Stability Analysis in Terms of Two Measures
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SMHIIYMIMHt ■ B U I IP
I M HNHIB V. Lakshmikantham Department of Applied Mathematics Florida Institute of Technology
X. Z. Liu Department of Applied Mathematics University of Waterloo
World Scientific Singapore • New Jersey London • Hong Kong
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Library of Congress Cataloging-in-Publication Data Lakshmikantham V. Stability analysis in terms of two measures / by V. Lakshmikantham, Xinzhi Liu. p. cm. Includes bibliographical references and index. ISBN 9810213891 : S48.00 1. Lyapunov functions. 2. Stability. I. Liu, Xinzhi. II. Title. QA871.L246 1993 003\85'0151535~dc20
93-14063 CIP
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Preface
The problems of modern society are both complex and multidisciplinary.
In spite of the apparent
diversity of
problems, tools developed in one context are often adaptable to an entirely different situation. As an example, consider the well-known Lyapunov's second method, which has gained increasing significance and has offered decisive impetus for modern
development
equations.
of
stability
theory
of
differential
A manifest advantage of this technique is that it
does not require the knowledge of solutions and therefore has great power in applications to the real world problems from many disciplines. It is now recognized that the concept of Lyapunov function can be employed to investigate various qualitative and quantitative properties of dynamic systems. v
Lyapunov
Preface
VI
function serves as a vehicle to transform a given complicated differential system into a relatively simpler system and as a result, it is enough to study the properties of solutions of the simpler system.
It is also being realized that the same
versatile tools are adaptable to discuss entirely
different
nonlinear problems, and these effective and fruitful techniques offer an exciting prospect for further advancement. The concepts of Lyapunov stability have given rise to many new notions that are important in applications.
For
example, eventual stability, partial stability, relative stability, conditional
stability,
total
stability,
and
corresponding
boundedness and practical stability notions to name a few. Relative to each concept, there exists a sufficient literature parallel to Lyapunov's theory of stability. It is natural to ask whether we can find a notion and develop the corresponding theory which unifies and includes a variety of known concepts of stability in a single set up. The answer is yes and it is the development of stability theory in terms of two measures. is in this spirit monograph.
we see the importance of our
It
present
Its aim is to present a systematic account of
recent developments in the stability theory in terms of two distinct measures, describe the current state of the art, show the essential unity achieved by wealth of applications, and provide a unified nonlinear problems.
general structure applicable to
several
Preface
vii Some important features of the monograph are as
follows: (i)
It is the first book that is dedicated to a systematic development
of stability
criteria in
terms
of
two
measures. (ii)
It demonstrates the manifestations of general Lyapunov method in terms of two measures by showing how this effective technique can be adapted to study various apparently diverse nonlinear problems.
(Hi)
It shows how in the present framework of two measures one can unify criteria of Lyapunov stability, practical stability
and
boundedness
properties,
and
provides
interesting applications of real world problems. In view of several excellent books on Lyapunov second method, we have not included any material of the usual Lyapunov stability criteria. different
measures
offers
Since the interplay of the two new insights into the
stability
problem and provides opportunity for new applications, and further advancement, we do hope that this monograph will stimulate further investigation of this fruitful technique of employing two different measures. The second author was supported in part by a grant from the Natural Science and Engineering Research Council of Canada, which is gratefully acknowledged.
viii
Preface
We are immensely thankful to Dr. B. Kaymakcalan for proofreading and Ms. Donn Harnish for the excellent typing of the manuscript. V. Lakshmikantham Xinzhi Liu
Table of Contents Preface
v
1. Basic Theory 1.0 Introduction 1.1 Definitions of stability 1.2. Basic Lyapunov theory 1.3 Comparison method 1.4 Converse theorem 1.5 Boundedness and Lagrange stability 1.6 Practical stability 1.7 (h0, h, Af 0 )-stability 1.8 Invariance principle 1.9 Notes 2. Refinements 2.0 Introduction 2.1 Several Lyapunov functions 2.2 Perturbations of Lyapunov functions 2.3 Several Lyapunov functions (continued) 2.4 Method of vector Lyapunov functions 2.5 Perturbed systems ix
1 1 2 11 25 39 48 55 68 79 86 89 89 91 107 125 131 153
x
Contents 2.6 2.7 2.8
Variation of Lyapunov's method Integral stability Perturbations of Lyapunov functions (continued) 2.9 Method of higher derivatives 2.10 Comparison systems 2.11 Cone-valued Lyapunov functions 2.12 Notes
169 180 192 200 207 214 221
3. Extensions 3.0 Introduction 3.1 Delay differential equations 3.2 Implusive differential systems 3.3 Stabilization of control systems 3.4 Impulsive integro-differential systems 3.5 Discrete systems 3.6 Random diffferential systems 3.7 Dynamical systems on time scales 3.8 Notes
223 223 225 255 279 296 310 319 330 350
4. Applications 4.0 Introduction 4.1 Holomorphic mechanical systems 4.2 Motion of winged aircraft 4.3 Models from economics 4.4 Motion of a length-varying pendulum 4.5 Population models 4.6 Angular motion of rigid bodies 4.7 Notes
353 353 354 361 364 369 373 380 386
Reference
387
Index
397
1. Basic Theory
1.0
Introduction. The concepts of Lyapunov stability have given rise to
several new notions that are important in applications. example,
partial
stability,
conditional
stability,
For
eventual
stability and boundedness to name a few. In order to unify a variety of known concepts of stability and boundedness, it is beneficial to employ two different measures and obtain criteria in terms of these measures. This chapter is therefore devoted to the development of the basic theory of Lyapunov in terms of two different measures.
1
Basic
2
Theory
In Section 1.1, we define various concepts of stability by means of two different measures which are general enough to include several known concepts in the literature.
Section
1.2 discusses results parallel to Lyapunov's original theorems which show the interplay between the two different measures. We
consider
in
Section
1.3,
the
use
of
general
comparison principle for obtaining stability criteria in the framework of two measures.
Section 1.4, offers a converse
theorem for uniform asymptotic stability in terms of two measures which is parallel to Massera's well known theorem. In Section 1.5, boundedness and Lagrange stability concepts are presented in the present set up, while in Section 1.6, practical stability considerations are investigated. Section 1.7 considers the stability behavior of a very general type of invariant set called M 0 -stability.
Finally, in Section 1.8, we
develop LaSalle's invariance principle relative to two different measures. 1.1
Definitions of stability. Consider the differential system x' = f(t, x), x{t0) = z 0 , tQ > 0,
where / € C[R smooth
+
enough
x Rn, Rn]. to
(1.1.1)
Suppose that the function / is
guarantee
existence,
uniqueness
continuous dependence of solutions x(t) = x(t,tQ,x0)
and
of (1.1.1).
Chapter 1
3
Let us first define the following classes of functions for future use:
% = {aeC[R + ,R + ]: a(u) is strictly increasing in u and a(0) = 0}. l={aeC[R
+
,R
+
}:
a(u) is strictly decreasing in u and Urn a(u) = 0}, %l = {aeC[R\,R
+
]:
a(t, s) G % for each s and a(t, s) G L for each t}, C% = {aeC[R\,R
+
]:
a(t,s) G 9G for each 0, t0 G R + , there exists
a positive function 6 — S(t0,e) that is continuous in t0 for each e such
that
h0(t0, x0) < 6 implies h(t, x(t)) < e,
t > t0, where x(t) = x(t, t0, x0) is any solution of system (1.1.1);
Basic
4
Theory
(5 2 )
(h0, /i)-uniformly stable if the 8 in (S^) is independent of
(53)
(h0,/i)-equi-attractive,
if for each e > 0
and < 0 Gi2 + ,
there exist positive constants S0 = S(t0) and T = such that hQ(t0, xQ) < 60 implies h(t, x(t)) <e, t>t0 (54)
T(t0,e) + T;
(hQ, /i)-uniformly attractive, if (5*3) holds with S0 and T being independent of t0;
(Ss)
(h0, /i)-equi-asymptotically stable if (Si)
and (5 3 ) hold
simultaneously; (56)
(h0, /i)-uniformly asymptotically stable if (S2) and (S^) hold together;
(57)
(h0, /i)-equi-attractive in the large if for each e > 0, a > 0 and
t0 € R +,
T = T(t0, e, a)
h(t,x(t))<e, (58)
t>t0
there
exists
a
positive
number
such
that
h0(t0, x0) < a
implies
+ T;
(hQ, A)-uniformly attractive in the large if the constant T in (5 7 ) is independent of £0;
(59)
(h0, /j)-unstable if (5j) fails to hold.
Remark
1.1.1:
Sometimes the notion of attractivity may be
relaxed somewhat as in (5 7 ) and (5 8 ) and the corresponding concepts of stability would be of global character. A few choices of the two measures (h0,h) given below will
demonstrate
the
generality
of the
Definition
1.1.1.
Furthermore, the concepts in terms of two measures
(h0,h)
Chapter 1
5
enable us to unify a variety of stability notions found in the literature, which would otherwise be treated separately.
