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Stability of the Zero Solution of Impulsive Differential Equations by the Lyapunov Second Method
Journal of Mathematical Analysis and Applications 248, 69᎐82 Ž2000. doi:10.1006rjmaa.2000.6864, available online at http:rrwww.idealibrary.com on
Stability of the Zero Solution of Impulsive Differential Equations by the Lyapunov Second Method M. U. Akhmetov and A. Zafer Department of Mathematics, Middle East Technical Uni¨ ersity, 06531 Ankara, Turkey E-mail: [email protected] Submitted by George Leitmann Received November 4, 1999
The paper is concerned with the stability of the zero solution of the impulsive system dx dt
s f Žt, x.,
⌬ x < ts iŽ x . s Ji Ž x . ,
t / i Ž x . i g N s 1, 2, . . . 4 ,
where ⌬ x < ts [ x Ž q . y x Ž ., x Ž q . s lim t ª qx Ž t .. The Lyapunov second method is used as a tool in obtaining the criteria for stability, asymptotic stability, and instability of the trivial solution. 䊚 2000 Academic Press Key Words: stability; instability; Lyapunov’s second method; impulse effect; variable moments.
1. INTRODUCTION AND PRELIMINARIES In recent years there have been intensive studies on the qualitative behavior of solutions of impulsive differential equations; see for instance w8, 14x and the references cited therein. The theory of stability for impulsive differential equations has also been well developed during the past several years. To the best of our knowledge, the first article devoted to the Lyapunov second method for impulsive differential equations is due to Milman and Myshkis w13x, where the authors considered the stability of the zero solution of differential equations with fixed moments of impulse points. Later, the Lyapunov second method was used for differential equations with impulses at variable times w4, 5x. 69 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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AKHMETOV AND ZAFER
Although there are several papers dealing with the stability of solutions of impulsive equations with fixed moments of impulse effects Žsee w11x and the references cited therein ., much less is known in the case of variable impulse actions since these equations exhibit more difficulties. One should even modify the stability definition of solutions of such equations since it is not possible to reduce the problem of stability of a nontrivial solution to that of the zero solution w1, 2, 10, 15x. No modification is necessary if only the zero solution is concerned. In this paper, by employing the Lyapunov second method, we investigate the stability, asymptotic stability, and instability of the zero solution of a system of impulsive differential equations with impulse actions on surfaces of the form dx dt
s f Ž t, x. ,
⌬ x < ts i Ž x . s Ji Ž x . ,
t / i Ž x . , i g N s 1, 2, . . . 4 ,
Ž 1.
where ⌬ x < ts [ x Ž q . y x Ž ., x Ž q . s lim t ª q x Ž t .. For our purpose we first introduce the notation G s Ž t , x . : t G 0, x g S 4 ,
S s x g R n : 5 x 5 - 4 ,
where ) 0 is a fixed real number and 5 x 5 denotes the euclidean norm of x g R n. Next, for each i g N we define i0 s i Ž0. and let Gi s Ž t , x . g G : t is between i0 and i Ž x . 4 . With regard to Ž1. we assume that the following conditions are satisfied. Ža. f Ž t, x .: Rq= S ª R n is piecewise continuous with discontinuities of the first kind at the boundary points of Gi , where it is left continuous with respect to t; f Ž t, 0. s 0 for all t G t 0 ; supŽ t, x .g G 5 f Ž t, x .5 s M - ⬁. Žb. i Ž x .: S ª Rq are continuous; i Ž x q Ji Ž x .. F i Ž x . and i Ž x . - iq1Ž x . for all x g S and i g N; lim iª⬁ i Ž0. s ⬁. Žc. Ji Ž x .: S ª R n ; Ji Ž0. s 0. As is well known, the solutions of differential equations with variable moments of impulse effect may experience pulse phenomena, namely, they may hit a given surface of discontinuity a finite or infinite number of times, causing rhythmic beating w8, 14x. This results in additional complications in studying such systems and therefore in most cases it is necessary to find conditions that guarantee the absence of beating.
STABILITY OF THE ZERO SOLUTION
71
There are various sets of sufficient conditions for the absence of pulse phenomena w1, 2x. For instance, if there exists an L ) 0 such that < i Ž x . y i Ž y .< F L 5 x y y 5 for all x, y g S and i g N, and i Ž x q Ji Ž x .. F i Ž x . for all x g S and i g N, then ML - 1 is sufficient for the absence of beating of solutions on a surface of discontinuity. These conditions are not necessary; see the remark after Example 3.1 at the end of this paper. Therefore, it is essential to assume that Žd. the beating of solutions of Ž1. on each surface of discontinuity is absent. It is also well known that the solution curve of a differential equation with discontinuous right-hand side may intersect a surface of discontinuity at a certain time and stay there for some period of time w3x. Therefore some care should be taken when such a system is under consideration. In our case, however, this phenomenon does not exist. We have two kinds of surfaces of discontinuities, namely, t s i0 and t s i Ž x .. It can be shown that no solution curve of Ž1. can stay on the surface t s i0 , for a period of time. In the case t s i Ž x ., this behavior is still not possible since each t s i Ž x . is also a surface of impulse points. In this paper we will state and prove three theorems. The first two theorems are about the stability and asymptotic stability of the zero solution of Ž1.. The third one is a Chetaev type instability theorem for the zero solution of Ž1.. Examples are also inserted to illustrate the results of the paper. We should point out that the arguments developed in w4, 5x were based on a comparison method. Specifically, the change of a Lyapunov function in the interval of continuity was compared with its changes at the moments of discontinuity. Our technique is also based on a comparison, but it is somewhat different. We compare the changes of a Lyapunov function in the vicinity of the moments where solutions meet a surface of discontinuity. Therefore, the results of this paper seem to be very useful for stabilization and controllability of impulsive systems w7x. The following definition is extracted from w3x. DEFINITION 1.1. A solution x Ž t . s x Ž t, a, b . of
˙x s f Ž t , x . ,
x Ž a. s b
Ž E.
