment. In the case IN < 0, Ii; > 0 the motion along the surface S is therefore unstable and does not occur in real systems. It can be easily calculated that in equation (4)
a
(5)
 1N
rN 
1+' N
o~ a
~
1,
where IN and Ii; have been defined above. Indeed, the segment joining the endpoints of the vectors 1 and 1+ is expressed by the first formula in (5), where a varies from 0 to 1; the value of a indicated in (5) is found from the condition If), = a/~ + (1 : a)IN = O. If the surface S is given by the equation ~(x) = 0 and V~ == grad ~:/: 0, then
(6)
r
('v~).
r
IV~I
N
'
(where the products of two vectors are the scalar products). If the surface of discontinuity of the function I(t, x) is given by the equation ~(t,x) = 0, then to find the vector 10(t,x) one must carry out the same construction in the (t, x) space with (n+ 1)dimensional vectors (1, t+) and (1, r). The the vector 10 will again be expressed by the first of formulae (5), but with
(7)
a
=
~t
+ (V~)· r
(V~). (1  f+)'
~t
a~
= at'
If the whole of the segment with the ends 1 and 1+ lies in the plane P, the velocity 1° of motion along the surface S is not determined uniquely.
Figure 5 On the basis of the definition (2), one can also find the velocity of motion along the intersection of surfaces of discontinuity (Fig. 5). This velocity is determined either uniquely or not uniquely depending on whether the set F(t, x) has one or more common points with the tangent to this intersection. If there are no such common points, then near the point under consideration there are no solutions lying on the intersection of these surfaces of discontinuity. How to determine which of these cases occur and how to find the velocity of motion along the intersection of surfaces of discontinuity is explained in 3. The above definition of solution may be varied ([94], p. 40). First, the set of the limit values of the function I(t, x*) for (t, x*) x, t = const may
Definitions oj Solution
§4
53
be replaced by the set of the limit values of the function J(t*, z*) for (t*, z*) ¢ M, t* + t, z* + z. In this case the definition of a solution turns out to be equivalent to the original one for a wide class of equations with piecewise continuous righthand sides (see below, 1, §6). The theory of such equations admits a simpler presentation, but such equations do not include CaratModory type equations. Second, one may assume that a given function f(t, z) takes prescribed values at its points of discontinuity and may not exclude these points in the determination of the set of limit values of the function f. In some cases one cannot define the set F(t, z) in (2) at the points of discontinuity of the function f if one knows only the values of the function f at its points of continuity. For instance, in a mechanical system with dry friction
u=
v,
mv =
g(u)  f(v)
+ e(t)
(where m is the mass of a body; u its displacement; g(u) an elastic force; f(v) the force of friction which is an odd function of velocity, discontinuous for v = 0; e(t) an external force) the rest friction f(O) may take on any value between its extreme values fo and  fo. If fo = limu++ o f( v) then the definition a) holds. If fo > limu++o f(v) the motion with a zero initial velocity depends not only on the values of the function f(v) in the domains of its continuity, but also on the value of fo. Then the definition a) does not hold. In both cases the system can be written in the form of the inclusion (2). For v oF 0 the set F(t, z) is a point, and for v = 0 it is a segment whose length depends on fo. The set F(t, z) is therefore not always determined by the limit values of the function I(t, z) from (1), and in the general case this set should be found using some information about the system. Consider another example of the kind ([5], p. 148; 195]):
where A is a matrix, b, c, z are vectors, ZI is the first coordinate of the vector z. Let, in a physical system, the functions !II and !l2 be produced by means of different relays, and for ZI = 0 the quantities !II and !l2 may take on any values from 1 to +1. Then, due to imperfection of the relay, the equality !/I = !/2 cannot be fulfilled at every instant. To emphasize this, one writes !II = sgnl ZI, !l2 = sgn2 ZI. If the system is written in the form (2), then for ZI = 0 the set F(z) is a set of points
If the vectors band c do not go in the same direction, this set is substantially wider than the set of points Az + (b + c)u (1 ~ u ~ 1) obtained under the definition a). The necessity of dealing with such systems leads to the following general method ([95], [96]; IS], p. 151) of constructing the set F(t, z). Consider a system
(8)
where x E RB, the vectorvalued function l(t,x,U1, ... ,Ur ) is continuous in the set of arguments, and the scalar or vectorvalued functions Ui(t, x) are discontinuous, respectively, on the sets M i , i = 1, ... , r, which have common points and may even coincide. At each point (t, x) of discontinuity of the function Ui a closed set Ui(t, x) must be given, which is a set of possible values of the argument Ui of the function I( t, X, U1, ••• , u r }. For i # :j the arguments Ui and Uj are supposed to vary independently of one another on the sets U.(t, x) and Uj(t, x}, respectively. This requirement is usually met if the functions Ui (t, x) and Uj(t, x} describe different, independent, components (blocks) of a physical system. At the points where the function Uj(t,x) is continuous, the set Ui(t,x} consists of one point Ui(t, x}. At the 'points of discontinuity of the function u.(t, x} it is necessary that the set Uj(t, x) contain all limit points for any sequence of the form Vk E Ui(t, x), where Xk + x, k = 1,2, ... (or Vk E Uj(tk, Xk), where tk + t, Xk + x, k = 1,2, ... ). The set Ui(t, x) is usually required to be convex (if Ui(t, x) is a scalar function, then Uj(t, x) is a segment or a point). Let
(9)
F1 (t, x) = I (t, x, UICt, x}, ... , Ur(t, x))
be a set of values of the function I(t, x, Ul, ••• , u r ), where t, x are constant and Ul, ••• , U r vary independently of one another on the sets Ul (t, x), ... ,Ur(t, x), respectively. Solutions of the differential equation (8) are solutions of the differential inclusion (2), where F(t, x} = Fl (t, x), as in [71, [971, or F(t, x} == co Fdt, x), as in [951. Particular cases of this method of constructing the function F(t, x) are both the definition a) and the definitions b) and c) presented below. b) Definition using the equivalent control method ([71, p. 37). This is applied to equations of the form (8), where I is a continuous vectorvalued function, Uj(t, x) is a scalar function discontinuous only on a smooth surface Sj (lPi(X) = 0), i = 1, ... , r. Intersections and even coincidence of these surfaces are allowed. At points belonging to one surface, or simultaneously to several surfaces, for instance, to surfaces Sl,"" Sm (here 1 ~ m ~ r), one assumes (if the solution cannot immediately leave such a surface or an intersection of such surfaces)
(10)
x = f (t, x, u~q (t, x}; ... , u~(t, x), um+ICt, x), ... , ur(t, x)) ,
where equivalent controls u~q, ... , u::f are defined so that the vector I in (10) is tangent to the surfaces Sl,"" Sm and the value u: q (t, x) is contained in the closed interval ui" (t, x), ut (t, x). Here ui", ut are limiting values of the function Uj on both sides of the surface Si, i = 1, ... , m. Thus, the functions u: q (t, x), i = 1, ... , m, are determined from the system of equations
i= 1, ... ,m. A solution is an absolutely continuous vectorvalued function, which outside the surfaces Si satisfies equation (8), and on these surfaces and on their intersections satisfies equations of the form (10) (for almost all t). For example, in the case r = 1 the endpoint b of the vector I(t, x, ueq(t, x)) lies on the intersection of the tangent to S at the point x with the arc abc which
Definitions 0/ Solution
§4
55
is spanned by the endpoint of the vector 1ft, x, u) when u varies from u (t, x) to u+(t,x) (Fig. 6). Equation (8) defined as above is reduced to the differential inclusion E F1 (t, x). The set Fdt, x) is defined in (9), where Udt, x) is a segment with ends u:(t,x), ut(t, x); for those which are continuous at the point (t,x), the set Us(t, x} is the point ut(t, x). The righthand side of (10) is a vector terminating at the point of intersection of the set F1 (t, x) with the tangent to the intersection of the surfaces 8 1 , ... 18m. In Fig. 6 the set F1 (t, x) is an arc abc, and the righthand side of (10) is a vector xb.
z
u.
Figure 6 For the cases where U1,"" U m enter equation (8) linearly, an explicit expression for the velocity of motion (10) along the intersection of surfaces of discontinuity 8 11 ••• ,8m will be obtained in 3. c) The general definition from [95J. This is applied to equations of the form (8), where the function / is continuous in t, x, U1l"" u r , and each of the functions u.(t,x) is discontinuous only on the surface 8. (V'.(t,x) = 0), i = 1, ...
,r.
Let U.(t,x) and F1(t,X) be as in b), and let F2(t,x) be the smallest convex closed set containing the set F1 (t, x). A solution of equation (8) is a solution of the inclusion
(11) On the surface of discontinuity S (V'(x) = 0) the motion may proceed only at a velocity E K(t, x), where K(t, x} is an intersection of the set F2(t, x) with a plane tangent to 8 at the point x. In the case shown in Fig. 6 the set F2(t, x) is the smallest convex closed set containing the arc abc. If this arc lies in one plane, the set F2 (t, x) is a segment between this arc and its chord, shown shaded in Fig. 6, and K(t, x) is an interval which is an intersection of this segment with the tangent to 8 at the point x. If the function I is nonlinear in U1, ••• , u r , then, generally speaking, the set K(t, x} contains more than one point, and the velocity of motion along 8 is not uniquely determined. Let us compare the definitions a}, b}, c}. One can write equation (8) in the form (1) and apply to it the definition a}. Since, in this case, the set F2(t, x} contains the sets F (t, x) and F1 (t, x) from (2) and (9), each solution in the sense
z
56
Solutions of Discontinuous Systems
Chapter 2
of the definition a) and each solution in the sense of definition b) is also a solution in the sense of definition c). The converse is, generally speaking, false (in Fig. 6 the set F is the chord ac, F1 is the arc abc, F2 is the shaded segl11ent). If the function / is linear in U1, ••• , u r , then F2 = F1 and the definitions b) and c) coincide. If, besides, all the surfaces 8 j are different, and at the points of their intersection the normal vectors are linearly independent, then the sets F, F1, F2 coincide and, therefore, all three definitions, a), b), c), also coincide. There exist also other definitions of the solution both for the general case and for the case of a righthand side discontinuous in one or several surfaces. The definitions proposed in the papers [97][99] are closely similar to those presented above. Some limit transitions that lead to one or another definition of solution are considered in [7], [95], [100][103]. The definitions from the papers [104][109] are inconvenient for one reason or another (in [104], [105] motion along surfaces of discontinuity is assumed to be impossible, and in [106][109] the concept of solution depends on the choice of directions of the coordinate axes Xb" ., xn). They do not reflect the character of motion in real systems and fail to find applications. In the papers [110], [111] the requirement of right continuity in x is weakened for the Caratheodory equation. Different definitions of solution are reviewed and compared in [112][117]. A survey of the history of the concept of a solution of a discontinuous system is presented in [95]. 3. We shall mention the main cases where the velocity of motion along an intersection of surfaces of discontinuity 8 1 , ••• , 8 m is uniquely determined and present the expression for this velocity for some cases. Consider the case ([7], p. 39). In equation (8) the controls U1, ••• , Urn, (which are discontinuous respectively on the surfaces 8i ( rank M. If from the matrix M* one can delete k  r columns such that in the matrix so obtained there are no numbers of different signs among algebraic adjuncts A~i of the elements of the first row, then the set K(t, z) is nonempty, and if there are no such k  r columns, the set is empty. Remark 1 and the first part of Remark 2 are proved similarly to Theorem 1. In this case we assume that the ai which correspond to the discarded columns are equal to zero. The second part is proved by using the properties of convex sets (see [118]).
§5
Convex Sets and SetValued Functions
We present here the known properties of closed and convex sets in ndimensional space, which are used hereafter, and the necessary information on setvalued functions. 1. Numbers and points of the ndimensional space Rn are denoted hereafter by small letters, while sets and matrices are denoted by capital letters. IT a and b are points with coordinates ai, ... ,an and bl , ... ,b n respectively, and "f is a number, then a + b, a  b, "fa are points with coordinates ai + bi, a,  bi, "fa, (respectively) where i = 1, ... , n. The closure of the set A is denoted by A, and the empty set is denoted by 0. The distance between points or sets is denoted by p:
pea, b) pea, B)
veal  bl )2 + ... + (an  bn )2, inf pea, b), peA, B) = inf pea, b).
= la  61 =
= beB
aEA,bEB
The set A is called closed if it contains all its limit points. The set A is called convez if for any two of its points a and 6 all the points of a segment joining a and b belong to this set, that is, if for any a E A, b E A we have aa + (1  a)b E A for all a, 0 ~ a ~ 1. The following known assertions are easily proved: 1) The union of a finite number of closed sets is closed.
60
Solutions of Discontinuous Systems
Chapter 2
2) The intersection of any set of closed (or convex) sets is a closed (corre_ spondingly, convex) set. 3) In a nonempty closed set A there is always a point a nearest to a given point b, i.e., such that p(b, a) = p(b, A). 4) p(b, A) = p(b, A), p(A, B) = p(A, B). 5) The function \O(x) = p(x, A) is continuous, Ip(x, A)  p(y, A) I ~ p(x, V)· LEMMA 1. If nonempty closed sets A and B do not have common points and B is bounded, then there exist points a E A, bE B, such that
p(A, B) = p(a, b) >
o.
PROOF: The function \O(x) = p(x, A) is continuous. Consequently, inf"'EB p(x, A is attained at some point bE B. By virtue of 3) there exists a point a E A such that p(b, A) = p(b, a) > 0 (since An B = 0). For any points x E B, yEA we have
p(x, y)
~
p(x, A)
~
p(b, A)
= p(b, a) > o.
Hence, p(B, A) ~ p(b, a) > o. But a E A, bE B, and, consequently, p(B, A) ~ p(a,b). Thus, p(B,A) = p(b,a) > o. If both the sets A and B are unbounded, the assertion of Lemma 1 does not hold. Example: A is one branch of a hyperbola, B is an asymptote of the hyperbola. LEMMA 2. In a non empty closed convex set A there exists only one point a nearest to a given point b, such that p(b, a) = p(b, A). PROOF: A nearest point exists (see 3)). Suppose there exist two such points and a2, and d is the midpoint of a segment joining these points. Then
al
dEA, since the set A is convex and bd is the height of an isosceles triangle Therefore, the points al and a2 are not closest to b.
al a2b.
The lemmas to follow can be found, for example, in [119J. LEMMA 3. Let b ¢ A, A being a nonempty closed convex set. Then there exists an (n  l)dimensional plane separating the point b from the set A. PROOF: Let a be the point of the set A nearest to b. Let us draw a plane P ..lab through any nonendpoint m of the segment abo If there existed a point c E A lying either on P or on the same side of P as the point b, then the angle bac would be acute and there would exist a point dE ac, which would be closer tol than the point a (Fig. 7). Since a E A, c E A, A is convex, then d E A. Thu contradicts the fact that a is the point of the set A nearest to b.
§5
Convex Sets and Set Valued Functions
61
~F.,IJ
c
aL~1J
Figure 7
Figure 8
LEMMA 4. A closed convex set A is an intersection of all closed half spaces that contain this set. PROOF: Let M be such an intersection. Then A c M. Let b ¢ A. By Lemma 3 there exists a plane P separating the space into two parts Q and S, A c Q, b E S. Then A is contained in a closed half space Q, and b ¢ Q. Consequently, b ¢ M. Thus,A=M.
LEMMA 5. H A and B are closed convex sets in Rn without common points, and the set B is bounded, then there exists an (n  l}dimensional plane separating A and B. PROOF: Let the points a E A and bE B be the same as in Lemma 1. The plane P1.ab intersecting the segment ab at a nonendpoint separates A and B. This is proved as in Lemma 3.
Lemma 5 does not hold if both the sets A and B are unbounded. Example: A is a convex set in a plane, which is bounded by one branch of the hyperbola, and B is an asymptote of the hyperbola. For a convex set A E Rn an (n  l}dimensional plane is called a support
plane if on one side of P there are no points of the set A, but they exist either on P or on the other side of P arbitrarily close to P. LEMMA 6. Through any point of the boundary r of a closed convex set A one can draw a support plane. PROOF: Let £IE r, points bi ¢ A, bi + a (i + 00). By Lemma 3, the point bi is separated from A by a plane Pi' Let Vi be a vector of length 1, vi..L~, Vi being directed from a to ~. Then for all z E A, y E ~ we have Vi . Z < Vi . Y < Vi . bi. From the sequence Vi we pick a convergent subsequence Vi + v. Passing to the limit in this subsequence, we obtain V . Z ~ V • a for all Z E A, that is, the set A lies on one side of the plane V • Z = V • a, and the point a lies on this plane. This plane is therefore a support plane. The smallest convex (convex closed) set containing the set A is denoted by co A (correspondingly, coA). Such a set co A (coA) always exists and is the intersection of all convex (correspondingly, convex closed) sets containing A. By virtue of Lemma 4, coA is also the intersection of all closed half spaces containing A.
Solutions of Discontinuous Systems
62
Chapter 2
EXAMPLES: 1) The set A consists of two points a and bj then co A is the segment abo 2) The set A consists of three points a, b, Cj then co A is the triangle abc. 3) The set A is as shown dashed in Fig. 8j then co A is the semicircle abc. Each point written in the form
(1) where
(2)
(i=O,1, ... ,k),
is called a convex combination of points Xl, X2, ••• , Xk. A convex combination is linear. Not every linear combination is convex but only those for which the coefficients satisfy the conditions (2). LEMMA 7. If a set A consists of a finite number of points, co A is the set of all convex combinations of these points. PROOF: It can be directly verified that a set B of points of the form (1) with the conditions (2) is closed and convex; B::J A, consequently, B ::J co A. Any closed half space Q can be written in the form c . X ~ 'Y (c is a vector). IT points Xi E Q, that is, c . Xi ~ 'Y, i = 0, 1, ... , k, then for any point x E B it follows from (1) and (2) that c . X ~ 'Y, that is, X E Q. Therefore, B C Q. The set co A is the intersection of all such half spaces Q, accordingly, B C co A. Thus, B = co A. LEMMA 8. Let c be a vector, A a set and let the inequality c . x for all x E A. Then this is also valid for all x E coA.
~
'Y be valid
PROOF: By hypothesis, the set A lies in the half space Q defined by the inequality c . x ~ 'Y. Since coA is the intersection of all closed half spaces containing A, then coA C Q, i.e., c . x ~ 'Y for all x E coA. THE CARATHEODORY THEOREM. For any bounded closed set A C R n any point x E co A can be represented in the form (1), where Xi E A, i = 0, 1, ... , k, the numbers Qi satisfy the condjtjons (2) and k ~ n. For the proof see [119] (p. 9). COROLLARY. If the set A is closed and bounded, then co A
= coA.
A closed eneighbourhood M~ of the set M is a set of points x such that p(x, M) ~ e. Obviously, MS is a closed setj (M)· = M·. For any point a ¢ M S we have p(a, M S ) = p(a, M)  e. LEMMA 9. If a set A is bounded, then
(3) PROOF: Let b ¢ (co A)·, that is, p(b, co A) = Q > e. There exists a point a E coA such that p(b, a) = Q. Let us place the origin at the point b and direct
§5
Contlez Sets and Set Valued Functions
63
the zlaxis from b to a. Fix any p, If < P < Q. As in Lemma 3, the set coA lies in the region ZI > p, so does the set A. Then Alies in the half space ZI ~ pe, so does co(A) (Lemma 8) and b ¢ co(A). Let b ¢ co(A). Let the ZIaxis go from the point b to the nearest point e of the set co(A). Then pCb, e) = 'Y > OJ co(A) lies in the region ZI > 0 (0 < 0 < 'Y)j so also does A. Therefore, A lies in the half space Zl ~ 0 + e, so does co A, and (coA) lies in the half space Zl ~ Hence, b ¢ (coA). Thus, the relations b ¢ (co A) and b ¢ co(A) are equivalent, and the result follows.
o.
COROLLARY. If the set A is convex, so is A. By virtue of Lemma 9, (co A) and co(A) can be written in a shorter form, as coA. 2. IT for all z EM a function I(z) is defined, then I(M) is a set of values of I(z) for all z E M. In particular, I(z) may be linear: I(z) = Az + b, A being a matrix. Then I(M) = AM + b. Similarly, if e is a number or a vector, then eM is the set of all values of the product ez, where z runs over the set Mj M + N (or I(M, N» is the set of values of the sum z + y (or of the function I(z, y)) where z runs over the set M, and y over the set N. LEMMA 10. If M is a bounded closed set and if a function fez) is continuous, then the set I(M) is closed. If M is convex, I(z) = Az + b, then the set I(M) = AM + b is convex. The lemma is proved directly from the definitions. LEMMA 11. If a set M is bounded and closed, then
co(AM + b) = AcoM + b. The proof follows from Lemma 10 and from the fact that co M is the intersection of all closed half spaces which contain M, and a linear transformation maps a half space into a half space. The necessity for using convex sets in the study of differential inclusions is seen, for instance, from the following lemmas. LEMMA 12 (on the mean value of a vectorvalued function). If M is a bounded
closed set, vet) E M for a ~ t
~
(4)
== b ~ a
tlmean
b, then
i"
v(t)dt E co M.
The same holds for the mean value of the vectorvalued function v(z) on any measurable set of finite measure. PROOF: Taking the Riemann or the Lebesgue partition of the domain of integration, we obtain Vmean = limS, £1i £1. b = Qi ~ 0, s = "" L, b 'a vetil, a Thus, the integral sum S is a convex combination of values vetil E M, and therefore S E coM, limS E coM = coM.
64
Solutions of Discontinuous Systems
Chapter 2
LEMMA 13. Let for a < t < b the vectorvalued functions Xk(t) be absolutely continuous, Xk(t) + x(t) and for each k = 1,2, ... the functions Xk(t) E M almost everywhere, M being a bounded closed set. Then the vectorvalued function x(t) is absolutely continuous and x(t) E coM wherever x(t) exists, that is, almost everywhere on (a,b). PROOF: Since
!Xk(t)!
~
I, then for t', til E (a, b)
(5) Letting k + 00, we conclude that the function x(t) satisfies the same inequality and so is absolutely continuous. By Lemma 12,
Hence,
(6)
M Iim qk = x(t+h)x(t) h E co .
k+oo
The function x(t) is absolutely continuous and so x(t) exists almost everywhere. By virtue of (6), x(t) E co M. REMARK: IT the set M is not convex, then under the assumptions of Lemma 13 one cannot be sure that x(t) EM. For example, for a sequence of "sawtoothed" functions (Fig. 9)
(2ik ~ t ~ 2ik+1) , (2i + 1 2i + 2) 2i + 2 Xk (t ) =  k   t  k  t  k  , 2i Xk(t) = t  k
~
~
i = 0, 1,2, ... , we have Xk(t) E M almost everywhere, the set M consisting of two points: 1 and 1. As k + 00
Xk(t)
+
x(t) == 0,
x(t) ==
of/. M.
Thus, for the differential inclusion x(t) E M in the case of a nonconvex set M, the limit of a uniformly convergent sequence of solutions may not be a solution. 3. The distance between two nonempty closed sets A and B in a metric space, in particular in Rn, may be characterized by the numbers [120]
,8(A, B) = sup p(a, b), aEA
,8(B, A) = sup p(b, a), bEE
a(A, B) = max {,8(A, B); ,8(B,
An .
Convex Sets and Set Valued Functions
§5
65
~'iPJvvs o 1
.
it
t
Figure 9
Figure 10
In Fig.10,.B(A,B) = p(a,b), .B(B,A) = p(c,d), a(A,B) = max{p(a,b)jp(c,d)}. If A and B are bounded sets, these numbers are finite. The inequality .B(A, B) ~ e is equivalent to the fact that the set A is contained in the closed eneighbourhood of the set B, i.e, A C B·, ·and the inequality a(A, B) ~ e is equivalent to the fact that each of the sets A and B is contained in the closed eneighbourhood of the other one. For any nonempty closed sets A, B, C,
o ~ p(A, B)
(7)
~
.B(A, B) ~ a(A,B), .B(A,B) = 0 ~ A c B, a(A, C) ~ a(A, B) + a(B, C).
a(A, B) = a(B, A), a(A, B) = 0 ~ A = B,
We shall prove (7). Let a(A, B) = 8, a(B, C) = e. Then A C B 6, B c C~, hence, A C 06+ e , .B(A, C) ~ 8 + ej B C A6, 0 c Be, consequently, C C A6+ e , .B( 0, A) ~ 8 + 6. Thus, a(A, 0) ~ 8 + 6, and the result follows. Thus, nonempty closed sets form a metric space in which the role of the distance is played by a(A, B) called the Hausdorff distance of the sets A and B. If A eRn, then sUPaEA lal will be denoted by IAI. Let to each point p of a set D C Rm there correspond a nonempty closed set F(p) C Rn. Then F(p) is a setvalued function. Its graph is a set of points (p, q) E Rm x R n such that p E D, q E F(p). Henceforth we denote setvalued functions by capital letters, singlevalued sCalar and vector functions by small letters. We use the notation
F(M) =
U F(p), pEM
IF(M) I =
sup !/EF(M)
Iyl·
A setvalued function F is called bounded on a set M if IF(M)I < 00, that is, if all the values of the function F at the points of the set M are contained in some ball. A setvalued function F(p) is called [120J continuous at the point p if a(F(pl), F(p)) + 0 as p' + pj a function F(p) is called upper semicontinuous (with respect to the inclusion) at the point p if .B(F(pl) , F(p)) + 0 as p' + p. A function F(p) is called continuous or upper semicontinuous on a set D if it is continuous or upper semicontinuous at each point of this set. Since always .B(A, B) ~ a(A, B), continuity of the function implies its upper semicontinuity.
Solutions of Discontinuous Systems
66
Chapter 2
LEMMA 14. Let a set D be closed, and a setvalued function F(p) be bounded in a neigh'lourhood of each point p ED. Then the function F(p) is upper semicontinuous on the set D if and only if its graph r is a closed set. PROOF: Let the function F(p) be upper semicontinuous and let (p, q) be a limit point of its graph. This implies that there exist sequences
i = 1,2, ....
Pi + P E D,
Then
p (q, F(p)) = O. Since the set F(p) is dosed, then q E F(p), that is, (p, q) E r. Therefore, r is a closed set. Let the function F(p) be not upper semicontinuous on D. Then there exist points p E D and Pi + P such that i = 1,2, ....
Hence, there exist points q. E F(p.) such that p(q., F(p)) ~ I!. By virtue of the assumptions of the lemma, the sequence {q.} is bounded. We pick from it a convergent subsequence q,,, + q. Then p(q, F(p)) ~ e. Thus,
that is, the set
r
is not dosed.
LEMMA 15. Let a function F be upper semicontinuous on a compactum K and let for each p E K the set F(p) be bounded. Then the function F is bounded on K. PROOF:
Otherwise there exist
(i=1,2, ... ).
Pi EK,
We choose a convergent subsequence Pi; tions of the lemml.l. that
IF(p)1 = a < 00,
qi; E
+ P E K.
It follows from the assump
F(p,;) c (F(pW
Then Iqi I ~ a + I!. This contradicts the assumption
Iq. I + 00.
LEMMA 16. If for each p E D the set H(p) is nonempty, closed, bounded, and the setvalued function H is upper semicontinuous (or continuous), then the function F(p) = co H(p) is upper semicontinuous (correspondingly, continuous). PROOF: For any Po ED and I! > 0 there exists 0> 0 such that for all p E (po)O we have H(p) c (H(poW. By Lemma 9
co H(p) c co [(H(poWl
=
[co H(poW ,
§6
Differential Inclusions
67
that is, F(p) C (F(po))·. The function F is upper semicontinuous. H the function H is continuous, then the above holds and, besides, H(po) C (H(pW. From this it follows, as in the previous case, that F(po) C (F(p))·. The function F is therefore continuous.
§6 Differential Inclusions We investigate here the properties of righthand sides of differential inclusions, to which differential equations with discontinuous righthand sides were reduced in §4. The connection between differential inclusions and contingent equations is established. Some properties of measurable setvalued functions are considered. 1. We will analyze the properties of setvalued functions obtained using the technique of §4. LEMMA 1. Let !(p) be a bounded singlevalued function, p E D c Rm, !(p) E Rn. Let for each Po E D the set H(po) be the set of all limit values of the function !(p) for p + Po, supplemented by the value !(Po) in the case Po E D. Then the functions H(p) and F(p) = co H(p) are upper semicontinuous.
PROOF: For each p E D the set H(p) is closed, H(D) is bounded. The graph of the function H is the closure of the graph of the function !, and is therefore closed. By virtue of Lemmas 14 and 16, §5, the functions Hand F are upper semicontinuous. LEMMA 2. Let !(p, Ul, ... , Ur) be singlevalued and continuous. If at the point Po the setvalued functions U1 (p), ••. , Ur (p) are bounded and upper semicontinuous, the function H(p) = !(p, U1(p), •.. , Ur(p)) is bounded and upper semicon
tinuous at this point. The proof is similar to the proof of the elementary theorem on continuity of a composite function at a point Poi in addition, we use uniform continuity of the function ! on a bounded closed seton a closed neighbourhood of a set of points with coordinates p = Po, Ul E Ut{po) , ... , U r E Ur(po). Consider a differential equation = !(t, x) with a piecewise continuous vectorvalued function !(t, x), as in 1, §4, or equation (8), §4. Each of the definitions a), b), c), §4, replaces this equation by the differential inclusion
z
(1)
3: E F(t,x),
The set F(t, x) is nonempty, bounded, closed, and in the case of the definitions a) and c)· is also convex. A solution of a differential inclusion is an absolutely continuous function defined on an interval or on a segment and satisfying this inclusion almost everywhere.
The setvalued function F(t, x) obtained under the definition a) is upper semicontinuous in x, and under the definitions b) and c) is upper semicontinuous in t, x. LEMMA 3.
PROOF: In the case a) this follows from Lemma 1 for p = x. In the case b) equation (8), §4, is equivalent to the differential iriclusion E F1(t,x) with the
z
68
Solutions 01 Discontinuous Systems
Chapter 2
function (9), §4, where the functions Ui(t, x) are upper semicontinuous in t, x, by virtue of Lemma 1, for p = (t, x), and the function Fl (t, x) by virtue of Lemma 2. In the case c), in formula (11), §4, the function F2(t,X) = coFl(t,x) is upper semicontinuous in t, x by virtue of Lemma 16, §5. We will show that in the case a), §4, the function F(t, x) in (1) can also be replaced by a function upper semicontinuous in t, x if for each of the domains Gi of continuity of the function 1ft, x) the following condition is fulfilled. CONDITION 'Y. For the domain G l , for almost all t, the crosssection of the boundary of the domain by a plane t = const coincides with the boundary of the crosssection of the domain by the same plane. The boundary 8M 01 a set M is a set of points, for each of which in an arbitrarily small neighbourhood there exist points of the set M and points that do not belong to M. A crosssection M t of the set M by the plane P (t = const) is a set M n P. In determining the boundary 8(Mt ) of the crosssection, the set M t is considered as a set in the plane P, that is, for the points from M t one considers neighbourhoods which lie in this plane. Using this notation, we write the condition 'Y as follows:
(2)
for almost all t
o Figure 11 Fulfillment of the condition (2) is usually easily verified. For example, for the domain shown in Fig. 11, (8G)t =/: 8(Gt ) only for t = tl, t2, ts, t4, and therefore the condition 'Y is fulfilled. The condition 'Y holds for a very wide class of domains, for instance, for all locally connected domains. Let H(t, x) be a set of limit values of the function 1ft, x') for x' + x, t = const, and Ho(t, x) is the same for 1ft', x') for t' + t, x' + x. LEMMA 4. If the domains Gi of continuity of the function 1ft, x) satisfy the condition 'Y (more briefly, if the function I satisfies the condition 'Y), then for abnost all t {t f/. To, p,To = 0, p, being the Lebesgue measure} we have Ho(t, x) = H(t, x). PROOF: For each domain G i the equality (2) is satisfied for all t ETa, Let us take a point (t, x) such that t f/. To = UiTi. If (t, x) E Gi, then at this point the function I is continuous and
Ho(t, x) = H(t, x) = 1ft, x).
P,Ta = O.
Differential Inclusions
§6
69
If (t, x) lies on the boundary of one or several domains Gi, each point v E Ho(t, x) is a lim f(tk, Xk) by some subsequence (tk' Xk) + (t, x) contained in one of the domains Gi. Since (t, x) E (8Gi)' = 8(Git), then in Git there also exists a subsequence (t, x~) + (t, x). The function f is piecewise continuous and therefore
Hence, Ho(t, x) C H(t, x). The inverse inclusion is obvious. COROLLARY.
(3)
Under the condition "f the differential inclusions
3: E F(t, x)
= coH(t, x),
:i; E
Fo(t, x) = coHo(t, x)
are equivalent, that is, have identical solutions. The function Fo is upper semicontinuous in t, x. Indeed, the solution must satisfy the inclusion almost everywhere, but by virtue of Lemma 4, Fo(t,x) = F(t, x) for almost all t. By Lemma 1 (where p = (t, x)), the functions Ho and Fo are upper semicontinuous in t, x. Thus, under the condition "f the definition a), §4, reduces the equation :i; = f(t, x) to the inclusion :i; E Fo(t, x) with the function Fo(t, x) which is upper semicontinuous in t, x. The change from the first of the inclusions (3) to the second is made for the reason that for the second inclusion the proof of the existence theorem and the investigation of the properties of the solutions are much simpler. 2. Under sufficiently wide assumptions the differential inclusion (1) is equivalent [120], [121] to the contingent equation [122] and to the paratingent equation [33]. The concepts of contingence and paratingence had originally a purely geometrical meaning (generalization of the concept of a tangent). In the theory of differential inclusions these concepts are interpreted as "manyvalued derivatives" of vectorvalued functions. In the definitions to follow, infinity is regarded as a limit point of a sequence {Yi} if a sequence {IYil} is unbounded. The set Cont x(to) of all the limit points of the sequences
(4)
X(ti)  x(to) ti  to
(ti
+
to, i = 1,2, ... )
is called contingence or. a contingent derivative [120][123] of the vectorvalued function x(t) at the point to. The set Parat x(to) of all the limit points of the sequences
(5)
ti 
tj
(ti
+
to, tj
+
to,
i,i = 1,2, ... ).
is called the paratingence [33] of the vectorvalued function x(t) at the point to. For any function x(t), Cont x(t) C Parat x(t) always since in (5) the case tj = to is not excluded. If at a given point to there exists a derivative x' (to), then Cont x(t o} = x'(to}, and if such a derivative also exists in the neighbourhood of the point to and is continuous at this point, then Cont x(to) = Parat x(to) = x, (to).
70
Solutions of Discontinuous Systems
Chapter 2
LEMMA 5. If IContx(t)1 ~ m on the interval (a,b) and if the vectorvalued function x(t) is continuous on the right at the point a and on the left at the point b, then on the interval [a, b] it satisfies a Lipschitz condition with a constant
m.
PROOF: If we assume the converse, there exist points aI, bl on (a, b), such that < bl •
al
(6) Then at least for one half of the closed interval [aI, bl ] there holds an inequality similar to (6), with the same mI' We shall denote this half by [a2' b2 ] and again divide it into two parts. Continuing this procedure, we obtain a sequence of nested intervals [ai, bi].
i = 1,2, .•.. For each of these there holds an inequality similar to (6), with the same mI. Let to be a point common to these intervals. Then for each i we have either
or
The length of the vector (4), where ti = ai or ti = bi, is therefore not less than mI' Hence, from the sequence of vectors (4) one can choose a subsequence which is either infinitely increasing or is tending to a finite limit v, Ivl ~ ml > m. Both the possibilities contradict the assumption of the lemma. THEOREM 1 [121]. Let, forany(t, x) from a closed domain Q, thesetF(t,x) be nonempty, bounded, closed, convex and let the setvalued function F be upper semicontinuous. Let (t, x(t)) E Q for a ~ t ~ b. Then the following assertions are equivalent: A. On a closed interval [a, bl a function x(t) is absolutely continuous and x/(t) E F(t, x(t)) almost everywhere. B. For all t E (a, b) the set Cont x(t) (or Parat x(t)) is contained in F(t, x(t))j for t = a the function x(t) is continuous on the right and for t = b it is continuous on the left. PROOF: In the case A, for each to E (a, b) and each e > 0, if It  tol ~ 5 = 5(to, e), we have F(t, x(t)) C F 6, where F6 is a closed eneighbourhood of the set F(to, x(to)). For any ti, ti E [to  5, to + 5] the vectors (4) and (5) belong to F6 by virtue of Lemma 12, §5. Therefore, the sets Cont x(to) and Parat x(to} are nonempty and are contained in F 6 , and since e is arbitrarily small, they are contained also in F(to, x(to)). Thus, B follows from A. Let B hold. The function x(t) is continuous on [a, bl since if for t = to it were discontinuous, the sequence (4) would have a limit point CX) outside
Differential Inclusions
§6
71
F(to,x(to)). By virtue of Lemma 15, §5, IF(t, x(t)) I ~ m for a ~ t ~ b. Then ICont x(t)1 ~ m for a < t < b. By Lemma 5, the vectorvalued function x(t) on a closed interval [a, bj satisfies the Lipschitz condition and is therefore absolutely continuous. Almost everywhere there exists
x'(t) = Cont x(t) c F (t, x(t)) , that is, A follows from B. 3. Next we present some properties of setvalued functions used for investigating differential inclusions with righthand sides not upper semicontinuous in t, x. A support function of a convex set A c Rn is a function of a vector vERn, defined by the equality
1,b(A, v) = sup v . x.
(7)
zEA
Since 1,b(A, AV) == A1,b(A, v) for any A ~ 0, it suffices to consider the function ..p(A, v) only for vectors v whose length is equal to 1. By virtue of (7), for any v =f 0 a plane V· x = "1, where "1 = 1,b(A,v), is a support plane for the set A, and a half space v . x ~ "1 contains the set A if and only if "1 ~ 1,b (A, v). From this and from Lemma 4, §5 've have the following result. LEMMA 6. A closed convex set A is funy defined by specifying its support function 1,b(A, v). The point a E A if and only if v . a ~ 1,b(A, v) for an v.
For a bounded convex set A the support function is continuous since from Ixl ~ m, IV1  v21 ~ 6, it follows that
It suffices, therefore, to know the values of the support function for an arbitrary countable set of vectors Vi, i = 1,2, ... , which is dense everywhere on a unit sphere Ivl = 1. Hence, a bounded convex closed set A is uniquely defined by specifying a countable set of numbers
(8)
1I'i(A)
= 1,b(A, Vi),
IVil
= 1,
i
= 1,2, ....
Any nonempty closed set A c Rn can be uniquely defined by specifying a countable set of numbers
(9)
"..(A) = p(ai' A),
i = 1,2, ... ,
that is, distances from this set to the points ai of a given set which is dense everywhere in Rn. The numbers Pi(A) may be considered as coordinates of the closed set A. The numbers (8) can also be used as coordinates of a closed convex set. Of course, for given points ai (or Vi) not every countable set of numbers is a set of numbers (9) (or (8)} for a certain set A.
Chapter 2
Solutions of Discontinuous Systems
72
The relations between the closed sets A and B impose particular constraints on the numbers pdA) and Pi(B), and for convex sets also on the numbers 1I";(A) and 1I";(B) or on the support functions of these sets. For instance,
(10)
A c B ¢=> p;{A) ~ p;{B), Pi (A U B) = min {Pi(A)i p;{B)} , {3(A, B) = sup (p;(B)  Pi (A)) ,
i = 1,2, ... , p;{A e ) = max {Oi p;(A)  e},
= sup Ip;{A)  p;{B) I,
a(A, B)
i
i
For convex closed sets
Ac B {3(A, B) = max
¢=>
1I"i(A) :::;; 1I".(B), i
{O;S~ph(A) •
1I"i(A)
= 1,2, ... ¢=> !JI(A, v) :::;; !JI(B, v),
1I";(B))} = max {OJ sup (!JI(A, v)  !JI(B,
= 1I"i(A) + e,
1"1=1 !JI(A, v) = !JI(A, v) + e Ivl.
V))},
If a set A is a function of the point pEG, that is, A = A(p), the numbers 1I"i and Pi also depend on p and they can be called the coordinate functions. From the relations (10) there follows the lemma [124][126]. LEMMA 7. For the function A(p) to be continuous (or upper semicontinuous) it is necessary and sufficient that all its coordinate functions p.(A(p)) , i = 1,2, ... , be continuous (correspondingly, lower semicontinuous).
A scalar function tp(p) is called lower semicontinuous at the point Po if for any e > 0 for Ip  pol < S(e) we have tp(p) > tp(po)  e. Let for each pEE C Rm the set A(p) c Rn be closed and non empty. A setvalued function A(p) on a set E is called measurable if the set E is measurable and if for each closed set BeRn a set of those pEE is measurable, for which A(p) n B is nonempty. (This is equivalent to the definition from [127].) If instead of closed sets B we take open sets or else only closed (or only open) balls, all of them or only those with the centres chosen from a given everywhere dense set {ai} and with rational radii, then we obtain a definition equivalent to the one given above. If B is an open ball with the centre and the radius r, the relation A(p) n B ¥: 0 is equivalent to the relation p(ai' A(p)) < r. Hence the measurability of the setvalued function A(p) is equivalent to the measurability of all the coordinate functions pi(A(p)) == p(a., A(p)), i = 1,2, .... If a setvalued function is continuous or upper semicontinuous, it is measurable. (This follows from Lemma 7.) If the functions A.(p) (i = 1,2, ... ) are measurable, so are the functions
a.
S(p) =
UA.(p), •
R(p) =
n •
Ai(P),
as well as the upper and the lower topological limits of the sequence Ai (p), i = 1,2, .... If the function A(p) is measurable, so is coA(p).
Differential Inclusions
§6
73
IT a singlevalued function !(p, Ul, ••• , u r ) is continuous, and setvalued functions AI(p) , ... , Ar(P) are measurable, then the composite function !(p, Adp), .,., Ar(P)) is measurable. For these and other assertions concerning measurability of functions consult, for instance, [128]. The theorem to follow extends the Lusin theorem ([64], p. 118) on measurable functions to setvalued functions. Let a set M be contained in the domain on which the function A(p) is defined. We say that on the set M the function A(p) is continuous with respect to this set if the function AI(p) , which is defined only on M and is equal there to the function A(p), is continuous on M. THEOREM 2 [127]. H a setvalued function A(p) is measurable on the set E, then for each e> 0 there exists a set E. c E, p,E. < e, such that the function A(p) is continuous on the set E\E. with respect to this set. PROOF [124]: By the Lusin theorem, for each coordinate function p, (A(p)) there exists a set E" p,E, < 2'e, such that on the set E\E, the function p,(A(p)) is continuous with respect to this set. Let E, = u,E., then p,E. < e and on the set E\E. all the functions p.(A(p)), i = 1,2, ... , are continuous with respect to this set. By virtue of Lemma 7, the function A(p) is continuous on E\E. with respect to this set. The theorem to follow is used for investigating the properties of the solutions of Caratheodory equations. THEOREM 3 [129]. Let a set D c Rn be bounded and closed and let for each zED a vector function I(t, z) be measurable in t and for almost all t E T [a, b] be continuous in z. Then for each II > 0 there exists an open set T. C T, IJ.T. < II, such that the function f,(t, z), which is defined only for t E T\T., zED and is equal there to the function I(t, z), is continuous in t, z.
=
PROOF: For almost all t E T the graph ret) of the function f(t, z), which is regarded as a function of z for t const, is a bounded closed set. We will show that the setvalued function ret) is measurable on T, Le., that for each open set G C D x Rn the set T(G) of those t is measurable for which ret) n G is nonempty. Take in D a countable everywhere dense set {Zi} and fix t E T, for which the function f(t, z) is continuous in z. The crosssections of the sets G and ret) by a hyperplane Z Zi are an open set Gi C Rn and a point IIi f(t, Zi). Since I(t, z) is continuous in z, the set of points (Zi,lIi), i 1,2, ... , is dense everywhere on ret). Therefore, if the set ret) n G is nonempty, it contains at least one of such points (Zi,lIi), that is,/(t,zi) C Gi for some i. The converse is obvious. Accordingly, T(G) = UiTi, where Ti is the set of those t for which I(t, Zi) E Gi. The function I is measurable in t and, therefore, the sets Ti and T(G) are measurable, and so is the function ret). By Theorem 2, for each II > 0 there exists a set T: c T, IJ.T: < II, such that on the set T\T; the function ret) is continuous with respect to this set. The set T; can be covered by an open set T", ~T. < II. By Lemma 15, §5, the function ret) is bounded on T\T., and by Lemma 14, §5, its graph is closed, that is, the set of points (t, Z,II), such that t E T\T" , zED, II I(t, z) is closed. Thus, the singlevalued function I(t, z) on the set t E T\T., zED is continuous with respect to this set.
=
=
=
=
=
COROLLARY. Under the assumptions of Theorem 3, the function I(t, z) is measurable on the set T x D. Theorem 3 makes it possible to specify on which set the solutions of a Caratheeodory equation have a derivative and satisfy the equation. THEOREM 4 [130]. Let the vectorvalued function I(t, z) satisfy the Carath~odory conditions (I, §1) in a closed bounded domain a ~ t ~ b, zED eRn. Then there exists a set To C [a, b] of measure liero, such that for all t E [a, b]\To each solution of the equation z I(t, z) has a derivative z(t) equal to I(t, z(t».
=
74
Solutions of Discontinuous Systems
Chapter 2
PROOF [131]: Let the set T. be the same as in Theorem 3, X(t)
=1
(t e T.),
X(t) = 0
(t
eQ
= [a, b]\T.).
By the assumption I/(t,z)1 ~ m(t), the function met) being summable. For almost all t
11.1+,. X(I)m(l)ds = d11 X(I)m(s)dl =
lim "0 h
(11)
dt ..
1
eQ
O.
Let Q" be the set of those points of density of the set Q at which (11) holds. Then for each solution of the equation z I(t, z) for each t E Q" we have
=
z(t + h)  z(t) h
= .!. /. h
1+,. I(s,z(s))ds
1
=h1 /.1+,. (1 X(s))ds. I(t, z(t)) 1
/.1+" (1  x(s» (f(s, Z(8))  I(t, z(t))) dt + 1 /.1+" X(I)/(s,z(s»ds. h
+ h1 1 t
=
Since X(I) 0 on Q and t is a point of density ofthe set Q, then the first term of the righthand side tends to I(t, z(t» as h ..... O. In the second term, for BeT. we have 1  xes) 0, and for s ~ T., because of continuity of the function I on Q, the absolute value of the integrand is less than 6(h) ..... 0 (as h ..... 0), and the whole second term is less than 6(h). By virtue of (11) the third term tends to lIlero as h ..... O. The whole righthand side thus tends to I(t, z(t)) as h tends to lIlero. Hence, for each t e q', p.Q" p.Q > b  a  s there exists z(t) = I(t, z(t)). Since s > 0 is arbitrarily small, the theorem is proved.
=
=
Theorem 3 and its corollary are extended to setvalued functions F(t, z) measurable in t and continuous in z. To achieve this, one can apply Theorem 3 to each coordinate function p;(F(t, z)) and then repeat the argument used in the proof of Theorem 2. A further extension to setvalued functions measurable in t and upper semicontinuous in z is impossible. Such a function may be not measurable in t, z.
Theorems on the choice of singlevalued branches (selectors) of setvalued functions are used in the theory of differential inclusions. set A is closed and convex, then the point a E A, a = a(A), which is closest to a given point b, depends continuously on the set A, that is, a(Ai) + a(A) as o:(A;, A) + O. LEMMA 8. If a
PROOF: Let o:(Ai' A) + 0) (i + 00) and let the points ai E Ai and a E A be the closest to the point b (see Lemma 2, §5). Tnen
(12)
p(b, ail = p(b, Ai)
+
p(b, A) = p(b, a).
If some subsequence ai~ + ao ::f a, then ao E A and from (12) it follows that p(b, ao) = p(b, a). This contradicts Lemma 2, §5.
Existence and Properties oj Solutions
§7
75
Let for each peE a set A(p) c Rn be nonempty, closed, convex. Then there exists a singlevalued function J(p) e A(p) which is continuous if the function A(p) is continuous and measurable if the function A(p) is measurable. THEOREM 5.
PROOF: Take any point a eRn. For each peE in a convex closed set A(p) there exists only one point closest to the point a (Lemma 2, §5). Denote this point by J(p). It depends continuously on the set A(p). Therefore, from continuity of A(p) on E there follows continuity of J(p), and from measurability of A(p) and from Theorem 2 there follows continuity of J(p) on the set E\E., where pE. < 6. Since 6 is arbitrary, J(p) is measurable on E. REMARK: IT we omit the condition of convexity of the set A(p), it is not always true that we can pick out a continuous branch, whereas we can always pick out a measurable branch. But to prove this assertion is more difficult [132J.
§'1 Existence and Properties of Solutions Here we prove the existence theorems for solutions of differential inclusions and differential equations with discontinuous righthand sides. The limits of convergent sequences of approximate solutions are shown to be solutions. We prove theorems on continuation of solutions, on compactness of solution sets. 1. Approximate solutions are often used in existence theorems (for example, Euler broken lines) and in studies of the dependence of a solution on initial data and on the righthand side of the equation. For a differential equation with a piecewise continuous righthand side it is natural to consider not only small variations of the righthand side in the domains of its continuity, but also small variations of the boundary of these domains. Therefore, as an approximate solution of equation 2: = J(t, z) one should consider, in particular, an absolutely continuous function yet), for which almost everywhere
(1)
Iy(t)  J(t,z(t))1 ~ 5,
Iz(t)  y(t)1
~
5,
where z(t) is some function, and the number 5 is sufficiently small. Denoting a closed 5neighbourhood of a set MOby M6, one can write condition (1) as follows
(2) IT we write z(t)  yet) = pet), condition (1) can be expressed as follows:
yet) = J(t, (y(t) + pet)) + q(t),
Ip(t)1 ~ 6,
Iq(t) I ~ 5.
In [117J, pet) are called inner and q(t) outer perturbations. Taking into account Lemmas 9 and 13 from §5, one can replace the condition (2) by a somewhat more general condition
(3)
yet) e [co J(t, (y(tW)]6 .
Solutions of Discontinuous Systems
76
Chapter 2
The same definition of an approximate solution is also suitable for a differential inclusion with the righthand side upper semicontinuous in x. If the righthand side of an inclusion is upper semicontinuous in t, x, then the definition of an approximate solution can be extended by replacing f(t, (y(t))6) by f(t 6, (y(t))6) in (3). In §7 and 8 the following definitions are used. A vector function y(t) is called a 0 solution (an approximate solution with accuracy 0) of an inclusion
(4)
:i: E F(t, x)
with a function F, upper semicontinuous in t, x, if on a given interval the function y(t) is absolutely continuous and almost everywhere
y(t)
(5)
E F6(t, y(t)),
Here and below F( t6, y6} implies a union of sets F( t1, Y1) for all t1 E t 6, Y1 E y6, that is, for It1  tl ~ 0, IY1  yl ~ o. REMARK: If a set F(t, x} is bounded and convex, if the function F is upper semicontinuous in t, x in a domain G and a compactum KeG, then for any e> there exists oo(e) > 0 such that for all 0 ~ oo(e) the graph of the function Fo (t, x) on K is contained in the eneighbourhood of the graph of the function F(t, x) on K. (Indeed, if for a certain e > 0 the points (ti' Xi, tli), i = 1,2, ... , lie on the graph of the functions FO;{Oi  0) outside the eneighbourhood of the graph of the function F, then, by choosing a convergent sequence of these points and using the upper semicontinuity of the function F at the limit point, we come to a contradiction.) 2. We shall say that in the domain G a setvalued function F(t, x) satisfies the basic conditions if for all (t, x) E G the set F(t, x) is nonempty, bounded and closed, convex, and the function F is upper semicontinuous in t, x.
°
Let F(t, x) satisfy the basic conditions in an open domain G. Then the limit x(t) of any uniformly convergent sequence of oksolutions Xk(t) (Ok 0, k = 1,2, ... ) of the inclusion (4) is a solution of this inclusion (if the graph of the limiting function x{t), a ~ t ~ b, lies within GJ. LEMMA 1.
PROOF: Take any to E [a, b] and any e > o. The function F is upper semicontinuous, and therefore there exists" > such that in a domain Go (It  tol < 217, Ix  x(to)1 < 317) we have
°
(6)
F(t, x) c F 0 and Yk(tk) ydt) > 0
d ( . rt:\
)
= 0, Yk(t)
Xk(tk)
t
0 for t > tk we have
d 2VYk(t) t = . rt:\ 1 ~ O. t V Ydt)
Existence and Properties of Solutions
§7
81
Therefore 2VYIo(t)  t does not decrease at t ~ tlo,
and YIo(t) cannot tend to zero on the interval 6 < t < (3. In the case YIo(tlo) < 0, for t < tlo, we have YIo(t) < 0,
Ylo (t) ~ Hence, 2VYIo(t)
VYlo (t),
~ (2V YIo(t) + t) ~ o.
+ t does not increase, (t 0 is small enough for DP C G. Then for 0./2 ~ p
(4) Now one repeats the reasoning of the proof of Lemma 5, §1, taking met) = m+ p. By Lemma 1, §7, the limit of the convergent subsequence will be a solution. THEOREM 1. Let F(t,:z;) satisfy the basic conditions (2, §7) in the open domain G; to E [a, bl, (to, :z;o) E G; let all the solutions of the problem
(5)
Z E F(t, :z;),
:z;(to) =:z;o
88
Solutions of Discontinuous Systems
Chapter 2
~ t ~ b exist and their graphs lie in G. Then for anye > 0 there exists a 8> 0 such that for any to E [a, bj, F*(t, x) satisfying the conditions
for a
(6)
It~
 tol
~ 0,
Ix~
 xol
~ 0,
da(F*,F)
~
Xo
and
0
and the basic conditions, each solution of the problem
±*
(7) exists for a ~ t more than e.
~
E
F*(t, x*),
x*(t~) = x~
b and differs from some solution of the problem (5) by not
This implies that each solution x*(t) of the problem (7) either exists on [a, bj or can be extended to the whole segment [a, bj, and that for this solution there exists a solution x(t) of the problem (5) such that max Ix*(t)  x(t)1 ~ e .
.. ~t~b
PROOF: By Theorem 3, §7, the set H of points (t, x), a ~ t ~ b, belonging to the graphs of solutions of the problem (5) is bounded and closed. By Lemma 1, §5, p(H, aG) = Po > O. Suppose the theorem is false. Then for some e (0 < 2e < po) there exists a sequence of solutions x.(t), i = 1,2, ... , of the problems
(8) for which, as O.
i = 1,2, ... , >
0 (i
> 00),
we have
and the solution Xi either does not extend to the whole segment [a, bj, or for each solution x(t) of the problem (5)
(9)
max IXi(t)  x(t)1 > e.
"~t~b
for all i. In both cases, for all i > i o , the point (toi' XOi) lies in H6 and the solution Xi(t) lies in H C for ai ~ t ~ (3., to. E (a., (3.), and the points
lie on the boundary aH" of the closed domain He (Theorem 2, §7). Let us choose a subsequence i = i l , i 2 , •.. > 00 such that
(10) Applying Lemma 1 to the subsequence x.(t), i = i l ,i2 , ..• , we obtain a new subsequence P converging to the solution x(t) of the inclusion (2) which passes through the points p and q.
§8
89
Dependence of Solution on Initial Data
Since the solution x.(t) passes through the point (tot,%o.) . (to,xo), then, by virtue of (4), for i > i* the solution x.(t) on a closed interval [to, to.) or [to., to) lies within H and
This implies that x(t) is a solution of the problem (5), and its graph for max{ajtp } ~ t ~ min{bjtq } lies in H. Then it follows from (10) that tp < a, tq > b. Now it follows from Lemma 1 that the subsequence P converges to x(t) uniformly on [a, b). This is in contradiction with (9). COROLLARY 1. Let F(t, x) satisfy the basic conditions in the domain G, let for t ~ to the problem (5) have a unique solution x(t) and let its graph on the segment [to, b] lie within G. Then for any IS > 0 there exists a 5 > 0 such that for any x F* (t, x) satisfying the inequalities (6) and the basic conditions in G each solution on the problem (7) on the segment [to, b) exists and differs from x(t) less than by IS.
to, o,
Thus, from the right uniqueness of the solution there follows a righthand continuous dependence of the solution on the initial data and on the function F. A similar assertion is valid for the segment [ao, to]. COROLLARY 2. Let F(t, x) satisfy the basic conditions in the open domain G and let all the solutions of the problem (5) with all possible initial data (to, %0) E M (M is "a given compactum) for a ~ t ~ b exist with their graphs lying in G. Then for any IS > 0 there exists 5 > 0 such that for any compactum M* C Mii and for any function F* (t, x) satisfying the basic conditions and the condition dG(F*, F) ~ 5 each solution of the problem (7) with an arbitrary (to, xo) E M* for a ~ t ~ b exists and differs from a certain solution of the problem (5) with some (to, xo) E M by not more than e. PROOF: IT the assertion is not true, there exists a sequence of solutions of the problems (8) for which the initial points go infinitely close to M, and the solutions themselves either satisfy the inequality (9) or fail to exist on the whole of the segment [a, b]. Let us choose a subsequence of solutions for which the initial points converge to a certain point (to, xo) E M. For these solutions there holds the assertion of Theorem 1. But this contradicts the above assumption. According to Corollary 2, the set of solutions of the problem (7) with
(to, xo) E M* lies in an ISneighbourhood (in the metric C[a, b]) of the set of solutions of the problem (5) with (to, xo) EM. Hence, the segment a ~ t ~ b of the funnel ofthe set M* for the inclusion :i; E F* (t, x) lies in the eneighbourhood of the segment of the funnel of the set M for the inclusion (2). Thus, a set of solutions with initial data from a given compactum and a segment of a funnel depend upper semicontinuously on this compactum and on the righthand side of the inclusion.
Solutions of Discontinuous Systems
90
Chapter 2
2. From Theorem 1 similar theorems are deduced for differential equations with piecewise continuous righthand sides if the solutions are understood in the sense of the definition a) or c), of §4. Let the vectorvalued functions f(t, x) and I*(t, x) be piecewise continuous in the domain G, as in 1, §4, and satisfy the condition 'Y of 1, §6. We will write dO(I*, f) ~ 5 if and only if for each point of continuity (t, x) of the function 1* there is a point of continuity (t', x') of the function I such that
It'  tl ~ 5, Ix'  xl ~ 5, If(t', x')  J*(t,x)1 ~ 5.