It is
easy to see that Definition 1.1.1 reduces to (1)
the well known stability of the trivial solution x(t) = 0 of (1.1.1) or equivalently, of the invariant h(t,x) = h0(t,x)=
(2)
set {0}, if
|| as ||;
the stability of the prescribed motion x0(t) of (1.1.1) if h(t,x) = h0(t,x) = || x - x0(t) ||;
(3)
the partial stability of the trivial solution of (1.1.1) if h(t, x) = \x\ s, 1 <s 0, there exists a r(e) > 0 t>ta>
such
that
x0 = 0
implies
|| x(i, tQ, 0) || < e
for
r(e). Recall also that x = 0 is said to be conditionally
stable if given e > 0 and t0 G R + there exists a 8 = 8(tQ, e) > 0 such that x0 e {x: || x || < 6} n M implies x(i) G {x; || x || < e}, t > t0. We remark that when we wish to discuss the notion indicated in (4), we need to restrict the initial time t0 to a suitable subset of R+
so that it is possible to have h0(t0, xQ) < S.
Similarly, when we intend to consider the concept defined in (7), we choose the initial data x 0 to be in the manifold M in order
that
h0(t0, x 0 ) < 8 n
S{h0,8) = {xe R :
implies
x0 £ S(hQ, 8) D M,
hQ= \\x\\ + d(x, M) < 8}.
where
We
note
further that several other combinations of choices are possible for h0,h in addition to those given in (1) to (8). Definition (i)
1.1.2: Let h0, h 6 I \ Then we say that
h0 is finer than h if there exists a p > 0 and a function V?eC3G
such
that
h0(t,x) < p
implies
h(t,x) 0 and a function tp € %l
such that h0(t, x) < p implies
h(t, x) < ip(h0(t, x), t). Definition
1.1.3: Let F G C[i? + x fl",R% ], JV > 1, V0(t,x) =
Y, Vi(t,x).
Then F is said to be
i= i
(i)
^-positive definite if there exists a p > 0 and a function b E% such that b(h(t,x)) < V0(t,x) whenever h(t,x) < p;
(ii)
/i-descrescent if there exists a p > 0 and a function a G % such that V0(t,x) < a(h(t,x)) whenever h(t,x) < p;
(Hi)
h-weakly decrescent if there exists a p > 0 and a function a G C9G
such
that
V0(t, x) < a(t, h(t, x))
whenever
h(t,x) < p\ (iv)
^-asymptotically decrescent if there exists a p > 0 and a function a £ 3G1 such that V0(t, x) < a(h(t, x), t) whenever h(t,x) < p. Note that in Definition 1.1.3, we have used the measure N
V0(t,x) = £ Vi(t,x).
We could use other convenient measures
» = i
such d,->0,
as
N
V0(t, x) = max i=l,2,...,JV,
V{(t, x); or
V0(t, x) = £ dy^t,
V0(t,x) = Q(V(t,x)),
x),
where
Q € C[R%, R + ], Q(u) is nondecreasing in u and Q(0) = 0. When JV = 1, it is understood that V 0 (*,x) =
V(i,x).
Basic
8
For any function V e C[R +xRn,R
+
Theory
], we define the
function D + V(t, x) = lim
supkv(t
s-o+ for (t,x) e R+ xR". Dii,i,i)V(t,x) system
+ S,x + 6f(t, x)) - V(t, x)}
°
(1.1.2)
Occasionally, we shall denote (1.1.2) by
to emphasize the definition with respect
(1.1.1).
One
could
also
use
other
to
generalized
derivatives, for example, D _ V(t, x) = lim_ infhvft s d ^°
+ S,x + Sf(t, x)) - V(t, x)]. (1.1.3)
We note that if V eC\R+x D_V(t,x)
= V'(t,x)
Rn,R
where V(t,x)
+
], then D + V(t,x)
= Vt(t,x) +
Let x(t) be a solution of (1.1.1) existing on [£0>°°)
=
Vx(t,x)f(t,x). an
d
V(t,x)
be locally Lipschitzian in x. Then, given t > tQ, there exists a neighborhood U of (t, x(t)) and an L > 0 such that |V(T,0-V(T,I7)1
< I | | C - i 7 | | for(r,C),(r,^)G£7.
Choose S > 0 sufficiently small so that (t + 6, x(t + 6))eU
and (t + S, x(t) + Sf(t, x(t))) G U.
Then we have V(t + 8,x(t + = V(t + 8,x{t) + 8f(t,x(t))
8))-V(t,x(t)) + 8e) -
V(t,x(t))
< V(t + 8, x{t) + 6f{t, x(t))) + L8 | e \ - V(t,
x(t)),
Chapter 1 where e tends to zero with 8. It then follows that lim supkv{t + 8, x(t + 8)) - V{t, x{t))] < lim supkV{t + 8,x(t) + 8f(t,x(t))) - V(t,x(t))]. On the other hand, we have V(t + 8,x{t +
8))-V{t,x{t))
> V{t + 8,x{t) + 8f(t,x{t))) ~L8\e\-
V{t,x(t)),
which implies lim supUv{t + 8, x{t + 8)) - V{t, x(t))] > lim sup±[V{t + 8, x(t) + 8f{t, x(t)) - V{t, x(t))]. 6->0
Thus we obtain lim sup\[V{t + 8, x(t + 8)) - V(t, x(t))} < sup\ [V(t + 8, x(t) + 8f(t, x(t))) - V(t, x(t))] = lim s--o + ° (1.1.4) Similarly, we can show lim infkv{t d s-> o+
+ 8, x{t + 8)) - V(t, x{t))}
= lim infl[V(t + 8, x(t) + 8f(t, x{t))) - V(t, x(t))]. 5-0 +
°
(1.1.5)
Basic
10 It should be noted that when V(t,x)
Theory
is not locally Lipschitzian
with respect to x, we do not necessarily have the above relations even if the solution x(t) is unique to the right.
For
example, consider a function V = y/x, x > 0, for the equation x' = 2r, t > 0. Then clearly D + V(0,0) = 0, but for a solution x(t) = t2 passing through (0,0), we have lim
sup\ [V(S, x(S)) —
V(0,0)] = 1. The following results are useful in the
subsequent
sections. Lemma
1.1.1:
Suppose m(t) is continuous
m(t) is nondecreasing
(nonincreasing)
on (a,b).
Then
on (a,b) if and only if
+
D m(t) > 0( < 0) for every t G (a, b), where D + m(t) = Urn supj[m(t 5_0+ ° Proof:
The condition is obviously necessary. Let us prove
that it is sufficient. If
+ S) — m(t)].
there
exist
Assume first that D + m(t)>0
such
that
m(a) > m(/3), then there is a /i with m(a) > /i > m(0)
and
some
Let
points
two
points
t G [a, /3]
( = sup{t;m(t)>fi,te[a,p}}.
a, /3 G (a, b), a < 0,
on (a,b).
such
that Clearly,
m(t) > /x. C £(«,/?)
m(C) = \i. Therefore, for every t £ (C,/?), we have
m(t)-m(C) < U^ t-C which implies D + m(() < 0. This is a contradiction.
and
Chapter 1
11
Assume now, as in the statement of the lemma, that +
D (t)>0
on
(a, 6).
For any e > 0,
D + [m(t) + et] = D + m(t) + e>0.
Hence
one gets
from
discussion m(t) + et is nondecreasing on (a, b).
t h e above Since this is
true for any e, m(t) is also nondecreasing on (a, b).
Similarly,
+
one can prove that m(t) is nonincreasing if D m(t) < 0. Thus the proof of the lemma is complete. 1.2
Basic Lyapunov theory. It is well known that Lyapunov's second method forms
the core of what he himself called his second method for dealing with questions of stability. The main characteristic of the method
is the introduction
of a function,
namely,
Lyapunov function which defines a generalized distance from the origin of the motion space.
It should be noted that
Lyapunov considered a Lyapunov function with continuous first order partial derivatives.
In this section, we shall
consider a more general case and develop the basic Lyapunov theory in terms of notions introduced in Section 1.1. Theorem 1.2.1: (i)
n
VeC[R+xR ,R Lipschitzian
(ii)
Assume
that heV,
V{t,x)
in x and h-positive
definite;
+
],
+
is
locally
D V(t, x) < 0, (t, x) E S(h, p), where S(h, p) = {(r, x) <E R+
xRn;h(t,x)0}.
Basic
12
Theory
Then (A)
if, in addition, h0-weakly
h0 £ T, h0 is finer than h and V(i, x) is
decrescent,
then the system
(1.1.1) is
(h0,h)-
equistable; (B)
if, in addition, h0 £ T, h0 is uniformly finer than h, and V(t,x)
is hQ-decrescent,
uniformly Proof:
then the system (1.1.1) is
(h0,h)-
stable.
Let us first prove (A).
decrescent, then for t0€.R
Since V(t,x) n
+
, x0ER ,
is /i0-weakly-
there exist constant
SQ = 8Q(tQ) > 0 and function a £ C3G such that
V(t0,x0)
< a(t0,h0(t0,x0)),
provided hQ(t0,x0) < 60. (1.2.1)
The fact that V(t,x)
is /i-positive definite implies that there
exist constant p0 £ (0, p) and function b £ 3G such that b(h(t, x)) < V(t, x), whenever h(t, x) < p0.
(1.2.2)
Also, by the assumption that h0 is finer than h, there exist constant 6X = S^tg) > 0 and function
C{8)T > a{8*),
*o which is a contradiction.