is called unique from the right if for each solution y Ž t . of Ž E . there exists a1 ) a such that x Ž t . s y Ž t . for all t g w a, a1 x. We will assume that for a given a g w0, ⬁., the solution x Ž t, a, 0. of Ž E . is unique from the right. We do not assume the uniqueness for all solutions since the system Ž1. is not a reduced system. In fact if we try to
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AKHMETOV AND ZAFER
reduce the problem of stability of a nontrivial solution to that of the zero solution, then because of the difference between the discontinuity points the resulting system becomes very complicated w10, 15x. Let us recall the following definitions. DEFINITION 1.2. The zero solution of Ž1. is called stable if for any given ⑀ ) 0 and t 0 g Rq there exists ␦ s ␦ Ž ⑀ , t 0 . such that 5 x 0 5 - ␦ implies 5 x Ž t, t 0 , x 0 .5 - ⑀ for all t G t 0 . DEFINITION 1.3. The zero solution of Ž1. is called asymptotically stable if it is stable and there exists ␦ ) 0 such that any solution x Ž t, t 0 , x 0 . with 5 x 0 5 - ␦ satisfies lim t ª⬁ x Ž t . s 0. DEFINITION 1.4. A continuous function a: Rqª Rq is said to belong to class K if a is strictly increasing and aŽ0. s 0. DEFINITION 1.5. A function V Ž t, x . is called positive definite on G if there exists a g K such that V Ž t, x . G aŽ5 x 5. for all Ž t, x . g G; it is called positive semidefinite on G if V Ž t, x . G 0 for all Ž t, x . g G. The function V Ž t, x . is called negative definite Žnegative semidefinite. on G if yV Ž t, x . is positive definite Žpositive semidefinite. on G.
2. MAIN RESULTS Let V Ž t, x . be a continuous real-valued function defined on G with V Ž t, 0. s 0 for t G t 0 . We assume that V Ž t, x . is locally Lipschitz in x and denote Dq V Ž t , x . s lim sup
V Ž t q h, x q hf Ž t , x . . y V Ž t , x . h
hª0 q
.
Note that if V is differentiable then Dq V Ž t , x . s V˙Ž t , x . s
⭸V ⭸t
q
n
⭸V
Ý
⭸ xi
is1
fi Ž t , x . .
The following basic comparison results are extracted from w9x. LEMMA 2.1.
Let V Ž t, x . be as abo¨ e, and DqV Ž t , x . F g Ž t , V Ž t , x . . ,
Ž t , x . g G,
Ž I1 .
where g g C w Rq= Rq, R x. Suppose that uŽ t . s uŽ t, t 0 , u 0 . is the maximal solution of the scalar differential equation uX s g Ž t , u . ,
u Ž t 0 . s u 0 G 0,
STABILITY OF THE ZERO SOLUTION
73
which exists to the right of t 0 . If x Ž t . s x Ž t, t 0 , x 0 . is any solution of Ž I1 . such that V Ž t 0 , x 0 . F u 0 , then V Ž t, x Ž t .. F uŽ t . for t G t 0 . Let V Ž t, x . be as abo¨ e, and
LEMMA 2.2.
DqV Ž t , x . G g Ž t , V Ž t , x . . ,
Ž t , x . g G,
Ž I2 .
where g g C w Rq= Rq, R x. Suppose that uŽ t . s uŽ t, t 0 , u 0 . is the minimal solution of the scalar differential equation uX s g Ž t , u . ,
u Ž t 0 . s u 0 G 0,
which exists to the right of t 0 . If x Ž t . s x Ž t, t 0 , x 0 . is any solution of Ž I2 . such that V Ž t 0 , x 0 . G u 0 , then V Ž t, x Ž t .. G uŽ t . for t G t 0 . Remark 2.1. By taking g Ž t, u. ' 0, one can easily verify that if Dq V Ž t, x . G 0 then V Ž t, x Ž t .. is nondecreasing, and if Dq V Ž t, x . F 0 then V Ž t, x Ž t .. is nonincreasing. In what follows we denote by A the set of all continuous functions f : R ª R such that f Ž0. s 0 and f Ž s . ) 0 for s ) 0. THEOREM 2.1.
Assume that the following conditions are fulfilled.
Ži. V Ž t, x . is positi¨ e definite on G. Žii. Dq V Ž t, x . is negati¨ e semidefinite on G. Žiii. Dq V Ž t, x . F y Ž V Ž t, x .. for some g A and for all Ž t, x . g Di g N Gi . Živ. V Ž i Ž x ., x q Ji Ž x .. F Ž V Ž i Ž x ., x .. for some g A and for all x g S and i g N. Žv. There exists an L ) 0 such that < i Ž x . y i Ž y .< F L 5 x y y 5 for all x, y g S and I g N. Žvi. There exists an L1 ) 0 such that < i Ž x . y i0 < G L1 5 x 5 for all x g S and i g N. Žvii. There exists a ␥ G 0 such that
HVŽVŽ x., xx. , x Ž Ž iŽ . i
..
ds
Ž s.
F Ž L1 y ␥ . 5 x 5
for all x g S and i g N. Then, the zero solution of Ž1. is stable if ␥ s 0 and is asymptotically stable if ␥ ) 0. Proof. Let x Ž t . s x Ž t, t 0 , x 0 . be a solution of Ž1. having discontinuities at t s i for i g N. It follows from Ži. and Remark 2.1 that ¨ Ž t . [ V Ž t, x Ž t .. is nonincreasing on each of the intervals w t 0 , 1 . and Ž i , iq1 .,
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AKHMETOV AND ZAFER
i g N. To obtain more information on the behavior of ¨ Ž t . we need to investigate its change in Gi . For each fixed i g N, there are two possible cases: Case 1. i G i0 . We let uŽ t . be the maximal solution of uX s y Ž u. on w i0 , i x such that uŽ i0 . s ¨ Ž i0 .. In view of Lemma 2.1 and Žvi., we have L i x Ž i . F i y i0 s
ds
ds
HuuŽŽ. . Ž s . F H ŽŽ. . Ž s . . 0 i
0 i
¨
¨
i
Ž 2.
i
Using Živ. and Žvii. we also have ds
Ž ¨ Ž i ..
Ž ␥ y L1 . x Ž i . F yH
Ž s.
¨ Ž i .
H Ž.q
Fy
¨Ž
¨
i
i
.
ds
Ž s.
.
Ž 3.
Summing Ž2. and Ž3. leads to ds
H ŽŽq.. Ž s . G ␥ 0 i
¨
¨
x Ži .
Ž 4.
i
and hence we obtain that ¨ Ž i0 . G ¨ Ž i q . .
Ž 5.
Case 2. i - i0 . We let uŽ t . be the maximal solution of uX s y Ž u. on w i , i0 x such that uŽ i . s ¨ Ž i q .. It follows that L1 x Ž i . F i0 y i s
ds
HuuŽ . Ž s . F H Ž q. Ž i. 0 i
¨Ž
¨
i
0 i
.
ds
Ž s.
.
Ž 6.