(11)
Note that it does not follow from (11) that under any of the definitions a)~),in §4, the values of the functions 1* and I on the surface of discontinuity differ by not more than 5. For example, if x = (X1,X2),
I (u(x))
= (2  1.£;0.1 0.11.£), 1.£
°
=
°
(X2
J* (u(x)) = (2  1.£;0.2  0.11.£), 1.£ = 3 (X2 > 0),
< 0),
then for X2 oF 11*  II = 0.1, and by virtue of each of the definitions a)~), §4, for X2 = we have 1= (1,0), 1*(0,0). The following theorem assumes that in the open domain G the vectorvalued functions I(t, x) and 1* (t, x) are piecewise continuous and satisfy the condition 'Y and that all the solutions are understood in the sense a), §4, to E [a, b], (to, xo) E
°
G. THEOREM 2.
Let all the solutions of the problem :i: = f(t, x),
(12)
x(tol = Xo
exist for a ~ t ~ b and let their graphs be contained in G. Then for any £ > there exists 5 > such that for any to E [a, b], Xo and I*(t, x) satisfying the conditions
°
(13)
Itotol~5,
°
Ix~
 xol
~
5,
each solution of the problem (14)
:i:'" = J*(t, x·),
x'"(t(j) =
x~
exists for a ~ t ~ b and differs for these t from a certain solution of the prob" lem (12) by not more than bye. PROOF: According to 1, §6" under the condition 'Y equations (12) and (14) are equivalent to the inclusions
:i: E F(t, x),
:i:'" E F* (t, x*).
In the domains of continuity of the function I we have F = I, and at the points of discontinuity F(t, x) = coH(t,x), where H(t,x) is a set of limit values for
Dependence 0/ Solution on Initial Data
§8
91
I(ti, Xi) for ti  t, Zi  z. This is also the case with F*. According to §6, 1, the functions F and F* are upper semicontinuous in t, z. IT the function f* is continuous at the point (t, z), we have from (11) r(t,z)
= f*(t,z)
c [/(t',z')J 6
C
[F(t6,z6)]6.
If, however, f* is discontinuous at the point (t, z) and continuous at the points (ti' z.)  (t, z), then
(15)
r(ti, :ii)
C
[f(t~, zmo ,
It~  til ~ 6,
Iz~  z. I ~ 6.
For sufficiently large i the points !(ti, zi) are contained in an arbitrarily small neighbourhood of the set F(t 6 , z6); otherwise, for a certain e > 0 there would not contained in [F(t 6 , z6W. It would be exist a subsequence of points f(ti, possible to choose a further subsequence from it (i = '1,'2,'"  00) such that
zD
(t~,zB_(t~,Z~)E(t6,z6),
!(t~,zD+u,
(i=i1coo).
But in this case U E F(t~,z~) C F(t6,z6). This is in contradiction with the choice of the first subsequence. From what has been proved and from (15) it follows that the set H*(t, z) of the limit values for f*(ti,Zi) is contained in [F(t 6,z6)J6. Since F*(t,z) = co H* (t, z) (1, §6) we obtain, using Lemma 9, §5,
r(t,z)
C co
([F(t 6,z6)t)
= [coF(t6,z6)t.
Thus (1) follows from (13), and the validity of the assertion of Theorem 2 follows from Theorem 1, We will formulate a similar theorem for the problem
(16) (17)
z = ! (t, Z, Ul (t, z), ... , ur(t, z)), z(to)
= Zo,
where (t, z) E G, ! is a continuous vectorvalued function, Ui (t, z) is a scalar function discontinuous only on a smooth surface Si, i = 1, ... ,r. A theorem similar to this is proved in [95J. As in the case c), §4, solutions of equation (16) are solutions of the inclusion
(18)
z E co Fdt, z), Fl (t, z) =
! (t, Z, U1 (t, z), ... , Ur(t, z)) .
At the points of continuity of the function ut the set U. (t, z) is the point ut (t, z), while at the points of discontinuity, i.e., on the surface Si, the set U. (t, z) is a line segment joining the points ui(t, z) and ut (t, z), which are the limit values for Ui(t', z') as t' + t, z, + z. It is assumed that (to, zo) E G, a ~ to ~ b. The same assumptions are made for the problem (19)
z = r (t, z, u~(t, z), ... , u~(t, z)), r is continuous, the function u; is either continuous, or discon
The function tinuous only on the surface
S;, i = 1, ... , r.
92
Chapter 2
Solutions of Discontinuous Systems
THEOREM 3. Let for a ~ t ~ b all the solutions (in the sense of the definition c), §4} of the problem (16), (17) exist and lie in the open domain C. Then for any ~ > 0 there exists 0 > 0 such that for any E la, bj, /*, ui satisfying the conditions
to
(20)
It~
 tol
~
S,
Ixo xol ~ S,
xo,
1/*  II ~ S,
i = 1, ... , T, each solution of the prqblem (19) exists for a ~ t ~ b and differs for these t from some solution of the problem (16), (17) by not more than e. PROOF: According to 1, §6, the function co FI (t, x) satisfies the basic conditions of 2, §7. By Theorem 3, §7, the set H of points (t, x), a ~ t ~ b, which lie on the graphs of solutions of the problem (16), (17), is bounded and closed. By Lemma 1, §5, p(H, aC) ~ 30' > o. Next, the function Ui and I are examined only for (t, x) E H
0 as
1, ... , T,
we have
If(t', x', u~, ... , u~)  I(t, x, Ul, ... , ur)1
(22)
~
1/(1]),
where 1/(1/l > 0 as I'] > o. Assuming the contrary and using the compactness of H t7 , we prove the following. For any 0 > 0 there exists S = S(O) > 0, such that for each set of indices N = (il, ... ,i.) and for each point (t,x) E Ht7 whose neighbourhood (t 8,x6) contains points of all surfaces Si, i E N, the set (to,xO) contains a point common to all these surfaces, more precisely, a point of the set niENSi. Obviously, 0(0) ~ o. Let 0 < 0 < 0', 0 < S < 0(0). By virtue of the last inequality in (20), for each point (T, e) close to the point (t, x) E Ht7 and for each i ~ T there exists a point (Tf, E (T'\ 6) such that
en
e
(23)
e)
If in (to, there are no points of surfaces S1, ... , Sr, then, by virtue of (23) and (21), as (T, e) > (t, x), all the limit values of the function UnT, e) are contained in the (S + J.L(S))neighbourhood of the value u;(t, x), i.e.,
(24)
ut(t, x) c (Ui(t, x)) 6+1'(0) ,
i = 1, ... , T.
Dependence 01 Solution on Initial Data
§8
93
If in (t D, x D) there are points of one or several surfaces S. (Le., surfaces S. with numbers i EN), in (t 9 , x9 ) there exists a point (t', x') common to all these surfaces. Arbitrarily close to this common point there exist points of each of the two domains into which the space is separated by the surface S., i.e., points of each of the domains of continuity of the function ttl' i E N. In both cases (i E Nand i ¢. N), arbitrarily close to the point (t', x') there are points (r?, belonging to the same domain of continuity of the function Ui, as the point (rI, in (23). One may assume that 7', e, 7'.0, e? differ from t, x, t', x' (respectively) by less than 0/2. Since (t',x ' ) E (t 9 ,x9 ), (rI, E (7's,e S ), I7'Ir,ol < f'J, le~e?1 < f'J, f'J = {} + 20. Now, by virtue of (21),
e?) en
en
(25) As
(7', e) + (t,x),
(rp, e?)
+
(t', x'), it follows from (23) and (25) that i
(26)
= 1, ... ,r.
Thus, for each point (t,x) E HtJ' there exists a point (t', x') e (t 8 ,x8 ) for which (26) is valid (in the case (24) t' = t, x' = x). Then, by virtue of (22) and of the inequality 11*  II ~ 0, the set
Ft(t, x) =
r (t, x, U;(t, x), ... , U;(t, x))
is containe:l in the (0 + v(f'J»neighbourhood of the set Fdt', x') C Fdt 9 ,x8 ). With the help of Lemma 9, §5, we obtain
(27)
co Fl*( t, x )
C
[co Fl (9 t ,x9)]D+v{rll .
As 6 + 0, we may take {} = 0(6) + O. Then f'J = {} + 26 + 0, v(f'J) + O. Hence, if 0 is sufficiently small, the number dD (co Fi, co F1 ) for the domain D = HtJ' is arbitrarily small, and from Theorem 1 there follows the assertion of Theorem 3 for solutions of the inclusions
X E coF1 (t, x),
X E co Fi(t, x),
which are equivalent to equations (16) and (19). COROLLARY. If under the assumptions of Theorem 2 (or Theorem 3) the problem (12) {correspondingly, the problem (16), (17») has a unique solution on the interva1[to, bj, as well as on any smaller interval [to, cj C [to, bj, then for to ~ t ~ b this solution depends continuously on the initial data and on the righthand side of the equation.
In this case, small variations of the righthand side are variations, similar to those in (13) and (20) for a small o. This means that it is not only small variations of piecewise continuous functions I in (12) and Ui in (16) in the domains of their continuity, but also small variations of the boundaries of these domains are admitted.
Solutions of Discontinuous Systems
94
Chapter 2
Figure 12
3. Differential equations with discontinuous righthand sides are often used as a simplified mathematical description of some physical systems. The choice of one or another way of definition of the righthand side of the equation on a surface of discontinuity, for instance, the definitions a), b) or c), §4, depends on the character of the motion of the physical system near this surface. Suppose that outside a certain neighbourhood of a surface of discontinuity of the function I(t, x) the motion obeys the equation :i; = I(t, x). In this neighbourhood the law of motion may be not completely known. Suppose the motion in this neighbourhood may proceed only in two regimes, and switching over from one regime to the other has a retardation, the value of which is known only to be small. Using these incomplete data, we should choose the way of defining the righthand side of the equation of the surface of discontinuity, so that for a sufficiently small width of the neighbourhood the motions of the physical system differ arbitrarily little from the solutions of the equation :i; = f(t, x) defined in the way we have chosen. The theorems proved above make it possible to motivate the choice of one or another way of definition in some frequently discussed cases. Let the piecewise continuous vector function f(t, x) and the solutions of the problem (12) meet the requirements of Theorem 2, the function yet) be absolutely continuous and /y(to)  xo/ ~ S. In each of the domains of continuity G. the function I is equal to some function I. continuous in G •. Let the function f. be continuously extended from the domain Go into its Sneighbourhood. 10 Let, outside the Sneighbourhood Mli of the set M, on which the function f is discontinuous, yet) satisfy the equation y = f(t, y), and in the neighbourhood itself let
/y(t)  f (t, z(t))/
(28)
~ 0,
for almost all t where z(t) is any function such that /z(t)  y(t)/ ~ o. In particular, at each point of Mli, which is at a distance not greater than S from the domains Gi, G j , Gk, ... , the motion may obey any of the laws
iI =
Mt, y),
y=fj(t,y),
if = !k(t, y), ....
The switching, i.e., the changeover from motion under the law iI = h(t, y) to motion under the law iI = fi(t, y), may occur at any point of the Mli which is at a distance not greater than 0 from the domains G i and Gj (Fig. 12).
Dependence of Solution on Initial Data.
§8
95
Then on a given interval a ~ t ~ b the function y(t) differs from some solution (in the sense of the definition a), §4) of the problem (12) by less than any 6 > 0 if 0 = 0(6) is sufficiently small. We shall show this. By virtue of the definition a), §4, the equation $ = f(t, $) is equivalent to the differential inclusion $ E F(t, $). According to (28), yet) E [F(t, (y(t))SjS, and the result follows from Theorem 1. 2° Let, for almost all t,
o ~ l' ~ o.
yet) = / (t, y(t  1')) ,
The retardation l' may have a constant value or may vary arbitrarily between 0 a.nd o. Then for y( t) there holds the same statement as in the case 1°. We shall show this. In a domain where If I ~ m, we have for 0 ~ l' ~ 0
Iy(t  1')  y(t)!
~
mo,
yet)
E F
(t, (y(t))mS) ,
the function F being the same as in the case 1°. Now the assertion follows from Theorem 1. 3° Let, in the oneighbourhood of the set of points of discontinuity of the function f, the derivative yet) differ for almost all t by not more than 0 from some mean (with any nonnegative weights) values of the function f(t, $) in the 6neighbourhood of the point (t, y(t)) , for example,
(29)
Iy(t)  (at!(t,zl) al
+ ... + am =
I,
+ ... + amf(t,zm))1
ai ~ 0,
~
6,
i
=
IZi  y(t)l ~ 6,
1, ... , m,
where the numbers ai and the vectors Zi may depend arbitrarily on t and yet). Another possibility is: Iy(t)  f*(t,y(t))! ~ 0,
ret, y) =
f(t, $)p(t, $, y)d$,
/
1"111 0 in the domain G. Then in these domains the differential inclusions
THEOREM 3.
(10)
X E F(x),
(11)
x E p(x)F(x)
have the same trajectories. PROOF: Let x(t) be a solution of the inclusion (11), that is, the function x(t) is absolutely continuous and
dx(t)
~
= v(t) E P (x(t)) F (x(t))
for almost all t. On the interval a
~
ret) =
t
~
b we put
r p (x(s)) ds.
Jto
The derivative r'ft) = p(x(t)) ~ c > 0 is continuous. There exists an inverse function t(r). The function x*(r) = x(t(r)) is absolutely continuous ([64], p. 264), and dx*(r) = dx t'(r) = v(t(r)} E F (x*(r)}
dr
dt
p(x*(r)) almost everywhere (since the functions r'ft) and t'(r) are continuous, the con
ditions "for almost all t" and "for almost aUr" are equivalent; see [64], p. 268), that is, x*(r) is a solution of the inclusion (10). Thus, the trajectory of any solution of the inclusion (11) is also the trajectory of some solution of the inclusion (10), and vice versa, since the function 1/p(x) > o is also continuous. REMARK: If at some points x = x(t) of the trajectory of the inclusion (10) the function pet) vanishes, this trajectory can be divided by such points into several (sometimes infinitely many) trajectories of the inclusion (11). The differential inclusion
dz dt
(12)
E
F(t, x),
where X= (Xl, ••• , x n ) is equivalent to an autonomous differential inclusion in an (n + I)dimensional space Xo, Xl, ••• , xn: (13)
dz o dt
=1
dx
'
dt
E
F(t,x).
After renaming t as zo, the graphs of solutions of the inclusion (12) in the (t, x)space coincide with trajectories of the inclusion (13) in the (xo, Xl, • •• , xn)space. The inclusion (13) is not of the general form because the set of admissible values of the derivative dz· Idt, where x* = (xo, Xl, ••• , xn ), lies in an ndimensional hyperplane of the (n+1)dimensional space. Hence a backward transition from an
Change of Variables
§9
103
autonomous differential inclusion to a differential inclusion in a space of smaller dimension is not always possible and must be considered separately. The set F in (10) and (12) is contained in an ndimensional space, which can be called a velocity space VI,"" Vn. Let, for all z E G (G being some region in Rn), the set F(z) be closed, bounded, and lie in the halfspace ~ 'Y > 0, the function F being upper semicontinuous. Projecting the set F{z) from the origin onto the plane VI = 1, we get the set H(z) (Fig. 15). In this case to each point tI = (VI,.'" vn) E F(z) there corresponds the point
til
tin)
V2 ( 1,,.",
(14)
til
til
( V2 , ... ,
j
VI
tin) E H(z). til
Figure 15 LEMMA 3. Under the above assumptions, the trajectories of the inclusion dz
(15)
dt E F(z)
coincide with the graphs of solutions of the inclusion
(16) in the domain G. PROOF: Let z(t) = (Zl(t), ... , zn(t)) be a solution of the inclusion (15). Then the function z(t) is absolutely continuous and (17)
dz(t) ( i t = (Vl(t),,,., tln(t)) E F (z(t))
almost everywhere. By assumption, vt{t) ~ 'Y > O. Hence for the function zt{t) there exists an inverse function t(zt}, which is monotone and absolutely continuous. For almost all Zl (this is equivalent to "for almost all t" [64], p. 268) we have from (17)
(18)
dZ(t(Zl)) _ dZ l

dzdt .~ _(1'tJl(t)"'" tJ2(t) dZ l

tJn(t)) tJl(t) .
Chapter 2
Solutions 01 Discontinuous Systems
104
Since x = (Xl, y), then (16) follows from (18) and (14). Conversely, let y(xt} = (X2(X1),.'" xn(xt) be a solution of the inclusion (16), a. ~ Xl ~ b. Then for almost all Xl (19) From the point (0, ... ,0) of the (VI, ••• , vn)space a ray passes through the point (1, tL2(Xl)"'" tLn(xt}) and crosses the set F(x). Let (Vl(Xt}, ... , vn(xt}) be the point of intersection with the smallest coordinate VI. The functions F(x) and F(x(xd) are upper semicontinuous and, accordingly, the function Vl(Xt} is lower semicontinuous and, therefore, measurable, 0 < "t ~ vt{xd ~ m. Hence the function
the inverse function X1(t), and the composite function x(t) are absolutely continuous. Almost everywhere, (20)
dx dt
dY ) dXl = ( 1, dXl dt =
= (Xl(t),
Y(Xl(t)))
(I, tL2,···, tLn)vdxt{t)) .
Since by virtue of (19) (tL2, ... , tLn) belong to H(x), the righthand side of (20) belongs to F(x), i.e., x(t) is a solution of the inclusion (15). 3. From the above theorems and lemmas there follow similar assertions for differential equations with discontinuous righthand sides
x=
(21)
I(t,x)
under the definition a), §4, and
x=
(22)
1 (t, x, tLt{t, x), ... , ttr(t, x))
under either of the definitions b) and c), §4. The condition formulated in §4 (piecewise continuity of the function 1 in (21) and tLi in (22), continuity of 1 in (22)) or in Theorem 8, §7 (measurability of the function 1 in t, X in (21) and the inequality I/{t, x) I ~ m{t) with the summable function m(t)), are assumed to be satisfied in this case. Indeed, in all these cases solutions of a differential equation are solutions of the differential inclusion x E F(t, x) in which the function F(t, x) is constructed in a certain way with the help of the set of limiting values of the function f (t', x') (or tLi(t', x')) for x' + X, t' = t or for x' + X, t' + t. Since all the transformations considered in 1, 2 are continuous in x (and, except Lemma 3, either retain the planes t = const or map them into the planes r = const), they map a set of limiting values into a set of limiting values, and a convex set of admissible values of the derivative is linearly transformed into a convex set. Therefore, from the results of 1, 2, there follow similar assertions for equations (21) and (22). In the assertion analogous to Lemma 3, in the case of the definition a), the condition "t of 1, §6, should be assumed to hold. By virtue of what has been said, we need only formulate these assertions.
x
Sufficient Conditions lor Uniqueness
§10
105
=
THEOREM 4 [931, [95]. After the transformation y 1Ji(t, :z:), where the function an inverse transformation :z: = 1Ji1(t, y) EO,
1Ji belongs to 0 1 and there exists
each solution of equation (21) or (22) is mapped into a solution of the equation
iI = 1Jiat,:z:) + 1Ji~(t, :z:)/(t, :z:)I",=.pl(t,y),
(23)
or, respectively, of the equation
Y=
(24)
1JiHt,:z:) + 1Ji~ (t,:z:) I (t,:z:, udt, :z:), ... , u,. (t,:z:)) I",=.pl (t,y)'
COROLLARY. After the change y = 1Ji(t, x), the equation :i; = fO(t, x) which determines the solutions of equation (21) lying on the surface of discontinuity or on the intersection of such surfaces is mapped into the equation iI = gO(t, y) which determines the same kind of solutions of equation (23). REMARK: IT 1Ji 1 (t, y) ¢ 0 1 , it may turn o~t that not every solution of equation (23) or (24) is obtained from a solution of equation (21) or (22). For instance, the equation :i; = 1 has only the solutions x = t + c, and after the change y = x 3 the derived equation iI = 3 y 2/3 has, besides the solution y = (t + c)3, also a solution y = 0, which cannot be obtained from the solutions x = t + c by the change y = x 3 •
t(r) be strictly monotone and t'(r) be piecewise continuous. Then, after the change t = t(r), each solution x(t) of equation (21) or (22) is mapped into the solution y(r) = x(t(r)) of the equation THEOREM 5 193], [951. Let the function
dy dr = J(t(r),y)t'(r) (the righthand side being equal to zero ift'(r) = 0) or
~~ =
f(t(r),y,u1 (t(r),y) , ... ,u,. (t(r),y))t'(r).
p(x) > O. Then the equations :i; = p(x)f(x) have the same trajectories in the phase space x. The
THEOREM 6. Let the continuous function
f(x) and
:i; =
same holds for the equations 3;=
f(x,ttdx), ... ,u,.(x)) ,
x
= p(x)f (x, ttl(X), ... , tt,.(x)).
REMARK: After the changes indicated in Theorems 46, the righthand sides of the derived equations satisfy the same conditions (out of those formulated at the beginning of 3), which they satisfied before the change. The conditions imposed on J(t, x) in Theorem 8, §7, hold also after the change with any absolutely continuous strictly monotone function t(r) in Theorem 5.
106
Solutions of Discontinuous Systems
Chapter 2
§10 Sufficient Conditions for Uniqueness
We present here sufficient conditions under which a solution lying on a surface of discontinuity of the righthand side of a differential equation or on an intersection of surfaces of discontinuity is uniquely continued in the direction of increasing t. 1. We say that for the equation
(1)
x = f(t,x)
right uniqueness holds at a point (to, xo) if there exists h > to such that each two solutions of this equation satisfying the condition x(to) = Xo coincide on the interval to ~ t ~ tl or on the part of this interval on which they are both defined. For equation (1) right uniqueness holds in a domain D (open or dosed) if for each point (to, xo) E D every two solutions satisfying the condition x(to) = x~ coincide on each interval to ~ t ~ tl on which they both exist and lie in this domain. Left uniqueness at a point and in a domain is similarly defined as uniqueness for tl ~ t ~ to. LEMMA 1. .From right uniqueness at each point of a domain D there follows right uniqueness in this domain D . .From right uniqueness in a domain D there follows right uniqueness at each interior point of this domain. Both these assertions are easy to prove by assuming the contrary. For the Caratheodory equations, Theorem 2 of §1 gives a sufficient condition for right and left uniqueness in a domain D, and the remark to this theorem gives the condition for right uniqueness. Both the theorem and the remark (as well as the proof presented in §1) remain valid for differential equations with discontinuous righthand sides if their conditions are satisfied not only for the values of the function f(t, x) in its domains of continuity, but also for those values which further define this function at its points of discontinuity. The next theorem gives conditions under which one may disregard these additionally defined values of the function f (t, x). THEOREM 1 [93]. Let a [unction f(t,x) in a domain D be discontinuous only on a set M of measure zero. Let there exist a summable function l(t) such that for abnost all points (t, x) and (t, y) ofthe domain D we have If(t, x)1 ~ l(t) and for Ix  yl < eo, eo > 0,
(2)
(x  y) . (f(t, x)  f(t, y))
:os;;
l(t) Ix _ Yl2 •
Then under the definition a), §4, equation (1) has right uniqueness in the domain D. PROOF: For almost all t we have l(t) < 00, and the inequality (2) holds for almost all x and y in this domain. Then with these t, we already have for all
x·,y·
(3)
(x·  y*) . (v  w) ~ l(t} Ix* _ y*1 2 ,
Sufficient Conditions for Uniqueness
§10
107
where v and w are arbitrary values of sets V and W of the limit values of the function I(t, x) for x + x· and, correspondingly, of the function I(t, y} for y + y •• By virtue of Lemma 8, §5, the inequality (3) is valid if one first replaces the condition v E V by the condition v E co V and then the condition w E W by the condition wEco W. The inequality (2) is therefore valid for almost all t for all x and y, Ix  tl < eo, if for (t, x) E M the value f(t, x) is replaced by any value of the set F(t, x) defined in §4; the situation is similar for (t, y) E M. But then for any two solutions x(t) and yet) in the domain D for almost all t
~! Ix(t) 
y(t)12 == (x(t)  yet)) (x(t) 
yet)) ~ let} Ix(t} 
yet) 12.
From this there follows right uniqueness (see the proof of Theor~m 2, §1). Theorem 1 is valid [93] also for any discontinuous function !(t,:Il) measurable in the domain D if its solutions are defined as in 6, S7.
2. Let a domain G c R n be separated by a smooth surface S into domains G and G+. Let I(t, x) and al lax" i = 1, ... , n, be continuous in domains
(a < t < b, x E G) and (a < t < b, x E G+) up to the boundary. The definition a), §4, is applied to equation (1) on the surface S. Let I(t, x) and 1+ (t, x) be the limiting values of the function I at the point (t, x), xES, from the regions G and G+, correspondingly. Let 1+ (t, x)  ret, x)
= h(t, x)
be a "discontinuity vector," Iii, I'J, hN be projections of the vectors 1,/+, h onto the normal to S directed from G to G+ at the point x. In the domain G and G+ uniqueness (right and left) of solution holds due to continuity of the derivatives aI lax" i = 1, ... , n. LEMMA 2. If at some point Xo E S we have I~(to, xo) > 0 (or I'J (to, xo) < 0), then in the domain G+ there exists a ,unique solution of equation (1) with the initial data x(to) = Xo. This solution is defined on some interval to < t < tl (respectively, tl < t < to). 'Similar assertions are also true for G in the cases Iii < 0 and Iii > O. PROOF: Let us extend continuously the function I(t, x) from G+ to a whole neighbourhood of the point (to, xo). A solution with the initial data x(to) = Xo will exist. For all such solutions the vector x(to) = I+(to, xo) is directed towards the domain G+ since I'J (to, $0) > o. Every such solution lies in S U G+ for to ~ t < h; also, I E 0 1 there and the solution is unique. COROLLARY 1. On the region of the surface S, where Iii > 0, I~ > 0 (or Iii < 0, Iii < 0), the solutions pass from G into G+ (correspondingly, from G+ to G), and uniqueness is not violated.
By virtue of the definition a), §4 (see explanation of Fig. 3), there are no solutions lying on the surface S.
108
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Solutions 01 Discontinuous Systems
COROLLARY 2. Onto each point of such a region of the surface S, where IN> 0, I~ < 0, for each t there comes exactly one solution from the domain G and one solution from the domain G+. In the case IN > 0, lJi < 0 as t increases, the solutions can escape from the surface S neither into the domain G nor into G+. They remain on Sand, according to a), §4, satisfy the equation :i; = 10(t, x), where the function jD is defined by formula (5), §4. If sEa l , the unit vector n( x) of the normal to S is a continuous function of the point x, so IN E a, lJi E a and, by virtue of (5), §4, 1° Ea. If I E 0 2 , then n(x) E 0 1 and, therefore, the vector 10(t,x) is a smooth function (of class 0 1 ) of the local coordinates on the surface. (If in the neighbourhood of the indicated point we have arp/aXi '" 0, then the equation rp(x) = o on the surface S is solvable with respect to Xi, and local coordinates are Xl, .•. , xil! xHl,"" X,..) Then through each point of the indicated region of the surface S there passes exactly one solution of the equation :i; = 10(t, x). This argument does not yet make it possible to prove right uniqueness at the points where IN > 0, lJi = 0, or IN = 0, lJi < 0 and cannot be applied to the case where, within a finite time interval, solutions arrive infinitely many times onto the surface of discontinuity and leave it. Lemma 3 and Theorem 2 given below include these cases also. LEMMA 3. Let SEa l j let, at the points of the open region So of the surface S for a < t < b the vector h = 1+  1 be directed along the normal to the surface (or be equal to zero), and hN ~ O. Then for equation (1) there holds right uniqueness in the neighbourhood of any point Xo E So for a < t < b. PROOF: We show that the inequality (2) with l(t) = const is fulfilled. If both the point x and the point y lie on one side of S, then (2) follows from boundedness of a I/ax. in the indicated neighbourhood for X ¢. S. IT x E G+ and y E G, then let z be a point of intersection of the line segment xy with the surface S, which is nearest to x. From the boundedness of al/axi it follows that
(4)
I/(t, x)  r(t, z)1 ~ lIx 
zl,
Ir(t,z)  l(t,y)1 ~ liz 
YI·
Since z E S, the remaining points of the segment zx lie in G+, and the vector h = 1+ (t, x)  1 (t, x) is directed towards the domain G along the normal to S or is equal to zero, then
(x  z) . (r(t,z)  r(t,z)) ~
o.
This inequality remains valid also if the vector x  z is replaced by the vector x  y which has the same direction. Adding the derived inequality to the one following from (4)
(x  y) . (J(t, x) 
r
(t, z)
+ r (t, z)
 I(t, y)) ~ llx _ Yl2 ,
Sufficient Conditions for Uniqueness
§10
we obtain the inequality (2). Theorem 1.
109
Now the assertion of the lemma follows from
REMARK: The sign of the number hN = ft;  fN is altered neither by renaming of the domains G and G+ nor by a differentiable transformation of coordinates. For instance, if hN < 0, the sum IXN I + IYN I of the distances from the tangent plane to S of the points x(t) E G+ and y(t) E G close to the point Xo E S decreases because XN > 0, YN < 0,
the number a being small. The property of this distance being increased or decreased and, therefore, the sign of hN(t,XO) is retained under differentiable transformations. The general case with the condition hN ~ 0 can be reduced to the case of Lemma 3 with the help of differentiable coordinate transformation. LEMMA 4. Let the function g(Z2, ••• ,Zn,t) E 0 1, and f(z2, ••. ,Zn,t) have continuous lirst and second derivatives except possibly 8 2 f / 8t 2 • Then there exists a function '7(Z1, ••• , zn, t) with continuous lirst and second derivatives, except possibly 82'7/8t2, which satislies the requirements '7 = f, 8'7/8z1 = g for Z1 = O. PROOF: The function '7(Zl, ••. , zn, t) = f(Z2, .•• , zn, t)
+ Zl
1 1
...
fo
g(Z2
1
+ U2Z1, ••• , Zn + UnZ1, t)dU2 ••• dUn
satisfies the conditions '7 = f, 8'7/8z1 = g for Zl = O. In the following integrals the arguments of the function g are the same as in the preceding one: 8'7 = 8 Zl
11 11 ...
0
gdU 2
••• dUn
+ Z1 Ln
i=2
0
11... 11 0
0
8g
U i  d U2 ••• dUn.
8
Zi
Integrating by parts, we obtain Zl
1 1
o
8g
U8i  dUi
Zi
= glu,=l 
11
gdUi.
0
Consequently, 8'7/aZ1 has first derivatives with respect to all arguments, which are continuous up to the plane Zl = o. The same is valid for a'7 /8zj, j = 2, ... , n. LEMMA 5. Let a nontangent vector h(X2, •• • , X n , t) E 0 1, Ihl ~ S > 0, be given on a surface S (Xl = e(X2, ••• , Xn) E 0 2 ). Then in the neighbourhood of the surface S there exists a transformation Xi
= ~i(Zl' ••. ' Zn, t),
i = 1, . .. ,n,
Chapter 2
Solutions of Discontinuous Systems
110
with a Jacobian J i 0, such that the surface Zl = 0 coincides with S, and the coordinate lines ZI (that is, lines on which Z2, •.• , Zn. are constant) are tangent to the vector h at the p.oints of the surface S j i,j = 1, ...
,n,
for Zl = 0 the function lPi do not depend on t. PROOF: The transformation Y2 = X2, ••• , Yn. = xn.
maps S into the plane Y1 = O. The direction of the vector h = (hlJ"" h n ) is expressed by the equations
in the old coordinates, and by the equations
(5)
dYl
dY2
dYn
=="'=, g h2 hn
in the new coordinates. The vector h is not tangent to S, therefore, g Now we pass over to Zl, ••• ,.%n by the formulae
(6)
(7)
Y1
= %1,
Tii
Yi = 77i(Zl"",Zn,t),
= Zi;
i o.
i = 2, . .. ,n,
i = 2, .. . ,n.
Such functions TJi exist by Lemma 4. Along the coordinate lines ZI we have dZ 2 = ... = dZ n = dt = 0, that is, by virtue of (6) and (7) at the points of the surface Zl = 0, aTJ' h· dYi = a• dYl = 2. dYl. %1 g From this and from (5) it follows that for %1 = 0 the coordinate lines Z1 are tangent to the vector h. Expressing Xl, ••• ,Xn through Z1, ••• , Zn, we obtain to the required transformation. By virtue of (7) the Jacobian of the transformation (6) is equal to unity for %1 = 0; the Jacobian of the transformation from X to Y is also equal to unity. THEOREM 2 [93]. Let the conditions of 2 be fulfilled, the surface S E 0 2 , and the vector h(t, x) = f+  f E Cl. If for each t E (a, b) at each point xES at least one of the inequalities fN > 0 or f"J < 0 (possibly different inequalities for
111
Sufficient Conditions lor Uniqueness
§10
different x and t) is fulfilled, then right uniqueness for equation (1) occurs for a < t < b in the domain G. PROOF:
a)
At least one of the three conditions
IN > 0,
I~ > OJ
b)
IN < 0,
I~ < OJ
c) I~
 IN < O.
is satisfied at each point of the surface S. In the cases a) and b) right uniqueness occurs in the neighbourhood of such a point by virtue of Corollary 1 of Lemma 2. In the case c), in some neighbourhood ofthe point the inequality I~ IN < 0 is satisfied also, and the equation of the surface S can be solved with respect to one of the coordinates, for example, Xl. After a transformation x = IP(z, t), as in Lemma 5, the surface S will be mapped into the plane Zl = 0, and the vector h = 1+  f into the vector orthogonal to this plane. By Theorem 4, §9, equation (1) is transformed into the equation
(8)
z = ,pHt, x) + ,p~(t, x)f(t, x) I:.:=rp(z,t) ,
where Z = ,p(t, x) is a transformation inverse to x = IP(z, t), ,p~ being a vector, and ,p~ being a matrix (see 1, §9). The function ,p possesses the same smoothness properties as in Lemma 5. Accordingly, the righthand side of (8) and its derivatives 8j8z, are continuous in the neighbourhood ofthe indicated point for Zl > 0 and Zl < 0 up to the plane Zl = o. We will show that the discontinuity vector of the righthand side of (8)
h*(t,x) = ,p~(t,x) (t+(t,x)  r(t,x» = ,p~h is orthogonal to the plane of discontinuity xspace, that is the lines
Zl
= O.
The coordinate lines
Zl
in the
(9) are tangent to the vector h at the points of the surface S. So, for vector IP~ is collinear with the vector h, that is,
Zl
= 0 the
(10) Applying to both sides of the equality (9) the transformation is inverse to '1', we have
Differentiating with respect to
Zl,
Z
= ,p(t, x) which
we are led to
(1,0, ... ,0) = ,p~ . IP~l' For Zl = 0 we deduce from this equation and from (10) that the vector ,p~h = h· is collinear with the vector (1,0, ... ,0), i.e., orthogonal to the plane Zl = O.
Solutions 01 Discontinuous Systems
112
Chapter 2
By virtue of the remark to Lemma 3 it follows from the inequality hN = that h'N < O. By Lemma 3, in the neighbourhood of the indicated point in the case c) right uniqueness takes place for equation (8), and therefore for equation (1).
I"J  Iii < 0
3. Consider the differential equation
(11)
x= I
(t, x, u(t, x))
with the definition b) on the surface of discontinuity. Let the functions I, a I / ax. (i = 1, ... ,n), ai/au be continuous and the function u(t,x) be discontinuous on a smooth surface S (s(x) = 0), which separates the domain G into domains G (s(x) < 0) and G+ (s(x) > 0), in which u(t, x) and aU/aXi are continuous up to the boundary. Suppose sex) E 0 1 , the gradient Vs i= 0 on S. In approaching S from the domains G and G+, the limiting values of the function u( t, x) will be denoted by u (t, x) and u+(t, x), and the limiting values of the function I(t,x,u(t,x)) by I(t,x) and f+(t,x). Let IN,lii,/~ be projections of the vectors 1,1,1+ onto the normal to'S, as in 2, that is,
( ) _Vs(x)'/(t,x,u) f Nt,x,U IVs(x)1 . Let U(t, x) be an interval with the endpoints u (t, x) and u+ (t, x). H xES and lii(t, x) I~(t, x) ~ 0, i.e., the function IN(t, x, u) does not retain the positive or negative sign under variation of u on the interval U(t, x), then there exists a solution ueq(t, x) E U(t, x) of the equation IN(t, x, ueq(t, x)) = 0, that is, (12)
v sex) . I (t, x, ueq(t, x)) = O.
Then the vector rq(t,x) = I(t,x,ueq(t,x)) is tangent to the surface S at the point x, and at such points equation (11) is further defined as follows:
(13)
x= I(t,x,ueq(t,x)).
LEMMA 6. If S E 0 2 , and at the point xES
(14)
lii(t,x)I~(t,x) ~ 0,
for all u E U(t, x), then on the interval U(t, x) equation (12) has a unique solution ueq(t, Xli the [unctions ueq(t, x) and au eq /aXi are continuous in t, x. PROOF: By virtue of (14), under the change of u on the interval U(t, x) the function IN(t, x, u) is monotone and reverses sign or is equal to zero at the end of the interval, so the solution ueq (t, x) exists and is unique. By the implicit function theorem, ueq (t, x) EO. H S E 0 2 , then aIN / aXi E and, hence, there exist continuous 8u eq /ax,.
a
Sufficient Conditions lor Uniqueness
§10
113
If, on the region 8 0 of the surface 8 E 0 2
COROLLARY.
Iii (t,x) > 0,
afN(t,x,U) =F au
lit(t,x) 0 or I"J < 0 (possibly, different inequalities for different t and x) is valid at each point x E 8 then for (I < t < b in the domain G a solution with the initial data x(to) = Xo E G exists and right uniqueness holds for equation (11). PROOF: As in Theorem 2, it suffices to consider the case I"J  Iii < 0. In the neighbourhood of a point where Iii > 0 I"J < 0, right uniqueness is guaranteed by the corollary of Lemma 6. It remains to consider those cases where at a given point Iii > 0, I"J = 0 or Iii = 0, I"J < o. The second case is reduced to the first one by renaming the domains G and G+. Let Iii (to, xo) > 0 and I"J (to, xo) = o. By the implicit function theorem, the solution ueq(t, x) of equation (12) exists and belongs to 0 1 in some neighbourhood of the point (to, xo) (It  tol < 8, x E 8, Ix  xol < 8) even in the region where I"J > o. Hence, in such a neighbourhood the function
(15)
tI(t, x)
=
1 aa 1
o
u
I(t, x, u+
+ u(ueq 
u+)) du
rq
= ueq 
1+ U
+,
r
q = I(t,x,ueq(t,x))) belongs to 0 1 • Since alN/aU retains its sign (here (possibly, in a smaller neighbourhood), so does tlN(t, :1:). Let (16)
sgntlN(t,x) = 0,
ret, x)
+ Otl(t, x) = f; (t, x).
r
Then I; (t,:I:) E 0 1 , the vector tI is collinear with the difference q  1+. and the endpoint of the vector I:: lies on the line which passes through the endpoints of the vectors 1+ and q (Fig. 16). We extend smoothly the function I:: from the surface 8 to its half neighbourhood which lies in G . We will show that the solutions of the equation
r
(17)
.  f .. (t ,x) _ { I (t, x, u(t, x)) xI:: (t, x),
xEG+, xEG,
114
Chapter 2
Solutions of Discontinuous Systems
o Figure 16 defined on S according to a), §4, coincide in the domain G+ and on S with the solutions of equation (11) defined on S according to b), §4. Indeed, by virtue of (16) (18)
r(t,x)  f;(t,x) = 8v(t,x) = h{t,x),
Hence, in the neighbourhood of the point (to, xo) the vector (t;)N > ,.,/2 > O. At points of this neighbourhood where I"J ~ 0, the vectors q = I(t, x, ueq(t, x)) and 12(t, x) are defined; 12(t, x) is the velocity of motion on the surface S for equation (17). The endpoints of these vectors lie on the straight line which passes through the endpoints of the vectors f+ and I;, and at the same time in the plane tangent to S. By virtue of (18) this straight line crosses the plane only at one point, which is the end point of the vectors q and I~. Hence, q q == 12. (The same result can be obtained in another way, by expressing in terms of 1+ and I; by means of formulas (15) and (16), and 12 by means of formula (5), §4.) Thus the solutions of equation (11) in G+ and on S coincide with the solutions of equation (17). For both the equations the solutions do not go off S into the domain G. For equation (17) with the definition a), §4, existence of the solution for any initial data is proved in 4, §7, and right uniqueness is proved in Theorem 2, §10. The condition that 1 belongs to C l in G provides existence and uniqueness of the solution of equation (11) in the domain G. Therefore, for any initial data x(to) = Xo E G the solution of equation (11) exists and for t ~ to it is unique. 4. We shall indicate the conditions for right uniqueness at points of intersection of several surfaces of discontinuity. Let a domain G c Rn be separated by smooth hypersurfaces sf< into domains Sj, j = 1, ... , r. The upper index denotes dimension, the lower index the number of the surface or of the domain; Sl are lines, S? are points. Suppose the edge of each hypersurface does not belong to this hypersurface and consists of a finite number of smooth hypersurfaces of smaller dimensions and points. For example, if G is a threedimensional space separated by three coordinate planes, then SJ (i = 1, ... ,8) are coordinate octants, Sf (i = 1, ... ,12) are quarters of coordinate planes, Sl (i = 1, ... ,6) are coordinate semiaxes, Sf is the origin.
r
r
r
r
§10
Sufficient Conditions lor Uniqueness
115
Let M be the closure of a set M. The vector V:f= 0 is called tangent to the set M at a point z E M if there exists a sequence of points at E M (i = 1,2, ... ) such that at  z,
(19)
(i  00). We shall consider the equation
z=
(20)
I(t,z)
in the domain G. Let the following condition be satisfied: 1*. The vectorvalued function I(t, z) is continuous in t, z for a < t < f3 in each of the domains Si up to the boundary, that is, I(t, z) = li(t, z) in Si, the function Ii is continuous in S'l. On some or all of the hypersurfaces sj, 0 =s;; k =s;; n  I, or on some of their closed subsets continuous vectorvalued functions INt, z) are given; the vector IlO(t, z) lies in the kdimensional plane tangent to Sf at the point z. (If the point z lies on the edge of the surface Si", then the vector lying in oa. tangent plane may be not tangent to Si" in the sense of (19).) At the point Sf either a zero vector If = 0 or no vector at all can be given. A function x(t) is a solution of equation (20) if it is absolutely continuous on an interval and satisfies the equation
z(t) =
(21)
Ii" (t, z(t)) .
for almost all t such that z(t) E Si". Hence, to the points of the surfaces Sf where the functions Ii" are not defined, the solution can get only on a set of t values of measure zero. These conditions are satisfied, in particular, in the case where for equation (20) on the surfaces of discontinuity the definition a) or b), §4, is applied provided only that all the vectors li"(t, z) are singlevalued where they are defined, that is, if for each z E Sf the set F(t, z) (or, correspondingly, FI(t, z)), defined in 2, §4, has not more than one common point with the kdimensional plane P;"(z) tangent to Sf at the point z. Indeed, the function F(t, z) is upper semicontinuous in t, z (1, §6) and, therefore, as well as the function (22) it has closed graphs (Lemma 14, §5). Thus, the function (22) is upper semicontinuous; if it is singlevalued, the function If(t, z) = Kf(t, z) is continuous. The same argument is valid for the function FI(t, z) in the case of the definition b). THEOREM 4. Let the condition 1* and the following three conditions be satis
fied:
1) The solutions of equation (20) cannot pass from one set S; to another in a finite time.
an infinite number of times
Solutions 01 Discontinuous Systems
116
Chapter 2
S:,
2) In each of the sets where the [unction Il· (t, x) is defined, right uniqueness holds for equation (21). 3) If the vector If (t, x) tangent to Sf or equal to zero is defined at the point x E Sf, then at this point we do not have a vector IJ (t, x) (for each S~. ¥= Sf) equal to zero or tangent to its SJ, except in the case If(r,x) = I~·(r,x) = 0 for all r ~ t. Then for the equation (20) in the domain G there holds right uniqueness. PROOF:
x(to)
Suppose the conditions of the theorem hold, but for some initial data
= Xo E G there exist two solutions x(t) and y(t), x(t) being not identical to
y( t) for to < t < to + h. Let tl be the greatest lower bound of those t E (to, to + h) for which x(t) ¥= y(t). Then x(tt} = y(tt) = Xl and there exists a decreasing sequence tm + tl + 0 for which x(t m ) ¥= y(t m ), m = 2,3, .... By virtue of the condition 1), for some m*
x(t) ESf, By virtue of the condition 2), Sf
y(t) E S~.
¥= SJ.
According to (21),
Since the function Il' is continuous in Sf, there exists a vector z(td = INtl. xd which is either tangent to Sik or equal to zero. Similarly, there exists a vector y(td = h(tl, xd which is either tangent to SJ or equal to zero. By virtue of the condition 3), this is possible only in the case where It(t, Xl) = IJ(t, Xl) = 0 for all t ~ ft. At least one of the functions x(t) and y(t) is nonconstant for tl ~ t < t + 5 for arbitrarily small 6, for instance, x(t). Then in Sf the problem z = lik(t, x), x(td = Xl has two solutions: x(t) ~ Xl and x"(t) == Xl since If(t, xd = O. This contradicts the condition 2). Hence, the supposition is false and the result follows. The assumption 1) of the theorem is far from being necessary. It can be weakened, for instance, in the following way: for any tl and for any solution there exists t2 > tl such that for tl < t < t2 this solution is contained in one of the sets But one must not discard the condition 1). This is seen from the following example. With the initial data x(to) = y(to) = 0 the system
Sr
(23)
z=
sgnx  2sgny,
y=
2 sgn x + sgn y
has a solution x(t) == y(t) == 0 and, besides, an infinite number of solutions, whose trajectories are untwisting spirals. Since each successive point of intersection between a trajectory and the coordinate axes is thrice as far from the point (0,0) as the previous one, and the velocity of motion is constant, the time intervals between the moments of these intersections form a geometric progression. The motion from the point (0,0) up to any point of the trajectory, therefore, takes only a finite time. Hence, for a
Variation 01 Solution"
§1l
117
certain finite to the initial data z = y = 0 are satisfied at least by two solutions: one is z(t) == y(t) == 0, while the other one has a spiralshaped trajectory. Right uniqueness does not hold. Note that by virtue of (23) (izi + Iyl) == 2. Hence all the solutions for which Iz(tdl + Iy(tdl = 2a left the point (0,0) at the same instant to = tl  a.
!
§1l Variation of Solutions Here we derive equations of first variation which are satisfied by the main part of the difference between two close solutions in the cases where solutions intersect the surface of discontinuity of the righthand side of a differential equation, enter this surface, and leave it. 1. Vectors z E Rn and I(t, z) E R n will be written as columns, for example z = (ZI,"" Zn) T, T being the transposition sign, and the vector rps = (81P/8z1,"" 81P/8zn) will be written as a rowi I~ = (8 J./8zi ki=I, ... ,n is a matrix. The products of vectors and matrices will be defined by the rectangular matrix multiplication rule (rows of the first matrix are multiplied by columns of the second one). In such products, factors can be grouped without permutation, for instance, a(Az) = (aA)z, where a is a row vector, z is a column, A is a matrix. Then az is a scalar product, za is a matrix (ztaik,.=I, ... ,ni E is a unit matrix. It can be easily proved that (za)2 = za . az.
(1)
Solutions and surfaces of discontinuity are considered in an (n+1)dimensional (t, z)space. All the surfaces and their intersections under consideration are smooth and all belong to class (J2. 2. Consider the variation of a solution in a region where the righthand side of the equation is smooth. Let z(t) and i(t) be solutions of the same equation Z = I(t, z) (z ERn) with the initial data z(to) = Zo, i(to) = Zo + ho, I and 81/8z,. being continuous. Then, as is known 19),
i(t)  z(t) = Y(t)h o + o(ho),
(2)
where the matrix yet) satisfies the firstvariation equation
(3)
yet)
= Is (t, z(t)) yet),
Y(to) = E.
The result remains valid if in some neighbourhood of a given arc of the graph of the solution z(t) the functions I and 81/8z,. are continuous in z and in their absolute value do not exceed some summable function m(t). If the same conditions are also satisfied by 8 21/8z,.8z1c, then the residual term in (2) can be replaced by O(h~), and one can write down the equation for the second variation. S. Consider the variation of a solution which intersects a surface of discontinuity 1156J. Let a vectorvalued function I(t, z) have a discontinuity on a smooth surface rp( t, z) = 0, I and 81/ 8z,. being continuous on both sides of this surface up to the surface. For t = t. let a solution of the problem
(4)
z=
I(t, z),
z(to) = Zo
Solutions of Discontinuous Systems
118
Chapter 2
pass from one side of the surface cp(t, x) = 0 to the other at the point x. = x(t.), and at this point let
(5) that is, intersection occurs without tangency. Here f± = f(t. ± 0, x(t. ± 0)). Then we represent the difference between two close solutions in the form (2) where the matrix Yet) satisfies equation (3) on both sides of the surface, and on the surface it has a jump
the values CPt and cplJ! being taken at the point (t., x.). According to the notation of 1, the numerator in (6) is a matrix ofrank 1 and the denominator is a number. In the case t. > to it is sufficient to require CPt + cplJ!f f:. 0 instead of (5), then the first of the qualities (6) is valid, but the matrix yet. + 0) can be degenerate. We shall prove formula (6). Near the point (t., x.) the equation of the surface cp(t, x) = 0 can be written in the form
(7) and the equation of the trajectory x(t) before its intersection with the surface can be represented as
(8) x(t) = x(t) + Y(t)ho
+ o(ho) =
x. + {t  t.)r
+ Y ho + 0 (It  t.1 + Ihol) ,
where Y = yet.  0). To find the point of intersection of this trajectory with the surface, we replace in (7) x by x(t) from (8). We obtain
From this we have for the intersection point
(9) tt. = 
cpzYh o f +o{ho), CPt + cplJ!
o x~( t ) =x. , CPIJ!Yhf+ Yh 0+0 (h 0 ) CPt + cplJ!
In the case t < t. on the interval (t, toO) we have dx/dt = f+ + O(h o), and therefore, x(t.)x(t.) = x(t.)x(t)+x(t)x. = (t.t)J++o(ho)+x(t)x.j using (9), we obtain (10)
i(t.)  x(t.) =
(r  r) CPtcplJ!Y~_ + Y ho + o(ho). + cplJ!
Since both the solutions have already intersected the surface cp(t, x) = 0, the lefthand side of (10) is equal to yet. +O)ho +o(ho ), and the first equality of (6) is thus proved. In the case t > t. the same result is obtained similarly.
Variation of Solutions
§11
119
The second equality in (6) is derived from the first if it is solved with respect to yet.  0) and the property (1) is used. 4. Consider the variation of a solution which lies on a surface of discontinuity. For t. ~ t ~ t· let solutions x(t) and i(t) lie on a smooth surface !pet, x) = 0 and satisfy there an equation :i: = fO(t, x). The vectorvalued function fa is continuous in t,x and is smooth in Xj the vector (1, fO(t, x))T is tangent to this surface at the point (t, $). We shall extend the function fO(t, x) from the surface into its neighbourhood, retaining continuity of fa and f~. Then for the variation of the initial data i(t.) = x(t.) +h., which does not lead the solution out of the surface (i.e., such that /p(t., x.)h. = o(h.), where $. = $(t.», the variation of the solution is written by analogy with (2), (3), that is, (11)
i(t)  $(t) = yO(t)h. + o(h.),
yO(t) = f2 (t,x(t» yO(t),
yO(t.) = E. The solutions i(t) and x(t) lie on the surface, therefore in (11) i(t)  $(t) does not depend on the way in which the function fa is continued. Hence, YO(t)h does not depend on this way either (for all vectors h such that !Plll(t., $.)h = 0), although the matrix yO(t) may depend on this procedure. These formulae are also valid for solutions lying on a smooth hypersurface of any dimension, for instance, on the intersection of a finite number of surfaces of discontinuity. In order that these formulae may be applied, the function fO must be smoothly continued from this hypersurface into its neighbourhood. 5. Consider the variation of a solution which reaches a surface of discontinuity and then remains on it. Let a solution x(t) of the equation :i: = f(t, x) lie in a domain G 1 (where f and fill are continuous) for to ~ t ~ t., reach the smooth surface S (!p(t, $) = 0) at a nonzero angle for t = t., and then lie on the region of the surface S, where the solutions do not leave S, and satisfy there the equation :i: = fO(t, x). Let the functions fa and f~ be continuous. To write the equation of first variation, we shall smoothly continue the function fa into its unilateral neighbourhood which does not belong to the domain G 1 • After this one can use the same argument as in 3, with the exception that the function f and f+ are replaced by f and fO. In this case !Pt + /PlIlf :F 0 at the point (t.,x.). As in (6), the matrix yet) for t = t. has a discontinuity
yet. + 0)  yet.  0) = (f0  j)!P1ll yet.  0), !Pt + !Plllf
(12)
the values fO,f,!Pt,!P1ll being taken at the point (t., x.). We will show that for t ~ t. the matrix yet) is degenerate. Solutions close to x(t) reach S at an instant close to t. and remain on S. Therefore in (2) for t > t. and for any (small) ho the vector Y(t)h o is tangential to an (n  1)dimensional crosssection of the surface S and the plane t = const, and rank
yet)
~ nl.
For t > t. the matrix yet) satisfies the equation
(13)
yet) = f2 (t,x(t» Yet).
We will show that yet) does not depend on the way in which the function fO is extended from the surface S into its neighbourhood.
120
Solutions of Discontinuous Systems
Chapter 2
If the surface 8 is a plane x,. = 0, then, as has been said above, for any small ho = x(to)  x(to) the vector Y(t)h o lies in the plane x,. = 0 for t > t •. Hence the last row of the matrix yet) consists of zeros. The remaining nl ro~s are uniquely determined from the (n  l)dimensional equation of first variation written in the plane x,. = 0; the initial value of yet. + 0) is known from (12). They do not therefore depend on the way in which the function fO is continued, and the matrix yet) is determined uniquely. The case of a smooth surface 8 is reduced to the case of a plane by a change of variables. 6. Consider the variation of a solution with the initial condition on a surface of discontinuity. Let us investigate the case where a variation of initial data is admitted which takes the solution out of the surface of discontinuity 8 (~(t, x) = 0). Let a solution x(t) with the initial condition x(t.) = x. for t > t. lie on 8 and satisfy the equation 3; = fO(t, x). In the region ~(t,x) > 0 near 8, let there hold an equation 3; = f+(t, x) and in the region ~(t, x) < 0 let there hold an equation 3; = f (t, x); let the vectors f+ and f be directed towards the surface 8, that is ~t + ~~f+ < 0, In the case ~~h. > 0 (~~h. < 0) the initial condition x(t.) = x. + h. of the solution x(t) lies in the region ~(t, x) > 0 (correspondingly, ~(t, x) < 0) if h. is sufficiently small (the required smallness of Ih.1 depends on the direction of the vector h.). Then the variation of the solution for t > t. is expressed by formulae (2) and (13), in which in the case lP~h. to the condition Y(t.) = E is replaced by the condition (14)
Y(t.)
= E + (f0  f%)~~ , ~t + ~~f%
where f+ (f) is taken for the case ~~h. > 0 « 0); the values f O, f+, f, ~t, ~~ are taken at the point (t., x.). We will prove formula (14). Let ~",h. < o. One can obtain this case from the one considered in 3 if one assumes to = t.  0 and denote the function fO continued to the region ~(t, x) > 0 by f+. Since on reaching the surface ~(t, x) = 0 the solution x(t) remains on it, formula (12) is valid. The jump of yet) at t = t. is described by the first equality in (6), where f+ is replaced by fO. From this equality there follows (14) with f instead of f%. The case ~~h. > 0 is obtained from the considered one by an obvious change of notation. 7. Consider the variation of a solution which passes from one surface onto another. Let solutions pass along a smooth surface 8 1 (,p1 (t, x) = 0) and satisfy there the equation:i: = fl(t,x). On reaching the surface 8 2 (,p2(t,x) = 0) at a nonzero angle (i.e, ,p~ + ,p;f1 to) they continue on this surface without leaving it and satisfy there the equation :i: = P(t, x). To write the equation of first variation of such solutions in the case of variations ho of the initial condition, which do not take the solution off the surface
§1l
Variation
0/ Solutions
121
8 1 , we continue the function /1 from the surface 8 1 onto its neighbourhood located on one side of 8 2 • Then the problem reduces to the one considered in 5. For the indicated values of ho the variation of the solution x(t) is expressed by formulae (2) and (13), where /0 is replaced by P for t < t. and by P for t > t. (the solution x(t) reaches 8 2 at t = t.); the jump of Yet) at t = t. is expressed by formula (12), where now one should replace /, /0 and 'I' by p, p, and ,p2. Consider the case where the surface 8 1 is tangent to the surface 8 2 along an (n 1)dimensional surface P. Draw an auxiliary surface rp(t, x) = 0 through P and consider the case where rpt + '1'11:/ 1 "1= 0 on P, i.e., solutions lying on S1 are not tangent· to the surface 'I' = O. Such a problem is analogous to the one considered in 3. The jump ofthe matrix Yet) is expressed by formula (12), where / and /0 are replaced by /1 and p. 8. Consider the variation of a solution which leaves a surface of discontinuity. Let solutions pass along a smooth surface ,p1(t, x) = 0 and satisfy there the equation z = P (t, x). Reaching the intersection P of this surface with a smooth surface rp(t, x) = 0 (which may be either a surface of discontinuity of the righthand side of the differential equation = P(t,x), like the surface ,p1 = 0, or an auxiliary surface serving only for defining the set P of points at which the solutions leave the surface ,p1 = 0), the solutions leave the surface ,p1 = 0 and continue into the domain G where they satisfy the equation z = /(t, x); the functions / and /11: are continuous in G. The set of points (t,x) lying on these solutions in the domain G forms a surface ,p2(t, x) = O. Then the problem of variation of the solution x(t), which first lies on the surface ,p1 = 0 and then in the domain G on the surface ,p2 = 0, is reduced to the problem considered in 7, but with the function / instead of p. As in 7, it is assumed here that rpt + 1jJ1I:/ 1 "1= 0, that is, on reaching the set P the solutions are not tangent to the surface rp(t, x) = O. In the case where /1 = / at the points at which solutions leave the surface ,p1 = 0, the matrix yet) does not have a jump. 9. Consider the variation of a solution which lies on a surface of discontinuity and passes over to the intersection P of this surface with another surface of discontinuity. Let solutions lie on a smooth kdimensional surface Sic (2 ~ k ~ n) and satisfy there the equation = /1 (t, x). Reaching a (k  1)dimensional intersection P of this surface with a smooth ndimensional surface S,. (rp(t, x) = 0), they lie further on P and satisfy there the equation = pet, x). It is assumed that rpt + '1'11:/ 1 "1= 0, i.e., that on reaching P the solutions are not tangent to the surface S". The variation of the solution is determined as in 5, 7, the jump of yet) at the moment t· of reaching P is expressed by formula (12), where now / and /0 are replaced by /1 and /2. As in 8, we consider the case where solutions lie on an intersection (of any dimension) of surfaces of discontinuity and then go off it onto one of these surfaces or into the domain of continuity of the righthand side of the equation. 10. Thus, the first variation of a solution can be written in the form (2) if the solution lies in the region of smoothness of the righthand side of the equation, if the solution crosses a surface of discontinuity or gets onto it and passes along it, if the solution gets onto the intersection of surfaces of discontinuity and passes
z
z
z
122
Solutions of Discontinuous Systems
Chapter 2
along it or goes off it, or if it goes off the surface of discontinuity. In this case it is assumed that: 1) All surfaces of discontinuity and their intersections (of any dimension) belong to class C2, and the functions determining the velocity of motion belong to class C1. 2) From each domain sn+l of continuity of the righthand side of the equation, from a surface sn of discontinuity or from an intersection Sk of such surfaces (the upper index shows dimension) the varied solution x(t) gets only onto a surface (or intersection of surfaces) pi which is by one dimension smaller and only at a nonzero angle; pi lies on Sk or on the boundary of Sk. 3) Then the solution either immediately leaves pi or goes along pi for some time; it can leave pi only having reached at a nonzero angle some smooth hypersurface Qi, :j = i  1 (i.e., dimQi = dim pi  1) which lies on pi or on the boundary of pi; leaving pi is possible into any of the sets sm adjacent to pi, the dimension m ~ i  1. 4) Leaving any Sk is uniquely determined by directions of the velocity vectors f (t, x) in the sets sm (m > k) adjacent to Sk. 5) In a finite time interval there may exist only a finite number of points where solutions pass over from one set Sk to another. For instance, if the surfaces of discontinuity are faces S2 of a cube S3, then one considers variations only of such solutions which do not get from S3 immediately on an edge S1 or onto a vertex So, as well as from S2 onto So. Going off the face S2 occurs only on reaching at a nonzero angle some line Q1 which mayor may not be an edge. Leaving a face is possible either onto an edge (but not onto a vertex) or into the interior or exterior of a cube. Under conditions 1}5) all solutions with initial data sufficiently close to x(to) get onto surfaces Sk in the same order as the solution x(t). The variation of the solution is determined by formula (2). The matrix Y(t) satisfies an equation of the form (3) or (13) when the solution lies in each set Sk but when a solution passes from one Sk into another, it undergoes jumps as indicated in 39. Variation of solutions of equations with discontinuous righthand sides is considered also in [207].1
1 Added
in translation.
