Stability analysis in terms of two measures

(1.2.3)

Chapter 1

13

Let

e G (0, pQ)

assumption

and r 0 e R +

be given.

By t h e

on a, there exists a 82 = S2(t0, e) > 0 that is

continuous in tQ such that a(t0,S2)
(1.2.4)

Then hQ(tQ,xQ)<8

(1.2.1)-(1.2.4), that b(h(tQ,x0)) < V(t0,x0)<

a(t0,h0(t0,xQ))

6(e), which in turn yields that h(t0, x0) < e. that

for every

solution

x(t) = x(t, tQ, x0)

implies, by <

We now claim of (1.1.1)

h(t,x(t))<e,t>t0.

with

(1.2.5)

If this is not true, then there would exist a tx > t0 such that h(tu x(ij)) = e and h(t,x(t))

< e, t 6 [t0,tj,

for some solution x(t) = x(t, tQ, x0) of (1.1.1). V(t,x(t))

for t G [
(1.2.6) Set m(t) =

is locally Lipschitzian in

x, it follows from (1.1.4) and assumption (it) that D + m(t) < 0, which implies, by Lemma 1.1.1, that m{t) is nonincreasing in [*0,
< V{h,x{tj)

< V(t0,x0)

< b(e),

which is a contradiction. Hence (1.2.5) is true and the system (1.1.1) is (h0, /i)-equistable.

Basic Theory

14

To prove (i?), note that if V(t,x) is /i0-decrescent and h0 is uniformly finer that h, then the functions a and

Assume that and h0 is uniformly finer than h; X Rn, R + ], V(t, x) is locally Lipschitzian in x, definite, h0-decrescent and

D + V(t,x) < -C{h0(t,x)),

(t,x) E S(h,p),

Then the system (1.1.1) is (h0,h)-uniformly

C€%. asymptotically

stable. Proof: Since V(t,x) is ft-positive definite and h0descrescent, there exist constants 0 < pQ < p, 0 < 80 and functions a, b G 3G such that b(h(t, x)) < V(t, x), (t, x) 6 S(h, p0)

(1.2.7)

V(t, x) < a(h0(t, x)), if k0(t, x) < 80.

(1.2.8)

and

Chapter I

15

It follows from Theorem 1.2.1 that the system (1.1.1) is (h0, /i)-uniformly stable.

Thus we let e = p0, there exists

81 = $i(p) > 0 such that ^o^o^o) < #i implies h(t,x(t)) < p0,t>

i0,

where x(t) = x(t,i 0 ,x 0 ) is any solution of (1.1.1). Let 0 < e < p0 and 8 = 8(e) be the same 8 as in the Definition (1.1.1) for (hQ, /i)-uniform stability. hQ(t0,x0)<8*

= min{80,8l}.

Assume that

Set T = T(e) = § § + l .

To

prove (/i0, /i)-uniform asymptotic stability, it is enough to show that there exists a t ' G [t0, tQ + T], such that

h0(t*,x(t*)) < 8. If this is not true, then there exists a solution x(t) — x(t, t0, x0) of (1.1.1) with h0(t0,x0)

< $* s u c n ^a,t

h0(t,x(t))>8,te[t0,t0 Let m(t) = V(t,x(t)).

+ T].

Then it follows from condition (it) that

D + m(t)<

-C(hQ(t,x(t)),

t>t0

which implies by (1.2.8) that t0 + T

J

(1.2.9)

C(h0(s, x{s)))ds < m(«o) < a{S*).

'0

On the other hand, from (1.2.9), we obtain

16

Basic

Theory

t0 + T

J

C(h0{s, x(s)))ds > 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,{t,x)eR+x

R3.

By Theorem 1.2.1, we conclude that the system (1.2.10) is (hQ, ft)-equistable. It should be noted that (h0, /instability in this example implies, by the choice of h and h0, that the trivial solution of (1.2.10) is partially stable with respect to xv

Chapter 1

17

Example 1.2.2:

Consider the differential system f |

X'= -y +

(l-XZ-y2)xe-\ 2

y' = x + (1 - x - y2)y sin x2.

Let V(x,y) = (x2 + y2-lf, see

h0 = h= \x2 +

(1.2.11)

y2-l\. Then we

h2(x,y) < V(x,y) < h2(x,y), (x,y) G R2, and D + V(x,y) = - 4(x2 + y2 - l) 2 (x 2 e~' + y2sin x2) < 0 , (t,x,y)eR+

xR2.

Thus (h0, /i)-uniform stability follows from Theorem 1.2.1, and from which it is easy to see that the periodic solution (x(t),y(t)) = (cos t , sin t) of (1.2.11) is uniformly orbitally stable. Example 1.2.3:

Consider the differential system

f

x\= — x x (l + sin2x3) — 2i 2 e ~', i x'2 = 2x1-x2et,

(1.2.12)

x3 = — xxe ~ * + x2cos t + x3sin t. Set V(t,x) = x\ + x\e-\ h(t,x) = d{x,B) and hQ(t,x) = d(x,A), where A = {(xu x2, x3) G R3; x1 = x2 = 0}, B = {(xx,x2,x3) G

Basic Theory

18

R3; Xl = 0). Clearly, A C B, h2(t, x) < V(t, x) < h&t, x) and D + V(t,x)< -2hl(t,x). Thus, by Theorem 1.2.2, the system (1.2.12) is (h0, ft)-uniformly asymptotically stable. Let us next discuss a result on (h0,h) equi-asymptotic stability that corresponds to Marachkov's result [1]. Theorem 1.2.3: Assume that (i) h0) h ET and h0 is finer than h; (ii) V G C[R + x Rn, R + ], V(t, x) is locally Lipschitzian in x, h-positive definite, h0-weakly decrescent and D + V(t, x) < - C(h(t, x)), (t, x) e S(h, P), C € %; (Hi) h e C^R +xR",R \h'(t,x)\

+

] and <M,

(t,x)eS(h,p),

where M > 0 and h'(t,x) = ht(t,x) + hx(t,x) • f(t,x). Then the system (1.1.1) is

(h0,h)-equi-asymptotically

stable. Proof:

It follows from assumptions (i) and (ii) that, by

Theorem 1.2.1, the system (1.1.1) is (h0, /i)-equistable. Thus let e = p, t0e R+ be given, there exists SQ = S0(t0,p) > 0 such that h0(to, *o) < S0 implies h(t, x(t))

tQ,

(1.2.13)

Chapter 1

19

where x(t) = x(t,t0,x0)

is any solution of (1.1.1).

Suppose that the system (1.1.1) is not asymptotically stable.

(h0,h)-equi-

Then for some e > 0 there exist a

solution x(t) = x(t, t0, x0) with h0(tQ, x0) < 60 and a divergent sequence {tk} such that h(tk,x(tk))>e,k

= 1,2,....

It then follows from assumption (Hi) that, on the intervals

**~2F-'-'* + 2fe'fc= 1'2v. we have h(t, x(t)) > | . We can assume that these intervals are disjoint and 11 — -^ > t0 by taking, if necessary, a subsequence of {tk}.

This, together with assumption (ii), implies

V(tk + ^

x(tk + ^ ) ) < V(t0, x0) - C ( | ) ^ f c - » - oo as

which is a contradiction.

Thus the system (1.1.1) is

k^oo, (h0,h)-

equi-asymptotically stable. Remark reduces

1.2.1: to

the

If

hQ = h = || x \\, well

known

then

Theorem

Marachkov's

result

1.2.3 [1] on

asymptotic stability of the trivial solution of (1.1.1). Remark

1.2.2:

If condition (m) of Theorem 1.2.3 is dropped,

then the system (1.1.1) may loose its (h0, /i)-equi-asymptotic stability property although it is still (hQ, /i)-equistable.

Basic Theory

20

Theorem 1.2.4: If the h-positiveness in Theorem 1.2.3 is weakened to h-positive semi-definiteness of V, that is, V(t, x) > 0, (t, x) G S(h, p), then the conclusion of Theorem 1.2.3 is still valid. Proof: Since V(t,x) is hQ-weakly descrescent and h0 is finer than h, then relations (1.2.1) and (1.2.3) hold. Let e > 0 and t0 G R + be given. Then by the assumptions on (p in (1.2.3) and a in (1.2.1), there exists a constant 82 = ^2(^0) > 0 such that ¥>(<„, 82) < § and a(t0,82) < C ( | ) ^ .

(1.2.14)

Choose 8 = min{60,S1,S2}. Let /i0(£0, x0) < £ and x(t) = x(t,t0,x0) be a solution of (1.1.1). Suppose that there are t2 > tx > t0 such that KhAh)) = f, ^2,^*2)) = e, h{t,x(t)) > §, and h(t, x(t)) < e, f G [tQ, t2).

t e [tut2], (1.2.15)

Since V(t, x) is locally Lipschitzian in x and £> + V(*,x)<

-C(h(t,x)\

it follows from (1.2.14) - (1.2.15) that 0 < V(t2,x(t2))<

V{t0,x0) + V(*a,*(*a)) - V(t lf *&))
Chapter 1

21
which is a contradiction.

Hence the system (1.1.1) is (hQ,h)-

equistable. The rest of the proof is similar to that of Theorem 1.2.3. Thus the theorem is proved. A stability property can be considered as a family of properties depending on some parameters.

Consequently,

when we employ a single Lyapunov function to prove a given stability property, the Lyapunov function used is assumed to play the role for every choice of these parameters. As a result, if we utilize a family of Lyapunov functions instead of one, it is natural to expect that each member of the family has to satisfy weaker requirements.

To illustrate this idea, we shall

next give two results which are improvements of Theorems 1.2.1 and 1.2.2. Theorem 1.2.5:

Assume

that

(i)

h0) h G T and h0 is uniformly finer than h;

(ii)

for every r\ > 0, there exists a function V^ G C[S(h, p) D Sc(h0, TJ), R + ] such that V^t, x) is locally Lipschitzian x and satisfies

b(h(t,x)) < v(t,x) < «(V*. *))»(*.*) e s(h,P)nsc(h0,T}), where a,b £ %; (in)

D + V(t, x) < 0, (*, x) G S(h, p) n Sc(h0, r,).

in

Basic Theory

22

Then the system (1.1.1) is (h0,h)-uniformly stable. Proof:

By assumption (i), there exist a constant S0 > 0

and a function

(1.2.16)

Let e € (0, p) and i 0 G R + be given. We choose 8 = 5(e) > 0, 8 < 5X such that ¥>(£) < e and a(S) < 6(e).

(1.2.17)

We let h0(t0, x0) < 8 and x(t) = x(t, t0, xQ) be any solution of (1.1.1). Then by (1.2.16) - (1.2.17), we see h(t0,x(t0)) < e. We claim that h(t, x(t)) < e, t > tQ, for any solution x(t) = x(r,r 0 ,x 0 ) of (1.1.1) with h0(t0,x0) < 8. If this is not true, then there exists a solution x(t) of (1.1.1) and tlt t2 > tQ such that fe0(i1}x(
(1.2.18)

Hence, by letting 77 = 8 and condition («), there exists a Vv(t, x) satisfying assumptions (ii) and (Hi), which implies 6(e) = b(h(t2,x(t2)) < V(t2,x(t2)) <
= a(8)
VfaXtJ)

Chapter 1

23

This is absurd. stable.

Thus the system (1.1.1) is (h0, ft)-uniformly

Theorem 1.2.6:

Let the assumptions

of Theorem

1.2.5 hold

except that condition (Hi) is strengthened to (in*) D + V(t, x)<-

C(h0(t, x)) for

(t, x) <E S(h, p) n Sc(h0, rj),

where C G 9G. Then

the system

(1.1.1) is (h0,h)-uniformly

asymptotically

stable. By combining the proofs of Theorems 1.2.5 and 1.2.2, we can construct the proof of Theorem 1.2.6.

We omit the

details. As we shall see later, due to the interplay of two different measures, obtaining a smooth converse theorem for (h0, /i)-uniform asymptotic stability is not possible unless we assume a larger domain of attraction.

This implies that we

need

than

to

have

a

stronger

asymptotic stability.

concept

(hQ, /i)-uniform

The following result is a direct theorem

of this type. Theorem 1.2.7:

Assume

that

(i)

h0) h E T and h0 is uniformly finer than h;

(ii)

V £ C[R . x Rn, R + ], V(t, x) is locally Lipschitzian h-positive definite, h0-decrescent D + V(t,x)<0,

and (t,x)ES(h,p);

in x,

Basic Theory

24

(Hi) W £ C[R + xRn,R + ], W(t, x) is locally Lipschitzian in x, W(t,x) < N for (t,x) e S(h,p) and D + W(t, x)< - C(V(t, x)), (t, x) e S(h, p), C <E 9G; (iv)

there exists a positive constant 7 such that 7 < p and D_h(t,x)<0

ifh(t,x) = ~f,

and h(t,x) is locally Lipschitzian in x for each t. Then the system (1.1.1) is (hQ,h)-uniformly stable and (h,h)uniformly attractive. Proof: (h0, h)-umfoim stability follows immediately from assumptions (i) — (ii). To prove (h, A)-uniform attractivity, we choose (t0,xQ) such that h(t0,x0) < 7 < p so that W(t0,x0) < N. Suppose that there exists a solution x(t) = x(t,t0,x0) of (1.1.1) with h(t0, x0) < 7 and tx > t0 such that h(tux(tj)

= 7 and h(t,x(t)) <j,te

[<0,
Then D _fc(t„x(tx)) = lim_ sup§h(tx + 6, x(tt + 6)) - fc(t„ xfa))] > 0, which contradicts assumption (iv). Thus the set S(h,f) is a positive invariant set of system (1.1.1). Now let e> 0 be given. Choose ^ = ^ ^ 4 - 1 . Then we claim that for any solution x(t) = x(t, tQ, xQ) of (1.1.1) with h(t0, xQ) < 7, there exists a t* G [t0, t0 + T] such that

V(t*,x(t*))
Chapter 1

25

If this is not true, then there would exist a solution x(t) of (1.1.1) with h(t0,xQ) < 7 such that

V(t,x(t))>b(e), te[t0,t0 + T]. It then follows from condition (in) that W(t0 + T, x{t0 + T)) < W(tQ, x0) - J
C(V(s,

x(s)))ds

T < 0,

Since condition (ii) implies that

V(t, x(t)) is nonincreasing, we get

V(t,x(t))t0 Since V(t,x)

+ T.

is /i-positive definite, it then follows that the

system (1.1.1) is (h, ft)-uniformly attractive.

The proof is

therefore complete. The

foregoing

result

indicates

that

stability

and

attractivity could be relative to different sets which shows that working with different measures is a natural way to investigate different concepts than known ones. 1.3

Comparison method. The concept of Lyapunov function together with the

theory of differential inequalities provides a very general comparison principle under much less restrictive assumptions.

Basic Theory

26

In this set up, Lyapunov function may be viewed as a transformation which reduces the study of a given complicated differential system to the study of relatively simpler scalar differential equation. Let us consider the following scalar differential equation tt' = 0(<,u)fu(to) = i*o>O,

(1-3.1)

where g £ C[R+ x R,R] and g(t,0) = 0. Definition 1.3.1: Let i(t) be a solution of (1.3.1) existing on some interval J = [t0, t0 + a), 0 < a < + oo. Then 7(f) is said to be the maximal solution of (1.3.1) if for every solution u(t) = u(t,t0,u0) of (1.3.1) existing on J, the following inequality holds u(<) < 7(0, t € J-

(1-3.2)

A minimal solution is defined similarly by reversing the inequality (1.3.2). We need the following known results for our discussion whose proof may be found in Lakshmikantham and Leela [1]. Lemma 1.3.1: Let g £ C[R +xR,R] and 7(1) = ~f(t, t0, u0) be the maximal solution of (1.3.1) existing on J. Suppose that m £ C[R + , R + ] and

Chapter 1

27 Dm{t) < g(t,m{t)),

where D is any fixed Dini

teJ,

derivative.

Then m(tQ) < u0 implies m(t)
1.3.2:

Let g G C[R+ xR,R]

the minimal

solution

m G C[R

+

+

,R

and p(t) = p(t,t0,u0)

of (1.3.1) existing on J.

Suppose

be that

] and Dm(t) > g(t,m(t)\

teJ.

Then m(t0) > u0 implies m(t) > p(t), t E J. We can now formulate the basic comparison result in terms of Lyapunov function V. Theorem

1.3.1: Let

locally Lipschitzian

V G C[R

+

x Rn,R

+

in x for each t G R+.

] and

V{t,x)

is

Assume further

that

+

the function D V(t, x) satisfies D + V(t,x)

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

where g G C[R+ xR,R]. solution

Let -y(t) = j(t,t0,u0)

of (1.3.1) existing

x(t) = x(t,t0,x0)

on J.

Rn,

(t,x)€R+x

be the maximal

Then for

of (1.1.1) existing on J, V(t0,x0) y(f,i(i))<7(t)(teJ.

(1.3.3)

any

solution

< u0 implies (1.3.4)

Basic

28 Proof:

Let

m(t) = V(t,x(t)),

x(t) = x(t, tQ, x0)

solution of (1.1.1) such that V(t0,x0)

< u0.

Theory

being

Since V(t,x)

a is

locally Lipschitzian in x, we get, by (1.1.4) and (1.3.3), the differential inequality D + m(t) < g(t,m(t)),

m(t0)
teJ

and Lemma 1.3.1 gives the desired result (1.3.4). Corollary g(t,u) = 0,

1.3.1:

If, in Theorem

then

V(t, x(t))

is

1.3.1, we suppose

nonincreasing

in

t

that and

V(t,x(t))
1.3.2: The trivial solution u(t) = 0 of (1.3.1) is said

to be equistable if for any e > 0 and t0 (E R + , there exists a 6 = 6(t0, e) > 0 that is continuous in i 0 for each e such that u0 < 8 implies

u(t, t0, u0) < e,

t > t0,

u(t, t0, u0)

being

any

solution of (1.3.1). Let us now establish some sufficient conditions for the (h0, /instability properties of the differential system (1.1.1).

Chapter 1

29

Theorem 1.3.2: (AQ) h,h0£T (Ax)

Assume

that

and h0 is uniformly finer than h;

V E C[R+

xRn,R

+

], V(t,x)

V is h-positive definite and (A2) (A3)

g e C[R +xR,R]

is locally Lipschitzian

in x,

h0-decrescent;

and g(t, 0) = 0;

+

D V(t, x) < g(t, V(t, x)), (t, x) € S(h, p).

Then, the stability properties

of the trivial solution of (1.3.1)

imply the corresponding (h0,h)-stability Proof:

properties of (1.1.1).

We shall only prove (hQ, /i)-equiasymptotic stability

of (1.1.1).

For this purpose, let us first

prove

(hQ,h)-

equistability. Since V is /i-positive definite, there exists a A £ (0,p] and b £ % such that b{h(t, x)) < V{t, x), (t, x) £ S(h, A). Let 0 < e < A and 10 6 R +

be given and suppose that the

trivial solution of (1.3.1) is equistable.

Then, given 6(e) > 0

and *0 € R +, there exists a function St = S^tg^) u0 < Si implies u(t,t0,u0) where u(t,t0,u0) uQ = V(tQ, x0).

