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ROCZN1KI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria 1: PRACE MATEMATYCZNE XXX (1990)

A. B. D

is h l ie v

(Sofia) and D . D . B

a i n o v

(Plovdiv)

Continuous dependence of the solution of a system of differential equations with impulses on the initial condition

Abstract. The initial value problem for systems of differential equations with impulses is considered. For the systems under consideration the impulses are realized at the moments when the integral curves meet some of previously fixed hypersurfaces in the extended phase space. Sufficient conditions for the continuous dependence of solutions on the initial conditions are found.

1. Introduction. Systems of differential equations with impulses provide an adequate mathematical description of numerous phenomena and processes studied by physics, chemistry, radiotechnics, etc. By means of such systems phenomena and processes subject to short-time perturbations during their evolution are studied. The duration of the perturbations is négligeable in comparison with the duration of the phenomena and processes considered, therefore they are regarded as “momentary” of the type of “impulses”.

The first publications on mathematical theory of systems with impulses were by V. Milman and A. Myshkis [5] (1960) and [6] (1963). This new and perspective theory was developed further in the works of A. Samoilenko [13]

(1961) and [14] (1962), T. Pavlidis [11] (1967), V. Raghavendra and M. Rao [12] (1974), S. Pandit [7] (1977), [8] (1980), etc. Recently the interest in systems of differential equations with impulses has grown considerably due to nume­

rous applications of these systems to mathematical control theory. The first two monographs dedicated to this subject appeared, by Halanai and Veksler [3] (1974) and by S. Pandit and S. Deo [10] (1982). Systems of differential equations with impulses are also the subject of the present paper.

2. Preliminaries. Consider the hypersurfaces

(1) Grf: t = tt(x), te R , x e D, i = 1, 2, ..., where D is a domain in Rn.

Denote by Pt a point with current coordinates (t , x{t)) such that the law of its motion is described by means of:

a) the set of hypersurfaces <rf, i = 1, 2 , . . . , defined by (1);

b) a set of functions ij: D->R", i = 1, 2, ...;

c) a system of ordinary differential equations

— = f ( t , x), dx t ф т(, i = 1, 2, ...,

( 2 )

(2)

where/: S -^ R n, S = {(t , x): t ^ 0 , x e D } . The points zt, i = 1, 2, satisfy­

ing the inequalities

(3) 0 < tj < i 2 <

are the moments at which the mapping point Pt meets the hypersurfaces (1).

Note that Pt meets the hypersurfaces (1) only at the moments тг;

d) the equalities

(4) х(т,. + 0 )-х (т г- 0 ) = 1]{(х( т,.)), i= 1 , 2 , . . . ,

where / is the number of the hypersurface met by Pt at the moment тг. In general it is possible that i Ф / (examples can be given such that for some i the inequality i < / holds, as well as examples for which i = / or i > /).

We assume further that х(т;) = х(т; —0), i.e. x is left continuous at тг, i= 1 , 2 ,. ..

The set of objects a), b), c) and d) which determines the motion of Pt is called a system of differential equations with impulses. The law of motion of Pt is called a solution of the system of differential equations with impulses and the curve described by this point is called an integral curve of that system.

In the present paper the initial value problem for such a system is considered with the initial condition

(5) x(0) = xo, x0eD.

We shall give a brief description of the motion of Pt along the integral curve of this initial value problem. The initial position of Pt is (0, x0). For 0 ^ t ^ i j , Pt moves along the integral curve of (2) with initial condition (5). At the moment i l5 Pt meets the hypersurface uh at the point (il5 x x) where xx = x C tj ) and “jumps instantly” from ( t 1# xx) to ( t 1, x 1 + Ijl(xi)). Further on it moves along the integral curve of (2) with initial point ( t 1? x 1 + IJi(x1)) until the moment t 2 when it meets the hypersurface ah , “jumps” again, etc. If after a “jump” the point Pt meets a hypersurface from (1), then a new “jump” at this moment is not realized.

The solution of such a system is a piecewise continuous function with points of discontinuity of 1st type at which it is left continuous.

We shall use the following notation: xt = х(тг); x f = х,. + / л(х{), i = l , 2 , . . . ; т0 = 0, j 0 = 0, £0(x) = 0 for x e D , X q = x0; by x(t, i*, x*) we denote the solution of the system with impulses with the initial condition

x ( t *, t *, x*j = x*; x(t) = x(t, 0, x0); <■, •) is the Euclidean scalar product in Rn; || • || is the Euclidean norm in Rn; q (A, В) is the Euclidean distance between the nonempty sets A and В in Rn. The set of points (t , x ) E R x R n satisfying t — x* ^ L||x — x*|| where ( t *, x *) is a fixed point in R x Rn and L is a positive constant will be denoted by K(z*, x*, L).

The qualitative theory of systems with impulses in the case when the

functions ti5 i = 1 , 2 , . . . , are constant, therefore tfx) = тг, xeD (i.e.' when the

(3)

impulses are realized at fixed moments), is comparatively better developed than in the case when the t( are not constant. One of the reasons is that in general it is possible for the integral curve of a system with impulses to meet repeatedly one and the same hypersurface from (1). This phenomenon is called “beating”.

When “beating” is present, the integral curve need not be defined for t sufficiently large. Hence one cannot claim that, e.g., the solution of the system depends continuously on the initial condition in a given finite interval since it need not be defined in this interval. In the following example “beating” is observed.