CHAPTER 3
BASIC METHODS
OF QUALITATIVE THEORY The basic methods of the qualitative theory of differential equations are applied to the study of differential equations with discontinuous righthand sides and differential inclusions. The general properties of trajectories of autonomous systems and the properties of trajectories in a plane (in particular,' if there holds only right uniqueness) are described. The existence conditions for bounded and periodic solutions, the methods of studying stability by means of Lyapunov functions and by a first approximation equation are presented.
§12 Trajectories of Autonomous Systems We will show that many properties of autonomous systems of ordinary differential equations hold also for differential equations with discontinuous righthand sides and for differential inclusions. Differences in the properties are due rather to lack of uniqueness than to the presence of discontinuity in the righthand side of the equation. 1. In a domain G of the space Rn we consider an autonomous differential inclusion $ E F(:z:)
or an autonomous system of differential equations with discontinuous righthand sides, which is reduced to (1), for instance, with the help of the definition a) or c), §4. It is assumed that there hold the basic conditions of 2, §7: for each z E G the set F(z) is nonempty, bounded, closed, convex, and the function F is upper semicontinuous. In this case for each bounded closed domain D c G the function F is bounded in D (Lemma 15, §5), and the solutions possess the following properties determined in 2, 4, §7: A0 For any initial condition z(t) = Zo, where Zo is an interior point of the domain D, the solution exists and continues, either unboundedly as t increases (and decreases), i.e., as t  00, or until it reaches the boundary of the domain D. B O All the solutions lying in D are equicontinuous. Co The limit of each uniformly convergent sequence of solutions is a solution.
123
124
Basic Methods of the Qualitative Theory
DO If x(t) (to ~ t ~ td and yet) (tl y(td then the function (1)
z(t) = x(t)
(to
~
t
~
~
t
~
Chapter 3
t2) are solutions and if x(td
=
ttl,
is also a solution. From the properties A o_Do there follow some other properties of solutions. In particular, if all the solutions with a given initial condition x( to) = Xo exist for to ~ t ~ tl then the set of these solutions is a compactum in the metric C[to, tll (Theorem 3, §7)j this compactum is an upper semicontinuous function (with respect to the inclusion) of the initial point (to, xo) (Corollary 2 to Theorem 1, §8). If a solution with the initial condition x(to) = xo is unique on the interval [to, tll and on any smaller interval [to, t'l c [to, tll, it depends continuously on to, xo and on the righthand side of the equation or of the inclusion (Corollary 1 to Theorem 1, §8). 2. For the differential inclusion (1) (or for an equation which can be reduced to this inclusion) the traiectory is a point or a line (in the x space) determined by the vectorvalued function x = \O(t), which is a solution of inclusion (1). Trajectories passing through the point p are denoted by T(p), Tdp) , T2 (p), L(p), R(p), etc. The part of the trajectory T(p) spanned for t ~ to (where \O(to) = p) is denoted by T+ (p), and for t ~ to by T (p) (positive and negative half trajectories). The point x = p is called stationary if it is a trajectory, that is, if x(t) == p is a solution of the inclusion (1). The term "singular point" is not used here since besides stationary points we also consider some other singular points, for instance, branching and joining points of trajectories. From the definition of solution and from tindependence of the righthand side of the inclusion (1) we have the following properties of solutions and trajectories. If x = \O(t) is a solution (for a < t < 13) then for any constant c the function x = \O(t + c) (a  c < t < 13  c) is also a solution, and these solutions have the same trajectory. A point p is stationary for the inclusion (1) if and only if 0 E F(p). A set of stationary points contained in a closed domain D is closed. As is known ([157], p. 30), in the case of uniqueness of solutions each trajectory of an autonomous system is either a stationary point or a closed curve or an open curve without selfintersections. In the general case the trajectories of the autonomous differential inclusion (1) may have any selfintersections. THEOREM 1. If there holds right uniqueness, each trajectory of the differential inclusion (1) pertains to one of the following types: 1) a stationary pointa trajectory of the solution x(t) == p; 2) a closed curve without selfintersectionsa trajectory of the periodic solution x(t) 1= const; 3) an open curve without selfintersections; 4) a trajectory coming into a stationary point, i.e., consisting of a simple arc without selfintersections, x = \O(t) (t < ttl, and a stationary point x = \O(t) == P
(t
~
ttl;
Trajectories of Autonomous Systems
§12
125
5) a trajectory joining a closed trajectory, i.e., consisting of a simple arc without selfintersections, z = ~(t) (t < t1), and a closed trajectory z = ~(t) == ~(t + I) (t ~ t1, 1= const > 0). In the cases 4) and 5) the parts t < hand t ~ t1 of the trajectory :r; = ~(t) do not have common points. PROOF: ff ~(t1) =F ~(t2) for the solution :r; = ~(t) (00 ~ a < t < f3 ~ 00) for any tlJt2' t1 =F t2, then we have the case 3). In the other cases let t1 be the greatest lower bound of those l' which are such that ~(1') = ~(O") at least for one value of 0" > 1'. Let 0"  l' = h. Then ,p(t) = ~(t + h) is a solution, ,p(1') = ~(O") = ~(1'). Hence, ~(t) == ,p(t) for t ~ 1',
~(t)
(2)
== ~(t + h)
(t
~
1').
ff ~(t) = ~(1') for l' ~ t ~ 0" then by virtue of (2) ~(t) = ~(1') for all t ~ 1', and as a consequence of continuity of the function ~ also for all t > t1 (t ~ t1 if t1 > a). Since by virtue of the choice of t1 the function ~(t) =F ~(td for t < t1 then there holds the case 4). ff h = a then ~(t) = const and there holds the case 1). Let ~(t) '¢ const for l' ~ t < 0" and l' + I be the greatest lower bound of those t > l' for which ~(t) = ~(1'). Then 0 < I ~ h (if I = 0 then, as in (2), ~(t) == ~(t + It) (t ~ 1') for arbitrarily small I. and since the function ~ is continuous, ~(t) == const (t ~ 1'), which contradicts the assumption). Therefore, ~(t) == ~(t+ I) (t ~ 1'), i.e., for t ~ l' the function ~(t) is periodic with a smaller period l> O. The closed curve :r; = ~(t) (1' ~ t ~ l' + l) has no selfintersections since otherwise ~(1'1) = ~(O"I)' 0 < 0"1  1'1 < I and as in (2), the function ~ would have a period 0"1  1'1 < I. ff in this case tl = a, there holds the case 2), and if tl > a, there holds the case 5). LEMMA 1.
Let D be a bounded closed domain, let
i
= 1,2, ... ,
be a sequence of arcs of trajectories of the inclusions (1) contained in D, ~i(ao)
= Pi + p,
(i
+
00);
~ "t > 0, i = 1, 2, •..• Then: 1) if lim._ oo li = 1 < 00 then the domain D contains an arc 0 ~ t ~ I of some trajectory z = ~(t), ~(O) = p, ~(l) = qj 2) if limi_oo Ii = 00 then D contains some half trajectories T+(p) (z = ~(t), 0 ~ t < 00) and T(q) (z = ,p(t), 00 < t ~ 0). The arc pq or the half trajectories T+(p) and T(p) are limits of a certain subsequence of arcs Piqi or of their parts.
bi

a; = Ii
PROOF:
(3)
The solutions
(0
~
t ~ It)
126
Chapter 3
Basic Methods of the Qualitative Theory
have the same trajectories as the solutions ~i(t). For arbitrarily large k we take Ak = l  2 k in the case 1) and Ak = k in the case 2). Fix k. According to BO, the sequence of solutions (3) for i = ik, ik + 1, ... , js compact in the metric C on the interval 0 ~ t ~ Ak. We choose from this sequence a subsequence, for which li + l in the case 1) or li + 00 in the case 2). From the latter subsequence we choose a uniformly convergent subsequence. Its limit is also a solution x = ~(t), 0 ~ t ~ Ak, whose trajectory is contained in D,
Now we consider solutions of the subsequence for 0 ~ t ~ Ak+1, and choose from them a new subsequence which uniformly converges on the interval 0 ~ t ~ Ak+l' The limit of this subsequence is a solution on 0 ~ t ~ Ak+1 which coincides with the solution ~(t) on 0 ~ t ~ Ak. Therefore, this solution can be denoted as before by ~(t). Proceeding similarly, we obtain a solution ~(t) for o ~ t < l (l = 00 in the case 2)) whose trajectory is contained in D. In the case 2) the half trajectory T+(p) is already constructed, T (q) is constructed in a similar manner. In the case 1) IF(x)1 ~ minD and, accordingly, Ix(t) I ~ m for all the solutions, and there exists limt_lo ~(t) = ~(l). For any s > 0 for k > ko(s) and i > io (k, e) we have I~(l)  ~(Ak) I ~ m(l  Ak)
ItPi(Ak) 
th (lo) I ~
= 2 km < s,
m(Ak lo) < s,
1~(Ak)
 tP.(Ak)1 < e,
ItP.(li) 
ql =
Iq. 
ql < e.
Hence, I~(l)  ql < 4s. Since s is arbitrary then ~(l) = q. 3. In the next theorem we consider the behaviour of trajectories near a nonstationary point without making the assumption of uniqueness. THEOREM 2. Let a point b be nonstationary for the inclusion (1), that is,
O¢F(b). Then there exists a vector v (Ivl = 1), a constant "1 > 0, and a closed neighbourhood K(lx  bl ~ eo) of the point b such that for all the trajectories lying in K we have
v . :i:(t) ~ "1 > 0,
(4)
and the angle between the vectors v and :i:(t) is not larger than
(5)
a =
11"
'2 
•
"1
arcsm m'
m=maxIF(x)l· .,EK
All the trajectories which have common points with the diametric crosssection S(v . x = v . b, Ix  bl ~ so) of the neighbourhood K intersect this crosssection in one direction, namely, in the direction of an increasing product v . x. PROOF: Since 0 ¢ F(b) and the set F(b) is convex then by Lemma 3, §5, there exists a plane P which separates 0 and F(b). The equation of the plane P can be written in the form v . x = "I, where Ivl = 1, "1 > O. The point 0
§12
Trajectories of Autonomous Systems
127
lies in the region v . x < "I, and the set F(6) lies in the region v . x > '1. By Lemma 1, §5, p(F(6), P) = Po > o. Since the function F is upper semicontinuous, there exists eo > 0 such that for Ix  61 ~ eo we have P(F(x),F(6)) ~ Po, i.e., F(x) C (F(6))Po. Then F(x) lies in the region v . x ~ "1 and for any solution of the inclusion (1) in the region Ix  61 ~ eo we have (4). Next, for Ix  61 ~ eo the set F(:z:) lies in a balllzi ~ m (Lemma 15, §5). For any y e F(x) we have IYI ~ m, '1 ~ v . Y = Ivl . IYI cos 01, 01 being the angle between the vectors v and y. Since Ivl = 1, COSOl ~ "IIIYI ~ "11m and therefore (5) follows. COROLLARY 1. Under the assumptions of Theorem 2, for anye > 0 there exists 5 > 0 such that each trajectory passing through the 5neighbourhood 66 of the point b intersects the transversal S (the crosssection S constructed in Theorem 2) without going out of its eneighbourhood b~, and then goes out ofb~.
PROOF: Let e be less than the eo specified in Theorem 2 and let 5 = "Ie/m. By virtue of (4), none of the trajectories passing through the point q e b6 can remain in bC , and therefore each of them reaches the boundary of the neighbourhood 6C both with a decreasing and with an increasing t. Figure 17 shows a cone with the axis directed along the vector v, with the base ppdlx  bl ~ e, V· x = v· 6), with the lateral surface tangent to the ball 66 • Then cos 0 = Ibr I : I6p I = ole = '1/ m, the angle between the element rp and the vector v being also equal to o. Since the angle between the vectors :i:(t) and t1 is not greater than 0 (see (5)), the trajectory going from the point q of the ball b6 will not pass out of the cone until it crosses its base PP1.
v
Pf Figure 17 COROLLARY 2. Let there hold right uniqueness on the arc ab of the trajectory T (x = ~(t), 0 ~ t ~ p, ~(o) = a, ~(P) = b) and let the point 6 be nonstationary. Then for any e > 0 there exists '7 > 0 such that each trajectory passing through the point al e a" intersects, at some point Cl. the crosssection S (as t increases) which passes through the point b, the whole arc alcl C (ab)·, and the
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time of motion along the arc aici differs arbitrarily little (for small '7) from the time of motion along abo PROOF: For a given e there exists 6 < e as in Corollary 1. Since right uniqueness holds, there exists '7 > 0 such that the arc alb l (a ~ t ~ ~) of any trajectory TI(ad (x = ,p(t), ,p(a) = ad is contained in the 6neighbourhood (ab)6 of the arc ab (Corollary 1 to Theorem I, §8). Since ,p(~) = bl E b6 then the trajectory Tdad intersects S at some point CI; in this case the arc blCI C be. Then the whole arc alcl C (ab)e. Since the arc blCl C be, Iv, (CI  bdl ~ 2e (if Ivl = 1) and from (4) it follows that the time of motion along the arc blCl is not greater than 2eh. The following two known theorems on trajectory straightening in the neighbourhood of a nonstationary point, in the case of uniqueness, are presented in the form required for further investigations. THEOREM 3. Let a continuous function Xl = ,p(tI), where v = (VI", .,vnd, give a onetoone mapping of the compactum K C Rnl onto the set M C RR. Let, for each Xo E M, the solution X = e(t; xo) of the inclusion (1) or of the equation :i; = f(x) with the initial data e(O; xo) = Xo on the interval I = [a, ~l (a ~ 0 ~ ~) exist and be unique, and let the arcs a ~ t ~ ~ of trajectories of such solutions have no common points L; e(t l , xo) i= e(t 2 , xo) iEtI i= t 2 • Then the function x = e(t; ,p(v)) maps topologically the set I X K onto the set Q C Rn filled with arcs a ~ t ~ ~ of trajectories with initial data x(O) E M. These arcs are images of the segments of straight lines v = const. PROOF: The mapping of I X K onto Q is a onetoone mapping since the considered arcs of trajectories have no common points. Continuity of this mapping follows from continuity of the functions ,p( v) and x = e(t; xo); the function e is continuous due to uniqueness of the solutions. The inverse mapping is continuous by Lemma 1, §9. THEOREM 4. Let the conditions of Theorem 3 be satisfied for the equation = f(x),' moreover, let K be a closed bounded domain in Rnl, fEel, ,p E . C l , and the vectors f(x), a,p/aVI,"" a,p/aVn_1 be linearly independent for all v E K, x = ,p( v). Then the mapping x = e(t, ,p (v)) of the set I X K onto Q and the inverse mapping are continuously differentiable. :i;
PROOF: By Theorem 3 the mapping x = e(t,,p( v)) is topological. It is continuously differentiable since e(t, xo) E C l by the theorem on solution differentiability with respect to initial data. The derivatives = aX/aVi' i = 1, ... , n I, satisfy an equation of first variation with initial data Ui (0) = a,p / av•. Since the equation :i; = fez) is autonomous, the solution x(t; to, zo) with initial data x( to; to, xo) = Xo depends only on t  to and on Xo. So the function Uo = ax/at == ax/ato satisfies an equation of first variation with initial data uo(to) = /(xo) ([131, p. 96). From the assumption of the theorem it follows that the vectors u.(t), i = 0,1, ... , n  1, are linearly independent for t = O. They satisfy a linear homogeneous equation of first variation and are therefore linearly independent for any t. Hence the determinant, which is composed of
u.
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129
these vectors and is the Jacobian of the mapping :I: = e(t, "'(11)), is not equal to zero, and the inverse mapping is also continuously differentiable. 4. Many properties of limit sets of trajectories of an autonomous system of differential equations are retained for limit sets of trajectories of a differential inclusion :i: E F(x) if the basic conditions of 1 are satisfied. In the first place, this concerns the properties inherent in limit sets of any continuous curves in the space Rn which are given in the form x = ~(t), to ~ t < 00. A point q E Rn is called an wlimit point for a curve L (x = ~(t), to ~ t < 00, ~ E C) if there exists a sequence tl, t2, ... , tending to 00 such that ~(ti) + q (i + 00). The set of all wlimit points of the curve L is called an wlimit set of the curve L and is denoted by O(L). For a curve :I: = ~(t) (00 < t ~ to, ~ E C) a point p E Rn is called an alimit if there exists a sequence of points ~(ti) + P (i = 1,2, ... ; ti + 00); the alimit set of the curve is the set of all its alimit points. H a given curve :I: = ~(t) is a trajectory or a half trajectory of an autonomous differential equation or of an inclusion, one speaks of limit points and sets of this trajectory or half trajectory. The following known and easily proved assertions are concerned not only with a limit set of trajectories but also with limit sets of any continuous curves in RA. The assertions are formulated for wlimit sets; alimit sets possess similar properties. 1) Let L be a curve x = ~(t), to ~ t < 00, and Lie be a part of this curve, tie ~ t < 00, tie + 00. (Ie + 00). Then
n 00
O(L) c L,
O(L) =
Lie.
Ie=l
2) The set O(L) is closed. 3) The set O(L) is empty if and only if 1~(t)l+ 00 as t + 00. 4) The set O(L) is bounded if and only if the curve L (:I: = ~(t), to
~
t < 00)
is contained in a bounded domain. 5) H the set O(L) is bounded then p(~(t),
O(L)) + 0
(t
+ 00)
6) Always (t + 00). LEMMA 2. The set O(L) is not a union of two disjoint nonempty closed sets if at least one of these sets is bounded.
PROOF: Suppose O(L) = Au B, An B = 0, A ::/: 0, B ::/: 0, A and B being closed, and B bounded. By Lemma 1, §5, p(A, B) = 2d > O. Take points a E A, b E B. These are wlimit points and so there exists an increasing sequence tb t2, ... , tending to 00 such that
Ie
=
1,2, ....
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130
Then p(fP(t2kd, B) > d, k = 1,2, ... (otherwise
which is a contradiction since p(A, B) = 2d). The continuous function p(fP(t) , B) is greater than d at t = t2kl and less than d at t = t2k, k = 1,2, .... Hence, for some t = Tk > t2kl k = 1,2, ... ,
T,. +
00.
A bounded sequence fP(Tk) has a limit point c E O(L). Since p(c, B) = d > 0, p(A, B) = 2d, then c ¢ B, c ¢ A. This contradicts the fact that c E O(L) = AuB. COROLLARY. If O(L) is bounded, it is connected. The formulation of Lemma 2 becomes simpler if the space RB is complemented with one extra point x = 00 to form a compactum [RBJ with the natural topology: Xk + x if IXk  xl+ OJ Xk + 00 if IXkl+ 00. Then Lemma 2 can be formulated as follows: the set O(L) is connected in [RBJ. LEMMA 3. If O(L) has no common points with the line L (x = fP(t), to ~ t < 00) then L is contained in one of the components G l of the open set Rn\O(L), O(L) is nowhere dense, O(L) = BG l , i.e., O(L) is the boundary of the domain G l . PROOF: The first assertion follows from the assumption of the lemma and from the connectedness of the line L. From L C G I it follows that O{L) c BG I , and since G I C Rn\O(L) then BG I C O(L). Thus, O(L) = aGl' By virtue of the definition of the boundary of a domain, in any neighbourhood of each point a E BGl there exist points of the domain G l • Hence, the set O(L) = BGl is nowhere dense. Next we consider limit sets of trajectories of the differential inclusion (1) in a closed domain D under the basic conditions formulated in 1 and the condition
IF(x)1
~
minD.
The condition IF{x) I ~ m is not a severe restriction since, by Lemma 15, §5, in each bounded part Dk = D n Bk (Bk being a ball Ixl ~ k) of the domain D we have IF(x) I ~ mk' The inclusion (1) can therefore be replaced by the inclusion :i: E p(lxI)F(x) which, by Theorem 3, §9, has the same trajectories as the inclusion (1), but whose righthand side is bounded in the entire domain D. To this end, it suffices to take as p( €) a continuous decreasing function such that p(k  1) ~ m;;l, k = 1,2, .... LEMMA 4. Through each point a E O(T) there passes a whole trajectory To (x = ,p{t), 00 < t < 00) contained in O(T). The same is true for an alimit set A{T). PROOF: The point a is an wlimit point for the trajectory T (x = fP(t)), that is, there exists a sequence ti + 00 such that fP(ti) + a. The functions (to  ti ~ t < 00)
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131
are solutions, and ,pi (0) + a. Since IF(:I:) I ~ m, on any finite intervalk ~ t ~ k the solutions ,pi(t), beginning from a certain one, are defined, equicontinuous, and uniformly bounded. Hence, from the sequence {,pi(t)} one can choose a subsequence uniformly convergent for 1 ~ t ~ 1 to the solution ,pet), and from it a subsequence uniformly convergent for 2 ~ t ~ 2 to the same solution continued onto the segment [2; 2], etc. We obtain the solution is ,pet) defined for 00 < t < 00. In this case, ,p(0) = lim ,pi (0) = a. The trajectory of this solution is contained in OCT), since for any t the point ,pet) is the limit of some subsequence of points ,pi(t) == ~(t + til, i = ij + 00. The case a e A(T) is reduced to the one just considered on replacing t byto 5. We will investigate in more detail the properties of limit sets in the case where the conditions listed before Lemma 4 are satisfied and, moreover, a solution with any initial data :I:(to) = :1:0 e D is unique for t ~ to. LEMMA 5. If a trajectory T (:I: = ~(t), a < t < 00) has a common point = ~(t.) with a set OCT) then either T c OCT) or some half trajectory T+ (:I: = ~(t), tl ~ t < 00) is contained in OCT), and the rest of the trajectory T (t < td has neither selfintersections nor common points with OCT). :1:.
PROOF: By Lemma 4, through the point :1:. there passes some trajectory To C OCT). Let To be the trajectory of the solution :I: = ,pet). Since:l: = ,p(t + c) is a solution for any c, one may assume that ,p(t.) = :1:. = ~(t.). Then, as a result ofright uniqueness, ,pet) = ~(t) for all t ~ t •. Let tl be the greatest lower bound of t. such that ~(t.) e O(t). From what has been proved it follows that ~(t) e OCT) for all t > tl. IT tl = a then T C OCT). IT tl > a then ~(T) ¢. OCT) for a < t < tl by virtue of the choice of tl, and ~(tl) e OCT) because ~(t) is continuous and OCT) is closed. The absence of selfintersections of the arc ~(t) for a < t < tl follows from Theorem 1. THEOREM 5. Let the conditions listed before Lemma 4 be satisfied and let right uniqueness hold in the domain D. If the trajectory T has a common point with OCT) then only the following cases are possible: 1) OCT) coincides with T or with some half trajectory T+ c T; in this case OCT) is a stationary point or a closed trajectory; 2) OCT) contains at least one point which does not belong to T; in this case OCT) consists of an uncountable set (continuum) of trajectories; in the neighbourhood of any point a e OCT) there exist points of the trajectory T and points of the set O(T)\T. PROOF: IT OCT) is a stationary point or a closed trajectory then there holds the case 1). IT the case 1) does not hold, the point bET n OCT) is nonstationary. By Theorem 2, through this point there passes a local crosssection S. Since the point b is an wlimit point of the trajectory T (:I: = ~(t», there exists a sequence ~(tn + b, ~~ + 00, i = 1,2, .... The trajectory T that reached some neighbourhood of the point b at t = t~ (i = il,i l + 1, ... ) must intersect the crosssection S at some instant ti close to t~ by Corollary 1 to Theorem 2. Thus, we have a sequence of points ~(ti)
= bi
+
b,
bi
E
SnT+(b) c SnO(T),
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since T+(b) C OCT) by Lemma 5. All the points b. are different since T+(bo) is not a closed trajectory. The point b is therefore a limit point for a closed set M = OCT) nS. The same is true for each point of the set M, except perhaps for the points lying on the boundary as of the crosssection S. Discarding from M the points which lie on as and are not limit points for the set M\aS, we obtain a nonempty closed set Mo containing no isolated points and having therefore the power of the continuum ([64], p. 58). Each trajectory which intersects S must come out of some neighbourhood of the point b (by virtue of (4)) before it crosses S once again. The time intervals between two intersections cannot therefore be arbitrarily small. Hence, each trajectory can intersect S in not more than a countable set of points. Thus, through the points of the set Mo C OCT) n S there passes, besides T, also an uncountable set (continuum) of other trajectories contained in OCT). This is true for any neighbourhood of any point a E OCT). Lemma 5 and Theorem 5 enable us to classify the trajectories in more detail than has been done in Theorem 1. We must take into account the absence or the presence of intersections of a trajectory with its wlimit set. In the cases 1) and 4) of Theorem 1, OCT) is a point, and in the cases 2) and 5) it is a closed trajectory. In the cases 1) and 2) OCT) = T, and in the cases 4) and 5) OCT) C T, but OCT) ¥ T. In the case 3) the trajectory T is an open curve without selfintersections, and there exist the following possibilities: 3a) OCT) is empty; 3b) OCT) is nonempty and has no common points with T; 3c) T C OCT), but T ¥ O(T); 3d) the part t < tl of the trajectory T(z = !pet)) has no common points with OCT), and the rest of the trajectory, tl ~ t < 00, is contained in OCT), but does not coincide with it. By Theorem 5, in the cases 3c) and 3d) the set OCT) contains, besides the trajectory T (or the portion tl ~ t < 00 of it), also an uncountable set (continuum) of other trajectories. Examples of trajectories of the types 1), 2), 4), and 5) are obvious. An example of the case 3a) is a trajectory z = t, 3b)a trajectory z = e t , 3ca trajectory from an irrational winding of a torus ([158], p. 70); for other examples see [158] (pp. 408 and 418), 3d)a trajectory which comes to a torus from its exterior and joins one of the trajectories of its irrational winding. Using arguments similar to those of Lemma 5 and Theorem 5, one can investigate the situation of a trajectory with respect to an alimit set A{T). The cases A(T) = 0 and A(T) ¥ 0, A(T) n T = 0 are possible. If the trajectory T has a common point a with the set A(T) then by virtue of Lemma 4 and right uniqueness, T+(a) C A(T). In this case, as in Theorem 5, there are the following possibilities: T = A(T); then T is a stationary point or a closed trajectory; T C A(T), but T ¥ A(T) then, besides T, A(T) contains a continuum of trajectories; another case is possible where T contains points a E A(T) and b ¢. A(T), then T(b) n A(T) = 0, T+(a) C A(T). The latter case holds, for instance, for the
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133
trajectory z = cos 6(t),
y = sin6(t)j
6. Let the basic conditions of 1 be fulfilled. Uniqueness is not assumed to hold. IT all 'the solutions of the differential inclusion (1) are defined for 00 < t < 00, they define [120] a generalized (i.e., without uniqueness) dynamical system. Such systems possess some properties of usual dynamical systems, in particular, properties concerning minimal sets and recurrent trajectories [159J, [160J. A set M is called minimal if it is nonempty, closed, consists of whole trajectories (i.e., through each point p E M there passes at least one whole trajectory z = !p(t), 00 < t < 00, contained in M) and contains no subset Mo =1= M which possesses the same properties. A trajectory T is called recurrent if for any e > 0 there exists 1"(e) such that the eneighbourhood of any ar~ of the trajectory T, which is passed through in the time 1"(e), contains the whole trajectory T. THEOREM 6 [159]. Any nonempty compact set consisting of whole trajectories contains a minimal set. COROLLARY. If the set O(T) is nonempty and bounded, it contains a minimal
'
~L
THEOREM 7 [159]. Each whole trajectory contained in a compact minimal set is recurrent. THEOREM 8 [159J. The closure of a recurrent trajectory contained in a bounded domain is a compact minimal set.
§18 The Properties of Trajectories in a Plane We now establish which of the wellknown properties of trajectories of the equation :i: = /(z) (z E R2, / E G1) are retained for trajectories of the differential inclusion :i: E F(z) (z E R2) and, therefore, for differential equations with discontinuous righthand sides in a domain of a plane under the definition a) or c), §4. In particular, for such equations and inclusions we formulate theorems similar to Bendixon's theorems on limit sets in a plane and on closed trajectories. 1. Fundamental properties of trajectories of differential equations in a plane in the case of uniqueness were investigated in [161] and in [158J (Chapter 2, §1), and in the absence of uniquenessin [162J and [13] (Chapter 7, §4). Many of these properties are retained with small modifications also for differential inclusions (see, in particular, [163J, [164]). The proofs are similar to those presented in [13J and, [158J with the following variations. Instead of a segment of a normal to a trajectory or of an arc without contact we consider a transversal, i.e., the crosssection constructed in Theorem 2, §12j instead of the theorem on continuous dependence of solutions on initial data we use the local compactness of a set of solutions (the properties BO and Co, 1, §12). In a closed domain of a plane we consider the differential inclusion
(1)
:i: E F(z),
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Basic Methods of the QualitatitJe Theory
134
A
Figure 18
which satisfies the basic conditions of 1, §12, and the conditions IF(x)1 ~ m. A segment S is called a transtJersal if it is intersected by trajectories only in one direction, more precisely, if in its neighbourhood for any x and any tt E F(x) we have v . tt ~ '1 > 0 (v being a given vector orthogonal to the segment S). For any nonstationary point the existence of a transversal passing through this point is proved in Theorem 2, §12. LEMMA 1. If on a transversal S there exists a point b E O(T) (or b E A(T)) then a trajectory T (x = ip(t)) intersects the transversal S for arbitrarily large It I, and among the intersection points one can choose a sequence
bi
= ip(t.) + b,
(or, correspondingly, ti
+
ti +
00
(i
+
00)
00).
PROOF: The point b is an wlimit point of the trajectory T (x = ip(t)) and, therefore, on T there exists a sequence of points = ip(to') + b, t~ + 00. By virtue of Corollary 1 to rheorem 2, §12, the trajectory T passing through any point sufficiently near b will intersect S at the point b. = ip(t.), and from + b there follows b. + b, and with an account of (4), §12, t.to' + 0, t. + 00. The case b E A(T) is reduced to that considered above on replacing t by to In Lemmas 26, which follow, a given trajectory T (x = ip{t)) is assumed to satisfy at least one of the two conditions: either a) ip(tt} t= ip(t2) for any tb t2, t1 t= t2 or b) at the points of the trajectory T there holds right uniqueness. Nothing is assumed concerning the other trajectories except T. Such an assumption is a severe restriction for differential inclusions. It is satisfied, in particular, for those differential inclusions, to which differential equations with discontinuous righthand side are reduced (by means of the definition a), §4) in the case of right uniqueness.
a.
a.
a.
LEMMA 2. If a trajectory T intersects a transversal S several times, the intersection points are placed on S monotonically (under the condition a) strictly monotonically), i.e., in the same order as on the trajectory. PROOF: If a trajectory T intersects a transversal S = ac at points hand b2 then a closed curve consisting of the arc h b2 of the trajectory T and of the segment b2 b1 of the transversal S separates the plane into two parts A and B. Let ab 1 C A, b2 c C B (Fig. 18). Having passed through the point b2 , the trajectory T remains in B and can reach neither the segment ab 1 E A of the transversal, nor the segment b1 b2 into which the trajectories come only from the region A. Therefore, having passed through the point b2 , the trajectory T can intersect the transversal ac only on the segment b2 c.
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135
LEMMA 3. For a trajectory T (x = ~(t)) a set O(T) can intersect the transversal S at not more than one point. If b is the point of intersection oIO(T) and S then T intersects S only at the points
(i = 1,2, ... ),
(2)
the points b. tend to b monotonically on S. A similar assertion is valid for A(T), but then t.+! < t., t, + 00. PROOF: Let be OCT) n S. By Lemma 1 there exists a sequence (2) of points of intersection of T and S. From the estimate tI • x(t) ~ 'Y > 0 it follows that after each intersection the solution must go out of a certain neighbourhood of the transversal S. The time intervals between successive intersections are therefore not less than a positive constant. We shall enumerate all the intersection points (beginning with a particular one) in increasing order of ti. By virtue of Lemma 1, a certain subsequence of this sequence converges to the point b. By Lemma 2, the points bi are disposed on S monotonically and, therefore, the whole sequence b. + b. IT the intersection OCT) n S contains, besides b, a point a then, according to what has been said above, bi + a. This is impossible if a "I b. For A(T) the proof is constructed in a similar way. THEOREM 1. If a trajectory T has a common point b1 with the set O(T) and if on T there holds right uniqueness, then O(T) is a stationary point or a closed trajectory and coincides with T or with its half trajectory T+ (bI). PROOF: IT b1 is a stationary point then, by virtue of right uniqueness, b1 = T+(b 1) = OCT). If 61 is a nonstationary point then, by Lemma 3, the points h, b2 , ••• of intersection of T and the transversal S drawn through h are disposed on S monotonically and tend to b1 • This is possible only in the case b1 = 62 = .... Hence, for the trajectory T (:.; = ~(t)) we have ~(t1) = ~(t2)' t1 "I t2' Then T+(b 1) is a closed trajectory (see the proof of Theorem 1, §12) and OCT) = T+(b 1). COROLLARY. If there holds right uniqueness, no trajectories olthe type 3c) and 3d) can exist in the plane (5, §12). THEOREM 2. If a trajectory T has a common point a with the set A(T) and if on T there holds right uniqueness then either T is a stationary point or a closed trajectory, or A(T) consists only 01 stationary points and OCT) = a E A(T). PROOF: Let a nonstationary point b E A(T). By Lemma 3, the transversal S passing through the point b intersects A(T) only at the point b, whereas it intersects the trajectory T (:.; = ~(t)) at the points
(3)
b.
= ~(ti) + b,
(i=l,2, ... ),
t,
+ 00.
IT bi = b for some i then, since the sequence {bi } is monotone on S, we have bi = b,. = b for aUk ~ i. Then the arc x = ~(t), tic+! ~ t ~ tic, of the trajectory T is a closed curve. By virtue of right uniqueness, T+ (blc+1) is a closed curve.
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Since blc = cp(tlc), tic + 00 (k + 00), the whole trajectory T is a closed curve and A(T) = T. IT bi =1= b for all i then by virtue of (3) and the monotony of the sequence {bi} we have bi =1= bHl for some i = i l and by virtue of right uniqueness this holds also for all i ~ il. For such i, the arc bH2 bi of the trajectory T and the segment bi bi +2 C S separate the plane into three parts, one of which contains T+(bi)\b i and the other contains T(bHd and, therefore, A(T) (Fig. 19). Taking a sufficiently large i such that bi =1= a and the point a E Tn A(T) lies on T+(bi), we reach a contradiction. Thus, either T is a closed trajectory or A(T} consists only of stationary points. In the latter case it follows from a E Tn A(T) and from the uniqueness theorem that T+(a) = a = O(T). Then either T = a or T(a) =1= a and A(T) consists either of one point a or of an infinite set of points.
o Figure 19
Figure 20
v Figure 21
The following examples (where p, 6 are polar coordinates, c > 0 is an arbitrary constant) show that both the latter cases are possible. 1) 6(t) = min{etj 21r}, p(t) = c. 2) 9(t) = min{etj 21r}, p(t) = c(2 + sin This trajectory is shown in Fig. 20. 2. Several assertions regarding the properties of wlimit sets containing nonstationary points are extended to differential inclusions with right uniqueness (in some cases without right uniqueness). In Lemmas 46 and in Theorems 35 proved below, the basic conditions of I, §12, the condition IF(x)1 ~ m, and at the points of the trajectory T at least one of the conditions a) and b) formulated before Lemma 2 are assumed to be fulfilled.
om)'
LEMMA 4. Each nonstationary point b E O(T) has a neighbourhood through which there passes only one simple arc of one trajectory L c O(T) and there are no other points of the set O(T}.
PROOF: By Lemma 4, §12, there exists a trajectory L c O(T) passing through the point b. By Theorem 2, §12, there exists a circle K(lx  bl ~ eo) in which, for all the trajectories v . :i; ~ '1 > 0, the angle between the vectors :i; and v is not greater than a, a < 1f/2, v being a constant vector with Ivl = 1. Let
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Properties 01 Traiectories in a Plane
137
= (Xl, :1:2), the :l:laxis being parallel to tI. Then for all the trajectories in K we have dXl/ dt ~ 'Y > 0 and each chord :1:1 = constant of the circle K is a
:I:
transversal. The trajectory L, which passes through the point b, intersects each chord :1:1 = c (el < C < 6) at one point (Fig. 21). By Lemma 3, in the part 6 < :1:1 < 6 of the circle K, there are no other points from O(T). LEMMA 5. H a closed trajectory (without stationary points) L c O(T), then
OCT) = L. PROOF: By Lemma 4, each point bEL has a neighbourhood containing no points of the set M = O(T)\L. The union of such neighbourhoods is an open set G:J L. Its complement R2\G = D is dosed, Me D. Since M = OCT) n D, M is closed. Hence, OCT) = LuM, Land M are closed, LnM = 0, L is bounded. By Lemma 2, §12, this is possible only in the case M = 0, O(T) = L. LEMMA 6. Let a trajectory L c O(T), the set O(L) or A(L) be nonempty and let there be no stationary points on L. Then either L is a closed trajectory and L = OCT) or all a and wlimit points of the trajectory L are stationary. PROOF: Let a nonstationary point bE O(L). Through the point b there passes a transversal S. By Lemma I, L intersects S at the points bl , b2 , ••• + b. Since L c O(T), then bi E O(T), and it follows from Lemma 3 that bl = b2 = ... = b. Hence, through the point b there passes an arc bl b2 of the trajectory L which is a closed curve Lo C L c OCT). By Lemma 5, OCT) = Lo. Hence, OCT) = L = L o. The case b E A(L) is considered similarly. REMARK: Lemmas 46 remain true if OCT) is replaced by an alimit set A(T). THEOREM 3. Let the conditions listed at the beginning of 2 be satisfied. H the set OCT) or A(T) is bounded and contains no stationary points then it consists of one closed trajectory. PROOF: Any trajectory L c OCT) is bounded, hence the set O(L) c L is nonempty. Since OCT) is closed then L c OCT) and in L there are no stationary points. By virtue of Lemma 6, OCT) is a closed trajectory. THEOREM 4. Let the conditions listed at the beginning of 2 be satisfied. Let the set OCT) be not a closed trajectory. Then 1) the set 0 0 of stationary points contained in OCT) is either empty or closed; 2) the set of nonstationary points contained in OCT) is either empty or consists of a finite or a countable set of nonintersecting arcs of trajectories Li C
OCT); 3) for these arcs L. the sets O(L.) and A(Li) are either empty (if L. tends to infinity) or consist only of stationary points and are contained in 0 0 , PROOF: 1) The set O(L) is closed (4, §12), so is the set M of all stationary points (2, §12) and the set 0 0 = OCT) n M is therefore closed.
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2) By virtue of Lemma 4, through any nonstationary point of O(T) there passes a single trajectory L., and those arcs of such trajectories which contain no stationary points either do not intersect or they coincide. We will show that the arcs L. amount to a countable set at most. By virtue of Lemma 4, for each arc L. one can construct a circle with a centre bEL. which has no common points with the other arcs L,.. Circles whose radii are twice less do not intersect. Such circles amount to a countable set at most, so do the trajectories L •. 3) Let :r; = !P.(t) (a. < t < 13.) be the maximal arc of the trajectory L. c O(T) which contains no stationary points. By Lemma 5, this arc (or a part of it) cannot be a closed trajectory since O(T) is not a closed trajectory. If 13. < 00 then there exists a limttJi !Pi (t) = q•. The point q. is stationary, otherwise the arc L. could have been extended beyond the point q•. We put !pi(t) = q. for Pi ~ t < 00. We may use the same procedure if > 00. The whole trajectory so obtained, :r; = !Pi (t) (00 < t < 00), is again denoted by L•. Then O(Li) (or A(Li)) is a stationary point. If Pi = 00 then in the case l!Pi(t) I + 00 (t + 00) the set O(Li) is empty, otherwise O(Li) is nonempty. The trajectory Li is open, and by Lemma 6, O(Li) consists of stationary points. Since O(Li) eLi c OCT), O(Li) cO o.
a.
COROLLARY. If the set OCT) contains a tinite or only a countable set M of stationary points then OCT) is either a stationary point or consists of the set M and a tinite or a countable set of arcs of trajectories in which each end either is one of the points of the set M or goes to intinity (Fig. 22).
Figure I!I! PROOF: By Lemma 2, §12, if the set OCT) is not a point, it cannot have isolated points. Then OCT) is a closed set without isolated points and has therefore the power of continuum ([64], p. 58). Hence, if M is not a single point then, besides points of the set M, the set OCT) contains also nonstationary points. By Theorem 4 these lie on a finite or a countable set of arcs of the trajectories Lij O(Ld is either empty (in which case L. goes to infinity) or is contained in M and by virtue of connectedness is a point. By virtue of the property 5),of 4, §12, the trajectory Li either comes close to this point as t + 00 (or as t + 00), or reaches it at a finite t, and the result follows. Under the conditions a) and b) (given before Lemma 2) the set OCT), which contains a nonstationary point b, possesses some properties of a stable limit cycle. If OCT) is a closed trajectory, two cases are possible: either the whole of the trajectory T lies on one side of OCT) (within or outside it), has no common points with OCT) and spirals round OCT), or the trajectory T joins the closed trajectory OCT) at some point (case 5) of Theorem 1 §12)j under the condition a) the second case is impossible.
§13
Properties of Trajectories in a Plane
139
We say that the trajectory T spirals round the limit set O(T) if 1) it has no points common with O(T) and, therefore, the whole of it lies in one of the domains G* into which the set O(T) divides the planej 2) there exist at least three simple arcs (j = i, ... ,mj m ~ 3) which have no pairwise common points, lie in G*, have the ends ai E O(T), and are always intersected by the trajectory T only in one direction; that is, the point
aiai
ai
remains on the left of T, and the point aion the right, i = 1, ... ,m (Fig. 23), or always vice versaj 3) beginning from some point, the trajectory T intersects these arcs alternately in the same order repeated infinitely many times.
THEOR.EM 5. Let the conditions listed at the beginning of 2 be satisfied. If a set OCT) contains a nonstationary point b then either 1° the trajectory T spirals round O(T), or 2° the trajectory T coincides with O(T) or joins O(T) at some point, and O(T) is a closed trajectory. Under the condition a) (see before Lefllma 2) the case 2° is impossible if neT) ¥: T . . If T is not a closed trajectory, A(T) has DO common points with O(T) in both these cases. PR.OOF: Draw the transversal S through the point b. By Lemma 3 the trajectory T intersects S at points bi which have the properties (2). IT bi = b for some i then by virtue of monotony of the sequence {bi} we have bi+1 = bi = h. Hence there holds, not the condition a) but the condition b), that is, right uniqueness on T. Then the arc bibi+1 of the trajectory T is a closed curve L without stationary points and T+(bi) = L = O(T), i.e., the case 2° holds. IT T ¥: L then from Theorem 2 it follows that T n A(T) = 0. Since OCT) C T, O(T) n A(T) = 0. Let bi ¥: h for all i. Then the trajectory T fails to pass twice through the same point not only under the condition a), but also under the condition b) (otherwise, by virtue of right uniqueness, the trajectory would have joined a closed trajectory, and we would have had hi ¥: h, i ~ it). Therefore, for all i we have bi ¥: hi+1' and the curve K i , which consists of the arc hi hi+1 of the trajectory and the segment hi+1hi of the transversal S, divides the plane into two domains: the domain Gi containing T(bi)\bi, and the domain Hi containing T+(hi+1)\bi+1' ThenGi C Gi+1 C "'j GinHi+2 = 0, neT) C Hi+2' T(bi ) C Gi, therefore, T (by n OCT) = 0. Since i is arbitrary then Tn OCT) = 0, and since A(T) C T(hi . then A(T) n OCT) = 0.
We now show that T spirals round OCT). By Theorem 4, OCT) contains infinitely many nonstationary points. Let Si (j == 1, ... , m) be transversals at some m of these points 0.1,,,., am. We shorten the transversals so that they do not intersect. Let the trajectory T (z = !pet)) intersect Sifor the first time at t = Ti and TO = mUTi, bi = !p(TO) E Sl. Then T(b.) intersects all of the transversals Sl,,,,,Sm' The points a1,,,.,a m E OCT) lie outside each of the domains G ic • One can therefore shorten the transversals so that the half trajectory T (hi) intersects each of them only once and so that all the points a~, (ends of the transversals) lie in the domain Gi1. Then for each k ~ i
... ,a:,.
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each segment aiai C 8i intersects the boundary of the domain GIc, that is, the arc blcblc+1 of the trajectory T. If the arc bi bi+l C T intersects all the transversals, for instance, in the order 8 1 , •.• ,8m then the arc bi+lbi+2 intersects them in the same order. Indeed, let the domain Q be bounded by an arc of the trajectory T (from the point bi of its intersection with the transversal 8 1 to the first point Ci of the intersection with the subsequent transversal 8 2 ), by the segments bial C 81, Cia2 C 8 2 of these transversals and by the part of the set O(T) bounded by the points a1 and a2 (Fig. 24). In Q there are no points of other transversals since other transversals intersect neither 8 11 nor 8 2 , nor the arc biC; c T. Entering the domain Q at the point bi+l, the trajectory T cannot remain within it, since before returning to 8 1 at the point bi+2, the trajectory T passes outside Q. It can go out of Q only after intersecting the segment Cia2 C 8 2 • Therefore, after each intersection with 8 1 , the trajectory T intersects other transversals in the same order.
Figu.re HS
Figure H4
3. The following theorems, which are similar to the known theorems in the qualitative theory of differential equations, hold for differential inclusions of the form (1) in a closed domain in a plane if the basic conditions of I, §12, and the condition IF(x)1 ~ m are satisfied without any assumptions concerning uniqueness of solutions.
THEOREM 6. H a half trajectory T+ is bounded then its limit set O(T) contains either a stationary point or a closed trajectory. PROOF ([13), Ch. 7, §4): The set O(T) is bounded and nonempty (property 3), in 4, §12). Let O(T) contain no stationary points. By Lemma 4, §12, through any point p E O(T) there passes a trajectory L C O(T). Since O(T) is closed, O(L) C L c O(T). Hence the set O(L) is bounded and contains no stationary points. If the trajectory L (x = ,,(t)) has no selfintersections, that is ,,(ttl :I: t/J(t2) for any t1, t2, fI :I: t2, then O(L) is a closed trajectory by Theorem 3. If ,,(ttl = ,,(t2) for some tb t2, tl < t2 then the part tl ~ t ~ t2 of the trajectory L is a closed trajectory. COROLLARY. H the half trajectory T+ is contained in a bounded closed domain in which there are no stationary points then in this domain there exists a closed trajectory.
§13
141
Properties of Trajectories in a Plane
Note that, as in the qualitative theory of differential equations, this is possible only in a ringshaped domain. THEOREM 7 [164]. In a closed domain D bounded by a closed trajectory L, let the conditions listed at the beginning of 3 be fulfilled. Then in this domain there exists a stationary point. The proof can be constructed by the same method as the one used in [158J (p. 54) for a system of two differential equations. Suppose that in D there are no stationary points. Through an arbitrary interior point p of the domain D there passes a trajectory T (z = !p(t)). IT this trajectory passes twice through an interior point q of the domain D, that is, !p(tl) = !p(t2) = q, tl < t2, then the arc tl ~ t ~ t2 of the trajectory T is a closed trajectory To which passes through the point q. IT To has selfintersections, one picks from it a smaller closed trajectory L, without selfintersections, which passes through the point q (some arc of the trajectory To containing the point q has no selfintersections by virtue of the estimate (4), §12). Since q ~ L then Ll :f: L. IT the trajectory T does not pass twice through any of the interior points of the domain D then each of its half trajectories either reaches the boundary L of the domain D or spirals round L or round a closed trajectory Ll :f: L. IT both half trajectories reach L at the points a and b then the arc ab of the trajectory T and the arc ba c L make up a closed trajectory which passes through the interior point p. IT one half trajectory reaches L and the other spirals round L, they intersect within D. IT they have no intersections within D, they cannot both spiral round L since by Theorem 5 A(T) :f: O(T). Thus, in all cases D contains a closed trajectory Ll :f: L. By the same arguments, the domain Dl C D bounded by the trajectory Ll contains a closed trajectory L2 :f: L 1. It bounds the domain D2 C D 1, etc. The sequence of embedded closed domains D :::> Dl :::> D2 :::> '" has a nonempty intersection D*. By the assumption, any point b E aD* is nonstationary. By Theorem 2, §12, there exists a circle K (Ix  bl ~ eo) in which, for the solutions, there hold the inequalities (4) and (5), §12. Let x = (Xl' X2) and let the Xlaxis be parallel to the vector tI from formula (4), §12. Then each chord Xl = const of the circle K is a transversal. For i > i*(6) the trajectory L. passes through the 6neighbourhood of the point b. By virtue of Corollary 1 to Theorem 2, §12, L. intersects the diameter Xl = /31 of the circle K and, therefore, all the chords near it, each at one point (by Lemma 3, since L. = O(L.)). By virtue of (5), §12, the equation of the trajectory L. in the circle K is written in the form X2 = "'.(Xl), where I"'~ I ~ tan Q. Since Dl :::> D2 :::> "', the sequence of the functions is monotone and for IXI  /311 ~ 60 it converges to the function "'(Xl), the graph of which passes through the point b and is the trajectory L* of the inclusion (1) (the properties BO and Co, 1, §12). IT for some i the part of the circle K lying in the strip IXI  /311 ~ 50 above the curve L,(X2 = "'.(Xl)) does not belong to the domain D. and the part lying below belongs to the domain D. then, since D. :::> DHI :::> ••• , the same will hold for all i. Therefore the part of the circle K lying in this strip above the
"'i
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trajectory L* does not belong to D*, and below L* it belongs to D*, that is, the set D* has interior points. The closure of any component of the set of these interior points will be denoted by Dw. According to the proof, the boundary of the closed domain Dw is the trajectory Lw. Now one can construct a transfinite sequence of the embedded domains and conclude the proof by using Baire's theorem, as in [158] (p. 55). THEOREM 8. Let a neighbourhood U of a stationary point P contain no other stationary points. Then either there exists a trajectory terminating at the point P (within some finite or infinite time), or in each neighbourhood of the point P there exist closed trajectories encircling this point. PROOF: In the circle K (Ix  pi ~ e, e being arbitrarily small) contained in U we take a sequence Pi + p. IT T+(Pi) c K then by Theorem 6 the set O(T) C K contains a stationary point, the point P (there are no other stationary points), or a closed trajectory To. In the first case either O(T) = P or by virtue of the corollary to Theorem 4, O(T) consists of the point P and one or several trajectories terminating with both ends at the point P (there are no other stationary points in K). Thus, there exists a trajectory, one end of which enters p. In the second case, within the domain bounded by the trajectory To there exists, by Theorem 7, a stationary point, the point p, since there are no other stationary points. IT T+ (Pi) leaves the circle K at a point qi, we select a convergent subsequence gi + g, i = i 1 ,i2 ,'" + 00. Since, for the solutions Ixl ~ IF(x)1 ~ m, the time of motion along the trajectory T from the point Pi to qi is not less than some '1 > O. Then, by Lemma 1, §12, K contains either an arc of the trajectory which joins the points P and q or a whole half trajectory T (q) c K. The latter case is considered as in the case T+ (Pi) c K. Take a sequence of circles K. (Ix  pi ~ e.), e. + O. By virtue of what has been proved, either at least one of the circles contains a trajectory terminating with one end at the point p, or in each circle there exists a closed trajectory surrounding the point p, and the result follows. Some results on the qualitative theory of differential inclusions can be found in papers on the theory of control systems. For instance, the regions in a plane which can be reached by going from a given point along the trajectories of a differential inclusion are investigated in [165J.
§14 Bounded and Periodic Solutions The concept of rotation of a multivalued vector field is formulated and the properties of rotation are pointed out. Using these concepts we establish sufficient conditions of existence of bounded and periodic solutions of differential inclusions similar to those known for ordinary differential equations. 1. In the whole of §14 we assume that a setvalued function F satisfies the basic conditions of 2, §7, in as open domain G and that the compactum KeG. We use the notation introduced in §5.
Bounded and Periodic Solutions
§14
143
LEMMA 1. For any 6 > 0 and for any compactum K there exists 00 > 0 such that for all 0 ~ 00 the graph of the function F6 (p) = [co F{p6) J6, P E K, lies in the eneighbourhood of the graph r of the function F(p), p E K. PROOF: In the contrary case there exists K, q. E F6; (P.), i = 1,2, ... , such that
p
(1)
«P., qi), r)
6
i
;;JI: 6,
> 0 and sequences O.
+
0, P' E
= 1,2, ....
Let maxo. = 01 < p(K, 80). By Lemma 15, §5, IF(p) I ~ m for p E K 61, and therefore IF,dp) I ~ m + 151 for p E K. As a consequence of this estimate and the compactness of K, one may assume that Pi + Po E K, qi + qo. From (1) it follows that p((po, qo), r) ~ 6 and, therefore, p( qo, F(po)) ~ e. By virtue of the upper semicontinuity of the function F there exists 0 > o such that F(p) c (F(pO))·/4 for all p E pg, that is, F(pg) c (F(pO))·/4. Since F(po) and (F(pOW/4 are convex then coF(pg) c (F(po))/4, and for O. < 15/2, c5i < e/4, IPi  Pol < 6/2 we have (p.)6; C pg,
qi
E
F6;(Pi)
=
[coF(pt,)t
C
[coF(pg)]_/4 c (F(po))·/2.
This contradicts the inequality p(qo,F(po))
~ 6
since q.
+
qo.
LEMMA 2. For a given setvalued function F(p) and for any 0, 6 > 0 there exists a singlevalued continuous vectorvalued function f(p), p E K, whose graph lies in the 6neighbourhood of the graph ofthe function F (p), P E K, and f(p) E co F(p6 n K). PROOF: For a given e > 0 take a number 150 as in Lemma 1 and any 6 < 60 • Cover the compactum. K with a finite set of balls Ip  Pi I < 0, Pi E K, i = 1, •.• , k. Take any qi E F(p.), i = 1, ... , k. Put 9i(p) = max{Oj 0 Ip  Pil}, 10
as(p) = 9i(P)/
E 9 (p) , 3,
;=1
10
f(p)
=
E as (p)q•• .=1
All the 9i(p) are continuous, E 9i(P) > 0, E (l'(p) = 1. Hence, the function f(p) is continuous. Since (li(P) ::f= 0 only for Pi E p6, f(p) E coF(p6). By Lemma 1, the graph f(p) is contained in the eneighbourhood of the graph of the function F(p). REMARK: The function f{p} satisfies the Lipschitz condition. LEMMA 3. Ito ¢ F(p) for all p E K, K being a compactum, then there exists 00 > 0 such that for all p E K, 0 ~ 00 we have
p (0, [coF(p 6W) ~ Po> O. PROOF: The graph of the function F(p), p E K, is a closed bounded set (Lemmas 14 and 15, §5). Hence, its projection F(K) is a closed set. Since o ¢ F(K), p(O, F(K)) = 2po > O. By Lemma 1 there exists 150 > 0 such that for all 6 ~ 150 the graph of the function F6 (p) = [co F (p6)J6, P E K, lies in the poneighbourhood of the graph of the function F(p). Then p(O, F6 (p)) ~ Po for
pEK.
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LEMMA 4. Let 0 ¥ F(p) for all p E K, K being a compactum, and let the vectorvalued functions f(p) and g(p) be singlevalued, continuous; let their graphs for p E K lie in the oneighbourhood of the graph of the function F(p), 0 < 00, 00 being the same as in Lemma 3. Then f(p) f 0 in K, g(p) f 0 in K, and in K there is no point p at which the vectors f (p) and g(p) have opposite directions. PROOF: The vectors f(p) and g(p) are contained in the convex set [co F(pDW. If for some p E K they had opposite directions, there would exist a E (0,1) such that af(p) + (1 a)g(p) = o. The sum belongs to the same convex set. But by Lemma 3 this set does not contain zero. This is a contradiction. LEMMA 5. IT bounded closed convex sets A and B in Rn contain neither zero nor oppositely directed vectors u E A, v E B, then 0 ¥ co(A U B). PROOF: Suppose 0 E co(AUB). By the Caratheodory theorem (§5) there exist points ao, al, ... , ak E A U B, k ~ n and numbers ao, a1,"" ak ~ such that
°
(2)
ao
+ ... + ak = 1.
