# Chapter 4

## Chapter 4

Chapter 4 4.0. Introduction As we have seen, using a single Lyapunov function, it was possible to study a variety of problems in a unified way. I t i...

Chapter 4

4.0. Introduction As we have seen, using a single Lyapunov function, it was possible to study a variety of problems in a unified way. I t is natural to ask whether it might be more advantageous, in some situations, to use several Lyapunov functions. T h e answer is positive, and this approach leads to a more flexible mechanism. Moreover, each function can satisfy less rigid requirements. I n this chapter, we attempt to obtain criteria for stability, instability, boundedness of solutions, and existence of stationary points, in terms of several Lyapunov functions.

4.1. Main comparison theorem Let us consider the differential system %(to)= xo,

x’ = f ( t , x),

(4.1.1)

to 2 0.

Let V E C [J x S , , R+N].We define the vector function D+V(t,x)

= lim

1 sup - [V(t

h-O+

h

+ h, x + /zf(t, x))

-

V(t,x)]

(4.1.2)

for ( t , x) E J x S o . T h e following theorem is an extension to systems of the corresponding theorem 3.1.1 and plays an important role whenever we use vector Lyapunov functions. Let V E C [ J x S o ,R+Y and V ( t ,2) be locally Lipschitzian in x. Assume that the vector function D+V(t,x) defined by (4.1.2) satisfies the inequality

THEOREM 4.1.1.

D+Jqt,x)

< g(t, V ( t ,x)), 267

( t , x) E

1 x so,

(4.1.3)

268

CHAPTER

4

where g E C [J x R, N, R N ] ,and the vector function g ( t , u) is quasimonotone nondecreasing in u , for each fixed t E J . Let r ( t , t, , u,,) be the maximal solution of the differential system U' = ~

( tu), ,

2 0,

u(tn) = U"

to

3 0,

(4.1.4)

existing to the right of t, . If x ( t ) = x ( t , 2, , x,,) is any solution of (4.1.1) such that (4.1.5) Y t " -2"") < U " , 1

then, as far as x ( t ) exists to the right of t o ,we have

v(t,~ ( tto,

7

xt.0))

< r(t, t o ,

(4.1.6)

~ 0 ) -

Proof. Let x ( t , to ,x,,)be any solution of (4.1.1) such that V(to,x,) Define the vector function m ( t ) by

< u,,.

m(t) == V(t,x(t, t o ,X")).

Then, using the hypothesis that V ( t ,x) satisfies Lipschitz's condition in x,we obtain, for small positive h, the inequality m(i

-th ) - m(t) 5; KII

x(t

+ h)

+ v(t + h,

-x(t) -

x(t)

hf(t, x(t))ll

+ hf(t, x(t)))

-

q t , x(t)),

where K is the local Lipschitz constant. This, together with (4.1.1) and (4.1.3), implies the inequality D+m(t)

Moreover, m(tJ

< At, m(t>).

< un . Hence, by Corollary 1.7.1, we have m(t)

< r ( t , to

7

uo)

as far as x ( t ) exists to the right of to , proving the desired relation (4.1.6). We can now state a global existence theorem analogous to Theorem 3. I .4.

4.1.2. Assume that b' E C [ J x Rn, R + N ] , V ( t ,x) is locally Lipschitzian in x, and Cr=l Vi(t, x) is mildly unbounded. Suppose that g E C [ J x R+N,R N ] ,g(t, u ) is quasi-monotone nondecreasing in u for each fixed t E J , and r ( t , t, , uo) is the maximal solution of (4.1.4) existing for t >, to . I f f € C [ J x R7L,Rn] and 'rlEOREhl

D+V(t,x)

< g(t, I'(t, x)),

( t , x) E J x R",

4.2.

ASYMPTOTIC STABILITY

269

then every solution x ( t ) == x ( t , to , x), of (4.1.1) exists in the future, and (4.1.5) implies (4.1.6) for all t 3 to . By repeating the arguments used in the proof of Theorem 3.1.4, with appropriate changes, this theorem can be established. On the basis of Corollary 1.7.1 and the remark that follows, we can prove the following:

THEOREM 4.1.3. Let V E C [ J x S, , R+N] and V ( t ,x) be locally Lipschitizan in x.Suppose that g, ,g, E C [J x R+N,RN], gl(t, u), g2(t,u ) possess quasi-monotone nondecreasing property in u for each t E J , and, for ( t , x) E J x S , ,

< D-'V ( t ,x) ,< gz(t, V ( t ,4).

gdt, V ( t ,4)

Let r ( t , to , uo),p(t, t o , vo) be the maximal, minimal solutions of u' = gz(t,

4,

v' = gl(C v),

4 t " ) = uo , v(t0) = U o ,

respectively, such that

Then, as far as x ( t ) = x(t, t o , xo) exists to the right of to , we have p(t, t o

,4

< V ( t ,4 t ) ) <

y(t,

t o , UO),

where x ( t ) is any solution of (4.1.1).

4,2, Asymptotic stability

An approach that is extremely fruitful in proving asymptotic stability is to modify Lyapunov's original theorem without demanding D+V ( t ,x) to be negative definite. As we have seen, Theorem 3.15.8 is a very general result of this nature, although it covers a particular situation of the function f ( t , x). T h e theorem that follows takes care of the general case of f ( t , x) and requires two Lyapunov functions.

THEOREM 4.2.1.

J

(i) f x Sp-

E

Suppose that the following conditions hold:

C [ J x S, , R"],f(t,0) = 0, and f ( t , x) is bounded on

270

CHAPTER

4

(ii) V , E C [ J x S, , R,], Vl(t,x) is positive definite, decrescent, locally Lipschitzian in x, and D+c;(t, x)

< w(x) < 0,

( t , x)

E

1 x s,,

where ~ ( x is) continuous for x E S, . (iii) V 2E C [ j x S, , R,], and V z ( t ,x) is bounded on J x S, and is locally 1,ipschitzian in x. Furthermore, given any number a , 0 < 01 < p, there exist positive numbers [ = ((a) > 0, 7 = ~ ( D I> ) 0, 7 < a , such that D+G',(t, x) > 8 for

tt

/ / x 11

p and d(x, E ) < 7, t

,

R

=

2 0, where

[x E s, : W(.)

= 01

and d(s, E) is the distance between the point x and the set E. Then, the trivial solution of (4.1.1) is uniformly asymptotically stable.

0 and t,, E J be given. Since V l ( t ,x) is positive definite Proof. Let t and decrescent, there exist functions a, b E .f such that A ;

We choose 6

6 ( ~ so ) that

:

h ( € ) > a(6).

(4.2.2)

Then, arguing as in the first part of the proof of Theorem 3.4.9, we can conclude that the trivial solution of (4.1.1) is uniformly stable. 1,ct u s now fix t p and define 6, = S(p). Let 0 'c E p, to E J , and S -= S(t) be the same 6 obtained in (4.2.2) for uniform stability. Assume that 11 A,, 11 .S,, . T o prove uniform asymptotic stability of the solution .T = 0, it is enough to show that there exists a T = T ( E )such that, for some t* E [t,,, t,, 7'1, we have ~

+

I1 x(t*, t"

9

x0)ll

< 8.

This we achieve in a number of stages: (1)

If d[x(t,), x ( t 2 ) ] > Y

> 0, t , > t , , then Y

< M?P(t,

-

tl),

(4.2.3)

4.2. where IIf(t, x)II

ASYMPTOTIC STABILITY

< M , ( t , x) E J

x S, . For, consider

I "Atl) - xi(tz)/< fz I x:(s)l

ds ,<

ti

< M(tz

and therefore

27 1

-

Itz tl

Ifds, x(s))l ds

( i = 1, 2, ..., n),

t,)

Let us consider the set

u = [x E s,: 6 < Ij x I/ < p, d(x, E ) < 171, and let sup Vz(t,x)

= L.

IIXII-CP

t2O

Assume that, at t = t , , x(tl) = x(t, , t o ,xo) E U. Then, for t we have, letting m ( t ) = V2(t,x(t)),

> t, ,

D+m(t) b D Vz(t,x ( t ) ) > 6,

because of condition (iii) and the fact that V2(t,x) satisfies a Lipschitz condition in x locally. Thus, m(t) - m(tl) =

j t D-Im(s)ds, tl

and hence m(t)

+ m(Q

2

it

D'-m(s)ds

fl

>at

2 f lD+V,(s, x(s))ds

- tl)

as long as x ( t ) remains in U. This inequality can simultaneously be realized with m ( t ) L only if

<

t

< t,

+ 2Llt.

272

CHAPTER

4

< +

It therefore follows that there exists a t, , t , < t , t, 2L/f such that x ( t 2 ) is on the boundary of the set U. I n other words, x ( t ) cannot stay permanently in the set U .

( 3 ) Consider the sequence t,&== t"

Set n(t)

~

{tJ

such that

2L

-t k -

5

(k = 0, 1, 2,...).

V,(t, x(t)). Thcn, by assumption (ii), we have /Pn(t)

We let =

<- DbV1(t, x ( t ) ) ,< 0.

s < I1 "X I1 < f ,

inf[l w(x)I,

and

d(x, E ) 2 11/21,

< <

Suppose that x ( t ) is such that, for t , t t,,, , 6 / / x(t)/l < p. If, for t, t t, , we have S -: /I ' x(t)ll < p and d(x,E ) >, 67, then, using assumption (ii) together with the definition of the set E , we obtain

< < ,,

j

-tk I 2

"(t,,,)

~

n(tn) =

/ _

D+n(s)ds

'Ir

(4.2.4)

On the other hand, if it happens that, for t ,

s < /I .Y(!Jl/

\

p,

< t, < t,,, ,

dlx(t,), B ]

< < +

< .ty,

then there exists a t, , t , t:, t , 2 L / f such that d[x(t,), El = 7, in view of (2). It follows that there also exists a t, , t, t, t, satisfying d[x(t4),El = 8.1. These considerations lead to d[x(t3),x(t4)] Bq, and hence we obtain, because of ( I ) ,

i-q =, Mn""(t,

<

~

t4),

% .

4.3.

