(Plovdiv) Stability of the general exponent of non-linear

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXX (1990)

D. D.

(Plovdiv), P. P.

(Minsk), S. I.

(Plovdiv) Stability of the general exponent of non-linear

differential equations with impulse effect in a Banach space

Abstract. In the paper a theorem is proved which guarantees stability under small perturbations of the general exponent of non-linear systems with impulse effect in a Banach space.

1. Introduction. Recently, in relation to numerous applications, the theory of differential equations with impulse effect has begun to develop rapidly. The first work dedicated to this subject is the paper by Millman and Myshkis [2].

Among other works in this field we mention [1], [3-5].

In the present paper the general exponent of equations with impulse effect of a special type is considered and conditions for its stability under quite natural assumptions are found.

2. Statement of the problem. Let X be a complex Banach space. Consider the equation with impulse effect

(1) dx

i =/(t> x)

(2) x(tn + 0) = Qnx(tn- 0)

where t i_ < t2 < ... are fixed moments of impulse effect satisfying the condition lim,,-^ tn oo.

We shall say that conditions (H) are fulfilled if

HI. The function/: [0, c o ) x X - * X is continuous and f ( t , 0) = 0.

H2. For f0 ^ 0 the Cauchy problem

(3) ^dx

⁴

I t =/(t’x)’

^*0

has a unique solution x(t) which is defined for

H3. The operators Q„ ( n = 1 , 2 , . . . ) are linear, bounded and map X into X.

D

1. A solution of the equation with impulse effect (1), (2) with the

initial condition (3) is a piecewise continuous functions x(t) (t ^ t0) with

discontinuities of first kind at the points t„ (n = 1, 2, . . . ) such that

— = / ( r , x ( 0 )

x{tn + 0) = Qnx{tn- 0) (n — 1 ,2 ,...).

We assume that at tn (n = 1 ,2 ,...) the function x(f) is left continuous.

L

1. Let conditions (H) be satisfied. Then for (t0, xo)e[0 , oo)x X the Cauchy problem for the equation with impulse effect (1), (2) with the initial condition (3) has a unique solution which is defined on [t0, со) and satisfies the integral equation

t

(4) x(t) = Q(t, t„)x„ + j Q(t, s)f{s, x(s))ds,

( ⁵ ) Q ( t . z ) = n e».

The proof of Lemma 1 is carried out on each interval [f„, t„ + i) by the standard methods for ordinary differential equations without impulse effect.

By means of the equality

(6) U(t, t0)x0 = x(t; t0, x0)

where x(t; t0, x 0) is the solution of the Cauchy problem for (1), (2) with initial condition (3) we can define a two-parameter family of operators U(t, t0) (0 ^ t0 < t < со) mapping X into itself. This family has the semigroup property (7) U(t,

)U(

, t0) = U(t, t0) (0 ^ t0 ^

^ t < со)

but is not a continuous operator-valued function since (8) U(tn + 0, i) = QnU(tn — 0,

) (tn >

) and, analogously,

(9) U (t,tn + 0) = QnU (t,tn- 0 ) (t > t„)

De f in it io n

2. A general exponent x(r) of the equation with impulse effect (1), (2) is the greatest lower bound of the numbers ô for which there exist numbers r and N such that for any solution x(t) satisfying

(10) \\x{t0)\\ ^ r

the following inequality holds:

(11) IlX(Oil ^ Ne5(t_x)i|x(T))| (t0 ^

^ t < oo).

If such numbers <5 do not exist, we set x(r) = oo.

Note that in the case considered the classical equality x{r) = sup lim

JC(to)|| § r T,t-T->0O

ljx(t)\\/\\x(x)\\

t — T

Stability of general exponent 21

is not always valid since the solutions of (1), (2) are piecewise continuous, therefore they are not continuously differentiable.

Note that the numbers N in (11) have to satisfy the inequalities

(12) l i e j ^ N (n = 1 ,2 ,...).

Let us consider the conditions under which (1), (2) has a finite general exponent.

L

2. Let the following conditions be fulfilled:

1. ||

(t,

MeS(t~z) (0 ^

^ t <

where M and Ô are constants.

2. II/(t, x)|| </x(t)||x||, where the function p{t) (t ^ 0) is such that t

(13) exp(M jg(s)ds) ^ Ney{t~z) (0 ^ т ^ t < oo) and N and у are constants.

3. Condition H2 holds.

Then the general exponent x(r) satisfies the inequality

(14) x(r)^<5 + y.

P ro o f. By Lemma 1, from (4) for 0 ^ т ^ t < oo we have t

x(0 = Q{t, т)х(т) + |б (г , s)f(s, x(s))ds.

X

From conditions 1 and 2 of Lemma 2 we obtain t

||x(t)|| ^ Мед{*~х)\\х(т)\\+М§e0(t~s)p,{s)\\x(s)\\ds.

X

Applying the lemma of Gronwall-Bellman to the last inequality we obtain t

||x(f)|| ^ Me*(t_t)||x(T)||exp(M§g{s)ds).

