Stability By Liapunov's Direct Method with Applications (Mathematics in Science and Engineering 4)

LStability by Liapunov's

Direct Methot!J With Applications]

JOSEPH LA SALLE

and SOLOMON LEFSCHETZ RIAS, Baltimore, Maryland (Research Institute [or Advanced Studies, The Martin Company]

1961

New YOrk

ACADEMIC PRESS

London

COPYRIGHT© 1961 BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS ACADEMIC PRESS INC.

111 FIFTH AVENUE NEW YORK, NEW YORK 10003

United Kingdom Edition Published by ACADEMIC PRESS INC. (LONDON) LTD. BERKELEY SQUARE HOUSE, LONDON

W.

1

Library of Congress Catalog Card Number: 61-18296

First Printing, 1961 Second Printing, 1963 Third Printing, 1964

PRINTED IN THE UNITED STATES OF AMERICA

Preface

In the development of the subject of differential equations one may distinguish two broadly distinct streams. On the one hand, there is the endeavor to obtain a definite solution, or one of definite type, either "in closed form," which is rarely possible, or else by some process of approximation. On the other hand, abandoning all, endeavors to reach an exact or approximate solution, one strives to obtain information about the whole class of solutions. This is the qualitative theory, initiated by Poincare around 1880 and pursued with varying degrees of energy ever since. A major problem in the qualitative theory is this: given a solution, what is its relation to its neighbors? A solution is a curve or trajectory C in some space. One asks then whether the trajectories D starting near C do tend to remain near C (C is then stable) or to depart from C (C is then unstable). This plants squarely the problem of stability within the qualitative theory and here it is not too much to say that the true creator of a stability theory is Liapunov. The starting point is his fundamental memoir (1892, in Russian; 1907, French translation: Probleme general de la stabiliU du mouvement. Photo-reproduced as Annals of Mathematics Study No. 17, Princeton Univ. Press, Princeton, New Jersey, 1949). This is, so to speak, the bible of the Soviet school, decidedly the outstanding school of "differential equationists" in the world today. In his paper Liapunov dealt with stability by two distinct methods. His so-called first method presupposes an explicit solution known and is only applicable to SOme restricted but important v

VI

PREFACE

cases. As against this the second or direct method of Liapunov is of great generality and power and above all does not require the knowledge of the solutions themselves - a manifest advantage. The purpose of the present modest monograph may now be described as expounding the main lines of Liapunov's stability theory and of his direct method, and making them accessible to technical people with some mathematical equipment. It will not be possible however entirely to run away from mathematics. To facilitate matters along this line, we have briefly recalled the inescapable mathematical essentials with which it is reasonable to suppose that most of our readers have a certain familiarity. We are especially pleased that our book appears in a series inspired and edited by an old and cherished friend, Richard Bellman. We would also like to acknowledge the ease of working with Academic Press and the fine job that they have done in presenting accurately and attractively what we have done.

ERRATA The lines below indicated by a minus sign are to be counted from the bottom.

Page vi line 8, should read: "possible however to run away entirely". Pag.e vi,. add after line 11: "In this regard it is quite possible and perhaps advisable to begin with Chapter 2 and to refer to Chapter 1 as required." Page 9, line 10 - replace "Ai;" by "A;i" Page 11, line -5 - replace "yx" by "y's" 1, Page 38, line -7 should read: "The function V(x) is bounded in each S(R), R < A. Take any r s R. Page 47, line -1 should read: "VI= 2(r,x,2 + ..." Page 47, line -10 should read: "V = 2(r,x,2 + ., ." Page 53, equation on line 7 should read: "V, = Y22 - By.2 = ci". Page 54, inequality on line -8 should read ; "A(a - b)(n-b) < 0" Page 55, line -4: " ... is a positive definite k-form." should read " ... is a negative definite k-form." Page 55, equation at bottom of page should read: "V = 2T2(p) - kWk(q) + 3T.(p) - (k+l) Wk+,(q) + ...." [2 "+" signs changed to "-" signs.] Page 56, line 1. "2T 2 + kWk" should read "2T 2 - kWk". Page 56, equation on line 18 should read: "V = - [- 2T2(p) + kWk(q) + ...] H." Page 60, line 4-replace "restricted" by "restrictive". Page 67, lines -5, -4: Transpose (2) and (3); i.e. should read: " ... ; (2) the damping is always positive; and (3) the potential energy approaches infinity with [z]." Page 79, last sentence of §15 should read: " ... f( a) satisfying ITf( IT) > 0 for IT '# O. The proof is greatly simplified by assuming (15.5) but is not actually required. It also turns out that the sufficient condition for absolute stability will imply that (15.7) is automatically satisfied, and therefore this inequality plays no role. The proof of this is a bit complex and is not given here." Page 83, line 8: "qolynomial" should read "polynomial". Page 83, line 13: "(17.3)" should read: "(17.2)". Page 85. Below (18.2) should read: / "LaSalle has shown that this inequality implies that T > - c' A x 2 , • • • , X n with the n X 1 matrix, or column vector

One refers to the transpose

as a row-vector, If x is as above and Y is the row-vector (Yl> Y2" .. , Yn) then the matr,ix product yx is the same as the inner product Y' x.' If A is an n X n matrix then the product A x is again an n X 1 matrix or column-vector. Thus A has the effect of transforming n-vectors into n-vectors. This transformation has two distinct interpretations.

12

1. GEOMETRIC CONCEPTS: VECTORS AND MATRICES

(a) Transformation of coordinates. This interpretation is only valid when A is a nonsingular matrix. There are in play two distinct sets of coordinates XII •.. , X n and Xl *, ... , X n * for the same space En and one passes from one system to the other by a set of n equations (we replace A by a matrix P): Xl X2

* = Pn Xl + Pl2 X 2 + ... + pIn X n * = P21 Xl + P22 X 2 + ... + p2n Xn

(3.1)

Here the point X or vector X is not changed but merely receives new coordinates x*. Let P = (PM)' A requirement of this situation is that one must be able to obtain also the Xh from the Xk*, that is, that one must always be able to solve the system (3.1) uniquely for the Xh' As is well known this requires that the determinant iPI =f= 0, that is, that P be nonsingular. This is why the condition was imposed at the outset. Here we think of X and x* as merely different representations of the same vector. These representations are related by X*

= Px,

X =

p-l

x*,

the first being merely "short hand" for the relations (3.1). One thinks then of a "vector world" with an existence independent of the coordinate system. A familiar case is the usual representation of velocities, accelerations, forces, and electric or magnetic fields by a pointed arrow, with a (tacit) existence independent of the coordinate system. (b) Transformation of vectors. In this interpretation we actually pass from a vector X to a new vector y in the same space by means of relations (i

=

1,2, ... , n)

or vectorially

y=Ax.

(3.2)

§ 3.

13

RELATION BETWEEN VECTORS AND MATRICES

This time we do not expect to be able to reverse the process, that is, to solve for x in terms of y. Hence A may well happen to be singular. \Ve have then an actual transformation of the vector world into itself. Suppose now that we make a change of coordinates represented by (3.1) so that x, y go into x*, y*. We have then

x* = Px,

y* = Py

and hence

y* = PAx = PAp-1 x*. Thus in the new coordinates the effect of the transformation is now represented by the new matrix A * = P A P:", which is said to be similar to A. Let us look at the characteristic equations. We have identically

PAp-I-rE = PAp-l_ P(rE)P-I = P(A -rE)P-I. Hence for the determinants

JPAP-I-rEi = IPp-II·jA -rEI = IA -rEI' Thus the two similar matrices A and PA p-l have the same characteristic equation. Suppose now that the characteristic roots of A are distinct, and at first let them be real. One may then show that the matrix A is similar to diag h, ... , rn ) . This means that in an appropriate coordinate system the effect of A will assume the form ... ,

That is, in this special system the effect of A is to multiply by r» the components of a vector in the direction of the axis x". Suppose now that the roots r" are not all real. If c = a ib let us agree to write c = a - ib, the complex conjugate of c. Since the characteristic equation has real coefficients, it is known from algebra that its roots occur in conjugate pairs. Let there be p conjugate pairs r l , i I ' r2 , i 2 , ••• , rp , i p and q = (n - 2P) real

+

14

1. GEOMETRIC CONCEPTS: VECTORS AND MATRICES

roots r2p + l , ••• , rno One may then choose a complex matrix P so that PAp-l = diag (rl,,1"'" r n ) . The associated transformation assumes then the form ..

"

•. 0'

Yn

=

r« x n ·

The real points have coordinates like

where X2P+l,' •• , may be written

Xn

are real. The transformation on these points

(c) Matrices and quadratic forms.

'.0'

yp = f p xp,

.• 0,

Yn

= rn x n .

Let

be a quadratic form and let Q = (qii)' If we denote as usual by x the column vector

(::) then we have F(x) = x' Qx. If one applies a transformation of coordinates x = Py, P nonsingular, then identically G(y) = F(Py) = y'(PIQP)y. Thus G(y) is a quadratic form whose matrix is PIQP. That is, the transformation of coordinates x --->- y, (x = Py) gives rise to the transformation of the quadratic form matrix Q --->- P'QP = Ql' The inverse transformation y --->- x, (y = p-l x), gives rise to Ql --->- P'< Ql P-l. We recall that by a suitable real transformation P one may reduce F(x) to the type

(3.3)

§ 3.

RELATION BETWEEN VECTORS AND MATRICES

15

where, if Pis the number of positive dh and q the number of negative dh , then p - q does not depend upon the special reduction to the above type, and p + q = m is the rank of Q.

Orthogonal transformations are of particular interest. They are characterized by matrices Q such that QQ' = E (unit matrix of order n) or Q' = Q-l. Their most important property is to leave the identity matrix E unchanged. That is, they do not change the form of H(x) = X 1 2 + ... + x n 2 • Indeed the matrix of H(x) is E and Q'EQ = Q-l EQ = E. Hence

H(x) = H(Qy) =

y12

+ ... + Yn

2

•

Orthogonal transformations are then those which do not modify distances. If we think in terms of dynamics, they are the rotations - or reflections - motions which preserve rigid bodies with a fixed point. We merely assert that there is always an orthogonal transformation reducing a given quadratic form to the type shown above (3.3). It may be noted that in two-dimensions, orthogonal transformations are those which reduce central conics (the origin is the center) to the form referred to the axes of symmetry: ( -Ax 2

+ By

2

= C

(x, yare scalars: the usual rectangular coordinates). The property is true for three dimensions, and, in fact, for any n (although for n :» 3 this may not be a familiar fact). A quadratic form F(x) is said to be positive definite or negative definite whenever F(x) > 0 or F(x) < 0 for all vectors x =1= O. The fact is often briefly indicated by writing for the matrix Q of F(x), 'Q > 0 or Q < 0, and stating: Q is positive or negative. Referring to the change of coordinates already utilized, x = 0 and y = 0 imply one another. Hence if F(x) is positive or negative definite so is G(y) and conversely. That is

Q >0,

P'QP >0

16

1. GEOMETRIC CONCEPTS: VECTORS AND MATRICES

are equivalent. Similarly, of course, with the "greater than" sign replaced by a "less than" sign. This means that in many questions involving quadratic forms one may freely apply transformations of coordinates. It is easy to construct positive matrices. If D is nonsingular, then D'D > 0. This is easily seen, since x'D'Dx = IIDxll2 is nonnegative and vanishes only if Dx = 0. The matrix D is nonsingular, and Dx = 0 implies x = 0. Therefore, D'D > 0. The converse of this statement is also true: if Q > 0, then there is a nonsingular matrix D such that Q = D'D. We know that there is an orthogonal matrix P such that P'QP = Q* = diag (d1 , d 2, ... , dn).

V

,V

Since Q> 0, the di are all positive. DefineD*=diag(Vd v d 2, . . . dn}. Then D* is nonsingular and symmetric (D*' = D*) and D*2 = Q*. Hence Q = PQ*P' = PD*D*P' = D'D, where D = D* P'. An immediate consequence of this result is that if Q is positive definite, then its inverse Q-l is also positive definite. The matrix Q is positive definite, and therefore Q = D'D for some nonsingular matrix D. Hence Q-l = D-l(D-l)', and by what we have just shown above this implies that Q-l is also positive definite. (d) Practical information. Let A = (aij) be an m x n matrix. It is often convenient to know that diag (b v ... , bm ) • A = (bi aij) ; A . diag (cv ... , cn) = (aij Cj).

That is, in the first case the terms of row i are multiplied by b., while in the second case the terms of column j are multiplied by Cj' Of course, if b1 = ... = b.; = b then all the terms are multiplied by b, while if C1 = ... = Cn = c, all the terms are multiplied by c.

§ 4.

17

A LITTLE GEOMETRY

§ 4. A Little Geometry Matrices are, after all, just an algebraic mechanism. Differential equations, however, cannot be ruled by pure algebra, and on the contrary, make an ample appeal to geometry. Above all the stability notions and related propositions of Liapunov are really of geometric nature. To understand them properly one must have recourse to a certain number of new concepts. We have referred to them as "geometric" but in fact they are, more or less, concepts borrowed from that extended geometry called topology. By the way, rather than say "figure" or "configuration" we shall use the more standard and simpler mathematical term "pointset" or even merely "set"; it denotes in fact any collection of points. Consider first a rather simple situation: the Euclidean plane £2 and in it a circle of center C(a, b) and radius r. Its equation is (4.1)

The points M nearer than r to C make up a circular region and their coordinates satisfy the relation d(M, C) < r. or equivalently d 2(M, C) < r 2• Analytically this relation assumes the form (x - a) 2

+ (y -

b)2 < r 2 •

(4.2)

In ordinary 3-sp?-ce we would have in place of the circle a sphere and in place of (4.2) the inequality (4.3) The analogue of the circle or sphere in n-space is the locus called hypersphere or (n - I)-sphere represented by (x 1 - a1)2+(x 2-a2)2+ ...

+ (x

n - an ) 2 = r

2

(4.4)

and the analogue of (4.2) or (4.3) is the spherical region made up of points x such that (4.5)

18

1. GEOMETRIC CONCEPTS: VECTORS AND MATRICES

If we think of x - a as a vector (4.5) may be replaced by

fix-ail < r.

(4.6)

One refers to a as the center of the spherical region, to r as its radius and denotes the spherical region in question by S(a, r). A set contained in some spherical region is said to be bounded.

s(C, r) ,

I

\

C '~V;'\ U I

,_ .......

I

I

o

FIG. 2.

We may now define a region of n-space. It is a point-set U of the space with the following two properties: (a) if C is a point of U then some spherical region of center C, S(C, r) is entirely in U; (b) any two points C, D of U may be joined by an arc which lies entirely in U (Fig. 2). Spherical regions will now enable us to define a certain number of basic point sets of n-space En. Open set: It is a set U with the u property that if a point C is in U, then a whole spherical region S(C, r) for some r is also in U [property (a) holds but not necessarily property FIG. 3. (b)]. Thus the interior of a square, or for that matter of any polygon in the plane E2 is an open set of E2.

§ 4.

19

A LITTLE GEOMETRY

Closed set: A closed set F is simply the outside of some open set U. Thus a straight line in any En, a plane in En are closed sets. Boundary BU of an open set U. The totality of the points C which are not in U but such that every spherical region S(C, r) contains points of U. Note that BU is a closed set. Compact set: A closed and bounded set. Simple examples are any bounded open set together with its boundary. Thus a spherical region plus its boundary is a compact set. This set is sometimes called a solid sphere. Compact sets are very important and have many remarkable properties. However, we shall only require the following: Let A be compact and let f(x) be a continuous (scalar) function on A. There can be found two numbers IX, (J such that IX < f(x) < (J for every point x of A. Moreover, if f(x) is positive at every point of A then one may select both IX and {J positive.

Application: Consider again the transformation of coordinates

We propose to compare the moduli Ilxkll and Ilxk*11 in the old and new coordinates. We shall actually compare their squares

Let us set

Xh

=

RUh

so that (4.7) n

RU =

n

2:(2: Phk UkY R2 = h=l

R2 f(u).

k=l

Notice now that since the determinant

IPhkl

#- 0 the system

n

h

=

1,2, ... , n

20

1. GEOMETRIC CONCEPTS: VECTORS AND MATRICES

has the unique solution: every Uk = O. Since this is incompatible with (4.7) one concludes that j(u) is always positive on (4.7). Therefore, there exist positive constants IY., fJ such that on (4.7) IY.