(1.3.5)

such that

< 6(e), t > t0,

is any solution of (1.3.1). Since V is h0-decrescent

(1.3.6) We choose

and hQ is uniformly

finer than h, there exists a A0 > 0 and a function a £ % such that for (tQ, x0) G S(h0, A0),

30

Basic h(to, xo) < A and V(t0, x0) < a(h0(t0, xQ)).

Theory (1.3.7)

It then follows from (1.3.5) that b(h(t0,x0))

< V(t0,xQ)

< a(h0(t0,x0)),

(t0,xQ) 6

S(h0,\Q). (1.3.8)

Choose 8 = 8(t0, e) such that 8 6 (0, A0], a{8) < 8X and let h0(t0,x0)<8.

Then

(1.3.8) shows that

h(t0,x0)<e

since

8X < 6(e). We claim that

h(t, x(t)) < e, t>t0 where

x{t) = x(t, tQ, xQ)

is

whenever h0(t0, x0) < 8, any

solution

of

(1.1.1)

with

hQ(t0, x0) < 8. If this is not true, then there exists a tx > tQ and a solution x(t) of (1.1.1) such that h(tlt x(tx)) = e and h(t, x(t)) <e,tQ
tl%

(1.3.9)

in view of the fact that h(t0, xQ) < e whenever hQ(t0, xQ) < 8. This means that (t, x(t)) € S(h, A) for [t0, i x ] and hence by Theorem 1.3.1, we have V(t, x(t)) < r(t, 1^, u 0 ), tQ < t < tu where r(t,t0,u0)

is the maximal solution of (1.3.1).

relations (1.3.5), (1.3.6), (1.3.9) and (1.3.10) yield b(e)
(1.3.10) Now the

Chapter 1

31

Suppose next that the trivial solution of (1.3.1) is equiattractive.

From the (/i0, /^-stability, we set e = A so that

^o — £(*o> ^)- Now let 0 < 77 < A. Then, by equi-attractivity of (1.3.1), we have that, given 6(77) > 0 and t0eR

+

, there exist

positive numbers 6^ = S^(t0) and T = T(t0, rj) > 0 such that u 0 < 61 implies u(t, <0, u0) < b(r)), t>t0

+ T.

(1.3.11)

Choosing u0 = V(t0, xQ) as before, we find a 8Q = £2(*o) > 0 such that 5* G (0, A0] and O(SQ) < 8\.

Let 60 = m i n ( ^ , #o)

an

d

/i 0 (t 0 , x 0 ) < <50. This implies that h(t, x(t)) < A, t > i 0 and hence the estimate (1.3.10) is valid for all i > t0.

Suppose now that

there exists a sequence {tk}, ijt ^ *o + ^ \ tk—>°° that rj
as

^—*°° such

where x(t) is any solution of (1.1.1) such

that hQ(tQ, x0) < 6Q. This leads to a contradiction b(rj) < V(tk,x(tk))

< r(tk,t0,uQ)

because of (1.3.10) and (1.3.11).

< b(rj)

Hence the system (1-1.1) is

(h0, /z)-equiasymptotically stable and the proof is complete. CoroUarv 1.3.2:

In Theorem 1.3.2,

(z)

g(t, u) = 0 is admissible to yield (h0, h)-uniform

(ii)

g(t, u) = X(t)u, A E C[R + , R], is admissible to yield (a)

(hQ, h)-equistability

stability;

if J \(s)ds < 00 for every t0 > 0,

Basic

32 (b)

(h0,h)-equiasymptotic

stability

if

Theory

f X(s)ds = — oo

for every t0 > 0; (Hi)

g(t, u) — —
Proof:

to yield (h0, h)-

stability.

The proof of (i) is immediate.

observe that the solutions u(t,t0,u0)

Concerning (ii), we

of (1.3.1) are given by

t

fx(s)ds u(t, t0, u0) = u0e° ,t>

tQ.

OO

If (a) holds, then letting N(t0) = JX(s)ds, we get u(t,t0,uQ) < *o uQe ° . As a result, for any e > 0, we have, choosing S(tQ) <ee~

° , u(t,t0,u0)

< e, t > t0.

Hence the null solution

of (1.3.1) is equistable, which, in turn, implies that system (1.1.1) is (h0, /i)-equistable. t

In case (b) holds, e °

is bounded for a l H > t0, from

which the equistability of the trivial solution of (1.3.1) follows oo

immediately. Since / \(s)ds = — oo, we have JX(s)ds lim u(t, t0, u0) = uQlim e ° =0. t-»oo

v

°

u/

u

t-»oo

Thus the trivial solution u = 0 of (1.3.1) is equiasymptotically stable, which implies, by Theorem 1.3.2, that system (1.1.1) is

Chapter 1

33

(h0, /i)-equiasymptotically stable. To prove (Hi), it is enough to show that the trivial solution u = 0 of (1.3.1) is uniformly asymptotically stable. It is evident that the trivial solution of (1.3.1) is uniformly stable. Let e > 0 be given. Define J(u) = J ^ if / ^ < oo, and J(u) = J-^rr for some small constant 6 > 0 if f-4*r = oo. Then we see that the solutions u(t,tQ,u0) of (1.3.1) are given by u(t,t0,u0) = J~a[J(u0) -(t-

to)], t > t0,

where J - 1 is the inverse function of J. Let u(t,t0,u0) be solutions of (1.3.1) with u0 < a, a > 0, and choose T > 0 such that T > J(a) - J(e). It then follows u(t, t0, u0) <e,t>t0

+ T.

Thus the trivial solution u = 0 of (1.3.1) is uniformly asymptotically stable. We have assumed in Theorem 1.3.2 stronger requirements on V, h, h0 only to unify all the stability criteria in one theorem. This obviously puts burden on the comparison equation (1.3.1). However, to obtain only nonuniform stability criteria, we could weaken certain assumptions of Theorem 1.3.2 as in the next result. The details of proof are omitted.

Basic Theory

34

Theorem 1.3.3: Assume that conditions (AQ) — (A3) hold with the following changes: (i) h0, h £ T0 and h0 is finer than h and (ii) V(t, x) is h0weakly decrescent. Then, the uniform or non-uniform stability properties of the trivial solution of (1.3.1) imply the corresponding non-uniform (h0,h)-stability properties 0/(1.1.1).

Recall that we have proved, in Theorem 1.2.5, (h0,h)uniform stability under much weaker conditions by employing a one parameter family of Lyapunov functions. We shall next give a general result on (h0, h)-uniform stability, using comparison principle, in the spirit of Theorem 1.2.5. Theorem 1.3-4: Assume that (i) hQ) h G T and h0 is uniformly finer than h; (ii) for every n>0, there exists a function Vn € C[S(h, p) D ^c(^o>v),R +) such that V^x) x and satisfies

is locally Lipschitzian in

Hh(t,x))
(t,x)<=S(h,p)nS%h0,r1)) where a,b 6 3G and D+

(1.3.12)

V^Kg&V^x)),

(t,x)es(h,p)ns°(h0,v),

(1.3.13)

Chapter 1

35

where g e C[R +xR Then the uniform (1.3.1)

imply

+

,R] and g(t,0) = 0.

stability properties

the

of the trivial solution of

corresponding

(hQ,h)-uniform

stability

(h0, h)- uniform

stability.

properties can be proved

similarly.

properties of (1.1.1). Proof: Other

We

shall

(h0, /i)-uniform

only

prove

Since hQ is uniformly finer than h, there exists a function ip 6 % and a constant 80 > 0 such that h(t, x) <
(1.3.14)

Let 0 < e < p be given and suppose that the trivial solution of (1.3.1) is uniformly stable.

Then for 6(e) > 0 there exists a

constant 8X = S^e) > 0 such that u0 < 6X implies u(t,t0,u0)

< 6(e), t > t0,

(1.3.15)

where u(i, £ 0 ,u 0 ) is any solution of (1.3.1). Because of the assumptions on ip and a, we can find a constant 8 € (0,60] such that
< 8 and x(t) = x(t,t0,xQ)

(1.3.16) be any solution

It is easy to see from (1.3.14) and (1.3.16) that If (hQ, /i)-uniform stability of (1.1.1) does not

hold, then there exists a solution x(t) = x(t,t0,x0)

of (1.1.1) tu

Basic Theory

36 t2 > t0 such that V i . s ( ' i ) ) = *» M*2i»('a)) = (t,x(t)) 6 S(h,e)r\Sc(h0,6)

e and

for i € [tut2].

(1.3.17)

Hence, setting TJ = 6 and using Theorem 1.3.1, we obtain V^(*,a:(*))<7(*»*i,«o)»*€[*i»*2]. where j{t,tuu0)

is the maximal solution of (1.3.1) through

(*i,«o) and u0 = V r r/ (i 1 ,x(t 1 ))<

a(fco(*i»*(*i))) < *i-

Jt

then

follows from (1.3.12), (1.3.15) and (1.3.17) that 6(c) = 6(A(*2,x(«a))) < V„(*a, «(<„)) < 7(*2.*i,«o) < 6 (0, which is a contradiction.

Thus the system (1.1-1) is (h0,h)-

uniformly stable and the proof is complete. Remark 1.3.1: The function g(t, u) = 0 is admissible in Theorem 1.3.4 to yield (h0, /i)-uniform stability and in this case Theorem 1.3.4 reduces to Theorem 1.2.5. As we have seen, the use of comparison principle provides a unified approach and generalizes several stability results into one framework. However, a direct analysis of the right-hand side of the comparison equation can sometimes yield sharper results. This can be seen in the following theorem.

Chapter 1

37

Theorem 1.3.5: Assume that (i) h0, h G T0 and h0 is uniformly finer than h; (ii) V G [R + x Rn, R + ], V(t, x) is h-positive definite, h0decrescent, locally Lipschitzian in x and D + V(t,x) < g{t,V{t,x)), (t,x) G S(h,p), where g G C[R + x R, R]; (iii) for every pair of numbers a, (3 such that 0 < a < ft, there exists constant 9 = 0(a, (3) > 0 satisfying g(t,u)<0, (iv)

hoEC^R+xR"^^ XeC[R + ,R + ],

a9; and

for

some

function

ftKfo *) + £ho(t, x) ■ f(t, x) < X(t)h0(t, x), (t, x) G S{h, p).

Then the system (1.1.1) is (h0,h)-uniformly stable. Proof:

By condition (i), there exist constant o~0 and

function

(1.3.19)

is /i-positive definite and /&0-decrescent, there

exist positive constants p0< p and ax < a0, and functions a, b G 3G such that

38

Basic Theory V{t, x) < a(h0(t, x)), if h0{t, x) < au

(1.3.20)

V(t, x) > b(h(t, x)), if h(t, x) < p0.

(1.3.21)

Let 0 < e < p be given. Choose St = £x(e) < ^ such that a(^) < 6(e) and
and

N = N(9) = sup

(1.3.22) X(t).

Choose

o< t <0 Ne

5 = 61e- . For (t0,x0) e S(h,p) such that /i0(*0, x0) < 6, let x(*) = x(i, *0, x0) be any solution of (1.1.1). Then (1.3.19) and (1.3.22) implies h(t0,x(t0)) < e. Defining m{t) = h0(t, x(t)), we obtain, by condition (iv) m'(t) < X(t)m(t), which implies, by the Gronwall's inequality, t

h0(t, x(t)) < h0(tQ, x0)exp[ I X(s)ds],

(1.3.23)

as far as h(t,x(t)) < p. By the definition of 5 and (1.3.23), we see that ho(t,x(t)) <6\,t0
if h0(t0,x0) < S

which implies from (1.3.21) that

V(t,x(t))tl> satisfying V(t1,x(t1)) = a(61),V(t2,x(t2))

= b(e),

9,

Chapter 1

39 a{5x) < V(t,x(t))

< 6( e ), t e [tut2].

(1.3.24)

Hence at t = tlr these results D + V^xit^^O.

(1.3.25)

On the other hand, as tx > 9 and (1.3.24) holds, we obtain, from condition (Hi), the inequality D+

V(t1,x(t1))
which contradicts (1.3.25). for

t > 6.

V(t,x(t))

It

therefore

This proves that V(t,x(t)) follows

that,

if

< 6(e)

h0(t0,xQ) < 6,

< 6(e), t > tQ, and consequently, in view of (1.3.21),

the (h0, /i)-uniform stability of the system (1.1.1) is proved. 1.4

Converse theorem. The importance of uniform asymptotic stability in the

investigation of stability properties of perturbed equations needs no emphasis.

differential

The converse theorem of

Massera which results from uniform asymptotic stability of the origin has been widely utilized in perturbation theory. A converse theorem of Massera type for uniform

asymptotic

stability in terms of two measures poses difficulties due to the interplay between two measures and hence it is not possible to construct a smooth Lyapunov function unless we pay a price. In this section, we shall prove such a converse theorem and we see that the price we have to pay to overcome the difficulty is reasonable.

Basic

40

Theory

Let us recall the following modification of a result of Massera which is useful in our discussion. Lemma

I.4.I:

Let /3 £ I and p. > 0.

Then, there exists a

function a £ 3G such that a(/3(s)) < exp( — (is),

s>0.

We shall assume that h0, h 6 T and h0 is finer than h, (h0,h) being the two measures discussed in Section 1.1. Recall that this means that there exists a A > 0 and a ip £ 9G such that h0(t,x) < A implies h(t,x) < (p(h0(t,x)). We are now in a position to prove our main result of this section. Theorem I.4.I: (i)

I h(t, x) - h{t, y)\
(ii) Then,

Suppose that for (t,x),

and || f{t, x) - f{t, y) \\ <

(t,y)£R+xRn,

(1.1.1)-is (hQ,h)-uniformly

uniformly

attractive.

a constant

p>0

there

exist

W £ C[S(h, p), R + ] which are Lipschitzian (a)

M>0

stable and

(h,h)-

constants;

the system for

where L,

U(t,x)>b(h{t,x)) a(h0(t,x))

for

for

two functions in x such that

(t,x)(=S(h,p)

(t,x) £ S(h0,p0),

U,

where

p0 £ (0, p) is a constant with
and a,

U(t,x)< b £ 3G and

Chapter 1

41

(6)

D + U(t, x) < 0 /or (*, x) e S(fc, p);

(c)

W(f,x)<JV /or ( i , i ) 6 5(/i,/)) and W(t,x) 0 is a constant; D + W(t,x)< -C(U{t,x)) for (t,x) £ S(h,p), where

(d)

ce%. Proof: We shall use some standard arguments. Choose a constant p > 0 and for any v > 0, a T = T(v) > 0, both associated with (h, /i)-uniform attractivity assumed in (ii). Obviously, the function T can be assumed to be decreasing. For (t,x) G S(h,p) and j = 1,2,..., define Uj(t, x) = sup{Gj(h(t + 9, x(t + 9, t, x))): e>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)\

<

\u-v\.

Because of (h, /i)-uniform attractivity and the continuity of Gj and h, Uj is well defined by (1.4.1) has a mapping from S(h,p) into R + . We have, Uj{t, x) = supiGjihit + 9, x(t + 6, t,»))): 0 < 9 < T(j ~ l)}exp[ - MT(j ~ %

(1.4.2)

Basic Theory

42

from which it easily follows that Uj is continuous in t. Moreover, by (1.4.2), (i) and Gronwall's inequality, we get \Uj(t,x)-Uj(t,y)\ < sup{ | h(t + 9, x(t + 6, t, x)) - h(t + 9, x(t + 9, t, y)) \: 0<9
a

}exp[ - MT(j ~x)]

(1.4.3)

< Lsup{ || x(t + 9, t, x) - x(t + 9, t, y) \\: 0<9
x

)}exp\ - MT(j " a )]

Thus, Uj G C[S{h, p),R + ]. Now, we set, for (t, x) G S(h, p), U(tfx)=i£2-iUJ{ttx).

(1.4.4)

i =i

Taking into account the decreasing character of T(u), it can be easily seen that Uj(t, x) < sup{h(t + 9, x(t + 9, t, x)): 0<9
*)}

< 1 + sup{h(t + 9,x{t + 0,i,x)):O < 9 < T(l)}, j = 1,2,..., which implies the uniform convergence of the series of (1.4.4) in any compact subset of S(h, p). Then U G C[S(h, p), R + ], By (1.4.3) we have \U(t,x)-U(t,y)\ < I | | x - y | | .

' (1.4.5)

Chapter 1

43

As a consequence of (1.4.1) and (1.4.4) we see that U is decreasing along the solutions of (1.1.1). From this and (1.4.5), it is easy to obtain the inequality D + U(t,x) < 0, for (*,i) £

S(h,p),

following the standard arguments. Now, given j

- 1

a £ (0,/>), 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 <5(e) > 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)) <
I W(t, x) - W(t, j/) | < LI C'W

-1)) || x(9, t, x) - x(9, t, y) \\
t oo


jexp[(9-t){M-n)}d9 t CO

Thus

< L || x - y || fexp[{M - n)9)d9. o |W(t,s)-W(*,y)| < L | | x - j / | | .

(1.4.12)

Here we have employed Gronwall's inequality, the relations (1.4.9), (1.4.11) and the choice fi>M + l.

Clearly W is

Basic

46

Theory

continuous in t and this fact, together with (1.4.12) proves that WeC[S(h,p),R + \. Because

of

(1.4.8)

and

(1.4.10),

we

get,

for

(t,x)eS(hQ,p0), oo

W(t,x)< Jc(p(ho(t,x))q(9-t))d0 t

< [C(p(h0(t, *)9(o))]1/2 / [C(P(p0)q(e))}^de o

sbM^x)), Now,

it

is

easy

to

bte%.

show,

using

(1.4.12)

that

D + W(t, x)= - c(U(t, x)) for (t, x) G S(h, p). The two functions U and W have all the desired properties. The proof is therefore complete. Conversely, it can be easily proved that the system (1.1.1) is (h0, /i)-uniformly

stable and (h,

ft)-quasiuniformly

asymptotically stable if the following conditions are satisfied: (i)

h0 is uniformly finer than h\

(ii)

for

every

p > 0,

every

solution

x(t,t0,xQ)

with

h(tQ, x0) < p exists for all t > t0; and (Hi)

there exist two functions U, W G C[S(h, p), R . ], which are locally Lipschitzian in x and satisfy the conditions (a), (6), (c) and (d) of Theorem 1.4.1. We have the following Corollary of Theorem 1.4.1.

Chapter 1 Corollary

47 1.4-1: Suppose

Theorem 1.4-1 hold. function

that the assumptions

(i), (ii) of

Then, for a constant p> 0, there exists a

V £ C[S(h, p), R + ], which is Lipschitzian

in x for a

constant and such that (a)

V(t,x)>b(h(t,x)) a(hQ(t,x))

for

(t,x)eS(h,p)

for (t,x) e S(h0,p0),

and

where a, b£%,

V(t,x) p0£

< (0,p)

is a constant with fp(p0) < p: (6)

D + V(t, x)<

- j{h(t, x)) for {t, x) 6 S(h, p), where 7 <E %.

Indeed, if U and W Theorem

are the functions obtained in

1.4.1, the function

V — U-\-W

has the desired

properties with a = b + bt, 7 = c. If h(t, x) = h0(t, x), Theorem 1.4.1 and its corollary become

two

equivalent

propositions.