E

x a m p l e

1. Let n = 1, D = (0, + oo), t f(x) = arc tan x + m , x e f i , i = 1, 2, .. . Then for every choice of functions / and I t, i = 1, 2, ..., such that:

a) / ensures the existence and uniqueness of the solution of (2) for t ^ 0;

b) Ifx ) > 0, x eD ,

the integral curve (t, x(t, 0, x 0)) “beats” on the hypersurface (in this case on the curve) t = tj(x) for every choice of the initial point (0, x0) such that x0 > 0.

What is more, in this case the solution of the equation with impulses is not defined for t ^ n/2.

3. Main results

1. Continuous dependence on the initial condition when “beating” is absent.

Denote by (A) the following conditions:

Al. For t ^ 0 the function f is continuous in its first argument and uniformly Lipschitz in x e fi with a constant L.

A2. There exists a positive constant M such that \\f(t, x)|| < M for {t,x)eS.

A3. The functions tt are Lipschitz in x e D with respective constants Lt < 1/M, i = 1 ,2 , .. .

A4. The inequalities

(6) 0 < ц(х) < t2(x) < ..., x e D hold and lim^*, tt(x) = + oo uniformly in x e D .

A5. tfa + Ifx)) ^ ti(x), x e D , i = 1 , 2 , . . .

A6. The integral curve (t , x(t)) of the system x(t) = x(t, 0, x0) does not leave the set S for t e l where

We shall show that conditions (A) are sufficient for the absence of

“beating”. For this purpose we shall use the following theorem.

T

h e o r e m

1 [ 1 ] . Suppose that:

1. Conditions A2, A3, A5 and A6 are satisfied.

i = 1

У (if-i, t J otherwise.

if there are finitely many moments xi?

(4)

Then the integral curve (t, x(t)) of the system with initial condition (5) meets each of the hypersurfaces (1) not more than once.

By means of Theorem 1 we get the following assertion:

T heorem 2. Let conditions (A) hold. Then the integral curve (t, x(t)) meets each of the hypersurfaces (1) not more than once and x(t) = x(t, 0, x 0) exists for all t ^ 0.

Proof. If (t, x(t)) meets no hypersurface from (1) for t > 0 then the proof is trivial. Suppose that (t , x(f)) meets at least one hypersurface from (1) for t > 0. Conditions A1 and A6 imply the existence and uniqueness of the solution x(t) for t e l .

Since for i f < t ^ Tj + 1 the solution x(t) coincides with the solution of the integral equation

t

(7) x(t) = x f + f f ( t , x ( t )) dr, i = 0 , 1 , . . . , ti

by A2 for t = 1 , 4-1 we obtain

(8) lk-+i-X i+ ll ^ М(т/+1- ^ ) ,

i.e. condition 2 of Theorem 1 is satisfied. Thus, according to Theorem 1, (t, x(t)) meets each of the hypersurfaces (1) not more than once.

Finally, we shall show that x(t) is continuable for t ^ 0. In fact, if (t, x(t)) meets a finite number of hypersurfaces from (1) for t > 0, then the claim follows directly from A1 and A6. Suppose that (t, x(t)) meets infinitely many hypersurfaces from (1). We shall show that j i+1 > j t, i = 1 , 2 , . . . {jt is the number of the hypersurface met by the integral curve at the moment i f Suppose that this is not true, i.e. j i + l < j { (the equality j i + l = j i is impossible since (t , x(t)) meets each of the hypersurfaces (1) not more than once). Then, according to (6),

Oi + iW < x e D - Hence

(9) Ъ + i = tJi + f x i + 1) < Гл(х{ + 1).

Condition A5 implies

(10) t j . ( x f ) ^ t j f X i ) = т{.

Let т'е(т;, т1 + 1). By (10) and A3

(11) tj.(x(0)-T i ^ ^(х(т'))-*д(*4+) < ^ ||х (т')-х ;+ || . By (7) and A2 we obtain

Then we have which implies

( 12 )

||x(T)-Xi+ || < М ( т '- Т 4).

*м(Х(т')) <

(5)

Consider the function q>(t) = £7i(x(t)) — t defined in the interval [т', t / + 1]. Since for x' ^ ^ t i + 1 the function x is a solution of the integral equation (7), it is continuous in this interval; hence tp is also continuous for x' ^ ^ xi + 1.

From (9) and (12) we deduce (p(x')(p(xi+1) < 0. Hence there exists x", xt < x' < x" < ij + i, which satisfies (p(x”) — 0, i.e. x" = tj.(x(i")). It follows that (t, x(t)) meets the hypersurface иh at the moment x". This contradicts the fact that for xt < t < xi+ ! the integral curve (£, x{t)) meets no hypersurface from (1).

Hence j i + 1 > j t, i.e.

(13) 1 < j 2 < ...

By (13), since the are positive integers, we obtain l i m ^ ^ = +co; hence, according to A4, we find

(14) lim xt = lim t,-.(xf) = + oo.

Since x(t) is defined in each of the intervals ( t ;, + i = 0, 1, ..., and (14) holds, we conclude that x(t) is continuable for all t ^ 0.

This completes the proof of Theorem 2.

It may happen that an integral curve of a system of differential equations with impulses, satisfying conditions (A), does not meet all hypersurface from (1), as is seen by the following example.