Let, for instance, ao, . .• , ai E Aj ai+l, ... , ak E Bj ao + ... + ai = a, ai+1 + ... + ak = {3. If {3 = 0 then 0 = aoao + ... + aiai E Aj if a = 0, then E B, which contradicts the assumption. Hence, a > 0, {3 > 0. Then
°
and, by virtue of (2), au + (3v = 0, that is, the vectors u and v are oppositely directed. This is impossible. Hence the assumption is false, and the result follows. 2. The definition of rotation of a continuous vector field in an ndimensional case for n > 2 is rather complicated. We therefore first define the rotation and describe its properties in the case n = 2, where this definition is very simple. Let f (x) be a singlevalued continuous vector field in a domain G in the plane R2, L be a continuous closed curve x = e(8) in G, So ~ 8 ~ 81' The direction in which the curve is described (i.e., the direction in which 8 is increasing) is assumed to be specified. Let f(x) f on L. Let 8(8) be a continuous function equal to the angle between the direction of the Xlaxis and the direction of the vector f(e(s)), So ~ IJ ~ 81. The angle is determined up to an additive constant 21Tk, where k is an integer. This constant is so chosen that the function 8(s) is continuous. The number
°
"1(/, L) = (8(8d  8(so)) j21T. is called the rotation of the vector field f(x) along the curve L. If the direction of the circuit is positive then the rotation is also called the index l of the curve L with respect to the vector field f [9, 13J. Since the curve is closed then 8(Sl)  8(so) is a multiple of 21T and the rotation is an integer. If 8G 1 is the 1The usual term in English mathematical literature
§14
Bounded and Periodic Solutions
145
boundary of a domain G l C G, and if this boundary consists of one or several closed curves L l , ••• , Lm then, by definition,
{3} the direction of motion on each curve L, being chosen so as to allow the domain G l to remain on the left (Fig. 25). 3:2
Q
Figure f5 A point z = a at which I{a} = 0 is called a singular point of the vector field yU, aH} of the field 1 on the boundary aH of any domain H containing this singular point and no other singUlar points either within it or on the boundary is called the indez yU, a) of the isolated singular point z = a in the vector field I{z}. (By virtue of the property 3° formulated below, the number yU, aH) is the same for all such domains H.) Let a vector field I{z, p} dependent on the parameter p be defined on L. If the vectorvalued function I(z,p) is continuous in (z,p), the vector field I{z,p) is said to vary continuously with p. The properties of rotation of a continuous vector field in a plane and the properties of the index of a singular point in such a field are presented, for instance, in [157] (pp. 205216) and in [9] (pp. 398400). Let L be either a closed curve without selfintersections or the boundary of a bounded domain and let I(z) oF 0 on L. 1° The rotation yU,L) of a vector field does not vary with a continuous variation of this field if 1 does not vanish on L. 2° If the vectors 1(z) and g( z) do not vanish and are not oppositely directed at each point z E L then yU, L) = y(g, L). 3° If in a closed domain D the vector I(z) oF 0 then yU, aD) = o. 4° The index of the point z = 0 in the vector field 1(z) = Az (det A oF 0) is equal to y(Az, 0) = sgn det A. 5° If in a domain Go there exists only a finite number of singular points al, ... , am and if on the boundary of this domain I(z) oF 0, then
I{z}. The rotation
Let F(z) be a setvalued vector function satisfying the basic conditions in a domain G in the plane, let L be either a boundary of the bounded closed domain D C G or a closed curve without selfintersection in the domain G, and let the direction of description (if not specifically indicated, the direction is positive) be given on L. Let 0 r!. F{z) for each z E L.
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The rotation '1(1, L) of any singlevalued vector field /(x) on L such that the graph of the function /(x) on L lies in a 50 neighbourhood of the graph of the function F(x) on L is called [166] a rotation 'Y(F, L) 0/ the multivalued vector field F(x) on Lj 50 is the same as in Lemma 3. Such a function / exists by Lemma 2.· From Lemma 4 and from the property 2° it follows that /(x) ¥: 0 on L and the number '1(1, L) does not depend on the choice of the function I. A point x = a such that 0 E F (a) is called a singular point of the multivalued vector field F(x). The index 'Y(F, a) 0/ a singular point is defined, like '1(1, a), through the rotation 'Y(F, aH), where a E Hand H contains no other singular points. We will show that 'Y(F, aH) does not depend on the choice of the domain H. Let Hl and H2 be domains with the same properties as H, and let the domain Ho be contained strictly within the intersection Hl n H2, a E Ho. Since o ¢ F(x) for all x in the closed domain K = (H 1 U H 2)\Ho one can construct in K a singlevalued continuous vector function I(x) ¥: 0 whose graph lies in the 50 neighbourhood of the graph of the function F(x) in K (Lemma 3). By the definition of rotation of the field F we have
Since I(x)
¥: 0 in K
then by virtue of the property 3° and formula (3)
0= 1(1, a (H.\Ho)) = 'Y(j, aH.)  '1(1, aHo),
i
= 1,2.
Consequently, 'Y(j, aHl ) = 'Y(j, aH2), 'Y(F, aHd = 'Y(F, aH2). For multivalued vector fields the properties 3° and 5° are retained when 1 is replaced by F and the condition I(x) ¥: 0 is replaced by the condition 0 ¢ F(x). This follows from the definition of 'Y(F, L) through '1(1, L) and from Lemma 4. The property 2° is replaced by the following: 2*. 1/ on L there is no point x such that either 0 E Fl (x) or 0 E F2(x) or the sets Fl(x) and F2(x) contain oppositely directed vectors u E Fl(x), v E F2(X),
then 'Y(Fl , L) = 'Y(F2,L). PROOF: The function F(x) = CO(Fl(X) U F2(x)) satisfies the basic conditions since for the function Ft{x) U F2(X) upper semicontinuity is obvious, and for the function F(x) it follows from Lemma 16, §5. By Lemma 5, 0 ¢ F(x) for all x E L. By Lemma 3, 0 ¢ toF(xO) for 5 ~ 50. By Lemma 2, for any 5> 0 there exist continuous singlevalued functions
ft, 12,
such that for x E L i = 1,2.
Since the vectors ft(x) and 12 (x) belong to the same closed convex set which does not contain the point 0, they are neither equal to zero nor oppositely directed. By virtue of the property 2°, 'Y(ft, L) = '1(12, L). From this and from the definition of 'Y(Fo, L) there follows the assertion 2*. 3. Let I(x) be a continuous vector field in Rn or on the boundary L = aGo of a bounded domain Go c Rn, I(x) ¥: 0 on L. The rotation 'Y(j, L) is defined, for instance, in [167] (p. 88) as the degree of mapping
Tx = l(x)/I/(x)1
(x E L)
§14
Bounded and Periodic Solutions
147
of the boundary L of the domain Go into a unit sphere. The rotation is an integer. At first the boundary is assumed to be smooth, but after the definition of the rotation is extended to the case of arbitrary boundary ([1671, 5.3). In the case n > 2 the degree of a mapping is not an elementary topological concept and the concept of rotation of a continuous vector field is therefore not elementary. In most applications one uses not the definition of rotation, but its properties presented below in line with the book [1671 (§5). The rotation ..,(/, L) possesses the properties 10 _5 0 , 2. The formulations of these properties and the definition of the index of a singular point remain unchanged for any n ~ 2. Below we point out some other properties of the rotation and of the index for the case of arbitrary n ~ 2. 60 ..,(I,L) = (l)"..,(/,L). 70 IT I(z) E 01, 1(0) = 0, 1'(0) is a matrix (81./8z,.).,;=I •...•" for z = a and if det I'(a) =F 0 then
..,(/,0)
= sgn det I'(a).
8 0 IT a domain Q is divided by surfaces into domains Ql, ... , Qm and if on their boundaries I(z) =F 0 then
The concept of rotation of a vector field is used in theorems on fixed points of a continuous mapping h(z), i.e., on solutions of the equation h(z) = z. Such points are singular points of a vector field I(z) == h(z)  z. THEOREM 1. Let a mapping h(z) be continuous in a bounded closed domain D c R" and ..,(/, aD) =F 0, I(z) == h(z)  z. Then there exists a point zED such that h(x) = z. PROOF: IT such a point does not exist, then I(x) =F 0 in D and by virtue of the property 30 the rotation ..,(/,8D) is equal to zero, which is false. THEOREM 2 (THE BROUWER THEOREM). Let a closed domain D c Rn be homeomorphic to a ball. Then for a continuous mapping h(z) of the domain D into itself there exists at least one lixed point Zo, such that h(zo) = zo. PROOF: First we prove the theorem for a ball K with the centre z = o. For any point x E 8 = 8K we have h(z) e K and, therefore, the vector h(z)  z cannot go in the same direction as the vector z. Hence, for any point z E 8 the vectors h(z)  z and z are not oppositely directed. IT h(z)  z = 0 for some z E 8 then z is a fixed point. IT h(z)  z =F 0 on 8 then, by virtue of the properties 20 and 4 0 ,
..,(h(z)  z,8) = '1(z,8) = (_1)" =F 0 By Theorem 1 there exists a point Zo E K such that h(zo) = ZOo Now let the closed domain D be homeomorphic to the ball K, i.e., let there exist a homeomorphism y = g(z), z = gl(y), zED, y E K, where g and g1 are continuous. To the points z and h(z) of the domain D there
Basic Methods of the Qualitative Theory
148
Chapter 3
correspond the points g(x) = y and g(h(x)) = g(h(gl(y))) = z(y) of the ball K, the mapping z(y) being continuous. According to what has been proved, there exists a point Yo E K for which z(yo) = Yo, that is, g(h(gl(yO))) = Yo. Hence h(gl(yO)) = gl(yo), that is, h(xo) = Xo, where Xo = gl(yo). Let F(x) be a setvalued vector function which satisfies the basic conditions in a bounded domain D c Rn with the boundary L. Let 0 ¢ F(x) for each x E L. The rotation "IU, L) of the multivalued vector field F(x) on L is defined, as in the case n = 2 (see 2), through the rotation of an auxiliary singlevalued vector field I(x). The proof of the nondependence of "IU, L) of the choice of the field I(x) for sufficiently small 6 (see 2), and the properties of the rotation "I(F, L), remain unchanged for any n ~ 2. For a detailed presentation of the theory of rotation of multivalued vector fields see [166]. 4. The' concept of rotation of a vector field makes it possible to establish several theorems on the existence of bounded and periodic solutions of differential equations ([167], §§68). Below we present some extension (obtained in [168], [169]) of these theorems to differential inclusions. THEOREM 3 [168]. Let W be a bounded closed convex domain in R'''', max{i 1; i 2 }) with the initial conditions Ix(O)1 ='1 we have
(12)
x(t)
of x(O),
0< t
~
l.
These solutions are contained in a ball Ixl < r2' IT Ix(t)1 ~ ro, 0 < t :;;;; to ~ I, then for these t, by virtue of (11), !p(x(t)) decreases and, therefore, x(t) of x(O). If to < I, Ix(to)1 ~ ro then, according to what has been proved, Ix(t)1 < r1, to ~ t ~ I and, therefore, x(t) of x(O). Since, by virtue of (10), grad !p(x) of 0 for Ixl ~ ro, the rotation of grad !p(x) on the spheres Ixl = r* and L(lxl = rd is the same (by virtue of the property 3°) and is not equal to zero. By virtue of (11), I.(x) '" 0 and the vectors li(x) and grad!p( x) are not oppositely directed for x E L. Hence,
(13) Let x = !/Ii(t; xo) be a solution of the equation x = Ii(t,x) with the initial condition !/I.(O; xo) = Xo E L. From (12), from Lemma 6 and (13) there follows
7(X!/Ii(l;x),L) =7(li,L) ",0 According to 3°, in the region Ixl ~ '1 there is a point XOi, at which !/I. (l;xo.) = xo.· From the sequence of solutions !/Ii(t; xo.) we choose a subsequence which converges uniformly for 0 :;;;; t ~ 1. By Lemma 1, §7, the limit x(t) of this subsequence is a solution of the inclusion (6), x(l) = x(O). Continuing the function x(t) with the period l, we obtain the required periodic solution.
Bounded and Periodic Solutions
§14
151
THEOREM 5 1169]. Let all the conditions of Theorem 3, except the condition
(4), be fulfilled. Then the inclusion (6) has at least one bounded solution
(00 < t < 00).
:I:(t) E W
(14)
PROOF: Taking any k = 1,2, ... and repeating the reasoning of Theorem 3, but on the interval k ~ t ~ k instead of 0 ~ t ~ I, we obtain the solution :l:1c(t) E W(k ~ t ~ k) of the inclusion (6). From the sequence {:l:Ic(t)} we choose a subsequence convergent for It I ~ I, and from it we choose 'a new subsequence convergent for It I ~ 2, etc. The limiting function :I:(t) satisfies (14) and, by virtue of Corollary 1 to Lemma I, §7, is a solution of the inclusion (6). Theorems on dissipative systems of differential equations, for instance Theorem 2.5 from 1170], also hold for differential inclusions. The differential inclusion (6), where the function F (t, :1:) is defined for t > :I: ERn, possesses the dissipation property if each solution can be continued up to arbitrarily large t and if there exists a ball 1:1:1 < b such that for increasing t each solution enters this ball and remains there.
t.,
THEOREM 6. Let a setvalued function F(t,:I:) satisfy the basic conditions of 2, §7, and the condition (4). Let there exist a function ip(t,:I:) E a 1 for 1:1:1 ~ a with the following properties:
ip(t + l,:I:) == ip(t, :I:)i
ip(t,:I:)
~ ipo (:1:)
 00
(1:1:1 00),
and for each y E F(t,:I:)
aip
at + (grads ip(t,:I:)) . Y < 0
(15)
Then the inclusion (6) possesses the dissipation property. PROOF: Let :I:(t) be a solution of the inclusion (6), :I:(to) = :1:0. The set Dl (0 ~ t ~ I, m ~ ip(t,:I:) ~ mlJ 1:1:1 ~ a), where m = m8.Xjel=a,OE;tE;lip(t,e)i ml > m, ml > ip(to, :1:0), is closed and bounded. On this set the graph of the function F(t,:I:) is a compactum K (Lemmas 14 and IS, §5). Hence, for (t,:I:, Y) E K the lefthand side of (15) reaches its maximum at some point, and this maximum is equal to '7 < O. Therefore, for each solution of the inclusion (6) which lies in the domain Dl and, by virtue of periodicity of the functions F and ip, also in the domain D (1:1:1 ~ a, m ~ ip(t,:I:) ~ mIl
!
ip(t, :I:(t)) ==
d: +
(grads ip(t,:I:)) . x(t)
~ '7 < 0
for almost all t. Consequently, ip(t, z(t)) decreases, and within a finite time the solution z(t) goes out of the domain D into a domain where ip(t,:I:) ~ m or 1:1:1 ~ a and remains there. This domain does not depend on the solution z(t) and is contained in some balllzi ~ b.
152
Basic Methods of the Qualitative Theory
Chapter 3
5. The question of the existence of periodic solutions of differential equations with discontinuous righthand sides is investigated in a number of papers, for instance, in [IJ (Chapter 8), [4J (Chapters 18, 19) by the usual methods of the qualitative theory of differential equations. To find periodic solutions, one uses the pointmapping method, solutionpatching method, [31 (Chapter 2, §4), [59J, [1711, [172], and approximate methods, in particular, the describingfunction method [172J, [173]. Stability of periodic solutions is considered, for instance, in [96], [159J, [172]. Bifurcations of periodic solutions are examined in [174J, [175J.
§15 Stability We present here several methods for investigating stability of differential equations with discontinuous righthand sides and of differential inclusions: the Lyapunov functions method, using the first approximation equation, the separation of fast and slow motions, the pointmapping method. Examples are given. 1. For differential inclusions there exist two types of stability: stability and weak stability [176J. A solution x = 0 there exists 0 > 0 which possesses the following property. For each Xo such that ]xo  0
(0
0 there exists 5 > 0 such that for to ~ t < 00 each solution x(t) with the initial data x(to) from the 5neighbourhood of the set M exists and satisfies the inequality p(x(t), M) < e. Obviously, for a closed set Me Rn to be stable, it is necessary that any solution with the initial data x(to) E M remain in M for to ~ t < 00. For a differential inclusion x E F(x} a stationary set may not possess this property (example: x E Rl, F(x) is a segment [x  1, x + 11, a solution x = et departs from a set of stationary points 1 ~ x ~ 1}. A stationary set is called stable in the large [S] if it is stable and if for t ..... 00 each solution comes infinitely close to this set. A stationary set is called pointwise stable in the large [S] if it is stable and if for t ..... 00 each solution tends to a stationary point. Sufficient conditions for stability, stability in the large,
Stability
§15
155
and pointwise stability in the large of a bounded stationary set of the differential inclusion :i; E F(x) are formulated in [5] (§2.3) by means of Lyapunov functions Systems of the form :i; = Ax
+ b1=0
(8)
Indeed, if for some t the derivatives :i;(t)
=
(y
= :i;(t)).
y and dv(t, x(t))/ dt exist then
d ( (» _ l' v(t + h, x(t + h))  v(t, x(t» d v t, x t  1m h
t
hO
= lim v(t + h, x(t) + hy)  vet, x(t» 1>0 h lim + 1>0
v(t + h, x(t + h))  v(t + h, x(t) h
+ hy)
.
Basic Methods of the Qualitative Theory
156
Chapter 3
The last limit is equal to zero since x(t+h) = x(t)+hy+o(h), and the function tI satisfies the Lipschitz condition. Hence the last but one limit exists also. It is equal to the righthand side of (8), and the result follows. EXAMPLE: Let v = It2xl and let it be known that Then, by virtue of (8), at this point
iJ= ddh tI(t+h,X+2h)!
x=
2 at the point t
= :hh2!
= x = 1.
=0.
h=O
t=.,=l,h=O
Note that it is impossible to express iJ through righthand derivatives of the function v with respect to t and x because at the point t = x = 1 we have ( the sign + implying a righthand derivative)
v; = 1,
vt = 2,
iJ+ =
°¥ vt + v; .X = 4.
By virtue of what has been said above, for instance, in order that the function tI(t, x(t)) should not increase, it suffices that the expression (8) be nonpositive. Thus, if the function tI(t, x) satisfies the Lipschitz condition then the upper and the lower derivatives due to the inclusion (1), iJ* and iJ*, of the function tI can be expressed by sup and inf of the righthand side of (8) for all y E F(t, x). Then Theorems 1 and 2 remain true, but the proof of Theorem 2 becomes more complicated [177J. Definitions of the derivative due to the differential inclusion (1) which are more general than (8) can be found in [176] and [177]. If v ¢ C 1 then one cannot neglect searching for dtl / dt on the lines and surfaces of discontinuity of the function f(t, x) even in the case of the definition a), §4. EXAMPLE: By virtue of the system the function v = Ixl + IYI is equal to
iJ =
x = sgn x,
tI.,x + VIIY =
1 2
Y=
2 sgn y
the derivative of
= 1 < 0.
for xy ¥ 0. This is insufficient for the use of Theorem 1 because the derivatives v., and till are discontinuous on the coordinate axes, that is, in the same place where the righthand sides of the system are discontinuous. Under the definition a), §4, we have :i: = sgn x, Y= 0, tI = Ixl on the xaxis and, consequently, iJ
=
!
Ix(t)1
= hgnx =
1> 0,
so that the conditions of Theorem 1 are not fulfilled. The same result is obtained by formula (8): iJ =
~ v(x + hsgnx,O)!
h=O
d~lx+hsgnxll h=O =
1.
Stability
§15
157
u
Figure 26 Since, on the xaxis, we have :i; = sgn x, Y = 0, the solutions depart from the point (0,0) along the axis with a velocity of 1, and the solution x == Y == 0 is unstable (Fig. 26). We will give an example of application of the Lyapunov function for obtaining sufficient conditions for stability of a zero solution of a discontinuous system. The conditions of the theorem that follows are not invariant under transformations of the form = i = 1, ... , n. The stability conditions invariant under such transformations are known [178] for n ~ 3, a'i = const.
x.
7.y.,
THEOREM 3 ([7], p. 85). For an asymptotic stability of a zero solution of the
system n
(9)
Xi
=
E ati(t, z) sgn Zi,
i = 1, ... , n,
i=l
defined according to a), §4, with continuous aii(t, z) it is sufficient that [or z = 0 and for all t the quadratic form n
lp(pl,···,Pn; t, z)
=
L
aii(t, Z)p.Pi
i.3=1
be positive definite (the condition ati
= aii is not necessary).
PROOF: Let J.'(t, z) = min Ip on the surface of the cube Ipi I ~ 1, i = 1, ... , n. The form Ip is positive definite, hence J.'(t,O) > O. Since the functions aii(t, z) and, accordingly, J.'(t, x) are continuous for z = 0, for some 00 > 0 in the domain Q(lxll + ... + IZnl ~ 00, to ~ t ~ to + 00) we have J.'(t, z) ~ h > O. Let 11 = IZ11 + ... + IZnl. In the domain Q for Zl i= 0, ... , z,. i= 0 ,.
iJ
n
= LXi sgn xi = i=l
E
a.oi sgn Xi sgn Zi ~ h < O.
i.i=1
Now we shall consider the point z, one or several coordinates of which are equal to zero. For instance, let
(10)
Z1 = ... = ZIc = 0,
ZIc+1
=f 0, ... ,Z,. i= 0,
1~k~n1.
158
Basic Methods 01 the Qualitative Theory
Chapter 3
According to the definition a), §4, at such a point the function x(t) may acquire only values from the smallest convex closed set which contains the limit values of the righthand side of (9), that is, for i = 1,.,., n k
Xi
=
(11)
n
n
E
EaiiPi aii sgn xi =  EaiiPb i=1 i=k+l i=1 1 ~ Pi ~ 1, j = 1, .. " kj 'Pi
= sgn xb
j
= k + I, ... , n.
If the solution x(t) satisfies the conditions (10) on a set of values t of measure zero then for these t the values Xi and may be disregarded. If the solution satisfies the conditions (10) on a set M of values t of positive measure then ahnost all these values t are nonisolated points of the set M. For almost all such nonisolated t there exists Xi, and from (10) and from the definition of derivative it follows that Xi = 0, i = 1, ... , k. From this and from (11) we have for such t
v
n
V=
E
n
n
E E aiiPi'Pi'
Xi sgnxi = 
i=k+l
i=k+l i=1
Adding to this sum the sum equal to zero k
k
n
0= Ep.x. =  EEaiiPiP;. i=1 i=1 i=1
we obtain the quadratic form u = !P(Pl, ... , Pnj t, x) ~ h < 0 (since k ~ nl, at least one of Pi is 1 or 1). Thus, in the domain Q we have
u(x(t))
~
JL(t, x(t»
~ h
< 0,
for almost all t as long as x(t) =I o. Consequently, if v(x, (to)) < min{50i Mo} then v(x(t)) decreases along the solution, and the solution does not leave this domain until t = to + 5 (5 < 50). At this instant tJ(x(t)) = 0, x(t) = O. The equality x(t) = 0 holds also for all t > to + 50 because in the neighbourhood of each point (t,O) we have ~ O. The asymptotic stability follows. REMARK 1 ([7], p. 86): If the condition of Theorem 3 is fulfilled and for all t, x the functions ati (t, x) are bounded and
v
det Ilaii
+ ai.ll..i=l, ...•n
~ const > 0,
then JL(t, x) ~ h > 0 for all t, x, and the zero solution of the system (9) is asymptotically stable in the large. (Indeed, JL(t, x) ~ '\1 (t, x), where ,\t{t, x) is the minimum of the quadratic form !P on the sphere p~ + ... + p~ = 1. This minimum is equal to the least eigenvalue of the matrix II (a'i + ai.)/21Ii.;=1 ..... n. Under the above conditions the roots of the characteristic equation of this matrix are bounded from below by a positive number.)
Stability
§15
159
REMARK 2: In order that the zero solution of the system
x.
10
E
= b.(t,z) 
as.,.(t,z)sgnzf,
i
= 1, ... , k,
.';=1
with continuous bi(t, z) and aii(t,z} be asymptotically stable, it is sufficient that at least one of the two conditions: a) Ibil + E~=I. '>F i laiil < aii, i = 1, ... , kj b) Ib1 1+ ... + (bioi < >'1, >'1 is the least root of the equation det II asi + 2 ai'  >.6·.,·11 . .
.,'=1 .... ,10
= 0'
be satisfied, where 6'i is the Kronecker symbol: 6.. = 1, 6ii = 0 (i =J:. i). Indeed, in both cases, because of continuity the difference between the rightand lefthand side of the inequality in some neighbourhood of the point (t,O) is not less than some h > O. Then for zi(t) =J:. 0 (j = 1, ... , k) in this neighbourhood in the case a) we have x.(t) ~ h < 0 if Zi(t) > 0 and Xi(t) ~ h> 0 if x.(t) < O. According to the definition a), §4, the same is also true for any Xi(t) (j =J:. i) if Xi(t) exists and Xi(t) is not equal to zero. Hence, those coordinates :es(t), which are not equal to zero decrease in absolute value at a speed not lower than h, and in a finite time the solution beaomes equal to zero. In the case b) we obtain
v ~ I' + Ib 1 1+ ... + Iblol ~ >'1 + Ib 1 1 + ... + Iblol ~
h < 0,
instead of the inequality v(z(t)) ~ I'(t, x(t)) ~ h < 0 and the proof is carried out as before. For other conditions of stability for the system (9), and for comparisons between them, see, for instance, [1781. 2. Below we present some methods for investigating stability of homogeneous differential inclusions and equations. If A is a set in R"', and e is a number, then eA implies a set of points of the formez for all x EA. A setvalued function F(x) is called homogeneous of degree a if F(ex) == eaF(x) for all e > O. The differential inclusion (12)
Z E F(x)
(F(ez) == ea F(z), e > 0)
is called homogeneous. It remains unchanged under the change x = eXt, t = etatt with any e > O. If X = !pet) is a solution of the inclusion (12) then for anye > 0 the function z = e!p(e a  1 t) is also a solution. The inclusions (12) and :i: E Fo(x}, where Fo(x) == IxlaF(x) is a homogeneous function of degree 0, have in the region x =J:. 0 the same trajectories, but different speeds of motion along those trajectories (Theorem 3, §9). Under any of the definitions a), b), c), §4, the homogeneous differential equation .
(13)
x=
f(x)
(f(ex) == ea f(x), e> 0)
160
Chapter 3
Basic Methods of the Qualitative Theory
also possesse'l the properties indicated above. All the surfaces of discontinuity of a homogeneous piecewise continuous function f(x) are cones with vertices at the origin. In 2 it is further assumed that the setvalued function F(x) satisfies the basic conditions of 2, §7, and that in (12) and (13) a ;?; O. LEMMA 1. H the inclusion x E F(x) has an asymptotically stable solution x(t) == 0 then there exists 00 > 0 such that all the solutions with the initial data Ix(O) 1 ~ 00 tend uniformly to zero as t + 00. PROOF:
If we assume the contrary then for any 6 > 0 there exist solutions
Xk(t), k = 1,2, ... , such that tk
k = 1,2, ... ;
+ 00.
The solution x == 0 is asymptotically stable and, accordingly, one take 0 sufficiently small that for all solutions with Ix(O)1 ~ 6 we have
(14)
Ix(t)! ~ e
(O~t 0 for all solutions with Ix(O)1 < IJ we have Ix(t)J ~ '7(0) for 00. Then for all Xk(t)
o~ t < (15)
for 0
~ t ~ tk,
k
=
1,2, ... ,
because in the case IXk(t·)1 < IJ, toO ~ tk for the solution z(t) = Xk(t + toO) the inequalities Iz(O)1 < IJ, Iz(tk t")1 > '7(0) would be fulfilled, and this contradicts the choice of IJ. From the sequence of the segments of the solutions (15) one can choose a subsequence convergent for 0 ~ t ~ tl, and from this subsequence, in turn, a new subsequence convergent for 0 ~ t ~ t2, etc. The limiting function x(t) is a solution for which
Ix(O) 1 ~ 6,
IJ~lx(t)l~e
(O~t 0 and a sequence of the solutions Xi(t) such that
Ix.(O)1 = 6.
+
0,
to> o.
Stability
§15
161
Let ai be the last of the points of the segment [OJ tiJ, at which IXi(adl = 0., and b. be the first point after ai, at which Ix.(bi)1 = e. Then Yilt) = Xi(t + ail is a solution, and Zi(t) = 0i 1 Yi(olat) is also a solution of the inclusion (I),
IZi (tt) 1 ~ o.le  00,
Iz.(O)1 = I,
tt
=
1 ~ Iz.(t)1 ~ 6. 1e
(0 ~ t ~
ttl.
orl (b.  ail.
All the solutions with Iz(O)1 = 1 exist for 0 ~ t < 00 because by hypothesis they tend to zero as t  00. By Theorem 3, §7, on any closed interval 0 ~ t ~ l the set of these solutions is compact. Hence, 00 (i  00). From the sequence {Zi(t)} we choose a subsequence uniformly convergent for 0 ~ t ~ I, and from it, in turn, a subsequence uniformly convergent for 0 ~ t ~ 2, etc. The limiting function z(t) is a solution, Iz(t) 1 ~ 1 for 0 ~ t < 00. This contradicts the fact that all the solutions tend to zero. Thus, the assumption is false, and the solution x(t) == 0 is stable. Since all the solutions tend to zero as t  00, the solution set) == 0 is asymptotically stable.
t; 
THEOREM 4 [144J. If the zero solution of the inclusion (12) is asymptotically stable and 0 ~ a < 1 then there exist constants Co and Cl such that for each solution set) with Ix(toll ~ a we have
Ix(t)1
(16)
x(t) =
0
~ coa (to + t*
(to ~ t
t ~ to + t*), t* = Cla1a
~
< 00),
PROOF: By Lemma 1 there exists r > 0 such that for all the solutions with Ix(O)1 ~ 00 we have I:z:(t) 1 ~ 00/2 for r ~ t < 00, and by Theorem 3, §7, Ix(t)1 ~ cooo for 0 ~ t ~ 'f. IT x(t) is a solution with Ix(O)1 ~ a then for C = 50a 1 the function so(t) = a cx(c  1 t) is a solution also, Ixo(O)1 ~ 50, and therefore
Ixo(t)1 ~ 50/2 (r ~ t < 00), Going back from xo(t) to x(t), we obtain for q = 5fI l a l a (17)
Ix(t) 1~
Co
Ix(O) 1 (0
~
t ~ qr),
Ix(t)l
~
Ix(O) 1/2
(qr ~ t < 00).
Since the substitution of t + const for t transforms a solution into a solution, it follows from (17) that if Ix( ti) 1~ ai then (18)
Ix(t)1
~
Ix(tdl/2 (t. +qiT
~
t
0, q > 0 let (21)
The function Fpq is a homogeneous function of the same degree the basic conditions, so does Fpq.
Q.
If F satisfies
THEOREM 5. If the zero solution of the inclusion (12) is asymptotically stable and Q ~ 0 then for sufficiently small p and q the zero solution of the inclusion
(22) is also asymptotically stable. In the estimates (16) and (20) for solutions of the inclusion (22), the constants co, Cl, C2, 'Y may differ arbitrarily little from the values of these constants for the inclusion (12) if p and q are sufficiently small and 0 ~ a ~ 1. PROOF: Let 0 < e ~ 1/2, the constants Co, Cl be the same as in (16) for the inclusion (12), 0 ~ a < 1. By virtue of Corollary 2 to Theorem 1, §8, there exists 5> 0 such that for 0 ~ t ~ Cl all the solutions of the inclusion;; E F'"(x) with Ix(O)1 ~ 1 differ by less than e from the solutions of the inclusion (12) with the same initial data, provided that d(F·,F) < 5(e). Then they are contained in a balllxl ~ Co + e for 0 ~ t ~ C2. For sufficiently small p and q the inequality
163
Stability
§15
+ 2e).
dD(Fpq,F)'< 6(e) is satisfied for the domain D(lzl ~ Co solutions of the inclusion (22) with Iz(O)l ~ 1 we have (23)
Iz(t) I ~
Co
Then for the
+e
Since the inclusion (22) is homogeneous, by means of the technique used in the derivation of the inequalities (18) and (19), we obtain from (23) for i = 1,2, ...
(24)
Iz(t)1 ~ cos' Iz(d,)1 ~ e'+l
(dol ~ t ~ do),
do = d,_1 + Cl e,(la).
Here Co = Co + e, do = Cl. Since do + d* = Cl + 0(e1  a ) then z(d*) = O. The assertion of the theorem follows by virtue of homogeneity of the inclusion (22). IT a > 1 then, using Lemma 1, we obtain for the solution of the inclusion (12) with Iz(O) I ~ 1 the estimate Iz(ctJ I ~ e/2 for some C1 > O. As in the case a < 1, we derive the inequalities (23) and (24) for the solutions of the inclusion (22). From (24) with d* = 00 there follows asymptotic stability of the zero solution. Let a = 1, and for the solutions of the inclusion (12) let there hold the estimate (20). For any e > 0 and f3 E (0,'1) we take 8 > 0 such that C2 + e ~ eb  fJ )'. As at the beginning of the proof, we show that for sufficiently small p and q all the solutions of the inclusion (22) with the initial data Ix(O)l ~ 1 for o ~ t ~ 8 differ from some solutions of the inclusion (12) with Ix(O)l ~ 1 by less than ee..,t and, on taking account of (20), they therefore satisfy the inequality Ix(t)1 ~ (C2 + e)e..,t (0 ~ t ~ B). Taking x(8),x(28), ... as initial data and applying the estimate obtained, we find
«i  1)&
~
t
~ is;
i
= 1,2, ... ).
By virtue of the ~hoice of 8, the righthand side is not greater than (C2 + e) e Pt , and the result follows. For the case 01= 1 this theorem is proved in [180]. Some results on stability of homogeneous differential. inclusions with or without perturbations is obtained in [181] on the basis of the principle of the absence of boundary solutions. The next theorem gives the stability condition for a homogeneous (a = 0) differential inclusion with a piecewise constant righthand side. It can be also applied to differential equations with piecewise constant righthand side under the definition a) or c), §4. Let some pieces of conic hypersurfaces (m denoting dimension, p the number of a piece), separate the space Rn into conic regions S; with a vertex x = O. The boundary of each piece consists of pieces of hypersurfaces of smaller dimensions and does not belong to
S;:
S;:
S;:.
THEOREM 6. Let a setvalued function F(z) satisfy the basic conditions and let this function depend on x in none of the regions S; and on none of the pieces
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164
S;:', that is, F(x) = F;:' for xES;:', m = 1, ... , n,' P = 1, ... , Pm. Let the solutions of the inclusion x E F(x) be unable to pass from one S;:' into another (1 ~ m ~ n) infinitely many times. The function x(t) == 0 is an asymptotically stable solution if and only if for each S;:' (1 ~ m ~ n) none of the vectors from the let F;: lie in S;:' or on its boundary as;:,.
F;:,
S;:,
PROOF. NECESSITY: IT v =f 0, v E and v E or v E S~ c as;;, then x = vt is a solution (in the case v E S~ c as;:, we have v E F;: c F: as a consequence of upper semicontinuity of the function F). If v = 0 E F;: then for any Xo E S;;' the function f(t) == Xo is a solution. In these cases the point x = 0 is not asymptotically stable. SUFFICIENCY: Let the solution x(t) not enter the point x = 0 for any finite t. Then for tl < t < 00 it remains in some S;:" and t
x(t)  x(tt} = _1_ / x(r)dr = y(t),
(25)
t h
t  tl
ly(t)1
~ c.
Since 5;(r) E F;: and the set F;: is closed and convex, by Lemma 12, §5, y(t) E F;:" for h < t < 00. On the other hand, x(t) E S;:, S;:" being a conic set with the vertex 0 and, therefore,
x(t)
E t  tl
sm P ,
p(y(t),sm) ~ p (y(t), x(t) ) = Ix(tdl _ 0 P
t
t  tl
tl
as t  00. Hence as t  00, any of the limit points for y(t) belongs both to F;:' and to S;:.. This contradicts the assumption. Thus the assumption is false, and each solution reaches the point x = 0 for some finite t. The solution cannot leave the point x = 0 since then x(tt} = 0, x(t) E S;:" (tl < t < t2), and by virtue of (25) y(t) E S;:", y(t) E F;:. This is impossible because n = 0. Consequently, each solution reaches the point x = 0 and remains there. By Lemma 2, x(t) == 0 is an asymptotically stable solution. REMARK: IT solutions may pass over from one set into another infinitely many times then the assertion of sufficiency is not true, Example: the system (23), §10. In the case where solutions may go over infinitely many times, stability must be investigated by other methods, for instance, by means of the Lyapunov functions, the frequency method ([5J, Chapter 3) or the pointmapping method ([3J, Chapter 2, §2). We will briefly present the pointmapping method on the assumption of right uniqueness of solutions. It can be applied not only to homogeneous equations, but also to others. In a neighbourhood of a point x = 0, let solutions intersect some surface P infinitely many times in one direction and let such intersections occur at each point of the surface. A trajectory going from any point x E P intersects the surface P next time at a point TIX. Since from right uniqueness there follows righthand continuous dependence of the solution on initial data, the point Tlx
F;: S;:
S;:
Stability
§15
165
depends continuously on the point X; that is, Tl is a continuous mapping of the surface P into itself. IT the mapping Tl has a fixed point a E P, that is T1a = a, then either a is a stationary point, i.e., an equilibrium position (if x(t) == a is a solution), or the solution passing through the point a is periodic. IT one succeeds in proving that for any point X E P sufficiently close to the point 0 the sequence
converges to 0 it is then usually easy to prove asymptotic stability of the zero solution. IT a given differential equation or inclusion is homogenous then a half plane or a plane passing through the origin or a conical surface with a vertex x = 0 is taken as P. The initial ndimensional problem is then reduced to investigation of the mapping Tl of an (n  I)dimensional surface P and then, by virtue of homogeneity, to investigation of the mapping of some (n2)dimensional surface (or a line in case n = 3). For examples of application of the points mapping method to discontinuous systems, see, for instance, 11J (Chapter 8). EXAMPLE: We will determine whether or not the zero solution of the system :i; = 2sgnx  6sgny  2sgnz,
y = 6sgnx  4sgnz,
(26)
Z = 12sgnx + sgny  9sgnz. is stable. In all the coordinate octants sgn:i;
= sgny,
sgny
= sgnz = sgnx.
So, trajectories make revolutions around the x3axis and pass many times into the plane x = 0, intersecting it for y > 0 in one direction, and for y < 0 in another direction. Construct a mapping Tl of the half plane x = 0, y > 0 into itself which is determined by the motion along the trajectories of the system. The solution with initial data Xo = 0, Yo > 0, Zo < 0 lies first in the region x < 0, Y > 0, fI < O. In this region :i; = 6, Y = 2, = 2, and the solution has the form z = 6t, Y = Yo  2t, Z = Zo  2t.
z
It intersects the plane y z*
= 0 at the moment t* = yo/2 at the point
= 3yo < 0,
y*
= 0,
z· = Zo  Yo
Next, it passes into the domain z < 0, y < 0,
< O.
Z
< 0, where
Z
= flO 
:i; =
6,
Y=
Z = 4 and has the form x
= 3yo + 6(t 
t*),
Y = 2(t  t*),
Yo  4(t  t*).
2,
Chapter 3
Basic Methods of the Qualitative Theory
166
It intersects the plane x
Xl
= 0 at the moment t1 = t· + Yo/2 at ·the point
= 0,
Y1
= yo < 0,
Z1
= Zo 
3yo < O.
The case 0 < Zo ~ 2yo (in this and in the next cases, one should take into account that the trajectory intersects the plane z = 0), 2yo < Zo < 13yo, and Z ~ 13yo are considered in a similar way. We find that from the point Xo = 0, Yo > 0, Zo the trajectory first goes back into the plane x = 0 at the point Xl, Yll Zl, where Xl = 0, (27) Y1
Zo
= Yo + 3'
Y1 =
17yo  14zo 33
Zl = Zo  3yo 7 Zl =  Zo  3yo 6
Zl =
2zo  26yo 33
Z1 = Zo 13yo
Y1 = 5yo,
(Zo ~ 0) (0 ~ Zo ~ 2yo) (2yo ~ Zo ~ 13yo), (zo ~ 13yo).
This is a mapping of the half plane Po (x = 0, y > 0) onto the half plane P1 (x = 0, y < 0). Next, from the point (Xl, Y1,Zt) e P1 through the region X > 0 the trajectory comes into the point (X2' Y2, Z2) e Po. Since the system (26) is not changed through a simultaneous replacement of x, y, z, respectively, by X, y, z, this mapping is expressed by formulae similar to (27) with the replacement of Yo, Zo, respectively, by Y1, Zl and Ylt Zl by Y2, Z2· Consequently, instead of the mapping T1 of the half plane Po into Po one may consider in this case the mapping T. obtained from (27) through the replacement of Y1 and ZlI respectively, by Y1 and Z1, T1 being equal to (T.)2. Using homogeneity of the mapping T., we reduce it to a mapping of a straight line into itself. Putting Zo = kyo, zl = f(k)Ylt we obtain from (27) (after the replacement of Yll Zl respectively by Yl, ztJ Y1 = cp(k)yo,
cp(k) = 1, cp(k) cp
k
= 1 3'
( k) = 14k  17 33'
cp(k) = 5,
f(k) = 3  k
(k
~
f(k) = 18 7k 6 2k f(k) = 262k 14k 17 f(k) = 13  k
(0
~ k ~
(2
~ k ~ 13),
(k
~ 13).
5
0), 2),
To study the iterations (T.)', i = 1,2, ... , of the mapping T., we consider for any ko a sequence of numbers k. = f(leod, i = 1,2, .... The function f(k) is continuous and decreasing. IT ko lies outside the closed interval 0 ~ k ~ 13 then after every two iterations the distance from the point leo to this interval decreases by more than a factor of 5, and after a finite number of iterations we are led to
o ~ ki ~ 13,
(j+2
~i
< 00).
Stability
§15
167
°
Then 1P(A1) < 0.8, hence, Y' + 0, Zi + as i + 00. Consequently, all the solutions of the system (26) tend to zero as t + 00 (we take into account the fact that when moving from the point (O,Yi,Zi) to the point (O,Yi+1,%>+1) the trajectory lies in the region Izl + lyl + Izi ~ const (ly.1 + IZil) and that by virtue of the definition a), §4, on the zaxis there exist solutions satisfying the equation =  ~ sgn z). By Lemma 2, the zero solution is asymptotically stable. We shall show that for the system (26) a Lyapunov function of the form
z
tI
= al Izl + a21yI + al Izi + 1hz + P2Y + Paz,
does not exist. Here ai > IPi I, i = 1,2,3, otherwise the function tI would not be positive definite. From the point (0, Yo, zo), where Zo > 13yo ;> 0, the solution enters the plane z = at the point (0, 5yo, Zo  13yo) and then the point (0, 5yo, Zo + 2yo). At the last point the value of any function tI of the indicated form is higher than at the point (0, Yo, zo). Such a function cannot, therefore, serve as a Lyapunov function. Using the methods presented in [182], one can construct the following Lyapunov function tI = Izl + 21YI + 10 Iz  2yl
°
for the system (26). The class of piecewise linear systems, i.e., variable structure systems [6] important for applications, is reduced to firstdegree homogeneous differential inclusions. The phase space of such a system is separated by switching planes (which pass through the origin) into regions in each of which the system is linear but has different coefficients in different regions. One of the frequently used methods for investigating stability of such systems is the following. H (1) each trajectory gets into the switching plane P or approaches it; (2) trajectories cannot depart from the plane P on either side; (3) all the solutions in this plane tend to zero as t + 00, then the solution z == is asymptotically stable. For more details see [6] (Chapters 2 and 3), [7] (Chapter 8). Stability of autonomous control systems has been investigated in many papers which cannot all be referred to here. One of the latest review papers is [183]. 3. To investigate stability of the zero solution of differential equations and inclusions close to homogeneous ones, one can replace a given equation or an inclusion by a "first approximation," i.e., by an equation or an inclusion with a homogenous righthand side, see [180] for the case a = 1 and [144]. These results are presented below in a more general form. We assume everywhere in 3 that setvalued functions F(z) and F(t, z) satisfy the basic conditions of 2, §7, and that a ~ 0. For each point z E Rn we define "polar coordinates": the number p = Izi and the vector w = z/Izl of length 1. Then z = pw. For z = the vector w is arbitrary or, to express this better, is multivalued and takes on all the values with Iwl = 1. Let a setvalued function F(z} (izi ~ Pl) satisfy the conditions: IF(z)1 ~ mllzlQ j there exists a sequence zi + such that IF(zi) I ~ rna IZiIQ, rna > O.
°
°
°
168
Basic Methods
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Qualitative Theory
Chapter 3
We define a homogeneo us main part H (x) of the function F (x). For each w, Iw I = 1, we consider all possible sequences Xi + 0 such that Xi :1= 0, Wi = X'; IXi I + w, and the sequences Yi E Ix. 1'" F(x.). Let H(w) = co HO(w), where HO (w) is a set of limit points for all such sequences {y.}. Let for each X :1= 0 the set H(x) = Ixl'" H(w) ,w = x/ Ixl; let H(O) = 0 if a> 0 and let H(O) = coHO(O), where HO(O) is a union of all the sets H(w) (Iwl = 1) and the set F(O), if a = O. Then the function H(x) is homogeneous, of degree a, IH(x) I ~ m1 Ixl"'. The graph of the function HO(w) is the set of all limit points of the sequences (w.,y", i = 1,2, .... It is t4erefore closed and the functions HO(w), H(w), and H(x) are upper semicontinuous (Lemmas 14 and 16, §5). EXAMPLE: Let a circle x! + x~ ~ P! be separated by smooth curves into a finite number of sectors, and in each sector S. let the function F(x), where x = (Xl, X2) be singlevalued and continuous up to the boundary. On the boundary Li between two sectors let the set F(x) be a convex compactum containing limit values of F(x' ) as x' + x. Let limF(x) = Fi for x E Si, X + 0; a(F(x), FJ~) + 0 for x ELi, x + 0; for the notation a(A, B) see 3, §5. Then H(x) is a piecewise constant function equal to Fi in each angular region whose boundaries are rays tangent at the point 0 to the boundaries of the sector Si and equal to F,": on a ray tangent to the line Li at the point O. If several lines L" Lk , ••• are tangent to one ray at the point 0 then on this ray H(x) is a convex closure of the union of sets F;, F;, ... . (In the latter case it would be incorrect to pass over from a given inhomogeneous differential equation with discontinuous righthand side to a homogeneous differential equation and then to a differential inclusion, because the limit values of the righthand side of the equation in infinitely narrowing sectors between such lines Li' Lk may be lost.) Let a setvalued function F(t, x) (Ixl ~ P1, a ~ t ~ b) satisfy the conditions IF(t, x) I ~ m1 Ixl'" and let for each tEla, bj there exist a sequence Xi + 0 such that F(t, Xi} ~ mo IXil"', mo > O. Let H~(w) be a set of limit points for all possible sequences Yi E IXir'" F(ti'X')' where t. + 5, Xi + 0, Xi :1= 0, Xi/ IXil = Wi + w. Then the graph of the function H~(w) (a ~ 5 ~ b, Iwl = 1) is closed. Next,
H.(w)
= coH~(w),
Ha(x)
= Ixl'" H.(w),
w = x/ Ixl;
H. (0) = 0 if a > 0, and if a = 0 then H. (0) is a convex closure of the set of limit points of all sequences Y' E F(t., x.), ti + 5, Xi + O. The function H.(x) is upper semicontinuous both in 5, X and in x for any 5 = const. Let a function H(x} be homogeneous, of degree a. We shall write d",(F, H) ~ S for Ixl ~ Po if for each P E (0, Po) the graph of the function p'" F(pw) , regarded as a function of w, Iwl = 1, lies in a 8neighbourhood of the graph of the function H(w), and F(O) C H(O).5. We shall say that a setvalued function F(x) is close to a homogeneous function H(x} of degree a if da(F, H) ~ 8(p) for Ixl ~ p, where S(p) + 0 as p + o. For a ~ t < b a setvalued function F(t, x) is close to a homogeneous function H. (x) (of degree a) dependent on the parameter 5 E [a, b) if for each
Stability
§15
169
E [a, b) and each 0 > 0 there exist l > 0 and t E (8 1,8 + l), t E [a, b) we have 8
'1
> 0 such that for each fixed
(28) LEMMA 3. If H(z) is the homogeneous main part ofa setvalued function F(z) then the function F(x) is close to the homogeneous function H(z). The same is true for a function F(t, x) and for the homogeneous function H, (z) constructed for F(t, x). PROOF: Suppose there exists 0 > 0 such that for some arbitrarily small numbers p < 0 the graph of the function pa F(pw) does not lie in the oneighbourhood of the graph G of the function H(w). Then there exist sequences Pi  0, wi(lwil = 1), 'IIi E piaF(Pi,Wi) such that the distance
P((Wi' 'IIi), G) ~ 0,
(29)
i = 1,2, ....
Since IYil ~ ml then for some subsequence {i1c} we have 'IIi"  '11, Wi"  w. But then'll E H(w) by the definition of the set H(w), that is, (w, '11) E G. This contradicts (29). From this the first assertion of the lemma follows. The second assertion is proved similarly. The next two theorems made it possible to investigate, using "first approximation," the stability of autonomous and some nonautonomous differential inclusions. The righthand sides of all the differential inclusions under consideration are assumed to satisfy the basic conditions of 2, §7. THEOREM 7. Let a function H(x) be homogeneous, of degree 0: ~ 0 and let there exist a function o(p)  0 (p  0) such that for each fixed t E [tl'oo) and P ~ Po
da(F(t, z), H(x))
~
o(p) for
Ixl ~ p.
If the inclusion
(30)
zE H(z)
has an asymptotically stable zero solution then the same holds for the inclusion
(31)
z E F(t, x).
PROOF: From the assumptions of the theorem it follows that for all the indicated t and P the graph of the function pa F(t, pw), which is regarded as a function of w, Iwl = 1, lies in the oneighbourhood of the graph of the function H(w) == paH(pw)i 0 = o(p). Then F(t,pw) C [H(pwSW as , that is,
(32) where 11
p=
= q = 0 = o(p)j for the notation Hpq
see (21).
Izl,
Chapter 3
Basic Methods of the Qualitative Theory
170
Since the zero solution of the inclusion (30) is asymptotically stable, for sufficiently small P and q (p, q ~ pt) the same is true, by Theorem 5, for the inclusion
(33) By virtue of Theorem 4 and the remark, for the solutions of the inclusion (33) with p = q = Pl we have Ix(t)1 ~ Co Ix(to)!
(34)
(to ~ t < 00),
x(t)
+
0
(t
+
00),
= 1 the number Co is replaced by C2 from (20). Let the number Pl > 0 be sufficiently small that for all P E (O,Pl] we have 6(2cop) ~ Pl. Then (32) with P = q = PI holds in the region Ixl ~ 2cop. Hence in this region the solutions of the inclusion (31) are solutions of the inclusion (33). The solutions of the inclusion (31) with Ix(to)1 = P ~ PI remain therefore in the region Ixl ~ Cop. Then (34) holds for these solutions, that is, the zero solution of the inclusion (31) is asymptotically stable. For
0
COROLLARY. If the function F{x) is close to the homogeneous function H{x) and the differential inclusion (30) has an asymptotically stable zero solution, then the inclusion 3: E F{x) also has an asymptotically stable zero solution.
For the case 0 = 1 a similar assertion is proved in [180]. The next theorem states that for investigating stability of a nonautonomous differential inclusion in the case 0 ~ 0 < lone can use the "freezingin coefficient" method.
Let a function F(t, x) for tl ~ t < 00 be close to a homogeneous function H. (x) of degree 0, 0 ~ 0 < 1. If for each value of the parameter s E ttl, 00) the inclusion THEOREM 8.
(35)
X E H.(x)
has an asymptotically stable zero solution then the inclusion (31) also has an asymptotically stable zero solution. Each solution of the inclusion (31) with sufficiently smalllx(to)1 reaches the point x = 0 within a finite time. PROOF: Let to E ttl, 00). Fix 8 = to. The zero solution of the inclusion (35) is asymptotically stable. By Theorem 5 there exists 6 > 0 such that for P = q = 6 the zero solution of the inclusion
(36)
3: E H.,pq(x)
is also asymptotically stable (the function H.,pq is defined through the function H., like (21)). By Theorem 4 the solutions of the inclusion (36) with Ix(to)1 = a, a being arbitrary, satisfy the relations (16). Since the function F (t, x) is close to the homogeneous function H. (x) there exists ao = ao(6) such that for each fixed t E [to, to + claAa] and '7 = 2coao (co and Cl are the same as in (16)) the inequality (28) holds. From this inequality
Stability
§15
171
there follows the relation (32) in a cylinder to ~ t ~ to+cla~a, Izi ~ 2coao, but with the functions H.,pq instead of Hpq. Then all the solutions of the inclusion (31) which lie in this cylinder are solutions of the inclusion (36). Hence, the solutions of the inclusion (31) with Iz(to}1 = a, a ~ 0,0, for t ~ to are solutions of the inclusion (36) and by virtue of (16) leave this cylinder only at the point t
= to + claAa, Z = o.
The solutions of the inclusion (31) do not leave the straight line z = 0 since otherwise an analogous reasoning for a cylinder constructed near a point of departure leads to a contradiction with (16). Stability of a zero solution of a homogeneous differential inclusion of degree a = 1 under permanently acting perturbations is considered in [1841. If a vectorvalued function get) is absolutely continuous then the inclusion
z e F(z) + g'lt)
(37) is equivalent to the inclusion
11 e F(y + get))
(38)
(y = z  get)).
The inclusion (38) is meaningful not only for absolutely continuous functions get), but also for some other functions. In [1841, passing over to (38) serves to define the solution of the inclusion (37) and for investigating its properties, in the case where the function get) has bounded variation on each finite interval. If the function get) has jumps then (37) is a differential inclusion with pulse disturbances. The next theorem refers to the case where for a given differential equation or inclusion the solutions are divided in a natural way into solutions reaching an ldimensional hypersurface 8 c R n within a finite time and solutions going along this hypersurface. In particular, this includes the case where 8 is an intersection of surfaces of discontinuity of the righthand side of a differential equation or inclusion (but upper semicontinuity is retained on 8) and stability of sliding motion along the surface 8 is investigated under perturbations leading the solutions off 8. We consider the inclusion (31) for t E [tl' 00), z e G, where G c Rn is the neighbourhood of the hypersurface 8. Let the coordinates ZI, ••. ,Zn be so chosen that 8 is a hyperplane Zl = ... = ZnI = o. Denote (Zl, ••. , ZnI) = y, (Znl+l, .•• , zn) = z. Then Z = (y,z). (Let G(t,y,z) and H(t,y,z) be projections of the set F(t,z) from (31) into the subspaces y and z. Then each solution of the inclusion (31) is a solution of the system
11 E G(t, y, z),
(39)
z e H(t, y, z).
The converse is not always true. For such solutions z(t} = (0, z(t)) of the inclusion (31) which lie on 8, the functions z(t} are solutions of the inclusion (40)
z e Ho(t, z)
(Ho(t, z)
= F(t, z) n 8
for
Z
= (0, z)} .
Let G '" (y) be the convex closure of the set of all limit values of the function G(t.,r.Yi,z.) for t. + 8, + v, !Ii + y, + o. Then G,u(Y) is a function homogeneous in
y,
Z.
of degree a
= o.
r,
172
Chapter 3
Basic Methods of the Qualitative Theory
THEOREM 9. Let xo(t) = (0, zo(t)), (to ~ t < 00) be a solution of the inclusion (31). For any constant s, tI (8 E [to, 00), I t I  zO(8)1 < EO)' let the inclusion if E G." (y) have an asymptotically stable solution y == 0, and let the solution zo(t) of the inclusion (40) be stable (or asymptotically stable). Then the solution xo(t) is stable (respectively, asymptotically stable). PROOF: Let x(t) = (y(t), z(t)) be a solution of the inclusion (31),
Iy(to) 1
0, and for small enough b = b(6) > in the domain B(b), we have
°
G(t, y, z) C G •. ,,; pq(y),
(42)
where 8 = to, tI = zo(to), P = q = 6, and the function G•. ,,; pq(y) is defined through the function G.,,(y), as in (21). By Theorem 5 the number 6 can be taken sufficiently small that the zero solution of the inclusion (43)
if
e G •. ,,; pq(y)
is asymptotically stable. Then by Theorem 4 with a = 0, for all the solutions of the inclusion (43) we have (44) where a (45)
y(t) =
= Iy(to) I.
Let
° (t
~
to
+ cia).
°be sufficiently small that
1]0
>
(co
+ 1 + CI + mctl'1o < b(6) < boo
On some segment to ~ t ~ t2 the solution of the system (39) with the initial data (41), where", < "'0, lies in the domain B(b(6». Let t2 > to be the first instant at which the solution reaches the boundary of this domain. While the solution lies in the indicated domain, its component y(t) is a solution of the inclusion (43) and, therefore, satisfies (44) for to ~ t ~ t2, and Iii ~ m. Hence for such t, (46)
ly(t)1 < co'" < b(6),
Iz(t)  zo(to)1 < '" + m(t  to).
By virtue of (45), for t  to ~ CI"'O the righthand side is less than b(6), so the solution goes out of B( b( 6)) only for t2 > to + Cl "'0, i.e., after it reaches, by virtue of (44), the plane 8(y = 0) at the moment t* ~ to + Cl '" < t2 at the point (O,z(t*)). By virtue of (46)
(47)
Iz(t*)  zo(to)1 < '" + mCl'1·
For t ~ t* the solution (y(t), z(t)) has not already left 8. This assertion is proved with the same reasoning as at the end of the proof of Theorem 8.
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173
Let z*(t) be a solution of the inclusion (40) (t ~ t*. Since IHo I :s:; m then
~
to) which coincides with
z(t) for t
Iz*(to)  z*(t*)1 :s:; m(t*  to) :s:; mCl'1. From this and from (47) we obtain
(48) Since the solution zo(t) of the inclusion (40) is stable by hypothesis from (48) there follows
(to :s:; t
k, the functions bi and aoi are continuous, and the definition a), §4, is used. We apply Theorem 9, taking y = (Xl, ... ,XIe)T, Z = (XIe+l, ... ,xn)T. Denote
lIaii(t, x) Ilid=1, ... ,1e = Al , (bt{t, x), ... , ble (t, x)) T = c(t, y, z), (sgnx1, ... ,sgn X,.)T
(50)
11
= sgny.