273

INSTABILITY

which implies (4.2.5)

Moreover, n(t,)

-

n(tl)

<[

t4

D+V,(s, x(s)) ds

tl

+

D+V,(s, x(s)) ds

[13 64

< -A(t3

-

t4)

-A7 < 2Mn1I2 ~

-A,.

-

Since n(t)is a nonincreasing function, we have fi(t,+,)

<

fi(t3)

<4tl) < n(tJ

-

A,

-

A,

.

Also, on the basis of (4.2.5), we obtain from (4.2.4) that 4t,+2)

G n(td

-

4

*

Thus, in any case,

<

l71(4C+27 +€+3))

>

4tk))

-

A,

> ~ ( 6 , )and T = T ( E )= 4k *L/f ( € ) .

Choose an integer k* such that A\$* Assume that, for to t to T,

< < +

I/ 4 4 t o

L’dtlC

7

x0)Il

2 6.

I t then results from the preceding considerations that Vl(t0 f-1’7 4 t ”

+ T ) ) e Vl(t0 < [email protected],) < 0,

3

Xo)

-

-

k*Al

K*A,

which is incompatible with the positive definiteness of V,(t, x). Thus, there exists a t* E [ t o , to TI satisfying

+

II “ (t* , to , X0)ll < 6 , and the proof is complete.

4.3. Instability I n Sect. 3.3, we proved a theorem on instability by means of a single Lyapunov function. We give below an instability theorem in which two Lyapunov functions are used.

274

CHAPTER

‘I’HEOREnI

4.3.1.

4

Suppose that the following conditions hold:

(i) f E C [ J Y S,, R”],f ( t , 0) = 0, and f ( t , x) is bounded on S,. (ii) V , E C [ J x S o ,R,], V,(t, x) is locally Lipschitzian in x, decrescent, and, for any t >- 0, it is possible to find points x lying in any given small neighborhood of the origin such that Vl(t,x) > 0. 0, (iii) D+V,(t, x) 3 0, ( t , x) E J x S o , and, in each domain t 11 .T /I . p, D+V,(t, x) 2 +a(t)w ( x ) , where w(x) 0 is continuous for x E S o and +Jt) 3 0 is continuous in t such that, for any infinite system S of closed, nonintersecting intervals of J of an identical fixed inter1 al, we have

J x

(4.3.1) and V z ( t x) , is bounded on J x So and is (iv) V , E C [ J x S o , R,], locally Lipschitzian in A . Furthermore, given any number 01, 0 < 01 < p , U , and a continuous function f a ( t )> 0 it is possible to find 7 : 7 ( ~ )7, such that

J

and, in the set z

11 x 11

00

t

p,

G..

L(t>

d(x, E )

=

J

(4.3.2)

a, 7, t

E

J, (4.3.3)

where Then, thc trivial solution of (4.1.1) is unstable.

Proof. T h e proof of this theorem closely resembles that of Theorem 4.2.1, and hence we shall be brief. Suppose that, under the conditions of thc theorem, the trivial solution is stable. T h a t is, given 0 E p, t , E J , there exists a 6 > 0 such that I] x, I/ < 6 implies 6, t 3 to . II x ( t , t,, , %,)I1 According to assumption (ii), a point ( t o ,x\$) can be found such that I/ .x(,’ / / 6 and tT1(t,,, x\$) > 0. We shall consider the motion x(t) = a ( t , t,, , x;) and its properties: I

( I ) d ( A ( t ) , x ( T ) ) 3 7 , t > T ; then t from ( 1 ) in the proof of Theorem 4.2.1. (2)

For every t

- T

2 q/Mn1/2.This

is clear

3 t, , there will be a positive number 01 such that 01

< [I x(t)ll < E < p .

(4.3.4)

4.3. INSTABILITY

275

This is compatible with the assumption of stability, that is, 11 x(t)ll t 3 to . However, since D+V,(t, x) 3 0, it follows that V1(4

< E,

40) 2 Vl(t,, .,*I > 0.

Since Vl(t,x) is decrescent, for numbers V l ( t o x\$) , > 0, a number a: > 0 can be found such that, for all t 3 t o , (1 x 11 a, we shall have

<

vlct, x) < V1(to

7

x,*).

Consequently, 11 x 1) < a is not possible. According to (iv), there exists ) 0 such a number 7 = ~ ( aE),, 7 < a , and a continuous function ( & ( t > that (4.3.2) and (4.3.3) hold. ) ,) < 7, then a t* > 7 can be found such that (3) If ~ ( x ( T E d(x(t*),E )

=

(4.3.5)

7.

Suppose that d(x(t), E ) < 7 for all t 3 7. Letting m ( t ) = Vz(t,x(t)), we obtain, using the Lipschitzian character of V2(t,x) in x,the inequality D+m(t) 2 D'Vz(f,

44) >, U t ) ,

and hence

Since Vz(t,x) is assumed to be bounded, the relation (4.3.2) shows that d(x(t),E ) < 7 cannot hold for all t 3 7. Hence, there exists a t* > T such that (4.3.5) is satisfied. ) ,) < 7 / 2 , then, for t = t*, when d(x(t*), E ) = 7, (4) If ~ ( x ( T E we have Vl(t*,x ( t * ) ) 2

where 7

= inf[w(x),

~

01

< 11 x /I < p, d(x, E ) 2 \$71 > 0.

In fact, under the given conditions,

T

d(x(f*),E )

and, for t ,

ds,

rl < t** === t* - 2Mn1I2

and E

4.))+ 6 J t** C&) t*

Vl(T,

< t < t*, we shall have 4,

< t , < t* can be found such that = \$7,

< d(x(t),E ) < 7.

216

4

CHAPTER

Hence, by (iii), it follows that 4 t ) ) 3 4At) w ( x ( t ) )3 E # b ( t ) ,

D"'l(t,

using the fact that V,(t, x) is locally Lipschitzian in x, and, consequently,

Observing, however, that d(x(t*),x(t*)) >, \$77, we get, in view of (I), that

( 5 ) There is no number t,

have

t,, such that, for all t

d(x(t),E ) 3

=

would

11.

> t, , we should have

Indeed, if such a t , exists, then, for all t 1 71(t,x(t))

> t, , we

+ J" n+Vl(s, x(s)) ds

V,(tl , ~ ( t , ) )

tl

Ry (4.3.1), this implies that V,(t, x ( t ) )+ co as t -+ co, which is absurd because of the relation (4.3.4) and the fact that V,(t, x) is decrescent. Thus it follows that, for any t:, a T ~ > , t: can be found such that

,

d(x(72+1),

< frl,

and, according to ( 3 ) , there corresponds a tz, > T ~ + ,satisfying d(x(tz*,,),E )

= rl-

1,et us consider the infinite sequence of numbers to

< T I < t: < '.' < Tz < t,* < "'.

I n view of assumption (iii) and (4), we have

<

where rj tF* = t,* ~ / 2 M n l /T ~ h. e infinite system of segments [tT*, t:] satisfies condition (iii), and therefore the last sum increases ~

4.4.

CONDITIONAL STABILITY AND BOUNDEDNESS

277

indefinitely with i. In other words, Vl(tF,x ( t 7 ) )-+ co as i+ CO. This is not compatible with the boundedness of Vl(t,x(t)). T h e contradiction shows that the assumption of stability is wrong, and the theorem is proved.

4.4. Conditional stability and boundedness Let, for k < n, M ( n p ! ~denote ) a manifold of (n - k) dimensions containing the origin. Let S(a), S(O)represent the sets S(a) = [x E s,: / / x // S(01) =

[x E

s, : 11 x 11

< 011,

< a],

respectively. Suppose that x ( t ) = x ( t , t o ,xo) is any solution of (4.1.1). Then, corresponding to the stability and boundedness definitions (Sl)-(S8)and (Bl)-(B8),we shall designate the concepts of conditional stability and boundedness by (C,)-(C,,). We shall define (C,) only, since, on that basis, other definitions may be formulated. DEFINITION 4.4.1. T h e trivial solution of (4.1.1) is said to be (C,) conditionally equistable if, for each E > 0 and to E J , there exists a positive function 6 = 6 ( t o ,6) that is continuous in to for each E such that x ( t , to , xo)

provided

c

S(C),

t

3 to ,

x0 E S(8)n MG-k) .

Evidently, if k = 0 so that M(7L-!L) = R",definitions (Cl)-(C16)coincide with the stability and boundedness notions ( S1)-(S,) and (B1)-(B8). Analogous to the definitions (C,)-(C,,), we need some kind of of conditional stability and boundedness concepts with respect to the auxiliary differential system (4.1.4). Perhaps the simplest type of definition is the following. DEFINITION 4.4.2. T h e trivial solution of the system (4. I .4) is said to be (CF) conditionally equistable if, for each E > 0, to E J , there exists a positive function S = S ( t o , 6) that is continuous in t,, for each E such that the condition

C uio < 6, N

i=l

and

uio = O

(i = I , 2,..., K)

218

CHAPTER

implies

c N

% ( t , t o , %)

4

< 6,

t

2=1

> to

'

Definitions (Ca )-( C&) are to be understood in a similar way.

THEOREM 4.4.1.

I'\ssume that

(i) g E C[/ x R+N,K N ] ,g(t, 0) = 0, and g(t, u) is quasi-monotone nondecreasing in 11 for each t E 1; (ii) V E C [ J x S o ,R+N], V ( t ,x) is locally Lipschitzian in x, CL, Vi(t,x) is positive definite, and

C T'i(t, x) N

--f

0

as

/ / x / ---f (

0 for each t~ J ;

i-I

(iii) Vi(t, x) = 0 (z' = I , 2,..., k), k < n, if x E M(n--k), where M(,,_,, is an ( n -- k) dimensional manifold containing the origin; (iv) f E C [ J x S o ,R " ] , f ( t ,0) = 0, and I ) ; q t , x)

< g(t, V ( t ,x ) ) ,

( t , x) E

J x

s,.

Then, if the trivial solution of (4.1.4) is conditianally equistable, the trivial solution of the system (4.1.1) is conditionally equistable. Proof. Let 0 :' E < p and t , , J~ be given. Since positive definite, there exists a b E .X such that b(l/x 11)

N

<: 1 Q t , x),

( t , x) E J x S, .