E xample 2. Let n = 1, D = (0, +oo), up. t = x + i, i = 1 , 2 , . . . , dx — = 0, Ax{f)\t=ti = - x / 2 , i = 1, 2, ...

at

Conditions (A) are satisfied for M = 1/2, Lt = 1, i = 1 , 2 , . . . Nevertheless for x0 > 2 the integral curve (t , x(f)) does not meet the hypersurface (in this case the curve) u2.

T heorem 3. Suppose that : 1. Conditions A1 and A6 hold.

2. The functions t ^ i — 1 , 2 , . . . , are continuous in D and the integral curve (t, x(t)) for t > 0 meets at least one hypersurface from (1).

Then

(^ + i - t s(^ i+ i))(^ -ts(^+)) ^ 0, i = 0, 1 , .. ., s = 0, 1,...

Proof. From A1 and A6 it follows that the solution x(t) with initial condition (5) exists and is unique in each of the intervals (тг, т[ + 1], i = 0, 1,...

Consider the function (p{t) = t — ts(x*(t)) defined in [ if, if+i] where x*(t) = K ’ t = Ti’

jx(£), i , < f < T i+1.

From the continuity of x* it follows that q> is continuous for xt ^ t ^ r[ + 1.

(6)

Suppose that

(Ti+ i - t s(xi + 1))(Ti- t sxi+)) < 0.

Then <р(т{+1)<р(т{) < 0, which implies that there exists T\T{< T ' < T i+1, such that <р(т') = 0, i.e. %' = £ s ( x ( t ')), which means that (t , x(f)) meets the hypersur­

face <7S at the moment x'. This contradicts the definition of r f, i = 1, 2, ... (for T; < £ < тг+1 the integral curve (f, x(£)) meets no hypersurface from (1)).

This completes the proof of Theorem 3.

Denote by (B) the following condition.

B. For each x e D and i = 1, 2, tt(x) < £,+ f x + fix)).

T heorem 4. Let conditions (A) and (B) be satisfied. Then the integral curve (t, x(t)) meets each of the hypersurfaces (1) precisely once.

Proofi First we shall show by induction that if at the moment xt, i = 1, 2, the integral curve (£,oc(£)) meets the hypersurface (jj{, then j. = j, i= 1,2, ...

Suppose that the integral curve meets oy, at the moment xx. Then Ti = 0i(x(Ti))- Suppose that j 1 > 1. Then by (6)

(15) Tt = th (x(tj)) = tjfxj) > t f x j . By the first inequality of (6) we find

%o = 0 < t f xo) . Together with (15), this contradicts Theorem 3.

Suppose th a tjs = s for s = 1, 2, . . . , i and the integral curve (t , x(t)) meets oji + 1 at the moment t i + 1, i.e. i i+1 = tji + 1(x{Ti + 1)). According to (13) we have

; i + 1 ^ i + 1, Suppose that j i + l > i + l . By condition (B) and the inductive assumption (jt = i) we obtain

(16) ti + 1{x ? )-T i = ti + 1(xi + I i{xi) ) - t i{xù> о.

The assumption j i + 1 > i-t-1 yields

( 1 ”7) £i+ l(-^ i+ l)

t i

+ 1 ^ i+ l(-^ i+ l)

t j i

+1

i p ^ i +

1) ^

Inequalities (16) and (17) contradict Theorem 3.

Now we shall show that the integral curve (t , x(t)) of the system with impulses with initial condition (5) meets cr1. Suppose that (t , x(t)) does not meet cr1. Then for t > 0 the integral curve (t , x(t)) meets no hypersurface from (1) at all (above we have shown that if an integral curve does meet hypersurfaces from (1), then the first of them is ox). Hence for t ^ 0

(18) t < t^xit)).

In fact, if we suppose that there exists x' > 0 such that x' ^ £1(х(т')), then for the

function (p{t) — £1(x(£)) — t we find <p(0) > 0 and (p(x')^O. It follows that

(7)

there exists x" such that <p(x") = 0, i.e. (t , x(t)) meets By (18), using A2 and A3, we obtain

t — t f x 0) < t ^ x f ) ) — î ^ X q ) < L1Mt, i.e.

(19) t i(*0)

\ - L xM const.

(19) contradicts the fact that (18) is satisfied for all t > 0 (and therefore for t > const as well).

Now suppose that (t, x(t)) meets the hypersurfaces <r1, . . . , at the respective moments x1, . . . , xt and that for t > тг it meets no hypersurface from (1). Then t < ti + 1(x(t)) for all t > xt hence

^ ^ 1_L- M +1 ~~^i+1 ’ which leads to a contradiction.

This completes the proof of Theorem 4.

Let T be an arbitrary positive constant.

D efinition 1. We shall say that the solution x(t) of a system of differential equations with impulses with initial condition x(0) = x 0, defined for 0 ^ t ^ T, depends continuously on the initial condition if for every choice of г > 0 and ц > 0 there exists Ô = <5(г, rj) > 0 such that each solution у = y(t) of the system with initial condition y(0) = y0, where ||x0 — y0|| < <5, satisfies ||x(t) — у (Oil < e for 0 < t ^ Ta nd \t — t ,-| > rj, where t 1? t 2, ... are the moments at which the integral curve (t, x(t)) meets the hypersurfaces (1).

Denote by (C) the following condition.

C. The functions It, i = 1, 2, ..., are continuous in D.

T heorem 5. Let conditions (A), (B) and (C) hold. Then the solution x(t) of the system with impulses depends continuously on the initial condition x(0) = x 0for 0 ^ t ^ T where T is an arbitrary positive constant.