Thus,

when

h(t, x) = h0(t, x) = || x || , Theorem 1.4.1 reduces to the wellknown Massera's converse theorem on uniform asymptotic stability (actually, in Massera's theorem further about smoothness of V are h0(t, x) = || x || , where II x II $ — V x\ H

|| • ||

made).

If h(t,x)=

assertions || x \\ 3, and

is the Euclidian norm

and

l"x*> 5 < n, then Theorem 1.4.1 yields a

converse theorem for partial uniform asymptotic stability.

It

is clear that various choices of h and h0 are possible and thus Theorem 1.4.1 offers a unified result that is flexible enough to warrant its use in several applications.

Basic

48

Theory

If we carefully examine the proof of Massera's theorem, we notice that it is the domain of attraction which plays the prominent role in obtaining a smooth Lyapunov

function.

Consequently, when two measures are employed, this same feature shows that the price we had to pay to prove Theorem 1.4.1 is reasonable and natural. 1.5

Boundedness and Lagrange stability. Corresponding

to

the

different

types

of

stability

notions, there are different types of boundedness concepts which we shall define below. Definition

1.5.1: Let h0, h £ T.

Then the differential system

(1.1.1) is said to be ( 5 j ) (h0, /i)-equibounded if for each a > 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) <

a

implies h(t, x(t)) t0

+ T;

(2?4) (A 0 ,/i)-quasi-uniform-ultimately bounded if T in (B3) is independent of t0; (B 5 ) (h0, /i)-equi-ultimately bounded if (Bj) and (B3)

hold

together; (B 6 ) (/i 0 ,/i)-uniformly ultimately bounded if (B2)

and (f?4)

hold together; (B7)

(h0, A)-equi-Lagrange

stable

if

(B x )

and

(£7)

hold

together; (2?8) (/i 0 ,/i)-uniformly Lagrange stable if (B2) and (Sg) hold together. Corresponding

to

Definition

1.5.1,

we

need

the

boundedness definition for the comparison equation (1.3.1). We merely state one of the concepts. (Bi)

The comparison equation (1.3.1) is said to be equibounded, if for any a > 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,(t,x)eR+xRn. Then the system (1.1.1) is (h0,h)-equibounded. Proof: Let a > 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))<(3,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)),

V £ C[S (h0, p), R + ], V{t,x)

is locally Lipschitzian

there exist functions a £%, q E C[R + , R + ] such a(h(t,x))

< V(t,x)

< q(h0(t,x)),

in x,

that

(t,x) G Sc(h0,p),

(1.5.4)

where 0(7)—»oo as 7—*oo; (in)

D + V(t, x) < 0, (t, x) e 5 c (/i 0 , />)•

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)<

Since

of

that

Then the system (1.1.1) is (h0,h)-uniform-ultimately Proof:

(ii)

the

system

(1.1-1)

is

bounded.

(h0, h)- uniformly

bounded, then there exists a positive number N such that ^o^o^o) < P implies h(t,x(t))

< N,t>

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

0
(1.5.8)

Chapter 1

53

Thus, in view of (1.5.7), h0(t0, x0) < a implies h(t,x(t)) t>tQ

< N,

+ T, where T depends only on a. Therefore, the system

(1.1.1) is (h0, /i)-uniform-ultimately bounded. Theorem,

1.5-4: Assume

that

conditions

Theorem 1.5.1 hold. Suppose further (Hi)*

D + V(t,x)

< - C(h(t,x)),

(iv)

heC1[R+xRn,R

+

bounded on

(i)

and

of

that Rn;

(t,x)eR+x

] and for

any p>0,

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) <
(ii)

V G C[R

+

h-positive

x R", R + ], V(t, x) is locally Lipschitzian definite

and h0-decrescent

with the

in x, function

b G % occurring in the definition 1.1.3 satisfying b(u)—*oo as u—*oo; (Hi)

g e C[R +xR,R] D+

and for (t,x)eR+x V(t,x)
Rn

Basic

54 Then boundedness imply

the

and Lagrange stability properties

corresponding

(h0,h)-boundedness

Theory

of (1.3.1)

and

Lagrange

stability properties of (1.1.1). Proof:

We

shall

only

indicate

the

proof

of

(h0,h)-

boundedness of (1.1.1) since the proof of other concepts are similar.

The

relations

(1.3.5)

and

(1.3.7)

_1

A0 =
+

.

hold

with

Let 0 < a < A0 and

Set al = a(a) and suppose that (1.3.1) is bounded.

Then, given ctx > 0 and tQ € 72 + , there exists a 01 =

Pi(t0,a)

such that u 0 < o^ implies u(t,t0,u0) where u(t,t0,u0)

< /3lt t > t0,

is any solution of (1.3.1). Choose (3 = /?(< 0 ,a)

such that b(/3) > $x and let ^(^o^o) < athat h(t0, xQ) < (3. x(t) = x(t,tQ,xQ)

(1.5.9)

Then (1.3.7) implies

If possible, let there exist a solution

of (1.1.1) and tx > t0 such that

h(tltx(*i))

= P and h{t,x{t)) <0,t€

Then by Theorem 1.3.1, we get, setting u0 = V(t,x(t))

[t^].

(1.5.10)

V(tQ,xQ),

< 7(Mo»«o)» * € [*0,*i],

(1-5.11)

where 7(£,uo) is the maximal solution of (1.3.1). Hence, the relation (1.3.5), (1.5.9) and (1.5.10) lead to the contradiction b{0) < Vfo,*(*,)) < 7(*i»«o»Uo) < &(/?),

Chapter 1

55

proving the theorem. 1.6

Practical stability. In the stabilization of nonlinear systems, interesting set

of problems deals with bringing state close to certain sets rather than to a particular state.

From a practical point of

view, a concrete system will be considered stable if the derivation of the motions from the equilibrium remain within certain bounds determined by the physical situation, in case the initial values and/or the disturbances are bounded by suitable constrains.

The desired state of a system may be

mathematically unstable and yet the system may oscillate sufficiently near this state and its performance is acceptable. For example, an aircraft or a missile may oscillate around a mathematically unstable course yet its performance may be satisfied with the predefined requirements.

Many problems

fall into this category including the travel of a space vehicle between two points and the problem, in a chemical process, of keeping the temperature within certain bounds. To deal with such situations, the notion of practical stability is more useful, which we define in a general set up below.

Basic

56 Definition

1.6.1: Let h0, heT.

Theory

Then the system (1.1.1) is said

to be (P5a)

(h0, /i)-practically 0 < A < A, h(t,x(t))

we

< A,

if,

have

t>t0,

x(i) = x(t,t0,xQ) (PS2)

stable for

given h0(t0,x0)

some

(A, A) <X

tQeR

with implies

+,

where

is any solution of (1.1.1);

{h0, /i)-uniformly practically stable if (PSJ

holds for

every t0 G R +; (PS3)

(/i 0 ,/i)-practically quasi-stable if given

(A,B,T)>0

and some t0 6 R + , we have h0(t0, x0) < A implies

h(t,x{t))
t>t0 + T;

(h0, /i)-uniformly

practically

quasi-stable

if

(PS3)

holds for all t0 £ 72 + ; (PS5)

(h0, /i)-strongly practically stable if (PSj)

and ( P 5 3 )

hold simultaneously; (PS6)

(h0, /i)-strongly uniformly practically stable if

(PS2)

and ( P ^ ) hold together; (PSy)

(/i 0 ,/i)-practically asymptotically stable if (P5 X ) and (£9) hold simultaneously with a = A;

(P£8)

(/i 0 ,/i)-equi-asymptotically practically stable if

(PS^

and (5 10 ) hold together with a = A; {PS9)

(h0, /i)-uniformly asymptotically practically stable if (PS2) and ( S n ) hold at the same time with a = A;

(PS 1 0 )

(h0, /i)-practically unstable if (PS^) does not hold;

Chapter 1 (-P'S'ii)

57

(^0> ^)-eventually practically stable if given (A, A), 0 < A < A,

there

exists

a

x

^o(*07 o) < ^ implies h(t,x(t)) (PS12)

r = r(A, A) <

such

that

A,t>t0>T;

(h0, /i)-eventually strongly practically stable if (-P5 n ) and (PS3) hold together;

(P5 1 3 )

(/i 0 ,/i)-eventually stable

if

uniformly

{PSU)

and

strongly (PS4)

are

practically satisfied

simultaneously. In (PSS) and ( P 5 6 ) , if 0 < B < A < A, then we say that the system (1.1.1) is (h0, /i)-contractively practically stable, while for 0 < A < B < A, the system is then said to be (h0,h)expansively practically stable. Sometimes, in physical problems, one is interested in the behavior of systems within specified bounds during a fixed time interval.

The concept of "finite time stability" is

appropriate to cover such situations. For example, the notion (PS5) would be translated into the following: the differential system (1.1.1) is (h0, /i)-strongly practically stable if, given positive numbers A, A, B and T we have h0(t0,x0)

< A implies h(t,x(t)) < A, t0
and h(t0 + T,x(t0 + T))

Basic Theory

58

The notion of practical stability is neither weaker nor stronger than Lyapunov stability. As is well-known, sometimes even asymptotic stability, in the sense of Lyapunov, is not sufficient in practice since the domain of attraction may not be large enough to allow the desired deviations to cancel out. As a result, the system may be asymptotically stable in theory but is actually unstable in practice. To illustrate the notions of practical stability, we consider some examples. Example 1.6.1:

Consider the system of equations

x' = n(t)y + m(t)x(x2 + y2), y'= - n(t)x + m{t)y{x2 + y2), where n, m G C[R + , R]. given by

x(t0) = x0, y(t0) = y0,

(1.6.1)

The general solution of (1.6.1) is

x0cos\ / n(s)ds + y0sin

/ n(s)d

x(t)

-2(4 + yl) Jm(s)di

Chapter 1

59

y 0 coa

/ n(s)ds

— x0siru

/ n(s)ds

m=2(x20 +

y20)jm(s)dsy

which reduces to -l 2 7

( t ) = x\t) + y\t) = 1 U l -

2
where 7J5 = XQ + J/Q. Suppose that Jm(s)ds get from (1.6.2)

(1.6.2)

= /? > 0. Then, we

*° Umol2(t) = ll(l-2^)-\

Let h0 = h = || x || and .A = 2A.

(1.6.3)

It then follows from (1.6.3)

that the system (1.6.1) is (h0, /i)-practically stable if /? <

^

8A"

and (/i0, /i)-practically unstable if /? > - ^ . 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) <
V € C[R

+

x R", R + ], V(t, x) is locally Lipschitzian

in x

and satisfies b(h(t, x)) < V(t, x), if h(t, x)
inequality

(1.6.6) (1.6.7)

Basic Theory

62

D + V(t, x) < g(t, V(t, x)), (t, x) G S(h, A),

(1.6.8)

where g G C[R + x i2 + , R\; (iv) (p(\) < A and a(X) < b(A) hold. Then the practical stability properties of the equation (1.3.1) imply the corresponding (h0,h)-practical stability properties of the system (1.1.1). Proof: Suppose that the equation (1.3.1) is practically stable with respect to (a(\),b(A)) so that we have u0 < a(X) implies u(t,t0,u0) < b(A), t > t0.

(1.6.9)

Then we claim that the system (1.1.1) is (h0, /i)-practically stable with respect to (A, A). If this is not true, there would exist a tl>tQ and a solution x(t) = x(t,t0,x0) of (1-1.1) such that ^o^o^o) < ^> ^(
tv

(1.6.10)

In view of (ii) and (iv), we have fc(*o>3o) ^
o = V(tQ, x0).

tlt

(1.6.11)

is the maximal solution of (1.3.1) such that It now follows from the relations (1.6.6)-

Chapter 1

63

(1.6.11) that b(A) = Khituxfa)) since u0 = V{t0,x0)

< V(tux(ti))

< l(h,to,V(t0,x0))

< b(A)

< a(hQ(t0,x0)) < a(X).

This is a contradiction.

Thus the system (1.1.1) is

(h0, /i)-practically stable. We shall next prove that (1.1.1) is (h0, /i)-strongly practically stable for (A, A,B,T) suppose

that

(1.3.1)

is

(a(A), b(A), b(B), T) > 0. only (A0, /^-practical

> 0.

strongly

To prove this, let us practically

stable

for

This means that we need to prove quasi-stability

of the system (1.1.1).

Since (1.3.1) is practically quasi-stable, we have u0 < a(X) implies u(t, t0, u0) < b(B) for t > tQ -f T, (1.6.12) where u(t,t0,u0)

is any solution of (1.3.1).

Suppose that

h0(t0, xQ) < X so that we have h(t, x(t)) < A for t > t0, because of the practical stability of (1.3.1). As a result, it follows that the estimate (1.6.11) is true for all t > t0, that is, !/(*,*(*)) < 7 ( i , i 0 , " o ) , t>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
(h0,h)-

t

> 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
Thus, the equation (1.3.1) is eventually practically

stable. Hence Theorem 1.6.1 implies the desired properties.

Chapter 1

65

To obtain only (h0, /i)-nonuniform practical stability criteria, we could weaken certain assumptions of Theorem 1.6.1 as in the next result, which we merely state. Theorem 1.6.2:

Assume that conditions (i) — (iv) of

Theorem

1.6.1 hold with the following changes in (ii) — (iv) (ii*)

h0,

h 6 T0

ip(t,h0(t,x)),

and

h0

is finer

tp e C%, ifh0(t,x)

that

V(t, x) < a(t,h0(t,x)),

(iv*)

a(t0, X) < b(A) for some t0 £ R + .

(1.3.1) imply

i.e.

h(t,x)<

< X;

(in*)

Then uniform

h,

a £ C%, provided h0(t,x) < X;

or nonuniform

practical stability properties

the corresponding

(hQ,h)-nonuniform

of

practical

stability properties of the system (1.1.1). As before, if we desire only uniform properties of practical stability of the system (1.1.1), we can relax the assumptions of Theorem 1.6.1 by employing Lyapunov-like functions which satisfy less restrictive conditions. Theorem 1.6.3: (i) (ii)

Assume

that

0<X
h£T

and

h(t,x) <

whenever

h0(t, x) < X; (Hi)

VeC[S(h,A)C\Sc(h0,X),R Lipschitzian

+

],

V(t,x)

is

in x and for (t, x) 6 S(h, A) 0 S°(h0, X),

locally

Basic

66 b(h(t, x)) < V(t, x) < a(h0{t, x)), a,b£%,

(1.6.14)

D + V(t,x)
+

Theory

(1.6.15)

x R +, R].

(f(X) < A and a(X) < b(A) hold.

Then uniform

practical stability of (1.3.1) implies the

(h0,h)-

uniform practical stability of the system (1.1.1). Proof:

Suppose that the equation (1.3.1) is uniformly

practically stable. Then, given (a(\),b(A)), uQ < a(X) implies u(t,t0,uQ) for

all

h(t,x(t))

t0 (E R +.

Let

< A, t> t0.

we have

< b(A), t > tQ,

h0(t0, xQ) < X.

We

(1.6.16)

claim

that

If this is not true, then, in view of (ii)

and (iv), there would exist a solution x(t) = x(t, <0,x0) of (1.1.1) and t2 > tx > t0 such that

K^iMU)) = ^ Kh^ih)) = A, and (*, x(t)) 6 S{h, A) D 5 c (/i 0 , A), t £ [tu t2).

(1.6.17)

By Theorem 1.3.1, we now have

v(t,x(t)) < i{trh,v(hAh)), h
(1.6.18)

where 7(2,£ 1 ,u 0 ) is the maximal solution of (1.3.1) with 7(*i,ti,Uo) = u 0 .

In view of assumption (m), the relations

(1.6.16) — (1.6.18), we are led to the contradiction b(A) < V(t2,x(t2))

< 7 ( ^ ^ , a ( A ) ) < b(A),

Chapter 1

67

which proves the (/i0, /i)-uniform practical stability of system (1.1.1). Corollary

1.6.2: The function

g(t, u) = 0 is admissible

Theorem 1.6.3 to yield (h0,h)-uniform

in

practical stability of the

system (1.1.1). Theorem 1.6.4'- Assume of

Theorem

1.6.3

0 < n < A, c

S {h0lrj),R

hold.

there +},

that the hypotheses (i), (ii) and (iv)

exists

VJtyX)

Suppose

further

firi>0

and

is locally Lipschitzian

that for

each

V^ 6 C[S(h, A) (1 in x and for a,

be% b{h(t,x))
the

system

asymptotically Proof:

-iin,

(1.6.19)

(t,x)eS(h,A)nSc(h0,r]).

(1.1.1)

is

(h0,h)-uniformly

(1.6.20) practically

stable.

Since (i), (ii) and (iv) of Theorem 1.6.3 hold,

taking n — A and g(t, u) = 0, we get (h0, /&)-uniform practical stability of the system (1.1.1), by Corollary 1.6.2. 0 < e < A and t0 G R + be given.

Let

Choose a 8 = 8(e) > 0 such

that a(8) < b(e) and (p(8) < e. Then, under the assumptions, it easily follows that h0(t0, x0) < 8(e) implies /i(i, x(t)) < e, t > t0.

Basic

68

Theory

Now, let us show that there exists a f £ [t0, t0 + T], where

T = &P±,

x(t) = x(t,t0,x0)

rj = 6(e), such that h0(t*,x(t*)) < 8, where

is any solution of (1.1.1) with h0(t0,x0)

Suppose that this is not true. t0
+ T.

h0(t0,x0)

Since we have h(t,x(t))

< ^

Then 8 < h0(t, x(t)) for < A, t>tQ

whenever

< A, we get from (1.6.20), Vn(t0 + T, x(tQ + T)) < Vn(t0, x0) - /i„r, r, = 8(e),

and consequently we arrive, from (1.6.19), at the contradiction

Thus, h0(t0,x0)<\

implies h(t,x(t))<e,

t>t0

+ T and the

proof of the theorem is complete. 1.7

(ft 0 ,/i,M 0 )-stability. Since in many concrete problems such as adaptive

control systems, one needs to consider the stability of sets which are not invariant, the notion of eventual stability was introduced to deal with such situations.

It is subsequently

recognized that although the set which is eventually stable is not invariant in the usual sense, it is so in the asymptotic sense.

This observation leads to a new concept of asymp­

totically invariant sets, which form a special subclass of invariant sets. A natural generalization of the above concepts

Chapter 1

69

is the notion of M 0 -stability, which

describes a very general

type of invariant set and its stability behavior.

In this

section, we introduce a very general type of stability called (h0,h,M0)-stability

by combining M 0 -stability and

(hQ,h)-

stability. This notion naturally leads us to consider the initial values on surfaces that critically depend on initial time and to use different topologies in the definitions of stability. Consider the generalized initial value problem x> = f(t, x), x(t0) = 0( 0, where / ,

i[> £ C[R

+

x Rn,Rn]

and /

(1.7.1)

is smooth enough to

ensure existence of solutions of (1.7.1).