= c(t,y,z) 
Ila.At, x) Ili=k+1, ... ,n = 3"=1, ... ,1e
A2,
(b le+1(t,x), ••• ,bn (t,x))T = d(t,y,z).
The system (49) is of the form
A 1 (t,y,z) sgny,
i
= d(t,y,z) 
A2 (t,y,z) sgny,
analogous to the system (39). The inclusion 11 E G~,tI(Y) from Theorem 9 takes here the form (51)
11 = c(s, 0, tI)  Al (s, 0, tI) sgn y
with the definition a), §4. Equation (51) is a vector notation of a system similar to that considered in Remark 2 to Theorem 3; sand tI are parameters. Let, for all t ~ to, x = (0, z), z < 60, the coefficients aii(t, x) and bi(t, x) (i,i = 1, ... , k)
174
Basic Methods of the Qualitative Theory
Chapter 3
satisfy one of the conditions a) or b) of the indicated remark. Then for any 8 ~ to and Ivl < eo the solution y = 0 of equation (51) is asymptotically stable and, moreover, det Al # O. The equation of motion along the hypersurface y = 0 is obtained if, as in the proof of Lemma 3, we replace in (50) sgn y by the vector p, determine it from the first of the equations (50) for iI = 0, and substitute it into the second equation. We derive an equation of the form (40): (52)
H the zero solution of equation (52) (with continuous righthand side) is asymptotically stable and one of the indicated conditions is satisfied for equation (51) then, by Theorem 9, the zero solution of the system (49) is also asymptotically stable. Theorem 3 and the remarks give only sufficient conditions for asymptotic stability of the zero solution to the systems (9) and (51). In the case of constant coefficients the necessary and sufficient conditions for the system (51) are known only for k ~ 2 (see 3, §20 below) and for the system (9) for n ~ 3 (see [182]). In the case n = 3 these conditions have a very complicated form. The sufficient conditions for n = 3, which in the case of constant coefficients strengthen Theorem 3, are given in [178J.
CHAPTER 4
LOCAL SINGULARITIES OF
TWODIMENSIONAL SYSTEMS Singularities in the pattern of trajectories of twodimensional autonomous systems with piecewise continuous righthand sides are investigated. Singularities on lines and singularities at points are topologically classified. All types of structurally stable singular points lying on a line of discontinuity of the righthand sides of a system, singular points of first degree of structural (of codimension 1) instability and their bifurcations are indicated. Singular points lying on intersection of linea of discontinuity are examined.
§16 Linear Singularities Singularities on lines are topologically classified, and the analytical conditions for a singularity on a line to belong to one or the other class are given. 1. Consider an autonomous system
(1) in a finite domain G of the vector form
(2)
;;=
Xl,
x2plane. The system (1) can be written in a
/(X)
It is assumed that continuity of the righthand side of equation (2) and uniqueness of the solutions can be violated only on separate piecewise smooth lines and at isolated points. In the cases where a solution, which has reached a discontinuity line, cannot leave this line as t increases (or decreases), the righthand side of the equation must be defined on this line in order that the solution may exist for any initial data x(to) = (xo) and that the limit of each uniformly convergent sequence of solutions be a solution. The vectorvalued function /O(x) which determines the velocity of motion along the line of discontinuity must be singlevalued and continuous, except at isolated points. Under these assumptions solutions depend continuously on initial data in domain of uniqueness.
175
174
Basic Methods of the Qualitative Theory
Chapter 3
satisfy one of the conditions a) or b) of the indicated remark. Then for any 8 ~ to and Itli < eo the solution y = 0 of equation (51) is asymptotically stable and, moreover, det Al =I o. The equation of motion along the hypersurface y = 0 is obtained if, as in the proof of Lemma 3, we replace in (50) sgn y by the vector p, determine it from the first of the equations (50) for iJ = 0, and substitute it into the second equation. We derive an equation of the form (40): (52) IT the zero solution of equation (52) (with continuous righthand side) is asymptotically stable and one of the indicated conditions is satisfied for equation (51) then, by Theorem 9, the zero solution of the system (49) is also asymptotically stable. Theorem 3 and the remarks give only sufficient conditions for asymptotic stability of the zero solution to the systems (9) and (51). In the case of constant coefficients the necessary and sufficient conditions for the system (51) are known only for k ~ 2 (see 3, §20 below) and for the system (9) for n ~ 3 (see [182]). In the case n = 3 these conditions have a very complicated form. The sufficient conditions for n = 3, which in the case of constant coefficients strengthen Theorem 3, are given in [178].
CHAPTER 4
LOCAL SINGULARITIES OF
TWODIMENSIONAL SYSTEMS Singularities in the pattern of trajectories of twodimensional autonomous systems with piecewise continuous righthand sides are investigated. Singularities on lines and singularities at points are topologically classified. All types of structurally stable singular points lying on a line of discontinuity of the righthand sides of a system, singular points of first degree of structural (of codimension 1) instability and their bifurcations are indicated. Singular points lying on intersection of lines of discontinuity are examined.
§16 Linear Singularities Singularities on lines are topologically classified, and the analytical conditions for a singularity on a line to belong to one or the other class are given. 1. Consider an autonomous system
(1) in a finite domain G of the vector form
(2)
:i;
ZIl
z2plane. The system (1) can be written in a
= /(z)
It is assumed that continuity of the righthand side of equation (2) and uniqueness of the solutions can be violated only on separate piecewise smooth lines and at isolated points. In the cases where a solution, which has reached a discontinuity line, cannot leave this line as t increases (or decreases), the righthand side of the equation must be defined on this line in order that the solution may exist for any initial data z(to) = (zo) and that the limit of each uniformly convergent sequence of solutions be a solution. The vectorvalued function /o(z) which determines the velocity of motion along the line of discontinuity must be singlevalued and continuous, except at isolated points. Under these assumptions solutions depend continuously on initial data in domain of uniqueness.
175
176
Local Singularities of TwoDimensional Systems
Chapter 4
A point x = a is called stationary if the vectorvalued function x(t) == a is a solution. At a point x = b uniqueness is violated if there exist two solutions which satisfy the same initial condition x(to) = b but which are different on an arbitrarily small interval to  S < t < to + S. If uniqueness is violated at a stationary point then some solutions enter this point (as t increases or decreases) within a finite time, and if uniqueness is not violated, there are no such solutions. Domains, all points of which are points where uniqueness ceases to hold, may be encountered for equations and systems with nondifferentiable righthand sides (example in [13], p. 31) and also for differential inclusions (example: = 1, llil ~ 1). We do not consider such cases. Under these assumptions one can distinguish between two types of topologically homogeneous domains: 1) domains which contain no stationary points and through each point of which there passes a single trajectory; 2) domains consisting only of stationary points, i.e., domains in which
x
!(x) == o. Topological homogeneity of a domain (or a line) means that each two points a and b of this domain have neighbourhoods Va and Vb for which there exists a topological mapping (that is, a onetoone mapping continuous on both sides) from one neighbourhood onto the other, which carries the point a into the point b and trajectories into trajectories; the inverse mapping also carries trajectories into trajectories. By Theorem 3, §12, each point in a firsttype domain has a neighbourhood for which there exists a topological mapping onto a rectangle such that trajectories are carried into lines parallel to a side of the rectangle. A firsttype domain is therefore topologically homogeneous. Topological homogeneity of a secondtype domain is obvious. Since it is assumed that points where uniqueness is violated cannot fill domains, there are no other types of topologically homogeneous domains for the class of systems under consideration. Boundaries of topologically homogeneous domains consist of points of nonuniqueness and of stationary points. For equations with piecewise continuous and piecewise smooth righthand sides it is typical that such a boundary can be divided into a finite number of topological homogeneous lines. A maximal topologically homogeneous line which does not lie within a topologically homogeneous domain is called a linear singularity. The maximality requirement implies that this line be not a part of another line which possesses the same properties. The line must be a nonclosed simple arc x = p(t) (0: < t < f3) or a closed curve. The absence of selfintersections follows from topological homogeneity. This line does not lie within a topologically homogeneous domain, otherwise its points will not be topologically distinguished among the other points of the domain. Not each linear singularity has a half neighbourhood (a onesided neighbourhood) which is a topologically homogeneous domain. For instance, let trajectories spiral round a circle L (approach a circle L like spirals) from the exterior and from the interior. Among these, let there exist a finite number of trajectories at each point of which uniqueness is violated, i.e., trajectories which are lines of trajectory confluence. On the other trajectories and on L uniqueness is not violated.
§16
177
Linear Singularities
e
(A concrete example can be obtained by taking a system r7 = (sin '7 )1/3, = 1, making the transformation = 8, '7 = 8 + In Ip  11, and interpreting p,8 as polar coordinates.) Then L is a topologically homogeneous line, that is, a linear singularity. In any half neighbourhood of each point of the line L uniqueness is violated only at the points of an infinite set of arcs convergent to L. Hence, such a half neighbourhood is not a topologically homogeneous domain. In what follows such cases are eliminated by the conditions imposed in 2. We will give examples of linear singularities belonging to different topological classes. In all the examples a linear singularity is a straight line v = o. Firstkind linear singularities, i.e., singularities containing no stationary points: 1) u = 1, iJ = sgnvi 2) u = 1, iJ = 0 (v ~ 0), iJ = 1 (v> O)i 3) u = 1, iJ = 3V 2/ 3 • Secondkind linear singularities, i.e., singularities consisting only of stationary points: 4) u = 0, iJ = sgnvi 5) u = 0, iJ = V (v ~ 0), iJ = 1 (v> O)i 6) u = v, iJ = 0 (v ~ 0), u = 0, iJ = 1 (v> O)i 7) u = 0, iJ = 0 (v ~ 0), iJ.,= 1 (v> O)i 8) u = 0, iJ = Vi 9) u = v, V = 0i 10) u=O, iJ=v(v~O), u=v, iJ=O(V>O)i 11) u=O, v=O (v~O), iJ=v (v>O); 12) u = 0 (v ~ 0), u = v (v> 0), iJ = 0i 13) u = 0, iJ = 3v 2 / 3 i 14) an example of irregular linear singularity:
e
the trajectories are the lines u = c + sin (1/v) and the stationary points are u = c, v = o. Irregular linear singularities are eliminated by the conditons of 2 and are not considered in detail. Those of the above examples where the righthand sides are discontinuous can be replaced by other examples in which righthand sides are continuous and which belong to the same topological classes.· For instance, the equation v =  sgn v can be replaced by the equation iJ = _v 1/ 3 , etc. Thus, the topological classification of linear singularities presented below in 2 is also appropriate, but with some extension (due to examples 3) and 13)), to differential equations with continuous righthand sides, which are nondifferentiable only on some separate lines. All points which belong neither to linear singularities nor to topologically homogeneous domains are called pointwise singularities. Thus, pointwise singularities include endpoints of linear singularities and points of nonuniqueness, limit points for the abovementioned, and stationary points which belong neither to the singularities nor to the domains mentioned above.
178
Local Singularities 01 TwoDimensional Systems
Chapter 4
2. We shall give the conditions sufficient for the system (1) to have linear singularities of one or the other kind and list the local topological classes of linear singularities. Let L be a smooth line, for instance, a discontinuity line of the vectorvalued function I(x) = (h (Xl, X2), h(X1' X2}) or a line on which I(x) = o. Let r (x) and I+(x) denote limit values of the vectorvalued function !(X') when x' tends to the point x E L from the domains G and G+ of continuity of the function I, which are adjacent to the line L, and let IN(x) and I;(x) denote projections of the vectors 1 (x) and 1+ (x) onto the normal to the line L at the point x directed from G to G+. 1° In a finite domain G, let the vectorvalued function I (x) be piecewise continuous and piecewise smooth. This implies that the domain G is separated by a finite number of smooth finitelength lines (which may have common ends) into a finite number of subdomains, in each of which I, a I I aX1, and a f/ aX2 are continuous up to the boundary. 2° At the points x of the discontinuity line L, where IN(x)/;(x) ~ 0, except, possibly in the case
(3)
IN(x)
= M(x) = 0,
a continuous vectorvalued function 10(x) is given which determines the velocity of motion :i; = 10(x) along the line L. The vector 10(x) is tangent to L at the point x. If IN (x) = 0 then 10(x) = r(x); if I~{x) = 0 then IO(x) = r(x). The condition 2° is satisfied, in particular, if the vector 10(x) on L is defined according to a), 2, §4. 3° The case (3) can occur only at a finite number of points. 4° If r(x) = 0 (or I(x) = 0) on the line L then near each point of the line L, except possibly a finite number of points, in G+ (respectively, in G) either I(x) :f 0 and the function
(4)
g(x) = l(x)/l/(x)1
satisfies the Lipschitz condition, or I(x)
== O.
LEMMA 1. If L is the line X2 = ,p(xt}, ,p E 0 1, and f+ (x) = 0 on L, then for the Lipschitz condition to be satisfied for the function (4) in G+ it sullices that there exist m ~ 1 such that on L the onesided (towards ~he domain G+) derivatives satisfy the conditions
(5)
aiel
ax2Ie
=0,
k
= 0, 1, ... , m
 1;
and that near L in G+ the derivative am! lax'; satisfy the Lipschitz condition. The same is valid also for G if I(x) = 0 on L.
179
Linear Singularities
§16
The prool follows from the representation of the function I(X1, X2) by the Taylor formula with an integral residual term
since for small z (on L and in G+ near L) the last integral does not vanish and satisfies the Lipschitz condition. It follows from Lemma 1 that the condition 4° holds also for the case where near L the function I is analytical separately in G+ and in G up to L.
°
LEMMA 2. Let the condition 1° and lit{x) ~ (or ~ 0) on L be satisfied. Then none of the trajectories from the domain G+ can approach any point of the line Last increases (respectively, decreases) up to a finite limit and if, moreover, I+(x) ::I: on L, then also as t + 00 (respectively, as t + 00).
°
PROOF: Suppose a trajectory· Zl = 'P1(t), Z2 = 'P2{t) from the domain G+ (Z2 > tP(Zl)) reaches the point (0,0) of the lin~ L (Z2 = tP(zI), tP E e 1 j we may suppose that tP(O) = tP'{O) = 0) as t + t1  0. Then (to~t
z = h(t,z) ~ h(t,O) 
kz
~
kzj
z(t)
~
°
z(to)ek(tt o )
for to ~ t ~ t1' This contradicts the assumption z(td = 0. IT lit(z) ~ 0, ::I: on L then 11+(0,0)1 = m > 0. Therefore, either ft(O,O) = I~(O,O) > m/2 or 111(0,0)1 >m/2. At least one of the inequalities h > m/2 or 111 I > m/2 is also satisfied in some onesided (in G+) neighbourhood of the point (0,0). Thus, the solution goes out of this neighbourhood within a finite time and cannot approach the point (0,0) as t + 00.
rex) °
180
Local Singularities of TwoDimensional Systems
Chapter 4
LEMMA 3. Let [unctions I(X1,X2) [or X2 ~ I/I(xt} and I/I(xt) satisfy the Lipschitz condition. Then the function I can be continued into the region X2 < I/I(xd, the Lipschitz condition being preserved.
l(xlJ X2) = I(X1, 1/1 (x11) for X2 < I/I(xt). Consider all possible dispositions of the trajectories of the system (1) in a onesided neighbourhood of the line L under the conditions 1° and 4°. A. IT Iii (x) :/= 0 on L then into each point of the line L there comes exactly one trajectory from the domain G+ either when t increases (if Iii < 0) or when t decreases (if Iii > 0) (Fig. 27). The trajectories from the domain G+ reach L at finite values of t. For the proof, one can continue the function I from G+ to G by Lemma 3 and apply the existence and uniqueness theorem. PROOF: We can take
I.
Figure
~7
Figure
~8
Figure
~9
B. IT Iii (x) == 0, 1+ (x) :/= 0 on L then none of the trajectories comes onto the line L from G+ (Fig. 28). This follows from the fact that after the function I is continued from G+ into G by Lemma 3, the line L itself is a trajectory. IT I+(x) = 0 on L, but I :/= 0 in G+ near the line L then for the function (4) we define g+(z) and 9k(x) as r(x) and lii(z). Then: a} if 9k (x) :/= 0 on L then the trajectories in G+ are arranged as in the case A but can approach the points of the line L only as t + +00 (if 9k < 0 on L) or as t + 00 (if 9k > 0) on L}j b) if 9k{Z) = 0 on L then the trajectories in G+ are arranged as in the case B. Indeed, in the domain G+ the trajectories of the equations :i; = I(x) and :i; = g(x} coincide, and the function equal to g(x) in G+ and to g+(x) on L satisfies the Lipschitz condition by virtue of 4°j we always have Ig{x) 1 = 1; different trajectories therefore have no common points either in G+ or on L. c) IT I == 0 in G+ near L then some onesided (in G+) neighbourhood of the line L is filled with stationary points. In the cases B, a), b), c) some functions vanish on a line or in a domain. Such cases are therefore exceptional and rare. They are presented here for completeness of classification. Any of the cases A, B, a), b), c} may also arise in the domain G near L. Combining each of the cases in the domain G+ with each case in the domain G , we obtain the following classification. Case AAo. IN Iii > O. Trajectories intersect the line L, and this line is not a linear singularity. Example: :i; = 1, 1; = 2 + sgn y (Fig. 29).
Linear Singularities
§16
181
Case AA 1 • IN I"J < 0, 1° :j:. O. Trajectories join (flow into) the line L on both sides at finite values of t, the line L is a trajectory also. See 1, example 1 (Fig. 30). Case AA2. IN I"J < 0, 1° E: O. Trajectories reach the line L on both sides at finite values of t; the whole of the line L consists of stationary points. See 1, example 4 (Fig. 31). . Case AB. I"J :f 0, IN = O,r :f 0 (or IN :f 0, I"J = 0, 1+ :f 0). On one side trajectories join the trajectory L for finite t, and on the other side none of them does so. See 1, example 2 (Fig. 32).
l
L
Figure 90
~~".
_ _ _.......... _ _ l
., •
Figure 91
Figure 9S
r
Case Aa. IN :f 0, 1+ = 0, g1i :f 0 (or I'J :f 0, = 0, gil = 0). Trajectories approach the points of the line L from both sides, from one side for finite t and from the other side for t  00 (or t  00); the line L consists of stationary points. See 1, example 5. = 0, g'N = 0). On Case Ab. IN :f 0, 1+ = 0, g1i = 0 (or I~ :f 0, one side the trajectories approach the points of the line L for finite t, and on the other side none of them does so; the line L consists of stationary points. See 1, example 6. Case aa. 1 = 1+ = 0, gil:j:. 0, g1i :f O. Trajectories approach the points of the line L on both sides as t  00 or as t  00; these points are stationary. See 1, example 8. Case abo 1 = 1+ = 0 and, moreover, either g'N :f 0, gl; = 0 or g'N = 0, gl;:f O. On one side trajectories approach the points of the line Last  00 (or t  00), and on the other side they do not; these points are stationary. See 1, example 10. Case bb. 1 = 1+ = 0, g'N = g1i = O. The line L consists 'of stationary points, and no trajectory on either side approaches these points. See 1, example 9. In the cases to follow, trajectories on one side of L are arranged as in the case A, a), b) (see above), and on the other side of L all the points are stationary. Case Ac. I{x) E: 0 in a near L, I~ :f 0, (or I(x) == 0 in a+ near L, IN:j:. 0). See 1, example 7. Case ac. I{x) == 0 in a near L, 1+ = 0, gl; :f 0 (or I{x) E: 0 in a+ near L, 1 = 0, g'N :f 0). See 1, example 11. Case bc. Near L in a I(x) == 0, in a+ I(x) :f 0, on L 1+ = gj{ = 0 (or in a+ I(x) == 0, in a I{x) :f 0, on L = g'N = 0). See 1, example 12. In the case cc, L is not a linear singularity by virtue of 2° . If we weaken the condition 3° by admitting the case (3) not only for a. finite number of points but also at each point of a finite number of arcs then
r
r
182
Chapter 4
Local Singularities 01 TwoDimensional Systems
there appear the possibilities BB, Ba, Bb, Bc. Their classification depends on the method of defining the righthand side of a differential equation on the line L. In this case the definition a) §4, is multivalued. Under this definition the following cases can be specified. Case BBl' IN = I'J = 0, the vectors 1 and f+ have the same direction. Case BB 2 • fN = f'J = 0, the vectors f and 1+ have opposite directions. Cases Ba, Bb, and Bc do not require explanation. Cases where the velocity of motion on L is not uniquely defined are not considered below. We will establish sufficient conditions for existence of a linear singularity of the kind considered in 1, example 3. The derivatives of the function f(x) will not be continuous up to the line L, that is, the condition 10 does not hold. Let X2 = !/I(xt} (a < x < (3) be the equation of the line L, !/I E 0 1 • The change Xl = X, x2  ""(xd = y maps the line L into a portion a < x < P of the straight line y = 0 and the system (1) into the system
(6)
:i;
= p(x,y),
iJ = q(xy).
In a neighbourhood of the line L, Jet the functions tinuous, p ::f 0,
LEMMA 4.
(7) IS
q(x, y) = tp(y)h{x, y),
> 0, the functions tp, h,
h~
/
be continuous,
•
p,p~,
q be con
dy
 () ::f ±oo,
. tp Y
Ihl
~
c > 0, tp(O) = 0 and Jet for
o < IYI < IS either tp(y) > 0 or tp(y) < O. Then the line L (y = 0) is an arc of the trajectory of the system (6), and from each point of the line L a positive half trajectory goes to one side and a negative half trajectory goes to the other side; that is, the line L is intersected by a trajectory at each point. PROOF: Since p{x, y) ::f 0, tp(O) = 0 then y = 0 is a trajectory. Dividing the second equation in (6) by the first one and making the change of variables
(!J dy
10
tp(y) = z,
dy = tp(y)dz,
h(x, y) = p(x, y)k(x, y)
we obtain dz/dx = k{x, y(z)). Since the functions k and 8k(x, y(z))/8z = k~ . tp are continuous, through each point (xo,O) there passes a unique solution z(x). Since dz / dt = h, the solution goes from the region z < 0 into a region z > 0 (or vice versa) if h ~ c > 0 (respectively, if h :;;; c < 0). Going back from z to y, we come to the assertion of the lemma. REMARK: We may assume that the function tp(y) has an ordinary discontinuity (a jump) for y = 0 under the condition tp(+O) tp(O) = O. The linear singularities described in the assertion of Lemma 4 will be ascribed to class AA3 (1, example 3, Fig. 33). If the line L consists of stationary points, and if at each point this line is intersected by a trajectory, then L is a linear singularity of class AA4 (1, example 13). The sufficient conditions for the existence of such a singularity are
Linear Singularities
§16
183
l
Figure 99
obtained from Lemma 4 if we discard the condition p =F 0 and require that 0 and that the functions p and h belong to 0 1 for y > 0 and for y < 0 up to the line y = o. The next theorem gives the conditions under which the system (1) or equation (2) has only a finite number of linear and pointwise singularities.
p(z, ±O)
=
THEOREM 1. Let, in a finite domain G, the righthand side of equation (2) satisfy the condition 1° and be able to vanish only in a finite number of domains, the boundaries of which consist of a finite number of smooth lines and, moreover, in a finite number of points and smooth lines. All these lines are of finite length. On lines of discontinuity of the functions /, a 1/ az 1 , a 1/ az2 , let the vectors r(z), I+(z), 10(z) be able to vanish and the vectors I(z) and I+(z) be tangent to the lines of discontinuity only in a finite number of points and possibly also at each point of a finite number of arcs. Let the conditions 2°4° be satisfied on lines of discontinuity and on lines where fez) = o. Then the domain G can contain only a finite number of linear and pointwise singularities. PROOF: In any subdomain, where I, al/az 1 , al/az2 are continuous and I(z) =F 0, only one trajectory passes through each point. In these sub domains and in those where I(z) == 0 there are no linear or pointwise singularities. The boundaries of those sub domains where I(z) == 0 consist of a finite number of lines on which the conditions 2°_4° are satisfied. Such lines consist of a finite number of linear singularities of classes Ac, ac, bc. The lines of discontinuity consist of a finite number of smooth pieces, on each of which each of the functions IN and I'J either preserves sign or is identically zero, and each of the vectorvalued functions 1, 1+, and 1° either does not vanish or is identically zero. By virtue of the conditions 2°4° and of the reasoning given in the classification, such pieces (except in the case AAo, where there are no singularity) are linear singularities of classes AA 1 , AA2 , AB, Aa, Ab. By virtue of the condition 4° the remainder of the lines where / = 0 consist of a finite number of pieces which are linear singularities of classes aa, ab, bb. Thus, there is a finite number of linear singularities. Their endpoints and isolated points where I(z) = 0 are pointwise singularities; hence, there is a finite number of them. REMARK: If the assumptions of Theorem 1 are fulfilled and the function I(z) is continuous in G then there may exist linear singularities only of classes aa, ab, ac, bb, bc. 3. We will show that the classification of linear singularities presented in 2 under the conditions 1°4° is a local topological classification. We say that the trajectories of two systems in open or closed domains G 1 and G2 (respectively)
184
Local Singularities
0/ TwoDimensional Systems
Chapter 4
have the same topological structure ([157], p. 125) or, in short, that domains G t and G 2 have the same topological structure if there exists a topological mapping of the domain G t onto the domain G 2 which carries, as does its inverse mapping, trajectories into trajectories. This means that the mapping of the domain G l onto the domain G 2 carries from G l each arc of a trajectory (or stationary point) of the first system into an arc of a trajectory (respectively, stationary point) of the second system, and the inverse mapping carries each (contained in G 2 ) are of a trajectory (or stationary point) of the second system into an arc of trajectory (or stationary point) of the first system. This mapping does not necessarily retain the direction of motion along trajectories or change the direction simultaneously on all trajectories ([157], p. 128). Example: the identity mapping x = x, 11 = 11 carries trajectories of the system x = x, 11 = 0 into trajectories of the system x = x 2 , y = 0 and vice versa; the direction of motion along trajectories in the half plane x > 0 remains unchanged, whereas in the half plane x < 0 it changes. If the domain G 1 is closed and bounded then from continuity of onetoone mapping of G l onto G2 there follows continuity of the inverse mapping ([155], p.321). The requirement that the inverse mapping should also carry an arc of a trajectory into an arc of a trajectory does not follow from the other requirements. REMARK ([157], p. 126): An identical mapping x = x, 11 = 11 carries each arc of trajectory of the system
y=o
(8) into an arc of trajectory of the system
(9)
x= 1,
y=o
but does not carry the arc 1 < x < 1 of the trajectory 11 = 0 of the system (9) into an arc of a trajectory of the system (8). The additional requirement that a stationary point of a first system should be mapped into a stationary point of a second system is not sufficient, for instead of the system (9) one can take the system (10)
y=o,
for which the same arc 1 < x < 1, 11 = 0 is an arc of the trajectory of the solution x = t 3 , 11 = o. The requirement that a stationary point of the first system should be mapped into a stationary point of a second system does not follow from the other requirements. This is seen by considering the mapping x = x, 11 = 11 of trajectories of the system (9) into trajectories of the system (10) and from the inverse mapping of (10) into (9). Instead of the requirement that each arc of a trajectory should be mapped into an arc of a trajectory, it is required in [157J, (p. 125) that each two points lying on one trajectory should be mapped into two points lying on one trajectory.
185
Linear Singularities
§16
ef
e
mf
Figure 34
Figure 35
These requirements are equivalent in the case where for any initial conditions x(o) = :Z:o, y(O) = Yo a solution is unique, but they are not equivalent for systems where uniqueness is violated, as is seen from the following example (Figs. 34, 35). In both cases the :z:axis is a trajectory at each point of which uniqueness is violated; solutions reach the stationary point 0 within a finite time. There exists a topological mapping, which carries any two points of one trajectory into two points of one trajectory. This mapping may, for instance, be identical in the region y > 0 and below the trajectory mOn and continued in an obvious way (along trajectories) into the regions mOd and nO I. Points a and b are mapped into points a1 and bll but the arc acOdelb of the trajectory is not mapped into an arc of a trajectoryj the points 41 and b1 are joined by the arc a1k10cli1b1 of a trajectory. The index of the stationary point 0 is equal to 0 in Fig. 34 and is equal to +1 in Fig. 35. We will say that linear singularities L and K belong to the same topological class if any (nonend) points :z: E L, :z:* E K have neighbourhoods of the same topological structure. Thus, we make a local topological classification of linear singularities. THEOREM 2. Under the conditions 10 _4 0 , 2 there exist only eleven topological classes of linear singularities: AAlI AA 2 , AB, Aa, Ab, Ac, aa, ab, ac, bb, bc. PROOF: We will show that each nonendpoint :Z:o of a linear singularity L has a neighbourhood which can be topologically mapped onto a neighbourhood of any point of the uaxis in the respective example considered in 1, 2, so that trajectories are carried into trajectories. Let :z: = ,p( s) (5 t;;;; S t;;;; 5) be an arc I of the linear singularity L, ,p Eel, ,p1(S) ::j; 0, ,p(0) = ::Co. If, on this arc, ft(::c) > 0 (for the notation see 2) and:z: = ~(t,s) is a solution with the initial condition tp(O, s) = ,p(s) which lies in G+ for 0 < t t;;;; t1, then by Theorem 3, §12, the function ~(tI,s) maps topologically a rectangle  5 t;;;; S t;;;; 5, 0 t;;;; tI t;;;; t1, onto some onesided (in G+) neighbourhood of the point :Z:o of the line L. The existence of a number t1 > 0 common to all such solutions ~(t, s), lsi t;;;; 5, follows from the condition It (:z:) > 0 and from the uniform continuity of the function I(:z:) in G+ near the arc l. The case It < 0 is reduced to the above case by the substitution of t for t, and the cases f+ = 0, gj; > 0 and f+ = 0, gj; < 0 is handled by passing over to the equation = g(:z:) which has the same trajectories in the domain G+ by virtue of (4).
z
186
Local Singularities
0/ TwoDimensional Systems
Chapter 4
If on the arc 1 E L we have Ii; = 0, /+ # 0, then we draw a sufficiently small contactless segment :c = X('7), ~ '7 ~ 6' (X Eel, X' # 0, X(O) = xo) from the point :Co E 1 into the domain a+. We will define the function f on L by the values f = f+. Let :c = \l'l(t, '7) be a solution of the equation x = f(:c) with the initial condition \1'1 (0, '7) = X('7). For '7 = 0 this solution runs through a part of the arc 1 for the time interval t1 ~ t ~ t 2 , where tl < 0 < t2' By virtue of continuous dependence of solutions on initial conditions the function \1'1 is defined and continuous in the rectangle P (tl ~ t ~ t2, ~ '7 ~ S) for sufficiently small S. By Theorem 3, §12, this function maps topologically P onto some onesided neighbourhood of the point Xo. The case where f+ = 0, gj{ = 0, g+ # 0 on 1, is reduced to the one considered above if we pass over to the equation :i: = g(:c). Thus in the cases A, B, a), b), 2, there exists a topological mapping of some onesided neighbourhood of the point :Co E L onto a rectangle, under which trajectories are carried into parallel straight lines. In the case c) one can take an arbitrary topological mapping of such a neighbourhood onto a rectangle if the arc l c L is carried into a side of the rectangle. Combining each of the cases A, B, a), b), c) in a+ with each one in a and noticing that a topological mapping of two onesided neighbourhoods (in a+ and a) can be continuously joined along the arc l, we obtain the mapping of the whole neighbourhood of the point :Co onto some rectangle. The cases BBl. BB 2 , Ba, Bb, Bc are eliminated due to the condition 3°; in the case cc there is no linear singularity. The case AA is divided into subcases, as in 2. In the cases AA 2, Aa, Ab, Ac, aa, ab, ac, bb, bc the constructed topological mapping of the neighbourhood of the point Xo onto a rectangle is the desired one (up to a change of the labelling of coordinates). In the case AA1 it remains only to map linearly both halves of the rectangle (one half, in the case AB) onto parallelograms bounded by the lines tI = 0, tI = ±S and by two trajectories from example 1 (or 1, example 2). Thus, all linear singularities satisfying the conditions of the same (anyone) of the eleven cases AA 1 , ••• , bc, 2, belong to the same topological class. These eleven topological classes ar different. For example, in the case AA2 each point of a linear singularity is stationary and, moreover, belongs to two other trajectories which enter this point within a finite time, in the case Aa it belongs to one such trajectory, and in the case aa, to none of such trajectories. The difference in the other cases is also proved very simply. The conditions 2°4°, 2, admit of the requirements imposed in each of the cases AAlo ... being violated in a finite number of "exceptional" points of the line L. If the structure of the neighbourhood of such a point differs from the structure of the neighbourhoods of other points of the line L then such a point does not belong to a linear singularity, but is a boundary between two linear singularities lying on L. If the structure of the neighbourhood of such a point is similar to the structure of the neighbourhoods of all nearby points on the line L then this point belongs to the linear singularity. The belonging of a linear singularity to one or another topological class is established by the fulfillment of requirements imposed on the functions f+, f, ••• at non"exceptional" points. Since in 2 we consider all the cases which under the conditions 1040 can be sat
°
°
187
Linear Singularities
§16
isfied on a whole arc of the line L, under these conditions each linear singularity belongs to one of the eleven indicated topological classes. The proposed local topological classification of linear singularities can be made more detailed. For instance, if one considers only topological mappings which retain the direction of motion along trajectories or change the direction simultaneously on all trajectories then each of the classes Aa, aa, bb is divided into two classes, and the other classes remain unchanged. 4. Now we extend the topological classification to classification with respect to diJleomorphi",." i.e., topological mappings which are continuously differentiable along with their inverse
mappings. The condition 4°,2 must now be strengthened by requiring that g(z) E 0 1 up to the line L. Note also that if the vectors 1+ and 1 (or 1+ and g, or 1 and g+, or g and g+) are collinear at some point z then they retain this property after any differentiable mapping. In addition to the conditions 1°4° we therefore require that the line L consist of a finite number of arcs, on each of which these vectors are collinear either at no point or at every point of the arc. Next, in the case bb, after passing over to the equation z·= g(z), the vectors g(z) and g+(z) are tangent to L. If they are oppositely directed, we take t instead of t in G. Since 1, we obtain the equation z g(z) with a continuous function g(z). If on always Ig(z)1 some arc of the line L we have ikgii(z) ¢ ikgt(z) then after any diffeomorphism these derivatives turn out to be different for the tran90rmed equation, that is, the discontinuity of the derivative &gN(Z) along the normal to L cannot be eliminated using diffeomorphism. One should therefore impose another condition: each line of discontinuity must consist of a finite number of arcs on each of which the equality
=
=
(11) holds either at all points or at none of the points of the arc. For such arcs one can give a classification with respect to diffeomorphisms. 3. Under the above assumptions there exist eighteen classes 01 linear singularities with respect to diil'eomorphismB. Out 01 the eleven classes mentioned in Theorem 2, class AA1 is divided into three classes and each 01 the classes AA2, Aa, aa, bb is divided into two classes, and one more class AAo with noncoIIinear vectors 1 and 1+ is added. lion a line 01 discontinuity the deJinition a), §4, is applied then instead 01 these eighteen there exist only Jllteen classes (AA1 and AA2 remain undivided).
THEOREM
Let L be an arc of a linear singularity of class AA1 and at each point z E L the vectors I(z) and I+(z) be noncollinear, Iii ¢ 0, Iii ¢ O. We will construct a diffeomorphism which maps trajectories of the system
PROOF:
z = I(z)
(12)
(z
~
L),
near L into trajectories of one of the two systems (IS) (14)
u= 1, u= 1,
v=sgntl
(tI¢O),
v=sgntl
(tI¢O),
Consider the case Iii > 0, Iii < 0 (the case t instead of t). For z E L we consider the function
lii(z)
'Y(z)
=  lii(z)
u= 1, u= I,
v =0
(tl
= 0),
=0
(tl
= 0).
V
Iii < 0, Iii > 0 is reduced to this one by taking
> 0,
Extend this function to the neighbourhood of the line L so that the conditions 'Y(z) > 0, 'Y(z) E 0 1 be satisfied (for instance, as in Lemma 3). Replace equation (12) by the equation
(15)
z = h(z)
(h(z)
= I(z),
z E G+; h(z)
='Y(z)/(z), z E G) ,
188
Local Singularities of TwoDimensional Systems
Chapter 4
which has in G+ and in G the same trajectories as in (12). Since h'j:, = fii = h'N for z E L put
< 0 then
(16) The vector hO(z) is tangent to L at the point z. Since the vectors f and f+ and, therefore, h and h+ are noncollinear, it follows that hO(z) #; O. Let Zo E L, and z = 1/I(t) be a solution of equation (16) with the initial condition 1/I(z) = zoo For some t > 0 the arc Z = 1/I(t), It I ~ t, lies on L. Let z = rp(t, s) and z rp+(t, 8) be solutions (respectively, in G and G+) of equation (15) with the common initial condition rp (0,8) rp+(O, s) = 1/1(8). As in the proof of Theorem 2, the functions rpand rp+ are defined for lsi ~ t, tl ~ t ~ O. We will show that
=
=
(17)
Z=rp+(II,U+II)
(tI~O)
is the unknown diffeomorphism. Indeed, as in the proof of Lemma 2, formulae (17) define the homeomorphism of the closed domain 1111 ~ tl, Iu + Itlil ~ t onto a neighbourhood of the point zoo The trajectories of equation (15) are expressed by the fonnulae z = rp±(t, s), where 8 = constant and t is a variable. Hence, the trajectories of the system (13) for tI #; 0 are mapped by formulae (17) into trajectories of equation (15). By virtue of the theorem on differentiability of a solution with respect to the parameter we have rp, rp+ E a 1. Obviously, the derivative 8z/8u in (17) is continuous also for II O. It remains only to prove continuity of the derivative 8z/811 for II = O. Since rp(t, 8), rp+(t,s), and 1/I(t) are solutions of the above equations, for tI = 0
=
Hence for the mapping (17)
By virtue of (16), these expressions coincide. Thus, (17) is a mapping of class 0 1 • For tI = 0 the vector 8z/8u = 1/I'(s) hO(z) is tangent to L, while 8z/8t1 = h  h O is not tangent to L. Consequently, the Jacobian of the mapping (17) is not equal to zero on the line L and in its neighbourhood. . Thus, (17) is a diffeomorphism which maps trajectories of the system (13) into trajectories of the system (15), (16). If the vectors fO(z) and hO(z) for z E L have one direction then the trajectories of the systems (12) and (15), (16) coincide and the diffeomorphism (17) maps trajectories of (13) into trajectories of (12). If tho vectors fO(z) and hO(z) are oppositely directed then the diffeomorphism (17) maps trajectories of the system (14) into trajectories of (12). If fO(z) == 0 (the case AA2) and the vectors f+ and f are noncollinear, then the diffeomorphism (17) maps trajectories of the system
=
u= I,
iJ=sgnll
(11#;0),
u=v=O
(tI=O)
into trajectories of the system (12). The cases AAo and Aa, as (for non collinear vectors f+ and f or g+ and g) are reduced to this case, the former through replacement of fez) by  fez) in G+ or in G, the latter two through a replacement of the function fez) by the function g(z) from (4). If for each z E L the vectors f+(z) and f(z) (or g+(z) and g(z» are collinear but not tangent to the line L, then in the case AAo the trajectories have no singularities and are mapped onto a family of parallel straight lines, and in the cases AAl, AA2, Aa, aa the trajectories are mapped into parts of the straight lines u = constant in half planes tI > 0 and tI < 0 and into the trajectory II 0 (in the case AAJ) or into stationary points on the straight line tI O. To construct a diffeomorphism in these cases, one takes the equation z g(z) instead of (12) (see (4» and in the case gtig'N < 0 replace in G+ (or in G) the
=
=
=
189
Linear Singularities
§16
vector g(z) by the vector g(z). After this g(z) Eel in the neighbourhood of the line L and the diffeomorphism is constructed as in Theorem 4, §I2. If on a line of discontinuity one uses the definition a), §4, then the vectors /0 in (12) and hO in (16) always have one direction, whereas in the case of collinearity of f+ and f we have /0 0, hO o. Therefore, in each of the cases AA1 and AA'J there remain only one of the abovementioned possibilities. In the cases AB, Ab, ab the vectorvalued function z tp( t, s) is constructed on one side of Land z = tp1(t,'7) is constructed on the other side, as in the proof of Theorem 2, but the function 9 from (4) must be first substituted for the function f. Since on L neither the vectors tp~ and tp~ nor the vectors tp~t and tp~1J are collinear then the Jacobians of the vectorvalued functions tp and tp1 do not vanish. Therefore, near L the functions s B(Z) Eel, '7 '7(z) E e 1 up to L. Extending each of the functions B(Z) and '7(z) onto the other side of the line L (in ell, we obtain a diffeomorphism B B(Z), '7 '7(z) under which L is mapped into the straight line '7 0 and the trajectories of the system (12) on one side of L are mapped into straight lines 8 constant, and on the other side into straight lines '7 constant; that is, in the cases Ab and ab, into trajectories of examples 6 and 10, 1. In the case AB, to obtain traJectories of example 2, 1, one should make a linear transformation. In the case bb, after passing over to the equation :i: = g(z) (and provided that g(z) g+(z) after t is taken instead oft in 0), we obtain an equation with a continuous function g(z). Since Ig(z)1 1 and since in the case bb on the line L the vector g(z) is tangent to this line then on the arc , C L, where (11) is satisfied, the derivatives og/oz; are continuous, i 1,2. In the neighbourhood of each point of this arc g(z) Eel, and by Theorem 4, §I2, there exists a diffeomorphism which maps traJectories of the equation:i: g(z) onto the family of parallel straight lines. It mapa trajectories of the initial system :i: fez) into trajectories of example 9, 1. Let the equality (11) be satisfied at none of the points of the arc , C L. As in the case ab. using diffeomorphism, we map treJectories in the domain 0 into parallel straight lines. The system will take the form
=
=
=
=
=
= =
=
=
=
=
=
=
=
e=
(18)
1,
t1 =0
e=
('7 ~ 0);
=
1,
=
Here q(e, 0) 0 (we have the case bbl. Since the equality (11) does not hold then by virtue of what has been said before the formulation of the theorem, q~ (e, 0) p( e) ¢ O. We shall make the change
=
We obtain the system du dT
du dT
= 1,
= 1.
d'7 dT
= q(e(u),'7)
('7 ~ 0).
p(e(u))
The last fraction will be denoted by I(u, '7). By virtue of the choice of the function pee), we have oe/O'7 = 1 for '7 = O. Let u = '1', '7 = tp(T, c) be a solution with the initial conditions u(O) = 0, '7(0) = 0 ~ O. Let us put
(II ~ 0),
(19)
(II
~ 0);
u=u.
Then the treJectories u = ", '7 = tp(",o) are mapped into the lines u = ", II = ce r for 0 ~ O. Consequently, the system is mapped into the system (here u du/dT, etc.).
=
u = 1 v =0
(20)
(II ~ 0);
u = 1,
iJ
= II
(II ~ 0).
We will show that 0'7/011 is continuous for II = O. From (19) 0'7/011 = e"otp/oc. The derivative atp/at: w satisfies the firstvariation equation
=
til
= woe/O'7;
w(O)
= 1.
190 For fJ
Local Singularities of TwoDimensional Systems
Chapter 4
= 0 we have aD/OfJ = 1; consequently,
=
=
For v ~ 0 we have fJ v, lJfJ/Ovl.=o 1. Hence (19) is a diffeomorphism. Thus in the case bb trajectories of a system are mapped either into trajectories of example 9, 1 or into trajectories of the system (20). In the cases Ac, ac, bc, a diffeomorphism which maps trajectories into parallel straight lines is constructed on one side of the line L as in the case abo Then this diffeomorphism is arbitrarily extended through the line L into a domain filled with stationary points.
§17 Topological Classification of Singular Points The known topological classification of singular points by the number and disposition of hyperbolic, parabolic, elliptic sectors extends to singular points of systems with piecewise continuous righthand sides. This enlarges the number of topological classes of sectors. 1. The concept of separatrix must be extended to autonomous systems with nonuniqueness. Not only stationary points (equilibrium positions) but also points where uniqueness is violated, and other pointwise singularities, may have separatrices. For instance, if at some point a bundle of trajectories flows together to become one trajectory then the two boundary half trajectories of the bundle and the half trajectory formed after the confluence are separatrices because they separate trajectories with different behaviour (Fig. 36). These separatrices are similar to the three separatrices of a saddlenode and differ from them only in that they are not whole trajectories.
Figure 36
Figure 37
A singular point may have orbitstable separatrices. For instance, in the system (Fig. 37)
3:
= x,
y=
{O Y =xsgny
Iyllt (Ixl ~ Iyl) , (x ~ Iyl) (x
OJ
y
=x < OJ
y= x > 0
Topological Classification of Singular Points
§17
191
are linear singularities and three half trajectories y= 0,
x < OJ
y = x
> OJ
y
=
x < 0
are separatrices. These six half trajectories separate the neighbourhood of the point (0,0) into six parabolic sectors, two of which belong to one topological class and the other four to another topological class. Below we define a separatrix for a system with a finite number of linear and pointwise singularities (I, §16). In all the cases the ends of an arc are considered as not belonging to this arc. What we call a separatriz I of a pointwise singularity b is either a trajectory tending to the point b or an arc of a trajectory with endpoint b such that unique· ness on I is not violated and for any point a E I there exist arcs of trajectories l., i = 1,2, ... , such that b ¢ I. and, besides, either 1) I. = a.b.c., a,b. + ab, p(c., b) ~ eo > 0, i = 1,2, ... , or, 2) to = tlib. + €lb, the point b. lying on a linear singularity. We distinguish between separatrices of the first and second kinds, for which the condition 1) or 2), respectively, is satisfied. A separatrix may be simultaneously of first and second kinds, for instance, the separatrix ab in Figs. 38 and 39 (in both figures, bc is a linear singularity).
b£
ci
ai~·~ai~
Figure 98
Figure 99
The separatrix of a point b is called a whole separatrix if it is extended from the point b either infinitely or up to a nearest (along this separatrix) pointwise or linear singularity. For a system without uniqueness violations and with only isolated singular points there can exist 'only firstkind separatrices. In this case the above definition of a separatrix is equivalent (by virtue of Theorem 38 from [157]) to the definition of separatrix as an orbitunstable half. trajectory tending to a singular point [157] (p. 277). 2. Consider a system in vector notation
(1)
:i: = !(x),
which satisfies the conditions 10 _4 0 , 2, §16. By Theorem I, §16, in a finite domain G this system may have only a finite number of linear and pointwise singularities. Let a point 0 (x = 0) be a stationary point or a pointwise singularity and let it have a neighbourhood containing
192
Local Singularities 01 TwoDimensional Systems
Chapter 4
no. o.ther statio.nary po.ints, po.intwise singularities, who.le linear singularities o.r who.le separatrices. If there exist so.lutio.ns entering the po.int 0 (x = 0) within a finite time then instead o.f equatio.n (1) we take the equatio.n
x = Ixl I(x),
(2)
which has the same trajecto.ries in the regio.n x ¥: o. Since I/(x)1 ~ m fo.r Ixl ~ 5 then fro.m (2) it fo.llo.ws that Idlxl/dtl ~ Ixl ~ mlxl, lIn IX(t2)1 In Ix(tdll ~ mlt2 tll. The so.lutio.n o.f equatio.n (2) may therefo.re enter the po.int x = 0 o.nly after an infinite time and the po.int x = 0 is a statio.nary Po.int (in §17 we call it a singular Po.int). It is sho.wn belo.w that under these co.nditio.ns the po.int x = 0, if it is neither a centre no.r a centrefo.cus, has a neighbo.urho.o.d separated by trajecto.ries (tending to. this Po.int) into. a finite number o.f secto.rs o.f ten classes (Fig. 40). As is kno.wn ([157], §19), fo.r the system (1) with 1 E 0 1 these secto.rs may belo.ng o.nly to. three classes, that is, it may be elliptic (E), hyperbo.lic (H), o.r parabo.lic (P).
E
H
I(
p
l
Figure
40
In each secto.r o.ne o.f the Po.ints o.f the bo.undary (the lo.wer o.ne in Fig. 40) is a singular Po.int, and there are no. o.ther singular Po.ints o.r po.intwise singularities in the secto.r o.r o.n its bo.undary. The directio.n o.f mo.tio.n o.n all the trajecto.ries may be simultaneo.usly reversed. In each secto.r there passes o.nly o.ne trajecto.ry thro.ugh each interio.r po.int, and in secto.rs E, H, P also. thro.ugh each bo.undary po.int. Secto.rs E, F, G are elliptic (all the trajecto.ries, po.ssibly excepting tho.se go.ing alo.ng the bo.undary o.f a secto.r, tend at bo.th ends to. a singular po.int), secto.rs H, K, L are hyperbolic (bo.th ends o.f all the trajectories, except trajectories go.ing alo.ng the bo.undary o.f a secto.r, leave the neighbo.urhoo.d o.f a singular po.int), secto.rs P, Q, R, S are parabolic (all the trajecto.ries at o.ne end reach a singular po.int and at the o.ther end leave the neighbouro.o.d o.f this Po.int). Each sector is bounded by a simple clo.sed curve which contains a singular po.int and a finite number o.f arcs of trajecto.ries and arcs witho.ut co.ntact intersected by trajectories o.nly in o.ne directio.n. A bo.undary o.f a secto.r E contains o.ne trajecto.ry bo.th ends o.f which tend to. a singular Po.int, a bo.undary o.f each of the o.ther secto.rs co.ntains two. half trajecto.ries which tend to. a singular po.int and are called lateral bo.undaries o.f a sector. In secto.rs F and R these two. half trajecto.ries have a co.mmo.n po.int and clo.se the secto.r. Secto.rs G, L, S are closed by o.ne arc o.f a trajecto.ry with its ends o.n lateral bo.undaries, a secto.r P is clo.sed by a co.ntactless arc, secto.rs K and Q by a contactless arc and by o.ne arc o.f a
Topological Classification of Singular Points
§17
193
trajectory, and a sector H by two contactless arcs and by one arc of a trajectory. In Fig. 40 these contactless arcs are shown by a dashed line. In sectors F, K, Q, R only one boundary trajectory is a linear singularity (or a part of it), whereas in sectors G, L, S both of them are. All the trajectories passing through interior points of the sector flow into these linear singularities (respectively, with one or with two ends). In a sector Q on the arc of a trajectory joining any interior point of the sector to a singular point there is a point of confluence of this trajectory with the linear singularity, and in a sector R there is no such point. In a sector E all the trajectories are loops entering a singular point at both ends, one of each two loops lying within the other. Note also that the whole neighbourhood of a singular point may consist of one sector P, Q, or S. Such a sector will be denoted by Po, Qo, or So. A sector of class So contains a linear singularity of class AA3 and may therefore be encountered only in the cases where the condition 10 , 2, §16, is weakened, for instance as in Lemma 4, §16. Mter we have cut a sector along one of its trajectories, it will have the same structure as a sector P, Q or S. A "centre" type singular point has a neighbourhood filled with closed trajectories surrounding this point and bounded by a closed trajectory. Such a neighbourhood may be interpreted as a sector of class 0 0 • We will prove that the classes of sectors described above are topological. LEMMA 1. In a sector of any of the classes F, R, G, L, S, on a lateral boundary oe, which is a linear singularity, we take a monotone sequence of points
ai, i = 1, 2, ... , convergent to a singular point o. The trajectories joining oc at
a.
the points separate domains Di on the boundaries of which there is a point and no point e. Then the diameter of the domain Di tends to sero as i + 00. PROOF:
sequence
0
domain D i , let Pi be a point farthest from the point o. The a.In isa closed monotone and, therefore, Ipi 
01 = di,
Suppose Pi does not tend to o. Then among the limit points of the sequence Pi there is a point P =1= 0 on which belongs to all the domains Di. Since the part of the line oc contained in Di is an arc oat arbitrarily small at large i then P fj. oe. Through each point of a sector of the considered classes there passes an arc of a trajectory passing within the sector from this point to some point on oe different from the point o. Hence, from a point P there passes an arc of trajectory pq, q E oe, q =1= o. It cannot intersect the trajectory which goes from the point at along the boundary of the domain Di and is, therefore, contained in Di and can reach oe only at points of the arc oa.. Consequently, q E oat for all i, q =1= o. This contradicts the fact that at + 0 (i + 00). Thus, the supposition is not true, and the result follows. LEMMA 2. Let an arc zob o of a trajectory, which goes from the point Zo and lies in the domain of uniqueness of solutions reach a linear singularity I at the point boo Then 1) all the trajectories going from points z of some neighbourhood of the point Zo also reach I at points b = b(z)j
194
Local Singularities of TwoDimensional Systems
2) the point b and the time of motion ously on the point x.
t:llb
Chapter 4
along the arc xb depend continu
PROOF: Through any point ao of the arc xobo we draw a transversal lo (a contactless arc) intersected by trajectories only in one direction. By virtue of Corollary 2 to Theorem 2, §12, all the trajectories going from points x of some neighbourhood VI of the point Xo intersect lo, the intersection point a and the time of motion t:ll4 along the arc xa depending continuously on x. By virtue of Theorem 2, §16, the point bo E I has a onesided (on the side of I where there passes the arc xob o ) neighbourhood U which is topologically mapped onto a rectangle filled with straight lines, images of trajectories. A trajectory T(x) going from any point x of the neighbourhood V2 of the point Xo lies near the arc xob o and enters U. Then it reaches 1 at a point b near the point boo Hence, for x E VI n V2 with points a(x) and b(x) of intersection of the trajectory T(x) with the lines 10 and I are definedj a(x), b(x) and t:ll4 depend continuously on X. IT we take x E 1 then b(x) = x and, therefore, the time of motion tb4 depends continuously on the point b. Then t:llb = t:ll4 + t 4 b (or t:llb = t 4 b  t 4 :1l' depending on the order of the points a, x, b on the trajectory) depends continuously on X.
LEMMA 3. Each of the sectors K and Q can be topologically mapped on a rectangle so that all the trajectories lying within the sector is mapped onto straight lines parallel to a side of the rectangle. PROOF: A sector is bounded by an arc boo of a linear singularity, by arcs of trajectory aob o and co, and by a contactless arc aoc (x = I/;(v), 0 ~ v ~ Vo, c = 1/;(0)) (Fig. (1). By the definition of the sector K (or Q), through each point x of the sector, except a singular point 0, there passes a trajectory, one end of which reaches the arc aoc at a point a, and the other end reaches the arc boo at a point bj and such trajectories have no common points. By virtue of Corollary 2 to Theorem 2, §12, the point a and the time of motion r(x) from x to a (or from a to x) depend continuously on x for x :j: O. This holds also for x E boo, that is, for x = b. Hence, a depends continuously and monotonically (since the trajectories do not intersect) on b, and b on a (by continuity of the inverse function).
DO\
61...Q6_ _ 
Figure 41
Let the motion along trajectories be directed from the arc aoc to the arc boo, and x = !p(t, a) be a solution with the initial condition !p(0, a) = a. Then b = !p(r(b), a). The point b and the number r(b) depend continuously on the point a E aoc, but as a + c we have r(b) + 00 (otherwise, by Lemma 1, §12, the time of motion from c to 0 would be finite), b + 0 (since each point of the arc
Topological Classification of Singular Points
§17
195
boo is reached by a trajectory the other end of which leaves the sector through the arc aoc). For 0 ~ v ~ Vo, 0 ~ u ~ I, let u(v) = 1'(b) be the time of motion from the point a = ,,(v) to the point bE boo, s(tI) = 1 e to, sectors of classes a, L, 8, So cannot exist; in sectors of classes F, F, Q, Q, Qo, Qo motion along a linear singularity is possible only in the direction towards a singular point, and in sectors of classes K, K, R, R only away from a singular point; this imposes new restrictions on the order of sector sequence. Now we consider the case where among linear singularities which end at a given singular point 0 there may also exist secondkind linear singularities, that is, those consisting of singular points. Then a singular point 0 is not isolated. We assume, as before, that in some neighbourhood of this point there may exist only a finite number of pointwise and linear singularities and separatrices, and the linear singularities belong to the classes considered in 2, 3, §16. In this case the above results on the structure of the neighbourhood of a singular point are mainly preserved. A finite number of topological classes of sectors is added which differ from those described in Fig. 40 only in that one or two lateral boundaries are linear singularities consisting of singular points. Circular sectors similar to
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Structurally Stable and Structurally Unstable Systems
205
sectors Po, Qo, So, but with a linear: singularity consisting of singular points, are added also. There appear obvious limitations on the classes of adjacent sectors. 4. Investigation of the isolated pointwise singularity z = 0 of equation (1), which is not a singular (stationary) point, reduces to investigation of the singular point z = 0 of equation (2). Such a point cannot be a centre of a centrefocus since then it would be a singular point also for equation (1). Therefore, by virtue of Theorem 1, its neighbourhood is divided into a finite number of sectors belonging to classes pointed out in 2. THEOREM 3. Let the righthand side of equation (1) be piecewise continuous, let the definition a), §4, be applied, and the pointwise singularity 0 be an isolated and a nonstationary point. Let there exist in some neighbourhood of the point 0 only a finite number of linear singularities. Then this point has a neighbourhood containing exactly two hyperbolic sectors and no elliptic sectors. PROOF: By Theorem 2, §12, for all trajectories in the neighbourhood of the point 0 there holds the inequality (4), §12. Then this neighbourhood does not contain stationary points, closed trajectories, or trajectories both ends of which tend to the point 0, and therefore it does not contain elliptic sectors. The crosssection S (Theorem 2, §12) is a diameter of some circle. Trajectories intersecting the crosssection S cannot enter the point 0 by virtue of (4), §12, and, hence, fill two hyperbolic sectors. These trajectories are separated by trajectories passing through the point 0 (and filling parabolic sectors if there are more than one of such trajectories). There are no other hyperbolic sectors since any trajectory in the neighbourhood of the point 0 must intersect the crosssection S.