From Ž3. and Ž6. we get ds
H Ž . Ž s . G ␥ ¨ Ž i.
¨
0 i
x Ži .
Ž 7.
and so ¨ Ž i . G ¨ Ž i0 . .
Ž 8.
Define ⌫ [ D⬁is1 Ž i , i x and ⌳ [ w t 0 , ⬁._ ⌫, where i s i and i s i0 if i F i0 , i s i0 and i s i if i G i0 . From Ž5. and Ž8. we may write ¨ Ž q . F ¨ Ž i ., and hence we conclude that ¨ Ž t . is nonincreasing on ⌳.
STABILITY OF THE ZERO SOLUTION
75
Let 0 - ⑀ - and t 0 be given. Without any loss of generality we may assume that t 0 g ⌳. It is now clear that ¨ Ž t . F ¨ Ž t 0 . for all t g ⌳. Set
⑀1 s
⑀
and
2 Ž 1 q ML .
s inf t G t 0 , 5 x 5 G ⑀ 1 V Ž t , x . .
Since V Ž t, x . is continuous and V Ž t, 0. s 0, it is possible to find a positive real number ␦ such that ␦ - ⑀ 1 and sup V Ž t 0 , x . - ␥ - .
5 x 5F ␦
We first claim that if 5 x 0 5 - ␦ then 5 x Ž t, t 0 , x 0 .5 - ⑀ 1 for all t g ⌳. Suppose that this is not true. Then there would exist a tU g ⌳ such that 5 x Ž tU , t 0 , x 0 .5 G ⑀ 1. But this leads us to the contradiction that F ¨ Ž tU . F ¨ Ž t 0 . - . Next suppose that t g Ž i , i x for some i. Clearly, x Ž t . s x Ž i . q
t
H f Ž s, x Ž s . . ds
if i ) i0
Ž 9.
if i - i0 .
Ž 10 .
i
and x Ž t . s x Ž i . q
t
H f Ž s, x Ž s . . ds i
In view of Žv., we easily obtain from both Ž9. and Ž10. that x Ž t . F ⑀ 1 Ž 1 q ML . - ⑀ . Therefore, the zero solution of Ž1. is stable. We shall now show that if ␥ ) 0 then lim t ª⬁ x Ž t . s 0. We first observe that since ¨ Ž t . is positive and nonincreasing on ⌳, there is a nonnegative real number such that lim ¨ Ž t . s ,
tª⬁
t g ⌳.
Ž 11 .
We claim that s 0. Suppose on the contrary that ) 0. Because of Ž11., there exists a positive real number 1 such that x Ž t . G 1
for all t g ⌳ .
Ž 12 .
If i0 ) i then, since i g ⌳, Ž12. implies that x Ž i . G 1 .
Ž 13 .
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AKHMETOV AND ZAFER
Suppose that i0 - i . In this case, i0 g ⌳, and therefore by Ž12. we have x Ž i0 . G 1 .
Ž 14 .
i Ž x Ž i . . y i0 F L x Ž i . .
Ž 15 .
Using Žv., we also have
In view of Ž14. and Ž15., we easily obtain from x Ž i . s x Ž i0 . q
i
H
0 i
f Ž s, x Ž s . . ds
that x Ž i . G 1 y ML x Ž i . and hence x Ž i . G 1r Ž 1 q ML . .
Ž 16 .
Thus, we see from Ž13. and Ž16. that x Ž i . G 2
for all i g N,
Ž 17 .
where 2 s 1rŽ1 q ML.. On the other hand, since Ž i , iq1 x g ⌳ for all i g N, lim iª⬁ ¨ Ž i q . s lim iª⬁ ¨ Ž i . s and ¨ Ž i . G ¨ Ž i q . G . Letting ms
min
FsF¨ Ž t 0 .
Ž s. ,
it follows from Ž4., Ž7. and Ž17. that ¨ Ž i . y ¨ Ž i q. G ␥ m 2
for all i g N.
Ž 18 .
Using ¨ Ž iq1 . F ¨ Ž i q . in Ž18. and then summing the resulting inequality over i from 1 to k we get ¨ Ž 1 . y ¨ Ž kq1 . G Ž ␥ m 2 . k
for all k g N.
Ž 19 .
It is clear from Ž19. that if k is sufficiently large, then the function ¨ takes on negative values. But this contradicts the fact that ¨ is positive definite. Thus, we must have s 0. As in the classical case, it follows that lim t ª⬁ x Ž t . s 0, and hence we may conclude that the zero solution is asymptotically stable.
STABILITY OF THE ZERO SOLUTION
77
COROLLARY 2.1. Let all conditions of Theorem 2.1 except Žv. be satisfied. In addition, suppose that the family i Ž x .4 is equicontinuous at x s 0 and that i0 G i Ž x . for all x g S . Then the conclusion of Theorem 2.1 remains ¨ alid. Proof. We proceed as in the proof of Theorem 2.1 until ⑀ 1 is picked. Now since the family i Ž x .4 is equicontinuous at x s 0 and i0 G i Ž x . for all x g S , given any ⑀ 2 , 0 - ⑀ 2 - ⑀rM, we can find ⑀ 3 ) 0 such that i0 y i Ž x . - 2 for all 5 x 5 - ⑀ 3 and i g N. Fix ⑀ 1 ) 0 such that ⑀ 1 min ⑀ 3 , ⑀ y M⑀ 2 4 . Then it follows from Ž9. and Ž10. that x Ž t . F ⑀ 1 q M⑀ 2 - ⑀ . Clearly, Ž11., Ž12., and Ž13. hold and by our assumption the case i0 - i does not exist. Thus Ž17. is satisfied with 2 s 1. The rest of the proof is the same as that of Theorem 2.1. In the next theorem we do not require that Dq V Ž t, x . be negative semidefinite on Di g N Gi . THEOREM 2.2.
Assume that the following conditions are fulfilled.
Ži. V Ž t, x . is positi¨ e definite on G. Žii. Dq V Ž t, x . is negati¨ e semidefinite on G_Di g N Gi . Žiii. Dq V Ž t, x . F Ž V Ž t, x .. for some g A and for all Ž t, x . g Di g N Gi . Živ. V Ž i Ž x ., x q Ji Ž x .. F Ž V Ž i Ž x ., x . for some g A and for all x g S and i g N. Žv. There exists an L ) 0 such that < i Ž x . y i Ž y .< F L 5 x y y 5 for all x, y g S and i g N. Žvi. There exists a ␥ G 0 such that
HVŽVŽ x., xx. , x Ž Ž iŽ . i
ds
..