For convenience, let

us introduce the following notation: M = M(R

+, n

R . to R

R") is the space of all measurable mappings from such that x G M if and only if x(t) is locally

integrable on R + and «+i

sup / || x(s) || ds < oo; t >oJ MQ = M0(R

+,

Rn) is the subspace of M(R

+,

R") consisting of

all x(t) such that

/ || x(s) || ds—>0 as t—too; t

the set S(MQ,e) is the subset of M(R

+

,Rn) defined by

Basic

70

Theory

t+ i

S(MQ,e) = {x£ M:\im

sup f \\ x(s) \\ ds < e}. t

By x € S(M0, e), we mean that for each e > 0, there exists a r(e) > 0 with the property that T(C)—»OO as e—»0 such that t+ i

j

|| z( 5 ) || ds < e, t > r(e). t

Let us now give the definitions for (h0, h, M 0 )-invariant set and the various types of (h0, h, M0)-stability. x(t) = x(t,S,I/J(S,x*)),

t>s

represent

As usual, let

a solution of

(1.7.1)

starting at (s, xf>(s, x*)). Definition invariant

1.7.1:Let with

AcR",

respect

h0, heT.

to system

A is

(1.7.1)

(hQ,h,MQ)-

if x* £ A

and

h0(s, V>(s, x*)) G M 0 , we have h(t, s, ^>(s, x*)) € MQ. Definition

1.7.2: With respect to system (1.7.1), the set A is

said to be (M x ) (/i 0 ,/i,M 0 )-stable

if for each e > 0

r^e)—>oo as e—>0 and S^t^e),

there exist

r^e),

62(t0,e) such that

t0 + i

/ h(t, x(t, s, V>(s, x*))ds < e for all t > tQ + 1 «0 + i

whenever to > 7-x(£);

x* G S{A,81)

and

/

h0(s,V>(s,x*))ds < 62,

Chapter 1

71

(M 2 ) (h0, h, M 0 )-uniformly stable if 61 and S2 in (Mx)

are

independent of t0; (M 3 ) (/to5^5-^o)" e c l u i" a t t r a c tive if for every e > 0, there exist positive numbers 6w(t0),

^2o(*o)> r o

an

d ^O)6)

sucn

a t

^

t0 + i

I

II x(t, a, rl>(s, a*)) \\ds<e,t>t0

+ l + T^

e), t0 > r 0 ,

*o provided x* e S(A,610)

and /i0(.s, V>(s, x*)) G SC-M,,, 52o);

(M 4 ) (/i0,/i, M 0 )-uniformly attractive if 610, S20 and T in (M 3 ) are independent of t0; (M 5 ) (/i 0 ,h, M 0 )-equi-asymptotically stable if (M a ) and (M 3 ) hold together; (M 6 ) (h0, h, M0)-uniformly

asymptotically stable if (M2)

and

(M 3 ) are valid simultaneously; (M 7 ) (/i0, h, M 0 )-unstable if (M a ) fails to hold. Consider also the comparison equation u' = g(t, u), u(t0) = y>(<0, "*), to > 0, where g E C[R

+

M 0 -invariant

if

xR,R],
C[R +xR,R

u(t, s, ip(s, 0)) G MQ

+

(1-7.2)

]. The set u = 0 is

whenever

(p(s, 0) E M 0 .

Concepts analogous to (Mj) —(M 7 ) can be defined.

For

example, the set u = 0 is {MQ M 0 -stable, with respect to (1.7.2), if for each e > 0, there exist r ^ e ) , S^t^e) such that «0 + i

/ u(t,s, tQ + l,

and

S2(t0,e)

Basic

72

Theory

provided u* < 8X and / r x (e). The remaining notions (Mj) - (M7) corresponding to (M 2 ) — (M 7 ) can be easily formulated. To understand the notion of M 0 -stability, we consider the following examples. Example 1.7.1:

Consider the differential equation x' = e ~ ', x(<0) = lK 0

(1.7.3)

where ^(s,x*) = x* + \. The solution of (1.7.3) is given by x(t, s, rp(s, x*)) = x* + | + e ~ * - e _ t , and it is clear that the set x = 0 is M 0 -uniformly stable.

For

this example, the set x = 0 is also eventually uniformly stable. If, on the other hand, we choose i/>(s, x*) = x* + A(s) where A:[0,oo)—>i? is a C 1 function coinciding with e - t except at some peaks where it reaches the values 1. There is one peak for each integer value of t and the width of the peak corresponding to abscissa n is smaller than (|) n . Then x = 0 is not eventually stable, but it is M 0 -uniformly stable.

This

example shows that stability behavior depends also on the initial values.

Example 1.7.2:

Consider the initial value problem

x'= - \'{t), x(t0) = x*,

(1.7.4)

Chapter 1

73

where A is defined by when t = n, an integer,

n, 2n\t -n)

X(t) =

+ n,

- 2n\t -n)

when n —^3 < £ < n, 2n

+ n, when n
0,

for all other t > 0.

Then A'(i) exists except on a set of measure zero. Considering only positive solutions of (1.7.4), we obtain x(t, s, 0(5, x*)) = x* + X(t0) - X(t) <x* + X(t0). t0 + i

The set x = 0 is M 0 -uniformly stable since / X(s)ds is at most '0

-^2 for n € [t0,t0 + 1].

But x = 0 is not eventually uniformly

stable since A(i0) does not approach zero as t0—>oo. We convenient

need

the

notations

following before

we

known

result

and

some

can

proceed

to

prove

(h0, h, M 0 )-stability criteria. Lemma 1.7.1 (Jensen inequality): and y integrable.

Let (p be a convex

function

Then

ip(Jy(t)dt)< Jip(y(t))dt. Definition

1.7.3:

Let h0, heT.

Then h0 is said to be

integrally finer than h if for every e > 0, there exists a

Basic

74

Theory

constant 8(e) > 0 such that x* G A and h0(s, xp(s, x*)) G S{MQ, 8) implies h(s,i/)(s,x*)) G S(M0,e). 9GC=

[aG C[R + , R

+

]:er £% and cr is convex].

C%'3>= [aeC[R+xR e > 0,

+

there

,R

+

exists

h0(s, XJJ(S, x*)) G S(M0,8)

]:aeC%

and

5(e) > 0

for

every

such

that

implies a(s, h0(s, ^ ( s , x*)))

eS(M0,e)}. We shall present results concerning (h0, h,

M0)-uniform

stability and (h0, h, M 0 )-uniform asymptotic stability.

Based

on these, one can construct proofs for other cases. Theorem 1.7.1:

Assume

that

(i)

h0t h € r and h0 is integrally finer than h;

(ii)

V G C[R

+

x Rn, R + ], V(t, x) is locally Lipschitzian

in x

and there exist functions a G C3G^P and b G 9GC such that b(h(t, x)) < V{t, x) < a(t, h0(t, x)), (t, x) G R + x Rn; (Hi)

D + V(t, x) < g(t, V(t, x)), (t, x) G R + x Rn.

Then the M0-uniform (h0,h,M0)-uniform

stability

of the set u = 0 implies

stability of the set A, that is, (M$)

the

implies

(M2). Proof:

Let 0 < e < p and t0 G R + be given. Suppose that

the set u = 0 is M 0 -uniformly stable with respect to (1.7.2). Then, given 6(e) > 0, there exist 6x(e), 82(e) and r ^ e ) such that

Chapter 1

75

I u(t, s, tp(s, u*))ds < 6(e), t > tQ + 1, «o

whenever

t0 + i

0
and

/ <^(s,u*)c?,s < 82,

(1.7.5)
Assumption (i) implies that there exist 83(e), 84(e) and r 2 (e) such that t0 + i

/ ft(s, V>(5, 2*))ds < e, f > t0 + 1,

(1.7.6)


provided x* £ S(A,83),

J h0(s^(s,x*))ds < 84 and t0 > r 2 (e). (s,x*))d.s<£ 2 , < 0 > T 3( e )>

(1-7.7)

(0 + i

when a;* G S(MQ, 85) and / /i 0 (s, ^>(s, x*))c?5 < 86. tQ

Let

Ji(e) = mm{^ 1 (e),^ 3 (e),^ 5 (e)},

86(e)] and f(e) — max{Tl(e),T2(e),T3(e)}. that x* G S(A,S1) that

and /

82(e) = min{84(e),

If we choose x* such

h0($,ij)(s,x*))ds < 82, then we claim

'° tQ + i

/ h(s, x(t, s, if)(s, x*))ds <e, t > t0 + 1, t0> r(e), f

o

where x(t,s,^(s,x*))

is any solution of (1.7.1).

Basic

76 If this is not

true,

then

there exists

Theory

<x > tQ + 1 ,

to > T(C), such that «n + i

|

/ l (i 1 ,x(i 1 , 5 ,^( 5 ,x*)))(f5 = e,

(1.7.8)

and t0 + i

/ /i(f, x(*, 5, ^(s, x*)))ds < e, <0 + 1 < i < tx, t0 > f (e).
Then,

by Theorem 1.3.2, we have V(t, x(t, s, i>{s, x*))) < 7 (r, s, tp(s, u*)),

(1.7.9)

since V(s,il>(s,x*) < a(s,hQ(s,il>(s,x*))) = tp(s,u*) letting u* = d(x*,A).

It then follows from assumption(n), relations (1.7.5),

(1.7.7) - (1.7.9) and Lemma 1.7.1, that t0 + i

b(e)
h(tux(tus,il>{s,x*)))ds

t0 + i

<J

V^XhisMstX*)))


< J

l{h,s,(p{s,u*))ds

Chapter 1

11
since u* < S1 and / ip(s,u*)ds < S2.

This is a contradiction

!°

and thus the proof is complete. CoroUarv

1.7.1:

1

M0DC [R

+

,R

+

Corollary 1.7.2:

The function

g(t,u) = — \'(t)

where

Ae

] is admissible in Theorem 1.7.1. The function

g(t, u) = 0 is also admissible in

Theorem 1.7.1. Theorem 1.7.2: hold.

Assume that the conditions of Theorem

Then if the set u = 0 is M^-uniformly

stable with respect to (1.7.2), the set A is

1.7.1

asymptotically (h0,h,M0)-uniformly

asymptotically stable with respect to the system (1.7.1). Proof:

By

Theorem

1.7.1,

the

set

A

is

(h0,h,M0)-

uniformly stable and we need to prove that (M 4 ) holds.

It

follows from (Ml) that there exist positive numbers 8iQ, ^o? TQ and T(e) such that t0 + i

J u(t, 5, t0

+ l+ r(e), t0 > r j ,

*o provided u* < £*0 and / (p(s,u*)ds < S^.

Basic

78

Theory

As in the proof of Theorem 1.7.1, we can find positive numbers 6j 0 , £20, r 0 which satisfy the inequality to + i / a(s, h0(s, ip(s, x*)))ds < 8*2Q, t > f0, «o if x* eS(A,S10)

and

/

h0(s,il>(s,x*))ds < S20.

Since £20 is

*°

independent of e, it is evident that r 0 does not depend on e. Let 610 = min{E1Q,<5j0} and r 0 = max{rj$,T 0 }. Then, we claim that *0 + i

j h(t, x(t, s, tfts, x*)))ds <e,t>tQ

+ l+ T(e), t0 >

0)

t0+l

when x* is chosen so that x* £ S^-A, S1Q) and / h(s, V>(s, x*))ds *o < ^20- ^ * m s i s n ° t true, there exists a sequence {f*}, ^—>oo asfc—>oo,such that t0 + i

/

Hh^k^^(s,x*)))ds > e .

*o This leads to a contradiction

(

o

79

Chapter 1 t0 + i

'o


Invariance principle. We shall discuss in this section, autonomous systems

only.

By employing some special properties of autonomous

systems, we shall prove some results on asymptotic stability through Lyapunov functions under weaker assumptions. Consider the initial value problem x' = f{x), x(0) = x 0 , where / E C[Rn,Rn]. uniqueness on R+

(1.8.1)

For simplicity, we assume existence and of solutions of (1.8.1). To avoid repetition,

we shall use, instead of defining the corresponding ones for (1.8.1), notions given in previous sections.

It is understood,

however, that all functions defined earlier are independent of t in

this

section.

For

example,

V = {h (E C[Rn, R + ]:

infh(x) = 0}. We denote by x(t,x0) x(0, x0) = x0.

the solution of (1.8.1) such that

A point y is said to be a positive limit point of

(1.8.1) through x0 if there is a sequence tn£R+

such that

Basic

80

Theory

<„->oo and x(tn, x0)—>y as n->co. The set of all such positive limit points is called the positive limit set of (1.8.1) through x0 and denoted by f2(x0).

One could define, similarly, negative

limit point and negative limit set, but we shall only consider solutions of (1.8.1) defined on R+

since we are interested, in

general, in the "future behavior", i.e., as t—► + oo. 7(x 0 ) = {x(t,xQ);t

The set

G R + } is called a positive trajectory

of

(1.8.1) through x 0 . A set H is said to be positively invariant if 7(x 0 ) C H whenever x 0 G H. Lemma 1.8.1:

If the solution x(t,x0)

of (1.8.1) is bounded on

R + , then (i) fi(x 0 ) ^ 0; (ii) fl(x0) is positively Proof:

(i)

Weierstrass

follows

theorem.

immediately

invariant.

from

To prove (ii),

the

Bolzano-

we need to

show

7(2/0) C fi(x0) for any yQ G fl(x 0 ). By the definition of y0, there exists a sequence tn G R + such that x(i n , x0)—>i/0 as n—►00. Let t G R + . Then we have x(t,y0) = IVTU^X^^X^^XQ)). uniqueness x(t,x(tn,x0))

of

solutions

= x(t + tn,x0),

of it

(1.8.1)

follows

Since

implies

that,

by

that letting

t'n = < + *„, *(*, j/ 0 ) = ( j ™ ^ ^ ^ci)- Thus x(t, j/ 0 ) G fi(x0). Since * is arbitrary, we have j(y0) C O(x 0 ), which shows f)(x 0 ) is positively invariant. Thus the proof is complete. Let VeC[G,R], define

GCR"

is an open set.

Then we

Chapter 1

81

D + V(x) = Km s u p i6 [V(x + Sf{x)) s-*o + +

£ = {x;£ Lemma 1.8.2: (i)

Assume

V € C[G, U],

V(x)],

V(x) = 0, x £ G } .

that

V(x)

is locally

Lipschitzian

in x

and

bounded from below; (ii)

D+

V(x)
(iii)

x(i,x 0 ) is a solution

of (1.8.1) such that x(t,x0)

EG,

*<E[0,oo). Then for V-\c)

some

real number

= {xEG;V(x)

Proof:

c, fl(x0) C E(~)V~1{c),

where

= c}.

By assumptions (i) — (ii), V(t, x(t, x 0 )) is bounded

from below and nonincreasing.

Thus lim V(t, x(t, x 0 )) = c for

some real number c. If lim || x(2,x 0 ) || = + oo, then £2(x0) = 0 and the lemma is proved.

Otherwise, there exists a sequence

tn G R+ such that {x(fn, x 0 )} is bounded. Then it follows from Bolzano-Weierstrass theorem that Cl(x0) ^ 0.

Let y £ fi(x 0 ).

Then by definition of fi(x 0 ), there exists a sequence tk(ER such

that

x(tk, x0)—>y

as

&—>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, (Hi)

all solutions

x(t,x0)

in x,

h-

and x€S(h,p);

of (1.8.1)

with

h0(x0) < a

are

bounded; (iv)

for any c > 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) <
h0 is uniformly finer than h. This, together with condition (ii) of Theorem 1.8.1 implies that the system (1.8.1) is (hQ,h)uniformly

stable.

Thus

for

e = p > 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))<0, V(x(t,x0))

as i—>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))
(r,,rtt).

(2.1.1)

Of course, we could have, instead of (2.1.1), W(ti,x(ti)) = *1W(tlx(t;))=Z W{t,xM) €{$,*)■

(2.1.2)

93

Chapter 2

Suppose that D + W(t,x)<M.

Then it is easy to

obtain, using (2.1.1), the relation t* — i,- > -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

'-» {
Thus, W(t,x(t))—*0

as i ^ o o and

hence h(t, x(t))—>0 as t—>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{ < /?,•
+ 1,

and /?,- - 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 t0a>0

and tk—>oo as k—+oo, such that WfaXtk)) > l > 0,

(2-1.4)

W,-(*fe,*(**))< - * < 0 .

(2.1.5)

or

Since ^,(<,x(i)) = VF,(fJt,x(ijfc))+ f D + Wi(s,x(s))ds, it follows from condition (iv) that there exists a constant 8, 0 < 8 < a, such that \WJt,x{i))\ >^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<;

= l,2,..„

(2.1-7)

follows from condition (Hi) that oo

UmV(t, x(t)) < V(t0, x0) - JD

+

V(s, x(s))ds

to
is

/i-positive

definite,

this

in

Since turn

»= i

implies lim h(t,x(t)) = 0. Hence the system (1.1.1) is (h0,h)asymptotically stable. Example 2.1.1:

Consider the nonlinear differential system 2/1 = - 2/i + 2y 2 + 2/3 + 2/12/36*, 2/2 = ~ 22/i ~ 2/2 + 2/4 - 2/22/4et>

(2.1.8)

2/3 = - 2/i + 2y 4 - 2/12/3^, 2/4 = - 2/2 - 22/3 + yl2/4 et -

Let

V(t,y) = \{y\ + v\ + y\ + y\),

h{t,y) = yl + y22

and

>»o(<,2/) = 2/? + 2/2 + 2/3 + 2/4- Then we see that V{t,y) is /ipositive definite, /i0-decrescent and D + V(t, y) = — y\ — y\. Let Wx = \y\ and W2 = \y\. then D + W,(t,y)=

-y] + 2y1y2 + y1y3 + y}ylet> ma.T <4 |y,.| < p,

-4p\

97

Chapter 2 and D + W2{t, y) = - 2yxy2 - y\ - y2y4 - y\yY

< Ap2,

It then follows by Theorem 2.1.2 that the system (2.1.8) is (h0, /i)-asymptotically stable. The next result is a generalization of a result due to Matrosov [1, 2] in which loss of negative definiteness is considered for nonautonomous systems. Theorem 2.1.3:

Assume

that

(i)

h0, h 6 T and h0 is uniformly finer than h;

(ii)

V1 EC[R+

xRn,R

+

], V-i(t,x) is locally Lipschitzian,

positive definite, h0-decrescent

h-

and

D + Vx{t,x) < W(x) < 0, (t,x) e

S(h,p),

where W{x) is continuous on Rn; (iii)

xRn,R

V2€.C[R+

+

] and V2{t,x)

is bounded on

and locally Lipschitzian

in x.

Furthermore,

number

there

exist

a,

0 < a < p,

S(h,p)

given any

positive

numbers

£ = £(a) > 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
Refinements

100

(2)

the set U. In other words, x(t) cannot stay permanently in the set U. Consider the sequence {tk} such that ^fc = ^o + k~r~j » = 0,1,2, — Set n(t) = V1(f,x(i)). Then, by assumption (ii), we have £> + n ( i ) < I ? + ^i(<,x(t))<0. We let.

A = inf{ | W(x) |, d(x, E) > | (t, x) £ S(fc, p) n S^fo, 5), t e iZ + } and ||/(*,*) ||

<M,(t,x)eS(h,P).