< j(u) < fJ,

and hence «u»

< R*2
"" X n by the relations Xl =

X,

This makes it possible to replace the system (*) by

in = j(x l , x 2 '

••• , X n,

t)

which is a special case of the system (**). As an example the well known van der Pol equation

x+k(x 2 - 1 ) i + x = O may be replaced by

which is of the second type. Regarding the second type, if one views xl>' • " X n as components of an n-vector x, and Xl" .. , X; as those of an n-vector X, it assumes the simpler aspect

=

i

X(x, t)

(F)

and this is the fundamental system discussed in the sequel. It may well happen that X depends upon X alone and not upon the time. The equation (F) assumes then the simpler form i

=

X(x).

(FA)

A system of this nature is known as autonomous. For instance the system deduced from the van der Pol equation is autonomous.

§ 5.

23

GENERALITIES

It is only fair to say that for a long time mathematicians were not concerned with the fundamental problem of the very existence of a solution of a system such as (F). Such outstanding men as Lagrange and Laplace appear to have taken the existence of such solutions as granted. Even recently astronomers accepted the existence of very special solutions with definite periodicity properties. At the beginning of the last century the great French mathematician Cauchy proved for the first time an adequate theorem regarding the existence of appropriate solutions for a broad class of systems. Observe that the problem is actually a little vague. Experience with the simplest systems with constant coefficients, for instance .~\

= aXI

+ bx

2,

x2 =

cXI

+ dX

2

shows that the solutions occur with a certain number of arbitrary constants. One may then ask for a solution doing this and that at a certain time. Since Cauchy, various ever sharper existence theorems have been developed. We are going to state one, without proof, which is a particular case of a classical proposition given the name of the Theorem of Cauchy-Lipschitz. It is not the best such theorem but quite ample for our purpose. Let E~.tl denote the space with the coordinates Xl>' •. , X n , t and let Q be a region of that space.

Suppose that at every point of Q there exist continuous partial derivatives aXh/aXk' Let (XO, to) be a point of Q. Then there exists a unique soluti~n x(t) ojthe system (F) such that x(to) = xc, and it may be extended throughout Q. This solution is a continuous function of (XO, to) as this point varies in Q. Geometric interpretation. A solution x(t) determines Xl>' . " Xn as functions of the parameter t. Write for a moment EXISTENCE THEOREM:

.. . ,

Xn

= fn(t) .

(5.1)

In E~.tl this expresses the first n coordinates as functions of the last coordinate t. For n = 1 we would have, for instance, Xl =

fl(t)·

24

2. DIFFERENTIAL EQUATIONS

If one writes x for t, and y for x, this takes the familiar form

y

=

t(x),

representing a curve in the (x, y) plane. Similarly and naturally the system (5.1) represents a curve in the space E:,tl (Fig. 4).

Xz

/------·x,

FIG. 4.

Such a curve will be referred to as an integral curve of the system (F). One may however consider the system (5.1) as defining a curve

Xz

FIG. 5.

in the space En of the variable x alone (Fig. 5). One will then refer to it as a trajectory or a motion, and the space En as the phase space (terms inspired by dynamics).

§ 5.

A utonomous system. be written

GENERALITIES

25

The autonomous system (FA) may also

with dt eliminated. This means essentially that one gives up the parametrization by t. Going one step further one may write it as

... , which is a system analogous to (F) but with t replaced by xn . The trajectories are now conveniently called paths, and as before, one and only one passes through a given point of a suitable "existence region" [,)1' The variable t is now merely a parametrization variable and if one changes it by,a constant amount (replacing t by t + C) one does not modify the paths. In dealing with stability we shall constantly consider a system (F) with the special property X(a, t) = 0 for all t > O. Thus x = a will be a solution. It is a ray (half-line) in the space E~,tl and a point, x = a, in the space En (of x). Such a point is called a critical or equilibrium point. By replacing x - a by x*, then again x* by x, the system continues to be of type (F) but with X(O, t) = 0 for t > 0: the equilibrium point is the origin. This is the general assumption prevailing later. It may, and often does, happen that one may represent X(x, t) as a function of x, near the origin, in the form X(x, t)

=

Ax

+ q(x, t)

where A is a constant matrix and Ilqll is small relative to Ilxli when the latter is small. To simplify matters let us assume that Ilq(x, t)II/IIxll tends to zero with Ilxll for t > O. This is expressed by means of a standard mathematical symbolism as

Ilq(x, t)l[ =

o(llxllJ·

26

2. DIFFERENTIAL EQUATIONS

Assume, in addition, that the characteristic roots Y v . . " r; of A are all distinct. There exists then a nonsingular matrix P such that P-lAP

=

diag (Y v "', yn ) .

Assume first that the characteristic roots are all real. Then the matrix P can be chosen real. Apply now the transformation of coordinates Y = Px to the system i

=

Ax

+ q(x, t).

(5.2)

In view of (djdt) (Px) = Pi, (5.2) yields

y=

Pi

= FAl>:! y + Pq,

= diag h, ... , yn)y + ql(Y' t), where ql

=

(5.3)

Pq and it may easily be shown that

Thus (5.3) is of the same nature as (5.2) save that A has been replaced by a diagonal matrix. If some of the characteristic roots are complex, the situation will be the same save that there will occur some conjugate pairs such as Y1, f v Y2, f 2 , • • • , and that for real vectors the coordinates Yv;lv Y2' )1 2 , ••• , will be conjugate complex. Linear equations with constant coefficients. These are the systems in the expanded form

i = 1,2, ... , n

where the coefficients are constants. (5.4) may be written more compactly i = Ax,

A

=

(5.4)

In vector-matrix notation (aij).

(5.5)

We recall briefly a few of the more salient properties of these systems. The proofs are well-known and are found in most texts on the subject. I. Let AI,"" l.n be the characteristic roots of A: the solutions of

1(.1) = IA -.1£1 =

o.

§ 5.

27

GENERALITIES

Each root is counted of course as often as its multiplicity. Every solution of the system (5.5) has for components sums of at most n terms of the form gj(t) exp (Ajt) where gj is a polynomial of degree < n. II. There exist n linearly independent solutions; that is, n vector solutions X(l), ••. , x(n) which satisfy no relation

where the c, are constants which are not all zero. III. Let Xlj, • • " Xnj be the components of xU) and let X be the matrix whose jth column consists of these components. The matrix X(t) is nonsingular for each t; that is, IX(t) I#-O whatever t and X(t) always has an inverse. IV. One verifies immediately that X satisfies the matrix equation

X=AX. V. If X is any solution of (5.6), so matrix C.

(5.6) XC for any constant

IS

VI. The particular solution X*(t) = X(t) . X-l(O), often written eAt, has the property that X*(O) = E, and it is the only solution with this property. One refers to it as the principal matrix solution. X*(t)c is the solution of (5.5) satisfying x(O) = c.

The adjoint system. Side by side with (5.4) one may conveniently consider its adjoint system (i=1,2"",n)

(5.4)'

or in matrix form (y is now a row-vector)

y=

-yA.

(5.5)'

The treatment of (5.4)' is the same as that of (5.4) save that rows and columns are interchanged and right matrix products replaced

28

2. DIFFERENTIAL EQUATIONS

by left matrix products. All the previous properties hold and will be written I', ... , VI'. The matrix equation is Y=-YA.

(5.6)'

The characteristic roots are the solutions of I-A -

AEI = (-I)"IA

+ AEI =

so that they are just - Av . . " -

A".

(-I)" j(- ).)

=

0,

Hence

I'. Every solution has for components sums of at most n expressions gj(t) exp (- Aj t) where gj(t) is a polynomial of degree < n, Notice that the n independent solutions of (5.5)' are now the rows of Y. Let X be the principal solution of (5.6). One may show that X-I is the principal solution of (5.6)'. In fact, if Y(t) = X-l(t) then at all events Y(O) = X-l(O) = Er? = E. In addition, from XY

=

E follows

O=.KY +XY =AXY +XY=A +XY,

Y = - X-I A = - YA and this proves the asserted property. X-l(t) solution of the adjoint equation (5.6)'.

=

e-A t is the principal

§ 6. General Considerations on Stability The term "stability" is so expressive that it almost tells its own story. A device of some sort operates under certain general conditions. These conditions are slightly changed. Does the change have little or considerable effect upon the device? In the first case it is thought of as stable, in the second as unstable.

§ 6.

GENERAL CONSIDERATIONS ON STABILITY

29

How does this apply to a physical system? The system will depend, let us say, upon a certain number of physical parameters Xl" .. , X n positions and velocities - and this will be represented by a point or vector X in some space E"n. The state of a system at time t will be x(t). The point x(t) will produce a trajectory g in the space E"n. And now the question is this: how do the trajectories g* which start near g behave with respect to g? Do they, as time goes, remain very near g, which is stability; or do they drift away from g, which is instability?

9

FIG. 6.

The analytical treatment of the problem assumes the following aspect. First one restricts the treatment to situations governed by differential equations. That is, one supposes at the outset that the trajectory g and its neighb~rs are solutions of an equation of type (F) i

=

X(x, t).

(F)

Then if j(t) is the special solution corresponding to g, that is, to the fundamental situation whose stability is under discussion, one makes the change of variables (Fig. 6) y=x-j(t),

X =

y

+ j(t).

This replaces (F) by

Y + t'(t)

=

X(y

+ j(t), t)

30

2. DIFFERENTIAL EQUATIONS

which takes the same general form

y = Y(y, t),

Y(O, t) = 0,

°

(F)

save that now g is the special solution y = and the stability to be dealt with is that of the origin. The most practical and also most frequent case before us deals with an autonomous system

x=

X(x)

(FA)

in which the parameters Xl" •• , X n have reached permanent (fixed) values av . . . ,an' In other words, the point X = a is a solution: X(a) = 0. If initially the system is in an equilibrium state a, it remains in that state. This is a mathematical statement. A real system is subject to perturbations and it is never possible to control its initial state exactly. This raises the question of stability: under a slight perturbation will the system remain near the equilibrium state or not? This is the problem to be discussed extensively below.

§ 7. Stability in Autonomous Systems We have then the autonomous system (FA) and the equilibrium state X = a whose stability is to be treated. It is convenient at the outset to take the fixed point under discussion as the origin. This merely requires us to make the simple transformation of coordinates x* = x-a. We do this, replace afterwards x* again by x and thus have to deal with the basic system

x=

X(x),

X(O) = 0,

(FA)

and the matter under discussion is the stability of the origin. It is pertinent to make a few simple assumptions. Generally speaking we shall denote by S(R) the spherical region Ilxll < R and by H(R) the sphere I!xll = R itself. The closed spherical annular region r < Ilxll < R will be written 5/. We shall suppose that in a certain open spherical region £2: Ilxll < A (it is really S(A)) the basic existence theorem holds

§ 7.

31

STABILITY IN AUTONOMOUS SYSTEMS

for (FA), and in particular we note that the partial derivatives oXi/oxi all exist and are continuous in Q. We also recall that through each point x of Q there goes a unique path g of the system

STABLE

--- --- ........

ASVM PTOTICALLY STABLE

H(A)

SeA}

FIG. 7.

(FA). We shall designate by s: the part of g described by x(t) when t > 0, and by g- the part described by x(t) when t < 0. We shall say that the origin is:

stable whenever for each R < A there is an r < R such that if a path (a motion) g+ initiates at a point xO of the spherical region S(r) then it remains in the spherical region S(R) ever after; that is, a path starting in S(r) never reaches the boundary sphere H(R) of S(R) (Fig. 7);

32

2. DIFFERENTIAL EQUATIONS

asymptotically stable whenever it is stable and in addition every path e: starting inside some S(R o), R o > 0, tends to the origin as time increases indefinitely (Fig. 7); unstable whenever for some R and any r, no matter how small, there is always in the spherical region S(r) a point x such that the path g+ through x reaches the boundary sphere H(R). EXAMPLE

1. Take the planar system (x and yare now ordinary

scalar coordinates):

x = y, We have

xx + yy =

y=-x.

0, so that the paths are

or circles centered at the origin. Moreover, the origin is the only critical point. If we take r = R then any circle through a point of S(r) remains in S(r) and hence in S(R) and so we have stability. However, since the paths do not tend to the origin, the latter is not asymptotically stable (Fig. 8). Consider now the system

x=

-x,

y=-y.

The solution is x = Ae- t , y = Be- t • The paths are yfx = BfA = k: rays through the origin (Fig. 9). We may take again r = R. Any path from a point x in S(R) remains in it and tends to the origin. Hence we have asymptotic stability. In fact, we have asymptotic stability "in the large" - every solution tends to the origin. Take finally the system

x=

x,

y=

y.

The solution is x = Ae t , y = Bet. The paths are again yfx = k: rays through the origin (Fig. 10), but they are followed in reverse: given R and no matter how small r, a path from any point in S(r) reaches the sphere H(R), and so we have instability.

§ 7.

STABILITY IN AUTONOMOUS SYSTEMS

33

The theorems of Liapunov aim to reduce the properties just discussed to those of certain functions which must first be dealt with.

o FIG~

FIG. 8. Stability.

§~.

A

Special Type

9. Asymptotic stability.

FIG. 10. Instability.

of Function

A very important role then will be played by scalar functions V(x) said to be positive definite with the following properties: (a) V(x) is continuous together with its first partial derivatives in a certain open region Q about the origin. (b) V(O)

=

o.

(c) Outside the origin (and always in Q) V(x) is positive. In other words, V is non-negative and vanishes only at the origin. The origin is an isolated minimum of V. Since V has first partial derivatives it has a gradient: grad V, and we recall that along the paths g of (FA) we have V=X'grad V. If in addition

V < 0 in

Q, V is called a Liapunov function.

34

2. DIFFERENTIAL EQUATIONS

Let us describe the geometrical meaning of the function V(x). To that end it is convenient to introduce a new coordinate z = V(x) and consider the nature of this locus in the space (Xl"'" X n , z) or (X, z). To see more effectively what happens take first the case n = 2 and instead of denoting the coordinates by Xl> X 2 write them in the more convenient manner x, y. Thus we have to say something about the surface

z = V(x, y) near the ongm when V is positive definite. There is no difficulty here. Since V > for x, y small and V = only for X = Y = 0, the surface (Fig. 11) has the general aspect of a par-

°

°

'raV (x,y) FIG. 11.

FIG. 12.

abolic mirror pointing upward or of a cup on a table. We will refer to it briefly as a cup. If V is negative definite, the cup is just upside down, or seen as .if the table were a mirror. In the n-dimensional case the situation is exactly the same and we have n-dimensional cups. A second geometrical representation of the function V(x) is likewise interesting. Taking again n = 2 and x, y as the usual cartesian coordinates, the level curves V(x, y) = k represent a

§ 8.

35

A SPECIAL TYPE OF FUNCTION

set of ovals surrounding the origin (Fig. 12). They may be thought of as the sections of the cup by horizontal planes projected on the x, y plane (the plane z = 0). For n > 2 the situation is identical. Some special Liapunov functions. Let us suppose that in the neighborhood of the origin V may be represented as a power series in the Xi' Then

V = Vp(x)

+ VP+l(x) + ...,

where Vk(x) is a homogeneous polynomial or form in Xv"" XnThe form Vp(x) is simply the collection of terms of the lowest degree in the series V(x). Since for X small the terms VP+l' V p+ z,"" are completely dominated by V p, the sign of V in a suitable neighborhood Q of the origin is the same as the sign of V p• A simple, but highly useful property is this: if p is odd, V cannot be a Liapunov function. Taking Xn-l

=

XnUn-l,

one obtains V p = xnP Vp(u 1 , Uz,' .. Un-I, 1). Now - keeping the u, fixed - the sign of V p will be that of xl or of - xl (one of the two but not both). Since p is odd x/ (or - xl) may assume both positive and negative values near the origin, and so V is not positive definite. We are assuming tacitly here that the u, have been so chosen that Vp(U 1 , •

• •,

Un-I, 1) -=1=

o.