Proof. If for 0 ^ t ^ T the integral curve (t , x(f)) of the problem under consideration meets no hypersurface from (1), then the proof follows directly from Theorem 2.1 (cf. [4], Ch. 5, §2). Suppose that for 0 t 5^ T the integral curve meets at least one hypersurface from (1). Then from (14) it follows that there exists a nonnegative integer p such that

0 =

t

0 < Tj <

t

2 < . . . <

tp

< 7 4

t p

+ 1 .

According to Theorem 4 the integral curve (t, x(t)) meets successively, precisely once, each one of the hypersurfaces (1). Hence for 0 ^ t < T the integral curve meets successively the hypersurfaces cr1,o’2, ..., op. From (7) for x{ < t ^ тг+1 we deduce that the solution x(t) satisfies the integral equation

t

x ( t ) = -x(Ti) + /,(x(Ti))+ f/(T, x ( x \ ) d t .

(20)

(8)

By (20), we obtain inductively

t

(21) x(t) = x 0+ £ /;(хг) + |/(т , x{x))dx, 0 ^ t ^ T.

t i < t 0

Let y(t) be the solution of the system with impulses with initial condition y(0) = y0, and let the moments at which the integral curve (t , y(t)) meets the hypersurfaces (1) for 0 < t < T be rçl5 ц2, ..., rjq, q = q(T' y0). Analogously to (21) for the solution y(t) we obtain

(22) y(t) = y0 + Z ! i(yù + № > У(т)) àx, O ^ t ^ T ,

4 i < t 0

where yt = yfaj, i = 1, 2, ...

Introduce the notations: 0\ = тт(т,-, ^), 9" = тах(т,-, rçf), = (9-, 9"], i — 1, 2 ,. . ., Q = [0, Г] \( J i Qt. It is easily seen that for t e Q the number of summands on the right-hand sides of (21) and (22) is one and the same. Hence by (21) and (22) we get

IM 0-y(0ll ^ ll*0-J'oll + Z

X i ^ r i i < t

+ j ||/ ( i , *(т) ) - / ( т> у ( т ) ) | | t eQ,

whence 0

(23) M 0 -y (0 ll ^ И*о-Уо11+ Z lki(-x(Ti)) — ^i(y(Ti))||

x i

<

m < t

+ Z ||/ i(^(^-))-A-(y(^))|| + Jb||x(T)-y(T)|| dx

V i < X i < t 0

+ Z 1МЖ))-*,-( у ( ъ ))|| + Z I |^i Л (Ti)) ~ h (x (*/;))| I » t eQ.

X i ^ t j i < t V i < X i < t

Suppose that x1 ^ rix. We shall show that

(24) Цх('П)-у(т1)|| < ехр^т^Цхо-уоЦ.

In fact, by (23) for 0 ^ t ^ xt we get

l|x(t)-y(OII < l!*o-yoll + l L ll*(T) - y ( T)MT, • о

from which, using the lemma of Gronwall-Bellman, we deduce l|x(0-y(0ll ^ lUo- ^oll exp(Lt).

By the last inequality for t = %x, we obtain (24).

Put hx = *i(y(Ti))~ Ti anc^ ^2 = *h ~ L(y(Ti))- From A3 we deduce К = fi(y(Ti))-fi(^ (Ti)) < lly^i) —^(^i)ll-

Therefore, according to (24), we obtain

(25) h1 ^ Lj exp(Li1)||x0 —y0||.

(9)

Since

(26) lly(Ti) —y(?7i)lt < M O b - i J = M{hl + h2), the following estimate for h2 holds:

h2 = ti(y^h))-h(y(^i)) < L iM n i)-y (? i)\\ < L ^ i h ^ h J , which implies

, L , M ,

(27) к> * Т Г Ц м к>-

Having in mind (25) and (27), for we find the estimate (28)

(29)

By (26) and (28) for Цу^) — y(^/i)|| we obtain L l M exp(LTx)

lly(^i) —y(^7i) 1 —LjM *o 3^o 11

We shall estimate the expression |[x(i / 1 +0) — y ^ + 0)|| provided that

*?i < t 2 (further on we shall show that if Цх 0 — y0Ц is sufficiently small, then 0'/-i < O'i, i = 1 , 2 , . . . , 0'o = во = 0, which implies < t 2). We have

l|x(*7i +0) —y(i/i +0)|| ^ || x (?71+ 0 ) - x ( t 1 + 0 )||+ ||x(i1+ 0)-y(i?1+0)||

<

M ( r i 1

— т х) + || x ( t 1) + / 1( x ( t 1)) —y(^j) —/ 1(y(i/1))||

^ M(//1- t 1) + || x ( t 1) - j (0 i )II+ ||/ i (^( h ) ) - / i (>’(0 i ))||

^ M{rjl - Tj) + ||х(т 1 ) - у ( т 1)|| + ||y(T 1 )- y (i/1)||

+ ||/1(х(т1) ) - / 1(у(т1))|| + ||/1(у(т1) ) - / 1(у(1/1))||.