Thus the proof of the theorem is
complete. As applications of the above results, we discuss some examples below. Example 1.2.1:
Consider the differential system
x'2 = - e%, X
3
Choose
=
\X2 ~
X
3J
(1.2.10) e
•
V(t, x) = x\el + [x2 — x3)2,
h(t,x) = \x1\
and
hQ(t, x) = Jx\ + x\ + x\. Then we have {h{t,x)f
< V(t,x)
< 2e\h0(t,x))\
(t,x)
eR+xR3,
and D + V(t, x)=
- 2e\x2 - x3f 0}exp[-MT{j-1)],
(1.4.1)
where Gj(u) = u — j ~ x for u> j - 1 and Gj(u) = 0 for 0 < u < j ~ l . Clearly, for every u,v > 0, \G£u)-G,{v)\
), we choose j > l
< a. We have, for (t,x) £ U{t,x) > 2 " jUj(t,x)
such
that
S(h,p)\S(h,a),
> 2">[h(t,x)-j~ l )exp{
-
MT(j~a)]
>/?>0, where follows
/? = 2~j(athat
j~x)txp[
there
exists
U(t,x) > a(h(t,x)) for (t,x) €
- MT{j~x)\. a
From
function
this,
a £ 3G such
it
that
S(h,p).
Let e £ (0,/?). Since j = 1,2,... and (t,x) £ S(h,p), Uj(t,x) < sup{h{t + 9,x(t + 6, t,x)): 9 > 0}, then
the
(h0, fo)-uniform stability
of
(1.1.1) implies
(1.4.6) the
existence of a 0 such that h0(t,x) < 8(e) implies U(t,x) < e, which is equivalent to the statement that there exist a constant p0 E (0, p) with ip(p0) < p and a function a £ 3G such that U(t, x) < a(h0(t,x))
for (t,x) £ S{h0,p0).
Thus, U satisfies
Basic Theory
44
condition (a). Next we consider the function W: S(h, p)—>R + defined by oo
W(t, x) = J C{U(6, x{0, t, x)))d9,
(1.4.7)
t
where C G % is to be chosen later. By (1.4.4), (1.4.6) and assumption (ii), the system (1.1.1) is obviously (h0,U)uniformly asymptotically stable and (h, l/)-quasiuniformly asymptotically stable. Then, there exists a constant "PQ 6 (0, p0] such that U(6,x(8,t,x)) < p(h0(t,x))q(6 - t), {t,x) 6 S(hQ,p0), (1.4.8) where p G 9G, q G 1. that
Also, there exists a function (3 € i. such
U(9, x(9, t, x)) < /3(6 -t)ioi9>t
and (r, x) € S(h, p). (1.4.9)
We can assume pQ — p0. We now choose c G 3G such that the integrals oo
jCW)d6)
oo
and j[C{p{pQ)q(d))]xl2d9
0
(1.4.10)
0
converge and c' exists and belongs to class 3G with c'W))
< ™p( ~ p0),
(1.4.11)
Chapter 1
45
where (i>M + l. Such a choice is possible by Massera's Lemma 1.4.1. As a consequence of (1.4.9) and (1.4.10), W is well defined and bounded in S(h, p). Now for (t,x), (t,y) G S(h,p), we have
\W(t,x)-W(t,y)\ oo
<j
\c(u(9,x(9,t,x)))-c(U(e,x(8,i,y)))\de
t CO
= / [c*(01 u(e, x(e, t, X)) - u(e, x($, t, y)) \ ]de, t
where U{9,x(0,t,x)) < 0, t0(=R exists
a
positive
function
+
/? = j3(t0, a),
, there
which
is
continuous in t0 for each a such that V ^ o ) ^
a
implies h(t,x(t))
where x(t) - x(t,t0,x0) (B2)
and
ft)-quasi-equi-ultimately t0 e R +,
T — T(t0,a)
t0,
is any solution of (1.1.1);
(h0, /i)-uniformly bounded if /3 in (BJ
(£3) (^0)
is independent of
bounded if, for each a > 0
there exist positive numbers
such that
N
and
Chapter 1
49 ^o( ^o)
0 and t0(ER f3(tQ, a) > 0 such
that
t > tQ, where u(t,t0,uQ)
u0 < a
+
, there exists a
implies
u(t, tQ, u0) < /?,
is any solution of (1.3.1).
Observe that if /? in (J5J is such that 0(to, •) G 3G, then (h0, /i)-boundedness implies (hQ, /instability, since given e > 0, there exists a 6 = S(t0, e), which is continuous in t0 for each c, such that f3(t0, a) < e whenever a < 8. We begin by proving a result on equiboundedness.
Basic Theory
50
Theorem 1.5.1: Assume that (i) h0)heT and h(t, x) < ip(t, hQ(t, x)), (p G C%; (ii) V € C[R + x Rn, R + ], V(t, x) is locally Lipschitzian in x, and there exist functions a £% and p € C[R + xR + , R + ] such that a(h(t,x)) < V(t,x) < p(t,h0(t,x)),
(t,x)eR+xRn, (1.5.1)
■where 0(7)—>oo as 7—►00; (m) D + V(t,x) 0, t0ER+ be given and x{t) = be any solution of (1.1.1) with h0(t0,xQ) 0 such that P > max{ip(tQ,a) a~\p(t0,a))}.
x(t,tQ,x0) Choose
(1.5.2)
It is easy to see from (1.5.1) and (1.5.2) that h(t0,x0) < j3. We are going to show h(t,x(t))t0.
(1.5.3)
If this is false, then there would exist a tl>tQ such that h(t1,x(t1)) = fl. Since V(t,x(t)) is nonincreasing by assumptions (ii) and (Hi), it follows from (1.5.1) that a(fi) < V(t,,x(ti)) < V(t0,x0) < p(t0,a),
Chapter 1
51
which contradicts (1.5.2).
Thus (1.5.3) is true and (1.1.1) is
(hQ, /i)-equibounded. Theorem 1.5.2: (i) (ii)
hQ, h£T
Assume
that
and h(t,x) < ip(h0(t,x)),
)•
T/iera 4/ie system (1.1.1) is (h0,h)-uniformly Proof:
bounded.
For any a > 0, we choose /? = /3(a) > 0 so that a(/3) > max{q(a),q(p),a-\ z o) < a-
Let t0€z R+ solution
x(t) = x(t,tQ,x0)
of
(1.5.5)
Now suppose that for some (1.1.1)
and
t*
such
that
ft(i*,a;(r)) > ft. Then there exist tx, t2, t0 < tx < t2 < t* such that V ^ i . ^ C i ) ) = max{a,/j}, h(t2,x(t2))
= (3,
(t,x(t)) g 5(A, ) 9)n5 c (A 0 ,max{a,p}), * G [ M 2 ) By
(1.5.4),
we
have
max{g(a),g(/>)} and V(t2,x(t2))>
(1.5.6)
V ^ . ^ D ^ V ' i . ^ ) ) a(h(t2,x(t2))
the other hand, we have V(t2,x(t2))
=
< V(tx,x(tx))
a(p). by
5
On the
Basic
52
Theory
assumption (Hi). Hence we have a(/3) < max{q(a), q(p)}, which contradicts (1.5.5). t>t0,
Thus h0(t0, x0) < a implies h(t,x(t))
< f3,
and therefore the system (1.1.1) is (h0, /j)-uniformly
bounded. Theorem
1.5.3: Assume
that
conditions
Theorem 1.5.2 hold. Suppose further (Hi)*
(i)
and
- C(h0(t, x)), (t, x) G Sc(h0, p), C € %■
D + V(t, x)
t0.
(1.5.7)
Now we consider solutions x(t) = x(t, tQ, x0) of (1.1.1) with ^o(^O) xo) < ai
where a is an arbitrary number and a > p.
Then there exists a positive number /3 — /3(a) such that h(t, x(t)) < / ? , £ > t0.
We are going to show that there exists a
t* G [t0,t0 + T], where T =
q
-^-,
such that hQ(t\x(t*))
< p. If
this is not true, then we have h0(t, x(t)) > p, t G [t0, t0 + T], Thus from assumption (m)*, V(t0 + T,x(t0 + T)) < V(t0,x0)
- C(p)T,
which, together with (1.5.4), leads to the contradiction
00,
h'(t,x)
is
S(h,p).
Then the system (1.1-1) is (h0,h)-equi-Lagrange Proof:
(ii)
stable.
(h0, /i)-equiboundedness follows from Theorem 1.5.1
and (h0, /i)-attractivity may be proved using arguments similar to that used in the proof of Theorem 1.2.3. Finally, we shall prove a general result using the comparison principle which includes several special cases. Theorem 1.5.5:
Assume
that
(i)
h0, h G T and h(t,x) < - ^ . 8A
Example 1.6.2:
Consider the differential system
I x' = - x - y + k(x-
y' = x-y
y)(x2 + y2), 2
2
+ k(x + y)(x + y ),
x
[*o) — xoi
y('o) = 2/0,
(1.6.4) where k > 0 is a constant.
Basic
60
Theory
The general solution of (1.6.2) is given by x(t) = - 4 = {x0cos9 - y0sin6), y(t) = - 4 = (x0sin6 + yQcos6) with
6 = 2{t - t0) - |/n/i
and
/x = 7o + (£ ~ 7 o W ( 2 ( * ~ *o))-
This reduces to
(1 6 5)
tf-bi-fc*
--
It is clear that if 7o = Xo + 2/o < £> a n d h0 = h = \Jx2+ y2, then the system (1.6.4) is (/i 0 ,/i)-asymptotically stable. However, if A, A are given such that -7= < A < A, then for the initial v fc values (x0>2/o) "with £ < 70 < A2, the system (1.6.4) is not (h0, /i)-practically stable which shows that asymptotic stability is not sufficient for practical stability to hold. In other words, the presence or absence of practical stability in a system does not depend on asymptotic stability of the system. It should be noted that practical stability is somewhat similar to uniform boundedness.