Vi(t,x) is (4.4.1)

i=l

Assume that the trivial solution of the auxiliary system (4.1.4) is conditionally equistable. Then, given b(6) > 0 and t,, E ], there exists a 6: 6 ( t , , E ) that is continuous in to for each E , so that (4.4.2)

provided

c u,o < 8, v

u,o

=0

(i

=

1 , 2 )...,k ) .

(4.4.3)

a=1

Let us choose ui0 = V , ( t o ,xo) (i = I, 2, ..., N ) and x,,E M(n-fc)so that 0 (z' = 1, 2, ..., k), by condition (iii). Furthermore, since Cf=l V L ( tx,) + 0 as 11 x 11 + 0 for each t E J , and V ( t ,x) is continuous,

uio =

4.4.

279

CONDITIONAL STABILITY AND BOUNDEDNESS

it is possible to find a 6, verifying the inequalities

=

S,(t, , e ) that is continuous in to for each

E,

(4.4.4) simultaneously. With this choice, it certainly follows that x0 E S(8,) n M(n-k)

>

implies x(t, to , xo)C S(E),t to . If this were not true, there would exist a t, > to and a solution x ( t , t o ,xo) of (4.1.1) such that, whenever xo E s(S,) n M(n-lc), we have x(t, t o , xo) C S ( E ) , t E [to , t,), and x ( t , , to , xo) lies on the boundary of S ( E ) This . means that

II 4 4

to

7

> x0)Il

<

II x ( t , to > .")I1

= €7

t

P7

E

[to

9

t119

and, consequently, N

(4.4.5)

Moreover, for t E [to , t J , we can apply Theorem 4.1 .I to obtain V ( t,x ( t , t o , xo))

< r ( t , to ,uo),

[to

t

7

tll,

where r(t, to , uo) is the maximal solution of (4.1.4), which implies that N

N

1 V,(t,x ( t , t ,

7

xo))

< 1 rdt, to ,uo),

t

i-1

i-1

6

[4J,tll.

(4.4.6)

Notice that, from the choice uio = V i ( t o ,xo) and the relation (4.4.4), xo E s(6,) n M(n-k)assures that (4.4.3) is satisfied. Hence, (4.4.2) and (4.4.6) yield the inequality

c N

Vi(t1 9 X(t1

1

i=l

to > xo))

ri(t1

i=l

, t"

7

uo)

which is incompatible with (4.4.5). Thus, x ( t , t o ,xo) C S ( E ) ,t >, t o , provided xo E s(6,) n M(n-k), and the theorem is proved.

THEOREM 4.4.2.

Let assumptions (i), (ii), (iii), and (iv) of Theorem 4.4.1 hold. Suppose further that

c Vi(t, N

i=l

x) + 0

as

11 x 11

--t

0 uniformly in t .

(4.4.7)

280

CHAPTER

4

Then the conditional uniform stability of the solution u = 0 of (4.1.4) guarantees the conditional uniform stability of the trivial solution of (4.1.1).

Proof. By definition ( C z ) , it is evident that 6 occurring in (4.4.3) is independent of t o . I n view of (4.4.7), this makes it possible to choose 6, also independent of t o , according to (4.4.4). Noting these changes, the theorem can be proved as in Theorem 4.4.1.

THEOREM 4.4.3.

Under assumptions (i), (ii), (iii), and (iv) of Theorem 4.4.1, the conditional equi-asymptotic stability of the trivial solution of (4.1.4) implies the conditional equi-asymptotic stability of the trivial solution of the system (4.1.1).

Proof. Assume that the trivial solution of the auxiliary system (4.1.4) is conditionally equi-asymptotically stable. Then, it is conditionally equistable and conditionally quasi-equi-asymptotically stable. Since, by Theorem 4.4.1, the conditional equistability of the trivial solution of (4.1.1) is guaranteed, we need only to prove the conditional quasi-equi0 of (4.1.1). For this purpose, asymptotic stability of the solution x suppose that we are given 0 < E < p and to E J . Then, given 6 ( ~ )> 0 and t,, E J , there exist two positive numbers 6, = S,(t,) and T = T ( t o ,6) such that, if the condition

c N

1

=I

U,"

.< 8" ,

U," = 0

(2-

=

1 , 2 ,..., k)

(4.4.8)

As previously, the choice uio = Vi(t,, x,,) and x,,E implies uio == 0 (i = 1, 2, ..., k). Also, there exists a So = S,(t,) satisfying

II xo /I <:

-

an,

c K(to N

i=l

so)

< 6,

(4.4.10)

at the same time. Let 8,, = min[S,, 6\$], where S\$ = 8(t,, p) is the n MG-/,-), we number obtained by taking E = p. Thus, if x,)E notice that the condition (4.4.8) is fulfilled. Furthermore, since (C,) holds, the inequality (4.4.6) is valid for all t 3 t o . We can now assert n M(n-jt). For otherthat x ( t ) C S ( E ) ,t >, to -1 T whenever xo E s(8,) wise, suppose that there exists a sequence {tx-},t, > to + T , and t,<-+ 00

s(8,)

4.4.

CONDITIONAL STABILITY AND BOUNDEDNESS

28 1

as k -+ 00 such that, for some solution x ( t , t o , xo) of (4.1.1) with xo E S(\$) n M(n-k), we have

II 4t7C

to

7

%)I1 2 E .

t

< i=lc Vi(tl,,X ( t , N

b(c)

<

N Yi(t,

9

i-1

I

to,

t o , uo)

&I))

< b(c),

in view of relations (4.4.1), (4.4.6), and (4.4.9). We thus have (C3),and consequently the theorem is established.

THEOREM 4.4.4.

Suppose that assumptions (i), (ii), (iii), and (iv) of Theorem 4.4.1 hold, together with (4.4.7). Then, the conditional uniform asymptotic stability of the trivial solution of (4.1.4) implies that the trivial solution of (4.1.1) is conditionally uniformly asymptotically stable. Proof. We proceed as in Theorem 4.4.3, observing that the uniformity of conditional stability is assured by Theorem 4.4.2. As the numbers 6, and T are independent of t o , resulting from (4.4.10) is certainly independent of t o , because of (4.4.7).

so

THEOREM 4.4.5. Assume that (i) g E C[J x RtN, RN]andg(t, u ) is quasi-monotone nondecreasing in u for each t E J ; (ii) V E C [J x R?',R N ] , V(t,x) is locally Lipschitzian in x,and

411x 11)

N

< i=l 1 Vi(4 4 6 .(it

where a, b E .X on the interval 0 b(u) + co

x ID,

(t,

4E J

x R",

< u < 00, and as

u+

00;

(iii) Vi(t, x) = 0, i = 1, 2,..., k, k < n, if x E M(n-k), where M(n-k) is an (n - k) dimensional manifold containing the origin; (iv) f E C[/ x Rn, R"], and D'V(t, x)

< g(t, V ( t ,x)),

( 4 ).

E

J x

Rn.

282

CHAPTER

4

Then, if the auxiliary system (4.1.4) satisfies one of the definitions (C\$)-(C&), the system (4.1.1) verifies the corresponding one of the definitions (C9)-(C16).On the basis of parallel theorems of Sect. 3.13 and the proofs given previously, the proof of the respective statements of this theorem can be constructed. Let us now indicate the modifications necessary in order to obtain the usual stability and boundedness results, using several Lyapunov functions. Designate by (Cf*)-(C\$*) the parallel definitions obtained by dropping the conditional character in (CT)-(C\$). For example, the definitions (CF*) would run as follows:

8

(CT*) For each E > 0 and to E J , there exists a positive function 8(t,) e ) that is continuous in to for each E such that the inequality

=

implies

As a typical example, we shall merely state a theorem that gives sufficient conditions, in terms of any Lyapunov function, for the equistability of the trivial solution of (4.1.1). THEOREM 4.4.6.

Suppose that

(i) g E C [ J x R+N,R N ] g(t, , 0) = 0, and g(t, u) is quasi-monotone nondecreasing in u for each t E J; (ii) V E C [ J x S o ,R + N ] ,V ( t ,x) is locally Lipschitzian in x, C;"=, Vi(t,x) is positive definite, and N

L'*(t, X)

+

0

as

(1 x 11 -+ 0

for each t

E

J;

2=1

(iii) f

E

C [ J x So,R n ] , f ( t0) , = 0, and

D+F7(f,).

< g(t, V ( t ,x)),

( t , x)

E

J x

s,.

Then the definition (Cf*) implies that the trivial solution of (4.1.1) is equistable. T o exhibit the fruitfulness of using vector Lyapunov function, even in the case of ordinary stability, we give the following example.

4.4.

283

CONDITIONAL STABILITY AND BOUNDEDNESS

Let us consider the two systems

Example.

+ y sin t y’ = x sin t + e+y x’ = e+x

+ xy2)sin2 t,

-

(x3

-

(x2y

+ y 3 )sin2 t .

( 4 . 4 . 1 1)

Suppose we choose a single Lyapunov function V given by V ( t ,x)

= x2

+

y2.

Then, it is evident that D+V(t,x)

< 2 ( r t + j sin t 1)

V ( t ,x),

< u2 + b2 and observing that [x2 + y2I2sin2 t 3 0.

using the inequality 21 ab I

Clearly, the trivial solution of the scalar differential equation u’ = 2(e+

+ I sin t I) u,

u(to) = uo 3 0

is not stable, and so we cannot deduce any information about the stability of the trivial solution of (4.4.1 1) from Theorem 3.3.1, although it is easy to check that it is stable. On the other hand, let us attempt to seek a Lyapunov function as a quadratic form with constant coefficients V(t,X)

=

+[x2

+ 2Bxy + Ay2].

(4.4.12)

Then, the function D+V(t,x) with respect to (4.4.1 1) is equal to the sum of two functions q ( t ,x), w,(t, x), where q ( t , x) = x2[e+

+ B sin t] + xy[2Be+ + (A + 1 ) sin t ]

+ y2[Aect+ B sin t ] , wz(t,x) = -sin2 t [ ( x 2+ y2)(x2+ 2Bxy + Ay2)].