X

The assertion of Lemma 2 now follows from condition (13).

Estimate (14) is rather rough. It does not always allow one to establish when the general exponent is negative. Note that this is possible if, for instance, it is possible to pass from the initial equation (1), (2) to the equation with impulse effect

(15) Yt = cz + ec'f( t, e~“z) (t Ф f„)>

(16) z(l„ + 0) = Q„z(t„) (n = 1,2 ,...)

with a positive parameter c, the solutions of which are related to the solutions of (1), (2) by means of the equality

z(t) = ectx(t).

The verification of condition 1 of Lemma 2 is in general a rather complicated problem. In the particular case when the sequence {£„} is an arith­

metic progression t„ = a + nh (n = 1 ,2 ,...; 0 ^ a ^ h) and Qn = Q (n = 1 ,2 ,...), condition 1 of Lemma 2 is satisfied for an arbitrary

<5>/i_1ln

(Q), where

(Q) is the spectral radius of the operator Q.

Consider the more general case when the sequences {£„} and {Qn} are periodic, i.e., for some constants к and T for « = 0 ,1 , . . . the conditions tn+k = tn+T, Qn+k = Qn hold. In this case condition (1) holds for any Ô > T ~ 1lnQ(Q1...Q k)1/k. For the general case simple estimates of Ô are unknown to the authors. Condition 2 of Lemma 2 is verified by means of standard methods of mathematical analysis.

R em ark 1. If the conditions of Lemma 2 are satisfied, the general exponents of all solutions of (1), (2) are estimated by one and the same number.

Le m m a

3. Let conditions (H) hold. Assume, moreover, that there exists a positive number h such that for some r > 0, the operator-valued function U(t,

) satisfies the conditions

(i) IIU(t,

)

|| ^M\\x\\ (|£-т| ^ h, ||x|| ^ r), (ü) \\U(t + h,t)x\\ < C ||x || (0 ^ t < oo, ||x|| ^ r), where С < 1 and M are constants.

Then the general exponent

of the equation with impulse effect (1), (2) satisfies the condition

(17) *(<?)</ Г 1 In C.

P ro o f. If ||x0II < r and t = T + nh + 0 for some positive integer « and 0 ^ в < h, then by the semigroup property (7) and conditions 1 and 2 of Lemma 3 we obtain the estimate

II ^U(t,

^{^} MCn\\x0\ \ ^ - e h- llnC^ \ \ x 0\\ M

which implies (17).

Lemma 3 is proved.

R e m a rk 2. Note that if S > x(r), then for some N

II [ / ( t , T)JC0 ||

If we set C = Meôh, then we obtain conditions 1 and 2 of Lemma 3,

moreover, if we choose h large enough, then the corresponding estimate for

x(r) will be close enough to <5.

Stability of general exponent 23

3. Main results. We shall consider the question of existence of a finite general exponent of the equation with impulse effect

(18) dx — = f { t , x ) + g{t,x) ( t i ^ t n)

(19) *(t„ + 0) = Qnx(tn),

where g: [0, oo)xX -»X .

De f in it io n

3. The function g(t, x) belongs to the class G(f, v, r) if for equation (18) the assertion of the local theorem for existence of a solution is valid (see e.g. [5]) and if, moreover,

(20) \MtiX)\\ ^ v||x|| (||x ||< r).

Th e o r e m

1. Let the following conditions be satisfied:

1. Conditions (H) hold.

2. The function f ( t , x ) satisfies the Lipschitz condition with a constant p > 0:

X2)\\

^ I K - x J ( ||x j , ||x2|| < r), 3. ||<2(£,

< Med<t~z), where Ô < 0.

4. There exists a general exponent x 0(r) of the equation with impulse effect (1), (2), moreover, x 0(r) < 0.

Then for any s > 0 there exist numbers p(0 < g ^ r) and v depending only on the function f ( t , x) and on the operators Qn (n = 1 ,2 ,...) such that the equation with impulse effect (18), (19) for g(t, x )e G (f, v, r) has a finite general exponent x(p) satisfying the estimate

(

) ^ x 0(r) + e.

P ro o f. In the proof of Theorem 1 we shall use Lemma 3. Let e > 0 be so small that X= x0(r) + e < 0 and x 0(r) < X < X. By Definition 2, there exists a number N such that each solution £(t) of (1), (2) satisfying ||£(

)|| < r also satisfies

(21) Ш011 s ;N e * '-,,K(T)|| ( T < t < o o ) . Let h > 0 be such that

(22) Ne(X~1)h< 1.

We shall show that the equation with impulse effect (18), (19) satisfies the conditions of Lemma 3 for C = ekh, whence the proof of Theorem 1 will follow.

Let v > 0 be chosen in such a way that

c ( ô - l ) h c M u h ( c M ( n + v ) h 2 ) ^ i _ j V g < A -3 t>*

jU + V

(23)

and the number

be chosen so that

(24) g N h ^ r,

where

N h = N + - ^ e M»h{em>l + v)h- \ ) . g + v

We shall show that conditions 1 and 2 of Lemma 3 hold.