This is always possible since V p is not identically zero. Thus for a Liapunov function the lowest degree terms III V must be even. This is, however, only a necessary condition but far from sufficient. Thus

36

2. DIFFERENTIAL EQUATIONS

is neither positive nor negative definite since it is > 0 for Xl = 0 and < 0 for x 2 = o. The simplest positive definite function is a quadratic form

i,

i=

1, 2, ... , n;

The necessary and sufficient conditions in order that V(x) be positive definite were given by J. J. Sylvester. They are that the successive principal minors of the symmetric determinant

be positive, or h, k = 1,2,3; h, k

=

1,2,3,4;

... ,

lal >

O.

We accept this result without proof. One should not take for granted that one must have power series with lowest degree terms even to have a positive definite function. Thus the function V(x) of the scalar variable X represented as follows V(x)

=

x2

for

V(x)

=

x4

for

x O. Since V is positive, V can only increase along g+ and so s: will not tend to the origin. For this reason, as before V > m > 0 along r: Hence again V cannot tend to a fixed value in S(R). It must increase definitely and hence t: must reach the boundary H(R) of S(R), and so we have instability.

§ 9. IV.

LIAPUNOV'S STABILITY THEOREMS

39

Under the same assump-

SECOND INSTABILITY THEOREM.

tions for V as in III with

V = lV + ~* where V*(x) is non-negative in Q and l > 0, the origin is again. unstable. Choose as before XO in 5(r) with V(XO) > O. Let x(t) be the solution x(t) satisfying x(O) = xc, and let i: be the path initiating from xc. The condition on V implies that

;t V[x(t)J

=

AV[X(t)J

+ V* [x(t)J

or d

dt (rAJ V)

=

e- AJ V*

> O.

Hence along g+

V increases indefinitely along g+, and as before this means instability. The two instability theorems of Liapunov have the deficiency that they take in the whole region Q. A proposition embracing both but resting upon a smaller region has been given in the early thirties by Cetaev.

Let Q be a neighborhood of the origin. Let there be given a function V(x) and a region Q 1 in Q with the following properties: (1) V(x) has continuous first partials in Ql' (2) V(x) and V(x) are positive in a; (3) At the boundary points of Q 1 inside Q, V(x) = O. (4) The origin is a boundary point of Ql' Under these conditions the origin is unstable. V.

INSTABILITY THEOREM OF CETAEV.

40

2. DIFFERENTIAL EQUATIONS

It is not difficult to see that any path starting in Q 1 must leave Q since it cannot cross the boundary of Q 1 inside Q. Since the origin is on the boundary of Q1' we can find points arbitrarily near the origin from which paths leave Q. Again this is instability.

V=k.

FIG. 15.

Intuitively, Fig. 15 illustrates the situation rather well. The loci V(x) = k in Q 1 can only behave as indicated in the figure and k decreases as one tends to the boundary B of Q1' Since it can only increase along g+, the latter must behave as indicated in the figure.

§ 10. Stability and the Theorems of Liapunov for Nonautonomous Systems We are dealing then with a system

x=

(F)

X(x, t)

where the existence and unicity theorem hold in a set

Q:llxll O.

§ 10.

41

STABILITY FOR NONAUTONOMOUS SYSTEMS

Except for the precaution of always initiating the trajectories for the fixed value t = 0, the definitions -of stability, asymptotic stability, and instability are unchanged. Let us denote by W(x) a positive definite function as previously understood. We will now define more generally V(x, t) to be a positive definite function if the following conditions are fulfilled: (a) V(x, t) is defined in Q for all t > 0; (b) V(O, t) = for t > 0; (c) V(x, t) dominates a certain W(x); that is, W(x) for all x in Q and all t > O~

°


0 for x =1= 0 and g(O) = o. Set also x

G(x) =

Jg(x) dx. o

§ 12.

SOME EXAMPLES

43

The kinetic energy of the mass is y2J2 and its potential energy G(x). Thus the law of conservation of energy yields V(x)

=

y2J2

+ G(x) =

k 2,

which may also be derived directly from (12.1). The system (12.1) has the origin as sole critical point. Now V(x) is a Liapunov function since we find from (12.1) that V = O. Hence the origin is stable (Theorem I). As a matter of fact the paths are V(x) = k 2 . Using the representation y=

± V2(k 2 -G(x))

we infer readily that they are ovals surrounding the origin, and so the latter is not asymptotically stable. The critical point just considered is a simple example of a center (Fig. 16).

CENTER

FIG. 16. EXAMPLE

2. Take again a planar system

X=-Ax+ ... ,

y = -,uy + ...,

where A, ,u > 0 and the ... indicate convergent power senes

44

2. DIFFERENTIAL EQUATIONS

beginning with terms of degree at least 2. y 2 • Then V = X2

+

11 =

-

2 (Ax2

Choose once more

+ fly2) + ...

where ... are terms of degree at least 3. For x and y small enough the sign is governed by that of the parenthesis and so 11 < 0 (except at the origin): 11 is negative definite. Since V is positive definite and the conditions of Theorem II are fulfilled, we have asymptotic stability. This is, in fact, the well-known stable node

"

FIG. 17. Stable node (2) fl

> 0).

(Fig. 17). If 2, fl > 0, 11 > 0 (except at the origin). The conditions of Theorem III being fulfilled this is the unstable node: Fig. 17 with arrows reversed. EXAMPLE

3. The system is now

x = Ax

+ ...,

y=

-

flY

+ ...,

2 and fl > O. = (Ax2 + fly2) + '" with

This time taking V = x 2 - y2, then 11 omitted terms of degree at least 3. Thus arbitrarily near the

§ 12.

SOME EXAMPLES

45

origin there are points where V > O. Since V has, near the origin, the sign of the parenthesis, it is positive definite. Since V = x 2 along y = 0, the conditions of Theorem III are fulfilled and so the origin is unstable. This is the well-known saddle point case. The paths are represented in Fig. 18.

FIG. 18. Saddle point.

FIG. 19. Stable focus.

4. In the three preceding cases the characteristic roots A, fl were real. Consider now a case with conjugate complex roots:

EXAMPLE

+ bi)x + '" x= (a - bi)x + --::: x=

(a

where a, b of 0, and ... is a power series in x, x and ... its conjugate, both beginning with terms of degree at least two. Suppose first a < O. Take V = xi so that

V = xx +

xx =

2aV

+ ...

where. " begins-with terms of degree > 3. Here then V and - V are positive definite functions. The other conditions of Theorem II are fulfilled and so we have asymptotic stability. In fact, this is the well known stable focus illustrated in Fig. 19. If a > 0 both V and V are positive definite functions and the origin is unstable. This corresponds to Fig. 19 with arrows reversed.

46

2. DIFFERENTIAL EQUATIONS

There is an interesting application to the standard closed electric RLC circuit (R is the resistance, L the inductance, C the capacitance) with nonlinear elements. The equation of the charge x (i being the current) is

Lx

+ Ri + ~ x + g(x, i) =

0,

where g represents nonlinearities - terms of at least second order. The equivalent x, y system is .

.

1

x= y,y= -LC x L

FIG. 20.

-

R

L y-g(x, y).

The origin is a critical point and its characteristic roots are the roots of

Since R, Land C are positive, the roots have negative real parts. They are both complex with negative real part - R/L, if R2/V < 4/LC or R2 < 4(L/C). When this happens we have spirals as paths and the origin is asymptotically stable. The origin is a stable focus. If R2 > 4(L/C) the origin is a stable node and is asymptotically stable. That it is asymptotically stable in view of a positive resistance near the origin is fairly obvious; the current must necessarily die down. The examples just considered correspond to the well known ordinary planar critical points. We proceed now with some ndimensional examples.

§ 12.

47

SOME EXAMPLES

EXAMPLE 5. (Stability determined by the linear approximation.) Once more let x be an n-vector and consider the system

i = Px

+ q(x, t)

(12.2)

where P is a constant nonsingular matrix and the nonlinear term q is quite small with respect to x for all t > O. This would be the case for instance if the components of q were convergent power series in those of x beginning with terms of degree at least 2 and with coefficients bounded functions of t for t large. For simplicity we also assume that the components of q have in some region Q and for t > 0 continuous first partial derivatives in the Xk and in t. Thus in Q and for t > 0 the fundamental existence theorem (see Section 5) is applicable. Let us assume that the characteristic roots r1> ••• , rn of the matrix P are all distinct and consider first the case where they are all real. One may therefore choose real coordinates in which the system preserves its initial form but P = diag h, ... , r n ) . There are now two cases. (a) The roots r» are all negative. Take V =

X1

2

+ ... + x

n

2

and hence

V ~(rl X1 2 +

... + r; xn + s(x, t) 2

)

where s is small with respect to the parenthesis. Thus in a sufficiently small region Q both V and - V are positive definite functions. Hence the conditions of Theorem II are fulfilled and the origin is asymptotically stable. (b) Some of the rk say r1>' .. , rp (p < n) are positive, and the rest negative. This time take

Here then

V= (r1x1 2 +

... +rpxp2-rp+1X;+1- ... - r

2

nXn )

+s(x,t}

48

2. DIFFERENTIAL EQUATIONS

where s is as before. At some points arbitrarily near the ongm (those where XP+l = ... = X n = 0) V is positive. As for V, since rp+l,' .. , r; < 0, it is positive definite. Hence by Theorem III the origin is unstable. Suppose now that some of the rh are complex. Let say r v . . . , rp be real and r p+ 1 , f p+ 1 , • • • , r p+ m, f p+ m be complex, so that p + 2m = n. If the r 1 , . . • , r p are negative and the rp+h' fp+h have negative real parts, we form V

=

X1

2

+ ... + Xp2 + XP+l XP+l + ... + xp+m xp+m

and everything proceeds as under case (a) with the origin asymptotically stable. If, on the other hand, some of the r v' .. , r p are posi ti ve or some of the rp +h have positi ve real parts we make an elementary adaptation of case (b) and find, once more, instability. To sum up we may state that a sufficient condition for the origin of the nonlinear system (12.2) to be asymptotically stable is that the characteristic roots all have negative real parts. If there is a characteristic root with positive real part the origin is unstable. Observe that often the explicit solution of (12.2) is wholly out of the question, yet the Liapunov procedure provides very precious stability information.

[Remark: It will be shown later in connection with controls that when all the rh have negative real parts, whether they are distinct or not, the origin is asymptotically stable.] EXAMPLE

6. (Critical cases.)

Suppose that the basic system

i = Px

IS

+ q(x)

where P is a constant matrix and q(x) is a vector whose components qv q2' ... , are convergent power series in those Xi of X which begin with terms of degree at least 2. The critical cases of the system are those for which several of the characteristic roots of the matrix P of the linear terms are zero or pure complex in pairs.

§ 12.

SOME EXAMPLES

49

Let us think of the terms Pij of P as coordinates of some Euclidean n 2-space (there are n 2 coordinates). In this space the points corresponding to all characteristic roots with negative real parts, fill up a certain region R - the region which corresponds to asymptotically stable systems. Now as one approaches a boundary point M of the region R some of the characteristic roots will become zero or pure complex; M will be a critical point. Liapunov has discussed at length the case of one characteristic root zero or of two pure complex. We shall deal rapidly with these two special cases. ONE CHARACTERISTIC ROOT OF THE MATRIX P IS ZERO. It is convenient to designate by n 1 the order of the system. We will suppose that the nonzero characteristic roots of P are distinct and all have negative real parts.

+

By means of a certain number of transformations of variables, which are complicated but not really difficult, one may reduce the basic system to the following form: one of the coordinates will be written y, the others ZI>"" z; and designated jointly by an n-vector z. Then the system is of the form

+ lo(z) + 11(Z)Y + ... = Qz + H(y) + ho(z) + h + ...

y = F(y) i

(12.3)

1(z)y

where the designations have the following meanings:

F(y) is a power series whose lowest degree term is gyN, N > 2; the /;, hi are power series whose lowest degree terms are of degree

> 3 for > 2 for > 1 for

10; ho' 11' ... , IN hI> h 2 , •••

1;

;

Q is a constant matrix whose characteristic roots are the same as those of P which are not zero, and therefore have negative real parts; H(y) is a power series which begins with terms of degree N

+ 1.

50

2. DIFFERENTIAL EQUATIONS

Our next step will consist in applying the theorems of Liapunov ; but first let us make a preliminary observation. Consider the linear system

z=Qz. Since the characteristic roots of Q are distinct and all have negative real parts the system possesses a Liapunov function W(z), which, as we have found, is in fact a quadratic form whose time derivative TV = U(z) along the paths is negative definite; in fact

and it is also a quadratic form. We shall utilize the property of U presently. We turn then our attention to the stability problem of our system and here one must treat separately N even and N odd. N even.

Take the function

V(y, z) = y -gW. At once along the paths

V = Y- g TV =

g{yN -

U} +

...,

where ... stands, here and below, for terms small compared to those in the brackets, terms such as yN+l, yz2, Z3, etc. Thus near the origin the sign of V is that of g{. . .}, i.e., the sign of g. On the other hand, arbitrarily near the origin, for z = 0, the sign of V is that of y and hence it is not fixed, Hence arbitrarily near the origin V and V may have the same sign. Hence the origin is unstable. N odd.

This time take

V=t y 2 _ gW so that

V=

yy - gTV

= g {yN + 1 - U} + ....

Once more V has the sign of g.

§ 12. SOME EXAMPLES

51

If g > 0 we will have near the origin V > 0 and arbitrarily near it at some points V > 0 also. Hence the origin is unstable. If, on the other hand, g < 0 near the origin (origin excepted) V > 0, V < 0; hence the origin is asymptotically stable.

To sum up we have this result of Liapunov: Necessary and sullicient conditions lor the origin to be stable lor the system (12.3) are N odd and g < O. THE MATRIX P HAS Two PURE CONJUGATE COMPLEX ROOTS. The treatment of this case, far more involved than the preceding one, was fully dealt with by Liapunov. We will merely show by a two dimensional (planar) example that "things" may happen. Consider namely the system x

= y - xit», y),

(12.4)

y=-x-YI(x,y), where I is a convergent power series and 1(0,0) = O. The characteristic roots are easily shown to be ± i. If V = t(x 2 + y2) then

V=

-

(x 2 + y2)/(x, y).

Hence when

I > 0 arbitrarily near the origin, the origin is stable; I < 0 arbitrarily near the origin, the origin is unstable; I is positive definite in some neighborhood of the origin,

the

origin is asymptotically stable. The example just discussed illustrates also the important fact that in some instances stability cannot be decided by examining only the linear terms. It may be necessary to take into account the nonlinearities of the system.

EXAMPLE 7. One of the simplest examples of the application of Liapunov's first stability theorem is of systems with first integrals V(x) = c. Then V = 0 and the origin is stable if V is either positive or negative definite. (If V is negative definite, then - V is a positive definite integral.) In this example we shall

52

2. DIFFERENTIAL EQUATIONS

illustrate the study of stability making use of the linear approximation and the existence of such integrals. A nonlinear system of differential equations of the form Xl = AX2(X3 -a), X2 = BX l(X 3 - b),

occurs frequently in the study of the motion of rigid bodies; A, B, a, and b are constants. There are three types of equilibrium states £1:

Xl

= l,

x 2 = 0,

x 3 = b,

£2:

Xl

= 0,

x 2=m,

X3 = a,

£3:

Xl

= 0,

X2 = 0,

X3 = n,

where l, m and n are arbitrary constants.

(i). Stability

01

£1'

Let us first transfer the equilibrium state £1 to the origin by the change of coordinates Y3 = X3 -b. The system of differential equations in the new coordinates is )\ =

A(b - a)Y2

+ AY2 Y3'

+ l)Y3' (Yl + l)Y2'

Y2 = B(YI Y3 =

The characteristic equation -A

A(b-a) -A

IS

°

lB

A(l2 B - A2)

° -AI ° The characteristic roots are 0, l VB, - l VB. =F °since in this case £1 and £3 coincide.

l

=

=

0.

We may assume that If B > 0, then one of

§

53

12. SOME EXAMPLES

the roots is positive, and we can conclude that E 1 is unstable. If B < 0, we have the critical case of one zero root and two pure imaginary roots. Thus, if B < 0, we can draw no conclusions by examining. only the linear approximation and must take into account the nonlinear terms. Fortunately, there are two obvious integrals

and

°

VI = and V 2 = 0. If B < 0, then VI is positive definite in Y2 and Ya' and this implies stability in these two variables. If initially Y2 and Ya are small, then they remain small. Now from the second integral we can conclude that E 1 is stable. If Yl> Y2 and Ya are all small initially, they must remain small. Hence we have shown that

E 1 is stable if B
0. (ii). Stability oj E 2 • By symmetry we see immediately that E 2 is stable if A


0.