It is, however, not merely
that a bound exists but that the bound be pre-assigned. Also, Lagrange stability is somewhat similar to practical asymptotic stability and ultimate boundedness is a necessary condition for the system to possess strong practical stability.
Chapter 1
61
Sometimes, the interdependence of (X,A,B,T) useful in practice.
may be
For example, (PS3) may be weakened as
follows. The system (1.1.1) is said to be (PS3)
(hQ, /i)-practically quasi-stable if given (A, B) > 0 and t0 £ R + , there exists a T = T(t0, A, B) > 0 such that hQ(t0,x0) < A implies h(t,x(t)) < B, t>t0
+ T.
If {PSx) and (PS3) hold together, we can identify that as (PS5„)
and other similar concepts may be introduced.
Occasionally, it is advantageous to restrict the initial times t0 to a given set T 0 C R +, instead of allowing the initial set (set T 0 ) to be the whole real line. Let us now establish some sufficient conditions for the (h0, /impractical stability of the system (1.1.1). Theorem 1.6.1:
Assume
(i)
0 < A < A;
(ii)
h0)
h6T
and
h0
that is
uniformly
finer
that
h,
i.e.
h(t,x) < tQ,
(1-6.13)
which implies b(h(t, x(t)) < V(t, x{t)) < 7(f, t0, u0) < b(B), t>t0 Thus
we see that
h(t, x{t)) < B
for
t > t0 + T
+ T. whenever
h0(t0, x0) < X and therefore (h0, ft)-strong practical stability of
Basic
64
Theory
the system (1.1.1) is proved. One can similarly prove other (h0, A)-practical stability properties of the system
(1.1.1) and hence the proof is
complete. Corollary 1.6.1: (i)
In Theorem 1.6.1,
g(t, u) = 0 is admissible to yield (hQ, h)-uniform
practical
stability; (ii)
g(t,u) = —au + k, a,k>0 uniform strong practical
(iii)
g(t,u) = — cr'(i),
a 6 i.
is admissible to imply stability; and
a
is
admissible to guarantee (h0,h)-eventual Proof:
differentiable, practical
is
stability.
The proof of (i) is immediate. Concerning (ii), we
observe that the solutions u(t,t0,u0)
of (1.3.1) are given by
u(t, t0, no) = u0e ~ °{t ~ f°> + 1 ( 1 - e " a ( t " ')), Suppose that B tQ.
(A, A, B, T) > 0 are given such that
Then, if o(A) + 1 < 6(A) and a(A)e "
aT
X < A,
+ & < b(B), we
have (h0, /i)-uniform strong practical stability of (1.1.1). To prove (iii), note that u(tf, t0, u0) < u0 + o-(t0), and hence it is enough to have a(t0) < b(A) — a(X).
t>t0 Since
a G 1, there exists a T = T ( A , A ) such that (oo.
V(/im x(tk, x0)) — lim V(x(tk, x 0 )) = c. k—*oo
k—+oo _1
Hence O(x 0 ) C V ( c ) .
Thus
+
V^j/) =
Evidently, Cl(x0) C G.
By Lemma 1.8.1, (l(x0) is positively
invariant.
It then follows that D + V(x) = 0 on fi(x0), i.e.
O(x 0 ) C E.
Thus we conclude f2(x0) C EC\ V _ 1 ( c ) and the
proof is complete.
Basic
82
The following
theorem
Theory
is an easy consequence of
Lemma 1.8.1 and Lemma 1.8.2. Theorem 1.8.1:
Assume
that
(i)
h0, h G T and h0 is uniformly finer than h;
(ii)
V G C[Rn, R + ], V(x)
is locally Lipschitzian
positive definite, h0-decrescent D + V(x) 0 the set EC\V~l(c)
contains
no
complete
positive trajectory of (1.8.1). Then the system (1.8.1) is (h0,h)-asymptotically Proof:
Assumptions
(i) — (ii)
imply
stable. that
the
system
(1.8.1) is (h0, /i)-uniformly stable. Thus for p > 0, there exists & 80 = S0(p) > 0 such that M^o) < ^o implies h(x(t,xQ)) < p,t> Choose 6 = min{50,a}.
0.
(1.8.2)
Then by assumption (Hi) and (1.8.2)
we see that /i0(zo) < S implies that x(t, x0) is bounded and h(x(t,x0))
< p, t>0.
Since V(x(t,x0))
is nonincreasing and
bounded from below, it follows that Urn, V(x(t, x0)) = c > 0. Suppose, for the sake of contradiction, that c > 0.
Since
83
Chapter 1 x(i,x 0 ) is bounded,
Lemma
1.8.1
implies
nonempty and positively invariant. x(t,y) e £l(x0), D + V{x(t,y))
iG[0,oo).
fl(x 0 )
V(x(t,y))
= c
and
Hence >y(y) C £ n V
which contradicts assumption (iv).
is
Then for y £ fi(x0),
Thus
= 0 for <e[0,oo).
that
_1
(c),
So we must have c — 0.
Since V(x) is k-positive definite, this shows Urn h(x(t, x 0 )) = 0. Thus the system (1.8.1) is (h0, /i)-asymptotically stable and the proof is complete. If we remove the condition (Hi) in Theorem 1.8.1, i.e. without demanding the boundedness of solutions of (1.8.1), then we have the following result. Theorem 1.8.2: (i)
Assume that
h, h*, h° € T and h(x) + h*(x) = ip(h0(x)),
where (p £ 9G
and ip(h0(x))-+oo as \\ x \\ —►oo; (ii)
assumptions (ii) and (iv) of Theorem 1.8.1 hold.
Then
there
exists
a constant
60 > 0 such that h0(x0) < S0
implies that either V(x(t, x0))—>0 or h*(x(t, x0))—»oo as t—*oo. Proof:
By condition (i), h(x) < 0,
there
exists
a
60 = S0(p) > 0 such that ^(^o) < ^o implies h(x(t,x0))
< />, t > 0.
(1.8.3)
Basic
84 Let x(t,x0)
Theory
be a solution of (1.8.1) such that h*(x(t,x0))
-/>oo as
2—xx>. Then there exists a sequence tn G R +, t„—>co as n—>oo such that {h*(x(tn, x0))} and {h(x(tn, x0))} are bounded, which implies, by condition (i), that {x(tn,x0)} tt(xQ) is nonempty.
is bounded.
Thus
Since V(x(t, x0))—>c > 0 as t—KX>, then
there exists a y £ f)(x0) such that V(j/) = c. But the set Cl(xQ) is positively
invariant,
consequently,
the
contains a complete positive trajectory.
Er\V~x{c)
set
This, together with
assumption (iv) of Theorem 1.8.1, implies c = 0.
Thus the
proof is complete. Employing
Theorem
1.8.2,
one
can
get
(h0,h)-
asymptotic stability as follows. Theorem
1.8.3:
Suppose further (A)
V(x)^0
Let the assumptions
of Theorem
that as D + V(x)^0
and
h*(x)^oo.
Then the system (1.8.1) is (h0,h)-uniformly asymptotically Proof:
1.8.2 hold.
stable and
(hQ,h)-
stable.
(fr0,/i)-uniform
stability of (1.8.1) is immediate.
Thus there exists 60 = S0(p) > 0 such that h0(x0) < ^0 implies h(x(t,x0)) x
0. Let x(t,x0)
be a solution of (1.8.1) with
6
h0( o) < o- Then by Theorem 1.8.2, V(x(t, x o ))-*0 as t->oo or h*(x(t,x0))-^>oo +
D V(x(t,xo))oo. t>0,
Since V(x) it
follows
is nonnegative that
and
Urn supD
+
= 0, which implies that there exists a sequence
85
Chapter 1
tnER+ such that D+ V(x(tn,x0))—>0 as n—>oo. Suppose that lim V(x(i,x0)) = c ^ 0. Then h*(x(tn, x0))—>oo as n—>oo, which implies, by assumption (A), V(x(tn,xQ))—*0 as n—«x>. This is absurd. Thus we must have V(x(£, x0))—>0 as t—*oo. Since y(x) is /i-positive definite, this in turn implies that h(x(t,x0))—*0 as t—»oo and the system (1.8.1) is (h0,h)asymptotically stable, completing the proof. Example 1.8.1:
Consider the differential system Xj =
— Xj(l + X3J,
x ' 2 = - x2, Xg = Xg
(1.8.4)
X-}.
Let V (Xi,x2,x3J = Xy + x2 + x2x3, ft(x1,x2, x3J = Xj + x2, /i*(xj, x2, x3) = x 3 and hQ{xl, x2, x3) = x\ + x2 + x3. Then V(x 1 ,x 2 ,x 3 ) is /i-positive definite, ft0-decrescent, h + h* = h0 and D + y(x l t x2, x3) = - 2[zJ(l + x\) + x22 + x\x\\ < 0. Clearly, D + V(xl,x2,x3)—»0 implies xt—>0, x2—»0 and x2x|—»0. If /t*(xa,x2,x3)—>oo, i.e. x3—>oo, then we must have x\x\—*0. Thus V(x1?x2, x3)—>0 and assumption (A) of Theorem 1.8.3 is satisifed. It is easy to verify that all other conditions of Theorem 1.8.3 are met. Hence system (1.8.4) is (h0,h)uniformly stable and (hQ, /i)-asymptotically stable.