Suppose that x(t) is such that, for ffc < i < i^. + 2, (t,x(i))eS(h,p)r\Sc(h0,8). If, for < f c \q, then, using assumption (ii) together with the definition of the set E, we obtain 'fc + 2

**(<* + 2) ~ n(*t) = J

D + n s ds

()

(2.1.12)

*fc + 2

< y


jD + v t (j,z(i))d3

D + V1(s,x(s))ds+

J D + V1(s,x(s))ds <-k + l

101

Chapter 2 <-A(*fe+1-y=-A^.

On the other hand, if it happens that, for tk < tx < tk + x, 5 C (/J 0 , 8),

(tlt sft)) € S(h, p) n

«*(*(*,), E) < \rj

then there exists a t3, tx < i 3 < fa

(2-1-13)

Since '3

'3

I *i(*3) - *i(t4) I < / I X;(S) I ds < J I /,<*,*(«)) I ds U

*4

<M(t3-t4),i

= 1,2,...,n,

it follows from (2.1.13) that lv<Mn^2(t3-t4), which implies V 2Mn^

^{ ~h

t U

~

^2L r

(2.1.14)

Moreover <4

n ( i 3 ) - n ( 0 < JD

<3 +

V1(s,x(s))ds+

j D+

V^x^ds

Refinements

102 -AT?

< -Kt3-U)<^ZJj2= 2Mnx where A, = —ft-

_

~Ai'

2MnX'i

Since n(£) is a nonincreasing function, we have n(tk

2)
+


2)
Thus, in any case, ^i(t fc + 2,x(tfc + 2 ))<7 1 (f f o a : (t f c ))-A 1 . Choose an integer k* such that Axfc* > 0.(6^ and T = T(e) = 4k*L/((e). Assume that, for *0 < * < i 0 + T /i 0 (i,x(<))>5. It then follows from the preceding considerations that yx(<0 + r , x(t0 + T)) < Fx(*0, x0) - fc*Aa < a(*J - fc*Aj < 0, which

is

a

contradiction.

Thus

** £ [*oi *o + T] satisfying h0(t*,x(t*))<6, and the proof is complete.

there

exists

a

103

Chapter 2

As we can see from the proof of Theorem 2.1.1, the assumption imposed on W(t,x), namely, D + W(t,x) is bounded from above or from below, can be weakened to D + W(t, x) < p{t), (t, x) e S(h, p),

(2.1.15)

D + W(t, x)> - q(t), (t, x) e S(h, p),

(2.1.16)

or

where p,q £ C[R + ,R + ] and uniformly continuous on R +.

t

/ p(s)ds o

and

t

fq(s)ds o

are

The following theorem is in this spirit which offers a better conclusion. Theorem 2.1.4: Assume that (i) h0, h £ r o and h0 is finer than h; (ii) Vx, V2eC[R+ xRn,R], V^x) and V2(t,x) are locally r Lipschitzian in x, V 1(i,a;) is h-positive definite, Vx{t,x) +■ V2(t,x) is h0-weakly decrescent and

D+v(t,x)< -xitpiv^x)), (t,x)es(h,p), where V = V-y + V2, A(<) is integrally positive and C € 3G; (Hi) for every solution x(t) = x(t, t0,x0) of (1.1.1), the function t J[D + V2(s,x(s))]±ds 0

Refinements

104 is uniformly

continuous

on R + l where the symbol [ • ] ±

means

that either the positive part [ • ] + or the negative part [■]_ is considered for all s G R+ . Then the system

(1.1.1) is (h0,h)-asymptotically

and V2(t,x(t))

has a finite

x{t) = x(t,t0,x0)

o / ( l . l . l ) such that (t,x(t)) €

Proof:

limit as t-*oo for any

stable solution

S(h,p).

By Theorem 1.2.1., system (1.1.1) is

(h0,h)-stable.

Choosing e = p and designating by S0 = S(t0,p), it is clear that we have ^ o . ^ o ) < so implies h(t,x(t)) where x(t) = x(t,t0,x0) functions

V\(£, x(t)) + V2(t, x(t)). and

Urn m(t) = a < 00.

t0,

is any solution of (1.1.1).

rn^t) = Vx{t,x(i)),

nonincreasing



m2(t) = V2(t,x(t)),

Define the m(t) =

Assumption (ii) yields that m(t) bounded

from

Clearly Urn infract)

below = 0.

and

is

therefore

For otherwise,

we would have, in view of (Hi), m(t)—* — 00 as t—►00. Suppose now that lim mJt) ^ 0. Then there exists a 7 > 0 such that lim supm^t)

> 37.

Since

lim m(t) = a

and

m(t)

is

nonincreasing, there exists a M > 0 such that a < m(t) t0

+ M.

(2.1.17)

For definiteness, suppose that assumption (in) holds with [ • ] + . Since m x (f) is continuous, we can choose a sequence

Chapter 2

105

t0+ M < <*x < px < ... < a,- < ft; < ..., such that for i = 1,2,..., ™i(«.) = 37, ra^ft) = 7 and 7 < mx(i) < 37, * € [au Pi].

(2.1.18)

From (2.1.17) and (2.1.18), it is easy to see that m(a t ) — m1(a,-) < <x — 27, m(/?,.)-m1(^)><7-7.

(2.1.19)

Since m(<) = mx(i) + m2(£), it follows from (2.1.19) that

fii 0 < 7 < m2(/3,-) - m 2 (a.) < | [ l > + V2(a, x(s))]+ ds, which implies, in view of assumption (in), that there exists a constant d > 0 such that Pi-ai>d,i

= l,2,...

(2.1.20)

By (2.1.18), (2.1.20) and condition (ii), we then get lim m(t) < m(t0) — C(f) / X(s)ds = — 00, /

where

OO

/ = (J [a-, pA.

This

contradiction

implies

that

n = l

h'm mx(i) = 0 and since V x (i,x) is /i-positive definite, we get in turn lim h(t, x(t)) = 0. Thus we conclude that the system (1.1.1) is (h0, /i)-asymptotically stable. To prove the last

Refinements

106 assertion

of the theorem,

note that

Urn m(t) = a

and

t—»oo

lim mj(i) = 0, consequently the definition of m(t) lim m2(t) = a. The proof is therefore complete. Example 2.1.1:

yields

Consider the generalized Lienard equation x" = a(t)g(x, x')x' + b(t)f(x) = 0,

(2.1.21)

or the equivalent system f I I

x' = V, (2.1.22) y' = - a(t)g(x,y)y

-b(t)f(x),

2

where aeC[R + ,R + ], geC[R ,R + ], f£C[R,R], b£ l C [R + ,(0,oo)], and F(x) = f f(u)du > 0 for all x£R-{0}. Define Va(x,y) = \y\ v/t,x,y) = b(t)F(x), V = V1 + V2. Then D + V(t, x, y) = - a{t)g(x, y)y2 + b'(t)F(x) = -2a(t)g(x,y)V1(x,y) D + V2(t, x, y) =

+

h

Mv2(t,x,y)

h

^V2(t, x, y) + b(t)f(x)y.

Let h = | y |, h0 = yjx2 + y2. Then we see that Vx(xty) is /i-positive definite, V(t,x,y) is V w e a k l y decrescent. Suppose that (i) b'(t) < 0, t(=R + ; (ii) f(x) is bounded on R and (m) for every 0 < -y0 < 7, there is a #0 > 0 such that

107

Chapter 2 g(x,y) > g0 if 7 0 < h < 7.

Then application of Theorem 2.1.4

yields that system (2.1.22) is (/i 0 ,/i)-asymptotically stable and the function b(t)F(x(t)) 2.2

has a finite limit as <—>oo.

Perturbations of Lyapunov functions. We shall discuss, in this section, (h0, /instability of

(1.1.1) under weaker assumptions, which shows that in those situations when the Lyapunov function chosen does not satisfy all the

desired

conditions, which

is often

the

case

in

applications, it is fruitful to perturb it than to discard it. We begin with the following result which

proves

nonuniform stability under weaker assumptions than Theorem 1.3.2 and also includes several special cases. Theorem 2.2.1:

Assume that

(i)

h0, h £ T 0 and hQ is finer than h;

(ii)

Vx 6 C[R

+

x Rn, R + ], Vx(t,x)

is locally Lipschitzian

in

x, h0-weakly decrescent and D + Vx{t, x) < 9l(t, V^t, x)), (t, x) € S(h, p), where gx G C[R+ x R,R] with ^ ( i , 0 ) = 0; (Hi) for

every

n>Q,

there

exists

a

V2T)€C[S(hyp)r\

S (h0,7/), R + ], V2Tt{t,x) is locally Lipschitzian satisfies the following

inequalities:

in x and

Refinements

108 (A)

b(h(t,x))< c

S (hQ,r)), (B)

V2r,(t, x) < a(h0(t, x)),

(t, x) 6 S(h, p) n

where a,b G % and

D + V1(t,x) + D + V2r)(t,x) < g2(t,V^t,x) V2r)(t,x)),

where g2£C[R

+

xR,R]

+

with g2(t,0) =

0; (iv)

the trivial solution is equi-stable relative to

differential

equation u' = g1{t,u),u(to)

= uo>0>

(2.2.1)

and uniformly equi-stable with respect to differential w' = g2(t, w), w(tQ) = wo>0.

equation (2.2.2)

Then the system (1.1.1) is (hQ,h)-stable. Proof:

By condition (i), there exist constant o~0 > 0 and

function ipo £ C3G such that h(t, x) <
(2.2.3)

decrescent, we have for some

constant a1 > 0 and function (px 6 C3G that h0(t,x) < (Tj, implies V^x)

< v?x(<,/io(<,x)).

Let 0 < e < p and t0 G R + be given.

(2.2.4)

Since the trivial

solution of (2.2.2) is uniformly stable, given 6(c) > 0, there exists & S0 = 60(e) such that

109

Chapter 2 w0 < 80 implies w(t, t0, w0) < 6(e), t > t0,

(2.2.5)

where w(t, t0, wQ) is any solution of (2.2.2). By the definition of a and y>0, there exists a 8t = S^t^e)

> 0 such that

a{8r) < ^ and
(2.2.6)

The stability of the trivial solution of (2.2.1) implies that given -^ > 0 and tQ G R + , there exists a, 82 = 82(tQ, e) > 0 such

that r

uQ < 82 implies u(t, t0, u0) < -£, t > t0, u(t,t0,u0)

(2.2.7)

being any solution of (2.2.1). Choose u0 = V 1 (i 0 ,a; 0 ).

Since ipx G C3G and (2.2.4) holds, there exists a 83 = 83(t0,e), 0 < 83 < <7l5 such that V ^ o ) < ^3 implies V^XQ) We set 8 = min{a0,a1,81,83}

< 62.

(2.2.8)

and suppose that h0(t0,x0)

< 8.

We note that, because of (2.2.3) and (2.2.6), h(t0,x0) < e. Let x(t) = x(i, tQ, x0) be any solution of (1.1.1) with h0(t0, x0) < 8, we are going to show h(t, x(t)) < e for all t > t0.

If this is not

true, then there exists a solution x(t) = x(t,t0,x0)

of (1.1.1)

and t2> t1> t0 such that /J 0 (£I,X(£I))

= 8, h(t2,x(t2))

= e and

(*, x(t)) e S(h, e) n sc{h0,8), t e [*!, t2].

(2.2.9)

Refinements

110

Setting r) = 6, we see by (Hi) that there exists a V2r) satisfying (A) and (B). Hence, letting m(t) = 1^(<,*(*)) + ^2„(<> *(*)), f o r £ G [^i,^], w e obtain the differential inequality

D + m(t)
(2.2.10)

where 72(^1, m (^i)) is the maximal solution of (2.2.2). Also, we can obtain similarly the estimate V^xit))

< 7i(Mo,^i(*o,*o)), < G [*o,*il,

where 7i(Mo> V\(toixo)) *s the maximal solution of (2.2.1). Hence by (2.2.7) and (2.2.8), we have ^i(*i, * ( * i ) ) < | .

Also, by (A) and (2.2.6), we get

V,n(tiXh))


Hence it follows that m(fj) < S0 and therefore (2.2.5) and (2.2.10) implies m

(^)<72(^,
But m{t2)>V2r,(t2,x(t2))>b(h{t2,x(t2)) = b{e), which leads to a contradiction. Hence the proof is complete.

111

Chapter 2 Remark

2.2.1:

1.3.2 relative g2(t,u) = 0,

Theorem 2.2.1 is a refinement of Theorem to nonuniform

Theorem

relaxing conditions.

stability.

2.2.1 improves

Vl(t,x)

Theorem

g(t,u) =

Theorem

1.2.1

by

If, on the other hand, V2fJ(t,x) = 0,

<7x(i, u) = g2(t, u) = 0 and Vx(t,x) definite,

When

is assumed to be A-positive

2.2.1 reduces

to Theorem

1.2.1.

If

= 0 and <71(<,u) = 0, then Theorem 2.2.1 reduces to

Theorem 1.3.4. The following result on (h0,h)-asymptotic

stability is in

the same spirit as that of Theorem 2.2.1 which improves Theorem 2.1.4. can

be

We merely state the theorem since its proof

constructed

using

a

combination

of

arguments

employed in Theorem 2.2.1 and Theorem 2.1.4. Theorem 2.2.2:

Assume

that

(i)

h0, h 6 T0 and h0 is finer than h;

[ii)

V1 EC[R+

xRn, R + ], V^tfX)

is locally Lipschitzian

in

x, h0-weakly decrescent and D + Vx{t,x)< where

X(t)

-\{t)C(Wx{t,x)), is

integrally

n

WleC[R+xR ,R (Hi) Wx(tyx)

is

+

(t,x)eS(h,p), positive,

C G %,

\;

h-positive

definite,

W2(t, x) is locally Lipschitzian

V1(t,x) — W1(t,x) in x,

and for

solution x(t) = x(t, t0, x0) of (1.1.1), the function

=

every

112

Refinements t f[D +

W2{s,x{s))]±ds

to is uniformly continuous on R + ; (iv)

for

every

r)>0,

c

S {h0,ri),R

+

there

], V2ri(t,x)

exists

a

V2r) £ C[S(h, p) D

is locally Lipschitzian

in x and

satisfies (A)

b(h(t, x)) < V2t}{t, x) < a(h0{t, x)),

(t, x) E S(h, p) D

(B)

SC(h0,v), a,b<E%, D + V1(t,x) + D + V2r,(t,x) < g(t, V,(<,x) +

v2ri{t, x)), {t, x) e S(h, p) n sc{h0, n), where g e C[R+ x R,R] with g(t,Q) = 0; (v)

the trivial solution of (1.3.1) is uniformly

Then the system W2(t,x(i))

stable.

(1.1.1) is (h0,h)-asymptotically

approaches a finite

stable and

limit as t tends to infinity

any solution x(t) of (1.1.1) with (t,x(t)) €

for

S(h,p).

We shall consider, again in the spirit of Theorem 2.2.1, a result on (h0, h)-umiorm Theorem

2.2.3:

In addition to the assumptions

2.2.1, suppose further (a)

asymptotic stability. of

Theorem

that

either (a,)

D + V1(t,x) + C{h0(t,x))
(a2)

is nondecreasing

where in u or

+

D + Vx{t, x) + D V2n(t, x) + C(h0(t, x)) < g2{t,V1{t,x)

+ V2r]{tix)),

is nondecreasing in u;

where C 6% and

g2(t,u)

Chapter 2

113

(b)

h0 is uniformly finer than h and V\(2, x) is

(c)

the trivial solution of (2.2.1) is uniformly

Then

the system

hQ-decrescent;

stable.

(1.1.1) is (h0,h)-uniformly

asymptotically

stable. Proof:

Suppose

that

(a a )

holds.

Then

because

of

condition (b), the functions
of t, i.e. <£>0,^ € 9G.

Consequently,

(2.2.4), (2.2.6), (2.2.7), (2.2.8) hold with

(2.2.3),

ff0,
being

independent of t. As a result, 8, in the proof of Theorem 2.2.1 can be chosen to be independent of t0. Thus system (1.1.1) is (h0, A)-uniformly stable.

Thus, we get, letting e = p and

8* = 6(p), hQ(tQ, x0) < 8* implies h(t, x(t)) < p, t> for any solution x(t) = x(t,tQ,x0)

f0,

of (1.1.1).

Now let 0 < e < p be given and S = 6(e) be the same S as in the definition 1.1.1 for (h0, /i)-uniform stability. To prove (h0, ft)-uniform asymptotic stability, it is enough to show that there exists a T = T(e) > 0 such that for some t* G [i 0 , t0 + T], we have h0(t*, x(t*)) < 8. T > 0, x

h(io, o)

there s

< *

exists such

If this is not true, then for any a

that

solution h0(t,x(t))

x(t) = x(t, tQ, x0) >8

for

with

t £ [*0,*0 + T].

Choose T = T(e) > 0 such that

T

>wk)

(22 n)

-

114

Refinements

Then setting m(t) = V1{t,x(t))+

Jc(h0(s,x{s))ds, to

and using condition (ar) and the monotonic character of gt, we get D + m{t) < 9l(t, m(i)), t £ [*0, *0 + T] using arguments similar to the proof of Theorem 1.3.1. Hence by the same theorem, we obtain, choosing u0 = V1(t0,x0),

the

estimate Vt{t,x(t))

t + JC(h0{stx(s)))ds

< 7i(Mo,^i('o,Zo)),

to

te[t0,tQ + T], where 7i(i,f 0 > u o)

1S

the maximal solution of (1.3.1). This then

implies, by (2.2.7), that C(6)T < | , which contradicts (2.2.11).

It therefore follows that there

exists &t*e [t0,t0 + T] such that h0(t*,x(t*)) < 6(e). If (a 2 ) holds, we proceed as before with modifications to get a contradiction.

suitable

Hence the proof is

complete. The next result offers only asymptotic stability.

115

Chapter 2 Theorem 2.2.4'-

Assume

that

(i)

h0, h £ T and h0 is uniformly finer than h;

(ii)

for

every

Sc(hQ,rj),R

rj>0, +

there

], Vlr,(t,x)

exists

a

VXr) € C[S(h,p) (~)

is locally Lipschitzian

in x and

satisfies (A)

b(h(t,x))
S (h0,rj), (B)

(t,x)eS{h,p)n

a,be%,

D + Vlr](t, x)<-

\(t)
(t, x) 6 S{h, p) n

c

S (h0,V), where ipE%, integrally (Hi) W(t,x)

W £C[S(h,p)nSc(h0,T]),R

+

] and X(t) is

positive;

is locally Lipschitzian

in x and for every

solution

x(t) of (1.1.1) the function t J[D +

W(s,x(s))]±ds

to is uniformly continuous on R + ; (iv)

for

every

rj > 0,

there

exists

a

V2ri G C[S(h, p) fl

S (^o> V)J R + ]> ^2IJ(*> X ) *s locally Lipschitzian bounded on S(h, p) 0

in x and

c

S (h0,n).

Moreover, for every a > 0, there exists a f3 = /3( 0 and a function TER

+

(:R

+

—>R + ) which has the property

,

t / ((s)ds—>oo as t—>oo, r

that for

any

Refinements

116 such that for all (tf, x) € S(h, p) D Sc(hQ, a), W(t,x) < fi implies D + V2r}(t,x) > C(<).