Ž s.
G Ž L q ␥ . 5 x5
for all x g S and i g N. Then, the zero solution of Ž1. is stable if ␥ s 0 and is asymptotically stable if ␥ ) 0. Proof. Let uŽ t . be the maximal solution of uX s Ž u. on w i , i x such that uŽ i . s ¨ Ž i q ., where i and i are as defined in the proof of Theorem 2.1. Proceeding as in the proof of Theorem 2.1 we easily obtain ds
H Ž . Ž s . F L ¨ Ž i.
¨
0 i
x Ži .
for i G i0
Ž 20 .
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AKHMETOV AND ZAFER
and ds
H ŽŽq.. Ž s . F L 0 i
¨
¨
x Ži .
for i - i0 .
Ž 21 .
i
Using Živ. and Žvi. we also have
H Ž.q ¨Ž
¨
i
i
.
ds
Ž s.
F yŽ L q ␥ . x Ž i . .
Ž 22 .
It follows from Ž20., Ž21., and Ž22. that ¨ Ž i q . F ¨ Ž i . for all i g N. The remainder of the proof is similar to that of Theorem 2.1 and hence is omitted. Our last result in this paper is a Chetaev-type instability theorem, see page 216 in w12x, for the zero solution of Ž1.. THEOREM 2.3.
Assume that the following conditions are fulfilled.
Ži. For e¨ ery ⑀ ) 0 and for e¨ ery t G t 0 there exist points x g S⑀ such that V Ž t, x . ) 0. The set B of all points Ž t, x . such that x g S and such that ¨ Ž t, x . ) 0 is called the ‘‘domain ¨ ) 0.’’ The set B is bounded by the hypersurfaces 5 x 5 s and ¨ Ž t, x . s 0. We assume that ¨ is bounded from abo¨ e in B and 0 g ⭸ B for all t G t 0 . Žii. Dq V Ž t, x . is positi¨ e semidefinite on B_Di g N Ž Gi l B .. Žiii. Dq V Ž t, x . G y Ž V Ž t, x .. for some g A and for all Ž t, x . g Di g N Ž Gi l B .. Živ. V Ž i Ž x ., x q Ji Ž x .. G Ž V Ž i Ž x ., x . for some g A and for all Ž t, x . g Di g N Ž ⌫i l B ., where ⌫i s Ž t, x . : t s i Ž x .4 . Žv. There exists an L ) 0 such that < Ž x . y i0 < F L 5 x 5 for x g S and i g N. Žvi. There exists a ␥ ) 0 such that
HVŽVŽ x., xx. , x Ž Ž iŽ . i
..
ds
Ž s.
G Ž L q ␥ . 5 x5
for all x g S and i g N. Then the zero solution of Ž1. is unstable. Proof. Fix ⑀ ) 0 and t 0 , Ž t 0 , x 0 . g B, and let x Ž t . s x Ž t, t 0 , x 0 . be a solution of Ž1. having discontinuities at t s i for i g N. We shall show that x Ž t . must leave the ball S⑀ in finite time. In view of Žii. and Remark 2.1 we see that ¨ Ž t . is nondecreasing on each interval of its continuity in ⌳. We need to prove that ¨ Ž t . is nondecreasing for all t g ⌳. So we let
STABILITY OF THE ZERO SOLUTION
79
uŽ t . be the minimal solution of uX s y Ž u. on w i , i x such that uŽ i . s ¨ Ž i q ., where i and i are as defined in the proof of Theorem 2.1. By using Lemma 2 and Žvi. we see that
H Ž q. ¨Ž
i
0 i
¨
.
ds
Ž s.
G ␥ x Ži .
if i ) i0
Ž 23 .
if i - i0 .
Ž 24 .
and ds
H ŽŽ. . Ž s . G ␥ 0 i
¨
¨
x Ži .
i
From Ž23. and Ž24. we may deduce that ¨ Ž i . F ¨ Ž i q .. Therefore, ¨ Ž t . G ¨ Ž t 0 . for all t g ⌳, implying that Ž t, x Ž t .. g B_Di g N Gi for all t g ⌳. Let M ) 0 be a real number such that V Ž t, x . F M for all Ž t, x . g B, which is possible by Ži.. Since ¨ Ž t . G ¨ Ž t 0 ., there is a 1 ) 0 such that 5 x Ž t .5 G 1 for all t g ⌳. If we now define ms
min
¨ Ž t 0 .FsFM
Ž s.
then it follows from Ž23. and Ž24. that ¨ Ž i q. y ¨ Ž i . G ␥ m 1 .
Ž 25 .
Since ¨ Ž iq1 . G ¨ Ž i q . we get ¨ Ž iq1 . y ¨ Ž i . G ␥ m 1
for all i g N.
Ž 26 .
Summing Ž26. over i from 1 to k we see that ¨ Ž kq 1 . y ¨ Ž 1 . G Ž ␥ m 1 . k
for all k g N.
Ž 27 .
But Ž27. leads to a contradiction that ¨ Ž t . is unbounded in B. This completes the proof.
3. EXAMPLES EXAMPLE 3.1. Let i Ž x . s i y x 12 q x 22 so that Gi s Ž t, x . g G : i y x 12 q x 22 - t F i4 . We define S s D⬁is1 Gi and consider the impulsive
'
'
80
AKHMETOV AND ZAFER
system
½ ½
˙x 1 s ˙x 2 s
yx 2 ,
Ž t , x . f S, Ž t, x. g S
yx 1 ,
Ž t , x . f S, Ž t, x. g S
x1 , yx 2 ,
⌬ x 1 < ts i Ž x . s y␣ x 1 q  x 2 , ⌬ x 2 < ts i Ž x . s  x 1 y ␣ x 2 . We choose V Ž x . s x 12 q x 22 and make the following observations: Ža. V˙Ž x . s 0 if Ž t, x . f S and V˙Ž x . s y2V Ž x . if Ž t, x . g S. Since V Ž x q ⌬ x . s ŽŽ1 y ␣ . 2 q  2 .Ž x 12 q x 22 . y 4 Ž ␣ y 1. x 1 x 2 we have VŽ x q ⌬ x. F l Ž ␣ ,  .VŽ x. , where l Ž ␣ ,  . s Ž<  < q <1 y ␣ <. 2 . Žb. 5 x q ⌬ x 5 2 s ŽŽ1 y ␣ . 2 q  2 .Ž x 12 q x 22 . y 4 Ž ␣ y 1. x 1 x 2 and so 5 x q ⌬ x 5 2 G Ž<1 y ␣ < y <  <. 2 5 x 5 2 . It follows that if < <1 y ␣ < y <  < < G 1, then
i Ž x q ⌬ x . F i Ž x . . Žc. < i Ž x . y i0 < s x 12 q x 22 s 5 x 5. Žd. Let Ž s . s 2 s and Ž s . s l Ž ␣ . s, and fix a positive number ␥ - 1. If l Ž ␣ ,  . F 1, then ln l Ž ␣ ,  . F 2Ž1 y ␥ .5 x 5 and hence
'
lV
HV
ds 2s
F Ž 1 y ␥ . 5 x 5.