Basic Theory
86 1.9
Notes.
The stability notions listed in Section 1.1 are the natural generalizations of those with respect to the trivial solution. See Lakshmikantham and Leela [1]. The idea of stability in terms of two measures was initiated by Movchan [1], developed by Salvadori [1], and Lakshmikantham and Liu [1,2,4-6]. Lemma 1.1 is taken from Rouche, Habets and Laloy [1]. Theorems 1.2.1 and 1.2.5 are taken from Lakshmi kantham, Leela and Martynyuk [1]. Theorems 1.2.3 and 1.2.4 are due to Liu and Sivasundaram [1]. Theorems 1.2.2, 1.2.3 and 1.2.7 are new. For Lemma 1.3.1 and Theorem 1.3.1 in Section 1.3, see Lakshmikantham and Leela [1]. Theorems 1.3.2 and 1.3.3 are adapted from Lakshmikantham, Leela and Martynyuk [1]. Theorem 1.3.4 is due to Lakshmikantham and Liu [2], and Theorem 1.3.5 is from Liu and Sivasundaram [1]. Section 1.4 is due to Lakshmikantham and Salvadori [1]. The concepts of boundedness and Theorem 1.5.5 in Section 1.5 are adapted from Lakshmikantham, Leela and Martynyuk [1]. Theorems 1.5.1 to 1.5.4 are new. The concepts of practical stability in terms of two measures are given for the first time. Theorems 1.6.1 to 1.6.3 are adapted from Lakshmikantham and Leela [1] while Theorem 1.6.4 is a result in Lakshmikantham, Leela and Martynyuk [2]. The contents in Section 1.7 are new. The notion of M 0 -stability was first given by Moore [1]. Section 1.8 is new. For related results see
Chapter 1
87
Hatvani [1-3], Abdullin and Yu [1], Aminov and Sirazetdinov [1], Kulev and Bainov [1], Makay [1], Oziraner and Rumiantsev [1], Rumiantsev [l] and Vorotnikov [1].
2.
Refinements
2.0
Introduction.
This chapter is devoted to refining several results of Chapter 1 by utilizing more than one Lyapunov functions. As we shall see, this approach of employing several Lyapunov functions offers a better mechanism to obtain results under much weaker assumptions. Since, in the application of Lyapunov theory to concrete problems, the difficulty is always to find a suitable Lyapunov function which verifies the assumptions of Lyapunov theorems, weakening the requirements of the Lyapunov functions and enlarging the class of Lyapunov functions to be utilized, is of great interest. 89
Refinements
90
We begin, in Section 2.1, with the results that prove asymptotic stability in terms of two measures using two or more Lyapunov-like functions.
We then discuss refinements
that have resulted in the loss of negative definiteness in Lyapunov theorem on asymptotic stability. This we do, in the general set up of two measures and nonautonomous systems. Section 2.2 investigates nonuniform stability properties under weaker assumptions by utilizing the idea of perturbing families of Lyapunov functions.
Such results are also extended to the
study of boundedness and practical stability considerations. In Section 2.3, we continue the use of several Lyapunov functions and give a different approach to discuss the case of loss of negative definiteness mentioned above. Section 2.4 is devoted to the development of the method of vector Lyapunov functions in the framework of two measures. We also provide global results in terms of arbitrary sets which can be utilized to prove several results on stability and boundedness.
Section 2.5 discusses the
perturbation
results employing the converse theorem as well as coupled comparison systems.
In Section 2.6, a variation of Lyapunov
method is considered which blends the method of variation of parameters and the Lyapunov second method.
Section 2.7
deals with integral stability while Section 2.8 continues the study of perturbing
Lyapunov functions
to provide
directions in the method of vector Lyapunov functions.
new In
91
Chapter 2 Section
2.9, we consider
the technique of using
higher
derivatives of Lyapunov function and in Section 2.10, we analyze the comparison systems that can be employed in the study of concrete problems.
Finally, in Section 2.11, the
method of cone-valued Lyapunov functions is indicated which shows that employing suitable cones other than R1^ is more advantageous in applications. 2.1
Several Lyapunov functions. As we have seen, it is possible to study stability
properties in a unified way by using a single Lyapunov function and the theory of differential inequalities.
It is
natural to ask whether it might be more advantageous, in some situations, to use several Lyapunov functions.
The
answer is positive and the approach leads to a more flexible mechanism.
We
begin
with
a result
which uses
two
Lyapunov-like functions. Theorem
2.1.1: Assume
that
(i)
h0) h G T 0 and hQ is finer than h;
(ii)
V 6 C[R
+
x Rn, R + ], V(t, x) is locally Lipschitzian
h-positive definite and h0-weakly (Hi)
W £ C[R
n
+
in x,
decrescent;
x R , R + ], W(t, x) is locally Lipschitzian +
x, h-positive definite, D or from below on S(h,p)
in
W(t, x) is bounded from above and
Refinements
92
D+v(t, x) < - c(w(t, x)), (t, x) e s(h, p), c e %. Then, the system (1.1.1) is (hQ,h)-asymptotically stable. Proof: By Theorem 1.3.1, it follows that the system (1.1.1) is (h0, /i)-equi-stable. hence it is enough to prove that given t0 E R +, there exists & SQ = S0(t0) > 0 such that ho(t0, x0) < 60 implies h(t, x(t))—>0 as t—>oo, for any solution x(t) = x(t,tQ,xQ) of (1.1.1). For e = p, let SQ = 5(tQ, p) be associated with (hQ, h)stability. We suppose that h0(t0,x0) < 60. Since W(t,x) is hpositive definite, it is enough to prove that lim W(t, x(t)) = 0 for any solution x(t) of (1.1.1) with h0(t0, x0) < S0. We first note that lim in fWit, x(t)) = 0. For otherwise, in view of (Hi), we get V(t, x(t))—> - oo as t—KXX Suppose that HmJV(t, x(t)) ^ 0. Then, for some a > 0, there exist divergent sequences {*„}, {t*n} such that t{ < t*( < fi + 1 , i = 1,2,... and W(thx(ti))
= %,W(t*i,x(t;)) = a
and | < W(t,x(t)) -g^. In view of (in), we have for large n,
n
0 < V(fm x(t*n)) < V(tQ, * 0 ) - £
[D
+
V(S, x(s))ds
'-» { oo. The argument is similar when D + W is bounded from below and we use (2.1.2). The proof is therefore complete. For a generalization of Theorem 2.1.1, we need the following definition. Definition
2.1.1:
Let \:R
+
—*R+ be a measurable function.
Then X(t) is said to be integrally positive if / X(s)ds = oo OO
whenever / = (J [a,-, /?,-], a{ < /?,• 8 > 0.
A function X(t) is integrally positive, roughly speaking, means that it can not be small in average in any period as
Refinements
94
*—►oo. It can be formulated also in the following way: for every 8 > 0
t+6 lira inf / X(s)ds > 0. t Theorem 2.1.2: Assume that (i) h0, h G T0 and h0 is finer than h; (ii) V G C[R + x Rn, R + ], V(t, x) is locally Lipsckitzian in x, h-positive definite and hQ-weakly decrescent; {Hi) D + V{t,x)< -\(t)C{h(t,x)), {t,x)eS(h,p), where X(t) is integrally positive and C G %; {iv) Wu..., Wm G C[R +xRn,R + ]. For each i = 1,2,..., m, Wi(t,x) is locally Lipschitzian in x, D + Wi(t,x) is either bounded from above or from below on S(h,p) and there are functions a, b G % such that m
b(h(t,x)) < £ W f a x )
< a(h(t,x)), (t,x) G S(h,p).
i= \
Then the differential system (1.1.1) is stable.
(h0,h)-asymptotically
Proof: By Theorem 1.2.1, it follows from conditions (i)(iii) that the system (1.1.1) is (h0, /i)-equi-stable. It remains to show that the system (1.1.1) is (/i0, ^-attractive. Let e = p and tQ G R +, then because of (h0, ^-stability, there exists a
95
Chapter 2
80 = S(t0,p) > 0 such that h0(t0,x0) < 80 implies h(t,x(t)) < p, t>t0, for any solution x(t) = x(t,tQ,x0) of (1.1.1). We claim that
lira £)W?Mi)H0,
(2.1.3)
i=I
for any solution x(t) = x(t,t0,x0) of (1.1.1) with h0(t0,x0) < 80. Suppose that (2.1.3) is not true, then there exists an i, 1 < i < m, such that lim W,-(i, x(t)) ^ 0. Thus we can find a sequence t0 0,
(2-1.4)
W,-(*fe,*(**))< - * < 0 .
(2.1.5)
or
Since ^,(^te[tk-S,tk],k
= l,2,....
Since £ W2i(t,x) < a(h(t,x)), we have from (2.1.6) i=i
. .
h(t,x(t))>a-^),
(2.1.6)
96
Refinements te[tk-6ttk],k
Let / = U [h ~ *»**]■
T h e n i»o( ma.T 0, T] = 77(a) > 0, 77 < a, such that d(x,E) < rj, where
E = {x G i?", w(x) = 0}
\\x — y\\,
and
d(x, E) = inf
implies
D + V2(t, x) > i, (i, x) € S(h, p) n Sc(h0, (iv)
f(t,x)
is bounded on
S(h,p)nSc(h0,a).
a);
Refinements
98 Then
the system
(1.1.1) is (h0,h)-uniformly
asymptotically
stable. Proof:
Since
descrescent,
V-i(t,x)
there
exist
is
/i-positive
constants
definite
0 < p0 < p,
and
h0-
8Q> 0
and
functions a, b £ 9G such that b(h(t, x)) < V(t, x), {t, x) € S(h, p0),
(2.1.9)
V(t, x) < a{h0(t, x)), if h0(t, x) < 8Q.