Then the system (1.1.1) is (h0,h)-asymptotically stable. Proof: It follows from conditions (i) — (ii) that, by Theorem 1.3.4, the system (1.1.1) is (h0, /i)-uniformly stable. Hence for p > 0, there exists a SQ = 6(p) > 0 such that ho(to,x0) < S0 implies h(t,x(t))

t0,

(2.2.12)

x(t) = x(t,tQ,x0) being any solution of (1.1.1). To prove the theorem, it is enough to show, because of (h0,h) uniform stability, that for any solution x(t) of (1.1.1) satisfying (2.2.12) inf ho(t,x(t)) = 0.

(2.2.13)

If this is not true, there exists a n > 0 and a solution x(t) = x(t,tQ,x0) of (1.1.1) such that h0(t,x(t))>T],

t>t0.

For this 77 > 0, there exist functions V lt? and V2„ such that hypotheses (ii)-(iv) hold. By (ii) and (Hi), we see that UmW(t,x(t)) = Q for otherwise we would get Vlr)(t,x(t))^ — 00 as t—>oo. On the other hand, by condition (iv), there exists a /? = P(rj) > 0 and a function ((t) with the properties as described in (iv) such that W(t,x) < (3 implies D + V2ri(t,x) > C(t),

Chapter 2

117

{t,x)eS(h,P)nSc(hQ,T)). Since lim Wit, xit)) = 0, there exists a,T>tQ such that W(t,x{t))T.

(2.2.14)

Let G = [(*, x) € 5(fc, p) n 5c(/i0,77); W(*, x) < /?]. Assume that at < = tx, we have, letting m(t) = V2T1(t,x(t)), D + m(t)>D

+

V2t)(t,x(t))>((t),

because of condition (iv). Thus we obtain t m(t)>m(t1)+

f((s)ds,

which implies, since V2n(t, x) is bounded and Jti((s)ds —» 00 as t —> 00, that (£, x(*)) cannot remain in G forever. This is a contradiction to (2.2.14). Hence (2.2.13) is true and the system (1.1.1) is (hQ, /i)-asymptotically stable completing the proof. As an application of Theorem 2.2.4, we consider the following system of differential equations X1 = '

x

=

x

=

2

3

Ji\t,Xi,X2,X3), J 2V'ixlix2ix3)i

(2.2.15)

J3\ttxlix2ix3,)i

where x1 e R*1, x2 £ R"2, fiEC[R+xRnixRn*xRnz,Rni]

x3 G R"3, n1 + n2 + n3 = n, and /,(*, 0,0,0) = 0 for

Refinements

118 i = 1,2,3.

The following result is an easy consequence of

Theorem 2.2.4. Corollary 2.2.1: Assume that (i) for every n>0, there exists a Vlr) € C[R+ xR",R + ], V1 (t, x), x = (x1,x2,x3)) is locally Lipschitzian in x and satisfies (A) 6( || y || ) < V x , ( t , x ) < 0(11*11), t£R + , \\y\\

TJ, where a,b E%, y = (x1,x2)^ d (B) D + Vln(t,x)< - A f r V d l s J ) , t£R + ) \\y\\

n, where ip £ % and X(t) is integrally positive; (ii) for every solution x(t) of (2.2.15), the function t J II fi(s,x1(s),x2(s),x3(s))

|| ds

to is uniformly continuous in R + ; (Hi) for every 7? > 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))oo as 7—*oo; H

D + y a ( i , x) + D + V2{t, x) < g2(t, V^t, x) + V2(t, x)), (t, x) £ Sc(h0, p), where g2 £ C[R

(v)

the

solutions

differential

are

equibounded

+

x R + , R + ]; with

respect

to

the

equation

u' = gl(t,u),u(to)

= uo>0

and uniformly bounded with respect to

(2.2.16)

120

Refinements W = g2{t, w), w(t0) = w0>0.

(2.2.17)

Then the system (1.1.1) is (h0,h)-equibounded. Proof: given.

Let a > 0 (we may assume a > p), t0E R+ Let al = ^{t0,a).

be

Since the solutions of (2.2.16) are

equibounded, there exists a /?i(<0?Q;i) > 0 such that «(Mo.«o) < fiu * > *o> i f "o < «i,

(2-2.18)

where u(tf,20,u0) is any solution (2.2.16). Set a2 = a(a)-f/?j. Then the uniform boundedness of solutions of (2.2.17) implies that there exists a /?2 = /?2(a2) > 0 such that w2(t, tQ, w0) < f32, t> i 0 , whenever w0 < a 2 ,

(2.2.19)

w2(t,tQ,uQ) being any solution of (2.2.17). Since 6(7)—»oo as 7—>oo and y> 6 9G, we can choose a /? = /3(t0, a) > 0 such that P3(a3) < &(/?), tp(Q) < /?.

(2.2.20)

We claim that h0(t0,x0)t0. If this is not true, then, in view of (2.2.20), there would exist a solution x(t) = x(t,t0,x0) of (1.1.1) with h0(t0,x0)tx> t0 such that h0(t1,x(tl)) = a,h{t2,x(t2))

= /3,

Chapter 2

121 (*, x(t)) G S(h, fi) n Sc(hQ, a), t e (*lf i 2 ).

Letting m(i) = V ^ a f t ) ) + V2{t,x(t)), the differential inequality

(2.2.21)

tx < t < t2, we obtain

D + m{t) < g2{t,m(t)), tx
t2.

Hence by Lemma 1.3.1, we have m

C ) < 7a(Mnm(*i))> *i ^ * ^ *2>

where 7 2 (Mi> u; 0 ) i s * n e maximal solution of (2.2.17) such that 72(ia, 0) = w0. Hence we have

vl(hXh))+v2(t2,x(h)) <72(Mi, V^xit^ + V^x^))).

(2.2.22)

Also, we can obtain similarly the estimate Vri(f1,a:(i1))<7i(*,uo) *s the maximal solution of (2.2.16). Since u0 = V1(t0,x0)(t0,h0(t0,x0))
< a(/i0(i1,x(t1))) = a(a),

V1(tux(tt)) +

V2(tux{t1))
= a1, it

(2-2-24) which

Refinements

122 This, together with (2.2.19) - (2.2.22), implies b((3) = b(h(t2,x(t2))) < V^xik))

+ V2(t2,x{t2)) <(32< b(/3),

which is a contradiction. Thus the system (1.1-1) is (h0,h)equibounded and the proof is complete. Theorem 2.2.6: Assume that (i) 0 < A < A; (ii) h0, h G T and h(t, x) <

V2(t,x) < a(hQ(t,x)), ifhQ(t,x) > X, a G 3G, and D + V1(t,x) + D + (v) (vi)

V2(t,x)<0;

cp(X) < A and i>(tQ, X) + a(X) < b(A); V3, V4 G C[S(h, A),R + ] such that V1 = V3 + VA, V3(t, x) is h-positive definite and D + Vx(t,x)<

-v(t)C(V3(t,x)),

(t,x)eS(h,A),

where C G % and v(t) is integrally positive;

123

Chapter 2 (yii) for every y E C[R + , Rn] with h(t, y(t)) < A, the

function

t [D + V4(s, y(s))] ± ds is uniformly continuous on R

+

.

0 Then the system is (h0,h)-practically Proof:

asymptotically

stable.

We shall first show that the system (1.1.1) is

(h0, /i)-practically stable, which means hQ(tQ, x0) < X implies h(t, x(t))

t0.

(2.2.25)

We note that wheneverfo0(^o>xo) < ^> w e have, because of (ii) and (iv), h(t0,x0) < t1> t0 such that h(t2,x(t2))

= A, holt^xfa))

(t,x(t)) e S(h,A) (1 Sc(h0,X) Setting m(t) = V1(t,x(t)) + V2(t,x(t)),

= A and (2.2.26) for tx < t < t2. we obtain, in view of

(iv), D + m(t)<0, which implies

fj


124

Refinements m(t) < mfo), tx < t < t2.

(2.2.27)

Similarly, (vi) gives V1(t,x(t))
< WoA(*o.*o)) < ^ o , A ) .

(2.2.28)

Also by (iv), we have V2(t1,x(t1))
= a(X).

Hence it follows that m(tl){t0,\)

+

a(\)
because of (2.2.28) and (v). But m(t2)>b(h(t2,x(t2))) = b(A) by (2.2.6) and (iv). Consequently, we arrive at a contradiction b(A) < m(t2) < mfa) < b(A) because of (2.2.25). Thus (h0,/^-practical stability of the system (1.1.1) is established. The (h0, /i)-attractivity of (1.1.1) can be proved using a similar argument to that used in the proof of Theorem 2.1.4. Thus the system (1.1.1) is (h0,h)practically asymptotically stable and the proof of the theorem is complete.

125

Chapter 2 2.3

Several Lyapunov functions (continued). It is known that

when the time derivative of a

Lyapunov function along the solutions of a autonomous system satisfies a weaker assumption, say, semi-negative definiteness, one may get asymptotic stability by applying the invariance principle. impose

For nonautonomous systems, however, we need to additional

compensating

conditions.

considered this situation and using another

Matrosov Lyapunov-like

function with suitable properties in the vicinity of the set where the time derivative of the first Lyapunov function became zero, proved asymptotic stability. 2.1.3).

(See Theorem

In this section, we give a different approach in which

the new ingredient is that when the time derivative of a Lyapunov function is negative definite with respect to a larger set containing the desired set, it is beneficial to choose another convenient set relative to which we may impose suitable assumptions to yield asymptotic stability of the desired set.

Theorem 2.3.1:

Assume

that

(i)

h0, h £ r and h0 is finer than h;

(H)

V} EC[R+

xRn, R + ], V\(£,x) is locally Lipschitzian

in

x, h-positive definite, h0-weakly decrescent and D + Vx{t,x)< where

X(t)

-mCi{W{t,x)),

is

integrally

n

W(EC[R+xR ,R

+

);

(t,x)eS(h,p), positive,

Cx £ %

and

Refinements

126 (Hi)

W(t, x) is locally Lipschitzian

in x and for every

solution

x(t) o / ( l . l . l ) the function t f[D

+

W(s,x(s))]±ds

to is uniformly continuous on R (iv)

V2 £ C[R

+;

n

+

x R , R + ], V2(t, x) is locally Lipschitzian

in

x, V2(t, x) + W(t, x) is h-positive definite and D + V2(t,x)<

-C2(V2(t,x))

where C2e%,ij>e

C[R

+

+ ^(W(t,x)), ,R

+

(t,x)eS(h,p),

] and ^(0) = 0.

Then the system (1.1.1) is (h0,h)-asymptotically

Proof:

It

follows

from

conditions

stable.

(i) — (ii)

that

by

Theorem 1.2.1, the system (l-l-l) is (hQ,h) -equi-stable. Thus for e = p and £0 G R +, there exists a S0 = 60(p) > 0 such that ho(to,Xo) < & implies h(t,x(t)) for

any

solution

" i ( 0 = V\{*Xt)\

x(t) = x(t,t0,x0)

< p,t> of

t0,

(2.3.1)

(1.1.1).

Let

"2(0 = V2{t,x{t)) and u(t) = W(t,x(t)).

It is

easy to see from conditions (ii) — (Hi) that lim u>(t) = 0.

We

t—»oo

shall next show lim infv2(t)

= 0. By condition (iv), we obtain

D + u2(t) < - C2(v2(t)) + xp(u(t)), t > t0.

(2.3.2)

127

Chapter 2

We claim that lim v2(t) = 0. If this is false, then there exists a d > 0 and a T\ > 0 such that u2(t) >dioTt>t0

+ Tv

(2.3.3)

Since lim u»(i) = 0, there exists a T2 > 0 such that # ^ ) ) < ^

f o r < > r 0 + T2,

because of the assumptions on ^ and C2. Choose T = max{T1,T2}

so that by (2.3.2)-(2.3.4)

D + u2(t)<-^-,t>t0

+ T,

which implies lim supv2(t) = — oo. This is a contradiction. Suppose that lim u2(t) ^ 0, then there exists a 7 > 0 and a sequence t0 < ati < 0! < ... < at;
(2.3.5)

Since /im w(£) = 0, there exists a T 3 > 0 such that

#^))<^,*>*o + 7Y

(2.3.6)

Thus from (2.3.2), (2.3.5) and (2.3.6), we obtain, for sufficiently large i,

D + u2(t)<-^l,te[aiM

Refinements

128 1S

which implies that v2(t) v

v

a

iifiii < 2( i)i

which is

a

decreasing on [a,-,/?,•].

Hence

contradiction to (2.3.5).

Thus

lim v2(t) = 0. Since V2(t, x) + W(t, x) is /i-positive definite, it follows that lim h(t,x(t))

= 0. Therefore the system (1.1.1) is

(h0, /i)-asymptotically stable and the proof is complete. Example 2.3.1:

Consider the nonlinear system x' = y, y'=

Let Vi(x,y)

(2.3.7)

- cosH{l + x2 + y2)y3 - x3.

= \xA + \y2.

Then the time derivative of V-^x^y)

along (2.3.7) is D + V(x,y)<

-cosHy4,

which implies that the trivial solution of (2.3.7) is uniformly stable and all solutions of (2.3.7) are uniformly bounded and consequently defined for all t > tQ. Let M > 0 such that

yfx2(t) +

y2(t)<M,t>t0

and set W(x,y) = y2. Then

D + W(x,y)<M\

Jx2 + y2<M.

Choose V2(x,y) = \{x + y)2. Then it is easy to obtain D + V2(x,y)< where

-2M2V2(x,y)

+ 4>(W(x,y)\

y/x2 +

0(u) = (3M 3 + M)y/u + (M 2 + l)u + M u 3 / 2 ,

y2<M, ^(u)

is

129

Chapter 2

continuous on R + and ip(0) = 0. Set hQ = h = y x2 + j / 2 . Then applying Theorem 2.3.1, we conclude that the trivial solution of (2.3.7) is asymptotically stable. As another application of Theorem 2.3.1, we split the system (1.1.1) as

{

xl =

fi{t,x1,x2), (2.3.8)

X

2

=

J2V'ixl>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)<-

C2(V2(t,«))

because of conditions (ii), (A) and (B).

+ i V I || x 2 | | ,

Chapter 2

131

Defining

il>(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\

| <

2( | xx - xx \ + \ x2 - x2 \)

Thus the trivial solution of the is uniformly

asymptotically

stable.

Hence all conditions of Theorem 2.3.2 are satisfied and we conclude that the trivial solution of (2.3.10) is asymptotically stable. 2.4

Method of vector Lyapunov functions. As we have seen in the previous sections, employing

more than one Lyapunov function is more advantageous in improving several results. In this section, we shall develop the

Refinements

132 method of vector Lyapunov functions.

A function F £ C[Rn, Rn] is said to be quasimonotone nondecreasing in x, if x < y and a;,- = j/,- for 1 < i < n implies Fi{x) < F{{y) for all i. The inequalities between vectors are understood to be componentwise inequalities. We shall need the following comparison result for systems, see Lakshmikantham and Leela [1]. Theorem 24.1: Let V E C[R + x Rn,R*l ], V(t,x) Lipschitzian in x. Assume that

is locally

D + V(t, x) = Urn+ sup \ [V(t + h,x + hf(t, x)) - V(t, x)} h-+o

"

0,to€R

+

,

(2.4.1)

existing for t >t0. Then V(t,x(t))t0i where x(t) = x(t,t0,x0) is any solution of (1.1.1) existing on [<0,oo), provided V(tQ,x0) < u0.

Chapter 2

133

Corresponding to the stability and boundedness notions given in Section 1.1, we need similar notions relative to the comparison system (2.4.1).

We merely state one of the

concepts. Definition 2.4-1:

The trivial solution of (2.4.1) is said to be

stable if given e > 0 and t0 G R +, there exists a 8 = S(t0, e) > 0 such that N

N

Y^u0.<6 i=1

implies J2u,(i,tQ,u0)

where u(t,t0,u0)

<e,t>

t0,

1=1

'

is any solution of (2.4.1).

As a typical result, we shall prove a theorem that gives sufficient conditions in terms of vector Lyapunov functions for the (hQ, /instability properties of system (1.1.1) which is an extension of Theorem 1.3.2. Theorem 2-4-2:

Assume

that

(A0) h0, h (E T and hQ is uniformly finer than h; (Aj) V G C[R+ X i 2 " , i ? ^ ] ; V(t,x) h-positive definite and {A2) g G C[R

N

+

N

x R , R ],

is locally Lipschitzian

h0-descrescent; g(t, u)

is quasimonotone

creasing in u and g(t, 0) = 0; (A3) D + V(t,x)

< g{t,V{t,x)),

in x,

(t,x) €

S(h,p).

nonde-

Refinements

134 Then the stability properties

of the trivial solution

of (2.4.1)

imply the corresponding (hQ, h)-stability properties of (1.1.1). Proof:

We shall only prove (h0, /i)-equiasymptotic stability

of (1.1.1). For this purpose, let us first prove (h0, /instability. Since V is /i-positive definite, there exists a p0 £ (0,p] and b 6 % such that b(h(t, x)) < VQ(t, x), (t, x) e S(h, p0).

(2.4.2)

Let 0 < e < p0 and t0 £ R + be given and suppose that the trivial solution of (2.4.1) is stable.

Then given 6(e) > 0 and

t0 £ R +, there exist a constant 61 — S^t^e) N

such that

N

J2 u0. < #i implies ^2 ui(*> *o> "o) < He)> * > *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) <

a{h0(tQ,x0)),

{to,Xo)£S{hQ,a0).

(2.4.4)

It then follows from (2.4.2) that W o , s o ) ) < Vofo.xo) < a{hQ(tQ,xQ)), {t0>x0) £ S(hQ,
(2.4.5)

Chapter 2

135

Choose 8 = 8(t0, e) such that 8 € (0, cr0), a(8) < 8t and let ^o(*o> xo) < <^- Then (2.4.5) shows that /i(f 0 ,x 0 )<e since ^ < 6(e). We claim that h(t, x(i)) <e, t>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 *(
e and

*(*»*(')) < e>
(2.4.6)

in view of the fact that /i(i 0 ,x 0 )<e whenever h0(tQ, x0) < 8. This means that (t,x(t)) £ S(h,p0) for t G [£0>*i] a n d hence by Theorem 2.4.1, we have V(t, x(t)) < 7 (i, *o, u0), <0 < < < ix,

(2.4.7)

where 7(Mo>uo) i s the maximal solution of (2.4.1). Now the relations (2.4.2), (2.4.3), (2.4.6) and (2.4.7) yield 6(e) < V0(tuxfa))

< f ^ M o ^ o ) < 6(0. i= i

a contradiction proving (/i0, /instability of (1.1.1). Suppose next that the trivial solution of (2.4.1) is equiattractive. From the (/i0, /instability, we set e = p0 so that 80 = 8(t0, pQ). NOW let 0 < TJ < p0. Then, by equi-attractivity of (2.4.1), we have that, given 6(77) > 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,( i=1

because of (2.4.7)-(2.4.8). Hence the system (1.1.1) is (h0,h)equi-asymptotically stable and the proof is complete. To exhibit the advantage in using vector Lyapunov functions, we consider the following example. Example 2.4.I:

{

Consider the differential system x\ = e ~ lxx + x2sint — (if +

x^DsinH,

x'2 = x^mt + e lx2 - {x\x2 + xfjsinH.