For arbitrary A and B, the functions V ( t ,x) defined in (4.4.12) does not satisfy Lyapunov’s theorem (Corollary 3.3.2) on the stability of motion. Let us try to satisfy the conditions of Theorem 3.3.3 by assuming w,(t, x) = h(t) V(t,x). This equality can occur in two cases:

+

+

1, B , = 1, h,(t) = 2[c1 sin t] when V,(t, x) = &(x Y)~. 1, B, = -1, h,(t) = 2[ect - sin t] when V,(t, x) = +(x - y)2.

(i) A, (ii) A,

=

=

284

CHAPTER

4

T h e functions V, , V, are not positive definite and hence do not satisfy Theorem 3.3.3. However, they do fulfill the conditions of Theorem 4.4.6. I n fact,

x:=l

(a) the functions Vl(t,x) >, 0, V,(t, x) >, 0, and Vi(t,x) = x2 + y2, and therefore C Z , Vi(t,x) is positive definite as well as decrescent;

< +

(b) the vectorial inequality D+V(t,x) g(t, V ( t ,x)) is satisfied with the functions g l ( t , u1 , uz) = 2(e+ sin t ) u1 , g2(t,uI, u2) = 2(e+

-

sin t ) uz .

I t is clear that g(t, u ) is quasi-monotone nondecreasing in u, and the null solution of u' = g(t, u ) is stable. Consequently, the trivial solution of (4.4.11) is stable by Theorem 4.4.6.

4.5, Converse theorems We shall consider the converse problem of showing the existence of several Lyapunov functions, whenever the motion is conditionally stable or asymptotically stable. T h e techniques employed in the construction of a single Lyapunov function earlier in Sect. 3.6 do not right away extend to this situation. As will be seen, the results rest heavily on the choice of special solutions of a certain differential system and the chain of inequalities among them, a kind of diagonal selection of the components of these solutions, and the quasi-monotone property. With a view to avoid interruption in the proofs, let us first exhibit some properties of certain solutions of the system (4.1.4) and its related system u' = g*(t, u ) , (4.5.1)

Assume that g E C [ J x R+N,A"], g(t, 0 ) E 0, ag(t, u ) / & exists and is continuous for ( t ,u)E J x R+N,and g(t, u ) is quasi-monotone nondecreasing in u for each t E /. Evidently, g*(t, u ) also satisfies these

4.5.

285

CONVERSE THEOREMS

assumptions. Moreover, since ui > 0 (i = 1, 2,..., N ) , it follows, in view of the quasi-monotone property of g(t, u), that

< g(t, 4.

g*(t, u )

(4.5.2)

Observe that the hypothesis on g(t, u ) guarantees the existence and uniqueness of solutions of (4.1.4) as well as their continuous dependence on initial values. Also, the solutions u(t, to , uo) are continuously differentiable with respect to the initial values. Furthermore, u s 0 is the trivial solution of (4.1.4). Clearly, similar assertions can be made with respect to the related system (4.5.1). If U ( t) = U(t,0, uo) and U*(t) = U*(t, 0, uo) are the solutions of (4.1.4) and (4.5.1), through the same point (0, uo),respectively, it follows, from Corollary 1.7.1, that U*(t)

< U(t>,

t

(4.5.3)

2 0,

in view of (4.5.2). Consider next the N initial vectors, with uio > 0 (i = 1, 2, ..., N ) defined by

o,..., 01, u20 o,..., 01,

Pl

= (u10 9

P,

=h10

Pi

= (u10 u20

,*-., uio, o,..., O ) ,

= (%I3

,..-,U N O ) .

P,

9

, 9

9

...

...

a20

<

It is easy to see that pi pi+l , for each i = 1, 2,..., N - 1. Let us denote the solution of the system (4.5.1) through the point (O,p,) by

Ui*(t)= Ui*(t,0 , p J

UiAt,

07 P i )

Uinr(t7

0, P i )

=

for each fixed i, i = 1, 2,..., N . By Corollary 1.7.1, it follows that

where UF(t), U\$l(t) are the solutions of the system (4.5.1) through (0, pi) and (0, pi+l), respectively.

286

CIIAPTER

This implies that, for each j 0, P I )

Ul,(t,

-2U Z i ( t ,

=

4

1, 2,..., N and t 3 0,

< ." <

0, p 2 )

UNj(t,

0, p N ) .

(4.5.4)

We may now have the following:

THEOREM 4.5.1.

Assume that

(i) the functionfg C [ J x So,R"],f(t,0) = 0, and af(t, %)/axexists and is continuous for ( t , x) E J x So; (ii) the solution x ( t , 0, xo) of the system (4.1.1) satisfies the estimate

Pi(1' xo ll) where

-c

ll x ( t , 0, xn)ll

< Pz(ll xn ll),

t

2 0,

xn E M(n--k)9

(4.5.5)

, PL E H ;

(iii) the function g E C [J x R t N ,R N ] ,g(t, 0 ) = 0, ag(t, u)/au exists and is continuous for ( t , u ) E J x R+N, and g(t, u ) is quasi-monotone nondccreasing in ii for each t E J ; (i~7) the solution U ( t ,O,p,) of the system (4.1.4) verifies the inequality

provided uiD 0 (i = 1 , 2,..., k), where y 2 E X ; (v) the solution U\$(t, 0, p N ) of the related system (4.5.1) is such that

if ziio =- 0 (i I , 2 ,..., k), where y1 E X . Then, there exists a vector function V ( t ,x) with thc following properties: ( I ) I/ E C [ J s S, , and V ( t ,x) possesses continuous partial derivatives with respect to t and the components of x for ( t ,x) E J x S,,;

4.5.

287

CONVERSE THEOREMS

Proof. Let us first observe that assumption (i) implies the existence and uniqueness of solutions of (4.1.I), as well as their continuous dependence on the initial values. Also, the solutions x(t, t o , xo) are continuously differentiable functions with respect to the initial values ( t o ,xo), and the system (4.1.1) possesses the trivial solution. Let us denote x(t, 0, xo) by x so that, by uniqueness of solutions, we have xo = x(0, t, x). Choose any continuous function p(x) E R + N , possessing continuous partial derivatives a p ( x ) / a x for x E S, , such that a1(II

where

JI)

, a2 E X and

N

< C pi(-%)< aAIJ x II),

(4.5.8)

i=l

( Y ~

pi(%)= 0

(i = 1, 2,..., K)

if

XE

M(n-k).

(4.5.9)

We then define the vector function V ( t ,x) as follows:

VI(4 x)

= Ull(t,

0, Pl(40,t , x)),

OY.9

01,

V d t , 2 ) = " z d t , 0, PcLI(4-2 4 x)), P2(40,

...

l"(4

2) = U N N ( t ,

0, PI(x(0, t , x)),...,

t 9

XI>,

CLN(X(0,

0,... 01, ?

(4.5.10)

t , x))).

Because of the continuity of the functions x(0, t , x), p(x), Up(t),..., U\$(t), with respect to their arguments, it is clear that Vi(t, x) (i = I, 2, ..., N ) is defined and continuous for ( t , x) E J x S, . Since the functions f and g (and hence g*) satisfy hypotheses (i) and (iii), the functions UT(t), U\$(t),..., U\$(t), and x(0, t, x) are all continuously differentiable with respect to their arguments. This, together with the choice of p(x), shows that V ( t ,x) possesses continuous partial derivatives with respect to t and the component of x. Thus, for each i = 1, 2,..., N , Vi'(t, x) = &(t, 0, p1(x(O, t , x)),..., CLi(X(0, t , x)), 0,..., 0 )

since, by relation (3.6.1 I),

288

CHAPTER

4

4.5.

289

CONVERSE THEOREMS

where u l , u2 ,..., uN are the components of the solution U(t,O,p,) of the system (4.1.4). In view of (4.5.9) and the fact that x,,E M(n--k), using the relation (4.5.6) and the upper estimates in (4.5.8) and (4.5.1 I), we get

c Vi(4 4 G [ c N

N

Yz

i-1

G

i=l

PLi(40, 4

Y 2 [ 4 40, t , x)ll)l x lD>l

G r2[.,(8;'(11 =

4)j

4 x Ill,

a E3 - 0

Finally, as the solution U\$(t) is nonnegative, we have

c vi(t, N

i=l

x)

2 ~ " [ t 0, , ~ i ( x ( 0t,, x)),--., E " N ( x ( O ,

t , x))I,

which, by using the inequality (4.5.7) and the lower estimates in (4.5.8) and (4.5. I I), yields

3 Y l [ ~ l ( I 1 4 0 ,t , 4l)l

T h e proof is complete.

x ll))l

2 Yl[%(P;l(ll = b(ll x II),

b E 3--

It is to be observed that the upper estimate in (4.5.5) and the inequality (4.5.6) ascertain the conditional stability of the null solutions of (4.1.1) and (4.1.4), respectively. The lower estimate in (4.5.5) and the estimate (4.5.7) are compatible with the conditional stability of the null solutions of (4. I. 1) and (4.1.4), respectively.

THEOREM 4.5.2. Let assumptions (i) and (iii) of Theorem 4.5.1 hold. Suppose further that (a)

the solution x( t , 0, xo) of (4.1.1) satisfies the inequality

Pl(ll xo 11) where

Ul(t)

< /I 4 4 0, %)I1 < Bz(ll xo 11) az(t),

, p2 E X and

u1 , u2 E 2;

t

3 0,

xu E M(7L-k) >

(4.5.12)

290

CHAPTER

4

(b) the solution U(t,0, p,) of (4.1.4) verifies the estimate (4.5.13) where y, E X , 6, E 27, and, whenever uio = 0, i = 1 , 2,..., I z ; (c) the solution U,\$(t,0, p,) of (4.5.1) is such that (4.5.14) and whenever ui0 = 0, i = I, 2,..., k ; where y1 E X ,6,E 9, y l ( y ) is differentiable, and y;(r) 3 m 3 0; (d) (e) 8,(t) and a2(t)are such that 6,(t) 3 mla2(t), m, > 0. Then, there exists a function V ( t ,x) with the properties (I), (2), (3) of Theorem 4.5.1 and

41x 11) ,<

c N

VZ(t>

z=1

4

< a ( t , II x ll),

( 4 x) E 1 x

s,,

where 0 t X and a(t, Y) belongs to class X for each fixed t E J and is continuous in t for each Y.