(2.1.10)
and
It follows from conditions (i)-(ii) that, by Theorem 1.2.2, the system (1.1.1) is (h0, /i)-uniformly stable.
Thus for pQ > 0
there exists a 8^ = S^po) > 0 such that hQ(t0, x0) < 8^ implies h(t,x(t)) for any solution x[t) = x(t,t0,x0)
< p, t> t0,
of (1.1.1).
(2.1.11)
To prove (h0,h)~
uniform asymptotic stability, it is enough to show that for any e > 0,
there
exists
a T = T(e) > 0 such
that,
for
some
t* e [tQ,t0 + T], h0(t*,x(t*)) < 6{e), where 8(e) is the same 8 as in Definition 1.1.1 for (h0,h) uniform stability.
We achieve
this in a number of stages: (1)
By assumption (Hi), given 8 = 8(e), 0 < 8 < p, there exist £ = £( e ). 1 = v(e), V < $ such that D + V2(t, x)>t,
(t, x) e S(h, p) n Sc(h0,8),
if d(x, E) < t).
99
Chapter 2 Let us consider the set
u = {x e Rn, d(x, E) < T/, (t, x) e S(h, p) n
Sc(h0,6),teR+},
and let L = sup
V2(t,x).
(t,x)es{h,p)
Assume that, at t — tu z(<x) = x(f 1 ,t 0 ,i 0 ) G U. Then, for t > ij, we have, letting m(t) = V2(t,x(t)), D + m(t)>D
+
V2(t,x(t))>£,
because of condition (Hi) and the fact that V2(t,x) satisfies a Lipschitzian condition in x locally. Thus t m(t) — m(t1)= D + m(s)ds, and hence m(t) + m(tx)>
x
t t ID + m(s)ds> fD
+
V2(s,x(s))ds
as long as x(t) remains in U. This inequality can simultaneously be realized with m(t) < L only if
t < tx + 2L/C. It therefore follows that there exists a t2, ti 0, there exists a V2r) G C[R + X R", R + ], V2n(t,x) is locally Lipschitzian x, bounded if t£R + ) || y || < p and \\x\\ > n. Moreover, for every a > 0, there exists a 3 = B(a) > 0 and a t
function ( = R + ->R +, f ((s)ds-+oo as TJ-»OO for any such that d(x,E) < 8 implies D + V2r)(t,x) > ((t), t£R || y || < p and || x || > n.
+
T£R
,
+
,
Chapter 2
119
Then the trivial solution
of (2.2.15) is asymptotically
stable
with respect to y. Proof:
Let h0 = || x ||, h = || y || and W - \\ xx \\.
Then
all the conditions of Theorem 2.2.4 are satisfied. We shall next give some results on boundedness and practical stability in the spirit of Theorem 2.2.1 and 2.2.2. Theorem 2.2.5:
Assume
that
(i)
h0, h £ T and h(t, x) < tp(h0(t, x)), tp £ %;
(ii)
V1(zC[R+ x
and
xRn,R
+
], Vx{t,x)
is locally Lipschitzian
Vx(t,x)<xl>{t,hQ{t,x)),
9i(t,Vi(t,x))>
r{>eC%,
D+
in
V^x)^
where gx E C[R+ xR + ,R];
(Hi) V2 G C[S (h0, p), R + ], V2(t, x) is locally Lipschitzian
in x
c
and for (t, x) G S (h0, p) b(h(t,x))x2)i
where x = col(x1,x2), f = col(f1,f2), xlERp, p + q = n. We also consider the subsystem
x2(=.Rq with
*'i = /i(*,*i,0).
(2.3.9)
Then we have the following result. Theorem 2.3.2: Assume that (i) V1 6 C[R + x R", R + ], Vx(t,x) is locally Lipschitzian in x, positive definite with respect to x, V^^O) = 0 for all t £ R+ and D(t.3.s)Vi(t>x) < ~ KWii
II x2 II) on R+ x S( 7 ),
where S(~f) = { i £ i? n , || x || < 7},
X(t) is integrally
positive, Cx E %; (n)
feC[R+
xS(-y),Rn],
/(i,0) = 0
and there
constants M,L > 0 such that for (t,x) € R+ x 5(7) x
2-
fi{i,Xi->x2)<M,
exist
130
Refinements \\f1(t,x1,x2)-f(t,x1,x2)\\
(in)
< £,( || xx - xx || + \\x2-x2\
the trivial solution of the subsystem asymptotically
|);
(2.3.9) is uniformly
stable.
Then the trivial solution
of system
(2.3.8) is
asymptotically
stable. Proof:
Since the trivial solution of system
uniformly
asymptotically
converse
theorem
that
stable, there
it follows
exists
(2.3.9) is
by Massera's
a function
V(i,x 1 )
satisfying the following conditions: (A) VeC[R+ xRn,R
+
], V{t,xx)
is positive definite with
respect to x1 and for some N > 0
\V{t,Xl)-V{t,x,)\
< N I I * ! - ^ ||,
t e R+, xuxY e 5*( 7l ) = {Xl e Rp; \\ *, \\ < 7 l } ; (B)
for some C2 € 3G
H**)V&xi) ^ -C2(V(t,Xl)), (t,Xl) e R+ x5*(7l). Let p = mm{7,7 1 } and V2(t,x) = V(t,x1).
W e then get, for
(t,x)eR+xS(p), D{t3.s)V2(t,
x)(u) = NLu,
W(t, x) = || x 2 1 | ,
h0(t,x)
=
h(t, x) = || x ||, then the conclusion of Theorem 2.3.2 follows from application of Theorem 2.3.1. Example 2.3.2: ( \ I
Consider the nonlinear system x
'\ = - hxi - (! + cos2t)x2, (2.3.10)
x'2 = xx— 1xxx\ — (5 + ^cosH)x\.
V^t, Xj, x 2 ) = 2x\ + 2x^2 + ( | + 2co5 2 i)x|.
Let
computing the time derivative of V1
Then
along solutions of
(2.3.10), we obtain D + Vi(t,xx, We see that Vl(t,x1,x2) Also,
x2) < -
is positive definite and V(i,0,0) = 0.
| / j ( i , xlt x 2 ) - ft(t, xvx2)
and / i ( f , x 1 , 0 ) = — \xx. subsystems x\ = —\xx
1cosHx\
|
* > *o> (2.4.3) i=I
'
where u(t,tQ,uQ) u0 = V(t0,x0).
»= I
is any solution of (2.4.1).
W e choose
Since V is /i 0 -descrescent and h0 is uniformly
finer than h, there exists a cr0 > 0 and a function a £ % such that for
{t0,x0)eS{h0,a0), h(t0,xQ) < p0 and V0(t0,x0)
x0) £ S(hQ, xo) < t0
whenever h0(t0, x0) < 8,
where x(i) = x(t, t0, x0) is any solution of (1.1.1) with ^o(*o> xo) < 6- ^ this is not true, then there exists a 11 > t0 and a solution x(t) of (1.1.1) such that *( 0 and t0£R + , there exist positive numbers 8\ — 8\(t^) and T = T(t0,77) > 0 such
Refinements
136 that
f; u 0 . < SI implies ^ u,(t, to, u0) < 6(77), i > t0 + T.(2.4.8) i=1
1=1
*
Choosing u0 = V(t0, x0) as before, we find a SQ = 5J(i0) > 0 such that 6Q e (0, <x0) and a(^) < 6[. Let S0 = min(So, 60) and h0(t0, x0) < S0. This implies that h(t, x(t)) < pQ, t> t0 and hence the estimate (2.4.7) is valid for all t > t0. Suppose now that there exists a sequence {tk}, tk > t0 + T, tk—*oo as k—»oo such that 77 < h(tk,x(tk)), where x(t) is a solution of (1.1.1) with hQ(t0, xo) < ^o- This leads to a contradiction
Kv) < V0(tk,x(h)) < E7,( 0, is not stable, and so we cannot deduce any information about the (h0, /instability of (2.4.9). On the other hand, let us seek a Lyapunov
function
as
a
quadratic
form
with
constant
coefficients given by V(t, x) = ^ [x\ + 2Bxxx2 + Ax]]. Then, the function D + V(t,x)
(2.4.10)
with respect to (2.4.9) is equal
to the sum of two functions ujx(t,x), u)2(t,x), where cja(tf, x) = x\{e ~' + Bsint) + x1x2(2Be ~ ' + (A + l)sint) + x\{Ae ~l +
Bsint),
io2(t, x) = — sin2t{x\ + xl) {x\ + 2Bx1x2 + Ax2). For arbitrary A and B, the function V(t,x) does not satisfy
defined in (2.4.10)
Lyapunov's theorem on the stability of
motion. Let us try to satisfy the conditions of Theorem 1.3.2 by assuming u>-y[t,x) = \(t)V(t,x).
This equality can occur in
Refinements
138 two cases: (i) A1 = l,
\1(t) = 2[e-t + sint]
J3i = l,
when
V^t.x)-
%* + 2/)2; (»)
A 2 = 1, B2 = - 1, A2(i) = 2 [ e " ' - sint] when V 2 (*,x) =
\{*-y)2The functions of Vu V2 are not /i-positive definite and hence, does not satisfy Theorem 1.3.2.