Let h0 = h = y x\ + x\. function V given by

(2.4.9)

We first choose a single Lyapunov

Chapter 2

137 V(t, x) = x\ + x\.

Then, it is evident that D + V{t, x) < 2(e " ' + | sin* | ) V(t, i ) 2 | aft | < a 2 -f 62 and observing

using the inequality 2

(x\ + xl)sin t > 0.

that

Clearly, the trivial solution of the scalar

differential equation u' = 2(e ~ J + | sint | )u, u(t0) = u0 > 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<\
and

h(t,x) <(p(h0(t,x))

if

h0(t,x)<X; (ii)

V G C[R + x Rn, R^ ], V(t, x) is locally Lipschitzian in x and D + V(t,x) < g(t,V(t,x)),

(t,x) € S(k,A),

where g E C[R + x 72 +, RN] and g(t, u) is quasimonotone nondecreasing in u; (iii) b(h(t, x)) < V0(t, x) if h(t, x) < A and V0(t,x) < a(h0(t,x)) ifh0(t,x)

< X,

N

where a, b & % and VQ = £ V,-(<, x); i= \

(iv) cp(X) < A and a(A) < b(A). Then the practical stability properties of (2.4.1) imply the corresponding (h0,h)-practical stability properties of the system (1.1.1). Proof: Let us first suppose that (2.4.1) is practically stable. Then, given (a(X),b(A)), it follows, because of (A3), that N

N

J2U0; < a(x) implies V«,-(«,t 0 ,u 0 ) < b{A), i = 1

i=1

t > t0-

(2.4.11)

Chapter 2

141

Let hQ(tQ,xQ) < A* Then by (i) and (iv), it follows that h(t0, *0) < v(Ao(«o» *b)) < V(A) < A.

(2.4.12)

We claim that /i(2, x(<)) < A, t > tQ, where x(t) = x(t, tQ, x0) is any solution of (1.1.1). If this is not true, then, because of any solution of (2.4.12), there exists a solution x(t) of (1.1.1) with h0(t0, x0) < A and a t1 > tQ such that &(
tv

By (Hi), this yields b(A)
(2.4.13)

Using (ii), we obtain by Theorem 2.4.1,

V(t, x(t)) < j(t, to, u0), t0
tx,

where 7(f,^0)uo) 1S the maximal solution of (2.4.1). relations (2.4.11), (2.4.13) and (2.4.14) imply

(2.4.14) The

b(A) < 70(*t,x(*x)) < 7o(*i»*o,«o) < KA), since VQ(t0, xQ) < a(hQ(tQ, x0)) < a(X). This is a contradiction which proves (hQ, /impractical stability of the system (1.1.1). We shall next prove that the system (1.1.1) is (hQ,h)strongly practically stable for (\,A,B,T) > 0. To do this, let us suppose that (2.4.1) is strongly practically stable for (a(X),b(A),b(B),T) > 0. This means we need to prove only

Refinements

142

(h0, /^-practical quasi-stability of the system (1.1.1).

The

practical quasi-stability of (2.4.1) means that N

N

^2uoi = 1

< aW implies J j u , ( £ , t 0 , u 0 ) < b(B), '

i= 1

t>t0

+ T.

(2.4.15)

Suppose that h0(t0, x0) < A so that by (h0, /impractical stability of (1.1.1), we have h(t,x(t))

< A, t>t0.

Consequently, the

relation (2.4.14) holds for all t > t0, that is, V{t, x(t)) < 7 ( t , t0, u0), t > t0,

(2.4.16)

which yields because of (2.4.15), (2.4.16), (Hi) and (iv), b(h(t,x(t)))

< V0(t,x(t))

< l0(t,to,uo)

Thus we have, whenever h0(t0,x0)

< b(B), t>tQ

< A, h(t,x(t))

+ T.

< B, t>t0

and hence the system (1.1.1) is (h0,h)-strong\y

+T

practically

stable. One can prove similarly other (h0, /^-practical stability properties of (1.1-1) and hence the proof is complete. If we desire only uniform practical stability properties, we can relax the assumptions of Theorem 2.4.4 considerably following Theorem 1.3.4. We shall not discuss such results to avoid monotony.

Chapter 2

143

The next result is on (h0, /i)-boundedness. As we have seen in Section 1.5, when we investigate boundedness properties, we also utilize class 3G functions. However, in this situation, the class 9G is defined by 9G = [a G (p, oo), R + ], oo as u—»oo. We shall employ the same symbol 3G for both situations, with this understanding. Theorem 2.4-5: Assume that (i) h0, h G T and h(t, x) <

V 6 C[R + x S°(h, p), R1^. ], V(t, x) is locally Lipschitzian in x and for (t,x) E R+ x Sc(h, p), D+

V(t,x)
where
V0(t,x) < a(hQ(t,x)), ifhQ(t,x) < p0. Then boundedness properties of (2.4.1) imply the corresponding (h0,h)-boundedness properties of (1.1.1).

Refinements

144

Proof: Let a > p0 and t0 e R + be given and let at = a(a). Suppose that (2.4.1) is uniformly bounded. There, given c*! > 0 and t0 G R + , there exists a /?x = ^(a) > 0 such that N

N

]Tu 0 . < ctx imply £ i=l

'

u

.(*> *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(<,
(2-4.18) It then

KP) < V0(t2,x(t2)) < l0(t2itx,V(tltx(h))) < W ,

Chapter 2

145

since V0(
con

tra-

We can prove analogously other (h0, /i)-boundedness properties and hence the proof is complete. We shall next consider a result on (h0, /i)-uniform ultimate boundedness under a different set of conditions. Theorem 2-4-6: Let the assumptions except that (ii) is strengthened

of Theorem 2.4.5 hold

to

(ii*) D + V p(t, x)<

- C(h0(t, x)), C G% and

D + Vi(t,x)

< gi(t,V(t,x)),

i^

Then uniform

boundedness

P)

(t,x) e

R+xSc(h,p).

of (2.4.1) implies

(h0,h)-uniform

boundedness of (1.1.1). Proof:

By Theorem 2.4.5 with gp(t,u) = 0, it follows that

(1.1.1) is (h0, /i)-uniformly

bounded.

Setting a = p0

and

B = (3(p), we have h0(t0, xQ) < p0 implies h(t, x(t)) < B, t> t0. Now let h0(t, xQ) < a for any a > p0. Then we claim that there exists a t* E [t0,t0 + T], where T = T(a) > ^ y , h0(t*, x(t*)) < p0. Po < h0(t,x(t)), a

^o(*oi ^o) < -

such

that

If this does not hold, then we would have t G [t0,to + ^]> f ° r

a

solution x(t) of (1-1.1) with

We then obtain using (ii*)

Refinements

146

0 < Vp(t, x(t)) < Vp(t0, x0) - C(Po)T < a(a) - C(pQ)T < 0, by the choice of T.

This contradiction proves the existence of

a t* e[tQ,t0 + T] with h(t, x(t)) < B, t>t*

h0(t*,x(t*))
which yields

that

>t0 + T. The proof is therefore complete.

We shall next discuss two global theorems of a very general character and utilizing them, we derive some results on stability and boundedness to demonstrate the effectiveness of the results obtained. We begin with the following theorem which provides a general set of conditions for preventing the solutions that start in a given set through any given part of the boundary. Theorem 2-4-7: (i)

n

E CR

Assume

that

is an open set, H C R" is such that H C E and

G C dH; (ii)

V G C[R

+

x E, RN], V(t, x) is locally Lipschitzian

in x

and D + V{t,x) < g(t, V(t,x)) for (t,x)eR+x where g G C[R

+

x R , RN] and g(t, u) is

E, quasimonotone

nondecreasing in u; (Hi)

V0(t,x)=£

Vi(t,x) >a{t)for(t,x)ER+xG » =

and

l

^o(*o>xo) < «(fo) ifxQEHQcE where a G C[R

+

,R];

such that H0C\H

^ ,

147

Chapter 2 (iv)

any solution u(t,t0,uQ)

of (2.4.1) satisfies

N

N

£M,-(',
< a M>* > *o» */ Z ) u o i < a(*o)-

«• = I

i= i

Then there exists no t* > t0 satisfying x(t) G H,t G [tQ,t*) and x(t*) G G, where x(t) = x(t,t0,x0) XQ

is any solution of (1.1.1) with

^ Ho-

Proof:

Suppose that there exists a t > tQ such

x(t) £ H,t (E.[t0,t*) x(t) = x(t,t0,x0)

and

i(<*) £ G

for

some

that

solution

of (1.1.1). Then assumption (in) yields V0(t*,x(t*))>a(t*).

(2.4.19)

Choose u 0 = V(t0, x0) so that condition (ii) gives, by Theorem 2.4.1, the relation V(t, x(t)) < r(t, tQ, u0), t G [to, t*),

(2.4.20)

where r(i,i 0 ,u 0 ) is the maximal solution of (2.4.1). We now let x0 G H0 which implies by (Hi) V0(t0, xQ) < a(tQ). N

that £ ui0 < a(tQ).

This means

It therefore follows readily from (2.4.20)

»= i

and (iv) that VQ(t*,x(t*))
where rQ(t,t0,u0)

= £ r,(i, 20, u 0 ), which contradicts (2.4.19).

Hence the theorem is proved.

Refinements

148

buppose that the solutions of (1.1.1) that start in a given set HQ are required to reach another set F C H in a finite time. The theorem that follows offers a set of sufficient conditions for such a behaviour. Theorem

2.4-8: Let the assumptions

except that (ii) is strengthened (if)

10,D

where

w G C[R

(t,x)eR+

xF ,

+

x E,R

FcH,G

2.4.7 hold

to

for

c

of Theorem +

+

],

Vp(t,x)<-w(t,x),

w(t,x)>i(t)

for

= dH and 7 6 C[R + , R + ].

Then there exists a t* > t0 such that xQ G HQ implies x(t*) G F for any solution x(t) = x(t,t0,x0) Proof:

Since

the

of (1.1.1).

assumptions

of

Theorem

2.4.7

are

satisfied with gp(t, u) = 0 and G = dH, if follows that x0 E H0 implies x(t) G H,t>

t0,

where x(t) = x(t, tQ, x0) is any solution of (1.1.1).

Choose a

T > 0 such that tQ + T j

1(s)ds>a{t0).

(2.4.21)

to We then claim that there exists a t* G [tQ,t0 + T] satisfying x(t ) G F.

If this is not true, there would exist a solution

149

Chapter 2 x(t) = x(t, tQ, x0)

with

xQ e H0

such

that

x(t) £ Fc,

t € [ M o + T]. Then we get by (it*).

t0 + T 0 < Vp(t, x(t)) < Vp(t0, x0) - J

i(s)ds

to

tQ + T
J

t0 + T -r(s)ds
which is a contradiction.

J

i(s)ds<0,

Hence the claim is true and the

proof is complete. Let us next apply Theorems 2.4.7 and 2.4.8 to derive stability and boundedness results under weaker assumptions. Let us first prove (h0, /instability. Theorem 2.4-9: (0

h0,h£T

Assume

that

and h(t,x) < ^>(h0(t,x)) if h0(t,x) < p0,<j>(p0) < p,

where 6 3G; (it)

V eC[R+

xS(h,p),RN],V(t,x)

is locally Lipschitzian

in

x and D + V{t,x) < g(t, V(t,x)) for (t,x)(=R+x

S(h,p),

where g € C[R + x RN, R ], g(t, u) is quasimonotone decreasing in u;

non-

Refinements

150

(Hi)

£ V,(i,x) = V 0 ( t , x ) - * - o o

as

hQ(t,x)-*Q

for

each

i = 1

teR

+

, V0(t, x) > b(t, h(t, x)) for (t, x) e R + x S(h, p)

where b £ C[R

+

x [/?, oo), R]; N

(iv)

for

every

r£(0,p),

if

T, uoi<°(toir)>

then

i = i

N

£ Ui(t,t0,uQ) < b(t,r) for t > tQ, where u(t,tQ,uQ)

is any

i = l

solution of (2.4.1). Then the system (1.1.1) is Proof:

For

any

(h0,h)-equistable.

(t0, e) G R + x (0, />),

there

exists

a

x

8 = S(t0, e) > 0 such that () < e and /*0(*o> o) < ^ implies V"0(<0,x0) < b(tQ,e) by (m). Note that, because of (i), we have h(t0,x0) < (h0(t0,x0)) <
= S(h,e),H0

= S(h0,6),G

= dS(h,e)

Then we see that all the assumptions of

Theorem 2.4.7 are satisfied and hence the conclusion follows. Next result considers (h0, /i)-boundedness. Theorem 2.4-10: Assume (i)

h0,heT

that

and h(t,x) < <j)(h0(t,x)) if h0(t,x) < p0,<j)(p0) < p

and G %;

151

Chapter 2 (ii)

V EC[R+xSc{h,p),RN],V(t,x) is locally Lipschitzian + in x and D V{t,x) < g{t,V{t,x)) for

{t,x)eR+x Sc(h,p), where geC[R+x RN,RN] and g{t, u) is quasimonotone nondecreasing in u; {Hi) V0{t,x) > 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))
Refinements

152

Hence all the assumptions of Theorem 2.4.7 are verified and the proof is complete. We shall finally give an application for (h0, /i)-uniform asymptotic stability. Theorem 2.4-11: Let the assumptions of Theorem 2.4.0 hold except that we strengthen (ii) by («*) for 1 0, D + Vp(t, x) <- w(t, x), where w e C[R + x S{h, />), R + ], w(t, x) > C(hQ(t, x)),

Ce%. Suppose also that b(t,r) = b(r). Then the system (1.1.1) is (h0,h)-uniformly asymptotically stable. Proof: Since gp(t,u) = 0 and b(t,r) = b(r), Theorem 2.4.9 yields (h0, h)-um{orm stability of (1.1.1). Fix e = p,6Q = 6(p) and let /lo^o^o) < 0, such that h0(t*,x(t*)) < S(n). Consequently, it follows that whenever h0(t0, x0) < 60, we have h(t, x(t)) < n, for t > *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) < <*i a n d II R(t,x) || < 82

Refinements

154 (ox (t,x) <E S(h,e),

(2.5.2)

imply h(t, y(t, t0, x0)) < e, t > 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»
U(t2,y(t2,tQ,x0)) = 6(e)

and a(Si) < U{t,y,t0,x0)) < 6(e) on [^,i2]Setting x-, = y(ti,to,Xo)i if *2 — *i ^ T? by Gronwall inequality we have I V{tx + r, yfa + r, t l5 z j ) -1/(*! + r, z(*j + r,
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))
Refinements

156 < a(6x) + k82r < 6(e), which is a contradiction. The proof is complete.

Theorem 2.5.2: Assume that (i) hQ, h G T and hQ is uniformly finer than h; (ii) V £ C[R + xR",R + ], V(t, x) is h-positive definite, h0descrescent and D + VllmlA)(t,x) < -C(hQ(t,x)),

(t,x) e S(h,p),

(Hi)

| V(t, x) - V(t, y) | < M || x - y ||, M > 0, (t,y)eS(h,p). Then the system (1.1.1) is (h0,h,T-^-totally stable.

C£%; (t, x),

Proof: Let us write D + V/25ml\(t,y) for the time derivative of V along the solutions of the perturbed system (2.5.1). Then it follows from (ii) and (Hi) that D + V{2,SA)(t, y)<-

C(h0(t, y)) + M || R(t, y) ||,

(t,y)eS(H,p).

(2.5.6)

Since U(t,x) is /i-positive definite and A0-decrescent, there exist constant />oe(0,/>), 8Q > 0 and functions a, b E% such that V(t, x) < a(h0(t, x)) if h0(t, x) < S0, and

(2.5.7)

b(h(t, x)) < V(t, x) whenever h(t, x) < p0.

(2.5.8)

Chapter 2

157

Let e £ (0, po) be given. Choose 6X 6 (0, S0) such that a

(*i) < Ke)

and

h(t,x) < e if ft0(t,x) <

tfj,

(2.5.9)

because of the assumptions on a,b and condition (i).

For

ike (0,1), choose S2 = kC(S1)/M. Let f„e.R + a1"1 y(t) = y(t,t0,x0) be a solution of (2.5.1). We claim that h(t0,x0) < 6, and || R(t,y) || < S2 for (2,y) 6 S(h,e) implies h(t,y(t))<e,t>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 . !>(<)) II <S*2,t>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. This contradiction shows the existence of t* and thus it follows from (h0, /i)-total stability of (1.1.1) that the system (2.5.1) is (h0, /i)-attractive. The proof is complete. A variant of the notion of total stability with respect to perturbations may be defined if we only require that the perturbations be bounded in the mean. Definition 2.5.2:The system (1.1.1) is said to be (h0,h,T2)totally stable if for every e > 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<62. t

(2.5.15)

(2.5.16)

Theorem 2.5.4-' Assume that (i) h0, h € T and h0 is uniformly finer than h; (ii) V € C[R + x Rn, R + ], V{t,x) is h-positive definite, h0descrescent and D + V{vlA)(t,x)< (m)

-C(V(t,x)),

(t,x)eS(h,p),Ce%;

\V(t,x)-V(t,y)\<M\\x-y\\,

(ttx),

(t,y)

£S(h,p),

M>0; Then the system (1.1.1) is (h0,h,T2)-totally stable. Proof: We proceed as in the proof of Theorem 2.5.2, and choose S1 = 61(e) by relation (2.5.9). Let hQ(tQ,x0) < St and m(t) = V(t, y(t)) where y(t) = y{t, r0, x0) is any solution of (2.5.1). It is evident that m(t0) < a ^ ) , which implies h(t0, x0) < e. We claim that m(t) < 6(e), t > tQ. If this is not the case, there exists a tl>tQ m(t) < 6(e) for / 0 < t < t „ which implies that h{t,y(t))
t0
such that

Chapter 2

161

Denote tl — t0 = T, and choose S2 = S2(e) < 6(e) - J ' l [ J(a(5 1 ))]/M,

(2.5.17)

where J(u) — J(uQ) = /577T and J " 1 is the inverse function of J. From (M) and (m), we obtain, for t G [^o>*i]> D + V(t, y(t)) < - C(V(t, y(t)) + M\\ R(t, y(t)) ||, if we now define Z(t) =

V(t,y(t))-u(t),

where t

u(t) = MJ\\R(S,y(s))\\ds, to we obtain the inequality D + Z(t)<

-C(Z(t)),

using the monotonic character of C(u) and the fact Z(t) > V(t,y(t)), which implies, in view of Theoreml.3.1 that Z(t) < J - V(V(t0, x0)) -(t-

to)], t G [t0, t,).