In view of Theorem 2.1 we deduce that the zero solution of Ž1. is asymptotically stable if < <1 y ␣ < y <  < < G 1 4
and
<1 y ␣ < q <  < F 1.
If we take i Ž x . s i y x 12 q x 22 , then it is easy to verify that condition Žv. is not satisfied and therefore Theorem 2.1 does not apply. However, since the additional conditions stated in Corollary 2.1 are true, we may conclude that the above conclusion is valid. We note that the beating is still absent in this case, since Ž drdt .5 x Ž t .5 s 0 for all Ž t, x . f S and Žb. holds.
'
STABILITY OF THE ZERO SOLUTION
81
EXAMPLE 3.2. Let i Ž x . s i q x 12 q x 22 . Clearly Gi s Ž t, x . : i - t F i q x 12 q x 22 4 . Define S s D⬁is1 Gi and consider the impulsive system
˙x 1 s
½
˙x 2 s
yx 2 , yx 2 q
½
x 13 ,
x1 , x1 q
x 23 ,
Ž t , x . f S, Ž t, x. g S Ž t , x . f S, Ž t , x . g S,
⌬ x 1 < ts i Ž x . s y␣ x 1 q  x 2 , ⌬ x 2 < ts i Ž x . s  x 1 y ␣ x 2 . We choose V Ž x . s x 12 q x 22 and make the following observations: Ža. V˙Ž x . s 0 if Ž t, x . f S and V˙Ž x . F 2V 2 Ž x . if Ž t, x . g S. Since V Ž x q ⌬ x . s ŽŽ1 y ␣ . 2 q  2 .Ž x 12 q x 22 . y 4 Ž ␣ y 1. x 1 x 2 , we have VŽ x q ⌬ x. F l Ž ␣ ,  .VŽ x. , where l Ž ␣ ,  . s Ž<  < q <1 y ␣ <. 2 . Žb. Let x, y g R n such that 5 x 5 F h and 5 y 5 F h, where h ) 0 is some real number. It follows that
i Ž x . y i Ž y . F 2 h5 x y y 5. Žc. If l Ž ␣ ,  . - 1 then i Ž x q ⌬ x . F i q l Ž ␣ ,  .5 x 5 - i Ž x .. Žd. Let g s Žyx 2 q x 13, x 1 q x 23 . and mŽ h. s max 5 x 5 F h 5 g 5. Clearly, mŽ h. ª 0 as h ª 0 and so there exists h 0 such that 2 hmŽ h. - 1 for all h F h0 . Že. Let Ž s . s 2 s 2 and Ž s . s l Ž ␣ ,  . s, and fix a positive real 3
number ␥ . Choose 5 x 5 F min h 0 , Ž 1 y l . r Ž 2 l Ž 2 h 0 q ␥ . . 4 . It follows that 1 y l G 2 l Ž2 h q ␥ .5 x 5 3 and so
'
V
ds
Hl V 2 s
2
G Ž 2 h q ␥ . 5 x 5.
By Theorem 2.2, the zero solution of Ž1. is asymptotically stable if <  < q < ␣ y 1 < - 1. In this case, 2 hM - 1 is sufficient for the absence of beating.
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AKHMETOV AND ZAFER
We end this paper by pointing out that the results of this paper can be generalized easily if Gi s Ž t , x . g G : t is between t s i Ž x . and i Ž x . 4 , where t s i Ž x . is any given surface like i Ž x . such that i Ž0. s i Ž0.. In this article, i Ž x . s i Ž0.. REFERENCES 1. M. U. Akhmetov, On motion with impulse actions on a surface, Iz¨ .-Acad. Nauk Kaz. SSR Ser. Fiz.-Mat. 1 Ž1988., 11᎐14. 2. M. U. Akhmetov and N. A. Perestyuk, A comparison method for differential equations with impulse actions, Differentsial’naya Ura¨ neniya 26 Ž1990., 1475᎐1483. 3. A. F. Filippov, ‘‘Differential Equations with Discontinuous Right-Hand Sides,’’ Kluwer Academic, Dordrecht, 1988. 4. S. I. Gurgula, Investigation of the stability of solutions of impulsive systems by Lyapunov’s second method, Ukrainian Math. J. 1 Ž1982., 100᎐103. 5. S. I. Gurgula and N. A. Perestyuk, On the second Lyapunov’s method in impulsive systems, Dokl. Acad. Nauk. Ukrainian. SSR Ser. A 10 Ž1982., 11᎐14. 6. P. Hartman, ‘‘Ordinary Differential Equations,’’ Wiley, New York, 1964. 7. H. Kwakernaak and R. Sivan, ‘‘Linear Optimal Control Systems,’’ Wiley, New York, 1972. 8. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, ‘‘Theory of Impulsive Differential Equations,’’ World Scientific, SingaporerNew York, 1989. 9. V. Lakshmikantham and S. Leela, ‘‘Differential and Integral Inequalities,’’ Vol. 1, Academic Press, New York, 1969. 10. V. Lakshmikantham and X. Liu, On quasi stability for impulsive differential systems, Nonlinear Anal. 13 Ž1989., 819᎐828. 11. X. Liu and R. Pirapakaran, Global stability results for impulsive differential equations, Appl. Anal. 33 Ž1989., 87᎐102. 12. R. K. Miller and A. N. Michel, ‘‘Ordinary Differential Equations,’’ Academic Press, New York, 1982. 13. V. D. Milman and A. D. Myshkis, On motion stability with shoks, Sibirsk. Mat. Zh. 1 Ž1960., 233᎐237. 14. A. M. Samoilenko and N. A. Perestyuk, ‘‘Impulsive Differential Equations,’’ World Scientific, Singapore, 1995. 15. A. M. Samoilenko and N. A. Perestyuk, Stability of solutions of impulsive differential equations, Differentsial’naya Ura¨ neniya 13 Ž1977., 1981᎐1992.