Proof. Let x ( t , 0, x"), U ( t ,0, p,), and U\$(t, 0, p,) be the solutions of (4.1.1), (4.1.4), and (4.5.1) satisfying (4.5.12), (4.5.13), and (4.5.14), respectively. Choose any continuous function p ( x ) E R,N possessing continuous partial derivatives with respect to the components of x, such that (4.5.9) and

all x II)

< c P Z ( 4 < 41x Ill? N

1 1

B2

1

01

E

.x,

(4.5.15)

hold. Using the same definition (4.5.10) for V(t,x) and proceeding as in Theorem 4.5. I , it can be easily shown that (l), (2), and (3) are valid. Assumption (d) implies that YI(YlY2)

2 mYlYZ.

(4.5.16)

T h e inequality (4.5.12), in view of the fact that x = x(t, 0, xo) and x,,= x(0, t , x), yields that (4.5.17) where p ~ ' ,1,3;1 both belong to class X .

4.5.

29 1

CONVERSE THEOREMS

As in Theorem 4.5.1, using the definition (4.5.10) and the nonnegative character of U\$(t), we get N

1vZ(t,

i=l

).

2 u N N [ t , O, P1(x(O,t ,

x))>***7

PN(x(o,

t , x))],

which, by virtue of (4.5.14), the lower estimates in (4.5.15) and (4.5.17), the relation (4.5.16), and the assumption ( e ) , gives successively

Again, as before, making use of the definition of V(t,x) and the relation (4.5.4) and (4.5.3), we obtain N

1

Z=l

N

vi(t,

< 1 uZ(t, i-1

Pl(x(o,t , x)),***, PN(x(o, t , x))],

which, in its turn, allows the following estimates successively,

because of (4.5.13) and the upper estimates in (4.5.15) and (4.5.17). The theorem is proved. Under the general assumptions of Theorem 4.5.2, it is not possible Vi(t,x) .(\I x 11). This can, to prove the stronger requirement that however, be done if the estimates (4.5.12). (4.5.13), and (4.5.14) are modified as in the following:

xr=l

<

292

CHAPTER

4

THEOREM 4.5.3. Let assumptions (i) and (iii) of Theorem 4.5.1 hold. Assume that the inequalities (4.5.12), (4.5.13), and (4.5.14) in assumptions (a), (b), (c) of Theorem 4.5.2 are replaced by

81 /I x’o /la

4 t ) < II 4 4 0, x’0)ll

< 8, II ~o I/= ~ ( t ) ,

/3, , p2 , 01 > 0 being constants, N

1

t

3 0,

i=l

O, \$ N )

(4.5.18)

> 0,

(4.5.19)

and u E 9; N

ui(t,

xo E M ( n - k ) ,

< y2 1

*iO

i=l

s(t),

where y 2 > 0 is a constant, 6 E 9, and, whenever uio = 0, i N

~ “ ( t 0, , pjv)

2 yi C uio s(t),

t

i=l

2 0,

=

1, 2 ,..., k; (4.5.20)

where y1 > 0 is a constant and uio = 0, i = 1, 2, ..., k; respectively. Furthermore, let the functions 6 ( t ) and a ( t ) be related by S=(t)

=

d(t),

for some constant p > 0. Then, there exists a function V ( t ,x) with the properties (I), (2), (3) of Theorem 4.5.1, and M I

/I x’ /I”

N

< 1 Vi(t,). < M , II x’ llP, 2=1

where MI = ylA1/3;”, M , = y2A,/3yp, p suitable positive constants.

=

Pla,

and A,,

A, are some

PToof. By choosing the continuous function p(x) E R,N that satisfies (4.5.9) and

4I1 x’ /ID <

c N

i-1

Pi(4

< A,

II x 1 °,

A,, A,, p being constants greater than zero and following the proof of Theorem 4.5.2, with necessary changes, it is easy to construct the proof of the theorem. It may be remarked that the conditional asymptotic stability of the null solutions (4.1.1) and (4.1.4) is expressed in terms of the upper estimate in (4.5.12) or (4.5.18) and (4.5.13) or (4.5.19), respectively. Also, the lower estimate in (4.5.12) or (4.5.18) and the inequality (4.5.14) or (4.5.20) are compatible with the conditional asymptotic stability of the trivial solutions of (4.1.1) and (4. I .4).

4.6.

STABILITY IN TUBE-LIKE DOMAIN

293

T h e conditional character of the stability notions in Theorems 4.5.1, and 4.5.2, and 4.5.3 are due to the requirements that xo E uio = 0, i = 1 , 2,..., k. By dropping these conditions and modifying the technique suitably, it is easy to get a set of necessary conditions for the stability concepts, in terms of several Lyapunov functions.

4.6. Stability in tube-like domain Lyapunov stability of the invariant set of a differential system does not rule out the possibility of asymptotic stability of the set, nor does the asymptotic stability of the invariant set guarantee any information about the rate of decay of the solution. Various definitions of stability and boundedness are, so to speak, one-sided estimates, and thus they are not strict concepts in a sense. It is natural to expect that an estimation of the lower bound for the rate at which the solutions approach the invariant set would yield interesting refinements of stability notions. We introduce below the concepts of strict stability and boundedness of solutions. represent the sets Let Z ( a ) and

z(a)

s : II x I/ > a ] , Z(a) = [x E s : [I x I/ 2 a], Z(a) = [x E

respectively, and let S ( a ) , S(ol), and M(n--k)have the same meaning as in Sect. 4.4. Let x(t, to , xo) be any solution of (4.1.1).

DEFINITION 4.6.1.

T h e trivial solution of (4.1.1) is said to be

(CS,) conditionally strictly equistable if, for any E , > 0, to E 1,it is possible t o find positive functions 6, = 6,(t0, el), 6, = ?&(to,E , ) , and E, = €,(to , el) that are continuous in to for each el , such that €2

provided

< 8,

< 6,

< €1 ,

x(4 t o , xo>c q . 1 ) n Z ( E 2 ) , xo E

W,) n Z(8,) n

t

2 to,

M(T2-k) ;

(CS,) conditionally strictly unijormly stable if 6, , 6, , and E, in (CS,) are independent of to ; (CS,) conditionally quasi-equi-asymptotically stable if, given E , > 0, 01, > 0, and to E 1, it is possible to find, for every 01, satisfying 0 < 01, < 01, ,

294

4

CHAPTER

positive numbers c2 , T , that Ti

<

r 2

~ ( tt,,, , .yo) c s(€,)

whenever

T I ( &, E , , al),and T ,

=

€2

j

c €1

nz ( E ~ ) ,

to

€2

=

T2(t0, c2 , a,) such

a2

+ T , e t G to + T , ,

x0 E S(al)n Z(.(,) n M ( ~ - ;~ )

(CS,) conditionally quasi-uniform-asymptotically stable if T , and T , in (CS,) are independent of t, ; (CS,) conditionally attracting if it is conditionally equistable and, in addition, (CS,) holds; (CS,) conditionally uniformly attracting if it is conditionally uniformly stable and, in addition, (CS,) holds. T h e system (4.1.1) is said to be (CS,) conditionally strictly equi-bounded if, given a1 > 0, to E J , it is possible to find, for every rx, satisfying 0 < a, a1 , positive functions PI = &(t,, , q),p2 = &(tn, q) that are continuous in to for each al , such that

<

P2

x(t, t"

9

XO)

provided X" E

< Pl

P2

< 012

t

c S(P1) n Z(P2)>

t

3 to ,

S(al) n Z(a2)n M ( ~ -;~ )

(CS,) conditionally strictly uniform bounded if p1 , p, in (CS,) are indepcndent of to . We observe that the foregoing notions assure that the motion remains in tube-like domains. I n order to obtain the sufficient conditions for the stability of motion in tube-like domains, we have to estimate simultaneously both lower and upper bounds of the derivatives of Lyapunov functions and use the theory of differential inequalities. We are thus led to consider the two auxiliary systems uo

u'

=: g l ( t , u),

up,)

v'

= g2(t, ?I),

v(to)= uo

=

3 0, 2 0,

(4.6.1)

(4.6.2)

<

where g , > g , E C [ J x R+N,R N ) ,g2(t, 4 g d t , u), and g1(t, u), gz(4 u> possess the quasi-monotone nondecreasing property in u for each t E J. Then as a consequence of Corollary 1.7.1, we deduce that p(t9 to

7

< ~ ( tto,

~ 0 )

7

uo),

t

>, to

9

4.6.

295

STABILITY IN TUBE-LIKE DOMAIN

provided where r ( t , t o , uo), p ( t , to , vo) are the maximal, minimal solutions of (4.6.1), (4.6.2), respectively. Corresponding to definitions (CS,)-( CS,), we may formulate (CSf)-(CSZ) with respect to the system (4.6.1) and (4.6.2). For example, (CSF) would imply the following:

(CSf) Given el > 0, to E J , there exist positive functions 6, = a1(t0, el), 6, = S,(to , el), eg = e z ( t o , el) that are continuous in to for each el such that €2 < 6 , e 6, < €1 , €2

<

c N

i=l

P i ( 4 t o , vo>

N

yz(t9

i=l

t o , uo> < €1

,

t b to

7

if uio = vi0 = 0 (i = 1, 2, ..., k) and

Let us restrict ourselves to proving conditional strict equistability only. Similar arguments with necessary modifications yield any desired result.

THEOREM 4.6.1. Assume that (9 g,

gz(t,0)

, g2 E C [ J x R+N,RN1,

gz(4

4 < gdt, 4,

gdt, 0) = 0,

= 0, and g l ( t , u), g2(t,u ) possess the quasi-monotone nondecreasing property in u for each t E J ; (ii) V E C [ J x S,, , R+N],V(t,x) is locally Lipschitzian in x, and, for ( t , x) E J x S , ,

(iii) Vi(t,x) = 0 (i = 1, 2,..., k), k < n, if x E M(n-k), where M(n-k) is an ( n - k) dimensional manifold containing the origin; (iv) f E C [ J x S,, , R n ] , f ( t ,0 ) = 0, and, for ( t , x) E J x S,, ,

<

gz(4 Jqt,4) D+V(t,4

< gAt, V ( t , 4).