Using successively (28), (24) and (29), we now obtain (30) ||x(»h+0) —y(»h + 0)|| ^ (1 + LtM) exp (L t ^

Ьо-Уо11 1 —LjM

+ ||/ i (^( ti ) ) - / i (>;( ti ))|| + ||/ i ();( ti ) ) - / i ();(0 i ) If tj > rjt , we obtain inequalities analogous to (24), (28), (29) and (30) with the only difference that the roles of the moments t 1 and 17 x and of the functions x and у are changed. More precisely, for t e [ 0 , T] define the functions zt, i = 1, 2 , . . . , by

fx(r) if т,- > щ,

1 у (0 if b ^ Щ■

(10)

Then (31)

(32) (33) (34)

L , exp(L0'i)

& { - e \ =

I ^ - T j ^ ; ; \ 7 "

\ \ *о- у о\\>

I Zl(0ï)-Z i(0i

1 — LXM L t Mexp(L0i)

^ ——--- I x f l —L^ M

Ы9\)-у(9\)\\ < exp(L0i)||xo —y0||, y0ll>

|x(0'i' + 0)—y{9'{ + 0)|| ^ ( l + L 1M)exp(L0'1)

l^o-УоИ 1 - L j M

+ ||/1(x(^1) ) - / 1(3;(^1))|| + ||/1(^i(0Y))-/i(^1(^i)).

Assume that 9ï - t <9\, i = 1,2, ..., p + l (further on we shall show that these inequalities are fulfilled for sufficiently small values of ||x0 — y0||).

Consider the solutions x(r, 0"_l5 x(9"-l +0)) and y(t, 0"_15 у(9’/~1 +0)), i = 1, 2, ..., of the system with impulses which coincide for t > 0-'-i with the solutions x(t) and y(t), respectively. Analogously to inequalities (31 )— (34), we obtain

(35) || O','-0i || g L‘exp ~f - ||x(fl"_ t +0) - , + 0)||,

(36) |z(e;.')-z(e;.)ii ^

1 -L .M

L iMexp(L(9’i- 9 '/ - 1j)

\x (9 ^ 1+ 0 )-y (9 f/ - 1+0)\\, 1- Ц М

(37) М0;-)-у(0'ОИ ^ exp(L(9'i — 9'/- 1 ))|| x(0"_x + 0) — y{9'{-i + 0) ||, (38) || x (9" + 0) — у (9" + 0) ||

« (1+LjM ^ g ()i- 9i- l) llx w -. + 0) - , w . , + 0)||

+ ||/f(x(0D)- /,(у(0У)|| + HMz.-W)) - It{z,№) II-

Let £ be an arbitrary positive constant. Then from (35)— (38) for i = p + l and from condition (C) it follows that there exists a positive constant ô p = ô p (e) such that ||x(0" + O) — y(0" + O)|| < ôp implies

Q'p + \ — 0 p + i

< e>

\\z p + i ( 6 p + i ) ~ z p + i ( 6 p + i ) \ \

< e?

Ы9'р^ ) - У(9'р+1)\\ < 8, ||x(0"+1+ O )-y(0"+1+O)|| < £.

Analogously, for each i = p, p —1 , . . . , 1 we define successively positive constants ôp- 1, ô p- 2,...,<50 such that ||x(0'/-i +0) — y{9”- 1 +0)|| < im­

plies (39)

9"—9'i < £, i|zi(0;/)-zl.(0;)ii < £ ,

^(0D -y(^)ll < e, IM0" + O)-y(0" + O)|| < min(£, Ô,).

From (39) it follows that if ||x0 — y0|| < e then for every i = 1; 2 , . . . , p + l the

(11)

left-hand sides of each of the inequalities (35)— (38) are smaller than the arbitrarily chosen positive constant.

Using these results we conclude that if rj is an arbitrary positive constant, then for sufficiently small values of ||x0 — y0|| the following relations are fulfilled:

a) The following inequality holds:

p

+1

(40) I to,—

i = 1

b) 97 ^ 0'i + 1, i.e. max(тг, rj^ ^ min(Ti+1, rji + l), i = 1, 2, p.

c) If zp < T< zp+1, then the number of the meetings of the integral curve (t, y(t)) with the hypersurfaces (1) for 0 < t ^ Г is precisely p. In fact, if we put

rj = min(T—i p, i p+1 — T),

then from (40) it follows directly that p = q. Note that for T= t p+1 both equalities p = q and p +1 = q are possible.

d) Directly from (40) we obtain

{t; 0 ^ t ^ T, \t — Tt\ > rj} c Q.

From (23), using (39), condition (C) and the above conclusions it follows that for arbitrarily chosen positive constants г and ц, for sufficiently small values of ||x0 — y0|| the inequality

t

MO-yWII < s + $ L\\x{x)-y(T)\\dx, O ^ t ^ T , \t Tj| > ц 0

holds, which, by the inequality of Gronwall-Bellman, gives

||x(t) — y(OII ^ eexp(LT), 0 ^ t ^ T, \t — xt\ > r\, i = 1,2, — Theorem 5 is proved.

In the following theorem we shall prove the continuous dependence on the initial condition of the solution x(t) of the system with impulses substituting for condition (B) the less restrictive condition (D):

D. ts{x?) ф т„ i = 1, 2, ..., s = i, i + 1, ...

Note that under the assumptions of the following theorem it is possible for the integral curve (t , x{t)) of the problem considered not to meet each one of the hypersurfaces (1), i.e. the claim of Theorem 4 is no more true.

T

h e o r e m

6. Let conditions (A), (C) and (D) be fulfilled. Then the solution x{t) of the system with impulses depends continuously on the initial condition x(0) = x 0 for 0 ^ t ^ T where T is an arbitrary positive constant.