§18 Structurally Stable and Structurally Unstable Systems The concepts of structural stability and degrees of structural instability extend to systems of differential equations with discontinuous righthand sides. Necessary and sufficient conditions for structural stability of a system are given. 1. Structurally stable systems are systems which preserve their topological structure under any sufficiently small admissible perturbations (that is, variations in righthand sides of the system). An exact definition will be given below. First we consider examples. The system z = y, iJ = z has a singular point z = y = 0 (a "saddlepoint"), and in the neighbourhood of this point is structurally stable under perturbations of class C 1 ([185J, §9). This implies, in particular, that any system
z = p(z,y),
iJ = q(z, y),
whose righthand sides in a given neighbourhood of the point (0,0) are close enough in the metric C1 (that is, closeness of functions and of their firstorder derivatives) to the righthand sides of a given system, has in this neighbourhood only one singular point; this point is close to the point (0,0) and is also a saddlepoint. The system = x, iJ = y2 is structurally unstable since it has one singular point (0,0), and the system = z, iJ = y2  a 2 , which is arbitrarily close to the
z
z
206
Local Singularities of TwoDimensional Systems
Chapter 4
first one if the number a is small, has in the neighbourhood of the point (0,0) two singular points: (0, a) and (0, a). Structural stability or structural instability of a system may depend on what perturbations are considered to be admissible. For instance, the system :i; = y, y = x is structurally stable under perturbations of class 0 1 • H we treat as admissible those perturbations which are discontinuous on a straight line y = 0 and smooth up to the boundary in each of the regions y < 0 and y > 0, then the same system will be structurally unstable because an arbitrarily close system :i; = :i; =
y+6, y6,
y=x y=x
(y < 0), (y > 0)
(8 > 0 being arbitrarily small) has three singular points (the saddlepoints (0, 6) and (0,6) and the centre (0,0)). In this chapter we treat as admissible these perturbations which are discontinuous on previously given lines. In a finite domain G of a plane we consider a system in vector notation (1)
:i:= I(x)
By means of a finite number of finitelength smooth lines which may have common endpoints, the domain G is divided into a finite number of sub domains Gi , j = 1, .. . ,1 in each of which I, al/ax1, al/ax2 are continuous up to the boundary. On the lines of discontinuity the definition a), §4, is used. Let Of be a class of systems of the form (I), in which the lines of discontinuity of class Op+l (1, §4) are the same for all systems of this class, and the functions I(x) along with the derivatives up to the pth order inclusive are continuous in each of the subdomains Gi up to the boundary. We say that the system (1) and the system
(2)
:i: = i(x)
IIi 
with the same lines of discontinuity are 6close in the metric or, that is, III:" ~ 6 if in each of the sub domains Gi the components of the vectorvalued and their partial derivatives up to the order m inclusive do not function exceed 6 in the absolute value. Next, in §§1820 we consider systems of class or, p ~ 1. Stationary points, at which I(x) = 0 or 10(x) = 0 (the notation is the same as in 2, §16), pointwise singularities (see 1, §16) and all points at which the vector I( x) is tangent to the line of discontinuity, i.e., IN (x) = 0 or (x) = 0, are called singular points. This definition is not purely topological because it exploits the concept of tangency. It is equivalent to the following definition. Stationary points, pointwise singularities, and points capable of bifurcation (this term means a point such that its arbitrarily small neighbourhood may change its topological structure for some arbitrarily small variations of the system) are called singular points. The latter implies that for any m ~ 1 and any arbitrarily small eo, e < eo, and 6 there exists a system of the type (2) 6close (in an eoneighbourhood Uo of a considered point to a given system (1)) in the metric 0:", and such that there
iI
IJ
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207
exists no topological mapping of the neighbourhood Uo which shifts each point by less than e and carries arcs of the trajectories of the system (2) into arcs of the trajectories of the system (1) so that the inverse mapping also carries arcs of trajectories into arcs of trajectories. The equivalence of these two definitions \..f a singular point follows from Lemmas 14. These lemmas provide information on which singularities can and which cannot undergo bifurcations. In these lemmas a "singular point" is a point satisfying the second definition. LEMMA 1. In a domain Gj of smoothness of the function
I the points at which
I(x) '" 0 are not singular. On discontinuity lines of the function I the points at which Iii (x) '" 0, I~ (x) '" 0 (and 10 (x) '" 0 if the function 10 (x) is defined} are not singular. PROOF: Let Xo E Gj, I(xo) '" o. From Theorem 3, §12, it follows that the point Xo is not a pointwise singularity. Let Z = ,p (8) (,p Eel, ,p' '" 0, lsi ~ h) be an arc without contact with the trajectories of the system (1), ,p(0) = Xo, and let Z = ~(t, s) (It I ~ r) be a solution of the system (1) with the initial condition ~(O,s) = ,p(s); let hand r be sufficiently small that these solutions be contained in Gi and ~(t1J Sl) '" ~(t2' S2) for (tlJ 81) '" (t2' S2). H the system (2) is sufficiently close to (1) then z = ,pes) is a contactless arc also for the system (2), and for the solutions of the system (2), z = ~(t,s) with ~(O,s) = ,p(s), we have
I~(t,s)
 ~(t,s)1 < e
(lsi
~ h,
It I ~ r).
A mapping under which a point ~(t, s) is assigned a point ~(t, s) is topological. The structure of the neighbourhood of the point Zo remains unchanged under transition from a system (1) to a sufficiently close system (2). The point Zo is not singular. In the case where the point Xo lies on a discontinuity line, a part of this line is taken as an arc z = ,pes). In other respects the proof is similar. The case where Zo is the endpoint of a discontinuity line (then lii(xo) = I~(zo) i= 0 is reduced to the previous case since the line can be smoothly continued beyond this endpoint. LEMMA 2. A common point Zo of several smooth lines of discontinuity is not singular if it is neither a stationary point nor a pointwise singularity and if for each of these lines of discontinuity we have Iii (zo) '" 0, I~(zo) i= O. PROOF: Let the point Zo not lie on a linear singularity. Then in some neighbourhood of this point there are no linear singularities and on the discontinuity lines we have lii(z) I~(z) > 0, that is, trajectories intersect the discontinuity lines at a nonzero angle. Hence, in the neighbourhood of the point Zo uniqueness is not violated. Neither is it violated at the point Zo because Zo is not a pointwise singularity. It is only in one of the sectors, into which the discontinuity lines separate the neighbourhood of the point Zo, that there exists a trajectory entering the point Zo as t increases, and only in one sector there is a trajectory leaving this
208
Local Singularities of TwoDimensional Systems
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point. The trajectories are not tangent to the discontinuity lines. All this holds also for any system (2) sufficiently close to (1). Through the point Xo we draw an arc, without contact with the trajectories of the system (1) which is not tangent to the discontinuity lines. It will be a contactless arc also for a close system (2). We map each point a E , into itself. The intersection points of the trajectory x = ~(t, a) (~(O, a) = a) of the system (1) with the discontinuity lines will be mapped into the intersection points of the trajectory x = ~(t, a) of the system (2) with the same discontinuity lines. In the neighbourhood of the point Xo we obtain a topological mapping of the boundaries of the sectors (the discontinuity lines and the arc ,). This mapping is continued along trajectories onto each sector. If the trajectory x = ~(t, a) intersects the sector boundaries for t = t1 and t = t2, and the trajectory x = 0 is arbitrarily small. Then at the point zo and hence in some neighbourhood of this point, IN > 0, I"J > OJ that is, in this neighbourhood the trajectories pass through L from a into a+. The structure of the eoneighbourhood of the point Xo has changed and, consequently, the point Xo is singular. Let IN(x) I"J ~ 0 in some eoneighbourhood of the point Xo on L, and at this point IN (zo) I"J (xo) = O. Then in this neighbourhood there are no linear singularities of class AAI (this follows from Lemma 2, §16). We put
(3)
i(x) = I(z)
+ ano
I(x)
= I(x) 
ano
where the vector no is the same as above, lal is arbitrarily small, and a > o if IN(xo) ~ 0, I~(zo) ~ 0, and a < 0 in the rest of the case. Then IN(xo) 1"J(xo) < 0 and the point Xo therefore lies on a. linear singularity of class AA 1 • The structure of the neighbourhood of the point Xo has changed, hence the point Xo is singular. LEMMA 4. A common point Xo of several smooth lines of discontinuity, in the case where for at least one of these lines IN(xo) 1"J(xo} = 0, is singular.
The prool is carried out in the same way as in Lemma 3, but with the following addition. H Xo is not an endpoint of a linear singularity 1 then near the point Xo the line 1 consists of arcs of two discontinuity lines (Li and L,.) and comes with both ends on the boundary of eoneighbourhood of the point Zo at points aj and a,.. H for the line L,. (or Lj) we have IN(x) ~ 0 (Iz  zol ~ eo), 1"J(xo) = 0 then in passing over to the function i(x) = I(x) + ano (see the proof of Lemma 3) part of the line L,. (or Li) is no longer a linear singularity. In a sufficiently small eneighbourhood of the line 1 the system (2) already has no linear singularities which join the points ai and a,. (or points close to them). Then there exists no topological mapping carrying trajectories of the system (l) in the eoneighbourhood of the point Xo into trajectories of the system (2) and shifting each point by less than e. Hence; Zo is a singular point. H for both lines L,' and L,.
(4)
IN (xo) =I: 0,
but for some third line of discontinuity Lm we have I~ (xo) = 0, then near the point Xo the line Lm contains no arcs of linear singularities (otherwise the point Xo would be a. pointwise singularity). Then in passing over to the function (3) on Lm there appears a linear singularity with the end Xo and for a sufficiently small a the lines Li and L,. remain linear singularities by virtue of (4). For the system (2) the point :1:0 is a common point of three linear singularities. The topological structure of the neighbourhood of the point :1:0 changes in passing over to the system (2), and Zo is a singular point. 2. Let A and A be systems (1) and (2). The system A in a domain H is eidentical ([185], p. 41) to the system A in a domain H, that is,
(H, A)
~ (H, A),
Chapter 4
Local Singularities of TwoDimensional Systems
210
provided that there exists a topological mapping of the domain fI onto the domain H under which each point shifts less than bye, and such that trajectories and singular points of the system A are carried into trajectories and singular points of the system A and provided that the inverse mapping possesses the same properties. Such a mapping will be called hereafter an emapping. Let the system A be of class C; in the open domain G, and W be a closed or an open sub domain, We G. The system A is called structurally stable ([185], p. 64) in the domain W if there exists a domain H, W c H c H c G, such that for any e > 0 there exists o > 0 such that for each system A which is oclose to A in C; (G) there exists a domain fI such that
(fI,A) £
(5)
(H,A).
It can be shown that structural stability or structural instability of the system A in the domain W does not depend on the choice of the domain G ::> W. REMARK: Let D c G. If for any e > 0 there exists 0 > 0 such that for each II system A oclose to A in C; (G) there exists a domain D such that (D,A) == (D, A) then the system A is not necessarily structurally stable in D (example in [185], p. 482)' but is structurally stable in any subdomain W which is strictly interior for D (indeed, if we take H = D, the condition (5) is fulfilled). If the system A is structurally stable in W then there exists a domain Hi, containing W strictly within it, such that the system A is structurally stable also in the domain Hi. From the definition of structural stability it follows that if a system is structurally stable in some domain then it is structurally stable in any subdomain. This makes it possible to give the following definition. A trajectory or a part of it (an arc or a point) is called structurally stable if it has a neighbourhood in which the system is structurally stable. An ordinary point (at which f(x) =1= 0) lying within a domain Gi of smoothness of the function f(x) in (1) is structurally stable. Any point or any arc of a smooth line of discontinuity is stable if at this point or on this arc fii (x) fit (x) =1= o (and fO(x) =1= 0 if fii(x) ft(x) < 0). This follows from the proof of Lemma 1. Singular points of "saddle", "node" and "focus" type which lie in Gj are also structurally stable if the matrix of a linearized (at a given singular point) system has Re ).1,2 =1= 0 ([185], §8 and §9). If there is at least one eigenvalue with Re). = 0, the singular point is structurally unstable ([185], p. 75, 103). On a discontinuity line a point may be either structurally stable or structurally unstable even if in its neighbourhood the trajectories are located topologically the same as in the neighbourhood of an ordinary point. For instance, for a system _
(6)
,..,
iJ = 1 (y < 0),
iJ
= x2
(y > 0);
,..,
N
x=l
through each point there passes a single trajectory. In the topological respect the point (0,0) is in no way distinguished. A system
(7)
iJ = 1 (y < 0),
3:=1
§18
Structurally Stable and Structurally Unstable Systems
211
which is arbitrarily close to (6) (for small a =I 0), has a linear singularity, namely, a. line a < z < a, y = 0, at the points of which the trajectories flow together. Hence, the system (7) is not eidentical to the system (6). Consequently, the system (6) is structurally unstable in any neighbourhood of the point (0,0) and this point is structurally unstable. Any point or any arc of a linear singularity of class AB for the system (1) is structurally unstable because the system (2) (arbitrarily close to (1)) with the function of the form (3) has· a linear singularity already of another class, namely, of class AA l • Structurally stable systems can be called systems of zeroth degree of structural instability. Among structurally unstable systems one can successively pick systems of first, second, etc. degree of structural instability. Let Ie ~ 1 and let systems of degrees 0, ..• , (Ie  1) of structural instability be already defined. A system A of class 0:"+1 in a domain G is called a system 0/ leth degree 0/ structural in.stability ([185], p. 217, 338) in a domain W, W c G, if this system A is not of a smaller degree of structural instability in Wand if there exists a domain H, W c H c H c G, such that for any e > 0 there exists 5> 0 such that each system A, 5close to A in the metric O~"+1(G), either has in W a degree of structural instability less than Ie or for this system there exists a. domain ii in which (ii,..4) ~ (H, A). A system A of class O~ has a degree of structural instability 00 in a domain W if any 5 > 0 and Ie there exist systems with a degree of structural instability ~ Ie in the domain W which. are O'close to A in the metric O~lo+l. It follows from these definitions that if a. system has a leth degree of structural instability in a domain W, its degree of structural instability in any subdomain of the domain W is not more than Ie. Let a sequence of domains Go :::> G 1 :::> G 2 • •• contract to a point a and let the system have a leath degree of structural instability in a domain Gi. Then leo ~ le l ~ ~ ~ ••• and there exists a j such that lea = lei for all i ~ j. For each domain G C Gi containing the point a there exists a domain Gi such that a E G. c G C G i and, therefore, the degree of structural instability of the system in G is equal to lei' This number lei is called the degree 0/ structural instability 0/ the point a. Thus, the degree of structural instability of the point a for the system A is the degree of structural instability of this system in each sufficiently small domain containing the point a. The degree of structural instability of a trajectory is defined similarly. The choice of the class of smoothness 0~"+1 in the definition of a system of leth degree of structural instability is determined by the properties of singular points of the type of complicated focus ([185], p. 264) and by the properties of singular points lying on discontinuity lines and their intersections and consisting only of sectors of classes G, L, S, So (2, 3, §17). In the absence of such points the class 0~"+1 can be replaced by the class 0:+1. 3. The number Zo is called a zero 01 multiplicity r of the function I(z) E or (a ~ z ~ .8) if
i
(8)
I(zo) = 0,
r
~
1.
212
Local Singularities of TwoDimensional Systems
Chapter 4
The following lemmas are closely similar to the statements of 3, §1 [185] and are proved using the same techniques. LEMMA 5. Let Xo be a zero of multiplicity r of the function f E C r , !( x) :f. 0 for a ~ x < Xo and for Xo < x ~ /3. Then for anye > 0 there exists 6 > 0 such that any function which satisfies the inequality (for some m ~ r)
i
Iii fllc
(9)
m
< 6,
may have not more than r zeros on the segment [a, /3Jj all the zeros lie on the interval (xo  e, Xo + e)j the sum of their multiplicities is equal to r or is less than r by an even number. LEMMA 6. Let f(x) E CP[a,bj, p
(10)
f(xo)
~ 1,
Xo E (a,b) and
= f'(xo) = ... = f(p) (xo) = o.
Then for any e > 0, 6> 0, m ~ 1, 1 ~ k ~ p there exists a function i{x) E CP satisfying the inequality (9) which coincides with f(x) for Ix  xol ~ e and has on the interval Ix  xol ~ e/2 the zero Xo of multiplicity p + 1  k and, besides, at least k different zeros co. REMARK 1: One can require any given number i ~ k of the zeros Co to lie on the interval (xo  e/2, xo). REMARK 2: The zeros can be made simple, that is, they may have a multiplicity 1.
c,
LEMMA 7 ([64J, p. 248). If f(x) is a continuous function of bounded variation on [a, bJ then for almost all c the function f(x)  c has only a finite number of zeros on [a, bJ. The assertion of the Lemma is true, in particular, for all functions of class C1 [a, bJ. LEMMA 8. Let the function f(x) E CP on an interval [a, b] have infinitely many zeros and let Xo be one of the limit points for the zeros. Then for any 6 > 0 and m ~ p there exists a function i{x) E CP which satisfies (9) and has on [a, bJ only a finite number of zeros, of which the zero Xo has a multiplicity p. PROOF: Applying Rolle's theorem to f, f', .. . , f(p1) , we deduce that f(i) = 1, ... ,p) has on [a, bJ infinitely many zeros with a limit point Xo. At this point there hold the equalities (10). For given 6 and m the function
(i
f(xj a) = f(x)
+ a(x 
xo)P
satisfies (9) and has a zero Xo of multiplicity p for all a E (all 2ad, where a1 > 0 is sufficiently small. By virtue of (10), for some '7 > 0 we have
f(xjad>O
(xo<x~xo+'7),
(1)Pf (Xj ad> 0
The same is true for !(Xj a) for all a E (al' 2ad.
(Xo  '7 ~ x < xo).
§18
Structurally Stable and Structurally Unstable Systems
213
By Lemma 7, there exists a E (aI, 2a1) such that the function
f(x; a) (x  xo)V == f(x) (:z:  :z:o)P + a has only a finite number of zeros on the intervals [a, Xo  '7] and [xo + '7, b]. Then j(z) == fez, a) meets all the requirements of the lemma. 4. When investigating structural stability of a system with smooth righthand sides one must establish whether or not its singular points and closed trajectories are structurally stable and whether there exist trajectories whose both half trajectories are separatrices ([185], p. 165). Structural stability of systems of class cl with piecewise continuous righthand sides are investigated similarly. The conditions for structural stability of singular points of such systems are obtained in [186] (see §19 and §20 below) and we now consider closed trajectories, separatrices, and linear singularities. It turns out that, besides trajectories, one must also consider lines composed of parts of trajectories. We used such lines in 3, §17, to construct a succession function and to determine the topological class of a singular point, the neighbourhood of which consists of sectors G, L, S. We will show that similar composite lines may play the role of separatrices which go from one singular point into another. Singular points are understood in the sense of the definition given in 1. The separatrix of a singular point is defined in the same way as the separatrix of a pointwise singularity in 1, §17. Figure 48 shows a system with two lines of discontinuity (mn and pq) which involves separatrices ab and cd of the points a and d. These separatrices, together with the arc bc of the trajectory, compose a line abcd which goes from one singular point to another and both of whose ends are separatrices. The presence of such a line leads to structural instability of the system. Indeed, by varying arbitrarily little the slope of trajectories between the lines mn and pq, one can obtain a system in which the endpoints band c of the separatrices ab and dc are not joined by a trajectory (Fig. (9). Such a system has already a different topological structure. The system shown in Fig. 48 is, therefore, structurally unstable.
Figure 48
Figure 49
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Local Singularities of TwoDimensional Systems
Chapter 4
The system shown in Fig. 49 is structurally stable if the singular points a and d are structurally stable. In this case the points a and d are joined by the line abecd which consists of arcs of trajectories and contains the part ec of the linear singularity. Thus we are led to the following definition [186]. A line composed of successively located arcs of trajectories is called a polytrajectory if it does not pass through singular points and contains no arcs of linear singularities. The lines abed and a'b'c'd' in Fig. 48 are poly trajectories, whereas the line abe cd in Fig. 49 is not. From this definition and from the existence, uniqueness, and solution continuation theorems it follows that for a system of class through each nonsingular point there passes a single poly trajectory. It is continued either infinitely or up to the boundary of a given domain or up to a singular point. A poly trajectory may intersect linear singularities of classes AA1 and AA 3 • According to the definition given in 1, the other linear singularities consist of singular points, and a poly trajectory, if it reaches such a point, terminates there. A poly trajectory may also be a closed curve without selfintersections. Each nonendpoint of a poly trajectory is either a uniqueness point or a point at which the poly trajectory intersects a linear singularity when passing from one side of it to the other. Two different polytrajectories (neither of which is a continuation of the other) may therefore have in common only the endpoints, which are singular points. A poly trajectory can be tangent to a discontinuity line of the righthand side of a system only at its end (since the tangency point of a trajectory with a discontinuity line is singular; see 1). A trajectory or a poly trajectory, both end arcs of which are separatrices, is called a double separatriz.
0:
LEMMA 9 [186]. A system which is structurally stable in a bounded closed domain W may have only a finite number of singular points in this domain. COROLLARY. Linear singularities of a structurally stable system may belong only to classes AA1 and AA 3 • Indeed, linear singularities of other classes consist of singular points. A structurally stable system has only a finite number of singular points. LEMMA 10 [186]. A system structurally stable in a domain W cannot have double separatrices contained in this domain. PROOF: Suppose a system A of the form (1), structurally stable in the domain W, has a double separatrix ab C W. Let H be such a domain as described in the definition of structural stability, W C H. Then for some Po > 0 a close poneighbourhood V of the separatrix ab is contained in H and contains no other singular points except a and b. Let a point Xo E ab be such that in its neighbourhood f(x} E 0 1 and f(zo} =I O. Then for some p E (O,Po) the angle between the vectors f(x} and /(xo) is less than 7r/4 for Ix  xol < p. Suppose that for the system (2)
(11)
;(x) = f(x)
+ 0;17 (Ix  xol) tI,
Structurally Stable and Structurally Unstable Systems
§18
215
where tI =F 0 is a vector orthogonal to the vector I(xo), the number a > 0 is arbitrarily small, and '7(e) is a function of the class 0 00 (0 ~ e < 00),
Then outside the pneighbourhood U of the point Xo the trajectories and polytrajectories of the system (2) coincide with those of (1), and within U trajectories of the system (2) intersect trajectories of the system (1), all in one direction, for instance, from left to right. Outside U, but in the same domain of smoothness of the function I(x), we draw a transversal S (a contactless arc) through the point P of the arc xob C ab (Fig. 50). Let us consider a polytrajectory T2 of the system (2) which coincides with the separatrix ab of a system (1) on the portion from the point a up to the point of entrance into U. Within U this poly trajectory does not already coincide with ab and intersects the transversal S at the point p(a) which depends continuously on a and tends to P as a  O.
Figure 50 Since ab is a separatrix of the point b then there exists a sequence of arcs aibici (or ~bi)' i = 1,2, ... , of trajectories such that aibi  aob cab,
P(Ci' b) > eo
(or the point bi lies on a linear singUlarity). These arcs, continued beyond the point as polytra,jectories, intersect S at points Pi  p. There exists a sequence ai + 0 (i  00) for which p(ai) = Pi. For a = ai, after the intersection with S at the point p( ai) = Pi the polytrajectory T2 , which is a separatrix of the point a, goes farther along the arc aibic. and out of the eoneighbourhood of the point b (or reaches a linear singularity at the point bi). By the definition of separatrix, on its arc aob there are no points of linear singularities and, therefore, in both cases, for small enough e, the emapping which carries trajectories of the system (1) into trajectories of the system (2) cannot carry the double separatrix ab into the polytrajectory T2 • On the other hand, this emapping must carry the separatrix axo of the point a into a separatrix of the point a. But in a structurally stable system the singular point a may have only a finite number of separatrices (this follows from [186]). Consequelitly, for sufficiently small e the emapping can carry the separatrix axo into no other separatrix except T2 which coincides with axo near the point a. This contradicts the above assertion. Hence, the assumption concerning the existence of a double separatrix is false, and the lemma follows. In structurally stable systems of class it is not only closed trajectories, but also closed poly trajectories that are structurally stable. Let a closed polytrajectory T intersect successively smooth discontinuity lines L 1 , ••• , L,. of the
a.
0;
216
Local Singularities of TwoDimensional Systems
Chapter 4
function f(x). By the property of a poly trajectory, the intersection takes place without tangency. Let, moreover, T intersect without tangency a smooth arc Lo which lies in the domain of smoothness of the function lex) or coincides with Lk. The lines L. are given parametrically by
The poly trajectory T departs from a point x· = 10 (0"0) E Lo and returns onto Lo at the same point. Using the theorem on solution differentiability with respect to the initial conditions and the implicitfunction theorem, one can easily deduce that from any point Zo = 10(0"0) E Lo sufficiently close to z· there goes a poly trajectory lying near T, intersecting the lines L 1 , ••• , Lk at points 1.(0".), i = 1, ... , k, and returning to Lo at a point Xk+l = 10 (O"k+d and that for these intersection points we have the following functions i = 0,1, ... , k.
(12)
A function "'(0"0) = O"k+t{O"k( .• . 0"t{0"0) ... )) E C1 is called a generalized succession function. The poly trajectories do not intersect one another and, therefore, "'(0"0) increases; ""(0"0) =I o. A poly trajectory is closed if an only if "'(0"0) = 0"0. ff a poly trajectory passes through a point common to several discontinuity lines, then this point is not singular (by the definition of poly trajectory). Therefore, in the neighbourhood of this point of vectors fez) are not tangent to discontinuity lines. As before, we can prove that they there exist lefthand tP~(O"o) and righthand "'~(O"o) derivatives of the function tP(O"o). LEMMA 11 [186]. For a closed poly trajectory intersecting an arc Lo at a point
x· = lo (0"0) to be structurally stable, it is necessary and sufficient that (tP~(O"o)
1) (tP~(O"o) 1) > O.
C:
REMARK: By contrast with poly trajectories, trajectories of systems of dass can join together and can pass through singular points. Hence, not all closed trajectories are poly trajectories. The assertions proved for poly trajectories do not always hold for such closed trajectories. For instance, in a structurally stable system a stable limit cycle or a part of it can be a linear singularity of class AA 1. In this case "'(0"0) = constant on some interval 11 < 0"0 < 12. Next, a structurally stable system may have an uncountable set of closed trajectories passing through a singular point. For instance, in a system :i;
= 2 + sgn x  2 sgn y,
iI = 2 + 2sgnx 
sgny.
defined on lines of discontinuity according to a), §4, closed trajectories are closed polygonal lines with three vertices (a, 0), (0,0), and (0, a) for any a > 0 (Fig. 51).
Singular Points on a Line 0/ Discontinuity
§19
217
Figure 51 THEOREM 1 [186J. For a system (1) of class C; to be structurally stable in a closed bounded domain, it is necessary and sufficient that it has no double separatrices, that it have only a finite number of singular points and closed polytrajectories (if they exist), and that all of them be structurally stable. For systems of class C1 a similar assertion is proved in [185J (p. 165).
§19 Singular Points on a Line of Discontinuity Singular points and pointwise singularities on a line of discontinuity of the righthand sides of a system of two differential equations are examined. Analytical criteria for ascribing singular points to one topological class or another are established. All structurally stable points and points of first degree of structural instability and some other points are specified. Bifurcation of singular points is analyzed. 1. To investigate a system, the righthand sides of which are discontinuous on a smooth line, one can make a smooth transformation under which this line is mapped into a segment of the zaxis. We therefore further consider systems with righthand sides discontinuous on the zaxis. In a domain G separat~d by the zaxis into parts G (y < 0) and G+ (y> 0), we consider a system
(1)
a: = P(z, y),
iI = Q(z, y).
Let P, Q e C!, k ~ 1. This implies that
P=P(z,y),
Q=Q(z,y)inG, P,Q e CIo (G),
P = P+(z,y), Q = Q+(z,y) in G+, P+, Q+ e CIo (G+).
Along the segments ofthe zaxis, where Q(z, O)Q+(z, 0) at a velocity
(2)
~
0, motion is possible
iI = O.
The function PO(z) is defined and belongs to C k everywhere on these segments except possibly at the points where
(3)
218
Local Singularities of TwoDimensional Systems
Chapter 4
We assume also that
(4)
pO(x) pO(x)
=
p (x, 0), = P+(x,O),
if Q(x,O) = 0,
Q+(x,O)
=1= 0,
= 0,
Q (x, 0)
=1=
if Q+(x,O)
0.
These conditions are fulfilled, in particular, under the definition a), §4. In this case
(5) 2. Let us investigate cases where the functions Q(x,O), Q+(x,O), and pO(x) can vanish only at isolated points. According to 1, §I8, only these points are singular. There exist six types of such points 1187] characterized by the following conditions (the values of all the functions are taken at a given point (c,O) of the xaxis. 1. QQ+ < 0, PO(c) = 0. 2. Q+ = 0, Q =1= 0, p+ =1= or Q = 0, Q+ =1= 0, p =1= 0. 3. Q = Q+ = 0, p =1= 0, p+ =1= 0. 4. p+ = Q+ = 0, Q =1= or p = Q = 0, Q+ =1= 0. 5. Q = Q+ = and only one of the functions P, p+ equals 0. 6. P = Q = p+ = Q+ = 0. For each of these types we investigate possible arrangements of trajectories near the point (c,O). Type 1. Let at the point (c, 0)
°
°
°
(6)
°
(the case Q < 0, Q+ > 0, pO = is reduced to the case (6) by replacement of t by t in the system (1)). In the neighbourhood of the point (c,O) we have ltil > constant> 0, Ixl < constant (for y =1= 0), therefore from some smaller neighbourhood of this point all the solutions reach the xaxis within a finite time. After this they remain on the xaxis and satisfy the system (2). Since x = c is an isolated zero of the function Po (x) three cases are possible for small Ix  cl (under the condition (6)). Ia. In the case (xc)PO(x) < (x =1= c) the solutions on the xaxis approach the point x = c from both sides. This point is a stable node (Fig. 52). Example: x = x, ti = sgny. lb. In the case (x  c)PO(x) > (x =1= c) the solutions on the xaxis move away from the point x = c on both sides. This point x = c is a saddlepoint (Fig. 53). Example: x = x, ti =  sgn y. Ic. In the case where PO(x) has the same sign for x < c and for x > c the solutions on the xaxis on the one side approach the point x = c and on the other side move away from it. This point is a saddlenode (Fig. 54). Example: = ~}, iJ =  sgn Thus, type 1 consists of three topological classes.
° °
x
y.
219
Singular Points on a Line of Discontinuity
Figure 5B
Figure 59
Figure 54
Class 1a is a node consisting of sectors QQQQ (the notation is the same as in 3, §17). Class 1b is a saddlepoint of sectors KKKK. Class 1c is a saddlenode of sectors KQQK. The topological equivalence of singular points belonging to the same class follows from Theorem 2, §17. If instead of isolation of singular points we require finiteness of the number of linear and pointwise singularities, then the function PO(x) can vanish both at some isolated points and on whole intervals. These intervals are linear singularities of class AA 2 , and their ends are pointwise singularities. In Fig. 55 the point a is a seminode, the point b is a semisaddle. Type 2. Let at the point (c, 0)
Q+ =0,
Q >0,
(7)
p+ >0
(the other cases are reduced to this by change of variables). In some neighbourhood of this point
iI > const > 0 for
11
< 0,
:i; > const > 0 for 11 >
o.
From the part of the neighbourhood where 11 < 0, solutions meet the xaxis. For y > 0 the following cases are possible: 2a. If (x  c)Q+ (x, 0) > 0 then into each point (x,O), where c  0 < x < c, there comes one trajectory from the region 11 > 0 and from each point (x,O), where e < x < e + 0, there departs one trajectory into the region 11 >" o. The limit of these trajectories is a trajectory passing through the point (e,O) and lying in the region y > 0 (Fig. 56). Along the closed interval e  0 ~ x ~ e of the xaxis there passes a trajectory for which :i; = pO(x) > 0 by virtue of (4). Example: :i;=O,
iJ = 1 (y < O)i
:i:
= 1,
y=x (1/>0).
2b. If (x  e)Q+(z,O) < 0 then trajectories depart from the points (x,O), where e  0 < z < e, into the region 1/ > 0 and come from this region into the points (x,O), wherec < z < c + o. Since near the point (c,O) for 11 > 0 we have :i; > const > 0, Iill < const, the above trajectories intersect some segment z = c, 0 ~ 11 ~ 1/1. Trajectories that come onto the xaxis with both ends pass through the points of this segment sufficiently close to (e,O) (Fig. 57). Along
220
Local Singularities of TwoDimensional Systems
Chapter 4
the closed interval c ~ x ~ c + S of the xaxis there passes a trajectory for which x = pO(x) > 0 by virtue of (4). Example:
x=o,
y=
1
(y < 0);
x= 1,
y = x (y> 0).
2c. If Q+(x,O) > 0 for 0 < Ix  cl < S then all the trajectories in the neighbourhood of the point (c, 0) pass from the region y < 0 into the region y> 0 (Fig. 58). Example:
:i: = 0,
y=
1
(y < 0);
:i:
= 1, Y= x 2
(y> 0).
2d. If Q+ (x, 0) < 0 for 0 < Ix  cl < S then all the trajectories in the neighbourhood of the point (c,O) join the trajectory that lies on the xaxis (Fig. 59). The xaxis is a linear singularity of class AA 1 • Example:
x = 0,
y=
1 (y < 0);
Figure 55
:i: = 1,
Y=
_x 2
Figure 56
Figure 58
(y > 0).
Figure 57
Figure 59
Thus, Type 2 consists of four topological classes: 2a (the neighbourhood of a singular point consists of sectors HQQH), 2b (sectors KK), 2c (sectors HH), 2d (sectors KQQK). In all these cases the point (c,O) is not stationary since it follows from (7) and (4) that if there exist solutions on the xaxis, they are such that :i: = pO(x) > o. This is, in particular, what distinguishes class 2d from class lc. If instead of isolation of a singular point we require finiteness of the number of pointwise and linear singularities then the function Q+ (x, 0) can vanish on whole intervals. They are linear singularities of class AB, and their ends are pointwise singularities (Figs. 6063, classes H K, HQQK, HQH, K QK). Type 3. At the point (c,O)
(8)
p
f.
0,
p+
f. o.
Singular Points on a Line of Discontinuity
§19
Figure 60
Figure 61
Figure 62
Figure 69
64
Figure 65
Figure 66
Figure 67
Figure 68
Figure 69
Figure
Figure 70
221
Figure 71
Therefore, in each of the halfneighbourhoods of this point the arrangement of trajectories can be the same as in the upper half neighbourhood of the point (c, 0) in any of Figs. 5659. Thus, there exist the following possibilities (Figs. 6471). Each figure corresponds to several topological classes of arrangement of
222
Chapter 4
Local Singularities of TwoDimensional Systems
trajectories depending on whether p and p+ are of the same or of opposite signs, and in some cases depending on the signs of pO(x) on both sides of the point (c,O). Under the definition a), §4, in the case P P+ > 0 the point (c,O) is nonstationary whereas in the case P p+ < 0 it is stationary. In Fig. 64, in the case P p+ < 0 we have a saddlepoint (class H H H H) and in the case P p+ > a quasisaddle (class QHQQHQ) with a trajectory lying on the xaxis. In Figs. 65 and 70, in the case P p+ > 0 the points do not differ topologically from ordinary ones (class HH), and in the case P p+ < o each of these figures presents three classes (GQHQ,LKHK,SQHK, and QQQQ, KKKK, KQQK, respectively) subject to the direction of motion along the xaxis on both sides of a singular point. In Fig. 66, in the case P P+ > 0 we observe one topological class, QHHQ, while in the case P p+ < 0 there are two classes HKKH and HQQH. In Figs. 67 and 68, in the case P p+ < 0 there are classes Po (focus) and 0 0 (centre), and in the case P p+ > 0 there are classes LL(,p) with ,pes) =F s (quasifocus) and with ,pes) == s (quasicentre)j the cases of centrefocus are not considered here. In Fig. 69, in the case P p+ > 0 we have class KK and, in the case P P+ < 0, classes QQ and KK. In Fig. 71, in the case P p+ < 0 we are dealing with class H H, and in the case P p+ > 0 there is no pointwise singularities and we are dealing with class KQQK. Each of these classes consists of one or several topological classes depending on whether the trajectories on the xaxis enter a singular point within a finite or an infinite time. There exists a total of thirtynine topological classes (in the case of analytical functions P, Q, P+, Q+ there are twentyfour classes). Type 4. In one half plane trajectories reach the xaxis without tangency, and on the boundary of the other half plane there exists a stationary point. This may be any stationary point admissible for a system of class C l • Type 4 therefore contains an infinite set of topological classes. The simplest of these are enumerated in 5. Type 5. In one half plane the vector (P, Q) is tangent to the xaxis at a given point, and the disposition of the trajectories is the same as in the region y > 0 in any of Figs. 5659. In the other half plane the point under consideration is a stationary point of an arbitrary type. Type 6. A given point is stationary both for the system = P, if = Qand for the system x = P+, if = Q+. The picture of the arrangement of trajectories near this point is obtained by gluing together two pictures of half neighbourhoods of any stationary points. Each of the types 5 and 6 contains an infinite set of topological classes. We will show that structurally stable points occur only in types 1 and 2, and singular points of first degree of structural instability occur only in types 14.
°
x
LEMMA 1 [188]. Isolated singular points of types 3 and 4 are structurally un
stable. PROOF: Let (c,O) be an isolated singular point of type 3 of a system A of the form (1), i.e., at this point the conditions (8) are fulfilled. We will consider, for any Cl close to c, a system A obtained from A by replacing Q+(x, y) by the function Q+ (x, y)Q+ (Cl' 0). In some 2eneighbourhood H of the point (c, 0) the system A has only one singular point, namely (c,O). The system A, arbitrarily
Singular Points on a Line of Discontinuity
§19
223
close to A if IC1 cl is sufficiently small, has two singular points, (c,O) and (Cl' 0), '"
~
c
in the eneighbourhood of the point (c,O) if ICl  cl < e. Therefore, (H, A) ~ (H, A) for any domain Hj hence, the point (c,O) is structurally unstable. Let (c,O) be a singular point of type 4 of the system A, and at this point p+ = Q+ = 0, Q :f:. 0, and let there be no other singular points in its 2eneighbourhood H. Take an arbitrarily small Yl > 0. The system A, obtained from A by replacement of P+(z,y) and Q+(z,y) by the functions P+(z,y)P+(C,Y1) and Q+(z,y)  Q+(C,Y1), is arbitrarily close to A and has a singular point (c, Yl). For Y > the system A belongs to class 0 1 , and therefore no trajectories enter this singular point within a finite time. In the system A one trajectory from the region Y < enters the singular point (c,O) within a finite time, that is, the singular point lies on this trajectory. For any domain H we
°
°
'"
_
c
have therefore (H, A) ~ (H, A), and the point (c,O) is thus structurally unstable. LEMMA 2 [188]. An isolated singular point of type 5 cannot have its degree of structural instability less than 2, nor can one of type 6 have its degree less than 3. PROOF: Let the point (c,O) be a point of type 5 of the system A and let, at this point, Q = Q+ = P+ = 0, P :f:. 0. In the neighbourhood of the point (C1,0), for C1 sufficiently close to c, the system A obtained from A by replacement of the functions P+(z,y), Q+(z,y) by
has two singular points: (c,O) of type 2 and (Cl,O) of type 4. The system A is therefore not eidentical to the system A. By Lemma 1, the system A is structurally unstable, and therefore the system A cannot have its degree of structural instability lower than 2. Let a point (c, 0) in the system A be of type 6. The system A obtained by the replace~ent (9) has two singular points of type 4: (c,O) and (C1,0). Hence, the system A is not eidentical to the system A. Shifting the singular point (C1' 0) into the region y > 0, as in the proof of Lemma 1, we obtain a system A· which is not eidentical to the system Aand has a singular point (Cl' yt} and the structurally unstable singular point (c,O). Therefore, the system A· is structurally unstable, the system A cannot have its degree of structural instability less than 2, nor can A have its degree less than 3. 3. We shall indicate all structurally stable and structurally unstable isolated singular points of types 1 and 2, and also some of their bifurcations. The system (1) in the domain G is assumed to satisfy the conditions of 1. On a line of discontinuity the definition a), §4, is used. LEMMA 3 [1871. If (c, 0) is a singular point of type 1 (or 2) of the system (1) then in some neighbourhood of this point any sufficiently close system (in the metric O!) can have singular points only of the same type.
°
PROOF: The result follows from the fact that inequalities Q(z, O)Q+(z, 0) < for type 1 and Q(z,O) :f:. 0, P+(z,O) :f:. for type 2 are also valid in the
°
Local Singularities of TwoDimensional Systems
224
Chapter 4
neighbourhood of the point (c, 0) i they hold for all sufficiently close systems also. Let only a finite number of singular points (Ci,O), i = 1, ... , m (m ~ 0) of a system A of the form (1) exist on a closed interval T (a ~ x ~ b) of the xaxis. Let all of these points be of type 1, and the points x = a and x = b be nonsingular. Then the signs of the functions Q and Q+ on this interval remain unchanged, and in some neighbourhood V of this interval we have IQ(x, y) I > const > 0 for y #: O. A closed domain in V bounded by four arcs of trajectories coming from the endpoints of the segment and by two straight line segments y = ±h (Fig. 72) is called a domain of type 1.
Figure 71! LEMMA 4 [187J. Let us consider a system A in a domain H of typP 1. Por any e > 0 there exist numbers 5 > 0 and '7 > 0 with the following property. Let a system A be aclose to A in the metric C!, and let, for the system A, the function PO(x) have zeros only at points Ci, Ic.  e.1 < '7, i = 1, ... , m, on the segment [a, bJ. Then there exists a domain H such that (H, A) == (H, A). PROOF: Let a
= Co < Cl
< ... Cm
0 and q ~ p there exists a function Pl(Z) with a zero Cj of multiplicity rj  1 and a nro Cj ~ Cj in V. This function coincides with PO(z) outside V and is such that II P 1  pOliot < 6'. By virtue of (5) there exists a system A which is 6close to the system A in ct and is such that for it the function PO(z) coincides with P1(Z). The system A has singular points (c;,O). i 1 ..... m. and (Cj.O). They are more numerous than in the system A. The system A is not. therefore. eidentical to the system A. In the case n = 0 this contradicts the assumption that the system A is structurally stable.
=
226
Local Singularities of TwoDimensional Systems
Chapter 4
In the case n > 0 the degree of structural instability of the system A is n, consequently for the system A it is not greater than n  1. By the inductive hypothesis, for the system A the sum S, similar to (11), is not greater than n  1. But for the system A the function has Zeros c" i = 1, ... m, of multiplicities r, ~ (i 't i), rj rj  1 and a zero Cj. Hence, for the system A
r,
m
=
pO
m
by virtue of (14). The contradiction shows that the inequality (14) is impossible. Suppose PO(z) has an infinite set of zeros on (a,b). By Lemma 8, §18, there exists a function Pl(Z) arbitrarily close to PO(z) and having on [a, bj only a finite numb~r of zeros, of which at least one, Zo, is of mUltiplicity ro = n + 1. There exists such a system A, close to A, for which PO(z) Pl(Z). Then A is not eidentical to the system A, the degree of structural instability of A is not greater than n  1, and by the inductive hypothesis, S :;;; n  1. This is in contradiction with ro = n + 1. Thus, for the system A of nth degree of structural instability, the function PO(z) has only a finite number of zeros and cannot possess the properties (14). This implies that the zeros have multiplicities and
=
r,
S
= (rl  1) + ... + (rm  1) :;;; n.
In the case S < n, by the inductive hypothesis the system has a degree of structural instability less than n. Hence, S n.
=
COROLLARY 1. For a system (1) of class 0: (k ~ s + 1) a singular point of type 1 has the degree of structural instability s if and only if at this point the function pO(x) has a zero of multiplicity 8 + 1. COROLLARY 2. If for a system (1) of class Or;" on some closed interval of the xaxis Q(x,O)Q+(x,O) < 0, PO(x) == then this interval is a part of a linear singularity of class AA2 (2, §16) and in any neighbourhood of this interval (or any of its points) the system has the degree of structural instability (Xl.
°
REMARK: From what has been said it does not follow that any physical system which has a whole interval consisting of equilibrium positions may lose it under arbitrarily small perturbations. Consider an oscillatory system described in the secondorder equation (15)
x+bsgnx+ g(x) = I(x,x),
where the function I, 9 E 0 1 • Any perturbations of these functions lead to perturbations only 01 the second equation of the system
(16)
:i;=y,
iJ = I(x, y)  g(x} 
b sgn y,
to which equation (15) is reduced. IT on the xaxis there exists a segment on which lI(x,O)  g(x)1 :;;; b it consists entirely of equilibrium positions. The part of the segment, where I/(x,O)  g(x) 1 :;;; b  e, is preserved under any perturbations of the function 1  9 which do not exceed e. Such a segment is a structurally stable singularity·
§19
Singular Points on a Line of Discontinuity
227
under perturbations of the second equation in (16) or of equation (15), that is, under perturbations of the physical system. By Corollary 2, under perturbations of both equations (16) this singularity has degree of structural instability 00 if f, g E 0 00 • But such perturbations are physically meaningless. Thus, in certain cases one should consider structural stability of a system under perturbations only of some equations of the system. In [51 (§3.1 and §3.2) many physical systems are considered which have an infinite set of equilibrium states occupying a whole segment in the phase space. Stability of these equilibrium states is investigated. In [1891 it is stated that for nonholonomic mechanical systems the case is typical where a set of equilibrium states is a manifold whose dimension is equal to the number of nonholonomic constraints. Bifurcations of singular points of type 1 are fully determined by bifurcations of the zeros of the function pO(z), having regard to the signs taken by this function on both sides of each zero. Hz = c is a simple zero for the function PO(z), the function PO(z) changes sign when z passes over this zero, and therefore the point (e,O) belongs either to class 1a or to 1b (Figs. 52, 53). By Corollary 1 to Theorem 1, this point is structurally stable, that is, it is not subject to bifurcations under small (in 0 1 ) variations of the system. Hz = c is a double zero of the function PO(z), the function PO(z) does not change sign when z passes over the zero. The singular point (e,O) therefore belongs to class 1c. Under small (in 0 2 ) variations of the function pO(z) three cases are possible: the zero of multiplicity 2 is preserved (possibly, it shifts along the zaxis); the zero disappears; the zero splits into two zeros of multiplicity 1 (Fig. 73). Therefore, under
Figure 73 a small variation of the system, a singular point of class 1c of first degree of structural instability (Fig. 54) either preserves its topological class or vanishes (in the case there remains a linear singularity of class AA 1 , Fig. 30) or splits into two structurally stable singular points of classes 1a and 1b (Fig. 72). Considering bifurcations of a triple zero of the function pO(z), one can obtain all possible bifurcations of singular points of type 1 of second degree of structural instability. Now consider a system A of the form (1) with singular points of type 2. Let, for instance, on a given segment T of the zaxis (17)
P+(z,O) > 0,
Here and below the inequalities for PO(z) refer to those points and intervals where the function PO(z) is defined. In the case (17) at the endpoints of each
228
Local Singularities of TwoDimensional Systems
Chapter 4
°
such segment we have Q+(x,O) = 0, and therefore by virtue of (4) PO(x) > at the endpoints and, accordingly, on the whole segment. What follows concerns a neighbourhood V of the segment T, in which for some IJ >
°
(18) Let H c V, let there exist in H a finite number of singular points (by virtue of (17) they are of type 2), and let H be a closed domain whose boundary is a simple closed curve which consist of a finite number of arcs of trajectories and arcs without contact, does not pass through the singular points, has only two common points (a,O) and (b,O) with the xaxis, and intersects the xaxis at these points (Fig. 74). The endpoints of these arc of trajectories will be called angular points. The points (a,O) and (b,O) must not lie on the chosen arcs without contact. The domain H must contain neither separatrices joining a singular point with an angular point nor arcs of trajectories lying within H (but not on the xaxis) and joining two angular points. Such a domain is called here a domain of type 2.
Figure 74 A separatrix of a singular point (c, 0) of the system A with the condition (17) will be called a left (right) separatrix if near this point on the separatrix x < C (x> c) and y > 0. LEMMA 5 [187]. Let, in a domain H of type 2 for a system A, there hold the conditions (18) and let there exist a finite number of singular points (Ci,O), i = 1, ... , n. Let a system A, be a 6close (in C1) to the system A, let it have only singular points (d., 0), i = 1, ... , n, and let 1) Id.  cil < 1'/ and the point (do,O) be of the same class as (Ci,O), i =
1, ... ,n;
2) from the point (do, 0) there departs a double separatrix (contained in H) of the system A which is left or right for the point (di' 0) if and only if from the point (c.,O) there departs a double separatrix (contained in H) of the system A, which is, respectively, left or right for this point. Then for ~ny e > there exist 6 > 0, 1'/ > such that (H, A) ~ (H, A) in some domain H. PROOF: Let Cl < C2 < ... Cn and 11 be so small that dl < d2 < '" < d n • If a double separatrix 8im of the system A passes from the point (c.,O) to the point (Cm,O), m > i, then by the
°
°
assumptions of the lemma, from the point (d.,O) there passes a right double separatrix S., of the system J... into some point (dl,O), 'I > i. We shall show that 1= m. Suppose, for instance, that I < m. From the point (Ct, 0) the left separatrix 8 jl of the system A goes to some point (cjo 0). The separatrices 8im. and Sjl cannot have common points for 1/ ~ 0; therefore i < i < I. From the point (dj, 0) the right separatrix Sjlo of the system J... goes to some point dlo, i < i < k < I since Sjlo and Sil cannot have common points. From the point (Clo, 0) there departs a left separatrix of the system A, etc. We obtain a sequence of
§19
Singular Points on a Line of Discontinuity
229
nested segments [e;, Cm] ::> [ci' e,l ::> ••• of the z.axis; the endpoints of each segment are joined by a separatrix of the system A. There is a finite number of singular points and, therefore, the process must ultimately stop. This contradicts the condition 2) of the lemma. Construct a domain H. Draw traJectories of the system A from the points (CI,O), (b,O) and from one of the endpoints of each remaining arc L; of the tn,jectories of the system A which pass along the boundary of the domain H (or from points suftlciently near those points) up to intersection with the same arcs without contact (or with arcs close to them) on which the arcs L; end. For a sufficiently smaH 6 the intersection points exist and are close to the endpoints of the arcs L;. These points and the points of intersection of separatrices and boundary trajectories with boundary arcs without contact (and with the zaxis) are arranged on these arcs in the same order for the systems A and A if 6 and '1 are sufficiently small. The drawn arcs of the trajectories of the system A and the arcs without contact bound the domain H of type 2 for the system A.
Figure 75
The trajectories of the system A (or ~, which pass through singular and angular points, and the zaxis, separate the domain H (or H) into elementary tetragons «~571, p..J6), generalized elementary tetragons Ri' and elementary segments T" (respectively, Ri and T,,) (Fig. 76). From what has been said about the order of the points of intersection between trajectories and arcs without contact it follows that for sufficiently small 6 and '1 these elementary domains in Hand H are arranged in a similar manner. For each domain Hi or T" one can construct an emapping (2, §18) onto a corresponding domain Ri or Tot (as for the elementary tetragons in [185], pp. 5153, and for the sectors of classes K, Q, L in Lemmas 3 and 4, S17). Mappings of adjacent elementary domains can be made coincident on their common boundary. To this end, the first to be mapped are those segments of a line of discontinuity, which are arcs of tra· jectories, then the domains Hi and Tot adj scent to these segments, and then all the remaining domains. Thus, we obtain an emapping of the domain H onto H. THEOREM 2
[1871. Let for a system A E
~
of the form (1)
W be a domain of type 2 intersected by the zaxis along an interval a ~ z ~ b. For the system A to have degree of structural instability h (0 ~ h ~ p  1) in the' domain W, it is necessary and sufficient that the function Q+ (z, 0) on [a, b1 have only a finite number of .eros Cl, ••• ,en (n ~ 0) and that their multiplicities rl, ... , rn satisfy the condition S = h, where (19) 8
S
= h  1) + ... + (rn  1) + 8,
is a number of double separatrices lying in the domain W.
PROOF: Sufficiency is proved by induction with respect to 8, and necessity is proved by induction with respect to h. Let an integer q ~ O. For q > 0 we assume that for all domains of type 2 sufficiency is proved for all 8 < q, and necessity for all h < q. SUfficienc/l. Let S q. Then by the inductive hypothesis the system A cannot have its degree of structural instability less than q. Consider a domain H ::> W which possesses the same properties as Wand contains no other singular points and separatrices. Take any e > 0 and a smaH '1 > 0 such that 31l.neighbourhood of each zero C; contain neither other zeros nor the points CI and b. Let a system A (2: 15(Z,II), iJ Q(z, II)) be 6cJose in C!+l to the system A;
=
=
=
Local Singularities
230
0/ TwoDimensional Systems
Chapter 4
=
let the number 6 < I' (see (18» be sufficiently small that sgnQ+(z,O) sgnQ+(z,O) outside the ,,neighbourhoods of all ,eoints Ci. Ca.tJe 1. The function Q+(z,O) has exactly one zero di in the ,,neighbourhood of each point Ci. Then for the system A there hold inequalities similar to (18), but with I'  6 instead of 1', and the signs on the left and on the right of each zero are the same at Q+(z,O) and at Q+(z,O). Then the condition 1) of Lemma 5 is fulfilled. If a left (or a right) double separatrix (contained in H) of the system A does not leave the point (Ci,O) then for small 6 the same is true also for the point (di,O) and the system A (because the solution depends continuously on the initial data and on the righthand side of the system). a) If, moreover, the condition 2) of Lemma 5 is satisfied then by this lemma (H,A) :.
(H,A). In particular, if S 0 then all ri I, 8 O. This implies that we are in Case 1 and the system A has no double separatrices. Then (H, A) :. (H, A), and the system A is structurally stable. b) If the condition 2) of Lemma 5 is not satisfied then by virtue of what has ben said, for small 6 the system A has fewer double separatrices than the system A. By Lemma 5, §18, the multiplicity of each zero di is not higher than the multiplicity of the zero Ci. Hence for the system A the sum S of the form (19) is less than q. Ca.tJe 2. In the neighbourhood of each point Ci there is not more than one zero of function Q+(z,O), whereas in the neighbourhood of some point Cj there is no zero of this function. By Lemma 5, §18, the Cj has multiplicity rj ~ 2. In passing over from the system A to the system A, the summand rj  1 ~ 1 is therefore discarded from (19) and as in Case 1 the rest of the summands do not increase. The number of double separatrices does not increase since they may cease to be double, or two (left and right) separatrices of the point Cj may merge into one. Then for the system A we have S < S = q. Ca.tJe S. In the neighbourhood of the point Ci there exist ni ~ 0 zeros dij (j = 1, ... , ni; i 1, ... , n) of the function Q+(z,O), where ni ~ 2 at least for one i. If for a certain i we do not have a left (right) double separatrix of the system A going from the point (Ci,O) then for a sufficiently small 6 we do not have a left (right) "external" double separatrix of the system A, going from each of the points (di;' 0), i 1, ... , ni i.e., there does not exist a separatrix entering a singular point (dkl,O), k :j: i. Indeed, if they did exist, external left separatrices of the points (di" 0), would lie for a smaH 6 in the small neighbourhood olthe left separatrix of the point (Ci' 0) of the system A, and by the assumption, this separatix passes by other singular points at a positive distance. On the interval Ii (Ci  ",Ci + ,,) the function Q+(z,O) either changes sign once, at the point Ci, or does not change sign at all. The number of sign reversals from "negative" to "positive" of the function Q+(z, 0) on this interval exceeds that of the function Q+(z, 0) by a number Pi ~ 0, and from "positive" to "negative" also by a number Pi; the number of zeros, passing through which the function Q+(z,O) does not change sign, is greater by ni  1 2Pi. According to I, under the condition (17) the point of sign reversal from "negative" to "positive" (or from "positive" to "negative") is a singular point of class 2a (2b). Such a point is reached by two Jor, respectively, by no) separatrices from the region 1/ > O. A lI:ero, in passing through which Q+(z,O) does not change sign, is a point of class 2c or 2d reached by one separatrix. Possibly, not all of these separatrices are double. Thus, if ni ~ 2, the number of endpoints of double separatrices in the interval Ii can increase by not more than 2Pi + (ni  1 2Pi) = ni  I, whereas for ni = 1 and ni :: 0, as in Cases 1 and 2, the number of such endpoints does not increase. Each double separatrix has two ends at singular points, and in passing over from the system A to the system A the number of all double separatrices can therefore increase by not more than a number Bl = 2:. (ni  1); summation is carried out only over those i for which ni ~ 2. The sum of multiplicities of the lI:eros of the function Q+(z, 0) in the interval Ii is equal to ri ~ ri (Lemma 5, SI8). In passing over from the system A to the system A, the sum 2:(ri  1) in (19) is replaced by the sum
=
=
=
=
=
!
R
ni
L L(rij i=1 j=1
n
1)
= L(ri  nil, i=l
Singula.r Points on a. Line of Discontinuity
§19
231
= E·(n. 
which is less than the sum ~)r..  1) at least by a number B2 1). Therefore in passing over from the system A to the system A the sum (19) acquires an increment ,.,
s  s ~ Bl 
B2
'\"" • = '21 LJ (ni 1).
The last sum is positive since in Case 8 at least one n .. ~ 2. The numbers Sand S are integers and hence. S  S ~ 1. that ia. for the system A we have S ~ q  1. Thus. for the system A in Cases lb. 2. 3 we have S < q. For the system A we construct a domain H of type 2 which has no singular points and double separatrices other than those lying in H. so that H c H. By the induction assumption. the degree of structural instability of the system A in and. accordingly. in H. is not greater than q  1. Taking into account Case la. we conclude that the degree of structural instability of the system A in the domain W is q. N«:e.nt,. Let the degree of structural instability of the system A in the domain W be q. Then there exists a domain H of type 2 such that W lies exactly within H. and the degree of structurally instability of the system A in H is q. and H has only those singular points and double separatrices which belong to W. Suppose that there exist numbers Ci E (a. b) and Pi ~ 1 such that for the function R(z) == Q+(z. 0) we have
11
(20)
(; = 0.1 •...• Pi 
(21)
(Pl  1) + ... + (Pn  1) + B ~ q + 1.
1; i
= 1•...• n).
where B is the number of double separatrices in the domain W. For some e > 0 and an arbitrary 6 > 0 we construct a system 1° the system A is 6close to A in the metric C:+ 1 ;
,..,
.
A such that
,..,
2° (H. A) ~ (H. A) for any domain H. if We H; 8° there exist numbers d .. E (a. b) and P.. ~ 1 such that for the function R(z) we have
== Q+(z.O)
(;=O.l ..... p.. l; i=I ..... k).
(22)
(Pl  1) + ... + (ps.  1) + i ~ q.