296

CHAPTER

4

Then, if the auxiliary system satisfies condition (CST), the trivial solution of (4.1.1) is conditionally strictly equistable.

< p and to E J be given. Assume that (CST) holds. Proof. Let 0 :c Then, given !I(€,) > 0, t, E J , there exist positive functions 8, = %Po 4, 8, = s2(t0, E ~ ) ,and i , = <,(to , el) such that 5

2,

< 8,

< 8,

< b(t,),

(4.6.3) provided uio = vin== 0

(i = 1, 2,..., K),

(4.6.4)

= Vi(to, xo) = uio ( i = 1, 2 ,..., N ) and x E M(%-,()so We choose that v?,,: uio = 0 (i : 1, 2, ..., k), by condition (iii). Let us make the following choice:

s, = b-1(8,),

6,

=

u-y81),

.(€,)

< i,,

€2

< 6,.

< 6, < and that < I/ xo 11 < 6, implies

Then, it is easy to verify that < 6, depend on to and . Furthermore, 6,

8,

E,

, a,, 6,

< c Vdto, ~XO) < 8 1 , N

z=1

and vice versa. With this choice of E, , 6,, and e l , the trivial solution of (4.1.1) is conditionally strictly equistable. Suppose that this is false. Then, there is a solution x(t, to , xo) of (4.1.1) satisfying xu E

S(8,) n Z(8,) n M ( ~ - ~ )

such that, for some t = t , > to , it reaches the boundary of S(el) n Z(e2). This means that either 11 x(t1 , t o ,xo)ll = or I/ x(t, , to , x,)lI = e2 . Also, I! x ( t , to , xg)lI -: p, t E [t,,, t l ] , and therefore, for t E [to, tl], we can apply Theorem 4.1.3 to obtain

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

297

where r ( t , t o ,u,,),p(t, t, , vo) are the maximal, minimal solutions of (4.6.1), (4.6.2), respectively, such that no = V(to, xo) = u, . This implies that

c Pi(4 to

N

N

i=l

* uo)

< i=l1 Vdt, 4 4 t o ,

.ON

(4.6.5)

for t E [ t o ,t J . I n the first instance, if 11 x( t , , t o ,xo)lI = e l , using the right side inequality in (4.6.3) and (4.6.5), we arrive at the contradiction N

b(4

< 1 Vztt, i=l

7

X(t,

7

to xo>> Y

because of the left side inequalities in (4.6.3) and (4.6.5). This shows that (CS,) follows from (CST), and the proof of the theorem is complete.

4.7. Stability of asymptotically self-invariant sets One has to consider, in many concrete problems like adaptive control systems, the stability of sets that are not self-invariant; this rules out Lyapunov stability, because those definitions of stability imply the existence of a self-invariant set. T o describe such situations, the notion of eventual stability has been introduced in Sect. 3.14. It is easy to observe that, although such sets are not self-invariant in the usual sense, they are so in the asymptotic sense. This leads us to a new concept of asymptotically self-invariant sets. Evidently, asymptotically self-invariant sets form a special subclass of self-invariant sets, and therefore it is natural to expect that their stability properties closely resemble those of invariant sets.

298

CHAPTER

4

Let zu E C[Rn,R"]. Define (4.7.1) We shall denote the sets

[x E R" : I/ 4411

< €1

by G, S(G, E ) , S(G, E ) , respectively. Suppose that x ( t ) = x ( t , to , xo) is any solution of (4.1.1). DEFINITION 4.7.1. A set G is said to be asymptotically self-invariant with respect to the system (4.1.1) if, given any monotonic decreasing sequence { ep} , e p ---f 0 as p + CO, there exists a monotonic increasing sequence {tJE)}, t , ( ~ -+ ) 00 as p + 00, such that xo E G, to 3 t p ( E ) , implies x(t)

C S(G, cD),

t

2 to,

p

=

1, 2,... .

be an (n - k) dimensional manifold containing the set G. Let We shall assume that G is an asymptotically self-invariant set with respect to the system (4.1.1). DEFINITION 4.7.2. T h e asymptotically self-invariant set G of the system (4.1.1) is said to be (AS,) conditionally equistable if, for each E > 0, there exists a tl(e), tl(e) + 00 as E + 0, and a S = S(t, , E ) , to 2 t , ( e ) , which is continuous in to for each e such that x(t)

provided

c S(G, €1, xo E

t

3 to 2 tl(.),

S(G, 6) n M n - - k )

*

On the basis of this definition, it is easy to formulate the remaining notions (AS,)-(AS,) corresponding to (Cl)-(C8) of Sect. 4.4. The following theorem gives sufficient conditions for the set G to be asymptotically self-invariant with respect to the system (4.1.1).

THEOREM 4.7.1.

Assume that

(i) g E C[J x R+N,R N ] , and decreasing in u for each t E J ;

g ( t , u ) is

quasi-monotone

non-

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

299

V I EC [ J x S(G,p), R+N],V ( t ,x) is locally Lipschitzian in x,

(ii)

< c Vi(t, x), N

b(ll w(x)ll)

( t , x)

E

i=l

and

c Vi(t,x) N

=

J

a(t)

X

S(G,p),

if

x E G,

6 E .X,

(4.7.2)

(4.7.3)

2-1

where u E 9; (iii) Vi(t,x) = 0 (i = 1, 2,..., k), k < n, if x E M(n--li), where M(n-k) is an (n - k) dimensional manifold containing the set G; (iv) f E C [ J x S(G,p), R"],and

o+v(t, x) < g(t, v(t,x)),

( t , x) E

J x S(G,p ) ;

(v) for any function P(t, u), which is continuous for t 2 0, u 2 0, decreasing in t for each fixed u, increasing in u for each fixed t such that lim lim/3(t, u ) t-m

u-0

=

0,

(4.7.4)

we have

provided uiO = 0 (i = 1, 2, ..., k), where u(t, t o , uo) is any solution of (4.1.4). Then, the set G = [x E Rn : 11 w(x)lI invariant with respect to (4.1.1).

=

01 is asymptotically self-

Proof. Let x0 E G. Since G C M(n-,c. , it follows that x0 E M(n--li). As a consequence, we have, by (iii), Vi(to, XJ = 0 (i = 1, 2,..., k), k < n. We choose uiO= V i ( t o xO) , (i = 1, 2 ,..., N ) . Then, because of (4.7.3), we obtain N

1

c N

ui0

i=l

=

i=l

Vi(t0 9

(4.7.6)

xo) = O(t0).

Consider the function y ( t ) = p(t, u(t)), which decreases to zero as t+ co because of the assured monotonic properties of the functions /3 and u. Let now {E,} be a decreasing sequence such that E , + 0 a s p -+ 00. Then, the sequence {b(e,)} is a similar sequence. Since y(to)+ 0 as to + m, it is possible to find an increasing sequence { t p ( e ) } ,t p ( e ) + co as p + 00, such that y(t0)

to

2 tD(E),

P

=

1 , 2,-

*

(4.7.7)

300

CHAPTER

4

We claim that xo E G implies that x ( t ) C S(G, el,), t 2 to 3 t p ( c ) , for each p = I , 2, .... Suppose, on the contrary, that there exists a solution x ( t ) of (4.1.1) such that xo E G, to >, t , ( ~ )for a certain p , (1 w(x(t,))I(= c p €or some t = t, > to >, t , , ( ~ )and ,

II 44t))ll G ~p < P , For t

E

t

E

[to 9 t i l e

[t,, , t J , we obtain, on account of Theorem 4.1.1, the inequality

where r ( t , t, , uo) is the maximal solution of (4.1.4). At t = t, , we arrive at the contradiction

< c Vdtl N

b(%)

N

7

+l))

i=l

< 1Y d t l < i=l

>

to

> %I)

P(t0 1 4 t O ) )

= At,)

( 4

making use of the relations (4.7.5), (4.7.6), (4.7.7), and (4.7.8). This proves that the set G is asymptotically self-invariant with respect to the system (4.1.1). If we assume that the set u = 0 is asymptotically self-invariant with respect to the auxiliary system (4.1.4), we have the following definition parallel to Definition 4.7.2.

DEFINITION 4.7.3. T h e asymptotically self-invariant set u = 0 of the system (4.1.4) is said to be (ACT) conditionally equi-stable if, for each E > 0, there exists a t , ( ~ ) t, , ( ~-+ ) 00 as E 4 0, and a 6 == S ( t o , E ) , to >, t l ( c ) , which is continuous in to for each E , such that

provided N

1uio < 8, i=l

ui0 = 0

(i = 1 , 2 )...,k).

T h e following theorem assures the conditional equistability of the asymptotically self-invariant set G.

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

301

THEOREM 4.7.2.

Suppose that hypotheses (i), (ii), (iii), and (iv) of Theorem 4.7.1 hold, except (4.7.3). Assume further that the set G is asymptotically self-invariant and

c N

i=l

< a(t, I1 w(x)II),

Vi(t,x)

(4 "4E J x S(G, P),

(4.7.9)

where the function a(t, r ) is defined and continuous for t >, 0, r 2 0, monotonic decreasing in t for each fixed r, monotonic increasing in r for each fixed t, and lim lim a(t, T ) = 0. t-m

r-0

Then (ACT)implies (AC,). Proof. Let 0 < E < p be given. Assume that the definition (ACT)holds. Then, given b ( ~ > ) 0, there exist a t , ( ~ ) t,l ( e ) -+ co as E -+ 0, and a 6 = S ( t o , E ) , to >, t , ( ~ such ) that N

1 udt, t o , uo) <

2 to 3 tl(4,

t

WE),

i=l

(4.7.10)

provided N

1 uio < 6,

uio = 0

(i

=

1, 2 ,...)k).