However, they do fulfill the
conditions of Theorem 2.4.2. In fact, (a)
the
functions
V1(t,x)>0,
V2(t,x)>0
£ Vt(t,x) = x2 + y2 and therefore V0(t,x) = £ Vfax) iIs 1
and is
i=1
/i-positive definite and /i 0 -descrescent; (6)
the
vectorial
inequality
D+ V(t, x) < g(t, V(t, x))
is
satisfied with the functions £fi(i,u 1 ,u 2 ) = 2 ( e _ t + g2(t,uuu2)
= 2(e"' -
sint)^, sint)u2.
It is clear that g(t,u) is quasi-monotone nondecreasing in it, and the null solution of u' = g(t, u) is stable.
Consequently,
the system (1.1.1) is (hQ, /i)-stable by Theorem 2.4.2. We shall next consider a result on (h0, /i)-asymptotic stability which generalizes classical results. Theorem 2-4-3: (i)
Assume
that
h0, h £ T0 and h0 is finer than h;
Chapter 2
139
(it)
V 6 C[R + x R", R*l ], V(t, x) is locally Lipschitzian in x, h-positive definite and h0-weakly decrescent; (Hi) W £ C[R + x R", R + ], W(t, x) is locally Lipschitzian in x, h-positive definite, D+ W(t, x) is bounded from above or from below on S(h,p); (iv) there exist C € % and 1 < p < N such that D + Vp(t,x) < -C(W(t,x)), D + V&x) < 9i(t,V(t,x)),
(t,x) e S(h,p) and (t,x) € S(h,p), i ^ p.
Then, stability of uniform stability of the trivial solution of (2.4.1) implies that the system (1.1.1) is (h0,h)-asymptotically stable. Proof: By Theorem 2.4.2 with gp(t,u) = 0, it follows that the system (1.1.1) is (h0, h)-stab\e. Hence it is enough to prove that given tQE R+ there exists a 8Q = SQ(tQ) > 0 such that ^o(^O) xo) < ^o implies h(t, x(t))—»0 as t—»oo. Since w(t,x) is ^-positive definite, it is enough to prove that Urn W(t,x(t)) = 0 for any solution x(t) of (1.1.1) with K(^xo) < ^o- However, this follows easily from assumption (iv) and the proof of Theorem 2.1. We shall next prove some typical results on practical stability and boundedness.
Refinements
140 Theorem 2-4-4■' Assume that (i) 0 *o» «o) < / M > *o»
(2.4.16)
i= i
where u(£,z0,u0) is any solution of (2.4.1). u0 = V(t0,x0) and let h0(t0,x0) < a- By (i), we have
Choose
Let /? = 0(a) > 0 be chosen such that /^(a) < &(/?) and a(a) < /?. We then claim that h(t, x(t)) < 0, t> f0, where x(t) = x(t,t0,x0) is any solution of (1.1.1). If this is not true, there would exist a solution x(t) = x(t, r0, x0) of (1.1.1) with h0(tQ,xQ) < a and i1? t2> tQ satisfying /i(ii,x(
t2.
By Theorem 2.4.1, we have, because of (n), the estimate V(t,x(t)) < 7((h0(t0,x0)) < b{h{t,x)) for {t,x)eR+x Sc{h,p),b € C[[p, oo), R] and for every {t, r) G R + X {p, oo), there exists a /3{r) > p such that h{t, x) = <j>{r) implies VQ{tix) t0, f X M ^ o ) < b{(3{r)),t > tx if JTuoi < b{/3{r)). t= l
i= i
Then the system (1.1.1) is {h0, h)-uniformly bounded. Proof: Let a(E(/9,oo) and t0£R + . By {i), we have if x a ^o(^O) o) < i then h{t0, x0) < {a) = a0. Also, by {iii), if h{t, x) = aQ, then V0{t, x) < b{p{a)). We now set c E = S {h,p),H = S{h,/3)\S{hQ,a), H0 = S{h0,a), G = dS{h,/3) and a{t) = b{/3). Then we claim that h0{t0,x0) < a implies h{t,x{t)) < /?, t > t0. To prove this statement, observe that we only need to consider those solutions x{t) of (1.1.1) with ^o(*0) xo) < a that reach S{h, a0) for some time tx > t0 and leave G remaining in H thereafter. For such solutions, we have VQ{tux{ti)) *o + 21, which proves the theorem.
153
Chapter 2 2.5
Perturbed systems.
When we model a physical system by means of a differential equation, it is not generally possible to take into account all the causes which determine the evolution. In other words, we have to admit that there are small perturbations permanently acting which cannot be accurately estimated. Consequently, the validity of the description of the evolution, as given by a corresponding solution of the differential equation, requires that this solution be "stable" not only with respect to the small perturbations of the initial conditions, but also with respect to the perturbations, small in a suitable sense, of the right-hand side of the equation. This kind of stability is called total stability which we shall define below in terms of the two measures. Let us consider the perturbed differential system x' = f(t, x) + R(t, x), x(t0) = x0,
(2.5.1)
where f,R e C[R+ x R",Rn], R(t,x) is perturbation term relative to unperturbed system (1.1.1). Definition 2.5.1: The system (1.1.1) is said to be (hQ,h,Tx)totally stable if given e > 0 and t0 G R + , there exist two numbers SuS2>0 such that M^o^o) < tQ, where y(t, t0, x0) is any solution of the perturbed system (2.5.1). Theorem 2.5.1: Theorem
Suppose that the assumptions
1.4.1 hold.
Then, the system
(i) and (ii) of
(1.1.1) is
(ho^hjT^-
totally stable. Proof:
Let
U, W G C[S(h, p), R + ]
be
two
Lyapunov
functions which satisfy the conditions of Theorem 1.4.1.
By
(b),
the
(d)
and
the
boundedness
of W,
we see that
unperturbed system (1.1.1) is (h, U)-uniformly
attractive and
that p is a constant associated with this property.
Then,
given v > 0, there exists a T{u) > 0 such that (t,x) € S(h,p) implies U(9, x(0, /, x)) < v for 0 > t + T{v). Let e € (0,p) and tQe R+
be given.
(2.5.3) Choose
Sx,82>0
so that 6t < Po and o(tfj) < 6(e),
(2.5.4)
kS2eMr < a ( ^ ) / 2 , a ( ^ ) + W 2 r < 6(e),
(2.5.5)
and
Chapter 2
155
where k > 0 is a Lipschitz constant for U and r = T( a - )■ Let x0 and i 0 (E i2 + be given such a way that (2.5.2) is satisfied. Then, let us suppose that for a solution y(t,tQ,xQ) of (2.5.1) and a t > t0, we have h(t, y(t, t0, x0)) > e. Since U(tQ,x0) < a(^) < 6(e) and Z7(i , t/(I, i0, x0)) > 6(e), it is clear that there exist tut2 > to,t2 > t\, such that U(*i>y(*i» which is a contradiction. Therefore, £2 ~~ *i < T- Because of the condition (6) of Theorem 1.4.1, we get for t G [ti,t2], Df2.5.x)U{t,y(t,tQ,x0)) t0.
(2.5.10)
If this is not true, there would exist a solution y(t) = y(t,t0,x0) of (2.5.1) with h0(t0,x0) < Sx and t2> ti> t0 such that *o(*i»y('i)) = 5i> Kk^ih))
= e> (2.5.11)
{t,y(t))eS{h,e)nSc(h0,S1)&nd
\\ R{t,y(i)) II < *2»
*e[tj,< 2 ). Then it follows from (2.5.6) and (2.5.11) that D + V(t,y(i)) < ~ 0(6,) + M ^ i l
< 0, t, < t < t2
which implies by (2.5.7)-(2.5.9) that
b(e) < V(t2,y(t2)) < V^yih)) < aft) < 6(e). This contradiction shows that (2.5.10) is true and thus the system (1.1.1) is (/^/^T^-totally stable completing the proof.
Refinements
158 Theorem 2.5.3:
In addition to the assumptions
2.5.2, suppose further
of
Theorem
that there exists a constant a > 0 such
that h(t, x) < a implies lim R(t, x) = 0 uniformly in y. Then the system (2.5.1) is Proof:
(2.5.12)
(h0,h)-attractive.
Because of (h0, /i)-total stability of system (1.1.1),
setting e = a0 = min{p0, cr}, there exist constants Sl0 and S20 such
that
(t,x) £ S(h,a0)
h(tQ, x0) < 610
and
||iZ(f,a;)|| < 820
for
implies h(t,y(t))t0,
where y{i) = y(t,t0,x0)
(2.5.13)
is any solution of (2.5.1).
Let e G (0,cro) and £j = ^ ( e ) , 82 = 82(e) be chosen as in cts \ the definition 2.5.1. Let 62 = min{62,-2-jl-}, it follows from (2.5.12) that there exist Tx = Tx(t0,xQ) > 0 such that
W . !>(T1 + t0.
(2.5.14)
To show (/i 0 ,/i)-attractivity of (2.5.1), it is enough to prove that there exists a T = T(tQ, x0) > 0 such that there exists a t* E [t0, T + tQ] satisfying h0(t*,x(t*)) < 8, and || R(t,y(t))
\\ < 6*2, t > t*.