(iii). Stability oj Ea. The change of coordinates

Ya

=

xa-n

54

2. DIFFERENTIAL EQUATIONS

transforms £3 to the origin and the new system of differential equations is

Yl = Y2 =

+ n), b + n),

AY2(Y3 -a BY1(Y3 -

The characteristic equation of the linear approximation is

-A

A(n-a)

B(n -b)

-A

o o

o

o

-AI

=

-A[A 2 - A B (n - a)(n - b)J = O.

If AB(n - a)(n - b) > 0, there is a positive real root, and £3 is unstable. If A B(n - a)(n - b) < 0, we have a critical case of one zero and two pure imaginary roots, and the linear approximation gives no information. An integral is

If the coefficients all have the same sign, the integral is either positive definite or negative definite, and hence, £3 is stable. We can then conclude that

£3 is stable if AB(n - a)(n - b) < 0

and

A(a-~n-b) < 0 and £3 is unstable if AB(n - a)(n - b) > O. If AB(n - a)(n - b) < 0 and A(a - b)(n - b) > 0, then we are unable to conclude by this analysis whether £3 is stable or unstable.

8. Consider a conservative dynamical system with n degrees of freedom. The state of the system can be described by n generalized position coordinates ql>"" qn and n generalized EXAMPLE

§ 12.

55

SOME EXAMPLES

momentum coordinates PI'" ., Pn- There will be a Hamiltonian function H (P, q), and the equations of motion in the n-vectors P and q will be

.

P=

-

oH

oq .

The positions of equilibrium correspond to the states where the partial derivatives of H all vanish. Take the origin to be an isolated equilibrium, and the free constant in H can be so chosen that H(O, 0) = O. For ordinary dynamical systems where H is the total energy of the system we may assume that H(P, q) = T(P) W(q) where T is kinetic energy and W is potential energy. The kinetic energy is positive definite with respect to P, and if the potential energy has an isolated minimum at the equilibrium q = 0, then W is positive definite with respect to q. Therefore, H is positive

+

definite, and since iI = 0, the equilibrium is stable. This is the well-known Lagrange Theorem (stated by Lagrange and proved by Dirichlet): a position where the potential energy is an isolated minimum is a stable equilibrium. Assume n0'Y that H is analytic. Then

and

W(q)

= Wk(q) + Wk+l(q) + ...

Ti(P) denotes the terms in P of order j and Wi(q) are the terms in q or order j; T 2 (P) is a positive definite quadratic form. Suppose that W(O) = 0 is an isolated maximum of the potential energy. Then Wk(q) is a ~ve definite k-form. This time take n

v = P' q =

I i= I

Then

Pi t-

56

2. DIFFERENTIAL EQUATIONS

*

Since the terms 2 T 2 kW which dominate near the origin are positive definite, V is positive definite. The Liapunov function V is the inner product of p and q, and therefore takes on positive values arbitrarily near the origin. Hence by Theorem II the origin is unstable. This is Liapunov's Theorem: at an isolated maximum of the potential energy the equilibrium is unstable. This theorem of Liapunov was extended by Cetaev. Assume only that W(O) = 0 is not a minimum of the potential energy. Hence there are points q arbitrarily near the origin where W(q) < O. Since H(O, q) = W(q), there are points (P, q) arbitrarily near the origin where H(P, q) < 0 for all p sufficiently near the origin. Therefore, there are points (P, q) arbitrarily near the origin where pi q > 0 and - H(P, q) > O. Let Q be some neighborhood of the origin, and let Q 1 be the region of points in Q where both of the above inequalities are satisfied. The origin is then a boundary point of Q1' Define V = the computation above

pi qH.

Since

iI =

0, we have as in

If we select the neighborhood Q to be sufficiently small, then T(P) > 0 within Q and therefore Wk(q) < 0 within Q1' Hence for Q sufficiently small the term in brackets is negative within Q1' and V is positive within Q1' On the boundary points of Q 1 that are in Q it must be that either pi q = 0 or H(P, q) = 0 and at these points V = O. All the conditions of Theorem IV are satisfied, and the equilibrium is unstable. If at an equilibrium position the potential energy is not a minimum, then the equilibrium is unstable.

§ 13. The Extent of Asymptotic Stability In applications it is moderately obvious that asymptotic stability is more important than stability. If one wishes to maintain say a certain temperature T in a system, it is clearly desirable that small deviations actually cancel out as the result of the operation

§ 13.

THE EXTENT OF ASYMPTOTIC STABILITY

57

of the system, and not desirable to maintain merely some temperature not too far removed from T. We must fix then our attention upon asymptotic stability. Here there arises another practical consideration. Suppose that an electrical system has been designed to operate at 110 volts. The system is so arranged that small deviations are cancelled out. But how large are the deviations that cancel out. The system may be asymptotically stable but yet not operate properly if deviations in excess of one millivolt occur. The system, while asymptotically stable in theory, is actually unstable in practice. To have true asymptotic stability one should allow for deviations, say, of several volts. This raises then the question of the extent of asymptotic stability. The desirable feature is asymptotic stability in the large, or as we shall have it complete stability. If one cannot secure that, one may have to be content with the assurance that when the perturbations are not too large the system tends to return to equilibrium. This means that one needs to know something about the size of the region of asymptotic stability. One must point out here a basic difference between linear and nonlinear systems: in determining practical stability, linear approximations are definitely unsatisfactory. In linear systems stability is always complete while it is only in nonlinear systems that it may fail to be so. In other words, to determine possible restrictions to asymptotic stability one must examine nonlinearities. In the present section we develop a variety of theorems related to the determination of the extent of asymptotic stability. Before proceeding with our topic it is necessary to introduce two preliminary notions. Take our usual autonomous system i = X(x),

X(O)

= o.

(FA)

Limiting sets. This is a highly important concept introducted by G. D. Birkhoff. Intuitively if x(t) is a solution of (FA) its positive limiting set F+ is whatever the curve x(t) tends to with

58

2. DIFFERENTIAL EQUATIONS

infinite time. Thus if x(t) spirals around 0, a limit-cycle, * 0 is its positive limiting set; if it tends to a point A, the point is its positive limiting set. More accurately, p is in F+ if there is an increasing sequence of times t; ---+ 00 with n and such that x(tn) ---+ p as n ---+ 00. If x(t) is bounded, then x(t) approaches its positive limiting set F+ as t ---+ 00; that is, given any e > 0, if N(e) is the e-neighborhood of F+ then there is a time T such that for t >1, x(t) lies in N(e). The negative limiting set of x(t) is defined in the same way, save that one must replace t by - t. However, these sets will not be required. Invariant set. An invariant set G is characterized by the property that if a point X o is in G then its whole path (forward and backward) lies in G. Thus a closed path is an invariant set; the collection of paths of all the points of an arc is an invariant set. Noteworthy property: If x(t) is bounded for t > then its positive limiting set F+ is a nonempty, compact, invariant set. (Admitted without proof.) From the definition of limiting set there follows also this property: if x(t) is bounded for t > and if a set M contains F+, then x(t) tends to M as t ---+ 00. After these preliminaries we are in a position to establish a proposition leading to criteria to determine the extent of asymptotic stability.

°

°

VI. THEOREM. Let V(x) be a scalar function with continuous first partial derivatives. Let Q 1 designate the region where V(x) < l. Assume that Q 1 is bounded and that within Ql:

°

V(x)

>

V(x)

< 0.

for

x i=- 0,

(a) (b)

Let R be the set of all points within Q 1 where V(x) = 0, and let M be the largest invariant set in R. Then every solution x(t) in Q 1 tends to M as t ---+ 00.

+

* In the plane a limit-cycle is an isolated closed path corresponding to a periodic solution.

§ 13.

THE EXTENT OF ASYMPTOTIC STABILITY

59

Proo]. The conditions on V imply that V(t) = V(x(t)) is nonincreasing as t -- 00 and that V(t) > within QI. Hence every solution x(t) starting in Q I must remain in it. Therefore V(t) has a limit 10 as t -- + 00, and 10 < I. By continuity one concludes that V(x) = 10 on the positive limiting set T» of x(t). Hence T» is in QI, and 11 = on T», Consequently, T» is in R, and since T': is an invariant set it isin M. Since x(t) remains in Q I it is bounded for t > 0, and so x(t) -- M as t -- 00.

°

°

From Liapunov's stability theorem and conditions (a) and (b) one concludes that in the situation envisaged by the theorem the origin is stable. To conclude asymptotic stability all that is required is to show that the set M is just the origin; that is, that no solution other than the trivial solution x = 0 can remain in M for all t > 0. For instance, if 11 is negative definite in QI, then R is the origin and so M is likewise the origin. Stated as a separate property we have: VII.

THEOREM.

II condition

(b) 01 Theorem VI is replaced by

1I(x) < 0 for all x =1=

°in

QI

(b)*

then the origin is asymptotically stable, and above all, every solution in Q I tends to the origin as t -- + 00. (The last conclusion goes beyond Liapunov's asymptotic- stability theorem.) There are interesting applications to the general second-order equation which generalizes the LRC equation of electricity:

x + I(x)x + g(x) =

0,

(I3.1)

an equation which has been extensively investigated and on which many papers have been written and continue to be written. In these general investigations the most varied assumptions are made. However, as great generality is not our purpose, we shall consider only frankly simple situations, and restrict most sharply the functions I and g. To simplify matters then we shall assume that I and g are polynomials, I even and g odd. Moreover, we shall

60

2. DIFFERENTIAL EQUATIONS

suppose that g(x) behaves more or less like a straight line through the origin; it is monotone increasing with x. Note that this general type includes Lienard's famous equation (his assumptions are however less restrict¢" than "polynomials") and definitely includes van der Pol's equation. It is convenient to introduce the integrals x

=

F(x)

j

j(x) dx,

o x

G(x)

=

1

g(x) dx.

o

We note that F is odd and G even and F(O) = G(O) = O. Instead of (13.1) we will discuss the equivalent system

x=

y=

y-F(x),

-g(x).

(13.2)

Note that because F and g are polynomials the conditions of the existence theorem are fulfilled for all values of x and y; one and only one path goes through each point of the plane. Under our assumptions the equations Y -F(x)

=

0,

g(x)

=

0

are satisfied by the origin. A convenient Liapunov function is V(x, y)

= t y 2 + G(x).

It is in fact the total energy when j = 0; that is, when there is no dissipation (positive or negative) in the system. We find at once

v=

-g(x)F(x).

Suppose that there can be found positive constants a and I such that the following two conditions are satisfied: g(x)F(x) > 0 G(x) < I

for implies

Ixl < a, Ixl < a.

x i= 0;

(13.3a) (13.3b)

§ 13.

THE EXTENT OF ASYMPTOTIC STABILITY

61

Then the region Q/, defined by V < l, behaves in accordance with Theorem VI. In fact, by (13.3b) and from V < l, it follows Ixl < a; y2 < 2l so that Q/ is bounded. From (13.3a) we conclude that 17 < 0 in Q/. Then 17 = 0 in the region when and only when x = O. The locus E is thus the y-axis. But at all points of the y-axis other than the origin the slope of the path

dy _ -g(x) dx - y -F(x) is finite. Thus this axis contains no arc of a path. Its only invariant set is the origin. Hence every solution initiating at an interior point of Q/ tends to the origin. Thus the origin is asymptotically stable and Q/ is a "measure" of asymptotic stability. \Ve shall now consider several examples. EXAMPLE

1. The van dey Pol equation.

We recall that this equa-

tion is

x + e(x

2

-I)x

+x=

0,

e>O

(13.4)

y=-x.

(13.5)

with its equivalent x

= y _ e (~3

_x),

The only critical point is the origin and it is unstable. In fact, it is an unstable node or focus depending upon the value of e. It is also known that there is a unique limit-cycle D, which must surround the origin. If we replace t by - t, the paths remain the same but their orientation is reversed. In particular, the limit-cycle Dis unchanged. However, the origin becomes asymptotically stable, and the region of asymptotic stability is just the interior of the limit-cycle D. This is rather small comfort since the actual location of D is not known. We shall obtain interesting information by the application of the general discussion.

62

2. DIFFERENTIAL EQUATIONS

Observe that instead of changing t into - t we may merely take 10 < 0 with the same result. Let this be done in (13.4) and (13.5). We have here

f(x)

,

= e(x 2 -

(x

g(x) = x,

1),

3

x2

G(x) =2'

) F(x)=e-y-x,

Thus we take

V(x)

.

= _Y

2

(x

3

2

+ G(x) =

x

2

+ Y, 2

2

(x

2

) V(x) = -ex 3 - x) = -ex 2 3-1.

Vi

Thus V < 0 for x 2 < 3. Take therefore a = One must now choose l so that G(x) = x 2 /2 < t implies x 2 < a 2 • This means taking l = 3/2. Thus the region x 2 + y 2 < 3 is interior to the region of asymptotic stability. In other words, the limit-cycle (~ is exterior,

whatever 10, to the circle of radius at least

2Vi

Vi

That is, its diameter is always

This is the only known result valid for all values of e.

Observe incidentally that our argument shows, that a priori the origin is asymptotically stable for the system (13.5) with e < O. This could not be determined easily in general since the first degree terms in (13.2) are not known. EXAMPLE

2. This time take the equation

x + ax + 2bx + 3x

2

=

0;

a, b

>

0,

(13.6)

or its equivalent

x= y,

y = -2bx-ay-3x 2

(13.7)

which is more convenient than the form previously considered. It does not fall under the scope of (13.1) but we will have no trouble in applying Theorem VI.

§ 13.

THE EXTENT OF ASYMPTOTIC STABILITY

63

There are two critical points: at the origin 0 and at P = (- lb, 0). The characteristic roots AI' A2 of the coefficients of the first degree terms satisfy

I-

1

A

A2 + aA + 2b = O.

=

-a-A

1-2b

We have the well known relations

Hence AI' }'2 are either real and both negative, or conjugate complex with negative real parts. Thus the origin is asymptotically stable. Transfer now the origin to the point P by the transformation

x

+ lb =

x = x* -lb.

x*,

The x*, y system reads i*

= y,

y=

-ay - 3(x* - lb)x*.

The characteristic roots AI' A2 are now the roots of - A

1

2b

-a-A

= A2

+ aA -

2b = O.

Since Al i' 2 = - 2b < 0, the roots are real, of opposite signs and hence P is a saddle point. The trajectories behave somewhat as illustrated in Fig. 21. The shaded area is the region of asymptotic stability. We proceed to the construction of a region of asymptotic stability for the origin. Take V(x, y) = y2/2

+ bx» + x 3.

This yields

v=

_

ay 2.

Thus V < 0 outside the z-axis: the set R will be a subset of this axis. Since on the z-axis dy/dx = 00 outside the origin, the only

64

2. DIFFERENTIAL EQUATIONS

possible invariant subsets are the origin and the point P. One must then select QI: V < l so as to exclude P. Take the curve V(x, y)

= y2j2 + bx 2 + x 3 = k

'X.

FIG. 21.

§ 13.

THE EXTENT OF ASYMPTOTIC STABILITY

such that it contains P. k = x 2(b

65

This yields

4 2 x1b 4 + x) = gb 3 = 27 b

3

.

Let us then plot the graph y2/2

+ bx 2 + x

3

=

4

27 b3 ,

Q

x

FIG. 22.

or y

= ±

V

87 2 b3 -

2bx 2 - 2x 3 .

We have

7x = 2

2(2bx

+ 3x 2)

so that maxima and minima occur for x = 0, - 2b/3, or at (0, V8/27 b3 ) , ( - 2b/3, 0). The first is a maximum, since beyond it y2 -+ - 00 for x large. The second is a minimum and the curve V = l = 4/27 b 3 is similar to the one shown in Fig. 22. The interior

66

2. DIFFERENTIAL EQUATIONS

of the oval is the region QI' The set E is the interval PQ of the x axis, and the set M is the origin. Hence all the paths initiating within the oval tend to the origin. Thus the open oval is a region of asymptotic stability. Where the whole space is the region of asymptotic stability we say that we have complete stability. The general arguments are valid here. Instead of Theorem VI we have the following whose proof is practically the same: Complete stability.