Note that the maximal solution of u' = — C{u), u(tQ) = V(tQ, x0) is just the right-hand side of the foregoing inequality. Thus, it follows that

V(t,y(i)) < J-V(V(t0,x0)) - (i - to)} + K*), * € [t0M

Refinements

162

From this, we derive a further inequality, using the facts that h{t,y(t)) <e for t0 < t < tQ + T, V(t0,x0)

< a f o ) and relations

(2.5.15), (2.5.16) and (2.5.17),

b(e) < V(t0 + T, y(t0 + T))<J~ VWi)) This

~T\ + MS* < W-

contradiction assures that m(t) < 6(e), t > t0, which in

turn implies (h0,h,T2)-tota\

stability of the system (1.1.1).

This completes the proof. In

order

to

unify

the

investigation

of

stability

properties of perturbed motion, it is sometimes useful to utilize coupled comparison functions. Of course, the use of coupled functions is also beneficial in the study of stability properties of unperturbed motion, since estimating D + V(t,x) by a function of t, x and V is more advantageous then by a function of t and V only. Consider the comparison equation u' = g(t, x(t), u), u(t0) = u 0 , where g G C[R+ x Rn x R + ,R]

and x(t) - x(t,t0,xQ)

solution of (2.5.1) existing on [*0,oo).

(2.5.18) is any

We define stability

concepts of the trivial solution u = 0 of (2.5.18) as follows. Definition

2.5.3: The trivial solution of (2.5.18) is said to be

/^-conditionally stable, if given e > 0 and t0 G -R + , there exist positive constants 8X = 8x{t0,t), 82 = 82(t0,e) such that

Chapter 2

163 h0(t0, xQ) < S2 and u0 < Sx imply u(t, t0, x 0) u0) <e,t>

where u(t, t0,x0,u0)

t0,

(2.5.19)

is any solution of (2.5.18), h0, h are

functions as defined in Section 1.1. The other stability concepts can be defined in a similar fashion. We state the following comparison result whose proof is similar to that of Theorem 1.3.1. Theorem

2.5.5: Let

locally Lipschitzian

V G C[R in x.

R] and for (t,x)eR+x

x Rn,R

+

] and

Assume that g(zC[R+

Let x(t) = x(i, t0, x0)

be any solution

of (2.5.1) existing

be the maximal

Then V(t0,x0)

V(t,x(t))

solution

< u0 implies

< i(t,t0,x0,u0),

t > t0.

Let us prove the following result.

(i)

xRnxR

is +,

V(t,x)
[£0,oo) and ■y(t,t0,x0,u0)

Theorem 2.5.6:

V(t,x)

R",

D+

existing for t > 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
*0

5+1

9

[J \(6)ds]d6

(2.5.21)

v —1

t

[J K0)d8]ds= J G(s)ds, 5


where G(i) = / X(s)ds. Let t

Q(f) = sup[G{s):t - 1 < s < oo] so that Q G £. Let 0 < e < /> and <0 G R + be given. Choose a £ = (5(e) > 0 such that 25 < e and aT = r(e) > 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<6 and if possible, suppose that u(ti) = u(i1,<0,x0,u0) > e. Then there would exist a t2 > i 0 s u c n ^ a *

Refinements

166

u(t2) = 6 and 5 < u(t) < e for t € [t2,
<8-c(5)(t1 -1 2 ) + J G{s)ds t2-\ < * - ( * ! - *2)[ - c(tf) + Q(T)] + Q(r) <£ + <3(r) 2 ^ 2 '

which is a contradiction. uniformly stable.

Hence u = 0 is /^-conditionally

Next, we shall prove /^-conditional uniform attractivity. Taking e = p, set S0 = S(p) and r 0 = r(p). Because of (h0, /i)-uniform stability of (2.5.1) it follows that ^o(*o>xo) < ^o implies h(t,x(t)) < p, t> t0. Let 0 < n < p and t0€ R+ be given. Choose 6 = 6(n) and r = T(T}) as before. Choose T = [c(6)T(r)) + 2Q(l) + 2p}/c(8)>T(n)

Chapter 2

167

and note that T = T{rf) only. Let us suppose that u0 < 60 but u(t) >6fort£

[t0 + r, tQ + T}. Then we get

0 < S < u(t0 + T) < u(t0 + r ) + [ - c(S) + Q(t0 + r)] X ( T - T ) + <3(*0 + T)

< / ) - i C ( 5 ) ( r - r ) + Q(l) = 0 which is a contradiction. Thus, there exists a tx such that u(tl7tQ + T,x07uQ)< S which implies u

(*, *o + r> xoi uo) <e for t>t0

+ T.

Hence u = 0 is /^-conditionally uniformly attractive, proving the claim. The proof is thus complete. We note that whenever either A(t)—»0 as /—>oo or *+ i

A E Lr[0, oo) we have / \(s)ds—>0 as £—>oo and hence these t

cases are included in the foregoing discussion.

The special

case discussed above results when system (1.1.1) is

(h0,h)-

uniformly asymptotically stable and u>(t,x) = M || R(t,x) || , M being the Lipschitz constant for the Lyapunov

function.

Consequently, one can conclude the /i 0 -conditional uniform attractivity of u = 0 of (2.5.18) by Theorem 2.5.6.

Refinements

168

Sometimes, it is more advantageous to discuss directly the inequality D + V(t,x)
(2.5.22)

rather than the corresponding comparison equation (2.5.18). In this approach, we can also weaken the assumption on the perturbation term. Let us assume that in (2.5.20), we have | u(t,x) | < \(t) whenever (t,x) G S(h,p) D Sc(h0,r)) for any 0 < n < p, t+i

A satisfying the condition / \(s)ds—*0 as t—►oo. Then we claim that (2.5.1) is (h0, /i)-uniformly attractive. Proceeding as before, let us choose 6 and r as follows: 2a(S) < 6(e) and 2Q(T) < mm(c(*),6(c)), where a, 6 £ 3G are the functions resulting from properties of V. If h0(t0,x0) < 6 and there exist tu t2 such that, for t0 > r, h0(t2,x(t2)) = 8, M'l.sfo)) = e>

(*,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 - <2) + Q(r) + a(<5)

169

Chapter 2


Proceeding

T = [0(6)^) + 2Q(1) + 2a(p)]/c(S) > r(rj) and suppose, if possible, that 8 < h(t, x(t)) for t €[t0 + r, t0 + T] whenever ^o(*o> xo) < ^oThen, setting y0 = x(tQ + r,r 0 ,i 0 ) and using (2.5.22), we are again lead to the contradiction 0 < b(S) < V(t0 + T,x{t0 + T,t0 + r,y 0 ))


Variation of Lyapunov's method.

In this section, we develop a new comparison theorem that connects the solutions of perturbed and unperturbed differential systems in a manner useful in the theory of perturbations. This comparison result blends, in a sense, the two approaches namely, the method of Lyapunov functions and the method of variation of parameters, and consequently

Refinements

170

provides a flexible mechanism to preserve the nature of perturbations. The results that are given in this section show that the usual comparison theorem (Theorem 2.4.1) in terms of a vector Lyapunov function is included as a special case and that perturbation theory could be studied in a more fruitful way. Consider the two differential systems y' = f(t,y),y(to)

= xo,

(2-6.1)

x' = F(t, x), x(t0) = xQ,

(2.6.2)

and

where f,Fe

C[R+ xRn,Rn\.

Relative to the system (2.6.1),

let us assume that the following assumption (H) holds: (H)

the solutions y(t,t0,x0)

of (2.6.1) exist for all t > tQ,

unique and continuous with respect to the initial data and | y(t, t0, x0) | is locally Lipschitzian in x0. For any V e C[R + x Rn,R1*}

and any fixed <€[0,oo), we

define D-V(s,y(t,s,x)) = lirn_ inf\[V(s

+ h, y(t, x + h, x + hF(s, x))

-V{s,y(t,s,x))\ for i 0 < s < t and x £ Rn.

(2.6.3)

171

Chapter 2

The following

comparison result which relates

the

solutions of (2.6.2) to the solutions of (2.6.1) is an important tool in the subsequent discussion. Theorem

2.6.1: Assume

that

the

assumption

(H)

holds.

Suppose that (i)

V EC[R+

xRn,R^],V(s,x)

and for t0 < s
is locally Lipschitzian

£ R,

D _ V(s, y(t, s, x)) < g(t, V(s, y(t, s, x))); (ii)

g EC[R+

x R + ,R

in x

n

],g(t,u)

is

(2.6.4) quasimonotone

nondecreasing in u and the maximal solution r(t, t0, u0) of u' = g(t, u), u(t0) = u0 > 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

<s
Then using the assumptions

Refinements

172 D _ m(s) < g(s, m(s)),

t0<s
which yields by Lemma 1.3.1 the estimate m(s)
m(t0) < u0.

V(t,x(t,t0,x0)),

Since

(2.6.7) m(t) = V(t,y(t,t,x(t)))

=

the desired result (2.6.6) follows from (2.6.7)

by setting s = t. Taking

u0 = V(tQ,y(t,t0,x0)),

the

inequality

(2.6.6)

becomes V(t, x(t, tQ, xQ)) < r(t, t 0 , V(i0, y(t, t0, x0))), t > t0,

(2.6.8)

which shows the connection between the solutions of systems (2.6.1) and (2.6.2) in terms of the maximal solution of (2.6.5). A number of remarks can now be made: (1)

The trivial function f(t,y)

= 0 is admissible in Theorem

2.6.1 to yield the estimate (2.6.6) provided V(t0lx0)

< u0.

In this case y(i, t0, x0) = x0 and the hypothesis (H) is trivially verified.

Since y(t, s, x) = x,

the

definition

(2.6.3) reduces to

D_V(s,x) = lim_inf\[V{s

+ h,x + hF(s,x)) - V(s,x)]

(2.6.9)

which is the usual definition of generalized derivative of the Lyapunov function relative to the system (2.6.2). Consequently, Theorem 2.6.1 reduces, in this special

Chapter 2

173

case, to Theorem 2.4.1. (2)

Suppose that

f(t,y)

continuous matrix.

= A(t)y

where A(t)

The solution y(t,t0,x0)

then satisfy y(t, t0, x 0 ) = $(£, t0)x0l fundamental $(t0,t0)

matrix

solution

= I (identity matrix).

clearly verified.

is a

nxn

of (2.6.1)

where $(£, t0) is the of

y' = A(t)y,

with

The assumption (H) is

Suppose also that g(t, u) = 0.

Then

(2.6.6) yields V{t, x(t, to, x0)) < V{t0, $(*, *o- (2-6.10) K, on the other hand, g(t, u) = Bu where B = (6 tJ ) is an N x N matrix such that 6tJ- > 0 for i ^ j , we get a sharper estimate V(t, x(t, t0, XQ)) < V(t0, *(*, t0)x0)exp(B(t

- to)),

t > tQ.

(2.6.11)

Clearly the relation (2.6.11) helps in improving the behavior of solutions of (2.6.2) relative to the behavior of solutions of (2.6.1). This is a great asset in perturbation theory F(t,x)

and = f(t,x)

it

can

+ R(t,x)

be

seen

where

R(t,x)

by

setting is

the

perturbation term. (3)

Suppose that f(t,y)

is nonlinear, fy(t,y) n

continuous for (t, y) £ R + x R . that

the solutions

y(t, t0,x0)

exists and is

Then, it is well known are differentiable

with

Refinements

174 respect t o (i 0 , a;0) a n d we h a v e f ^

# ( * , *o, xo) - ~ $(*, *o)/(*oi *o)» * > *o> UL o

,„ „ , n. (2.6.12)

%Hi,t0,xQ) = $(t,t0,x0) ux0 where $(/, i 0 , i 0 ) is the matrix solution of the variational equation z' =

fy(t,y(t,to,xo))z-

If V(s, x) is also assumed to be differentiable, then by (2.6.12), we have, for a fixed i, D _ V(s, y(t, s, x)) = V.(s, y(t, s, x))

(2.6.13)

+ Vx(s, y(t, s,x))- 9{t, s, x) ■ [F(s, x) - f(s, x)]. The relation (2.6.13) gives an intuitive feeling of the definition (2.6.3). (4)

When the solutions of (2.6.1) are known, a possible Lyapunov function for (2.6.2) is W(s,x)

= V{s,y(t,s,x))

where V(s, x) and j/(i, s, x) are as before. We need the following definitions.

(2.6.14)

Chapter 2 Definition

175 2.6.1.-The

system (2.6.1) is said to be

(h0,h0)-

practically stable at t0, if, given (A, A) with 0 < A < A, we have h0(tQ, x0) < A implies h0(t0, y(t)) < A, t>t0 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) <
(in)

g £ C[R

+

x R1^, RN],

g(t, u)

is

quasimonotone

nondecreasing in u and there exist a, b £ 3G such that b(h(t, x)) < V0(t, x), (t, x) £ S(h, A) and V0(t, x) < a(hQ(t, x)), if h0(t, x) < A where VQ(t,x) = £

(2.6.16) (2.6.17)

V{(t,x).

i = 1

(iv)

(p(X) < A and a(A) < b(A) hold.

Then the practical corresponding perturbed

stability

(h0,h)-practical

system

properties stability

(2.6.3) provided

(2.6.1) is (h0,h0)-practically

of (2.6.5) imply properties

the unperturbed

of

the the

system

stable at t0 with respect to (A, A).

Refinements

176 Proof:

Assume that (2.6.5) is strongly practically stable.

Then, we have, given (A, ArB,T) > 0 such that

\
£ ) ufa t0, tie) < b(A), t > t 0l if £ ) u0. < o(A), (2.6.18) i = l

i = 1

and JV

N

^ u 0 . < a(A) implies ^«,(*,t 0 ,u 0 ) < 6(B), i = 1

'

.' = 1

t > t0 + T.

(2.6.19)

Since (2.6.1) is (/i0, ft0)-practically stable at t0 with respect to (A, A), we have K(^ !/(<)) < A, < >
(2.6.20)

where y(t) = y(t,t0,x0) is any solution of (2.6.1). We claim that h0(t0, x0) < A also implies that h(t,x(t)) < A, t> 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))
tv

Then by Theorem 2.6.1, we have V(t,x(t)) < j(t,t0,V(t0,y(t))),

t0
tv

Consequently, we get

b(A) < VQ(t„x(ti)) < E 7 i ( < . . < o , # o > « i = 1

177

Chapter 2

<£7,(Mo»aW)<W i=i

This contradiction proves that h0(t0,x0) < X implies h(t,x(t)) < A, t> t0. To show strong practical stability, we see from the foregoing argument that b(h(t,x(i)) < V0(t,x(t)) < E 7 i ( M o ^ ( i o , y W ) ) , * > *o, i=l

if h0(t0,x0) < X.

From this, it follows that b(h(t,x(t))) <

f)7,(/,< 0 ,a(A)),<>< 0 .

This, together with (2.6.17) yields

i=1

(hQ, /i)-strong practical stability of the system (2.6.2) and the proof is complete. Setting F(t,x) = f(t,x) + R(t,x) in Theorem 2.6.2, we see that although the unperturbed system (2.6.2) is only practically stable, the perturbed system (2.6.2) is strongly practically stable, an improvement caused by the perturbing term. Let us present a simple but illustrative example. x' = e ~ *x2, x(t0) = x0, y' = - y, y(t0) = j / 0 , whose solutions are given by X(M

°'X°)

=

l+* 0 (e-°-e-V

(2.6.21)

178

Refinements y(t,to,y0) = y0e'{t~to\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(<,B,y)I(s,a:,y), where i?, L are perturbations so that the perturbed differential system is given by (

x' = e ~ lx2 + R(t, x, y), x(t0) = x0,

I

y'= -V + L(t,x,y),y(tQ) = y0.

(2.6.23)

2

2

Let R(t, x, y) = ■=£- and L(t, x, y) = —jjp-. Then it is easy to compute 9i(t,V1,V2)=

-V?,g2{t,Vx,V2)=

-VtV2,

so that the comparison system reduces to 3

(

if I

u[ = -Mi 5 ,ti 1 (« 0 ) = u 1 0 ,

, , u2= -u 1 u 2 ,u 2 (i 0 ) = u20.

< 2 - 6 - 24 )

179

Chapter 2

choosing tx10 = Vx(t0,z/(i,t0,a:0)),u20 = V2(t0,, y(t, tQ, x0)) we find the solutions of (2.6.24) are given by 4ujo

u 1 (i,i 0 ,u 0 ) =

[2+< 0 (<-g] 5 u2(t, t0, u0) = u20exp

2u10(t -10)

2+

,t>t0.

2

Ul 0(t-t0)

Thus by Theorem 2.6.1, the estimates for the solutions x (t, i0, x0, y0), ~y (t, t0, x0y0) (of 2.6.23), are of the form [H-x0((e-'-e-to)

+ ( l^o ) ) p

l ? ( M o , « o ) l 2 < l2/ol2e~2(t"'o) x exp[ [l+x0(e-<-e-'° + ( ^ % l

+ Xo(e-<-e-

f

°)]

for t > t0, which show that all solutions x" (t, i0, x0, j/ 0 ), 1/ (£,£0,x0,?/o)—>0 as 2—>oo, although from (2.6.22), it is clear that the solutions of the unperturbed system (2.6.21) do not enjoy this nice property. In fact, for tQ = 0 and x0 = 1, we get x(i,t 0 ,x 0 ) = e\y(t,tQ,y0) = e \t> 0.

Refinements

180 2.7

Integral stability.

We shall continue to study system (1.1.1) and its perturbed system (2.5.1). Corresponding to total stability, we shall introduce notions of integral stability which characterize boundedness and Lagrange stability of the solutions of (1.1.1) with respect to perturbations of both the initial conditions and the right-hand side of the equation. Definition 2.7.1: The system (1.1.1) is said to be (7X) (h0, h)-equi integrally stable if, for every a > 0 and to G R + , there exists a positive function /? = fi(t0, a), which is continuous in tQ for each a and /? G 9G for all each t0, such that, for every solution x(t) = x(t,t0,x0) of the perturbed system (2.5.1), the inequality

h(t,x{t))t0, holds, provided that V^o.Zo)^0 and, for every T > 0

t0 + T /

sup

\\R(s,x) II ds
(J2) (h0, h)-uniformly integrally stable if the ft in (7J is independent of t0.

181

Chapter 2

(J3) (h0, h)-equi asymptotically integrally stable if (Jx) holds and, for every e > 0, a > 0 and t0 G R + , there exist positive numbers T = r(£ 0 , a, e) and 7 = 7(r 0 ,a,e) such that, for every solution of the system (2.5.1), the inequality h(t,x(t))<e,

t>t0 + T,

holds, provided that h0(t0, xo) — a oo / J

sup

an

d

|| R(s, x) |j ds < 7.

(.,x)€5(fc,/J)

(74) (h0, /i)-uniformly asymptotically integrally stable if the T and 7 in (J3) are independent of tQ and (J 2 ) holds. In addition to the scalar differential equation (1.3.1), let us consider the following perturbed equation u' = g(t, u) + A(i), u{tQ) = u0, where g£C[R+

(2.7.1)

x R + ,R] and A G C[R + ,R + ].

Definition 2.7.2: The null solution u = 0 of (1.3.1) is said to be (Ji) equi integrally stable if, for every ax > 0, tf0 £ i2 + , there exists a positive function /?x = /?i(£0>ai) that is continuous in <0 for each a1 and /?x £ 3G for each tQ such that if uQ < ax and for every T > 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