159
Chapter 2 Choose
M^O
T =
+
T 1 .v( t o + T 1 ))
T h e n
.f
for
*o + r i < ' < fo + r , («,y(t)) <E 5(A,a0) n S c ( M i ) > we get by (2.5.6) and (2.5.7) £
+
V(i )2 /(t))< - ^ , i o + 7 \ < i < t o + 7\
which implies
v(t0 + r, yfo, + r)) < a(h0(t0 + rlt y(t0 + ro) - ^ ( r - i y 0, t0 6 R+ and T > 0, there exist two positive numbers Sx = S^e) and 82 = 62(e) such that for every solution y{t) = y(t,t0,x0) of the perturbed system (2.5.1), the inequality h(t,y(t))<e,t>i0 holds, provided that
Refinements
160 h0(t0,x0)<Su and
\\R(t,x)\\
<X(t)iovh(t,x)<e
t+T f \{s)ds t0.
+
Assume
that
h0, h £ T and hQ is uniformly finer than h;
on
of (2.5.18)
Refinements
164 (M)
V € C[R +xRn,R
+
], V(t,x)
h-positive definite and {Hi) g6C[R+xRnxR
+
is locally Lipschitzian
in x,
hQ-descrecent;
,R],
g(t,0,0) = 0
and
for
some
p>0 D + V{t, x) < g(t, x, V(t, x)), (r, x) <E S(h, p). Then any one of the hQ-conditional trivial
solution
(h0lh)-stability Proof:
stability properties
u — 0 of (2.5.18) implies
the
of the
corresponding
properties of the system (2.5.1).
The proof is very much similar to the proof of
Theorem 1.3.2 except that we now employ Theorem 2.5.5 instead of Theorem 1.3.1.
The relations (1.3.5), (1.3.7) and
(1.3.8) remain the same. Assuming /i 0 -conditional stability of the trivial solution of (2.5.18), we have Su (2.5.19) when tQ (E R+
52
satisfying
and 6(e) is given. We set 6* =
min(S,62)
where 6 is the same one defined in the proof of Theorem 1.3.2. With this 6*, one can show, as in Theorem 1.3.2, that {hQ,h)stability holds for the system
(2.5.1).
Based on
these
modifications, it is not difficult to construct proofs of other stability properties and hence, the theorem is proved. Let us now discuss an important
special case of
(2.5.18). Suppose that g(t,x,u)=
- c{u) + w(t,x)
(2.5.20)
Chapter 2
165
where c G 3G and | u>(t, x) \ < \{t) whenever h(t, x) < e and
*+l / \(s)ds—*0 as t—»oo. t We claim that u = 0 of (2.5.18) is /^-conditionally uniformly asymptotically stable. For this purpose, let us first prove h0conditionally uniform stability. Using the assumption of X(t), we note that t
t
J\{6)d6= j *0
t
<j 1 so that 2Q(T) < min(c(6), e). Let h0(t0,x0) < 5, t0>r and h(t,x(t)) <e for *i]- Also let u0 i 0 s u c n ^ a *
Refinements
166
u(t2) = 6 and 5 < u(t) < e for t € [t2,
(*,x(0)e5(fc,e)nsc(M).*e[*2,t1], then, using (2.5.22), we are lead to the contradiction 6(e) < Vfo.xfo)) < a(ft0(r2,x(r2))) + [Q(r) - c(5)](r1 - 0 exists for
(2.6.5)
t>t0.
Then, if x{t) = x(t, t0, x0) is any solution of (2.6.2), we have V(t, x(t, t0, x0)) < r(t, t0, u0), t > t0, provided V(t0,y(t,t0,x0)) Proof:
(2.6.6)
< u0.
Let x{t) = x(t,t0,x0)
be any solution of (2.6.2). Set
m(s) = V(s,y(t,s,x(s))),t0 so that m(tQ) = V(t0, y(t, t0, x0)). (H) and (i), it is easy to obtain
<st0 where y(t) = y(t,t0,x0)
for some t0 £ R
+,
is any solution of (2.6.1).
As an application of Theorem 2.6.1, we shall consider some results on practical stability of the system (2.6.2). Theorem 2.6.2: 2.6.1 verified.
Assume
that (H) holds and (i) of
Suppose further
Theorem
that
(i)
0 < A < A are given;
(it)
h0, h £ T and there exists ip £ 3G such that h(t,x) < 0 such that
\ t0 + T.
(2.6.19)
Since (2.6.1) is (/i0, ft0)-practically stable at t0 with respect to (A, A), we have K(^ !/( t0, where x(t) = x(t,t0,xQ) is any solution of (2.6.2). If this is not true, there would exist a solution x(t) = x(t,tQ,x0) of (2.6.2) with h0(t0,x0) < A and a tx > t0 such that fc(t„ «(*,)) = A and h(t,x(t)) tQ-
The fundamental
(2-6.22)
matrix solutions of the corresponding
variational equations are
* ( M °-*° ) = [l + x 0 (e->-e-°)P
■«*•*»*)-«"''" 0 -
Consequently, choosing Vi[t,x) = x2,V2{t,y) = y2, we see that V[(t,x) =
2x(t,s,x)$(t,s,x)R(s,x,y),
V'2(i,y) = 2y(«,s,y)0( 0, tf0 £ i2 + , there exists a positive function /?x = /?i(£0>ai) that is continuous in 0
182
Refinements
t0 + T /
X(s)ds < av
to then
u(t,t0,uQ) < f3u t>t0. The definitions (II) - (IX) may be formulated similarly. Theorem 2.7.1: Assume that (i) h0) hET and there exists
oo as u—+oo, and b(h(t,x)) < V(t,x) < a(h0(t,x)); (Hi) D + V{13A)V(t,x) 0 and t0 € R + be given and let a M^o^o) < - Let x(t) = x(t,t0,x0) be any solution of (2.5.1). Then, conditions (ii) and (in) yields
183
Chapter 2 D$.s.i)V(t,x) < g(t,V(t,x)) + M || R(t,x) ||.
Define r)(t) = M \\ R(t,x(t))\\ and a1 = max{Ma,a(a)}. Choose u0 = V(tQ,x0). An application of Theorem 1.3.1 shows that V(*,x(*))< 7 (Mo,"o),
(2-7.2)
where y(t, t0, u0) is the maximal solution of u' = g(t, u) + Tj{t), u(t0) = u0.
(2.7.3)
Assume now that (II) holds. Then, given ax > 0 and t0 G R + , there exists a ftx = /?i(ai) which is continuous in t0 for each ax and /3j 6 3G for each t0, such that, for every solution u(t,t0,u0) of (2.7.1), the inequality u(t,t0,u0) < 0U i x >fo, holds, whenever u0 < ax and, for every T > 0, *o + r / A(s)c?.s < a x . to Since assumption (ii) holds, it is possible to choose a /3 = fl(t0, a) satisfying the relation b(/3) > /?! and (p(a) < /?.
(2.7.4)
where /?j is the function occurring in (JJ). Evidently, /? is continuous in 20 for each a and /? £ 9G, for each tf0- We claim that, with this /?, definition (Ix) holds. If this is not true,
184
Refinements
there would exist a t1 > t0 such that hfaxfa)) For t G [Wi]>
ta,
= P> h(t,x(t)) < /?, * G [UM
(2.7.5)
ke A( i 2 . Let 7*(Moiuo) De t n e maximal solution of the perturbed equation (2.7.1) with X(i) chosen as before. Because of (JJ), it would follow from u0 < ax and
t0 + T /
X(s)ds < au
to for every T > 0, that 7*(*,< 0 ,ti 0 )i 0 . But, on [ b(0) < VfaMti)) < 7( 0 be given. It then follows that, for 6(e) > 0, there exists a pair of numbers 7x = 71(f0,a1,c) and T = T(t0,aue) such that, whichever be the function A £ C[R + , R + ] with
Refinements
186
oo jX(s)ds aie)
so
We now choose a positive number
that Mi = i!
(2.7.8)
and maintain that, with the positive numbers T and 7 so defined, (73) is satisfied. For otherwise, let {tk} be a sequence such that tk > t0 + T, £fc—>oo as k—>oo. Suppose that there is a solution x(t) = x(t,t0,x0) of the system (2.5.1) such that a an ^o(^O) ^o) ^ d h(tk, x(tk)) > e. As before, condition (Hi), in view of the fact that V(t, x) is Lipschitzian, gives D + V(t, x(t)) < g(t, V(t, x(t))) + M || R(t, x(t)) ||. (2.7.9) If we now define X(t) = M || R(t,x(t)) || , we have 00
00
J\(s)ds=
JM\\R(s,x(s))\\ds
00
<M I J '0
sup h(s,x) t0. Then the zero solution u — 0 of (2.4.1) is said to be equiuniformly stable if for given et > 0, e2 > 0 and t0 £ R + , there exists = £i( 0, S2 = S2(e2) > 0 such that QiiKlp) < &i implies (?1([u((tMt,x))> if h0(t, x)e C9G;
(2.8.1)
b(h(t,x)) 0
an
d ^20
=
be
given.
e2 > 0 and t0€.R e
^2o( 2) > 0
sucn
+
By , there
na
* *
Qi(["o]P) < *io implies uo)]P) < e n * > tQ,
(2.8.5)
and Q2(Iuo]9) < ^20 implies Q 2 ([ u ('i *o> u o)],) < e2> < > *„•
(2-8-6)
Since h0 is finer than h, there exist a function a2 G C9G and a constant a0 G (0, a) such that M'oi ^o) < a2(