VIII. THEOREM. Let V(x) be a scalar function with continuous first partial derivatives for all x. Suppose that V(x) > 0 for all x =F 0 and V(x) ~O. Let E be the locus V = 0 and let M be the largest invariant set contained in E. Then all solutions bounded for i;» 0 tend to M as t - 00. If, in addition, we know that V(x) _

Ilxll-

as

00

00,

then each solution is bounded for t > 0, and we can conclude that all solutions approach M as t - 00. If M is the origin, we have complete stability. For instance, if for equation (13.2) it is known that g(x)F(x) G(x) -

00

>

0

for as

x =F 0,

lxi-

00,

(13.8a) (13.8b)

then using the same function V as before we know that V - 00 as x 2 + y2 _ 00. Hence every solution is bounded for t > 0, and as before M is the origin. Conditions (13.8a) and (13.8b) imply therefore that the system (13.2) is completely stable. Often it turns out to be simpler and stronger results are obtained by handling the boundedness of solutions as a separate problem (cf., Section 24). Thus, in order to establish complete stability we need to show that every solution is bounded for t > 0 and that M is the origin. Suppose for instance that V(x) - 00 as Ilxll - 00 and that

§ 13. THE EXTENT OF ASYMPTOTIC STABILITY

67

V(x) < 0 for x i= O. Then certainly M is the origin, and it is also not difficult to see that every solution is bounded for t > O. Let x(t) be the solution through xo. Then for some r sufficiently large V(x) > V(XO) for all Ilx!1 > r. Since V(x(t)) decreases with t we see that Ilx(t) II < r for all t > O. Hence every solution is bounded and we have shown that

IX. THEOREM. Let V(x) be a scalar junction with continuous jirst partial derivatives [or all x. Suppose that: (i) V(x) > 0 [or x i= 0; (ii) V(x) < 0 [or x i= 0; and (iii) V(x) ~ 00 as Ilxll ~ 00. Then the system (FA) is completely stable. It does occur in many applications that one can .construct a Liapunov function V satisfying Theorem IX. Examples of this are given in the next chapter. However, it is often easier to find a Liapunov function whose time derivative is only nonnegative, and then to use Theorem VII. The next example is a simple instance of this.

EXAMPLE 3. For Lienard's equation.

x + j(x)x + g(x) =

0

we assume this time that (1)

xg(x) > 0

for all

x i= O.

(2)

j(x) > 0

for all

x i= O.

(3)

G(x) =

as

Ixl~

1" g(~) d~ ~

00

00.

° Thus we assume that: (1) the po~ntial energy G(x) is positive potential energy approaches definite, x = 0 is its mip:i~~m; infinity with Ixl; and"'~ the damping is always positive. An equivalent system is

0t'rrl

x= y,

y= -

g(x) -j(x)y.

68

2. DIFFERENTIAL EQUATIONS

We take the Liapunov function to be the total energy: V(x, y) =

t y 2 + G(x),

An easy computation gives V(x, y)

= - t(X)y2 < O.

+

V(x, y) ~ 00 as x 2 y2 ~ 00 and therefore all solutions are bounded for t > O. Now V vanishes only on the axes x = 0 and y = 0, and it is clear that excluding the origin, no solution remains on these axes. M is the origin and by Theorem VIII every solution approaches the origin as t ~ 00. By Liapunov's first theorem the origin is stable, and the system is completely stable. In the example above we were able to construct a Liapunov function V(x) with the property that V(x) ~ 00 as IIxll ~ 00, and hence conclude that all solutions are bounded for t > O. In the next example we illustrate that it may be easier to establish the boundedness of the solutions as a separate problem. EXAMPLE 4. We' continue our investigation of the complete stability of Lienard's equation with a weaker assumption on g(x) and a stronger assumption on the damping. Assume that

(1)

xg(x)

>

0

for

x =1= O.

(2)

t(x)

>

0

for all

x =1= O.

(3)

IF(x) \ =

"

IJt(~) d~1

~

00

as

Ixl~

00.

0

We use the same Liapunov function V(x, y) = t y2 + G(x) as before. However, since it may not be true that G(x) ~ 00 as Ixl ~ 00, we can conclude only that every solution bounded for t > 0 approaches the origin as t ~ 00 (Theorem VIII). Thus, to establish complete stability we need to show that all solutions are bounded for t > O. To do this, consider the region Q (Fig. 23) defined by V(x, y)

= t y 2 + G(x) < l

§ 13.

THE EXTENT OF ASYMPTOTIC STABILITY

69

and [y

+ F(x)]2 < a

2

•

For any 1 and a this is a bounded region. Let [x(t), y(t)] be any solution, and select 1 and a so large that the solution starts in Q. Then the solution cannot leave without crossing the boundary of Q.

~

+F(X)=a

FIG. 23.

+

+ +

It must cross either V = 1 or y F(x) = - a or y F(x) = a. We can select a sufficiently large that the part of y F(x) = a which is the boundary of Q corresponds to x > and the part of y F(x) = - a corresponds to x < 0. Since V < 0, a solution starting inside Q cannot cross V = l,

+

°

70

2. DIFFERENTIAL EQUATIONS

Now d· dt [y

+ F(x)]2 =

-

2[y

+ F(x)Jg(x).

Along that part of y + F(x) = - a or y up the boundary of Q, we have

:t

[y

+ F(x)]2 =

-

+ F(x) =

2aig(x)1


O. Thus, under somewhat different conditions we have again shown that Lienard's equation is completely stable. Usually it is not difficult to study the stability of second-order systems. There are two reasons for this. The phase space is a plane, and we have no trouble visualizing the qualitative behavior of the system. We are also accustomed to identifying the damping in systems with one degree of freedom. It is worthwhile then to illustrate the method for a third-order system. Equations of the type considered here have been studied by V. A. Pliss and A. 1. Ogurtsov. 5. We wish to determine conditions on the damping j(x) in order that EXAMPLE

x + j(x)x + ax + bx =

0

be asymptotically stable. We assume that a and b are positive constants. Letting y = x and z = x we obtain the equivalent system

x= y,

y =z,

z=

-j(y)z -ay - bx.

§ 13.

71

THE EXTENT OF ASYMPTOTIC STABILITY

Starting with a quadratic form as the Liapunov function, one soon arrives at y

V(x, y, z)

=

~ Z2 + byz + b) j(u)udu + t(bx + ay)2 o y

=

2~ (az + by)2 + t(bx + ay)2 + b) [j(U) -

:] udu.

o

and

V(x, y,z)

=

-a[j(y) -bfaJ z2.

If j(y) > c > bfa for all y, then it is clear that the conditions of Theorem IX are satisfied, and the system is completely stable. If one assumes only that j(y) > b]a, we are not sure that V -- 00 as x 2 + y2 + Z2 __ 00. However, it is possible to give a separate proof in the manner of Example 4 that all solutions are bounded for t > 0 and conclude complete stability.

§ 14. Stability Under Persistent Disturbances This is a problem amply discussed by Soviet writers, notably by Malkin from whom we borrow the theorem proved below. Return to the nonautonomous system

x=

X(x, t)

(F)

with the same assumptions as before and notably that X(O, t) = 0 for t > O. Now if one deals with a "practical" system, that is arising out of physics or technology, it is quite certain that perturbations will come up not only because of initial conditions other than x = 0 but also, and above all, due to the action of exterior and adventitious actions, for example fairly brusque impulses, etc. The system that one will have to deal with will be a modification of (FA):

x=

X(x, t)

+ R(x, t)

(14.1)

72

2. DIFFERENTIAL EQUATIONS

where one really has no information about R, save that "hopefully" it is not excessive. All that one may reasonably expect to happen then is some stability. In this connection a noteworthy property due to Malkin is expressed by:

Let there exist for the system (F) a Liapunov function V(x, t) concerning which it is known that in the basic region Q(A): Ilxll < A and for all t > 0 it behaves as required by Liapunov's asymptotic stability theorem. That is, there exist three positive definite functions W(x), W 1(x), W 2(x) such that in Q(A) and for t > 0 we have W 1(x) > V(x, t) > W(x); V(x, t). < - W 2(x). Suppose, in addition, that in Q(A) all the partial derivatives oVjox, are bounded for t > 0; that is, there exists an M > 0 such that in Q(A) X.

THEOREM.

i= 1,2, ... ,n;

t>

o.

Then the origin has the following kind of stability: Given any 0< e < A, ,there correspond to it two numbers 0 < 171(e), 0 < 172(e), such that if

Ilx(O)11 < 171(e); IIR(X,t)11 m on H. for all t > O.

§ 14.

STABILITY UNDER PERSISTANT DISTURBANCES

73

Let,u be a positive lower bound for W 2 (x) in S~,: 'YJl < Ilxll < E, and let 0 < k < 1, as close to 1 as we please. Set 'YJ2(E) = k,u/nM and suppose that IIR(x, t)11 < 'YJ2(E) for x in Q(A) and t > O. Let a trajectory of (14.1) start at XO in S('YJ1) and suppose that it gets into S~" Thus from a certain time to on we will have

v, < -

W 2 (x) + grad V . R

< -,u + nMllRl1 < -

(1-k),u

< O.

Hence V will decrease along any trajectory of (14.1) in S~" and therefore no trajectory of (14.1) starting in S('YJ1) can reach H e- This proves Malkin's theorem.

3. Application of Liopunov's Theory to Controls § 15. General Remarks

on Controls

The reader is, of course, well aware of the enormous importance of servo-mechanisms and controls of all kinds in the modern industrial world. The most advanced theory of stability of such systems has been developed in the U.S.S.R., with Lur'ye as the initiator. Other important authors are Letov, Malkin, and Yacubovich. The latter, and in a parallel way also Bass (unpublished work), have developed a treatment based on the extensive use of matrices. It is this treatment which we shall follow here. A real square matrix A whose characteristic roots all have negative real parts will be referred to as stable (modern usage). This alludes to the fact that the system ,x = A x is asymptotically stable when and only when A is stable in the sense just described. Let 5 be a natural system: mechanical, electrical, thermic, ... , or several of these combined, and let its state depend upon a finite number of parameters U v U 2, ••• , Un; that is, upon the n-vector u. The parameters may be positional or kinematic (velocities). Let the state of the system 5 be governed by a certain set of differential equations, in vector notation

u=

U(u).

(15.1)

This is quite an assumption, but assumptions must be made in order to bring the system within the range of reasonable mathematics. 75

76

3. LIAPUNOV'S THEORY AND CONTROLS

Let u = UO represent a stationary state.. That is, U(UO) = 0, so that UO is a critical point of (15.1) (an equilibrium position or point solution of the system). Let it be found desirable to maintain the system 5 as near as possible to the position u = u". That is, one wishes to have UO be a stable equilibrium position. Introduce the variable vector y = u - u". One may then replace (15.1) by the analogous system

y=

Y(y)

=

U(UO

+ y)

(15.2)

so that Y(O) = 0; in terms of the new parameters Yv"" Yn (components of y) the critical point has become the origin y = O. Under very general conditions, and assuming y to be and remain small, one may, for all practical purposes, replace (15.2) by a linear approximation

y=Ay

(15.3)

where A is a constant matrix. We assume henceforth that A is nonsingular, that is, that its determinant IA I =P o. To maintain the stability of the origin one utilizes a compensating mechanism (control). Let ~ be its (scalar) parameter with ~ = 0 corresponding to no control. The new equations of motion are y=Ay+~b,

~ = /(a),

a

(15.4)

= c'y -r~

where b, care n-vectors and r is a scalar. The scalar parameter a (signal) is introduced as an intermediate step. The scalar function /(a) is the characteristic of the servomotor (mechanism). One generally assumes that it has these properties.

a/(a) > 0

for

The form of /(a) may vary.

I-

• I

_ _I

--.

/

a =P 0,

.--

./

/(0)

=

O.

Generally it is of one of the types

1--

_1-·-'

§ 15.

GENERAL REMARKS ON CONTROLS

77

Actually, and to save more complicated arguments, we shall assume that j(a) is continuous and that ±oo

1

j( a)da =

+

00

(15.5)

(the integral diverges). We will need this property in connection with the Liapunov function. For the sequel it is convenient to pass to new variables x, a instead of y, ;. They are defined by

x

= y=

Ay

+ ;b,

a = e'y -r;.

(15.6)

The latter relation is the last equation in (15.4). It is clear that if y and; ---+ 0, so do x and C1. We wish to have the converse hold also: that is, that x, a ---+ imply that y, ; ---+ 0. This will insure that the stability situation in the new variables is the same as in the initial variables.

°

The requirement is that the system (15.6) have a unique inverse, that is, that one may solve (15.6) uniquely for y, ; in terms of x, a. This being a system of n 1 equations of the first degree in n 1 unknowns, the system must have a nonsingular matrix of coefficients. In other words, the matrix

+

+

must have a nonzero determinant. Now since A is nonsingular so is A -1, and hence likewise the matrix

Hence it is sufficient to arrange matters so that the product of the two matrices, which IS

(~

A b) -1

-r

78

3. LIAPUNOV'S THEORY AND CONTROLS

be nonsingular, or finally that its determinant be nonzero. Computing the determinant we obtain r

+ c'A-I b =1= 0.

(15.7)

This is a condition on the control parameters: the vectors b, C and the scalar r. We assume henceforth that it is satisfied and so we may freely apply the transformation (15.6). We find at once

x= (F) d'

Ax + f(a)b,

= c' x - rf(a)

and this is the basic system to be dealt with. In order that we not be troubled by this fact later let us note now that (15.7) is not affected by a change of coordinates x = Px* for the system (F). We then have x* = p-l APx* (F*)

+ f(a)P-l b

d' =c'Px*-rf(a),

and under this change of coordinates A C' ~ c' P, and r ~ r. Therefore,

~

p-l A P, b

~

p-l b,

c'A-I b ~ C' P(P-l A-I P)P-l b = c'A-I b.

Until further notice the matrix A will be assumed stable. This working hypothesis is justified by the following considerations. Suppose that f(a) = ko

+ fl(a)

where k is a constant =1= 0, and fl(a) is small relative to a for a small. That is, the curve f(a) has a nonhorizontal tangent at the origin. The matrix of the linear terms in (F) is now

(A c'

kb) -kr'

§

15. GENERAL REMARKS ON CONTROLS

79

and asymptotic stability requires that the matrix be stable. Now for k = 0 the characteristic roots of (15.4) are the roots of the polynomial A - AI l c' that is, zero and the characteristic roots of A. Hence for k small they will be: one root small and the remaining n very close to those of A. Hence, if A has roots with positive real parts, the same will hold for those of (15.4) when k is small. One infers then that some lower limit must be imposed on k to have a regulating mechanism insuring stability. Moreover, one would like to guarantee stability and even asymptotic stability for more or less arbitrary functions I(a) and arbitrary initial values (complete stability). This is called "absolute" stability. To avoid any complication it is simpler, therefore, to assume that the matrix A is stable. As a matter of fact the sufficient conditions for asymptotic stability to be given below will actually guarantee asymptotic stability of absolute type: whatever the initial values or function I(a); that" is, the system is completely stable for all continuous I(a) satisfying al(a) > 0 for a =f. 0 and such that a

j I(a)

do

---+

oo

o

as

lal

---+

co.

§ 16. Construction of a Special Liapunov Function Weare going to introduce a special type of Liapunov function due to Lur'ye, Conditions under which it behaves in accordance with the theorems in Section 13 will thus be sufficient conditions for the complete stability (asymptotic stability in the large) of the

80

3. LIAPUNOV'S THEORY AND CONTROLS

basic system (F). One must not forget, however, that they will he in no sense necessary conditions. \Ve endeavor then to take a Liapunov function a

+ JI(a) de

V(x, a) = x' . Bx

o

°

where B > and B ' = B. This is a positive definite function over the whole space (x, a): V(O, 0) = 0, and V and its first partial derivatives are continuous throughout the whole space. Moreover, V is the sum of two terms: the first positive for all x =1= 0; the second for all a =1= 0. Hence, the sum can only vanish if x = 0, a = 0, and is positive otherwise. Our assumptions on I(a) also imply that V(x, a) -+ 00 as IlxW + a 2 -+ 00. By means of (F) we find V(x, a)

=

x'Bx + x' Bx + I(a)&

=

x'(A' B

+ BA)x -

+ 2/(a){t(b' Bx

Since B ' = B we have at once b' Bx Setting also

rI 2(a)

+ x' Bb) + ie' x].