(28)
where i is the number of double separatrices of the system A in the domain W. Let, > o. Let tp E CPo tp(Z.II) > 0 in a small neighbourhood Vl of the point (Zl.1I1).1I1 > O. which lies on one of the double separatrices. tp 0 outside Vl. The system A obtained from A by replacement of the function Q by Q Q + >.tp is close to the system A for sufficiently small>' > O. has the same singular points (di Ci) and the numbers Pi Pi. but it has one double separatrix fewer than the system A. as in Lemma 10. §lS. Such a system A has the properties 1°8°. Let, O. Then. by virtue of (21). Pi ~ 2 for some i. Let the segment I· ICi  v. ci + vI c (a. b) contain no singular points c... i:# i. By Lemma 6. §lS. for any 6 > 0 there exists a function R(z) which has in I· a r;ero ci of multiplicity Pi Pi  1 and a r;ero ci '¢ c· and coincides with R(z) outside I· and is such that
= =
=
=
=
=
=
(24)
IIR(z)  R(z)llo9+ 1 < 6.
Then for sufficiently small 6 the system
(25)
Q(z.lI)
=
A obtained from A
= Q(Z.II) + R(z) 
by replacement of Q(x.y) by
R(z).
has the properties 1° and 8° for CS .. Ci (since Pi = Pi. i :# i. Pi = Pi  1). If the system A has a finite number of singular points. the system A has an extra singular point (ci.O) and possesses the property 2°.
232
Local Singularities of TwoDimensional Systems
Chapter 4
If the system Ahas infinitely many singular points, that is, the function R(z) has infinitely many zeros on [o,bl, then by Lemma 8, §18, there exists a function H(z) which satisfies (24) and has only a finite number of zeros on [0, bl, of which one zero Cl is of multiplicity p. Then the system A, obtained similarly to (26), has a finite number of singular points and possesses the properties 1° and 2°. Taking i Ie I, Pl p in (22) and noticing that, by assumption, p 1 is greater than or equal to the degree of structural instability of q, we obtain (23). Hence, the system A has the property So. Thus, in all cases there exists a system A with the properties 1°3°. In the case q 0 this contradicts the structural stability of the system A. In the case q ~ 1 for a system A sufficiently close to A, we construct a domain if of type 2 as in Lemma 6. We can make We if c H. It follows from 1° and 2° that the system A in the domain H, and accordingly, in the domain if has degree of structural instability less than q. Then, by the inductive hypothesis, the seros di of the function H(z) q+(z, 0) have finite multiplicities Fi, and
= =
=
=
=
(26)
(Fl  1) + ... + (Fj:  1) + i < q.
Since by virtue of (22) Fi ~ P., the inequality (26) contradicts (23). In a1l the cases of existence of the system A with the properties 1 0 _3 0 leads to contradiction. Hence, the assumption (20), (21) is false, i.e., the zeros Ci have finite multiplicities ri, and the sum (19) is equal to q (in the case S < q the degree of structural instability of the system A would be less than q).
COROLLARY 1. For a system (1) of dass C~(p ~ q + 1) the point (c,O) is a singular point of type 2 and has degree of structural instability equal to q if and only if at this point Q f; 0, p+ f; 0, and Q+ (x, 0) has a zero x = c of multiplicity q + 1 {or Q+ f; 0, P f; 0, and Q (x, 0) has a zero x = c of multiplicity q + 1). COROLLARY 2. If for a system (1) of class
C:,
on some interval of the xaxis
then this interval is part of a linear singularity of class AB, and in an arbitrarily small neighbourhood of this interval the system has infinite degree of structural instability. PROOF: By Corollary 1, the system obtained from (1) on replacing the function Q(x, y) by the function Q(x, y) = Q(x, y) + A(X  c)P (A > 0 being arbitrarily small) in the neighbourhood of the point (c, 0) has degree of structural instability p  1, the number p being arbitrarily large. Bifurcations of singular points of type 2 are determined by bifurcations of the zeros ofthe function Q+ (x, 0) (or Q (x, 0)), by the signs of this function on either side of each zero and by the disposition of separatrices. Let Q > 0, p+ > o. H x = c is a simple zero of the function R(x) == Q+(x,O) then the singular point (c,O) is of class 2a (if R'(c) > 0) or 2b (if R'(c) < 0) (Figs. 56 and 57); it is structurally stable and does not undergo bifurcations. H x = c is a zero of multiplicity 2, the function R(x) does not change sign and the singular point (c,O) is of class 2c (if R"{c) > 0) and of class 2d (if R"(c) < 0) (Figs. 58 and 59); it has first degree of structural instability. For small variations of the function R(x) in the metric (J2 the zero of multiplicity 2 may be preserved, may vanish or split into two simple zeros as in Fig. 73.
§19
Singular Points on a Line of Discontinuity
233
Therefore, under small (in 02) variations of the system a singular point of class 2c can either be preserved or vanish (in this case there remains such a singularity on the line y = 0 as in the case AAo, 2, §16, Fig. 29) or split into two structurally stable points of classes 2a and 2b (Fig. 76) joined by the line on which trajectories join together, i.e., by a linear singularity of class AA 1 • A singular point of class 2d also can either be preserved or be transformed into a nondistinguished point of a linear singularity of class AAl (Fig. 30), or split into two singular points of classes 2a and 2b; the line on which trajectories join together becomes discontinuous (Fig. 77).
Figure 76
Figure 77
If :r; = c is a triple zero of the function R(:r;) then the singular point (c,O) is of second degree of structural instability and belongs to class 2a (if Rill (c) > 0) or to class 2b (if Rill (c) < 0). Considering bifurcations of a triple zero, we obtain all possible bifurcations of such a singular point. For RIII(C) > 0, for systems close in 0 3 the following cases of the presence of singular points are possible: 1) one point of class 2a; 2) two points of classes 2d and 2aj 3) two points of classes 2a and 2c; 4) three points of classes 2a, 2b, 2a arranged in that order (here three topologically different arrangements of separatrices are possible). For RIII(c) < 0 the following cases are possible: 1) one point of class 2bj 2) two points of classes 2c and 2b (Fig. 78); 3) two points of classes 2b and 2d (Fig. 79)j 4) three successively ordered points of classes 2b, 2a, 2b.
Figure 78
Figure 79
REMARK: The concept of degree of structural instability introduced in 2, §18, changes essentially if in the definition of eoidentity of the systems A and A we do not require that singular points be mapped into singular points (or if we do not regard as singular those points in the neighbourhood of which trajectories are arranged topologically the same as in the neighbourhood of ordinary points or as in the neighbourhood of nonendpoints of a linear singularity, Figs. 58, 59). For instance, at bifurcation of the singular point considered the last (for R"'(c) < 0) in the cases 1),2),3) the system obtained is topologically equivalent to the initial one since singular points of classes 2c and 2d are topologically equivalent to nonsingular points (cf. Figs. 57, 78, 79) and in the case 4) the
Chapter 4
Local Singularities of TwoDimensional Systems
234
system obtained is structurally stable. Therefore, under the definition of eidentity modified as above, (or under the modified definition of a singular point) one would have to ascribe to such a singular point with R(e) = R'(e) = R"(c) = 0, RIII(e) < 0 the first degree of structural instability. This would complicate the formulation and the proof of Theorem 2. Moreover, for such singular points the degree of structural instability would not then coincide with codimension, whereas they do coincide for the remaining singular points considered here. 4. Among singular points of type 3 the "sewed focus" 1 (Fig. 67) has been investigated most thoroughly ([4J, p. 393; [190]). Let, for the system (1) of class e:" m ~ 2, at the point (0,0).
p < 0,
(27)
p+ >0,
and in the neighbourhood of this point
(0 < Ixi < po).
(28)
Under these conditions, for a sufficiently small Xo > 0, a trajectory from a point (xo,O) passes into the region y < 0, intersects the xaxis at a point (X1,0), Xl < 0, passes into the region y > 0 and goes back to the xaxis at the point (X2'0), X2 > 0. Instead of the succession function X2 = /(xo) it is more convenient to consider ([4J, p. 396) the functions
(29)
Xo
= 0' (Xl),
X2
= 0'+ (Xl),
X(xt)
= X2 
Xo
= 0'+ (Xl) 
0' (xt).
If X(xt) > 0 on some interval 01 < Xl < 0 then the focus is unstable (Fig. 80); if X(Xl) < 0 then it is stable (Fig. 81), and if X(xt) == 0 then the singular point is a sewed centre (Fig. 68). If on each interval of the form 01 < Xl < the function X takes both zero and nonzero values, the singular point is a sewed centrefocus.
°
Figure 80
Figure 81
In order to examine the properties of the functions 0'+ (x) and 0' (x) it is convenient to define them for X ~ 0, assuming 0'*(0) = 0, 0'+(X2) = Xl, O'(xo) = Xl, where X2 > 0, Xl < 0, Xo > are the same as above. Then the functions 0'* (x) are defined for P1 < X < P2, P1, P2 > 0, and
°
0'+(0'+ (x)) == X,
(30)
LEMMA 6. If P, P+,
Q, Q+
E
0' (O'(x)) == x.
em,
m ~ 1, at the point
tions (27) are fulfilled, and the derivatives Q; < 0, em for Ixl < 01, 01 > O. PROOF: For y
~~ =
(0,0) the condiX(x) E
Qt < 0 then 0',0'+,
> 0 we have from (1)
F(x, y)
F=
~: E em
(y ~ 0),
F.,(O,O) = 2rP < O.
lOther terms used are: "fused focus," "merged focus" and "stitched focus."
Singular Points on a Line of Discontinuity
§19
235
Putting y = f}2z2, we derive the equation 2f32z dz/dx = F(x,z2), from which we pass over to the system
dz dr
(31)
1 = 2f32
(
2_
F X, z )
= H ( x, z ) ,
Since HeC rn (x 2 +z2 < 5~), H(x,z) = [185J (p. 252).
LEMMA 7. If u(x)
e Cl, u(O) =
dx dr = z.
x+o(lxl+lzl) the result follows from
0, u(O'(x))
u'(O) = l.
= x,
xO'(x) < 0 (0
0 and y < 0 satisfy the equation (its coefficients are different for y> 0 and y < 0) dy dx
(34)
Q(x, y)
2
2
= P(x, y) = ax + by + ex + dxy + ey + rp(x, y),
rp(x, y) E C 2, rp(x,l/) = 0(x2 + y2). If P, Q E ct then we can write (35) We seek for a solution which goes from the point (p, 0). For this solution Y = O(p2). Making, in (34), the change 2y = aYl and then 2x + bYl = 2X1' we obtain x
= O(p),
(36)
Yl ddxl
= 2Xl + 3Ax~ + 0 (x~ + IYll) ,
Integrating from p to Xl and taking into account the fact that obtain (writing x instead of xd
yd p)
= 0, we
(37) The function (37) vanishes for X and cancelling X + p, we obtain
= p and for x = O'(p),
Equating (37) to zero
(38) Since we seek for the zero x = 0'( p) = O(p) then from (38) we have X  P = O(p2). Substituting this again into (38), we get (39) In the case where the function rp in (34) has the form (35), we similarly obtain
From (29) and (39) we derive for systems of class
(41)
C;
Singular Points on a Line of Discontinuity
§19
and for systems of class
C!
x(p) = a2p2 + asps as = (A+)2  (A)2,
(42)
237
+ a4p4 + O(p4), a4 = K+  K.
Here A+, K+ (A, K) are expressed by formulae (39) and (40) through the values of the coefficients a = a+ < 0, b = b+, ... from equation (34) in a region 1/ > (respectively, a = a > 0, b = b, ... in a region 1/ < 0). They can also be expressed immediately through the values of the functions P, Q and of their derivatives at the point (0,0), for instance,
°
(43)
Knowing the coefficients a2, a4, ••• of the function X( p), one can investigate stability of a sewed focus and its bifurcation. THEOREM 3. Let the conditions (27) and
Q;(O,O) < 0,
(44)
°
Q~(O,O) < 0.
°°
° °
be fulfilled. If a2 < or a2 = 0, a4 < or X(k) (0) < in (33) then the zero solution is asymptotically stable. If "2 > or a2 = 0, a4 > or X(k) (0) > 0 in (33) then the zero solution is unstable. In the case of stability the trajectory reaches the point (0,0) only after an infinite time, and solutions tend to zero with a characteristic exponent
(45)
 1 ln Vz2 (t) 'Y = lim t ..... oo
t
~ + 1/2 (t) = , 21"
1 1"
1
= P(O, O) + P+(o, 0)"
PROOF: A trajectory passing through the point (p,O) intersects the semiaxis Oz, z> 0, at points Zo = q(p) and Z2 = q+(p). IT X(p) < (0 < p < Po) then Z2 < Zo (Fig. 81) and after that the trajectory intersects this semiaxis at points Z4 > Z6 > ... > O. There exists .lim Z2i ~ 0. Since, as in (29),
°
...... 00
(46) it follows that as i . 00 we have X(Z2Htl . 0, Z2H1 . 0. Hence Z2i = (Z21+1) . and on the trajectory we have z(t) . and the asymptotic stability follows. IT X(p) > (0 < p < Po) then we similarly find that z(t),y(t) . as t decreases, and the zero solution is unstable. Let us estimate the speed at which the solutions approach zero in the case X < 0. By virtue of (32), Z2H1/ Z2i . 1 as i . 00, hence for any e > from (41) and (46) we have for all i ~ i1(~) (1'
° °
°
°
°
238
Chapter 4
Local Singularities of TwoDimensional Systems
Since lnx ~ x 1 (0 < x < 00), (47) The time of motion along the trajectory from the point (X2., 0) to the point t. = 2TX2.(1 + 0(1)), where T is the same as in (45). Summing (47) over i from some i = i(e) to any k, we obtain (X2H2' 0) is equal to
k
Tk =
Lt•.
'=i
From this, taking account of boundedness of dyjdx, (45) follows. We now consider bifurcations of a sewed focus under the conditions (27) and (44). In passing over from the functions P, Q to the functions P, Q, which are close to P, Q in C!, the singular point either remains isolated (if the functions Q (x, 0) and Q+ (x, 0) have a common zero) or splits into several singular points (if the zeros of these functions are distinct). The former case is considered in [4J (p. 398) and in [190J, and the latter one is analyzed in 5 below. In the former case, the conditions (28) are preserved after the origin is placed at a singular point. IT a2 =1= a in (41) then by virtue of (43) under small (in C:) variations of the functions P, Q, the sign of a2 is preserved and by Theorem 3 the singular point remains a stable (if a2 < 0) or an unstable (if a2 > 0) focus. Let P, Q E C! and a2 = 0, a4 < O. Then the singular point is a stable focus. Under small (in C!) variations of the functions P, Q the sign of a4 is preserved. Two cases are possible. IT it turns out that a2 ~ 0 then the focus remains stable. IT a2 > 0 then the focus becomes unstable and around it a stable limit cycle appears (Fig. 83).
Figure 89
Indeed, for small P the function (42) is negative in the case a2 = 0, a4 < 0 and has a zero P = 0 of multiplicity 4. For a4 < 0, and sufficiently small a2 > 0 the function remains negative for some p. < 0 and P" > 0 but becomes positive for small Ipl. Consequently, it vanishes for some PI < 0 and P2 > 0, and through the points (PI, 0) and (P2, 0) there passes a closed trajectory. By Lemma 5, §18, the sum of the multiplicities of the zeros of the function (42) (with a4 =1= 0) in the neighbourhood of the point P = 0 is 4 at most. Since P = 0 is a double zero then PI and P2 are simple zeros. Hence, the closed trajectory is a limit cycle. It is stable because as Ipi increases, the function X changes sign from "positive" to "negative."
§19
Singular Points on a Line of Discontinuity
239
5. By Lemma 1, singular points of types 3 and 4 are structurally unstable. We will select from them, points of first degree of structural instability. Suppose that P, Q e O~ and that the velocity of motion on the intervals of the xaxis where Q(z,O) . Q+(x,O) ~ is determined by formula (5). The proofs are briefly presented. Let (0,0) be a singular point of type 3, that is, at this point
°
(48)
P ",0,
Q =Q+ =0,
p+",o.
LEMMA 1 O. If (0,0) is a singular point of type 3 which has a nrst degree of structural instability then it is isolated and at this point
(49)
Q; (0, 0) '" 0,
Qt(O,O).", 0.
PROOF: Let the point (O,O}be a limit point for singular points (x.,O), that is, for points at which either PO(x.) = or Q(x.,O) = or Q+(x.,O) = 0, i = 1,2, .... The functions pO and f(x) are defined in (5). We will consider a system A obtained from a system A of the form (1) by taking the function
°
°
(50) instead of the function Q+(x,y). For the system A we have 1 then the singular point (Fig. 64) is either a saddle (for P p+ < 0) or a quasisaddle with a trajectory going along the xaxis (for P p+ > 0).
°
COROLLARY 2. If at the point (0,0) we have Q; P > 0, Q;t P+ < then near this point all the trajectories come onto the xaxis at both ends. If there are no closed curves composed of arcs of trajectories then the singular point (Fig. 67) is a focus (for P P+ < 0) or a quasifocus with the trajectory going along the xaxis (for P P+ > 0).
°
COROLLARY 3. If at the point (0,0) we have Q; Q;t P P+ > then for 11 0 the trajectories are convex on the same side (Fig. 65). In the case P P+ < there exists a trajectory going along the xaxis.
°
In all the three cases (Corollaries 13) near the point (0,0) there exists a trajectory going along the xaxis only if
Q;(O, O)Q;(O, 0) < 0.
(53)
At the point (0,0) it follows from (5) and (48) that
(54)
/'(0) = P+(O,O)Q;(O,O)  P(O,O)Q;(O,O).
In the cases 1 and 2 under the condition (53) we have /(0) = 0, 1'(0) :F 0, Q; :F Q;t, hence pO(x) in (5) does not change sign near the point x = OJ that is, the motion along the xaxis on both sides of the singular point proceeds in one direction. In the case 3 this holds if
that is, if at the point (0,0) the curvatures of the trajectories of the two systems (51) are different (this follows from (52)).
C:
THEOREM 4 [188]. If a system A is of class and at the point (0,0) the conditions (48) are satisfied then this point has a first degree of structural instability if and only if the following conditions are satisfied: 1° Q;t(O,O}:F 0, Q;(O,O):F 0. 2° If at the point (0,0) we have Q; P > 0, Q;t p+ < then at this point
°
PI
+ Qt _ Q;t., _ P., + Q; P+ 2Q% P
+
Q;., 2Q; 
:F a2
° •
Singular Points on a Line of Discontinuity
519
241
30 If at the point (0,0) we have P p+ < 0, Q; Q;!' < 0, then at this point
Q;
Q;!'
a) p_ :f. p+' o= 2 for Q; p+ > 0,
(55)
b) Q; ..J. OQ;!' p_ r p+' o= 1/2 for Q; p+ < O.
PROOF: SujJicienc". Let the conditions (48) and 1°_3° be fulfilled. Then in some neighbour
hood U of the point (0.0) the functions P .P+. Q;. Qt (and in the case Pp+ < 0 also /'(z)) do not change sign and exceed a constant" > 0 in absolute value. For /'(z) in the case 3° ~his follows from (66) and in the case pp+ < o. Q;Ql >,..,0 fr2m !,!i4). For _any system A sufficiently close in C~ to the system A. the functions p.P+.Q;.Qt (and /' if pp+ < 0) in U have the same signs as p.P+.Q;.Qt (and I'). Hence the system A may have singular points in U only of types 1. 2. and 3 (or only of types 2 and S if P p+ > 0). Since :j:. o. :j:. 0 (and :j:. 0). each of the functions q(z.O). q+(z.O) (and i(z) if pp+ < 0) in U has only one. necessarily simple. zero (Zl.Z2.Z0. respectively). If Zl Z2 then i(zl) OJ hence Zl Z2 zo and there exists in U only one singular point 0 of the system A. At this point the functions p:l:. q~ (and i') satisfy the same inequalities as p:I:. Q~ (and I'). The singular points (0.0) and 0 belong therefore to the same topological class. In the neighbourhoods of these points one can prove aidentity of the systems A and A. In the case Zl :j:. Z2 we have j(Zl) :j:. o. i(Z2) :j:. O. hence Zl :j:. Zo :j:. Z2. The system A has in U two singular points (ZI.0) and (Z2.0) of type 2 (and one point Zo of type 1 ifPP+ < 0). These points are structurally stable (Theorems 1 and 2) because the zeros z 1. Z2. Zo are simple. a) If P Q; < O. p+ Qt > 0 at the point (0.0) then the Bame ineqUalities hold for the system A in U. The tr~ectories are convex towards the zaxis (Fig. 84). The singular points (ZI.0) and (Z2.0) are of class 2a. (zo.O) is of class lb. Structural stability of the system A is proved as in Lemma 6. b) If PQ; > O. P+Qt < 0 then the tr~ectories of the systems A and A are concave towards the zaxis. The singular points (Zl. 0) and (Z2.0) are of class 2b. (zo.O) is of class 1a. By virtue of the condition 2°.112 :j:. 0 in (41). Hence. «1+)"  «1)":= X" :j:. O. and the graphs of the functions (1+ and (1 lie as shown in Fig. 82. Under small (in C~) variations of the system A the functions u+ and (1 change little in C 2 (this can be obtained by applying the results from [186). p. 262 to the system (31)). Then for the system A close to A the graphs are either also tangent to one another or have no common points or intersect at two points; that is. the equation X(z) = 0 has either a double zero or no zeros or two simple zeros (Lemma 6. §18). In the first case the system A is aidentical to the system A. In the second and third cases the trajectories of the system A are shown in Figs. 86 and 86 if P < O. p+ > O. 112 < 0 (for different signs the direction of motion along tr~ectories in one or two half planes is different; for P p+ > 0 the tr~ectorie8 on the zaxis lie only outside the interval (Zlo Z2)). Simple teros of the function X correspond to a structurally stable limit cycle. Double separatrices are absent. and the structural etability of the system A follows. c) If P p+ > o. Q;Qt > 0 at the point (0.0) then in U all the tr~ectories of the systems A and A are convex to one side. The system A has only two singular points (Zl.0) and (Z2. 0) of classes 2b and 2a (Fig. 87). Structural stability of the system A follows as in the case a). d) If P p+ < o. Q;Qt < 0 at the point (0.0) then in U all the trajectories of the systems A and A are convex to one side. For the system A the zaxis is a trajectory. the motion along it on both sides of the point z 0 has one direction by virtue of the condition SOa). The system A has two singular points (Zl. 0) and (Z2.0) of type 2. classes 2a and 2b. and one
q;
qt
=
i'
= =
=
=
Chapter 4
Local Singularities of TwoDimensional Systems
242
Figure 85
Figure 86
Figure 87
Figure 88
singular point (zo,O) of type 1 (Fig. 88). Closed trajeetories are absent. A separatrix of a sing'llar point of dass 2a may go into a singular point of type 1. We will show that under the eondition 3 0 b) this is impossible for systems A sufficiently dose to A. By virtue of (48) for trajectories of the system A we have h±
= Q~(O,O)
p±(O,O)
'10
,
'"I).
=
the positive or negative sign is taken for II > 0 (II < 0); O. Let (12 O. Then for the system A we have X(z) 0(z2) by virtue ofj41) and (4S). For the system A, which is obtained from the system A by the substitution of P+(z,SI) P+(Z,SI) + ~Z for P+(Z,SI), by virtue of (4S) the function X is equal to
=
=
"'() X Zjl'
(58)
=
21' 2 (2) = SP+(O,O)Z +0 Z .
If (0,0) is either a centre or a centrefocus for the system A then for any I' '# 0 this point is a focus for the system A by virtue of (58). If in some neighbourhood of the point (0,0) there are no closed tr~ectories of the system A then X(zt) '# 0 for an arbitrarily small ZI > 0, for instance, X(Zl) > o. There exists a small I' '# 0 such that X(Zl, 1') > 0 and for some Z2 e (0, ZI) we have X(Z2'~) < 0 by virtue of (58). Then i(zs, 1') 0 for some Zs e (Z2,ZI). Through the point (zs,O) there passes a closed tr~ectory of the system A. In both cases the systems A and A are not IIidentical for small II. For the system A the point (0,0) is of type S, hence, the system A is structurally unstable. The system A and its singular point (0,0) cannot, therefore, have a first degree of structural instability. So Let P pf < 0, Q;Q't < 0 at the point (0,0). If the condition So a) of Theorem 4 is not fulfilled then "(0) 0 (Bee (54». Let the system A be obtained from A by taking the function
=
=
instead of the function Q:!::(z, SI). The system A is close to the system A if a is small, and for it q+(O, O)q(O, 0) a 2p+(O,O)P(O,O) < 0, j(O) 0, j'(O) O.
=
=
=
Consequently, for the system A the singular point (0,0) is a structurally unstable point of type 1 (Theorem 1). It is topologically different from a singular point of type 3 under the conditions PP+ < 0, Q;Q't < 0 (the neighbourhood of this point consists of other sectors, see 2). Let the condition 3° b) be not fulfilled and, for instance, Q'tP+ > O. We will construct such a system A with a double lIeparatrix which is arbitrarily close to the system A. For this purpose we take 01+ p+ 0, O. The motion along the xaxis therefore proceeds only in a half neighbourhood (x < 0 or x > 0) of the point x = O. According to (5), if 1'(0) < 0 then this motion and the motion along the trajectory of the system A  , which passes through the point (0,0) for y ~ 0 (see (51)), are both directed either towards the point (0,0) or away from this point and if 1'(0) > 0 then one is directed towards this point and the other is directed away from it. In the case .6. < 0, the singular point (0,0) of the system A+ is a saddlepointj in the case 0 < 4.6. < 0'2 it is a node, and in the case 0 < 0'2 < 4.6. it is a focus. This reasoning gives the following eight topological classes of singular points. 1) If .6. < 0, Q~(O,O) =1= 0 then in the case 1'(0) < 0 we have the class HQQH (Fig. 89), and in the case 1'(0) > 0 the class HKKH (Fig. 90). 2) If 0 < 4.6. < 0'2, Q~ (0,0) =1= 0, then in the case I' (0) < 0 we have the classes Q PQ for O'Q (0, 0) < 0 (Fig. 91) and H P FQ for O'Q (0, 0) > 0 (Fig. 92), and in the case I' (0) > 0 we have the classes K P K for O'Q (0, 0) < 0 (Fig. 93) and HPRK for O'Q(O, 0) > 0 (Fig. 94). 3) If 0'2 < 4.6. then in the case 1'(0) < 0 we have the class QQ (Fig. 95), and in the case 1'(0) > 0 the class KK (Fig. 96). If a singular point is a node or a saddlepoint then ([157], p. 186) trajectories entering this point are tangent at this point to eigenvectors of the matrix (61)
M= (a !) = (P~ Q",
c
P~) ",=y=O
Qy
A nonzero vector (u, v) is an eigenvector if and only if (62)
au + bv
= .Au,
cu + dv = .AV,
that is, if
(63)
(au
+ bV)v = (cu + dv)u.
If c =1= 0 then the eigenvectors have the form (kl' 1) and (k2' 1), where kl and k2 are roots of the equation
(64)
~(k)
== ck 2 + (d  a)k  b = O.
LEMMA 12 ([185], §9). Let (0,0) be a singular point 01 a system A + E a!, and .6. =1= O. Then a) there exist p > 0, 00 > 0 such that for 0 < 0 < 00 any system A· oclose in a 1 to the system A+ has in a circle Kl (x 2 + y2 ~ p2) exactly one singular point (xo, Yo), where xo = 0(0), Yo = 0(5); b) if .6. < 0 or 0 < 0'2 =1= 4.6. > 0 then there exists 51 > 0 such that for o < 5 < 51 the singular point (:to, Yo) of the system A* is of the same type (node, saddlepoint, focus) as the point (0,0) of the system A+j
Singular Points on a Line of Discontinuity
§19
Figure 89
Figure 90
Figure 91
Figure 9£
Figure 99
Figure 94
245
!J
Figure 9S
Figure 96
Figure 97
c) if I:l. < 0 or 0 < 41:l. < 0'2 then for anye > 0 there exist Pl(e), 6'2(e) such that for 0 < P < Pl(e), 0 < 6' < 6'2(e) there exist smooth lines L 1 , L 2; each of these smooth lines consists of such two half trajectories of the system A· which enter the singular point (xo, Yo), and divides the circle K «z xO)2 + (y YO)2 ~ p2) into two parts (Fig. 97); at any point the tangent to L" i = 1,2, forms an angle less than e with the eigenvector (t4, tli) of the matrix (61). REMARK: IT Q;t :/: 0 then instead of the statement concerning the angle one can use the following: on L,
(65)
LEMMA 13.
(66)
IdZdy  kil < e ,
i = 1,2.
Let a system
z=
az+ by,
y= cz+dy
have a "focus"type singular point. Then the trajectory of the system tangent to the straight line y = 1 has a derivative dx/dy = ko at a first point (6,0) of
Local Singularities of TwoDimensional Systems
246
Chapter 4
its intersection with this straight line; O"(a  cko ) > 0, r 1 c[ck~+(da)kob] n 20" (a  ckO)2
+ arctg (2.6 a  cko
 0"
)
0" = 271"sgnO"  arctg, r
(67)
r
= V4bc
 (d a)2 > O.
PROOF: Dividing the first equation from (66) by the second one, we obtain a homogenous equation. We solve it by means of the substitution x = uy. An integral curve passing through the tangency point y = 1, u = d/ c is separated into arcs, on each of which the integration constant is determined separately. At the intersection point with the straight line y = 1 we obtain an equation for Uj expressing U through dx/dy = ko, we obtain (67).
[188]. Let a system A E C; and let the conditions (60) be satisfied at the singular point (0,0). In order that this point have a first degree of structural instability it is necessary and sufficient that the following conditions be satisfied: 1° Either .6 < 0 or t:.. > 0, 0" f: 0, 0"2 f: 4.6. THEOREM 5
2°
Q:t (0, 0) f: o.
3° P:(O,O)Q(O,O) f: P(O,O)Q~(O,O). 4° If.6 < 0 and (63) with U = P(O,O), v = Q(O,O) is fulfilled, then the condition
(68) must hold. 5° If 4.6 > 0"2 and for ko = P (0, O)/Q (0, 0) (67) holds, where a, b, c, dare as in (61), then the condition O"Q (0, 0) < 0 must be fulfilled. Nece,litll. For the condition 2°. necessity is proved as in Lemma 10. Let the condition 2° be fulfilled and the condition 1° be not fulfilled. If t:.. 0 or t:.. > O. (1 0 then the point (0.0) for the system A+ is structurally unstable ([185]. Theorems 11 and 15). We shift this point to the point (0.'1). that is. replace the functions P+(Z.II) and Q+(z.lI) by P+(Z.II) P+(Z.II") and Q+(Z.II) = Q+(Z.II ,,). The system so obtained is structurally unstable and, as shown at the end of the proof of Lemma 1, is not eidentical to the system A. Hence, the system A cannot have a first degree of structural instability. (Here and below we do not dwell on the choice of the domain H for which (H,..4) ~ (H, A) because it is obvious.) If (12 4t:.. > 0 then for the system A + the point (0,0) is a node. By virtue of 2°. the vector (1,0) does not satisfy (64), hence infinitely many trajectories from the region II > 0 enter the point (0,0). We make an arbitrarily small variation of the system, so that 4t:.. > (12. The point (0,0) becomes the focus of the system A +, and there will not be a trajectory from the region II > 0 entering this point. One trajectory will enter from the region II < 0 and one along the zaxis (see Figs. 95 and 96). The system so obtained is not eidentical to the initial system A and is structurally unstable by Lemma 1. Hence, the system A cannot have a first degree of structural Instability. Let the condition 8° fail. Then for the function /(z) from (5) we have 1'(0) = O. For the system A obtained from the system A through replacement of the functions P+, Q+ by PROOF:
=
=
=
=
P+(z,lI)
= P+(Z,II) 
aP(O,O)  azP;(O,O),
Q+(Z,II)
= Q+(z, II) 
aQ(O,O)  a 2 zQ;(O,O),
Singular Points on a Line of Discontinuity
§19
for an arbitrarily small a
247
> 0 we have i(o) = i'(O) =
o.
For the system A the point (0,0) is a singular point of type 1. It is structurally unstable by Theorem 1. Near this point QQ+ < 0, hence tr~ectories join together on both intervals p < z < 0 and 0 < z < p of the zaxis, whereas for the system A trajectories join together only one one interval because q(z, O)q+(z, 0) changes aign by virtue of 2° and (60). Then the singular point (0,0) of the system A and the singular point (0,0) of the system A are topologically different. Hence, the system A cannot have a ftrst degree of structural instability. Let, as in 4°,11 < O. Then for the system A+ the point (0,0) is a saddlepoint and its separatrices are tangent to the vectors (ki' 1), ki (i 1,2) being the roots of equation (64). For the system A +
=
(69)
z =P+(Z,II) =P+(Z,II) 
P+(O,,,),
iI =Q+(Z'II)
q+(O, ,,).
= q+(z, II) 
For small" > 0 the point (0,,,) is a saddlepoint by Lemma 12. Let (63) with u P(O,O), v q(O,O) hold. Then (u,v) is an eigenvector of the matrix (61). It is collinear with one of the vectors (ki' 1), for instance, with the vector (ki' 1). By Lemma 12 one separatrix T of the system A+ intersects the zaxis at a point (ZI,O), and at this point
=
=
(70) where 211 + 0, 1£(") :! 0 as " + O. _ Let the system A coincide with the system A + for II > 0 and for II < 0 let it be obtained from the system A through replacement of the function P(Z,II) by the function P(Z,II) P (Z,II) + v, where v is determined from the equality
=
(" +
0),
q
= q. Then a~ the point ,,(ZI' Ok. fC?! the system A the vectors (p+, Q+) and (P, q) are collinear and /(ZI) = p+q  Pq+ = O. If ~ is an eigenvalue of the matrix (61) which corresponds to the vector (u,v) (P(O,O),
aEd l~
=
q(O,O» then by virtue of (62) the lefthand side of (68) is equal to ~q(O, 0). If the condition (68) does not hold then ~q(O,O) > O. If ~ > 0 then q(O,O) > 0 and the motion along the separatrix T is directed away from the saddlepoint (0,,,) towards the point (ZIl0), hence Q+(ZI,O) < O. Then for small" and ZI we have q+(Zl,O)Q(Zl,O) < O. Since i(zI) 0, (Zl'O) is a singular point of type 1 of the system A and T is a double separatrix. The same holds in the case ~ < O. The system A is therefore structurally unstable and the system A cannot have a first degree of structural instability. Let, as in 5°, 411 > (72. Then (0,0) is a focus both for the system A + and for its linear part, namely the system (66). Now for small" the point (0,,,) is a focus for the system (69). After the change z "X, II = ,,(y + 1), the point (0,,,) is mapped into a point X Y = 0, and the straight line II 0 into a straight line Y 1. The system (69) is transformed into a system arbitrarily close to (66) in the region IXI ~ '"', WI ~ '"' if" is sufficiently small. The system so obtained has a trajectory which is tangent to the straight line Y 1 and then intersects it at the point Xl (close to 6) with a derivative dX/dY ko + 1£("), where 1'(,,) + 0 as " + 0 and 6 and leo are the same as in Lemma 13. Going back to 21,11, we deduce that the trajectory T of the system (69) which is tangent to the zaxis at some point (z·, 0) subsequently intersects it at the point (Zl' 0), Zl "X1, with a derivative dz/dll ko + 1£("). Next, as in the proof of the necessity of the condition 4°, we show that if the condition 5° is
=
=
=
=
=
=
=
=
=
248
Local Singularities of TwoDimensional Systems
Chapter 4
not satisfied then there exists a system A arbitrarily close to the system A, for which the point (ZI'O) is a singular point of type 1 and T is a double separatrix. Hence, the system A cannot have a first degree of structural instability. Sufficiencl/. Let the conditions 1°5° be satisfied and let V be a sufficiently small neighbourhood of the point (0,0). According to 1°, the point (0,0) for the system A+ is either a saddlepoint or a focus or an ordinary node. For a small 6 > 0, for any system A 6close in C~ to the system A, the function Q(Z,II) y!; 0 in V. The system A may therefore have in V singular points only of types I, 2, and 4 and not more than one stationary point (ZO,lIo) for II > O. IC 6 is small then by virtue of 2°,3°, and (60), for the system A the functions Q+(z,O) and . }(z) P+(z, O)Q  (z, 0)  P (z, O)Q+(z, 0)
=
have, on the segment of the zaxis in V, only one zero each, z· and ZI, respectively (since for the function 1 from (5) we have 1(0) 0, 1'(0) y!; O)i these zeros are simplei z·, ZI, Zo,II0 0(6), so the point (:CO.lIO) is of the same type as the point (0,0) for A+. A ° If 110 0 then P+(:co, 0) Q+(zo,O) 0, i(zo) 0, hence z· ZI Zo. Then in V there exists only one singular point (zo,O) of the system A. This is the point of type 4. For small 6 at this point there hold the conditions similar to the conditions 1°3° for the system A, and the functions Q, QT, A, (72  4A, (7 (for A 0) have the same signs as the corresponding functions for the system A. Then the neighbourhoods of these singular points consist of the same sectors (see cases 1)3), Figs. 8996) and are topologically equivalent. We
= =
=
=
=
=
= =
=
i',
(V, A) ~ (V, A)
for a small 6 > 0 in some neighbourhood V of the point (zo, 0). BO If I/O > 0 or if in the part II ~ 0 of the neighbourhood V there are no stationary points of the system A+ (2: P+(Z,II), iI = Q+(:C,I/)) then in V there exist exactly one singular point (:c., 0) of type 2, not more than one singular point iZl' 0) of type ~, and not more than one stationary point (:co, I/o) in the region II > O. Since QT(z·, 0) y!; 0, 1' (zI} y!; 0, by virtue of Theorems 1 and 2 and the condition 1°, these points are structurally stable. We will show that in V there are no double separatrices or structurally unstable closed poly trajectories. The functions Q presetves its sign in Vi hence a closed trajectory cannot lie, even partially, in the region II < O. If a whole closed trajectory lay in the region II > 0 then within it there would be a singular point (:co, 110) of index I, that is, with A > o. But then (7 y!; 0 for the system A and, therefore, for small V and 6 for the system A the sum Pi" + QT ,'close in V to the number (1, preserves its sign in V. Then in V for II > 0 there are no closed trajectories ([157J, p. 228). Consequently, a closed trajectory can lie only partially in the region 1/ > 0 and partially on the :caxis (Fig. 98). But then it contains a segment of the :caxis on which Q y!; 0, that is, a segment of linear singularity, and is not a poly trajectory. We will show that V contains no double separatrices. 1) Suppose that a separatrix T of a singular point (:c., 0) of type 2 goes into a point (ZI'O) of type 1. At (Zl,O) the function h(z) Q(z,O)Q+(:c,O) < Oi it changes sign only at the point (:c·,O). Hence h(:c) < 0 on the interval of the zaxis between these points, and along this interval there passes a trajectory L. The trajectories T and L bound a domain W. The second separatrix of the point (:c., 0) goes inside W since otherwise the function Q+(:c,O) would once again change sign on the indicated interval (Fig. 99), which is impossible. Then within W there is a singular point (zo, I/o). At small 6 it can be only a focus because, by virtue of Lemma 12, c), in the case of a node or a saddlepoint through the point (zo, 110) there passes a line which separates W into two parts and cannot be intersected by the trajectory T. Hence for the system A + the point (0,0) is also a focus. Then from the reasoning used in the proof of the necessity of the condition 5° it follows that if the condition 5° is fulfUled, there' are no double separatrices. 2) Let a separatrix depart from the point (:co, I/o), I/o > O. Then (:co, I/o) is a saddlepoint of the system A. Since QT(O,O) y!; 0 the separatrices of the singular point (0,0) of the system A + are not tangent to the :caxis at this point. By Lemma 12, for small 6 the separatrices of the point (:CO,IIO) of the system A are not tangent to the :caxis in the neighbourhood V either. Therefore, they do not enter the point (z·, 0) at which the trajectory is tangent to the zaxis. Neither do they enter the singular point (ZI' 0) if the condition 4° is satisfied. This is proved by reasoning similar to that used in the proof of the necessity of the condition 4°. Hence, V contains no double separatrices in this case either. Thus, in case BO for small 6 the system A has neither structurally unstable singular points nor structurally unstable closed poly trajectories nor double separatrices. Then, by virtue of have
=
=
§19
Singular Points on a Line 01 Discontinuity
Figure 98
249
Figure 99
Theorem 1, §18, the system A is structurally stable in V. From A ° and BO it follows that the system A in .the domain V and the singular point (0.0) have a first degree of structural instability.
By virtue of this theorem there exist eight topological classes of singular points of type 4 of first degree of structural instability (Figs. 8996). Let us consider bifurcations of these points. In the case f:j, < 0, 1'(0) < 0 (Fig. 89) the following bifurcations may occur. For a system A close to A, for 'Yo > 0 near the origin there exists (Fig. 100) a singular point (x*, 0) of class 2b, a singular point (Xl, 0) of class la, and a saddlepoint (xo, 'Yo). The point (x*, 0) lies on the interval between the points of intersection of the separatrices with the xaxis, and the point (XlJO) lies either on the same interval or outside it. IT 'Yo < 0, that is, the system A+ has no stationary points for 'Y ~ 0, then the system A has only a singular point of class 2a (Fig. 56). In the case f:j, < 0, I' (0) > 0 (Fig. 90 for the system A) the arrangement of the trajectories of the system A is similar to that in the previous case, but for 'Yo > 0 there is no singular point of type 1, and for 'Yo < 0 there exists a singular point (Xl, 0) of class 1b and a singular point (x*, 0) of class 2a. In the case 0 < 4f:j, < (12 (Figs. 91 and 92 for the system A), if 'Yo > 0 then the system A has a singular point (x*, 0) of class 2a and a node (xo, 'Yo), and if 'Yo < 0 then it has singular points of classes 2b and 1a. Bifurcations of the singular point shown in Fig. 91 are shown in Fig. 101. (Bifurcations of the singular point shown in Fig. 92 are considered in a similar way.) The bifurcations of singular points in the case 0 < 4f:j, < (12, I' (ot > 0 (Figs. 93 and 94) differ from the considered ones in that the system A has a singular point of type 1 only if Yo > 0, and this singular point belongs to class lb. In the case 0 < (12 < 4f:j" 1'(0) > 0 the trajectories of the system A are shown in Fig. 96. IT 'Yo > 0 then the system A has a singular point (X·, 0) of class 2a, a singular point (Xl; 0) of class lb, and a focus (xo, 'Yo). IT the condition 5° holds, the system A either may have a structurally stable limit
250
Local Singularities of TwoDimensional Systems
Figure 100
Chapter 4
Figure 102
Figure 101
cycle (Fig. 98) or may have none (Fig. 102). In the case O'Q (0, 0) < 0 there is no limit cycle. IT Yo < 0, the system A has only a singular point of class 2b (Fig. 57). In the case 0 < 0'2 < 4~, 1'(0) < 0 (Fig. 95 for the system A) the bifurcations of a singular point differ in that the system A has a singular point of type 1 only for Yo < 0, this point belonging to class 1a. The number of topological classes of isolated singular points of each type (excluding centrefoci because there are infinitely many of them) is tabulated [188]: Type Total number of topological classes Structurally stable First degree of structural instability
1
2
3 39
4
5
6
3
4
00
00
00
2
2
o
o
1
2
7
8
o o
o o
Note 1. For type 3 the definition a), §4, is used. Note 2. Bifurcations of vector fields with singularities on the boundary of a half plane were considered in [191], [192].
§20 Singular Points on an Intersection of Lines of Discontinuity A singular point lying on an intersection of any finite number of discontinuity lines is investigated qualitatively. Sufficient conditions for stability and instability of such a point and sufficient conditions for its structural stability are given. More complete results are presented for a singular point on an intersection of two lines of discontinuity.
§20
Singular Points on an Intersection
01 Lines 01 Discontinuity
251
1. Let a circle K (3:~+3:~ < r~) with a centre 0 be separated by smooth lines (simple arcs of class C1) L 1 , ••• , Lm into m domains S. (i = 1, ... , mj m ~ 2) called sectors. A sector S. lies between lines L. and L1+1j Lm+1 = L 1. Only endpoints as of the lines L. lie on the circumference of the circle K. The direction of the circuit a1a2'" a m a1 is positive. Each pair of lines L. and Li have no common points, except the point 0, which is their common endpoint. In the circle K we consider a system in vector notation
:i: = 1(3:)
(1)
In each sector S., the vectorvalued function 1(3:) is assumed to satisfy the Lipschitz condition. The function can be discontinuous only on the line L 1 , ••• , Lm. Let Io(x) be a continuous extension of the function 1 from the sector S. onto its closure S •. For 3: E L., let IN(3:) and 1~(3:) be projections of the vectors 1'1(x) and 10(3:) onto the normal to L. directed from S.1 to S •. On those arcs of the line L., where IN(X)/~(x) ~ 0 (or at least IN (3:) ~ 0, It(x) ~ 0) a continuous vectorvalued function If(x) is defined which is tangent to L. and determines the velocity of motion :i: = If (x) along such arcs. LEMMA 1. If 10 (x) =1= 0 in Si \0 then, when continued on both sides, a trajectory passing through an arbitrary point of the sector S, either comes onto the sector boundary or tends to the point O.
This is proved in the same way as the assertion 3 of Lemma 5, §17. be an angle (which is taken to be in the positive direction) between Let the positive direction of the xaxis and the ray tangent to the arc Li at the point OJ then we have
o.
If Ii (0) =1= 0 then !Pi is an angle between the 3:axis and the vector 10(0). Adding to !Pi, if necessary, a multiple by 211", we always assume that
(2) Let Kp be the circle x~
+ x~ < p2.
LEMMA 2. Let "(0) =1= 0 for some i. Then
a) if 0i < !Pi < at+! (or at < !Pi  11" < 0'+1) then there exists a p > 0 such that each trajectory passing into Si n Kp goes out of Si n Kp through the arc of the circumference Ixl = past increases (respectively, as t decreases); in Si there exists only one trajectory entering the point 0 as t decreases ( respectively, as t increases); at this point the trajectory is tangent to the vector li(O); b) if Oi+1 < !Pi < 0i + 11' (or Oi+1 < !Pi  11' < at + 11') then there exist k and p such that each trajectory passing through any point b E Si n Kp goes out of Si onto the line Li (respectively, LHd as t decreases, and onto the line Li+1 (respectively, Li) as t increases, and the arc of this trajectory from the ingress to the egress point in S, is contained in the region k 1 Ibl < Ixl < k Ibl. PROOF: Let 0i < !Pi < 0i+1' The vector "(0) is directed from the point 0 inside the sector Si. Due to continuity of I.(x) in Si and smoothness of the
252
Local Singularities of TwoDimensional Systems
Chapter 4
lines Li and L Hls there exists p> 0 such that at each point x of these lines, at which Ixl < P, the vector Ii(x) is also directed inside the sector. Into the sector there enter trajectories, one through each of these points x (due to the Lipschitz condition). They leave the sector Si n Kp only through the arc Ixl = p. Let aHl < 'Pi < ai +11". Fix '1 > 0 such that ai+1 +'1 < 'Pi < ai+ 1[' '1 and take Pi > 0 such that in the region Ixl < Pi, x E Si' the directions of the vector Ii (x) and of the tangents to the lines Li and Li+1 differ from the directions of the rays 'P = 'Pi, respectively, rp = ai, rp = a'+1, by less than e (0 < e < '1/4). Then for Ixl < Pi the line Li lies between the rays OCI and OC2 (rp = T e), and the line LHI lies between the rays OCs and OC4 (rp = ai+1 T e) (Fig. 103). Take P = Pi sin '11, '11 = '1  2e, and draw straight lines pq and rs through any point b E S. n K p , b =j:. 0, in directions with polar angles rp = rpi T e. These lines will intersect the rays OC1,"" OC4 in the region Ixl < Pi because, for instance, the difference of polar angles of the rays bp and OC4 is equal to
a.
Lopb > '71 and, by the law of sines for the triangle obp we have
(3)
1 1 sinobp lopl = lobi' . b ~ lobl. ~ P. = Pl· sm op sm'11 sm'11
Figure 109 Similarly we conclude that the distance from the point 0 to the nearest point of the straight lines pq and rs is at least lObi sin '11. The trajectory passing through the point b lies between these straight lines. From this there follows the assertion b) of the lemma for the case under consideration. The other cases are reduced to the previous ones by taking t instead of t or rp instead of rp. THEOREM 1. Let Ii(O) =j:. 0 for all i, and let the numbers rpi be the same as in (2). For the point :I: = 0 to be asymptotically stable it is sufficient that 1) for each i the inequalities
(4) be satisfied;
§20
Singular Points on an Intersection 01 Lines 01 Discontinuity
253
2) for 0 < Ixl < Po, the vectors I?(x), wherever defined, be directed along the tangents to the lines L. towards the point 0, and that I.O(x) =I OJ 3) if
i = 1, ... , m,
(5) then it is required that
(6)
q
1. H the boundaries of the sectors are straight lines, and if the direction of the vector I(x) in each sector is constant, then the conditions 1)3) are, moreover, necessary. REMARK: If the velocity of motion along the lines L. is defined according to a), §4, and if the condition 1) holds, then for the condition 2) to hold, it is sufficient that I~.  ~.+!I < 11' for those i for which
PROOF: First we consider the case where
(8)
i= 1, ... ,m.
Then for sufficiently small e > 0
(9)
i = 1, ... , m.
By virtue of (4) and (9), for Ixl < Po the vector "(x) is not tangent to L. if x E L. and is not tangent to Li+1 if x E L.+!. Then for each i, according to the sign of the difference fP.  a.  11', on the whole arc Ixl < Po of the line L. the trajectories either only go out of the sector S. onto the line L. or only go off the line L. into the sector S•. The same holds for Ixl < Po on the boundary Li+1 of the sector S•. In particular, in the case (5) there holds (8), and all the trajectories for small Ixl successively pass from Sl into S2,. .. Sm, then again into Sl, etc. For the points Xi of intersection of a trajectory with L., i = 1,2, ... , it follows from (3) that
In a sufficiently small neighbourhood of the point 0, this ratio of the sines is arbitrarily close to the one in (6). Hence for a sufficiently small IX11 the trajectory from the point Xl makes a revolution around the point 0 and returns
Local Singularities of TwoDimensional Systems
254
Chapter 4
onto Ll at the point Xm+1, where IXm+11 ~ ql IXII. Since q < 1 then ql < 1 if IXII < p •• But then the trajectory makes another revolution around the point 0, etc. After n revolutions, IXnm+d ~ q? IXll  0 and n  00. On the arc of the trajectory between the points Xnm+1 and X(n+1)m+1 we have Ix(t)! ~ c IXnm+11 by virtue of (3). From these estimates there follows the asymptotic stability of the point x = o. In the case (7) the reasoning is similar, but revolutions proceed in the negative direction. If (8) holds, but neither (5) nor (7) does, then a trajectory cannot return to a sector where it has already been. By virtue of (3), in each sector the value of Ix(t)1 may increase at most by a factor (sin'7d 1 and, therefore, Ix(t)1 ~ (sin '7J)m Ix(to) I for to ~ t < 00. Beginning from some instant tl the trajectory remains in one of the sectors S. (and then enters the point 0 by Lemma 1) or on one of the lines L. (and then tends to the point 0 by virtue of the condition 2) of the theorem). Thus, in the case (8), asymptotic stability follows. Let, for some i, the vector f.(x) be tangent to the line L. at some points x E L. arbitrarily close to O. Then 1 an unstable one. In the case (7),
o there are no singular points other than 0, and the point 0
§20
Singular Points on an Intersection of Lines of Discontinuity
255
motion proceeds in the negative direction, for q < 1 we have an unstable focus, for q > 1 a stable one. The case q = 1 is critical, the point 0 can be a focus, a centre, or a centrefocus. If all the lines L. are straight and the directions of the vectors 10(3:) are constant in each of the sectors S. and q = I, then the singular point is a centre. If for some (not all) values of i there holds the double inequality (5), and for the remaining values there holds (7) then there exists a line Li for which IN > 0, f"J < 0 and another line for which IN < 0, I"J > O. There is an odd number of such lines. They are linear singularities of class AAI and separate the neighbourhood of the point 0 into topological sectors of classes G, L, S (the sectors considered in §17, as distinguished from S., are called topological). Stability of a singular point and belonging of topological sectors to a class G, L, or S depend on the direction of motion along the lines L. which are linear singularities. If for some i we have a. < !P' < a'+l (or a. < tp.  1(' < aHl) then by Lemma 2 in the sector S. there exists a trajectory which enters the point 0 as t decreases (increases). In at least one sector S., let there hold the inequality a. < tp.  1(' < a'+l, and let the inequality ai ~ tpi ~ ai+l hold in none of the remaining sectors, and in each of the lines L. near the point 0 let there exist neither points where 1°(3:) = 0 nor motions directed from the point O. Then the point 0 is asymptotically stable. In its neighbourhood there may exist only topological sectors of classes Q and S. If a. < !P. < a.+l for at least one i then the point 0 cannot be stable. If a. < tp. < aHl for at least one i and ai < !Pi1r < ai+l for at least one i then in the absence of sectors S. with the inequalities (5) and (7) the point 0 is a saddlepoint; its neighbourhood may consist only of topological sectors of classes H, K, Q. If moreover, sectors S. with the inequalities (5) or (7) are present then there may also exist topological sectors of classes G, L, S. The point 0 may be either singular or nonsingular (of class H H or KQQK). LEMMA 3. If for i = 1, ..• , m the conditions (10) are satisfied, I. (0) =f 0, and f.O(O} =f 0 on the lines L. on which the function fP is defined and if q =f 1 in the cases where for each i either (5) or (7) holds (possibly, forsome i it is (5) and for other i it is (7)) then the point 0 is structurally stable. PROOF: Under sufficiently small (in 0 1 ) variations of the righthand side of a differential equation for each i there remains exactly that condition from the four from a) and b) of Lemma 2 which has been satisfied for that particular i. Then the behaviour of the trajectories in each sector and on each line L. remains unchanged. The condition q =f 1 guarantees the absence of closed poly trajectories (4, §18) near the point O. The smapping of each sector S. onto a corresponding sector of the phase plane of another, aclose equation is constructed by the methods used in §17. This proves structural stability of the point O. The conditions of Lemma 3 are only sufficient for structural stability. The necessary and sufficient conditions are obtained in [1861. Now let us consider equation (1) with the definition a), §4. All the previous results hold in this case also, but their formulations now become simpler. Along with the phase plane 3:lo 3:2 separated by the lines L. into sectors S., we consider a "velocity plane" til, tl2 separated by rays I. (tp = a.) into angles
256
Local Singularities of TwoDimensional Systems
Chapter 4
Si (ai < IP < aHd. The point 0 is considered as belonging to all the rays l.,. Let tti Je the endpoint of the vector "(0) constructed in the plane t11, V2. Let P be a closed broken line with vertices ttl, tt2,'" tt rn , ttm+l = ttl' If a segment ttitti+1 intersects a ray lHI (or its continuation beyond the point 0) at a point w then, according to a), §4, the vector fP+1 (0) = w is defined on the line L i +1' We denote the convex closure of the set of points ttl, .•• ,ttrn by M. The following assertions hold. 1) The point x = 0 is stationary if and only if 0 E M (2, §12). 2) If 0 E M, 0 ¢ P then x = 0 is an isolated stationary point (since in this case t.(x) :f 0, fP(x) :f 0 by virtue of continuity of the functions" for Ixl < pl. 3) If for i = 1, ... , m the point tLi lies neither on the rays li' lHI nor on their continuations, if 0 ¢ P, and if q :f 1 in the cases where for each i there holds either (5) or (7) then the point x = 0 is structurally stable (Lemma 3). 4) If for at least one i neither (5) nor (7) is satisfied and if the points tti ¢ Si and the segments ttitti+1 have no points in common with the rays lHI' i = 1, ... , m then the point x = 0 is asympotically stable. (This follows from Theorem 1.) 5) If the broken line ttl tt2 ... ttrn ttl does not pass through the point 0 then the indez of the point x = 0 is equal to the number of revolutions made by this broken line around the point 0 (2, §14). 6) If the lines L. are straight and in each sector Si the functions ''(x) are constant then the conditions formulated in 2) and 3) are not only sufficient but also necessary. Systems with righthand sides discontinuous on the lines Li which go from the point 0 have been investigated in the papers [104], [193][196], etc. 3. We now consider the case where righthand sides of a system are sums of two terms, one of which is discontinuous on one smooth line and the other is discontinuous on another smooth line and these lines intersect at a nonzero angle. If we make the change of variables so that these two lines become coordinate axes, the system will take the form 5;
= It (x, y) + gl (x, y),
the functions Ii are discontinuous only on the yaxis, the functions gi only on the xaxis. To investigate the behaviour of trajectories near the intersection point of discontinuity lines in structurally stable cases, one can replace the functions Ii, g. in each of the coordinate quadrants by their limit values for x + 0, y + 0 and obtain a system of the form [197]
(13)
:i:
=a+
bsgn:z:+ csgny,
iJ = d + e sgn x + f sgn y.
On the lines of discontinuity we use the definition a), §4. We will mention some properties of solutions of this system, which partially follow immediately from the results of 2 and partially have been obtained in [197]. If Ifl ~ Id + e sgn xl, x > 0 or x < 0 then there exists a solution going along
§20
Singular Points on an Intersection of Lines of Discontinuity
257
the indicated semiaxis Ox with the velocity
:i: =
7
(Dl
~ f 1'
D= 1
+ Dsgnx),
Dl = 1
y=
~ f 1'
0,
~ ~ I·
D2 = 1
IT Ibl ;;I?; la + csgn yl, y > 0 or 11 < 0 then there exists a solution going along the indicated semiaxis Oy with the velocity
:i:=O,
iJ = ~ (D2 + Dsgny).
The point (0,0) is stationary if and only if either D:/: 0,
or
D=D1 =D2 =0,
lal ~ Ibl + Icl ,
Idl ~ lei + III j
the point (0,0) is an isolated stationary point if and only if (14) In the case (14) the index I of the singular point (0,0) is equal to sgn Dj if the point (0,0) is nonstationary then 1=0. The point (0,0) is a saddlepoint if D < ID1I, D < ID21. The point (0,0) is a focus or centre if (15)
Icl > lal + Ibl ,
lei> Idl + III ,
ce < o.
Then in the case (16) the point (0,0) is a stable focus, if the reverse inequality holds then it is an unstable focus and in the case of equality in. (16) it is a centre. For x = 0, y = 0 to be an asympotically stable solution of the system (13), it is necessary and sufficient that in the case (15) there hold the inequality (16) and that in the case of failure of at least one of the inequalities (15) there hold the inequalities
(17)
1< IIdllell·
The point (0,0) is a stable node if the inequalities (17) hold and at least one of the inequalities (15) does not hold. IT the same conditions are fulfilled, but with b and  I instead of b and I in the last two inequalities (17), then (0,0) is an unstable node.