Proof. First we shall show that if ||x0 — y0|| is small enough, then for

0 ^ t < T the integral curves (t , x(t)) and (t, y(t)) either both do not meet, or

both meet the same hypersurfaces from (1).

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Assume that (t, x(t)) meets no hypersurfaces from (1) for 0 ^ t ^ T.

Denote by A and В the sets

A = {{t, x): 0 ^ t ^ T, x = x(t)}, B = {{t, x): t = tx(x), xeD}

where by D we have denoted the closure of D. Since A and В are closed, A is bounded and A n В = 0 , we have q (A, B) = ex > 0. Applying Theorem 2.1 (cf. [4], Ch. 5, §2), we conclude that if ||x0 — y0|| is small enough, then j|x(t) — y (f)!l < £l for 0 ^ t ^ T, which implies that the integral curve (t, y(r)) for 0 ^ t ^ T meets no hypersurface from (1).

Now assume that (t, x(t)) meets some hypersurfaces from (1) for 0 ^ t ^ T.

We shall prove that if

(41) ts(xf) < T{ < ts+1(Xi+),

then s +i = j i + l , i.e. the integral curve (t, x{t)) meets the hypersurface <rs+1 at the moment i i + 1. In fact, if we assume that j i + l > s + l , then by (6) (42) ta+1(xi+1) < tji + 1{xi+1) = t i+1.

Inequality (42) and the second of the inequalities (41) contradict Theorem 3.

If we assume that j i+1 < s, then the inequality

^•i+l tji+ifai+i) < ts(Xf + i)

and the first of the inequalities (41) contradict Theorem 3.

If we assume that j i+x = s, then by (41) we obtain

Ti+i-T < tji + i(xi + x) - t s(x?) = M*i+i) - * s(*i+) ^ j j W x i + i - x t II, which contradicts (8).

Having in mind that lim ^ ^ ts(x?) = + oo and by condition (D) we conclude that for any i = 1 , 2 , . . . there exists a nonnegative integer s such that inequalities (41) hold. Hence

(43) tJt + x _ ! (x^) < T; < tj. +, (x +).

In order to show that the integral curves (t , x(t)) and (t , y(f)) meet the same hypersurfaces for 0 ^ t ^ T it suffices to show that if ||x0 — y0|| is small enough, then

(44) tji + i- x{yf) <Vi < tJi + 1(y?), i = 1, 2 , .. ., p,

where у? = у{г}{) + (y(rç,)) and j\ is the number of the hypersurface met by (t, x(t)) at the moment zt.

The two integral curves meet oy at the moments ту and rjx respectively.

Assume that for i = 2, 3, ..., s — 1 inequalities (44) hold, i.e. the two integral

curves meet successively the hypersurfaces oy, oy2, ..., oys. We shall show that

for || x0 — y01| small enough inequalities (44) hold for i = s as well, which implies

that the two curves meet oys+1 at the moments i s+1 and rjs + 1, respectively.

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In fact, by the inductive assumption, analogously to (39) it can be shown that for ||x 0 — y 01 | small enough

(45) e:-Q's = |т,—f|J < e, ||x($' + O)-j;(0;4O)|| < 8 .

Assume that i s < 4s ^ б* (the case ts > r]s is considered analogously). Then

l|x,+ - y s+ ll « l|xs+ - x ( ,Î!)|| + ||x((;s) - y ,+ || M f e - T s)+ ||x ( 0 " + O )-y( 0 " + O)||, which, by (45), gives

(46) l|xs+ - y s+ || « (M + l)e.

The first inequality of (45) and inequality (46) imply (47) e((Ts, xs+), (f]s, y+)) ^ (M 4-2)г.

By (43) the point (is, xs+) belongs to the open set

G = {(t, x): ty. + 1 -i(x) < t < tjs+1{x), xeD}.

From (47) it follows that (rjs, ys+)eG for sufficiently small e, which implies (44) for i = s. By induction we see that for ||x 0 —y0|| small enough inequalities (44) are satisfied for i — 1, 2 ,..., p.

If (t, x(t)) meets no hypersurface from (1) for 0 ^ t ^ T, since for ||x 0 —y0||

small enough the integral curve (t , y(t)) also meets no hypersurface from ( 1 ), by Theorem 2.1 ([4], Ch. 5, §2) we conclude that the solution x(t) of the system with impulses under consideration depends continuously on the initial con­

dition.

Assume that (t, x(t)) meets some hypersurfaces from (1) for 0 ^ t ^ T. By Theorem 2 it meets each hypersurface no more than once. Then by A4 we conclude that for 0 ^ t ^ T the integral curve (t , x(f)) meets a finite number of hypersurfaces from (1). Let these be the hypersurfaces < 7 ^ , ah , ..., ajp (jx = 1).

For IIx 0 — y 0 II small enough the integral curve (t, y(t)) meets the same hypersurfaces for 0 ^ t ^ T.