+ x'

Bb

=

2b' Bx

=

2(Bb)' x.

(16.1)

A'B+BA =-C

we see that

C' = - (A'B

+ BA)' = -

(BA

+ A'B) =

C,

so that C is the matrix of a quadratic form. We have then

v= -

x'· Cx -rI 2 (d)

+ 2/(a)(Bb + ie)'x.

(16.2)

Thus V is a quadratic form in Xl' x 2 , • • • , x n , I(a). To have a behavior in accordance with Theorem IX of Section 13 we want this quadratic form to be negative definite. This implies first that V(x, 0) be negative definite and so that C > 0, then that V(O, a) < for all a =1= and r > 0. We must therefore first study the relation (16.1).

°

°

§

17. THE RELATION BETWEEN THE MATRICES

B, C

81

§ 17. The Relation Between the Matrices B, C The relation in question is (16.1). Basically it arises as follows. Take the equation of the system 5 without control i=Ax

(*)

and consider the function W(x) = x' Bx. Its time derivative along the paths of the system (*) is precisely

W = x'(A'B + BA)x = -x'Cx where C is given by (16.1). It is obtainable from (16.2) when one makes c = 0, f(a) = 0. The preceding interpretation given to (16.1) has the important consequence that this relation is preserved under a transformation of coordinates. For the relationship between Wand W is independent of the choice of coordinates; if one passes from coordinates x to coordinates y given by x = Py; W(x) goes merely into W(Py) and W(x) into W(Py), leading to the same ;elation (16.1) between the new matrices A, B, C. Returning to (16.1), it is trivial that B determines C uniquely, and this whatever A. This is, however, less important than the converse. Let AI"'" An be the characteristic roots of A each repeated as often as its multiplicity as root of , the characteristic equation. We shall prove: XI. THEOREM. If every sum Ai + Ak =1= 0, then the symmetric matrix B is uniquely determined by the symmetric matrix C (without reference to their being positive or not). Thus (16.1) is then a one to one relation between the two matrices (one and only one B for each C and conversely). Noteworthy special case: A is stable. (The sums Ai evidently =1= 0.)

+ Ak are then

82

3. LIAPUNOV'S THEORY AND CONTROLS

If one writes down the symmetry condition for B and compares the terms (i, k) on both sides of (16.1) one arrives at the following system of linear equations to determine the bhk in terms of the Cij'

(17.1)

.I

(ahi

bhk

+

ahk

bhi )

= -

i

Cik,

> k.

h

As observed at the beginning of the present section the system (17.1) is unchanged under a transformation of coordinates x = Py, P nonsingular. It will yield new matrices

A*

=

P-IAP,

B* = P'BP,

c*

=

P'CP,

(17.2)

and we will still have

A*'B*

+ B*A* =

-C*,

(17.3)

with relations (17.1)* being the same as (17.1) with all letters starred. Now a particular transformation of coordinates (perhaps complex) may be chosen such that

]. where the Gj are blocks of the form

all the terms not written in the two matrices being zero. Moreover, e is a fixed (but arbitrary) constant =I=- 0, and the characteristic roots of A * do not depend upon e. Let zl (e) be the determinant of the

§ 17.

THE RELATION BETWEEN THE MATRICES

B, C

83

linear equa tions (17.1)* with unknowns bi*,. . It is not difficult to see that Ll (e) is a polynomial in e. The special system (17.1)* corresponding to .0 = 0 assumes the form

i

> k.

(17.4)

Hence its determinant, which is zl(0) is the product of all the (Ai + Ak ) , i > k and therefore it is not zero. Since Ll (0) =F 0 the qolynomial zl(e) is not identically zero. It has a finite number of roots and by taking .0 distinct from anyone of them we will have Ll (e) =F O. As a consequence the bh~ of the general system (17.1)* (for general e) will be uniquely determined by the c;;' and so B* will be uniquely determined from C*. Since B*, C* are uniq uely determined from B, C by (17.3) and vice versa, we conclude that B is uniquely determined from (17.1) by C and our theorem follows.

XII.

Assume that A is stable. If C is positive then the unique solution B of (17.1) (or of (16.1) as a symmetric matrix) which exists, is likewise posi#ve. THEOREM.

The proof will consist in exhibiting an explicit solution of (17.1) as a symmetric matrix, and showing that it is > 0 when C > O. This explicit solution is obtained as follows. Let

Y=-YA

(**)

be the matrix equation associated with the adjoint to (*) (see Section 5). Then o

B

f

Y'(t)CY(t) dt.

= -

00

We must first show that B is a true matrix. Referring to (Section 5) property lithe terms of Yare finite sums of terms of type g(t) e: JJ, where g is a polynomial of degree < n, and A is a characteristic

84

3. LIAPUNOV'S THEORY AND CONTROLS

root of A. Hence the same holds for the terms of Y'CY. -}. has a positive real part

Since

o

Jg(t) e:

At dt

-00

exists and so does the matrix B. Since (Y'CY)' = Y'CY, we have B' = B: Consider now o

B

IS

symmetric.

0

JA'Y'CY dt J(- Y')CY dt,

A'B =

=

-00

-00

and o

BA

0

JY'CY A dt = - JY'CY dt.

=

-00

-00

Hence, with Y as the principal matric solution, o

A'B

+ BA

=

J~ [Y'CYJ dt = -

-

Y'CYJ~oo =

-C,

-00

so that B satisfies (16.1), and so it is its unique symmetric solution. Let now C > 0, and consider the quadratic form Q(x, t) = x'Y'(t)CY(t)x.

Since Y(t) is nonsingular, whatever t, one may take it as a matrix of transformation of coordinates y = Y x. Then Q(x, t) = y'Cy > 0 for all vectors y =1= 0 and arbitrary t. Since x = y-l y, y =1= 0 for arbitrary t implies the same for x. Hence Q(x, t) > 0 for all vectors x =1= 0 and arbitrary t. It follows that o

x' Bx

=

JY'(t)CY(t) dt)X > 0

x' ( -

for all x =1= 0 and so B

00

> O. This completes the proof of Theorem XII.

§ 18.

85

RETURN TO THE CONTROL PROBLEM

§ 18. Return to the Control Problem Take then any matrix C > 0 and let B > 0 be the unique symmetric matrix solution of (16.1), then form V(x, a) as in Section 16. Thus V(x, a) will be positive definite for all (x, a). We form then V as in Section 16, and, as already observed, it is a quadratic form in x, f(a). To have - V > 0 for all x, f one must satisfy the well known n + 1 determinantal inequalities of Sylvester. Since C > 0 the first n are already fulfilled. There remains then the last one - (Bb + tc)

I

r

I

(18.1)

, >0,

and it is a necessary and sufficient condition for - V > 0 for all x, f(a) =/=- O. Since IC- 1 1 = ICI-l> 0, the determinant of the matrix

is positive. Hence the determinant of the product C -(Bb + tc)'

- (Bb + }c) ) r

E

= ( -(Bb + tc)

- C-l (B b + tc) ) r

is up to a positive factor the same as that in (18.1). Hence (18.1) is equivalent to r> (Bb

+ tc)' C-l(Bb + tc).

(18.2)

To this relation one must add the inequality (15.7) or

r =/=- - c'A -1 b.

(18.3)

The two inequalities (18.2), (18.3) are our fundamental control inequalities. By Theorem IX of Section 13 they imply complete asymptotic stability of the control system for all allowable functions f.

86

3. LIAPUNOV'S THEORY AND CONTROLS

Since C > 0, we know (Section 15) that C-I > O. Consequently, the right hand side of (18.2) is non-negative, and this fundamental inequality implies (as we have assumed already) that r > O. Observe that the control efficiency actually increases with r, as this means moving away from the situation where V ceases to be negative definite, and so in substance to augment the efficiency of the control mechanism. A noteworthy special case. Let the characteristic roots of A be - f-tv' .. , - f-tn, where the f-tn are all distinct and positive. Choose coordinates x such that

A =- D,

D

= diag (f-ti' .. , f-tn).

The system (F) becomes i

= -Dx + j(a)b,

a = c' x - rj(a). The system (17.1) becomes bioi

=

b ik

i

Choose now C

=

> k.

diag (dv . . . , dn) where every d" > O. Then

1) C - I -_ d'lag (1 d;"'" dn ' B (Bb)'

· (d2f-t1"'" dlag i

=

= t

(b f-tid i

l , ... ,

dn ) 2f-tn '

bn.dn). f-tn

Hence the inequality (18.2) becomes 1 (b d )2 .L: 4d" ~" " + c, = .L: ~ (b" e" + !:.!:.-)2 ,

r>

4

f-t"

e"

e" =

Vd,,> O.

§ 18.

87

RETURN TO THE CONTROL PROBLEM

We look for the minimum of the sum as a function of the ek' This will occur when each parenthesis squared assumes its least value. Suppose first b k and Ck of the same sign. We may obviously assume that both b k, Ck > O. This being the case the hth parenthesis is the sum of two terms with constant product. Hence its extremum occurs when the two terms are equal or when ek 2 = ttk Ck/bk. For b k Ck < 0 the parenthesis square can assume its least value zero. Hence ' \ ' ek bk Ck

r > ..:::---' ttk

where ek = 1 if b k Ck > 0 and = 0 otherwise. This gives a lower bound in the present case for the value of r.

§ 19. Control with Some Characteristic Roots Zero Let several, but not all, the characteristic roots of A be zero. A comparatively simple situation arises when, by a suitable choice of coordinates one may put the basic system (F) in the form

u=

Au

Z=

~d,

+ ;b,

{= j(a),

a = c'u+ 2e'z -r~ where y, d, e are p-vectors and the other letters are as before. Introducing the new variable vector x by the relation x

=

u=

A~t

+ ~b,

Y = Z = ~c

and operating as in Section 15, it may be shown here also that the stability relations remain unchanged provided that (15.7) holds. This being as~umed the new system takes the form i = Ax

y=

+ j(a)b,

f(a)d,

0' = c'x

+ 2e'y -rf(a).

(19.1)

88

3. LIAPUNOV'S THEORY AND CONTROLS

Here x, A, b, a, j(a), r are the same as before (same dimensions), and y, d, e are p-vectors. Since y remains fixed in the uncontrolled systems its components Yh are called neutral parameters. We look for a Liapunov function (J

V(x, y, a) = y" My

+ {XI. Bx + f j(a) o

-!

Here the bracket is the same as before, and M is a positive matrix. Thus V is positive definite for the whole space (x, y, a). By direct calculation one finds

- V = {- x'Cx - rj2(a) + 2j(a)(Bb A'B+ BA

=

+ tC)'X} + 2(Md + e)'yj(a), -C.

If one treats {... } as earlier and choose (if possible) M, d, e so that

Md+e= 0,

'(19.2)

°

for one will obtain a Liapunov function V such that - V > x, j(a) =1= but zero for x, j(a) = 0, y =1= 0. Hence from the chosen type of V(x, y, a) one may at best, deduce stability by the first theorem of Liapunov, and the asymptotic behavior can be deduced by Theorem ViII of Section 13. The set E of Theorem Vl Il corresponds to x = 0, a = 0, Assume that a solution of (19.1) remains in E. Then

°

i= 0,

:Y = 0, a

= 2e' y ,

and we see that this can happen only if e'y = 0. The plane x = 0, a = 0, e'y = is the set M of Theorem VIII, and all solutions approach this plane as t ~ 00. Each point of this plane is a critical point (an equilibrium) of (19.1) and this is the best result that one could expect.

°

§ 19.

89

SOME CHARACTERISTIC ROOTS ZERO

To proceed a step further suppose that d' = (1, 1,.. " 1). Upon setting

m' = (m1 , • • • , mp), we find from (19.2) m=-e.

(19.3)

One may ask then whether it is possible to find an M > 0 such that (19.3) holds. Consider first the case p = 2. We have

and since M

> 0:

For simplicity set m 12

=

h.

Then

and h is subject to the sole inequality

(-e 1 -h)(-e2 -h) _h 2 = e1 e2 + (e1 + e2)h > O.

+

If e1 e2 = 0 this inequality becomes e1 e2 > 0 which is ruled out, and one cannot satisfy (19.3) in the present instance. If e1 e2 i= 0 one may always choose a suitable h. Of course the general case is more complicated. Let us endeavor to select

+

M=(;

~),

H=diag(h 1 , ••• ,hp _ 1) , g' = (gl" . " gp - 1), Thus H > O.

hi>O,

S is a scalar.

The condition that M > 0 reduces to (19.4)

90

3. LIAPUNOV'S THEORY AND CONTROLS

On the other hand (19.3) yields gi+h i = -ei,

S

+ .2: gi =

-

ep.

Hence (19.4) becomes S

>

'\"""'I (ei

~

i

1, then (19.5) will have a plane of critical points and the best that one can hope for is that every solution approach that plane as t ->- 00.

§ 20. Control Computation This side of the problem has two facets: strictly numerical and literal. That is, the first problem is the computation of controls given in a strictly numerical form. This problem may be viewed as solved. One selects a numerical matrix C > 0, and for simplicity in the normal form diag (dv' .. , d»), dh > 0, then one calculates B from the numerical linear system (16.1). We have

C-l = diag (1/dv' .. , lid»)

92

3. LIAPUNOV'S THEORY AND CONTROLS

and so all the elements entering in the inequality (y{2) are at hand. One may then consider the numerical problem as solved. We turn now our attention to the literal problem. This problem is, of course, of great practical importance. For in many questions there do enter, besides the strict numerical data, certain variable parameters whose range of variation for certain purposes must be determined. This justifies the attention which is to be devoted to the question. In order to avoid major complications we shall assume that the characteristic roots of A are distinct and nonzero. We must still distinguish between the two possible cases: all roots real; or some complex. FIRST CASE: The characteristic roots oj A are all real and hence negative (general assumption).

Let the characteristic roots be Av"" An, and set An = - fin, fin> o. Let also K = diag (fi1" . " fin). What we wish to accomplish is to derive explicitly the basic inequality (18.2). Whatever procedure one follows one must directly or indirectly use a transformation of coordinates x = Py reducing the matrix A to -K. We take then an arbitrary symmetric matrix C >0 and will determineB from the system (17.1). Set as in (17.2) A * = -K =P-1AP, B* = pi BP, C* = P'CP and also b* = p-1 b, c* = PiC. The system (F) assumes then the form

y = - Ky

+ j(a)b*,

(20.1)

a = c*' y -rj(a), and (17.4) becomes

i

> k.

Thus we have C -+ C* -+ B* -+ B. However, our inequality does not require the return of B to C. If C*-l = (Yii) , then Yii =

cofactor of Cii*

ICI

In

C*

§ 20, CONTROL COMPUTATION

93

and we have directly

"" . (1

r > ..:::... Y"

.*

2 C,

+ ..:::... "" c,~+b: ~,

)

~h

(1

.*

2 C,

+ ..:::... "" c/i.+b* ) ' k

~j

~k

(20.2)

If we choose C* = diag (d v . . . , d n ), d; > 0, then our earlier computations yields the much simpler expression

"" e,-b,* c,* r>..:::...

(20.3)

~i

where e, = 1 if b,* c,* is positive and = 0 otherwise. To complete the treatment there remains to calculate both P and t-».

Calculation oj P. We have P-IAP=-K, hence AP=-PK. Upon comparing (i, k) terms there follows

Hence the

Pik

for k fixed, are a solution of the vector equation

(A - Ak I)~ = O.

(20.4)

Now because the Ak are simple roots of the characteristic equation of A, it is known that: (a) the matrix A - Ak I is of rank n - 1; (b) the vector solution of (20.4) is unique up to a proportionality factor; (c) the components ~, of the solution are proportional to the cofactors not all zero of any row of the determinant IA - Ak II. In view of (a) one set of cofactors (for some r) has nonzero terms. Such a set is chosen as the solutiorr e. Thus the kth column of the matrix P will consist of a definite nonzero solution of (20.4), whose components may be taken as the cofactors of the elements of some one row of A - Ak I.