Let Ц-х(т)j| ^

Note that for all

such that the solution of (18), (19) lies in the ball ||л:|| < r the following integral identity holds:

t t

x{t) = Q{t,

)

(

) + JQ(t, s)f(s, x{s))ds + $Q{t, s)g(s, x{s))ds.

T T

By conditions 2 and 3 of Theorem'1/*since^(t, x )s G (f, v, r), we have

*

IHOII ^ Me0(t~T) ||x(

)|| + j Me0(t- s) (/r + v) || x (s) || ds.

X

Applying to the last inequality the lemma of Gronwall-Bellman, we obtain (25) ||x(r)|| ^ Me(ô + + v))(t“ T) ||x(

)||.

Denote by £(£) the solution of (1), (2) which satisfies the initial condition

£(

) =

(

). Then

x(t)-Ç(t) = fQ(t, s)(f(s, x(s))-f(s, Ç(s))ds + fQ(t, s)g(s, x(s))ds

X X

for which as above we obtain the inequality

t t

||x(£) —£(£)|| ^ jM e 0(t~s)g ||x(s) — £(s)|| ^5 + J’M ^ (t~s)v||x(s)|| ds.

T г

From the above inequality and (25) for £е[т, т + h] we deduce the inequality

||x(£)-£(£)|| ^ $Meô(t~s)g\\x(s)~Ç(s)\\ ds

+ —— e0{t ~ x)(eM{fl + v)h — 1) ||x(r)|| . g + v

Applying again the lemma of Gronwall-Bellman we obtain (26) l|x(0-£(t)|| < eâ«~z)^ - { e MUl + v)h-\)\\x{T)\\eM»h.

g + v From (21) and (26) it follows that

M m ^ \ m \ \

^ {NeX(t - z) + e0{t ~z) eMtlh + v)h - 1 )} || x(

) ||,

l g + v J

Stability of general exponent 25

i.e.,

From (27) and (24) it follows that

\\x(t)\\ ^

7Va^ (i~t)

^ t ^ т + /г),

which implies that the solution x(t) for t e [ i , т + h] lies in the ball ||x|| ^ r.

Inequalities (27) and (23) imply

+

< elh

, i.e., conditions 1 and 2 of Lemma 3 hold. The proof of Theorem 1 follows from Lemma 3, whence, in particular, we obtain x(g) ^ 1 ^ x0(r) + £.

Theorem 1 is proved.

Consider the perturbed equation with impulse effect

where the An are linear bounded operators mapping X into itself.

D

4. The sequence {An} belongs to the class D(L, 3, {Qn}) if

T

2. Let the assumptions of Theorem 1 hold. Then for any e > 0 there exist numbers

,

and L depending only on the function f ( t ,

and on the operators Qn so that the equation with impulse effect (28), (29) for g(t,

eG(f, v, r) and {An}

D(L, 3, {Qn}) has a finite general exponent x{g) such that

x(q) ^ x 0 ( r ) + e -

The proof of Theorem 2 is a modification of the proof of Theorem 1.

R e ma r k 3. In Theorems 1 and 2 the conditions 3 < 0 and x0(r) < 0 are not essential. By the exponential substitution

= eatz for

suitably chosen we can reduce the general case to the one when x 0{r) and 3 are negative.

[1] Ju. L. D a le c k ii and M. G. K rein , Stability o f solutions o f differential equations in Banach space, AMS, Books and Journals in Advanced Mathematics, 1974, 386 pp.

[2] V. D. M ill m an and A. D. M y s h k is , On the stability of motion in the presence o f impulses, Sibirsk. Mat. Zh. 1 (2) (1960), 233-237 (in Russian).

(29)

(28) — = f ( t, x) + g{t, x) ( t ^ t n), x(*„ + 0

(Qn + A ^ x if ) ,

(30) \\Q(t, s ) - Q ( t , 5)|| ^ Le0{t~s) { O ^ s ^ t ^ o o ) , where the operator Q(t, s) is defined by (5) and

(31) 6(t,s)= П

References

[3] A. D. M y s h к i s and A. M. S a m о i 1 e n к о, Systems with impulses in prescribed moments of the time, Mat. Sb. 74 (2) (1967) (in Russian).

[4] A. M. S a m o i le n k o and N. A. P e r e s t iu k , Stability o f the solutions o f differential equations with impulse effect, Differ. Eqs. 11 (1977), 1981-1992 (in Russian).

[5] P. S. S im e o n o v and D. D. B a in o v , Stability under persistent disturbances for systems with impulse effect, J. Math. Anal. Appl. 109 (2) (1985), 546-563.

PLOVDIV UNIVERSITY “PAISSII HILENDARSKI”

PLOVDIV, BULGARIA

BYELORUSSIAN STATE UNIVERSITY MINSK, BYELORUSIAN SSR