The rest of the proof is similar to the proof of Theorem 5. In fact, analogously to (23), (35)— (38) we obtain

(48) il* (t)-y(t )il < Ц Х 0 0 Н+ Z IMx(Ti))-/ji0;(?l-))||

V i , r i i < t

+ Z ||7;Д*(^))-^(у(^))|| + К|1*М-у(^)11^

r j i < V i < t 0

+ Z И / л ( у ( т Д-Ы у ( ъ ))И+ Z ||^(*( т Л-^(*(^))11’ teQ’

^ + + 0)|U

1 —LhM

u r n - m u « + 0 ) 1 ,

(14)

Ы в \ ) - у ( в д \ \ ^

е х р ( Ь ( 0 ; - 0 Г _ 1))||х (0 Г _ 1 + О ) - у ( 0 Г _ 1 + О )||,

\ \ х ( в "

+ 0) —

у { в "

+ 0) Il

(1 + Lj. M) exp (L(0- — 0"_ x))

<

1 - L JtM I x {&l-1 + 0)— y(6"-1 + 0) ||

+ \\iJt( x № ) - i j M W )\\+

where the points 0 - and 0 "', the function z and the set Q were introduced in the proof of Theorem 5. From (48), repeating the arguments from the proof of Theorem 5, it follows that the solution of the system with impulses depends continuously on the initial condition.

This completes the proof of Theorem 6 .

2. Continuous dependence on the initial condition when “beating” is present.

Denote by (E) the following group of conditions:

El. The inequalities т г ^ ||/Дх)|| ^ M {, x e D , i = 1 ,2 ,..., hold were the constants mt and M { are positive.

E2. The functions tt, i = 1 ,2 ,...:

(i) are Lipschitz on x e D with respective constants L < ___ ^ ___ •

г М (т; + М,.)’

(ii) satisfy the inequalities

ti(x + I i+1(x)) < ti+1(x), xeD .

E3. For any i = 1 ,2 ,... there exist neighbourhoods Ufx), Vt(x) and Wfx) of each point x e D such that:

(i) Vfx) is bounded;

(ii) Ufx) c= Vfx) c= J^(x) c D;

(iii) U,(x)) » m ,+M „ q (R"\W,( x ), В Д ) > m, + M (;

(iv) t,(z,) » t,(z2) for z 1eV,(x)\U,(x) and z2<=Wt(x)\Vt(x).

E4. For x e D and each i = 1, 2 ,... there exists a unit n-vector yf(x) and a function a£: D->R such that

6 i < « ; ( * ) <

Q i ( z ) ,

у Д х )>

пади ’ zeVfx), where MLiM i

т{ — МЬ{т^

As was shown by Example 1, if the integral curve (t, *(£)) of the system with impulses meets infinitely many times one and the same hypersurface, it is possible for the solution x(t) not to be continuable for all t ^ 0. The following theorem contains sufficient conditions under which the integral curve meets a finite number of times each of the hypersurfaces ( 1 ).

T

h e o r e m

7 (cf. [2]). Let conditions Al, A2, A4, A 6 and (E) be satisfied.

Then the integral curve (t , x(t)) meets each of the hypersurfaces (1) a finite

number of times.

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T

h e o r e m

8 . Let conditions Al, A2, A4, A 6 and (E) hold.

Then the solution x(t) of the system of differential equations with impulses is continuable for all t ^ 0 .

Proof! If (t, x(t)) meets a finite number of hypersurfaces from (1) then, by Theorem 7, it follows that there are finitely many moments i f; hence A1 and A 6

imply that the solution x(t) is defined for all t ^ 0 .

Assume that (t , x(t)) meets infinitely many hypersurfaces from (1). Then by Theorem 7 only a finite number of members of the sequence of positive integers j l , j 2> •• • can be equal to 1, 2, etc. Hence lim ;-»^ = + со, which, by A4, gives

(14) and shows that x(t) is defined for all t ^ 0.

Theorem 8 is proved.

T

h e o r e m

9. Let conditions Al, A2, A4, A 6 , (C), (D) and (E) be satisfied.

Then the solution of the system with impulses depends continuously on the initial condition for 0 ^ t ^ T where T is an arbitrary positive constant.

P ro o f. If for 0 < t ^ T the integral curve meets no hypersurface from (1), then the assertion is proved analogously to Theorem 6 .

Assume that the integral curve meets some hypersurfaces from (1) for 0 ^ t ^ T. We shall show that j i+l ^ j t for any i = 1, 2 ,... In fact, suppose that there exists a positive integer s such that

(49) js + 1 < j s.

Since for ts < t ^ ts+1 the solution x{t) of the system with impulses with initial condition (5) coincides with the solution of the integral equation

t

x(t) = x f + j/(T, x(T))dT, Ts

by condition A2 we obtain the estimate

(50) ||xs+ 1 - x s+ || ^ M( ts + 1 - t s).

By (49), ( 6 ) and condition (ii) from E2 we find

(51)

h s + i ( X S

) < 0s+ ! + l(*S+) < ••• < t j ' . f x f ) < tjs(xs) = V

By (51) and condition (i) from E2 we obtain

ïs+ 1 - ï s < tjs+i(xs+l) - t j s+1( x f ~ ) ^ ^ \ \ x s+l- x f \ \ , which contradicts (50). Hence

(52) Л *Zj2 ^

In the proof of Theorem 8 we showed that inequality (14) is satisfied. From (14), (51) and Theorem 7, we conclude that the integral curve (t , x(t)) for 0 < t ^ T meets successively a finite number of hypersurfaces o^, cr2, ..., crp respectively nlt n2, . . . , n p times, where 0 ^ щ < +oo, i.= 1 , 2 , ...,p, np > 0

and p < oo. Assume that the integral curve (t , y(t)) of the system with impulses

— Commentationes Math. 30.1

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with initial condition y(0) = y0 for 0 < t ^ T meets successively the hypersur­

faces ox, <r2, aq, q < oo, respectively m l9 m2, mq times, 0 ^ mx < + 00 , i = 1, 2, q, mq > 0. Let ap and aq be the last hypersurfaces from (1) met respectively by the integral curves (t , x(t)) and (t, y(t)) for 0 ^ t ^ T.