Calculation oj P':',

If p-l = (n,j) then

cofactor of nij =

IFI

Pi'

in P

.

Our problem is thus solved for the case under consideration.

94

3. LIAPUNOV'S THEORY AND CONTROLS SECOND

CASE.

Among the characteristic roots

0/ A some are

complex.

Since the characteristic equation IA-UI=O

has only real coefficients, the roots Av . . . , A.,. appear in conjugate pairs. We will assume that they are all distinct, with negative real parts and order them as follows

where the first 2k are the k conjugate complex pairs and the last 2k are real. Let

n -

s < k, s

>

2k,

where the Ph are all positive. A first transformation of coordinates x Pl-lAP l

= diag (A v

XV"" Ak'

= Pz is made such that

z, A2k+l,""

An).

The matrix P l is the same as the former P save that now the columns corresponding to Ah and Xh consist of conjugate elements. The coordinates Z will be so disposed that for real points the Zv Zv Z2' Z2" .. , Zk, Zk' Z2,+1>"" z; correspond to the Ah in the order already indicated. Thus the equation x = Ax, will have become (for real points)

h< k,

s

We will now make a second coordinate transformation as follows. If Zh = Zh' + is,", h < k, then

Zh = Yh'

+ iYh", Zs

= Ys,

in S

> Z =

2k.

P2 Y

= Yh' -iYh",

>

2k.

Thus P 2 transforms separately the conjugate pairs Zh, Zh and does not affect the real coordinates s; Set P = P 1 P 2' Thus P is again a matrix of transformation from real coordinates x to real coordinates Y and therefore, as readily proved, P is real.

§ 20.

95

CONTROL COMPUTATION

We will now operate solely in the coordinates y. Let A *, ... , have the same meaning as before. The intermediate matrix, A o = P 1 - l API multiplies Zk by Ak' Zk by .':k for h < k, and Zs by As (real) for s > 2k. Hence it sends Zk' iz," into (-,uk iVk)(Z,,' iz,,") or it sends z,,' into -,u" z,,' - V" z;", and z/' into v" z/ - ,u" z;", In terms of the y', y" we have then

+

(A *yh'

(A *yh" =

(A *Y)s

,u" Yk' -

= -

v" y,," -

+

+

v" y,,",

,u" y,,",

= - ,us Ys,

h

< k.

s

>

2k.

This means that A * has the following structure: G1

A*

0

=

- ,un where

Choose now Thus 1 C *- 1 -_ dilag (1 d;' d;

1 1 1 ) ,... , d; , d; , d2k1+1 ' ..• 'd;. .

To calculate B* we observe that we may consider C* as having the same structure as A * (with different blocks of course). It is readily shown that one may assume the same structure for B*, say HI

B* =

0

96

3. LIAPUNOV'S THEORY AND CONTROLS

Upon writing down the system (16.1) for the k, we find at once s

>

2k.

As for the H blocks to compute them we only need to take into consideration Gh and the analog (

0)

dOh

dh

of C*. Upon supressing the indices we arrive at the relation

G'H or with H - fl ( -v

= (; v -fl

+ HG =

-diag (d, d)

~)

)(rx

{3

(3)+(rx y (3

(3)(-fl -V)=_(d Y v -fl 0

0). d

Equating now terms in the same positions there follow the three equations - 2flrx ;J;;,2v{3

=

-d,

- 2flY - 2v{3

=

-d,

- 2fl{3

+ v(y -

rx) = 0,

which yield readily

Since G is nonsingular

and finally

flY

# 0 and hence {3 =

o.

Thus

§

With b* = P 2 the inequality

20. CONTROL COMPUTATION

1 P1-l

band c*'

=

97

c'P 1 P 2 we obtain from (18.2)

The minimum of the second sum is calculated as before and found to be "\"

.£.; s > 2k

bs* cs* es - fls

where e, = 1 if b,* c,* > 0 and e, = 0 otherwise. Regarding the terms of the first sum we wish to select each d h to minimize

Define

The above term is minimized by selecting

and the minimum value is

98

3. LIAPUNOV'S THEORY AND CONTROLS

The basic inequality for the case of 2k complex roots can now be written r

"" 21 (bh Ch + b" > ...:::.. h Ch h;;;;'k

+ bh

II

+...:::..

c« " )

""

flh

b,* c,* , e, -

s>2k

°

(20.5)

fls

°

where e, = I or according to whether b, * c,* > or < 0. Remember that flh is the negative of the real part of the complex root Ah'

§ 21. Application to an Actual Case The following question arose in a control problem. A system with a servo-mechanism is represented by

e + !Xe + fJe = re, g + aq = gi + he, C=

f(a),

a = kl q - k 2

e

e,

where the nonconstant parameters are e, q, a. Of course a, f(a) are the control elements. The system is to be examined for stability. Upon differentiating the first two equations and letting Xl = i, x 2 = e, X s = g there results the system i

= Ax + f(a)b,

(21.1)

a = c' . x - rf(a) which is really (F) but with dimension n are given by

A

~ (~p :. ~ c' = (0,0, k l ) ,

= 3. The new constants

J~ b

r = k 2•

( : )

§ 21.

99

APPLICATION TO AN ACTUAL CASE

+

'*

It is not difficult to show that if k and k i hy k 2 mfJ 0 then if x, a tend to 0 so do e, q. Hence the stability properties may as well be studied for the system (21.1).

The characteristic equation is

-fJ

-Cl-A

o o

h

g

-a-A

1

-).

=0

or

(A + a)(A(A + Cl) If we set b

+ fJ) = o.

= VCl 2 - 4fJ the roots are

FIRST CASE: A2 , A3 are real (and distinct). The requirement that they be real distinct and negative yields:

a >0,

a

>0.

We note also that

Calculation oj the matrix P. The elements of the first column are proportional to the cofactors of one of the rows (not all zeros) of the matrix 1

a-Cl

g Taking the cofactors of the elements in the second row we find: Pn

=

P21

=

0,

P3I =

h-ag.

100

3. LIAPUNOV'S THEORY AND CONTROLS

For the elements of the second column P2i we must take the cofactors of a row, not identically zero, of 1

g

Take the cofactors of the second row:

Using the equation satisfied by A2 we may write

The calculation is the same for the P3I =

A3

+ a,

P32

P3I

with A2 replaced by A3 • Thus

= (a -1X)A3 - fJ,

Thus a

+ A2

a

(a -1X)A 2 -

fJ

gA 2 + h

+A

3 )

(a-IX)A 3 - f J gA 3

.

+h

Since c' = (0, 0, kl ) we have

c*'

= c' P = [kl(h - ag), kl(gA 2 + h), kl(gA 3 + h)J.

We must now calculate b* = p-I b. Let p-I b* = p-I b, and b' = (0, y, 0) we have b*'

=

(ni,k)'

Since

= (yn I 2 • yn 2 2' yn 32 ) .

Hence it is only necessary to calculate the elements of the first column in P-I. If P ii is the cofactor of Pii in P then nii = Pii/IPI. Hence all that is required is to calculate the cofactors of the elements of the second row in P and the determinant IFI. We have at once

iPI = = =

+ A [(a - IX)A (h - ag){(a(a -IX) + fJ)(A - (h - ag) [a(a - at) + fJJo. (h - ag){(a

2

3 -

fJJ - (a + A3 ) [(a - IX)A2 - fJJ}

3 -

A2 )}

§ 21. APPLICATION TO AN ACTUAL CASE

101

The cofactor of P21 is: P 21 = - (a + A2)(gA3 =

+ h) + (a + 1.

3)

(gA 2 + h)

(ag -h)o = n121PI.

The cofactor of P22 is: P 22 = (ag -h)(a

+ 1.3) =

n221PI.

+ 1. 2) =

n321PI.

The cofactor of P23 is: P 23 = (h -ag)(a Hence

b*' = y(ag-h)(s IPI u, a

+1

11.3' -

a-

1)

11.2 •

Finally r

>

8 1 bl

* cl * + 8 2 b2 * c2* + 8 3 ba* ca* = - 1.2

- Al

.{8l(h -

ag)o 2a

where e, = 1 if bi *

Aa

-

2k l Y

-=-.,---~-=-o-

[a(a

+ 82(a + A:})(gA2+ h) _

Ci*

0

IX -

>

+ PJO 8a(a + A2)(gAa + h)}. +0

-IX)

IX

° and equals zero otherwise.

SECOND CASE: .1. 2 , Aa are complex. Here 1X 2
to and Ilxll < e then jlX - f31 < 'fJ.

°

°

§ 22.

GENERAL CONTROLS

105

The practical value of the relation -=- is simply that for large t and small x the sign of (J. is just that of p. Let us suppose then that the system (22.2) may be written i . Ax + t(a)b,

(22.5)

6' = c'x -rt(a)

where sufficiently small refers now to the vector (x, a) that is, to the smallness of Ilxll + lal. Of course, the various symbols: A, ... , are as in the earlier fundamental system (F). The meaning of (22.5) is simply that the right hand sides are just the "principal parts" of F and G. Now if we go through our earlier treatment in Sections 14, 15 and 17, we will find that essentially everything goes through with = replaced by '. In particular if the basic inequality (17.2) holds V will still be a Liapunov function for a certain region:

o < Ilxll + lal
o.

(F)

Let x(t) be a solution such that x(to) = xo. It is known that: (a) either the solution may be extended for all t > to' and then we shall say that x(t) is defined in the future; or else (b) there is a time T > to such that Ilx(t) 11-.. + 00 as t -.. T and then the solution is said to have a finite escape time. This property and the previous one are manifestly the opposite of one another. A third possibility is (c) the solution x(t) is bounded, which is entirely compatible with (a) but of course not with (b). The boundedness of all solutions is a kind of stability - it is, in fact, described as stability in the sense of Lagrange, or more simply as Lagrange stability. 107

108

4. EXTENSIONS OF LIAPUNOV'S METHOD

Throughout what follows V(x, t) will continue to denote a scalar function which is positive and has continuous first partial derivatives in some specified domain. In this domain

.

V

av

av

av

av

= aXl Xl + ... + aXn X 2 + at = at + X

grad V.

The first move is to consider the general differential inequality

v < G(v, t),

(23.1)

where G is a scalar function and v is a scalar function of t. We are only interested in positive functions v satisfying (23.1). There are two types of such inequalities: (I) those which have no positive solution with finite escape time; (II) those which have no positive unbounded solution. The first type includes the second. Besides the solutions ruled out under (I), (II) also rules out solutions which ->- + 00 with as t ->- + 00. If Q is any set, QC will stand for its complement; that is, for the set of all points outside Q. XIII. THEOREM. Let Q be a bounded set containing the origin, and let V(x, t) be defined throughout QC and for all t > O. Moreover, let V(x, t) ->- + 00 as Ilxll ->- + 00 and this uniformly on every finite time interval 0 < a < t < b. Furthermore let V < G(V, t) hold throughout QC and for all t > O. If (23.1) has no positive solution with finite escape time then every solution x(t) of (F) is defined in the future. This is almost obvious. If x(t) had a finite escape time t1 < T it would be and remain in Q'. Then v(t) = V(x(t), t) would be a positive solution of (23.1) with positive escape time and this is ruled out. In applying this and similar results later one usually finds it convenient to take a function G(v, t) = k(t)L(v), where k is continuous for all t > 0 and L(v) is positive and continuous for all positive v. The inequality (23.1) then becomes vIL(v)

< k(t).

(23.2)

§ 23.

FINITE ESCAPE TIMES; LAGRANGE STABILITY

109

(a) If J+<x> [dvjL(v)] = + 00 (the integral does not converge), then (23.2) has no positive solution with finite escape time T. For then I

v(l)

J L~~) = Jk(t) dt, v(I.)

I.

Here however the right-hand side is bounded as t -> T while the left-hand side would approach 00 if there were a positive solution with finite escape time T. Similarly:

+

(b) If J+<x> [dvjL(v)] = + 00 while J+<X>k(t) dt < + 00 (convergent integral) then (23.1) has no unbounded positive solution for t > o. Thus iJ = k(t)v where k is any continuous function for t > 0, is of type (I), while iJ < e:' v is of type (II). Of course the simplest inequality (23.2) is iJ < o. We shall now discuss systems with finite escape time. This time one has recourse to the inequality iJ

> G(v, t),

t> 0,

(23.3)

such that there are no positive solutions defined in the future. If G(v, t) = k(t)L(v), with k, L as before, we see that (c) If J+<x> [dvjL(v)]


k(t) dt = 00 then (23.3) has no positive solution defined in the future. (The argument is obvious.) As an example iJ > co", c > 0, ':I. > 1, can have no positive solutions defined in the future. The following proposition providing conditions for a finite escape time may be viewed as an instability result. 00

XIV. THEOREM. Let Q be a region such that if a solution x(t) starts in Q it remains thereafter in Q. Let V(x, t) be positive for all x in Q and all t > o. Suppose that V >G(V, t) holds for all t > 0 and all x in Q. If (23.3) has no positive solution defined in the future then, each solution x(t) of F with x(to) = X O in Q has a finite escape time.

110

4. EXTENSIONS OF LIAPUNOV'S METHOD

Suppose in fact that x(t) has no finite escape time. Then v(t) = V(x(t), t) satisfies (23.3), is positive and defined in the future in contradiction to assumptions. Linear systems are the best known example where the solutions are all defined in the future. Thus, if x

=

A(t)x

+ j(t),

t > 0,

where A (t) is an n X n matrix and j(t) is an n-vector and both are continuous for t > 0, then the solutions are all defined in the future. As an illustration of Theorem XIII we generalize this result for linear systems as follows: 1. Assume for some R > 0 and some scalar function k(t) continuous for all t > 0 that X(x, t) of (F) satisfies

EXAMPLE

IIX(x, t)1I < k(t)lIxli for all t > 0 and all Ilxll > R. Define V(x) = IIxl12 = x'x. Then by Schwarz's inequality (a'b < Iiali' Ilbll) we have for all Ilxll ~ R and all t > 0 that V(x)

=

2x 'X(t, x)

< 2k(t)llxI12 =

2k(t)V(x).

Since iJ < 2k(t)v has no positive solutions with finite escape time, we can conclude from Theorem XIII that all solutions for t > 0 are defined in the future. EXAMPLE 2. If the system is nonlinear and does not satisfy a linear inequality such as that in the example above, then sufficient conditions for all solutions to be defined in the future become more specialized. Consider the second-order system

x + j(x, X, t)x + g(x) =

e(t).

Assume that j(x, X, t) and g(x) have continuous first partials and that e(t) is continuous for all t > O. Assume also that j(x,

x, t) > 0

for

and

t > 0,

§ 23.

111

FINITE ESCAPE TIMES; LAGRANGE STABILITY

and that x

G(X)

= .\ g(U)

du

-+

+

Ixl

as

00

-+

+

00.

o

Sufficiently far out from the origin the damping is non-negative and for Ixl sufficiently large g(x) is like a restoring spring. An equivalent system is

y = - g(x) - j(x, y, t)y

i = y,

+ e(t).

Let

V(x, y)

t y 2 + G(x).

=

Here the set Q of Theorem XIII is the interior of the circle x 2 + y2 < r 2. Outside this circle

v=

- j(x, y, t)y2 + e(t)y

< le(t) I, Iyl Thus, V 0, and the system is Lagrange stable. This is a slight generalization of a result due to Antosiewicz. The condition in the example that <Xl

Jle(t) Idt o

be bounded restricts the forcing terms e(t) to those that die out sufficiently fast as t ->- 00. This rules out periodic forcing terms, and this is to be expected since the damping is assumed only to be non-negative. For instance, + x = cos t has unbounded solutions, and we see that in order to relax the restrictions on the forcing term e(t) it is necessary to place further restrictions on the damping. This is illustrated in the next example.

x

EXAMPLE 6.

For the second order differential equation

x + j(x)x + g(x) =

e(t)

§ 23.