(4.7.11)

i=l

Choose uiO = V i ( t o ,x,,), i = 1, 2,..., k, and xo E M(n-k)so that ui,, = 0 (i = 1, 2, ..., R ) , by condition (iii). If we now make the choice that CF=l uio = a(to , 11 w(xo)[l),the assumptions on a(t, r ) imply the existence of positive numbers tz(.) and 6, = Sl(t0 , E ) , to >, t Z ( e ) ,such that 4to

9

ll w(x0)ll)

< 8,

I/ w(x0)lI G 6,

9

(4.7.12)

provided to 3 t2(e). Let t3(c) = max[t,(E), t 2 ( c ) ] .I t can then be claimed that, if xo E S(G, S,) n M(n--k), we have x(t, to , xo)C S(G, E ) for t >, to >, t3(c), where x(t, to , xo) is any solution of (4.1.1). Let us assume that this is not true. Then, there exists a solution x ( t ) of (4.1.1) such that, , x(t) C S(G, c) for t E [ t o , t,], whenever xo E S(G, 6,) n t, > to 2 t 3 ( e ) , and x(tl) lies on the boundary of S(G, c). This implies that

I/ w(x(t))ll

and

<

€9

t

E

[to tll, Y

11 w(x(t,))ll = E. Thus, there results (4.7.13)

302

CHAPTER

Furthermore, for t

E

4

[to, t J , we obtain the inequality

in view of Theorem 4.1.1, r ( t , t o , uo) being the maximal solution of and the relation (4.7.12) guarantee that, (4.1.4). Since the choice of us,, whencver x,,E S(G, 8,) n M(%-,J, the condition (4.7.1 I ) is satisfied, it is easy to derive, from (4.7.10) and (4.7.14), the inequality

Thi s relation is incompatible with (4.7. I3), thereby establishing (AC,).

COROLLARY 4.7.1. Under the assumptions of Theorem 4.7.2, the conditional equistability of the trivial solution of (4.1.4) assures the definition (AC,). We can easily prove the statements corresponding to the definitions (AC,)-(AC,), on the basis of Theorem 4.7.2. T o show the close relationship between theorems of this section and Sect. 3.14, we shall merely state below a theorem parallel to Theorem 3.14.1.

THEOREM 4.7.3.

Assume that

(i) g E C [ J x R , N , R N ] g, ( t , u ) is quasi-monotone nondecreasing in u for each t E J , and the asymptotically self-invariant set u = 0 of (4.1.4) is conditionally uniformly stable; V ( t ,x) is locally Lipschitzian in x, and (ii) V E C [ J x S(G,p ) ,

4 441) <

c Vdt, x) < 4 N

w(x)II),

i=l

-, Y 11 w(x)ll < p and t 3 O(Y), where a, b E X and the function B(r) 2 0 is monotonic decreasing in Y for 0 < r < p ;

for 0

(iii) V,(t,x) 0 (i 1, 2,..., k ) , k < n if x E M ( l L - k,~where M(n-k) is an ( n k ) dimensional manifold containing the set G; ~

:

~

(iv) f~ C[/ >\ S(G,p ) , R"],the set G is asymptotically self-invariant with respect to the system (4.1. l), and D-i-V(t,x)

< g(t, q t , x)),

4.7.

STABILITY OF ASYMPTOTICALLY SELF-INVARIANT SETS

303

Then, the asymptotically self-invariant set G is conditionally uniformly stable. Analogous to the boundedness concepts (Bl)-(B8)defined in Sect. 3.13, we have the following weaker notions. DEFINITION4.7.4. The system (4.1.1) is said to be, with respect to the set G, (EB,) conditionally eventually equi-bounded if, given a 2 0, there exist tl(a) > 0 and /3 = ,f3(t,, a ) , to 2 tl(a), which is continuous in to for each a , such that x(t)

c S(G, 81,

provided x,, E

t

3 to 3 tl(a),

S(G,a ) n M(+-k) .

The remaining definitions (El?,)-(EB,) may be easily formulated. As previously, the definitions (EBf)-(EB\$) refer to the conditional boundedness concepts with respect to the system (4.1.4). A typical theorem on eventual boundedness is the following:

THEOREM 4.7.4.

Suppose that

(i) g E C [ J x R+N, decreasing in u for each t (ii) V E C [ J x R",

and g(t, u) is quasi-monotone

J;

E

non-

V ( t , x) is locally Lipschitzian in x, and N

b(II w(x>II)

< i=l1 vi(t, < 44 I1 w(x)II),

t

3 0,

x E R",

where a(t, r ) is continuous for t >, 0, r 2 0, montonic decreasing in t for each Y, monotonic increasing in r for each t , and lim lim a(t, r )

2-02

7-0

= 0,

<

and b E Z on the interval 0 Y < GO such that b(r) + co as r -+ co; ; (iii) Vi(t,x) = 0 (i = 1, 2,..., k ) , k < n, if x E (iv) f

E

C[J x R", R"], and D+V(t,x) ,< g(t, v(t,x)),

Then the condition (EBf) implies (EB,).

( t ,x)

E

1x

R".

304

CHAPTER

4

Proof. Let (L: 3 0 be given. Suppose that x" E S(G, a ) n M(n--h-) . Because of the assumptions on a ( t , Y), it is possible to find two positive numbers y = y(m) and t l ( m ) such that 4t" a) 9

if

to

3 tl(a).

(4.7.15)

Assume that (El?:) holds. Then, given y > 0, there exist two numbers and p = B ( t o , a), t , 3 t2(n),such that

whenever N

1 Ui" < y,

(i = 1 , 2)...)k).

Ui" = 0

i=l

(4.7.17)

t3(a)= max[t,(cu), tz(a)]. Choose uio = Vi(t,, xo), to 3 t3(n) = a(t,,,I] w(x,)ll).Since x, E S(G, a ) n M(n--k), this implies, in view of condition (iii), that ui0 = 0 (i = I, 2, ..., k). Moreover, the condition (4.7.17) is satisfied, in view of this choice, and consequently (4.7.16) is true. Since h(r) + 03 as Y -+ CO, there exists a p1 = & ( t o , a) such that

Let

(i = 1 , 2,..., A), and Cf=lui,

W l )

3 P*

(4.7.18)

We can now conclude that (EB,) holds with and t3(a).T h e assumption that this is false leads to the existence of a t , > t, 3 t3(a) and a solution x ( t ) with xo E S(G, m) n M(n-,~) , such that

/I w(x(t1))lI = P1 at t = t, > to 3 tB(a).By assumption (iv) and Theorem 4.1.1, we can infer that

1 Vdt, 4 t ) ) s c N

N

2-1

z=1

Y,(t,

4,

7

Uo),

> t" 3 t3(a),

t

which, because of the relation (4.7.16) and assumption (ii), shows that

-= c N

4Pl)

Z-1

VL(tl

4tl))

G

c N

2=1

rz(t1

7

to

3

Uo)

< P.

This is a contradiction to the choice of in (4.7.18), and hence we claim that (EB,) holds. T h e proof is complete.

4.8.

STABILITY OF CONDITIONALLY INVARIANT SETS

305

4.8. Stability of conditionally invariant sets We shall introduce in this section the concept of a conditionally invariant set with respect to a given set and consider the stability properties of such sets.

DEFINITION 4.8.1. Let A and B be any two subsets of Rn such that A C B. Then, the set B is said to be conditionally invariant with respect to the set A for the differential system (4.1.1) if xo E A implies that x ( t , t o ,xo) C B for all t 2 to 3 0. Let w E C[Rn,Rm],and let 11 w(x)ll mean the same norm of w defined by (4.7.1). Let us continue to use the sets G

= [X E

R" : 1) w(~)ll= 01,

S(G, c)

= [X E

R" : [I w ( ~ ) )< l €1,

S(G,6)

= [X E

Rn : 1)

ZU(X)~I

< €1,

and let us designate the set S(G, a) by B. Suppose that the set B = S(G, a) is conditionally invariant with respect to G, for some a > 0. Let M(npk)denote, as before, an ( n - k) dimensional manifold containing the set G. We define S(B, C)

=

S(G, 01

+ c),

E

> 0.

DEFINITION 4.8.2. T h e conditionally invariant set B with respect to the set G and the system (4.1.1) is said to be (CC,) conditionally equistable if, for each E > 0 and to E 1, there exists a positive function S = S ( t o , E ) , which is continuous in to for each E , such that

Evidently, on the strength of (CC,), we can define (CC,)-(CC,) corresponding to (Cz)-(C8).

REMARK4.8.1. We observe that the set B need not be self-invariant. If 01 = 0, these definitions coincide with (C,)-(C,), that is, the conditional stability concepts of the self-invariant G.

306

4

CHAPTER

T o define the corresponding definitions (CCF)-( CC:) for the auxiliary system (4.1.4), let us define the set, for some p > 0, N

1 ui < 81,

u E R+N:

(4.8.1)

iL1

and assume that B* is conditionally invariant with respect to the set 0 and the system (4.1.4).

zi =

DEFIKITION 4.8.3. T h e conditionally invariant set B* with respect to the set 11 - 0 and the system (4.1.4) is said to be (CCF) conditionally equistahle if, for each E > 0 and t,, E J , there exists a positive function S = S ( t , , E ) , which is continuous in t, for each E , such that N

1 ~ t ( ttn,

7

un)

z=l

provided N

1

Ui"

1-1

THPORERI 4.8.1.

< 6,

ui,

B

=

+ 0

t

E,

2 to

7

(i = 1 , 2 )...,K).

Assume that

, = 0, and g(t, a) is quasi-monotone (i) g E C [ J x R+N,R N ] , g ( t0) nondecreasing in u for each t E J ; (ii) V E C [ J x R", R . , N ] ,V ( t ,x) is locally Lipschitzian in x, and N

b(Il 4.z)lI)

< 1 Vdt, x) < 4w(x)II),

( t , x) E

J x R",

2-1

where a , h E f on the interval [0, a)and h(r) + co

(iii) f

E

as

Y + co;

C [ J Y R", R"], and D - l ' ( t , x)

< g ( t , V ( t ,x)),

( t ,x) E J x R".