258
D >
Local Singularities 01 TwoDimensional Systems
Chapter 4
The point (0,0) is stationary with an elliptic region (Fig. 51) if D > ID21 and, besides, either
Ie + al
< b< a
e 
< b
d of the surface P do not reach P. All the points of the surface P are stationary for the system (1), but not for the system (5). Example: u = w, V = til = 0 (w ~ 0). c) If f(x) == 0 in G+ near the surface P then a onesided (in G+) neighbourhood of the surface P is filled with station~ points of the system (1). Example: u = v = til = 0 (w ~ 0). If the condition 4° is not fulfilled, an irregular case is also possible, which can be seen from the example 14), I, § 16, with an additional equation z = O. Combining each of the cases A, B, a), b), c) in G+ with each of the same cases in G , we obtain the following possible disposition of the trajectories of the system (1) near the surface P under the conditions 1°_4°: AAo, AAlI AA2 , AB, Aa, Ab, Ac, aa, ab, ac, bb l , bb2 , bc. The description of these cases is similar to the description of the respective cases of the disposition of trajectories in a plane near a linear singularity in 2, §16. We therefore do not present it here. The case bb, however, is divided into the case bbl , where the vectors g"": (x) and g+(x) are collinear for each x E P, and the case bb 2 , where they are noncollinear for each x E P. If the condition 3° is not fulfilled, there may also hold the cases BBlI BB 2 , Ba, BblJ Bc, which are analogous to the respective cases in 2, §16, as well as the
264
Local Singularities of ThreeDimensional. .. Systems
Chapter 5
cases BB3 , Bb2, which differ in that the vectors f+ (z) and f (z) or g+ (z) and g(z) are noncollinear for each z E P. THEOREM 1. Under the conditions 1°_4° there exist only twelve local topological classes of twodimensional singularities: AA l , AA 2, AB, Aa, Ab, Ac, aa, ab, ac, bb l , bb 2, bc.
The proof is carried out in the same way as for Theorem 2, §16, and is now presented briefly. If for the system (1) in G+ there holds one of the cases A, B, a), b), c) then for an arbitrary point Zo of the surface P we construct, by the same method as in Theorem 2, §16, a topological mapping from some half neighbourhood U+ (xo) C G+ onto the half neighbourhood the point u = tI = W = 0, under which the trajectories of the system (1) are carried into trajectories from the respective example out of those given above .. If on one side of P there holds the case B or b) then one first maps this half neighbourhood and then the other half neighbourhood, both maps coinciding on P. In the case bb 2, on P there exist trajectories of the.. system :i: = g (x) which intersect trajectories of the system :i: = g+ (x). First these two families of trajectories are mapped into families of lines u = const and tI = const of the plane w = 0, and then the half neighbourhoods are mapped, the latter map coinciding on P with the already constructed one. In each case we obtain a topological mapping from a whole neighbourhood of the point Xo onto the neighbourhood of the point u = tI = W = 0, under which trajectories of the system (1) are carried into trajectories of the corresponding standard system. In any two systems, for which there holds the same case out of AA l , AA 2, ... , the neighbourhoods of any two points of the surface P have therefore the same structure. The difference in the structure of such neighbourhoods for systems belonging to different cases is proved as in Theorem 2, §16j the difference in the structures in the cases bb l and bb2 follows from the fact that in the case bb l on the surface P there exists only one family of lines which are limits of convergent sequences of trajectories of the system (1), whereas in the case bb 2 there exist two such families (trajectories of the systems :i; = g(z) and :i; = g+(z)). We will prove that under the conditions 1°4° each twodimensional singularity P belongs to one of these twelve classes. By virtue of 1°_4°, on P there exists a point Y in the neighbourhood U+ (y) C G+ for which there holds one of the cases: (A) f"J t OJ (B) f"J == 0 on P, J+ t OJ (a) J+ = 0 on P, gt. t OJ (b) J+ = 0, gt. = 0 on P, g E Lip(loc U+(Y))j (c) f == 0 in U+(y). Near the point Y there exists a point z on P for which on the side of the domain G+ there holds the same case as for y, while on the side of the domain Gthere holds one of these five cases, but with the function f and g j this point z is different from the points mentioned in condition 3°. The case AA is divided into subcases J{x)J+(x) > 0 (AAo, no topological singularity), r(x)J+(x) < 0 and JO(z) == 0 (case AA 2 ) or JO(z) ¢ 0 (then near z we take a point q E P at which JO(q) to, case AAd. The case bb is divided in a similar way. Then the neighbourhood of the point z (or of the point q) belongs to one of the indicated twelve topological classes. By the definition of a twodimensional singularity, any of its points z has a neighbourhood of the same structure as the
TwoDimensional Singularities
§21
265
neighbourhood of the point z (or q), and therefore belongs to one of these twelve classes. We will show that structural stability occurs only for singularities of class AAI in the case IN I"J < O. The following condition is assumed to be satisfied. IT INI"J < 0 on P in a closed neighbourhood V of a point Xo e P, then under sufficiently small variations (in 0;) of the function I the function 10 varies arbitrarily little in the metric 0 1 (V) j if 10 (x) == 0 in V then one can vary the function I in such a way (arbitrarily little in C;) that 10 (x) ~ 0 in each neighbourhood of any point Xl e V. Under the definition a), §4, this condition is fulfilled. LEMMA 2. If in some neighbourhood of a point Xo e P there holds the case AAo (IN I"J > 0) or AAI (IN I"J < 0, 10 i 0) then in some neighbourhood of the point Xo the system (1) is structurally stable.
PROOF: For small enough variations of the function I the indicated inequalities are preserved, and the twodimensional singularity retains its class. First we construct an eol mapping = "'(x), x e P, which carries the trajectories ofthe system (1) lying on P near the point Xo into trajectories of a aclose system. Let x = lP(t, x*) and x = ~(t, x*) be solutions (lying in G+) of these systems with initial conditions x* e P and x* e P at t = O. A mapping which assigns the point x = ~(t, ",(x*)) to the point x = lP(t, x*) for Ix*  xol ~ p, 0 ~ t ~ l' (or 1' ~ t ~ 0) is topological if p and l' are small. This follows from Theorem 3, §12. For sufficiently small a it shifts each point by less than e, by virtue of the continuous dependence of solutions and compactness of the neighbourhood Ix  xol ~ p. The same holds for trajectories in G. Hence, in the neighbourhood of the point Xo the system (1) is structurally stable.
x
LEMMA 3. If on a
twodimensional singularity P at some point Xo we have (or 1"J(xo) = 0, or 10(xo) = 0) then in any neighbourhood of this point the system (1) is structurally unstable.
IN(xo)
=0
PROOF: IT 10(xo) = 0, IN(xo)/"J(xo) < 0 then Xo is a stationary point. Since P is a twodimensional singularity then all the points on P are stationary. Let us change the system arbitrarily little so that 10(x) ~ 0 near the point Xo. Then not all the points there will be stationary. Hence, the system (1) is structurally unstable.
Let IN{xo) = 0, and h be the vector of the normal to P at the point Xo. The equations 3; = I'{x), i = 1,2, where
I'(x)
= I{x) + ",h
I'{x)
= I{x) + (I),,,,h
differ from equation (1) arbitrarily little for small ",. For small enough", > 0, in the neighbourhood of the point Xo, for one of these equations there holds the case AAo, and for the other the case AA 1 • In at least one of these cases the structure of the neighbourhood of the point Xo differs from the structure of this neighbourhood for equation (I), and the structural instability follows. 8. We will consider transformations of trajectories of the system (1) with the help of diffeomorphisms, that is, such onetoone mappings II "'(z) E C l , for which the inverse
=
266
Local Singularities of ThreeDimensional ... Systems
Chapter 5
mapping ",1(11) belongs to 0 1 • The definition a), §4, is assumed to be used on the surface of discontinuity. Let the righthand sides of the systems (1) and
(6)
iI
= 9(11)
of dass O! be discontinuous only on surfaces 8 and
Iii>
(7)
Iii < 0 in 8,
0,
8 of dass 0 2 , respectively, and let + 0,
ON
~
in 8.
THEOREM 2 [208]. Let the condition (7) be fuIfllled and let there exist a diffeomorphism II X(z), z E 8, II E 8, satisfying the condition
=
k(z) > 0,
(8)
and, therefore, transforming trl\iectories (z = 10(z» of the system (1), which lie on the surface hi 0°(11)) of the system (6), which lie on 8, the direction of motion along trl\iectories being preserved. Then in the neighbourhood of the surface 8 there also exists a diffeomorphism transfonning trl\iectories of the system (1) into trl\iectories of the system (6), the direction of motion being preserved.
=
8, into trl\iectories
PROOF: Since, by using a smooth transformation, one can map the smooth (of dass 0 2 ) surfaces 8 and 8 into planes we assume 8 and S to be finite domains in the planes Zs 0 and lis 0 (here z (ZI' Z2, zs), 1= (h, 12, fa), etc.). Let
=
=
=
p(z)
I(z)
0(11)
= IIs(z)I'
q(lI)
= 10a(II)I'
Then
PH
=pii = 1,
(9)
°
p=
Ph
= pt = 1,
p
+ p+
q; = 1,
°
2'
q=
=
=
q
+ q+ 2
and by Theorem 6, §9, the systems :i: p(z), iI q(g) have the same trajectories as the systems (1) and (6), respectively. In (8) we express 10 and gO by formula (6), §4, and so obtain x'(z)(p + p+) k(z) (q + ~+)( Ot)9;
(!t)!a = la  It
The functions
(Os 
ot)
I
11=,..(")
la, rt, 0; ,ot, k( z) belong to Oland do not vanish, hence the function m(lI) >
o can be so chosen that
there hold
(10)
=
Let z = e(a, t) and z e+(a, t) be solutions of the system z = p(z) with the initial conditions e(a,O) e+(a,O) = a E 8 which for 6 ~ t < 0 lie in the regions Zs < 0 and Za > 0, respectively. The solutions II '7(1., t) and II = '7+(b, t), b = x(a) E 8, of the system iI q(lI) are defined similarly. Each point z e(a, t) (or z e+(a, t)) on the trajectory of such a solution is mapped to a point
=
=
=
11= '7 (b,m(b)t)
=
=
(respectively, II
=
= '7+ (b,m(b)t)). =
The mapping II "'(z) so obtained coincides on 8 with the mapping II X(z) and carries trajectories of the system z p(z) into trajectories of the system if q(II), the direction of motion being preserved. For Z8 ¢ 0 we have "'(z) E 0 1 , and for Zs = 0 the derivatives
=
=
Linear and Point Singularities
§22
267
=
8'1/1/8zi, i 1,2, are continuous. For continuity of 8'1/1/8z8 it suffices that the derivative of '1/1 in the direction of the vector n = 1'0  1'+ = 1'  1'0 be continuous, i.e., that 'I/I'(z)n ('I/I'(z) is a Jacobian matrix) be continuous. Since 'I/I(z) = X(z) in the plane Za = 0 and the vector 1'0 lies in this plane then, by virtue of (10), for Za 0
=
(11) Next,
(12)
=
=
=
=
for t 0, b X(G), 'I/I±(z) lim"a_±o 'I/I'(ZI, Z2, za). Since V '1±(b, t) is a solution of the system iI = q(V), the righthand side of (12) is equal to m(v)q±(,,), where V X(G). Hence, for V X(z)
=
From this and from (11) it follows for V
=
=X(G) , G E S, that
= m(v) (qO(v)  q+(v») , 1'0«1») = me,,) (q(v)  qO(v») .
'I/I+(G) (1'0 (G)  1'+ (G»)
(13)
'I/I~(G) (p(G) 
The righthand sides coincide by virtue of (9). Then the limit values of the vectorvalued function 'I/I'(z)n on the two sides of the surface S coincidej that is to 8ay, this function is continuous on S. Consequently, 'I/I(z) E C 1 • Since V X(z) is a diffeomorphism on S, the vecton 'I/I'(z)el, 'I/I'(z)e2 (z E Sj el, e2 are vectors parallel to the Zl, z:zaxes) are linearly independent and lie in the plane za O. By virtue of (7) and (13), thevector;,'(z)(pO_p+) does not lie in this plane. The range of the linear transformation 'I/I'(z) contains three linearly independent vectors, hence det;,' ¢: 0 (z E S). Hence V 'I/I(z) is a diffeomorphism in the neighbourhood of the surface S.
=
=
=
Under the definition aJ' §4, in the neighbourhood of any point of a twodimensional singularity, where IN(z)/p(z) < 0, 10(z) ¢: 0, there exists a diffeomorphism carrying trl\iectories of the system (1) mto trl\iectories of the system
COROLLARY.
ill
= I,
il2
= 0,
ils
=  scn Va·
4. Most of the results of 2 hold also for autonomous systems of differential equations in an ndimensional space with righthand sides discontinuous on a smooth (n  1)dimensional surface. The cases bb 2 , Bb 2 , BBs (where the vectors g(x) and g+(x) are tangent to the surface P, but are noncollinear) are however divided into smaller topological classes. EXAMPLE: For a system
there exist twodimensional surfaces Xs = c, x, = 0, which lie in a three dimensional hyperplane P(x, = 0) and are filled with two families of trajectories, namely, trajectories of the system z = J(x) and x = J+(x). For the system obtained from that written above by replacing only one equation Zs = 0 by the equation Z3 = Xl for X, > 0, there are no such twodimensional surfaces.
268
Local Singularities 01 ThreeDimensional. .. Systems
Chapter 5
§22 Linear and Point Singularities on a Surface of Discontinuity Linear singularities and point singularities lying on a smooth surface of discontinuity of the righthand side of a system are topologically classified. Structurally stable and several structurally unstable singularities are specified. 1. Let a surface S E C 2 separate a finite domain G C R3 into parts Gand G+, and let a vectorvalued function I E ct be discontinuous only on S. Let the conditions 10 _40 , 2, §21, be satisfied. The notation 1+, It;, etc. means the same as in §21. We will consider the system
(1)
2: = I(x),
The cases Iii It; > 0 (involving no topological singularity) and Iii It; < 0, 1° t= 0 (involving a twodimensional singularity of class AAd are dealt with in §21. Therefore, other singularities may appear only on those subsets of the surface S, where either 1° = 0 or Iii = 0 or It; = o. One can specify six types of singularities lying on the surface S, characterized by fulfillment of the following conditions at the points of a given singularity:
Iii It; < 0, 1° = O. It; = 0, t+ t= 0, Iii t= 0 (or Iii = 0, r t= 0, It; t= 0). Iii = It; = 0, t= 0, t+ t= o. 4. t+ = 0, Iii =F 0 (or r = 0, It; t= 0). 5. t+ = 0, Iii = 0, t= 0 (or = 0, It; = 0, 1+ =F 0). 1. 2. 3.
6.
r
r = t+ =0:
r
r
The conditions characterizing the types 46 can be violated under arbitrarily small variations of the function I. Hence, in what follows, we analyze only singularities of types 13. In the ndimensional case, singularities of these types have been examined in [3], Chapter 2, §1, and in [100J, 1199J1202], 1208]. Suppose the functions 1 ,1+ ,1°, Iii, It; can vanish only at isolated points, on a finite number of piecewise smooth lines, and in a finite number of domains with piecewise smooth boundaries. This supposition is true, in particular, if the function I(x) and the surface S are piecewise analytic. We will consider the case Iii It; < O. Let, for instance,
(2)
Iii>
0,
It; < O.
The domains on the surface S, where 1° =F 0, are two dimensional singularities which belong to class AA 1 , and the domains, where 1° == 0, belong to class AA2 (2, §21). It remains to consider the boundaries of the domains where 1° = 0, and isolated lines and points where 1° = O. For the system
(3)
xES,
which describes motion along the surface S, these lines (or parts of them) and points are linear and pointwise singularities consisting of stationary points.
Linear and Point Singularities
§22
269
Since f O E C1 it follows from Theorem 2, §16, that such linear singularities may belong only to classes aa, ab, ac, bb, bc (if the function fO satisfies the condition 4°,2, §16). Considering different directions of motions along trajectories of the system (3) on the two sides of a particular line L and taking into account (2), we obtain ten different cases of behaviour of trajectories in the threedimensional neighbourhood of this line L. Class aa gives three cases, classes ab, ac, and bb give two cases each, class bc gives one case. We will show that these ten cases give ten topological classes of linear singularities. If fii > 0, fii < 0 on a closed bounded domain K lying on the surface S, then into each point a E K there comes one trajectory from a domain G and one from G+ at t = o. For r ~ t ~ 0 (with a certain r > 0) all these trajectories exist, do not have common points, except common endpoints a E K (see case A, 2, §21), and fill a closed domain Z(K, r). We will consider the system (1) and the system z = i{z) with the function E C; discontinuous on the surface Sj as in (2) and (3), on S
i
iii> 0, LEMMA 1 [2001. If there exists a topological mapping T from a closed bounded domain K c S onto a closed domain if c S under which trajectories of the system (3) are carried into trajectories of the system z = iO(z) (and inversely), the direction of motion along trajectories being preserved, then the mapping T can be continued onto Z(K, r), the indicated properties being preserved for trajectories of the systems (1) and z = i{z). PROOF: Let for r
~
t
~
0
be a solution of the system (1), with !p+ in G+ and !p in G. Similarly, let z = ",:t: (t, b) be solutions of the system z = i{ z), b E if, r1 ~ t ~ o. The point z = !p+(t, a) is mapped to the point i = 1/I+(r1 r 1t, Ta), similarly for !p. Since solutions depend continuously on initial conditions, and by virtue of (2), the point a and the time of motion t depend continuously on the point z = !p+ (t, a), it follows that mapping so constructed meets the requirements of the lemma. REMARK: For a given system (1), for the domain K, and for any e > 0 there exists a number 6 > 0 with the following properties. For any system z = i{z) such that i E C;, Ii  fl < 6 and ITz  zl < 6(z E K), the mapping T, continued by virtue of Lemma 1, shifts each point of the domain Z(K, r) by less than e. For all 6 from some interval (0,60 ) one can choose a number r > 0 which depends neither on 6 nor on the choice of the function PROOF: The possibility of choosing r follows from (2) and from the uniform continuity of the function f in both half neighbourhoods of the compactum K. The statement concerning the shift smaller than e follows from the theorem on continuous dependence of solution. The continuity is uniform on the compactum K. From Lethma 1 it follows that under the condition (2), and the condition 4°, 2, §16, for f O all the systems, for which there holds the same (any) of
J.
270
Local Singularities 01 ThreeDimensional. .. Systems
Chapter 5
the abovementioned ten cases, compose one local topological class. Thus, under these conditions there exist ten local topological classes of linear singularities. For investigating structural stability, the following condition is assumed to be fulfilled. 5° For some m ~ 1 we have S E om+l, 1 E Or:', and on each compactum, where 1/;;(x)I+l/it(x)1 =1= 0, for small enough variations (in Or:') of the function 1 the function f O also varies little (in am), and any small variation (in am) of the function 1° can be obtained by slightly varying the function 1 (in Or:'). This requirement is met, in particular, under the definition a), §4. LEMMA 2 !200]. Let the system (1) satisfy the condition 5° with m = 1 and the condition (2). Then the structural stability of the system (3) in a closed bounded domain W c S is a necessary and sufficient condition for the structural stability of the system (1) in the domain Z(W, r). The assertion is proved by using the standard reasoning concerning structurally stable systems and the remark to Lemma 1. REMARK: The statement similar to Lemma 2, but with m = 3, is true for systems of first degree of structural instability. COROLLARY 1. The linear singularities of the abovementioned ten local topological classes are locally structurally unstable, that is, the system (1) is structurally unstable in each neighbourhood of any point of such linear singularity. One can show that in this case the degree of structural instability is infinite. COROLLARY 2. Under the condition (2) there exist only three topological classes of structurally stable pointwise singularities of the system (1). On the surface S the system (3) has a structurally stable stationary point which is respectively: 1) a stable node or a focus; 2) an unstable node or a focus; 3) a saddlepoint. COROLLARY 3. Under the condition (2) there exist only four topological classes of point singularities of nrst degree of structural instability for the system (1). On the surface S the system (3) has either a saddlenode of multiplicity 2 ([185J, p. 236) or a complicated focus of multiplicity 1 ([185J, p. 253). Each of these cases for the system (3) corresponds to two topological classes of pointwise singularities for the system (1) which differ in the direction of motion along trajectories on the surface S. This statement follows from the remark to Lemma 2 and from the fact that only the two abovementioned types of singular points of the system (3) have a first degree of structural instability ([185, p. 382). Let us now consider equation (1) in an ndimensional space in the neighbourhood of a smooth (n l)dimensional surface S, on which the vectorvalued function I(x) is discontinuous. As in the case n = 3, under the condition (2) the topological classification of singularities of the system (3) (preserving the direction of motion along trajectories) completely determines the topological classification of singularities of the system (1), which lie on the surface S, and all such singularities consist of stationary points. Lemmas 1 and 2 remain valid
Linear and Point Singularities
§22
271
also in the ndimensional case. In line with Corollary 2 to Lemma 2 there holds the following assertion: For equation (1), where x ERn, untier the condition (2) there exist only n topological classes 01 structurally stable points lying on the surlace 8. 118 is a plane Xn = 0 then lor singular points 01 the kth topological class a matrix 01 order (n  1)
(:~:)
i.i=1 ..... n1
(the values 01 the derivatives being taken at a singular point) has n  k negative and k  1 posit?'ve Re Ai, II the conditions (2) are lulfiJled and Re Ai < 0, i = 1, ... , n  1, the singular point is asymptotically stable. 2. Let us consider the case where on a smooth line L c 8 only one of the functions Iii and Iii is equal to zero. Let, for instance,
Iii > 0, 1+ =1= 0, f"J
(4)
= 0 on L.
The part of the surface 8 lying near the line L is separated by this line into two domains: 8 1 and 8 2. By virtue of (4) we may assume that Iii > 0 in 8 1 and 8 2. In each of the domains 8 1 and 8 2 let the function f"J preserve sign or be equal to zero, and at points x E L let the vector f+ (x) be directed either from 8 1 to 82 or from 8 2 to 8 1 or along the tangent to L. Considering the various combinations of these possibilities, we obtain the basic classes of linear singularities lying on the surface 8 under the condition (4). The cases, where in an arbitrarily small neighbourhood of a point a E L the function fii changes sign in 8 1 or in 8 2 or the vector f+ changes direction, lead to pointwise singularities. Since there are very many such possibilities, we discuss only the basic ones, including all structurally stable singularities. a) Let the condition (4) be fulfilled and let the function fii have opposite signs in 8 1 and 8 2 , for instance,
(5) Then a twodimensional singularity of class AAl in the domain 81 adjoins the line L and in the domain 8 2 trajectories intersect the surface 8 and pass from Gto G+. For x E L, let n{x) be a nonzero vector tangent to 8, normal to L, and directed from 8 1 to 82. Consider the cases
f+ > 0 on L, f+ < 0 on L, n' f+ == 0 on L.
(6) (7) (8)
n' n'
LEMMA 3 1200j. In the neighbourhood of each point of the line L, all the systems satisfying the conditions (4)(6) have the topological structure (Fig, 106) similar to that of the system
ill (9)
= 1, Y2 = 0, Ys = Yl, (Ys> 0), Y1 = Y2 = 0, Ys = 1 (Ys < 0),
272
Local Singularities of ThreeDimensional. .. Systems
defined by a), §4, in the neighbourhood of the point 0
=
Chapter 5
(0,0,0).
PROOF: Let us construct a topological mapping carrying trajectories of a given system (1) in the neighbourhood of a point a E L into trajectories of the system (9). Let P be a smooth surface passing through the line L, tangent to the vector /(x) at none of the points x E L, and located in G+ near the line L. Let y = I/I(x) be a topological mapping carrying the point a into the point 0, an arc l (containing the point a) of the line L into a segment iY2i ~ p of the Y2axis, and those portions of the surfaces 8 2 and P which adjoin l and are intersected by trajectories of the system (1) only in one direction into those portions of the half planes Y3 = 0, Y1 > 0, and Y1 = 0, Y3 > which adjoin a segment of the Y2 axis . We continue this mapping along trajectories onto 8 1. If x = ~O(t, b) is a solution of the system (3) on 8 1 with the initial condition ~O(O,b) = bEL and Y = 0 on bc, ~(O) = b. The vector 1+(6) is tangent to L. Let the vector ~'(O) have the same direction as I+(b). We investigate the arrangement of trajectories in the general case, that is, under the conditions
VIi; =/: 0
(14)
p'(O) =/: O.
on L,
According to [203], in a domain G+ C RS with a smooth (of class 0 00 ) boundary 8 in the neighbourhood of any point of the boundary, the vector field I(z) E 0 00 of general position without singular points (that is, I(z) =/: 0) can be reduced by a smooth change of variables to one of the three forms
€l = I, €l = e2,
el =6,
€2 =0, €2 = I, €2 = es,
es = OJ €s = OJ es = 1.
In each of these cases there are two possibilities: the domain G+ is transformed into a domain > 0 or into a domain < O. The first case has been dealt with in 2, §21, (case A), and the second in Lemmas 3 and 4, §22. In the third case the change 26 = Ys, e~  2e2 = y, = Yl reduces the system to the form (Fig. 108)
el
(15)
ill
el es
= I,
•
2
Ys = Yl  Y2
(YS > O)j
276
Local Singularities of ThreeDimensional ... Systems
Figure 108
Figure 109
if however, the system is considered in the region by ys we have (Fig. 109)
(16)
Yl
= 1,
Y2 = 0,
Chapter 5
Ys
= Y2  Y~
6
0).
In both figures the trajectories in the region Ys > 0 have the same topological arrangement. Let the system (15) or (16) be obtained from the system (1), where the function I is defined in G+ and in G , and let Iii > O. The trajectories from the domain G reach the surface 8 as t increases, and the surface 8 is transformed into the plane Ys = O. Therefore, in the plane Ys = 0 there exist trajectories in the domain Y2 > y~ (Fig. 108) or in the domain Y2 < y~ (Fig. 109). This domain is the domain 8 1 (see (5)). On the line L (Y2 = y~, Y3 = 0) the vector 1+ (y) = (1, 0, 0) is tangent to the plane Y3 = 0, and at the point Yl = Y2 = Y3 = 0 it is tangent to the line L. By virtue of the condition 2°, 2, §21, we have IO(y) = J+ (y) = (1,0,0) on L. Making use of the fact that IO(y) E C t near the point Y = 0, one can show that the system (15) has there some trajectories which lie in the plane Y3 = 0 and reach the line L at both ends, whereas the system (16) has no such trajectories. Hence, these two cases are topologically different. Putting s = Yl and noting that the vector n(y) is directed from 8 1 to 8 2 , we conclude that for n(y) = (2Yl, 1,0) the function (13) is equal to p(yt) = 2Yl for the system (15). For the system (16) n(y) = (2Yl, 1,0), p{y) = 2Yl. 1 [200J. If I E Goo in G+ U 8, IE C 1 in G, 8 E Coo and on 8 the definition a), §4, is used then under the conditions (4), (5), and (14) there exist only two topological classes of point singularities on the line L. They are specified by the conditions p'(O) > 0 (Fig. 108) and p'(O) < 0 (Fig. 109). THEOREM
PROOF: Let systems A and A satisfy the conditions of the theorem and, for instance, p'(O) < 0, ii'(O) < O. Using the abovementioned statements from [203J, we reduce both systems to the form (16) in the region Ys > O. Both transformations are smoothly continued into the domain G, and the systems B and B so obtained are, generally speaking, different for Ys < OJ hence in the plane Ys = 0 their trajectories are also different. The trajectories of the system (16), which are tangent to the plane Y3 = 0 at points of the arc Kl (Y2 = y~, Yl ~ 0), return into the plane Y3 = 0 at points of the arc Ks (4Y2 = y~, Yl ~ 0) shown by dashed line in Fig. 109. Near the point 0 = (0,0,0) for the arc Ks we have dY2/ dYl = Yl/2, and for a trajectory
Linear and Point Singularities
§22 lying in the plane !/3
277
= 0, by virtue of the definition a), §4, we obtain on Ks k
(17)
= const.
In the neighbourhood of the point 0 on K 3 , the slope (17) of trajectories is less than the slope of the arc K 3 , and these trajectories there intersect the arc K 3 , once each. In the neighbourhood of the point 0 we construct a topological mapping y = t/I(!/) carrying trajectories of the system B into trajectories of the system B. On the arcs K 1 , K 3 , on the half line !/1 = !/3 = 0, !/2 ~ 0, and on the trajectories passing through these arcs and lying in the region !/3 > 0, let t/I(!/) = !/. Next, to the part !/2 ~ !/~ of the plane !/s = 0 the mapping is extended along trajectories lying in the plane !/3 = O. To the remaining part of the threedimensional neighbourhood of the point 0 the mapping is extended along trajectories passing in the regions !/3 > 0 and !/3 < 0, as in Lemmas 3 and 4. The case p'(O) > 0 is considered in a similar way. REMARK: The conditions f E 0 00 , 8 E 0 00 in Theorem 1 are not necessary. They can be weakened by excluding the reference to [2031 and changing the proof. Then the derivation of (17) becomes more complicated. Obtaining the estimate (17) (even with O(!/1) in the righthand side), uniform for all the systems sufficiently close to the initial one, makes it possible to prove structural stability of the pointwise singularity under the conditions (4), (5), and (14). Now we consider the case where the condition p'(O) =F 0 in (11) is replaced by the condition
(18)
p(i)(O)=O
(i=0,1, ... ,k),
In all the smoothness conditions the order of smoothness should be increased by
k. s
If k is odd, the function p(s) does not change sign and vanishes only for Then there is no pointwise singularity by virtue of the following lemma.
= O.
LEMMA 9. Let the condition (4) be fulBlled, L E 0
2,
and let none of the arcs lying on 8 be an arc of the trajectory of the equation :i: = f+ (x). Then L is a linear singularity of class L1 if n . f+ ~ 0 on L, and of class L2 ifn· f+ ~ 0 on L. PROOF: By virtue of the remark to the case A, 2, §21, it follows from the conditions of the lemma that in 8 1 there exists a twodimensional singularity of class AA 1 , and in 8 2 the trajectories intersect the surface 8. Let =f in G+, E 0 1 in a whole neighbourhood of the point bEL. Having made the change of variables, we may assume 8 to be a plane and L to be a straight line. With an appropriate choice of coordinates the function satisfies the condition (11). By Lemma 6, in the neighbourhood of the point b there exists a surface P passing through the line L, intersected by trajectories of the system
r
r
r
278
Local Singularities of ThreeDimensional . .. Systems
Chapter 5
x = /*(x) in only one direction, and having only one common point with each of these trajectories. By means of this surface one can construct, as in Lemmas 3 and 4, a topological mapping carrying trajectories of the considered system into trajectories of the system (9) if n· 1+ ~ 0, and into trajectories of the system (10) if n· 1+ ~ O. For an even k ~ 2 the function pes) changes sign, and the point 6 E L is an endpoint of linear singularities of classes Ll and L2 as in Theorem 1. For k ~ 1 the case (18) is structurally unstable since under an arbitrarily small variation of the function I in (1) the multiple root of the function pes) can split into several simple roots. Then the pointwise singularity splits into several other pointwise singularities. By similar methods one can also analyze those points of the line L on one side of which pes) == 0 on L and on the other side pes) :j: 0, as well as the case where Iii has the same sign in 8 1 and 8 2 under different assumptions concerning the function pes). e) Let the condition (4) be satisfied and at some point of the surface 8 (19)
Iii
= 0,
Viii
=
o.
Let us assume the surface 8 to be a plane Xs :::;:: 0, this point to be the origin 0 and Iii E C2. Then in the neighbourhood of the point 0
(20)
tp
= o(x~
+ x~).
If a > 0, ac > 62 (or a < 0, ac > 62 ) then the quadratic form is positive (negative) definite and It > 0 (It < 0) everywhere except at the point O. Then in the neighbourhood of the point 0 there are no topological singularities (respectively, on S there exists only a twodimensional singularity of class AAd. This follows from the remark to the case A, 2, §21. In both these cases, in any neighbourhood of the point 0 the system is structurally unstable, since an addition to the function (20) of an arbitrarily small linear term leads to the appearance of a linear singularity. If ac < 62 then the quadratic form is indefinite (signvariable) and It = 0 on two smooth lines intersecting at the point O. Two sectors of the plane Xs = 0 between these lines are intersected by trajectories, and the other two are twodimensional singularities of class AA I • Parts of the abovementioned two lines are linear singularities of classes Ll and L 2 , the point 0 is a pointwise singularity, structurally unstable as in the previous case. The case ac = 62 and other cases can be treated similarly. In the general case the set on which = 0 may have a more complicated structure. Under the assumptions of 1 it consists of a finite number of smooth curves K. which enter the considered point pES with one end and, possibly, of a finite number of domains between some of these curves. In a topological classification of such cases one must take note of which of the domains between the curves Kl the vector I+(p) is directed towards. If in the neighbourhood of the point p there exist arcs of trajectories with both ends on 8 then one must also consider the position of the lines described by one end of such an arc when the other end moves along the curve K. or along a trajectory lying on 8.
tJ
Linear and Point Singularities
§22
279
3. On the surface S at some point p, let
(21)
r
fii = f"J = 0,
¥: 0,
that is, at this point the vectors f and f+ are tangent to S. On each side of S trajectories may be arranged as in the domain G+ in 2. When each of the arrangements considered in 2 for the domain G+ is combined with each one in G , we are led to a large number of cases which are split, 'in turn, into topological classes. Hence, we consider only the case of general position, where each of the functions fii and f"J vanishes on some line, these two lines intersect at the point p at a nonzero angle, the vectors f and f+ are not tangent to these lines, the gradients Vfii(p) ¥: 0, Vf+(p) ¥: O. Then on each side of S trajectories are arranged as in Lemmas 3 and 4 in the domain G+ (Figs. 106 and 107). On the surface S, we use the definition a), §4. The point p is stationary only if the vectors f(p) and f+(p) are oppositely directed. In other cases, in the neighbourhood of this point there are no stationary points. We shall now use new coordinates, in which the surface S is a plane z = 0, and the aforesaid lines are the z and yaxes. By an appropriate choice of the direction and scale on the axes, we obtain the system (22)
(23)
:i: = e + .. " :i: = a + ... ,
y = b + ... , y = e+ + . ",
z = z + mz + .. . z = y + nz + .. .
(z < 0), (z> 0),
where IeI = le+ I = 1, and the omitted terms near the point 0 are infinitesimal as compared with those written down. In the plane S (z = 0), in the second and fourth quadrants trajectories intersect the plane S, while in the first and third quadrants the plane S contains trajectories for which
.
(24)
y=
e+z+by+ ...
z+y+'"
,
z = o.
The point z = y = 0 is always nonstationary, except in the case ab = e e+, ae 1 in (34) we have 1  ab + ... < const < 0, hence, those trajectories which have not reached Rl intersect the planes z = 0 alternately in R2 and R4 , ultimately moving away from the point O. Stability is absent (under any definition of a system in Rl and R3). The case a < 0, b < 0, ab = 1 is critical. The point 0 is stationary, it may be stable or unstable depending on the higher order terms in (22) and (23). For instance, for the system :i; =
1:z;  y,
:i:=lxy,
y=lxy,
y=
1 x  y,
z=:z;z z = y  z
(z < 0), (z> 0),
defined for z = 0 according to a), §4, the point 0 is asymptotically stable. This is proved using the Lyapunov function
Singularities on an Intersection
§23
285
It can be shown that in the case e = 9+ = 1 the system (22), (23) of class 0 1 defined in the quadrants z = 0; xy ~ 0 according to a), §4, for any values of a and b is structurally unstable in any neighbourhood of the point 0 and that there exist infinitely many topological classes of such systems. We will consider the case where, as distinguished from (22), (23), both the equalities I;; = 0 and IJ = 0 are satisfied, not on different lines but on the same line L. This line or parts of it can be linear singularities. They are investigated by the same methods as in the cases a)c), 2, depending on the signs at Iii and I~ in the domains 8 1 and 82 and on the direction of vectors I(x) and I+(x) for x E L. The most complicated case, where trajectories in G+ go from 8 1 to 8 2 , while in G they go from 8 2 to 8 1 , has been discussed in [201], [202] by means of the pointmapping method. Sufficient conditions for asymptotic stability of a stationary point lying on L have been obtained. In this case a system is considered in an ndimensional space, and its righthand side is discontinuous on an (n  l)dimensional surface 8, Iii = I~ = 0 on an (n  2)dimensional hypersurface L. 4. Singularities of types 46 (1, §22) consist of stationary points lying on a surface of discontinuity. If such a point is isolated then its structural instability is proved as in the twodimensional case (Lemmas 1 and 2, §19). §23 Singularities on an Intersection of Surfaces of Discontinuity Linear and point singularities on intersection lines of a finite number of surfaces of discontinuity and pointwise singularities at intersection points of surfaces of discontinuity are considered. In some cases, the topological structure of the arrangement of trajectories near a singularity is established. Methods are proposed for investigating stability. 1. Let the line L E 0 2 be adjoined by a finite number of surfaces 8 i E 01, i = 1, ... ,p, which have no common points outside L (Fig. 122). These surfaces divide the neighbourhood of the line L into domains G., i = 1, ... , p, in each of which I(x) E 0 1 up to the boundary. The line L may also be an edge of a smooth surface of discontinuity; in this case p = 1. We will investigate such linear and pointwise singularities of a system (in vector notation)
a: = I(x)
(1)
which lie on L. On surfaces and lines of discontinuity one uses either the definition a), §4, or any definition satisfying the conditions 10 and 20 , 2, §21 (with m= 1). It is supposed that no two surfaces 8 i and 8 3 are tangent to one another at points of the line L. One of them is permitted to be a continuation of the other, like 8 1 and 8 s in Fig. 122. Stability of a stationary point lying on the line L can be analyzed using Theorem 9, §15. Let the line L pass along the xsaxis. Then, as in Theorem 9, §15, we put x = (y,z), where y = (Xl' X2), z = xs,
I(x)
= (g(y,z), hey, z)),
286
Local S£ngularities of ThreeDimensional. .. Systems
Chapter 5
Figure lff The system (1) is written as follows
(2)
y = g'(y, z),
) y. = g0, (y,z,
z= z=
h'(y,z) hOi(y,z),
(y,Z)EG" (y, z) E So,
i = 1, ...
,p,
i E I,
where I is the set of those values of i for which arcs of trajectories lie on surfaces St. For solutions lying on the line L we have
(3) the function hO(z) is assumed to be continuous. The point y = 0, z = Zo of the line L is stationary if hO(zo) = 0. To investigate stability of this point, we consider a first approximation to the system (2) in its neighbourhood. The surfaces S, are replaced by half planes S? tangent to Si at the point (0, zo), the domains Gi are replaced by dihedral angles G~ with sides S?, and the functions gi(y,z) and gOi(y,z) by the constant 9'(O,zo) and gO,(O, zo). Let Vi and L. be crosssections of the domain ~ and the side Sp by the plane z = Zo. THEOREM 1.
(4)
Let the solution y =
if = g'(O,zo), ) Y. = gOi (0 ,Zo,
°of the twodimensional system YE YE
i = 1, ... ,p, i E I,
Vi, L.,
be asymptotically stable, the solution z = Zo of equation (3) be stable (or asymptotically stable) and
(5)
g'(O,zo) :fO,
i
= 1, .. . ,p,
g
0'
'(0, zo) :f 0,
i E I.
Then the solution y = 0, z = Zo of the system (2), (3) is stable (respectively, asymptotically stable). The theorem is proved similarly to Theorem 9, §15. Theorem 1 reduces the study of stability of a stationary point (0, zo) E L to the study of the onedimensional equation (3) and the twodimensional piecewise constant system (4). The system (4) is investigated by the methods considered in §20. The most essential restriction in Theorem 1 is the requirement (5). IT this requirement is not met then t.he system (4) has stationary points arbitrarily close to the point y = 0, hence its solution y = cannot be asymptotically stable.
°
287
Singula.rities on a.n Intersection
§23
The case where (5) fails is a critical one, like the case where zero roots are present in the usual theorem on stability by first approximation. It should be investigated by other methods, this time taking account of the values of gi (y, z) for y :f. 0, z :f. ZOo 2. We will indicate one of the classes of linear singularities lying on the line L and describe the structure of these singularities. The simplest examples of linear singularities are those in which the surfaces Si are planes or cylindrical surfaces with generatrices parallel to the zaxis, and the functions gi, hi, gOi, hOi, h O are zindependent. Then any shift parallel to the to the zaxis maps trajectories of the system (2) into trajectories. The neighbourhoods of any two points of the zaxis have, therefore, identical structure, and this axis is a linear singularity (naturally, provided it does not lie on a surface or within a domain, all points of which have neighbourhoods of identical structure). We will specify cases where the neighbourhood of each point of the line L is foliated into surfaces z = Z(Yi >.) (>. being a parameter of the family) filled with arcs of trajectories of the system (2). It is natural to require that the function Z(Yi >.) be continuous and that the foliation be preserved under small variations of the righthand sides of the system (2). For this purpose it is necessary that near the line L there should be no such lines consisting of arcs of trajectories of the system (2) which would be projected onto the plane Z = 0 into closed curves (if nonclosed, such a line has endpoints (y, Zl) and (y, Z2), Zl :f. Z2, not lying on one surface Z = z(y, >.), and if closed, it is destroyed under small variations of the functions hi in (2)). On such a surface trajectories must not therefore form sectors of classes E, F, G, L, R,S near a singular point Y = 0 (2, §17). The condition (5) is necessary for a similar reason. These arguments account for the conditions imposed below. LEMMA 1. Let surfaces Si be half planes adjacent to the zaxis, let, in the system (2), (3), each of the functions gi,hi,gOi,hOi,ho be constant, each vector gi be noncollinear with the sides of the angle V. where it is defined, and the neighbourhood of the singular point Y = 0 of the system (4) consist only of sectors of classes H, K, P, Q. Then the zaxis is a linear singularity. Its neighbourhood is filled with a family of surfaces
(6)
z
= tp(y) + c
(00 < C< 00),
tpE
0,
each of which is filled with arcs of trajectories of the system (2). PROOF: Let M o be a set consisting of a point y = z = 0 and points (y,z) lying on trajectories of the system (2) which lie along the surfaces Si and enter the point Y = z = 0 as t increases or decreases. In each sector of class H, we draw a ray in the plane z = 0 which goes from the point 0 to intersect the trajectories of this sector at a nonzero angle. Adding to M o, the points which lie on arcs of the trajectories with one end entering points of the set M o, and on trajectories intersecting the rays constructed in the plane z = 0, we obtain Ii. set M. The projection of this set into the plane z = 0 fills all the sectors of classes H, K, P, Q. From the assumption concerning the functions gi, hi, ... and from the presence of these sectors only it follows that into each point of the plane
288
Local Singularities of ThreeDimensional . .. Systems
Chapter 5
z = 0 only one point of the set M is projected, that is, the set M is given by the equation z = !p(y). In each domain Gi located between the planes Si and 8i+1' the points of the set M fill a piece of a plane. The function !p(y) is, therefore, continuous and even satisfies the Lipschitz condition. Since any shift parallel to the zaxis maps trajectories into trajectories then for any c the surface z = !p(y) + c is also filled with arcs of trajectories and, as at the beginning of 2, the zaxis is also a linear singularity. The result follows. In the next theorem the surfaces 8i and the system (2) satisfy the conditions enumerated at the beginning of 1, the line L being the zaxis. THEOREM 2. Let for each Zo E (0:,.8) the vector gi(O,zO) be noncollinear with sides of the angle Vi (i = 1, ... , p), let the neighbourhood of the singular point y = 0 of the system (4) consist only of sectors of classes H, K, Q, and hO(zo) =1= O. Then the interval Q: < z < .8 of the zaxis is a linear singularity (or part of it) for the system (2). The neighbourhood of any point of this interval is filled with a family of surfaces, each of which is filled (topologically similarly) with arcs of trajectories of the system (2). In some neighbourhood of an arbitrary point of this interval, the system (2) is structurally stable. PROOF: For any zo E (0:,.8) we construct a system Ao of first approximation, as in 1, also by replacing hO(z) by hO(zo). Then we construct a topological mapping of the neighbourhood of the point y = 0, z = Zo, such that the points of the zaxis are carried into themselves, and trajectories of the system Ao are carried into trajectories of the system (2). Let x = ei(tjb) and x = e~(tjb) be solutions of the system (2) and of the system Ao which go along the surface 8i and the plane 8~, respectively, to pass through the point x = b at t = O. For small enough t (of the sign for which these solutions go along the indicated surfaces) each point a = e~(tj bIl, where b1 = (O,zIl, is assigned a point ,p(a) = t(tjbd. We deal similarly with trajectories which go from the point (0, Z1) directly into the domain Gi . On the rays lh lying in the plane Ie = Zl and projected into the rays constructed in the proof of Lemma 1 in sectors of class H in the plane z = 0, we put ,p(a) = a. Through each interior point of the sector K or Q there passes a trajectory of the system (4) which reaches the lateral boundary of the sector only with one end, and in the sector H there passes a trajectory which has only one common point with the constructed ray. Such a trajectory is a projection of the trajectory x = eo(tj a) of the system A o, which passes along the surface M1 (z = !p(y) +zd constructed in Lemma 1. The point a lies on the surface 8~ or on the ray lh' and at this point the mapping ,p(a) is already defined. The point x = eo(tj a) is assigned a point ,p(x) = e(tj ,p(a)) on the trajectory of the system (2). In some neighbourhood of the point (0, zo), ,p(x) is a onetoone mapping carrying trajectories of the system Ao into trajectories of the system (2). The point a, at which the trajectory reaches the boundary of the sector or is intersected by the ray lh' depends continuously on the point x = eo(tj a) by virtue of the noncollinearity condition, and the point b1 = (0, zI) depends continuously on the point a. From this and from the theorems on the continuous dependence of solution it follows that the mapping ,p(x) is continuous both on the surface Ml and in a whole neighbourhood of the point (0, zo) (the point Zl runs
§23
Singularities on an Intersection
289
through some neighbourhood of the point zo). By virtue of Lemma 1, §9, the mapping ",(z) is topological in a closed neighbourhood of the point (0, zo). It carries the family of surfaces z = tp(y) + c into the family of surfaces which fill some neighbourhood of the point (0, zo) and consist of arcs of trajectories of the system (2). For the system Ao all the points of the zaxis have neighbourhoods of identical structure by virtue of Lemma 1. The mapping 1/I(z) is topological and, therefore, for the system (2) all the points of the zaxis lying in the neighbourhood of the point So also have neighbourhoods of the same structure. Any compact part of a given interval a < z < P of the zaxis can be covered with a finite number of such neighbourhoods. Hence any two points of this interval have neighbourhoods of the same structure, and for the system (2) this interval is a linear singularity (or part of it). II the system (2) varies little in C; (1, §18) then the systems Ao and (4) also vary little. By the noncollinearity condition, all the sectors for the system (4) preserve their topological classes. By virtue of Theorem 2, §17, the topological structure of the neighbourhood of the point y = 0 for the system (4) remains unchanged. The same is true for the system Ao due to the presence of the foliation (6), and hence for the system (2) in some neighbourhood of any point of the indicated interval of the zaxis, since a topological mapping similar to '" (z) also exists for a modified system. To construct an £mapping carrying trajectories of the system (2) into trajectories of the modified system, one should, as at the beginning of the proof, construct this mapping first for trajectories lying along the surfaces Si and then for the rest of the trajectories. For sufficiently close systems this mapping shifts each point by less than £ by virtue of the continuous dependence of solutions on the initial conditions and on the righthand sides of the system. The system is therefore structurally stable in this given neighbourhood. From this theorem and from Theorem 2, §17, it follows that the topological classification of those linear singularities which exist under the conditions of the theorem is completely determined by the number and the cyclic order of the sectors of classes H, K, Q for the twodimensional system (4). Let the topological structure of the neighbourhood of the singular point 11 = 0 for the system (4) be different for Zo E (a,p) and for Zo E (P,'1), or let it be the same, but with the function hO(z) in (3) having opposite signs on the intervals (a, P) and (P, '1). Then the intervals (a, P) and (P, '1) of the zaxis belong to different linear singularities, and their common endpoint z = P is a pointwise singularity. The arrangement of trajectories near the intersection line of surfaces of discontinuity has not been given a more detailed consideration. Several systems of this kind used in applications (for instance, in [204]) and automatic control systems with two and more relay functions (for instance, in [5J, pp. 197 and 249) have been analyzed by some authors. 3. The pointwise singularity at an intersection point of several discontinuity surfaces has been treated mainly from the point of view of its stability [182J, [178J, [205J. Under a natural assumption that discontinuity surfaces (possibly, curved ones) are arranged as sides of polyhedral angles, an autonomous system
290
Local Singularities of ThreeDimensional . .. Systems
Chapter 5
in the neighbourhood of such a point is close to a homogeneous system. Several methods for studying stability of such systems are dealt with in §15. The arrangement of the trajectories in different cases is examined in [182] in order to obtain necessary and sufficient conditions for stability of a system with three discontinuous functions (sgn x, sgn y, sgn z).
REFERENCES 1. Andronov, A. A., Vitt, A. A., and Khaikin, S. E., Vibration Theory. Fizmatgiz, Moscow, 1959. (In Russian). 2. Barbashin, E. A., Introduction to Stability Theory. Nauka, Moscow, 1967. (In Russian). 3. Neimark, Yu. I., Points Mapping Method in the Theory of Nonlinear Vibrations. Nauka, Moscow, 1972. (In Russian). 4. Bautin, N. N., Leontovich, E. A., Methods and Techniques of Qualitative Study of Dynamic Systems in a Plane. Nauka, Moscow, 1976. (In Russian). 5. Gelig, A. Kh., Leonov, G. A., and Yakubovich, V. A., Stability of Nonlinear Systems with Nonunique Equilibrium State. Nauka, Moscow, 1978. (In Russian). 6. Theory of Systems with Variable Structure. Ed. by Emelyanov, S. V., Nauka, Moscow, 1970. (In Russian). 7. Utkin, V. I., Sliding Regimes in Optimization and Control Problems. Nauka, Moscow, 1981. (In Russian). 8. Sansone, G., Equazioni Differenziali nel Campo Reale. Parte 2. Bologna, 1949. 9. Coddington, E. A., and Levinson, N., Theory of Ordinary Differential Equations. McGrawHill, New York, Toronto, London, 1955. 10. Bokstein, M. F., "Theorems on existence and uniqueness of solutions of ordinary differential equations." Uchen. Zapiski Mosk. Gos. Univers. Mathem. 15 (1939), 372. (In Russian). 11. Krasnosel'skii, M. A., and Krein, S. G., "Nonlocal existence theorems and uniqueness theorems for systems of ordinary differential equations." Dokl. Akad. Nauk SSSR. 102 (1955), 1316. (In Russian). 12. Kamke, E., "Zur Theorie der Systeme gewohnlicher Differentialgleichungen." Acta Math. 58 (1932), No.1, 5785. 13. Hartman, P., Ordinary Differential Equations. John Wiley, New York, London, Sydney, 1964. 14. Krasnosel'skii, M. A., and Krein, S. G., "On the averaging principle in nonlinear mechanics." Uspekhi Mat. Nauk. 10 (1955), No.3, 147152. (In Russian). 15. Kurzweil, J., "Generalized ordinary differential equations and continuous dependence on a parameter." Czechosl. Math. Journ. '1 (1957),418449. 16. Kurzweil, J., "On generalized ordinary differential equations possessing discontinuous solutions." Prikl. Mat. i Mekh. 22 (1958), 2745. (In Russian). 291
290
Local Singularities of ThreeDimensional . .. Systems
Chapter 5
in the neighbourhood of such a point is close to a homogeneous system. Several methods for studying stability of such systems are dealt with in §15. The arrangement of the trajectories in different cases is examined in [182] in order to obtain necessary and sufficient conditions for stability of a system with three discontinuous functions (sgn x, sgn y, sgn z).
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302
References
203. Vishik, C. M., "Vector fields in the neighbourhood of the edge of a manifold." Vestnik Mosk. Gos. UnilJers. ser. matem. i mekh. (1972),No. 1, 2128. (In Russian). 204. Kinyapin, S. D., "On the stability of equilibrium of a twostage relay system." IZlJestiya Vysshikh Uchebnykh ZalJedenii, RadioJizika. 3 (1960), 511525. (In Russian). 205. Lozgachev, G. I., "Sufficient conditions for stability of one class of discontinuous systems." Sbornik TrudolJ VNII sistem. issled. 4 (1980). 2124. (In Russian). 206. Pandit, S. G., and Deo, S. G., Differential equations with impulses. SpringerVerlag, Berlin, 1982. , 207. Glodde, B., "Numerische Behandlung von Zweipunktrandwertaufgaben fiir gewohnliche Differentialgleichungssysteme erster Ordnung mit unstetiger Rechterseite." Zeitschrift fur Angew. Math. und Mech. 63 (1983), 559568. 208. Shumafov, M. M., "Diffeomorphisms of threedimensional discontinuous systems." Vestnik MGU, ser. mat., mekh. (1984), No.6, 8588.
INDEX Control system, 80, 96, 152 Convex combination, 62 Convex set, 59 Crosssection, 16, 79, 82 of a funnel, 16, 79, 82
A
Approximate solution, 75, 76, 82 Asymptotic stability, 152, 252, 257 Averaging, 95, 99
D
B
Degree of structural instability, 211 Deltafunction, 18, 41 Diffeomorphism, 187,265 Differential inclusion, 67 Double separatrix, 214 Dry friction, 53, 97
Basic conditions, 76 Bifurcations, 227, 232, 238, 241, 249, 281 Boundary, 68
C
E
Canonical neighbourhood, 202 Caratheodory equation, 4 Change of variables, 99 Characteristic exponent, 237 Classification of twodimensional singularities, 264 of linear singularities, 180, 185, 268 of distributions, 29 of singular points, 204, 218, 268 Closed sets, 59 Compactness of set of solutions, 7, 79, 82,84 Contingence, 69 Continuation of solutions, 7, 78, 82 Continuity, 65 upper semicontinuity, 65 Continuity up to the boundary, 49 Continuous dependence of solutions, 9,89,93,97
Eidentity, 209 Equivalent control, 54 Existence of the solution, 4, 77, 82
F Frequency method, 164 Frozen coefficients, 170 G
Graph, 65
H Half trajectory, 124 Hausdorff distance, 65 Homogeneous differential inclusion, 159 Homogeneous setvalued function, 159 Hyperplane, 49 Hypersurface, 49 303
302
References
203. Vishik, C. M., "Vector fields in the neighbourhood of the edge of a manifold." Vestnik Mosk. Gos. UnilJers. ser. matem. i mekh. (1972),No. 1, 2128. (In Russian). 204. Kinyapin, S. D., "On the stability of equilibrium of a twostage relay system." /zlJestiya Vysshikh Uchebnykh ZalJedenii, Radiojizika. 3 (1960), 511525. (In Russian). 205. Lozgachev, G. I., "Sufficient conditions for stability of one class of discontinuous systems." Sbornik TrudolJ VNIlsistem. issled. 4 (1980). 2124. (In Russian). 206. Pandit, S. G., and Deo, S. G., Differential equations with impulses. SpringerVerlag, Berlin, 1982. , 207. Glodde, B., "Numerische Behandlung von Zweipunktrandwertaufgaben fur gewohnliche Differentialgleichungssysteme erster Ordnung mit unstetiger Rechterseite." Zeitschrift fur Angew. Math. und Meeh. 63 (1983), 559568. 208. Shumafov, M. M., "Diffeomorphisms of threedimensional discontinuous systems." Vestnik MGU, ser. mat., mekh. (1984), No.6, 8588.
INDEX A
Control system, 80, 96, 152 Convex combination, 62 Convex set, 59 Crosssection, 16, 79, 82 of a funnel, 16, 79, 82
Approximate solution, 75, 76, 82 Asymptotic stability, 152, 252, 257 Averaging, 95, 99
D
B
Degree of structural instability, 211 Deltafunction, 18, 41 Diffeomorphism, 187,265 Differential inclusion, 67 Double separatrix, 214 Dry friction, 53, 97
Basic conditions, 76 Bifurcations, 227, 232, 238, 241, 249, 281 Boundary, 68
C
E
Canonical neighbourhood, 202 Caratheodory equation, 4 Change of variables, 99 Characteristic exponent, 237 Classification of twodimensional singularities, 264 of linear singUlarities, 180, 185, 268 of distributions, 29 of singular points, 204, 218, 268 Closed sets, 59 Compactness of set of solutions, 7, 79, 82,84 Contingence, 69 Continuation of solutions, 7, 78, 82 Continuity, 65 upper semicontinuity, 65 Continuity up to the boundary, 49 Continuous dependence of solutions, 9,89,93,97
Eidentity, 209 Equivalent control, 54 Existence of the solution, 4, 77, 82
F Frequency method, 164 Frozen coefficients, 170
G Graph,65
H Half trajectory, 124 Hausdorff distance, 65 Homogeneous differential inclusion, 159 Homogeneous setvalued function, 159 Hyperplane, 49 Hypersurface, 49 303
Subiect Index
304
I
Index of a singular point, 145, 146 Integral funnel, 16
J Jumps of solutions, 18, 28, 41 L
Limit point, 129 Limit set, 129 Linear singularity, 176, 260 Lipschitz condition, 80 Lyapunov functions, 153157
M Metric C:', 206 Minimal set, 133 P
Paratingence, 69 Periodic solutions, 19, 148152 Piecewise smoothness, 178 continuity, 49 Pointmapping method, 164 Pointwise singularity, 177, 205, 260 stability, 154 Poly trajectory, 214
Setvalued function, 65 Sewed focus, 234 Singular point, 145, 146, 192, 206 Sliding motion, 51, 80 Solution, 4, 4956, 67 asolution, 76, 82 Stability, 152, 154, 169 Stationary point, 124, 154, 176 Structurally stable singular point, 210 system, 205, 210, 217 Succession function, 200, 216, 234 Support plane, 61 function, 71
T Tangent vector, 115 Topological homogeneity, 176 structure, 184 mapping, 176 Trajectory, 124 Transversal, 134 Twodimensional singularity, 260
u Uniqueness of the solution, 5, 106
v Variation of solutions, 117
Q
W
Quasitrajectory, 81
Weak stability, 152
R Recurrent trajectory, 133 Retardation, 95 Right uniqueness, 106 Rotation of the vector field, 144148
S Sectors, 192 Segment of a funnel, 9, 16, 79, 82 Semicontinuity, 65 upper semicontinuity, 65 Separatrix, 191
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