We shall show that for ||x 0 — y0|| small enough the integral curves (t , x(t)) and (t , y(£)) meet for 0 < t ^ T the same hypersurfaces, an equal number of times each. Suppose that this is not true, i.e. n1 = ml , n2 = m2, ...

..., ns_! = ms_i and ns < ms and xx < щ where l = X f= i?1r The remaining three cases: ns < ms and xx ^ t]t; ns > ms and xr < rjr; ns > ms and xr ^ цг where r = 1 mi’ are considered analogously.

Let e be an arbitrary positive number. For ||x 0 — y0|| small enough, similarly to (47), we obtain

(53) е((тг, хг+), (rj[t yx+)) ^ (M + 2)e.

From condition (D) and the fact that the integral curve (t, y(t)) meets the hypersurface crs at least once for t > rjx it follows that (r}t, y;+)eG where

G = {(£, x): £s-i(x) < t < ts{x), xeD}.

Since G is open, from (53) it follows that for e small enough (тг, хг+)е G, which contradicts the assumption that (t, x(£)) for t > xx does not meet the hypersur­

face as.

The rest of the proof is analogous to the proof of Theorem 5.

Theorem 9 is proved.

E xample 3. Let n = 1, D — ( — со, + o o ) and assume the hypersurfaces (in this case they are curves) are of the type

(54) t = tx(x) = 5i — |x|,

4i,

x e [ - i , i],

x< £[-i, i], i = 1 , 2 , . . . Consider the differential equation with impulses

(55) dx/dt = \, x(ij + 0) —x (if) = 5 •(—l)l/4.

This equation satisfies the conditions of Theorem 7 for M = 1/4, mx = M x

= 5/4, Ц - 1, Ux(x) = ( —* —|x|, i + |x|), K-(x) = ( — i — 3 — |x|, i + 3 + |x|), И/.(х)

= ( —г —6 —|x|, f + 6 + |x|), ax(x) = 1 , and if the unit vector y,(x) (in this case it is one-dimensional) has a coordinate ( — l)1. Hence, by Theorem 7, the integral curve (t , x(£)) meets each of the impulse curves (54) a finite number of times.

E.g., if x 0 = 0, i.e. the initial point of the integral curve (t , x(£)) is (0, 0), then (£, x(t)) meets cr1 consecutively two times at (4, 1) and (5, 0) and for any i = l , 2 , . . . it meets a2i once at (Юг, 0 ) and <x 2l+ 1 three times each at (10/ + 3, 2), (Юг+ 4, 1) and (10/ + 5, 0).

It is easy to verify that this system satisfies conditions (C) and (D) too.

Then from Theorem 9 it follows that the solution of the initial value problem

from Example 3 depends continuously on the initial condition.

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References

[1] A. B. D is h li e v and D. D. B a in o v , Sufficient conditions for absence o f “beating” in systems of differential equations with impulses, Appl. Anal. 18 (1984), 67-73.

[2] —, —, Conditions for the absence of the phenomenon “beating” for systems of impulse differential equations, Bull. Inst. Math. Acad. Sinica 13 (3) (1985), 237-256.

[3] A. H a l an ai and D. V e k s le r , Qualitative Theory o f Impulse Systems, Mir, Moscow 1974 (in Russian).

[4] Ph. H a r tm a n , Ordinary Differential Equations, Wiley 1964.

[5] V. D. M ilm a n and A. D, M y s h k is , On stability of motion in the presence of impulses, Sibirsk.

Mat. Zh. 1(2) (1960), 233-237 (in Russian).

[6] —, —, Random impulses in linear dynamical systems, in: Approximate methods of solution of differential equations, Publ. House Acad. Sci. Ukr. SSR, Kiev 1963, 64-81 (in Russian).

[71 S. G. P a n d it, On the stability of impulsively perturbed differential systems, Bull. Austral.

Math. Soc. 17 (1977), 423^132.

[8] —, Differential systems with impulsive perturbations, Pacific J. Math. 86 (1980), 553-560.

[9] —, Systems described by differential equations containing impulses: existence and uniqueness, Rev. Roumaine Math. Pure Appl. 26 (1981), 879-887.

[10] S. G. P a n d it and S. G. D e o , Differential Systems Involving Impulses, Springer, Berlin 1982.

[11] T. P a v lid is , Stability o f systems described by differential equations containing impulses, IEEE Trans. AC-12 (1967), 43^15.

[12] V. R a g h a v e n d r a and R. M. R ao, On stability o f differential systems with respect to impulsive perturbations, J. Math. Anal. Appl. 48 (1974), 515-526.

[13] A. M. S a m o ile n k o , Application of the averaging method to the investigation of oscillations stimulated by momentary impulses in 2nd order auto-oscillatory systems with a small parameter, Ukrain. Math. Zh. 13(2) (1961), 103-109 (in Russian).

[14] —, On one case of continuous dependence of the solutions o f differential equations on a parameter, ibid. 14(3) (1962), 289-298 (in Russian).

HIGHER INSTITUTE OF CHEMICAL TECHNOLOGY, SOFIA, BULGARIA PLOVDIV UNIVERSITY “PAISSII HILENDARSKI”

PLOVDIV, BULGARIA

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