Stability of the Zero Solution of Impulsive Differential Equations by the Lyapunov Second Method M. U. Akhmetov and A. Zafer Department of Mathematics, Middle East Technical Uni¨ ersity, 06531 Ankara, Turkey E-mail: [email protected] Submitted by George Leitmann Received November 4, 1999
The paper is concerned with the stability of the zero solution of the impulsive system dx dt
s f Žt, x.,
⌬ x < ts iŽ x . s Ji Ž x . ,
t / i Ž x . i g N s 1, 2, . . . 4 ,
where ⌬ x < ts [ x Ž q . y x Ž ., x Ž q . s lim t ª qx Ž t .. The Lyapunov second method is used as a tool in obtaining the criteria for stability, asymptotic stability, and instability of the trivial solution. 䊚 2000 Academic Press Key Words: stability; instability; Lyapunov’s second method; impulse effect; variable moments.
1. INTRODUCTION AND PRELIMINARIES In recent years there have been intensive studies on the qualitative behavior of solutions of impulsive differential equations; see for instance w8, 14x and the references cited therein. The theory of stability for impulsive differential equations has also been well developed during the past several years. To the best of our knowledge, the first article devoted to the Lyapunov second method for impulsive differential equations is due to Milman and Myshkis w13x, where the authors considered the stability of the zero solution of differential equations with fixed moments of impulse points. Later, the Lyapunov second method was used for differential equations with impulses at variable times w4, 5x. 69 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
70
AKHMETOV AND ZAFER
Although there are several papers dealing with the stability of solutions of impulsive equations with fixed moments of impulse effects Žsee w11x and the references cited therein ., much less is known in the case of variable impulse actions since these equations exhibit more difficulties. One should even modify the stability definition of solutions of such equations since it is not possible to reduce the problem of stability of a nontrivial solution to that of the zero solution w1, 2, 10, 15x. No modification is necessary if only the zero solution is concerned. In this paper, by employing the Lyapunov second method, we investigate the stability, asymptotic stability, and instability of the zero solution of a system of impulsive differential equations with impulse actions on surfaces of the form dx dt
s f Ž t, x. ,
⌬ x < ts i Ž x . s Ji Ž x . ,
t / i Ž x . , i g N s 1, 2, . . . 4 ,
Ž 1.
where ⌬ x < ts [ x Ž q . y x Ž ., x Ž q . s lim t ª q x Ž t .. For our purpose we first introduce the notation G s Ž t , x . : t G 0, x g S 4 ,
S s x g R n : 5 x 5 - 4 ,
where ) 0 is a fixed real number and 5 x 5 denotes the euclidean norm of x g R n. Next, for each i g N we define i0 s i Ž0. and let Gi s Ž t , x . g G : t is between i0 and i Ž x . 4 . With regard to Ž1. we assume that the following conditions are satisfied. Ža. f Ž t, x .: Rq= S ª R n is piecewise continuous with discontinuities of the first kind at the boundary points of Gi , where it is left continuous with respect to t; f Ž t, 0. s 0 for all t G t 0 ; supŽ t, x .g G 5 f Ž t, x .5 s M - ⬁. Žb. i Ž x .: S ª Rq are continuous; i Ž x q Ji Ž x .. F i Ž x . and i Ž x . - iq1Ž x . for all x g S and i g N; lim iª⬁ i Ž0. s ⬁. Žc. Ji Ž x .: S ª R n ; Ji Ž0. s 0. As is well known, the solutions of differential equations with variable moments of impulse effect may experience pulse phenomena, namely, they may hit a given surface of discontinuity a finite or infinite number of times, causing rhythmic beating w8, 14x. This results in additional complications in studying such systems and therefore in most cases it is necessary to find conditions that guarantee the absence of beating.
STABILITY OF THE ZERO SOLUTION
71
There are various sets of sufficient conditions for the absence of pulse phenomena w1, 2x. For instance, if there exists an L ) 0 such that < i Ž x . y i Ž y .< F L 5 x y y 5 for all x, y g S and i g N, and i Ž x q Ji Ž x .. F i Ž x . for all x g S and i g N, then ML - 1 is sufficient for the absence of beating of solutions on a surface of discontinuity. These conditions are not necessary; see the remark after Example 3.1 at the end of this paper. Therefore, it is essential to assume that Žd. the beating of solutions of Ž1. on each surface of discontinuity is absent. It is also well known that the solution curve of a differential equation with discontinuous right-hand side may intersect a surface of discontinuity at a certain time and stay there for some period of time w3x. Therefore some care should be taken when such a system is under consideration. In our case, however, this phenomenon does not exist. We have two kinds of surfaces of discontinuities, namely, t s i0 and t s i Ž x .. It can be shown that no solution curve of Ž1. can stay on the surface t s i0 , for a period of time. In the case t s i Ž x ., this behavior is still not possible since each t s i Ž x . is also a surface of impulse points. In this paper we will state and prove three theorems. The first two theorems are about the stability and asymptotic stability of the zero solution of Ž1.. The third one is a Chetaev type instability theorem for the zero solution of Ž1.. Examples are also inserted to illustrate the results of the paper. We should point out that the arguments developed in w4, 5x were based on a comparison method. Specifically, the change of a Lyapunov function in the interval of continuity was compared with its changes at the moments of discontinuity. Our technique is also based on a comparison, but it is somewhat different. We compare the changes of a Lyapunov function in the vicinity of the moments where solutions meet a surface of discontinuity. Therefore, the results of this paper seem to be very useful for stabilization and controllability of impulsive systems w7x. The following definition is extracted from w3x. DEFINITION 1.1. A solution x Ž t . s x Ž t, a, b . of
˙x s f Ž t , x . ,
x Ž a. s b
Ž E.
is called unique from the right if for each solution y Ž t . of Ž E . there exists a1 ) a such that x Ž t . s y Ž t . for all t g w a, a1 x. We will assume that for a given a g w0, ⬁., the solution x Ž t, a, 0. of Ž E . is unique from the right. We do not assume the uniqueness for all solutions since the system Ž1. is not a reduced system. In fact if we try to
72
AKHMETOV AND ZAFER
reduce the problem of stability of a nontrivial solution to that of the zero solution, then because of the difference between the discontinuity points the resulting system becomes very complicated w10, 15x. Let us recall the following definitions. DEFINITION 1.2. The zero solution of Ž1. is called stable if for any given ⑀ ) 0 and t 0 g Rq there exists ␦ s ␦ Ž ⑀ , t 0 . such that 5 x 0 5 - ␦ implies 5 x Ž t, t 0 , x 0 .5 - ⑀ for all t G t 0 . DEFINITION 1.3. The zero solution of Ž1. is called asymptotically stable if it is stable and there exists ␦ ) 0 such that any solution x Ž t, t 0 , x 0 . with 5 x 0 5 - ␦ satisfies lim t ª⬁ x Ž t . s 0. DEFINITION 1.4. A continuous function a: Rqª Rq is said to belong to class K if a is strictly increasing and aŽ0. s 0. DEFINITION 1.5. A function V Ž t, x . is called positive definite on G if there exists a g K such that V Ž t, x . G aŽ5 x 5. for all Ž t, x . g G; it is called positive semidefinite on G if V Ž t, x . G 0 for all Ž t, x . g G. The function V Ž t, x . is called negative definite Žnegative semidefinite. on G if yV Ž t, x . is positive definite Žpositive semidefinite. on G.