FINITE ESCAPE TIMES; LAGRANGE STABILITY

115

assume that j(x) and g(x)· have continuous derivatives for all x and that e(t) is continuous for all t > O. Define x

F(x)

=

Jj(u) du, o x

G(x) =

Jg(u) du, o

and t

E(t)

j

= e(r) da: o

Assume further that j(x) > c > 0 for all x and that 4j(x) [F(x) E(t)Jg(x) > e2(t) for all t > 0 and all sufficiently large 14 We will show in a moment that under these conditions the system is Lagrange stable. For this second order equation we have as an equivalent system

x= y,

y=

-

g(x) - j(x)y

+ e(t).

Defining VI =

Hy + F(x) -E(t))2 + G(x)

and

we obtain

171 = - [F(x) - E(t)Jg(x) and

172 = - j(X)y2 + ye(t). Now our assumptions above imply that 171 < - e2(t)j4j(x) for Ixl sufficiently large and that 17 2 < e2(t) j4j(x). Therefore for Ixl > a

116

4. EXTENSIONS OF LIAPUNOV'S METHOD

+

(a sufficiently large) and all y we have T\ V2 < o. Our basic 2(t) inequalities imply that e and IE(t) I are bounded for t > 0, and hence we see that V 2 < - A.(y) where A.(y) -.. 00 as Iyl -.. 00. Therefore, for Ixl < a and Iyl > b (b sufficiently large) VI V 2 < o. We have shown that VI + V 2 ~ 0 outside the region defined by Ixl < a and Iyl < b. It is also not difficult to see that VI V 2 -.. 00 uniformly for t > 0 as Ixl2 + lyl2 -> 00. The function V = Vi + V2 satisfies the conditions of Theorem XV, and this system is Lagrange stable.

+

+

§ 24. Ultimate Boundedness Continuing with our extension of the Liapunov method along the lines laid down by Yoshizawa, we present the extension as a generalization of another sort. Essentially we consider the stability of sets M, where M is a closed but otherwise quite arbitrary set of points in n-space. For any positive number r we use -,, -' \ ,, Me M; to denote the set of all / , ... , .. points whose distance from M / I ... I is less than r (Fig. 24). Thus Mr\ M \ I x in M, means that, for some I \ ,, / yin M, IIx - y!1 < r. M/ de-' ...... notes the set of points outside M r ; that is, M,' is the complement of Mr. If M is the set FIG. 24. defined by IIxll < R, then M; is the set IIxll < R + rand M,' is the set IIxll > R + r. We give first two quite elementary lemmas. Our fundamental system is

-- ------

;'

,

/

---

---

x = X(x, t),

t > O.

(F)

1. V(x, t) is a scalar function with continuous first partials for all x and all t > 0, and M is a closed set in n-space. If V(x,t) < 0

LEMMA

§ 24. ULTIMATE BOUNDEDNESS

117

for all x in Me and if V(x l , tl ) < V(x 2 , t2 ) for all t2 > tl > 0, all Xl in M and all X 2 in M,c, then each solution of (F) which at some time to > 0 is in M can never thereafter leave Mr. PROOF. Let x(t) be a solution of (F) which at time to > 0 is in M. Suppose at some later time T> to that x(T) were in M;". There would then exist a tv to < t l < T, with the property that x(t) is in Me for t l < t ::::;: T and that t l is the smallest number with this property. This would imply that x(t l ) is in M, and therefore V(x(t l ) , t l ) < V(x(T), T). But this cannot be since V(x(t), t) is nonincreasing for t l < t < T. The above lemma does not exclude the possibility that a solution starting in M might not have a finite escape time. If M is bounded as well as closed, then this cannot happen and every solution starting in M remains in the future in Mr. The set M, is also a bounded set, and the solutions which, at any time to > 0 are in M, are uniformly bounded; that is, there is a number b > 0 with the property that x(to) in M for any to > 0 implies Ilx(t) II < b for all t' > to' LEMMA 2. If, in addition to the conditions of Lemma 1, V(x, t) > 0 and "V(x, t) < - e < 0 for all t > 0 and all X in Me, then each solution of (F) that is defined in the future is ultimately inside Mr. [That is, if x(t) is a solution of (F) defined in the future, then there exists a T such that x(t) is in M, for all t ~ T.] PROOF. Since V > 0 and "V < - e < 0, it is clear that every solution that is defined in the future must eventually enter M. By Lemma 1 it must thereafter remain in Mr. The first theorems to be obtained as consequences of these two lemmas have to do with the ultimate boundedness (of the solutions) of the system (F). We say that the system (F) is ultimately bounded if there is a b > 0 such that corresponding to each solution x(t) of (F) there is « T::» 0 with the property that Ilx(t) II < b for all t > T.

118

4.

EXTENSIONS OF LIAPUNOV'S METHOD

XVI. THEOREM.

If in addition to the conditions of Lemma 2 the set M is bounded and V(x, t) - 00 uniformly for t > 0 as Ilxll- 00, then the system (F) is ultimately bounded.

PROOF. Since M is assumed to be bounded, M; is bounded for each r > O. Therefore, by Lemma 2 all that we have to show is that these additional conditions imply that all solutions of (F) are defined in the future. As we pointed out before, when M is bounded, all solutions starting in M are uniformly bounded, and hence defined in the future. Suppose that x(t o) is in Me. Because of the condition on V(x, t) as Ilxll approaches infinity we can select an R so large that V(x, t) > V(x(t o), to) for all t > to and all x in M R e • If x(t) remains outside M, then V(x(t), t) is decreasing and x(t) must remain in MR' If x(t) enters M, it follows from Lemma 1 that it remains thereafter in Mr. Hence all solutions are bounded, are consequently all defined in the future, and by Lemma 2 the system (F) is ultimately bounded. This theorem on ultimate boundedness suffers from the fact that there are a large number of conditions to be verified. There are some simple and quite useful instances where the statement can be simplified. The simplest theorem is for an autonomous Liapunov function V(x). Although V(x) does not depend upon t explicitly, V(x) does for nonautonomous systems (F). In this case inequalities on V are assumed to hold for all t > O. XVII. THEOREM. Let V(x) be a scalar function which for all x has continuous first partial derivatives with the property that V(x) - 00 as Ilxll - 00. If V(x) < - e < 0 for all x outside some closed and bounded set M, then (F) is ultimately bounded.

PROOF. We need to verify the conditions of Theorem XVI which contains also the conditions of Lemmas 1 and 2. Since V does not depend upon t, its approach to infinity is uniform in t. It is also clear that this implies that V(x) is bounded from below

§ 24.

ULTIMATE BOUNDEDNESS

119

for all x. Therefore, we can add a constant to V(x) and have it be nonnegative for all x. The set M is closed and bounded and V is continuous, and V(x) is bounded for x in M. Therefore, we can find an r sufficiently large that V(x 1) < V(x 2 ) for all Xl in M and all X 2 in Mr". All the conditions of Theorem XVI are satisfied, and (F) is ultimately bounded. We wish now to give an example illustrating the application of this theorem on ultimate boundedness. EXAMPLE.

For the forced Lienard equation

i

+ f(x)x + g(x) =

e(t),

t > 0,

assume that f and g have continuous derivatives and that e(t) is continuous. Define x

F(x)

=

j

f(u) du,

o t

E(t) =

je(T)

dT,

o

and x

G(x) =

jg(u) duo o

An equivalent system is

x= Y=

Y -F(x)

+ E(t),

-g(x).

Assume that xg(x) > 0, Ixl sufficiently large, that IE(t) I is bounded, and that g(x)F(x) -+- qo as Ixl -.. 00. Let V = t[y -:-h(x)J2 + G(x),

120

4. EXTENSIONS OF LIAPUNOV'S METHOD

where h(x) is to be determined.

V=

Then

-h'(x)[y-F(x) +E(t)J[y-h(x)J -g(x)[F(x) -h(x) -E(t)].

We use here the bounded closed set M defined by Iyl ~ b. For Ixl > a define h'(x) = O. Then

V=

Ixl < a

and

-g(x)[F(x) -h(x) -E(t)J,

and we see for a sufficiently large that V < - B < 0 for Ixl > a. Define h(x) = - c for x < - a and h(x) = c for x > a (c > 0). For Ixl < a, we want h'(x) > O. Define h(x) = cia x. Then

V= -

h'(x) [y -F(x) - E(t)J [y - h(x)J

-g(x)[F(x) -h(x) -E(t)J

and for b sufficiently large we have V < - B < 0 for jxl < a and Iyl > b. Therefore, outside M we have V < - B < O. The function h(x) is continuous but has a discontinuity in the derivative at x = a and at x = - a. It is not difficult to see that this does not matter. Clearly V(x) --+ 00 as Ixl --+ 00 and it follows from Theorem XVII that the system is ultimately bounded. For somewhat more general conditions which imply the ultimate boundedness of Lienard's equations with a forcing term, note that the above argument goes through if:

> 0;

(1)

IE(t) I < Eo

(2)

(sgn x)g(x)

> IX > 0

for

Ixl >a;

(sgn x)F(x)

> F o > Eo

for

Ixj >a.

for all

t

and (3)

In particular, this example shows that the solutions x(t) and i(t) of van der Pol's equation

x + B(X 2 - l ) i + x = e(t) t

are ultimately bounded if

If e('l") d'l"l o

is bounded for t > O.

§ 25.

121

PRACTICAL STABILITY

§ 25. Practical Stability In this section we wish to present some ideas on the question of practical stability with the hope that this discussion will stimulate new thinking on this important question. At the same time it becomes clear why some stability investigations should not be taken too seriously. Investigations that take into account only the linear approximation fall into this category. We have made the point in Section 13 that stability and even asymptotic stability by themselves may not assure practical stability. One needs to know the size of the region of asymptotic stability, and then based on estimates of the conditions under which the system will actually operate, requirements on its performance, etc., one can judge whether or not the system is sufficiently stable to function properly and may be able to see how to improve its stability. Having decided that asymptotic stability is not by itself sufficient for practical stability, we might be inclined to conclude that it is, however, always a necessary condition. This, too, is incorrect. The desired state of a system may be mathematically unstable and yet the system may oscillate sufficiently near this state that its performance is acceptable. Many aircraft and missiles behave in this manner. (Later in the section we discuss a system with an unstable equilibrium which is practically stable.) With these ideas in mind, let us formulate a definition of practical stability. The fundamental system is

x=

X(x, t),

t>

(F)

O.

The equilibrium state is at the origin: X(O, t) = 0 for all t > O. The perturbed system is

x=

X(x, t)

+ P(x, t),

t>

O.

(F*)

We are given a number 0 and all x. If for each p in P, each X o in Qo, and each to 2 0, x*(t, xo, to) is in Q for all t > 0, then the origin is said to be practically stable. The solutions which start initially in Qo remain thereafter in Q (Fig. 25). This concept of practical stability is relative to the number for a > 3/oc and F(a) - h(a) = for a = 3/oc, oc > 0. Let A = F(a) - h(a). By selecting a > 3/oc but as close to 3/oc as we please we have A > but as small as we please. For Ixl < 3/oc,

°

°

V"3

the maximum of IF(x) - h(x) I is Ao = 2 p/oc. Let us agree to select a > 3/oc and such that A = Ao· As Ao -+ 0, a -+ 3/oc. Then for Ixl > a, since h'(x) = 0,

V*
0

for z not in M

(c)

V(Z) < 0

for z not in M.

(i.e., x # y);

The total derivative V is relative to the product system (26.3). The above conditions on V state that relative to M the function V is positive definite and V is negative definite. Just as in Liapunov's second stability theorem it follows that every solution of (26.3) approaches M as t ~ 00. This means that (FP) is extremely stable. Since it was assumed that the solutions 'of (FP) were ultimately bounded, we can then conclude XIX. THEOREM. STEADY-STATE OSCILLATIONS. Under the above conditions the system (FP) has a periodic solution of period T and every solution approaches this solution as t ~ 00.

Literature We list here a few important books which have much bearing upon the topics dealt with in the Monograph.

Vectors and Matrices: PAUL HALMOS, "Finite Dimensional Vector Spaces." Van Nostrand, New York, 1959. A most lucid book by an exceptionally good writer. RICHARD BELLMAN, "Introduction to Matrix Analysis." McGraw-Hill, New York, 1960. An excellent book with a definite slant toward applications. Particularly pertinent are Chapter 10 (Matrices and Differential Equations) and Chapter 13 (Stability Theory). F. R. GANTMACHER, "The Theory of Matrices" (translated from Russian), 2 volumes. Chelsea, New York, 1959. A first rate and standard reference book on vectors and matrices; also contains many applications to differential equations.

Differential Equations: LAMBERTO CESARI, "Asymptotic Behavior and Stability Problems in Ordinary Differential Equations" (Ergebnisse der Mathematik und ihrer Grenzgebiete). Springer, Berlin, 1959. An indispensible source book on the general subject of ordinary differential equations, It contains an unequaled wealth of material, and, in particular, deals with the Liapunov theory and its application to control problems. Noteworthy also is a most extensive bibliography unapproached by any other work on the subject. E. A. CODDINGTON AND NOR~AN LEVINSON, "Theory of Ordinary Differential Equations." McGraw-Hill, New York, 1955. An exceptionally careful, well written, and complete treatise. SOLOMON LEFSCHETZ, "Differential Equations: Geometric Theory." Interscience, New York, 1957. The first chapter consists of a resume of introductory material - vectors, matrices, topology, functions of several variables - not generally found in a book on this topic.

131

132

LITERATURE

V. V. NEMICKII AND V. V. STEPANOV, "Qualitative Theory of Differential Equations." This Russian book, whose second edition came out in 1949, has just appeared in an edited translation issued by the Princeton Univ, Press. The original is in Russian. The translation will, no doubt, be considered for a long time as the standard work in this field.

Liapwnoo's Theory: WOLFGANG HAHN, "Theorie und Anwendung der direkten Methoden von Liapunov" (Ergebnisse der Mathematik und ihrer Grenzgebiete). Springer, Berlin, 1959. A thorough resume of the main points of the Liapunov theory. Quite an extensive bibliography. The best book on the subject in a western language. A. A. LIAPUNOV, "Problerne general de la stabilite du mouvernent." Photoreproduction as Annals of Mathematics Study No. 17 (Princeton Univ. Press, Princeton, New Jersey) of the 1907 French translation of the fundamental Russian paper of Liapunov, published in 1892. It is the bible on the subject of the present Monograph.

Controls: A. M. LETOV, "Stability of Nonlinear Regulated Systems" (in Russian). An English translation will be issued in the near future by the Princeton Univ. Press. A very good and detailed treatment of nonlinear controls. As yet nothing of the sort exists in book form in any western language.

Index Absolute stability, 79 Adjoint system, 27 Asymptotic stability, definition, 32 Asymptotic stability in the large, see Complete stability Asymptotic stability theorem, 37 Boundary of a set, definition, 19 Bounded set, definition, 18 Characteristic equation, 10 Characteristic root, 10 Closed set, definition, 19 Compact set, definition, 19 Complete stability, 57, 66 Critical case, 48 Critical point, 25 Equilibrium point, 25 Existence theorem, 23

Liapunov function, 33, 35, 41 Lienards equation, 60, 68, 114, 119 Limit-cycle, 58 Lim~ting sets, 57 Negative definite quadratic form, 15 Nonsingular matrix definition, 9 Open set, definition, 18 Orthogonal transformations, 15 Oscillations, forced, 126 steady-state, 127, 128, 130 Phase space, 24 Positive definite, function, 33, 41 quadratic form, 15, 36 Principal matrix solution, 27 Region, definition, 18

Finite escape time, 107, 109 Forced oscillations, 126 Fundamental control inequalities, 85 Inner product, definition, 3 Instability, definition, 32 Instability theorem, 38, 39, 56 Integral curve, 24 Invariant set, definition, 58 Inverse matrix (A -I), 9 Lagrange stability, 107, 113 Lagrange theorem, 55

Stability, absolute, 79 asymptotic, 32, 37 complete, 57, 66 of conservative dynamical system, 54 definition, 31 instability, 32, 38, 39, 56 Lagrange, 107, 113 by linear approximation, 47, 57 of motion of a rigid body, 52 under persistent disturbance, 71 theorem, 37, 55

133

134

INDEX

Stable matrix, 75 Steady-state oscillations, 127, 128, 130 Transformation of coordinates, 12 vectors, 12

Trajectory, 24 Ultimate boundedness, 117, 118 Unstable equilibrium, definition, 32 van der Pol's equation, 60, 61, 111, 120