Then, if the set B* is conditionally invariant with respect to the set 0 and the system (4.1.4), the set B = S(G, a ) , where N = b-l(P), is conditionally invariant with respect to the set C and the system (4.1.1).

zi =

PFoof. Assume that the set B* defined by (4.8.1) is a conditionally invariant set. This implies that, if ui0 = 0 (i = 1, 2 ,..., N ) , N

1 ~ , ( tt ,o , 0 ) < P,

7=1

t

2 t o 3 0.

(4.8.2)

4.8.

STABILITY OF CONDITIONALLY INVARIANT SETS

307

Let us choose uin = V i ( t o x ,,,) (i = 1, 2, ..., N ) . Then, it follows that xo E G and Vi(to, xo) == 0 (i = 1, 2, ..., N ) hold simultaneously. By Theorem 4.1.1, we obtain

where r ( t , to , u,,) is the maximal solution of (4.1.4) through ( t o ,u,,). Since b(ll w(x)ll) CC, Vi(t,x), we readily get the inequality

<

in view of (4.8.2) and (4.8.3). As a consequence, we deduce that, if xo E G, x(t, t,, , xo)C S(G, a ) , t 3 to , where 01 = h-l(P). T h e Conditional invariancy of the set B is immediate, and the proof is complete. REMARK 4.8.2. Notice that the p occurring in (4.8.2) may depend on to , in which case cx depends on t o , and, as a result, the set B depends on t o . This suggests that the invariant sets we generally consider are, in a sense, uniform invariant sets, and perhaps a classification of invariant sets and the study of their stability properties may be of some interest. Regarding the stability behavior of the conditionally invariant set B, we have the following:

THEOREM 4.8.2.

Assume that conditions (i), (ii), and (iii) of Theorem 4.8.1 hold. Suppose further that Vi(t,x) = 0 (i = 1, 2,..., A ) , k < n, if x E MG-,;) . Then, if one of the conditions (CCT)-(CC\$) is satisfied, the corresponding one of the conditions (CC,)-(CC,) is assured.

Proof. We shall only indicate the proof corresponding to the statement (CC,), that is, the conditional quasi-uniform asymptotic stability of the conditional invariant set B. Let E > 0, y > 0, and to E J be given. Suppose that

V i ( t o xo) , = 0 (i = 1, 2, ..., k). Choose uio = Vi(t, , x,,)(i = 1, 2,..., N ) . Then, we have by Theorem 4.1.2 that every solution x(t, t,, , x,,) of (4.1.1) exists for t 3 to and satisfies

so that we can infer that

"(4 x ( t , t o , xo))

< Y ( t , t" , 4,

t

2 to,

where r ( t , t o ,u,,) is the maximal solution of (4.1.4). Define y1

(4.8.4)

= a(y),

308

CHAPTER

4

+

and assume that (CC,") holds. Let 01 = h-'(P). Then, given b(oc C) > 0, > 0, and to E 1, there exists a positive number T = T ( y , e ) such that

y,

V

1

U , ( f , f,,

, UO)

t=1

< b(a

+

+ T,

(4.8.5)

(i = I , 2,..., k).

(4.8.6)

t

c),

2 4,

provided V

2 U," < y1 ,

/

u,,, = 0

I

Clearly, by the choice of y1 and u i O , the condition (4.8.6) is satisfied. Hence, wc obtain, using (4.8.4), (4.8.5), and the fact that

WI w(4Il)

c V,(f, N

d

4

7

2-1

the relation b(ll zu(.v(t, t o , ~ 0 ) ) l l )

< b(a

+

e),

t

3 f0

+ T,

whenever xo E S(G, y ) n M(,L--I,) . Evidently, this implies that the conditionally invariant set 13 is conditionally quasi-uniform asymptotically stable. T h e proof of the theorem is thus complete.

4.9. Existence and stability of stationary points This section is concerned with the conditions sufficient to assure the existence of yo satisfying (4.9.1)

f(Yo) = 0

and the stability of the solution x ( t ) = y o of the autonomous differential system x' = f ( x ) , x(0) = x0 , (4.9.2) where f E C[R",R"].

TIIEORENI 4.9. I . (i)

Assume that

I/ E C[R",K + N ] , V(x)is locally Lipschitzian in N

C

V ( ( x )4

as

11 x 11

---f

x,and

co;

t=1

, is quasi-monotone nondecreasing in u , and (ii) g E C [ R ,N , R N ] g(u) D+V(x) g( V ( x ) ) ,x E RtI;

<

4.9.

EXISTENCE AND STABILITY OF STATIONARY POINTS

309

(iii) Q E C[R+N,R+], Q(v) is monotone nondecreasing in w, and Q(v(x)) = 0 only if f(x) = 0; (iv) for a certain u o , the system U'

u(0) = u g

= g(u),

>0

(4.9.3)

possesses the maximal solution r ( t , 0, uo) defined for 0 that r ( t , 0, uo) is bounded and satisfies Q(r(t,0,U,]))

--f

0

as

t

+

co.

such (4.9.4)

Then, if x ( t ) is any solution of (4.9.2), it exists and is bounded for t E J , and every cluster point (a-limit point) yo of x(t) satisfies (4.9.1).

Proof. Let x ( t ) be a solution of (4.9.2). Then, by Theorem 4.1.2, x ( t ) exists for 0 t < m. Furthermore, if V(t,, xo) uo ,

<

<

< r(4

V(x(t))

O,UO),

t

2 0,

(4.9.5)

where r(t, 0, u,,) is the maximal solution of (4.9.3). T h e assumptions that r ( t , 0, uo) is bounded for t 0 implies, in view of (i), the boundedness of x(t). Also, the function Q being monotonic nondecreasing, we have, by (4.9.5),

8(Q ( t ) ) ) G Q(r(t, 0, uo)), which, on account of (4.9.4), guarantees that

-

lim O ( V ( x ( t ) ) )= 0.

t-m

Hence, every w-limit point y o of x ( t ) satisfies Q(V(YoN

= 0,

and (4.9.1) follows, because of assumption (iii). This proves the theorem.

COROLLARY 4.9.1. Let the hypotheses of Theorem 4.9.1 hold, except that Q( V ( x ) )= 0 only if f(x) = 0 is replaced by Q(V(x))= 0

only if

U ( x ) = 0,

where U E C[Rn,R n ] . T h e n the assertion of Theorem 4.9.1 remains valid if (4.9.1) is replaced by V Y n ) = 0.

310

CHAPTER

4

THEOREM 4.9.2. Suppose that the conditions of Theorem 4.9.1 are satisfied. In addition, assume that, for every uo > 0, the maximal solution r ( t , 0, u,,) of (4.9.3) exists on 0 < t -: a3 and is uniformly bounded for t 3 0 and bounded uo and satisfies (4.9.4) uniformly for bounded ZL,, . Then the set Z

=

[x : f ( x ) = 01

is nonempty and connected.

Proof. It is clear that, under the assumptions of the theorem, every solution x ( t ) of (4.9.2) exists for 0 t a,and, given any 01 > 0, there exists a P ( N ) such that

<

I1 x(t)ll

<<

t

&a),

2 0,

<

provided 11 xo I / n. Furthermore, every w-limit point y o satisfies (4.9.1). Ry ‘I’heorem 4.9.1, it follows that the set Z is nonempty. Hence, only connectedness remains to be proved. Let E, (Y be arbitrarily positive numbers. Then, it follows from (ii) that there exists a 6 = a(€, a ) > 0 such that

p(I+)) 2 6

if

d(x, Z )

2 E, /I x 1) < p(a).

Hence, by thc uniformity of (4.9.4) and by (4.9.5), it is possible to find a T - T ( E

if

t

3 T , 1) so(1

< a.

(4.9.6)

<

Since ail solutions ~ ( tof) (4.9.2), for arbitrary xo , exist on 0 t < GO, it follows by a generalization of H. Kneser’s theorem that the set Za(t) of points z reached by some solution of (4.9.2) at a time t 3 0, when 11 x,) j j 1, that is,

<

ZJt) = [z : z = x ( t ) , I/ XI) 1)

< a],

is closed and connected. We notice that the set

Zm= Z n [ x : I I . Y / ~

< a]

is contained in Zn(t)for t >, 0 [for, if yo E 2, then x ( t ) = y o is a solution of (4.9.2)j. I,ct s,, , xt,be two arbitrary points of .Z;, C Zm(T ) . Then, there exists a finite set of points x , ~= x,,,x1 ,..., xN+l xb in Z,(T) such that E, if i / I s, - s~~~/I 0, 1,..., N . I n view of (4.9.6), there is a point

4.10.

NOTES

311

xi E 2 satisfying 11 xi - xi 11 < E for i = 1, 2,..., N . Hence, there is a finite set of points x , = x , = xu, x l , ..., x N , xN+I = xN+l = xb in 2 such that /I xi - xi+l 11 < 3 ~ for , i = 0, I , ..., N . Since c is arbitrary and 2 = n 2, , 01 3 0, the set 2 is connected. This proves the theorem.

COROLLARY 4.9.2. Assume that the conditions of Theorem 4.9.2 hold. Suppose that the zeros off(.) are isolated. Thenf(x) has a unique zeroy, , and the solution x ( t ) = y o of (4.9.2) is completely uniformly asymptotically stable. 4.10. Notes

Section 4.1 introduces comparison theorems that are useful when several Lyapunov functions are employed (see Lakshmikantham [ 131). T h e results of Sects. 4.2 and 4.3 have been taken from the work of Matrosov [l]. For the results contained in Sect. 4.4, see Lakshmikantham [13] and Matrosov [2]. Converse theorems of Sect. 4.5 are due to Lakshmikantham et al. [I], whereas the results of Sect. 4.6 concerning the stability in tube-like domains are taken from the work of Charlu et al. [l]. T h e notion of asymptotically self-invariant sets is introduced by Lakshmikantham and Leela [l], and the contents of Sect. 4.7 are based on their work. T h e results of Sect. 4.8 dealing with the criteria for the stability of conditionally invariant sets are due to Kayande and Lakshmikantham [l]. Section 4.9 deals with the results due to Hartman [6]. For related work using several Lyapunov functions, see Antosiewicz [4], Bellman [4], D’Ambrosio and Lakshmikantham [l], and Lakshmikantham and Verma [I].