2. MAIN RESULTS Let V Ž t, x . be a continuous real-valued function defined on G with V Ž t, 0. s 0 for t G t 0 . We assume that V Ž t, x . is locally Lipschitz in x and denote Dq V Ž t , x . s lim sup
V Ž t q h, x q hf Ž t , x . . y V Ž t , x . h
hª0 q
.
Note that if V is differentiable then Dq V Ž t , x . s V˙Ž t , x . s
⭸V ⭸t
q
n
⭸V
Ý
⭸ xi
is1
fi Ž t , x . .
The following basic comparison results are extracted from w9x. LEMMA 2.1.
Let V Ž t, x . be as abo¨ e, and DqV Ž t , x . F g Ž t , V Ž t , x . . ,
Ž t , x . g G,
Ž I1 .
where g g C w Rq= Rq, R x. Suppose that uŽ t . s uŽ t, t 0 , u 0 . is the maximal solution of the scalar differential equation uX s g Ž t , u . ,
u Ž t 0 . s u 0 G 0,
STABILITY OF THE ZERO SOLUTION
73
which exists to the right of t 0 . If x Ž t . s x Ž t, t 0 , x 0 . is any solution of Ž I1 . such that V Ž t 0 , x 0 . F u 0 , then V Ž t, x Ž t .. F uŽ t . for t G t 0 . Let V Ž t, x . be as abo¨ e, and
LEMMA 2.2.
DqV Ž t , x . G g Ž t , V Ž t , x . . ,
Ž t , x . g G,
Ž I2 .
where g g C w Rq= Rq, R x. Suppose that uŽ t . s uŽ t, t 0 , u 0 . is the minimal solution of the scalar differential equation uX s g Ž t , u . ,
u Ž t 0 . s u 0 G 0,
which exists to the right of t 0 . If x Ž t . s x Ž t, t 0 , x 0 . is any solution of Ž I2 . such that V Ž t 0 , x 0 . G u 0 , then V Ž t, x Ž t .. G uŽ t . for t G t 0 . Remark 2.1. By taking g Ž t, u. ' 0, one can easily verify that if Dq V Ž t, x . G 0 then V Ž t, x Ž t .. is nondecreasing, and if Dq V Ž t, x . F 0 then V Ž t, x Ž t .. is nonincreasing. In what follows we denote by A the set of all continuous functions f : R ª R such that f Ž0. s 0 and f Ž s . ) 0 for s ) 0. THEOREM 2.1.
Assume that the following conditions are fulfilled.
Ži. V Ž t, x . is positi¨ e definite on G. Žii. Dq V Ž t, x . is negati¨ e semidefinite on G. Žiii. Dq V Ž t, x . F y Ž V Ž t, x .. for some g A and for all Ž t, x . g Di g N Gi . Živ. V Ž i Ž x ., x q Ji Ž x .. F Ž V Ž i Ž x ., x .. for some g A and for all x g S and i g N. Žv. There exists an L ) 0 such that < i Ž x . y i Ž y .< F L 5 x y y 5 for all x, y g S and I g N. Žvi. There exists an L1 ) 0 such that < i Ž x . y i0 < G L1 5 x 5 for all x g S and i g N. Žvii. There exists a ␥ G 0 such that
HVŽVŽ x., xx. , x Ž Ž iŽ . i
..
ds
Ž s.
F Ž L1 y ␥ . 5 x 5
for all x g S and i g N. Then, the zero solution of Ž1. is stable if ␥ s 0 and is asymptotically stable if ␥ ) 0. Proof. Let x Ž t . s x Ž t, t 0 , x 0 . be a solution of Ž1. having discontinuities at t s i for i g N. It follows from Ži. and Remark 2.1 that ¨ Ž t . [ V Ž t, x Ž t .. is nonincreasing on each of the intervals w t 0 , 1 . and Ž i , iq1 .,
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AKHMETOV AND ZAFER
i g N. To obtain more information on the behavior of ¨ Ž t . we need to investigate its change in Gi . For each fixed i g N, there are two possible cases: Case 1. i G i0 . We let uŽ t . be the maximal solution of uX s y Ž u. on w i0 , i x such that uŽ i0 . s ¨ Ž i0 .. In view of Lemma 2.1 and Žvi., we have L i x Ž i . F i y i0 s
ds
ds
HuuŽŽ. . Ž s . F H ŽŽ. . Ž s . . 0 i
0 i
¨
¨
i
Ž 2.
i
Using Živ. and Žvii. we also have ds
Ž ¨ Ž i ..
Ž ␥ y L1 . x Ž i . F yH
Ž s.
¨ Ž i .
H Ž.q
Fy
¨Ž
¨
i
i
.
ds
Ž s.
.
Ž 3.
Summing Ž2. and Ž3. leads to ds
H ŽŽq.. Ž s . G ␥ 0 i
¨
¨
x Ži .
Ž 4.
i
and hence we obtain that ¨ Ž i0 . G ¨ Ž i q . .
Ž 5.
Case 2. i - i0 . We let uŽ t . be the maximal solution of uX s y Ž u. on w i , i0 x such that uŽ i . s ¨ Ž i q .. It follows that L1 x Ž i . F i0 y i s
ds
HuuŽ . Ž s . F H Ž q. Ž i. 0 i
¨Ž
¨
i
0 i
.
ds
Ž s.
.
Ž 6.
From Ž3. and Ž6. we get ds
H Ž . Ž s . G ␥ ¨ Ž i.
¨
0 i
x Ži .
Ž 7.
and so ¨ Ž i . G ¨ Ž i0 . .
Ž 8.
Define ⌫ [ D⬁is1 Ž i , i x and ⌳ [ w t 0 , ⬁._ ⌫, where i s i and i s i0 if i F i0 , i s i0 and i s i if i G i0 . From Ž5. and Ž8. we may write ¨ Ž q . F ¨ Ž i ., and hence we conclude that ¨ Ž t . is nonincreasing on ⌳.
STABILITY OF THE ZERO SOLUTION
75
Let 0 - ⑀ - and t 0 be given. Without any loss of generality we may assume that t 0 g ⌳. It is now clear that ¨ Ž t . F ¨ Ž t 0 . for all t g ⌳. Set
⑀1 s
⑀
and
2 Ž 1 q ML .
s inf t G t 0 , 5 x 5 G ⑀ 1 V Ž t , x . .
Since V Ž t, x . is continuous and V Ž t, 0. s 0, it is possible to find a positive real number ␦ such that ␦ - ⑀ 1 and sup V Ž t 0 , x . - ␥ - .
5 x 5F ␦
We first claim that if 5 x 0 5 - ␦ then 5 x Ž t, t 0 , x 0 .5 - ⑀ 1 for all t g ⌳. Suppose that this is not true. Then there would exist a tU g ⌳ such that 5 x Ž tU , t 0 , x 0 .5 G ⑀ 1. But this leads us to the contradiction that F ¨ Ž tU . F ¨ Ž t 0 . - . Next suppose that t g Ž i , i x for some i. Clearly, x Ž t . s x Ž i . q
t
H f Ž s, x Ž s . . ds
if i ) i0
Ž 9.
if i - i0 .
Ž 10 .
i
and x Ž t . s x Ž i . q
t
H f Ž s, x Ž s . . ds i
In view of Žv., we easily obtain from both Ž9. and Ž10. that x Ž t . F ⑀ 1 Ž 1 q ML . - ⑀ . Therefore, the zero solution of Ž1. is stable. We shall now show that if ␥ ) 0 then lim t ª⬁ x Ž t . s 0. We first observe that since ¨ Ž t . is positive and nonincreasing on ⌳, there is a nonnegative real number such that lim ¨ Ž t . s ,
tª⬁
t g ⌳.
Ž 11 .
We claim that s 0. Suppose on the contrary that ) 0. Because of Ž11., there exists a positive real number 1 such that x Ž t . G 1
for all t g ⌳ .
Ž 12 .
If i0 ) i then, since i g ⌳, Ž12. implies that x Ž i . G 1 .
Ž 13 .
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AKHMETOV AND ZAFER
Suppose that i0 - i . In this case, i0 g ⌳, and therefore by Ž12. we have x Ž i0 . G 1 .
Ž 14 .
i Ž x Ž i . . y i0 F L x Ž i . .
Ž 15 .
Using Žv., we also have
In view of Ž14. and Ž15., we easily obtain from x Ž i . s x Ž i0 . q
i
H
0 i
f Ž s, x Ž s . . ds
that x Ž i . G 1 y ML x Ž i . and hence x Ž i . G 1r Ž 1 q ML . .
Ž 16 .
Thus, we see from Ž13. and Ž16. that x Ž i . G 2
for all i g N,
Ž 17 .
where 2 s 1rŽ1 q ML.. On the other hand, since Ž i , iq1 x g ⌳ for all i g N, lim iª⬁ ¨ Ž i q . s lim iª⬁ ¨ Ž i . s and ¨ Ž i . G ¨ Ž i q . G . Letting ms
min
FsF¨ Ž t 0 .
Ž s. ,
it follows from Ž4., Ž7. and Ž17. that ¨ Ž i . y ¨ Ž i q. G ␥ m 2
for all i g N.
Ž 18 .
Using ¨ Ž iq1 . F ¨ Ž i q . in Ž18. and then summing the resulting inequality over i from 1 to k we get ¨ Ž 1 . y ¨ Ž kq1 . G Ž ␥ m 2 . k
for all k g N.
Ž 19 .
It is clear from Ž19. that if k is sufficiently large, then the function ¨ takes on negative values. But this contradicts the fact that ¨ is positive definite. Thus, we must have s 0. As in the classical case, it follows that lim t ª⬁ x Ž t . s 0, and hence we may conclude that the zero solution is asymptotically stable.
STABILITY OF THE ZERO SOLUTION
77
COROLLARY 2.1. Let all conditions of Theorem 2.1 except Žv. be satisfied. In addition, suppose that the family i Ž x .4 is equicontinuous at x s 0 and that i0 G i Ž x . for all x g S . Then the conclusion of Theorem 2.1 remains ¨ alid. Proof. We proceed as in the proof of Theorem 2.1 until ⑀ 1 is picked. Now since the family i Ž x .4 is equicontinuous at x s 0 and i0 G i Ž x . for all x g S , given any ⑀ 2 , 0 - ⑀ 2 - ⑀rM, we can find ⑀ 3 ) 0 such that i0 y i Ž x . - 2 for all 5 x 5 - ⑀ 3 and i g N. Fix ⑀ 1 ) 0 such that ⑀ 1 min ⑀ 3 , ⑀ y M⑀ 2 4 . Then it follows from Ž9. and Ž10. that x Ž t . F ⑀ 1 q M⑀ 2 - ⑀ . Clearly, Ž11., Ž12., and Ž13. hold and by our assumption the case i0 - i does not exist. Thus Ž17. is satisfied with 2 s 1. The rest of the proof is the same as that of Theorem 2.1. In the next theorem we do not require that Dq V Ž t, x . be negative semidefinite on Di g N Gi . THEOREM 2.2.
Assume that the following conditions are fulfilled.
Ži. V Ž t, x . is positi¨ e definite on G. Žii. Dq V Ž t, x . is negati¨ e semidefinite on G_Di g N Gi . Žiii. Dq V Ž t, x . F Ž V Ž t, x .. for some g A and for all Ž t, x . g Di g N Gi . Živ. V Ž i Ž x ., x q Ji Ž x .. F Ž V Ž i Ž x ., x . for some g A and for all x g S and i g N. Žv. There exists an L ) 0 such that < i Ž x . y i Ž y .< F L 5 x y y 5 for all x, y g S and i g N. Žvi. There exists a ␥ G 0 such that
HVŽVŽ x., xx. , x Ž Ž iŽ . i
ds
..
Ž s.
G Ž L q ␥ . 5 x5
for all x g S and i g N. Then, the zero solution of Ž1. is stable if ␥ s 0 and is asymptotically stable if ␥ ) 0. Proof. Let uŽ t . be the maximal solution of uX s Ž u. on w i , i x such that uŽ i . s ¨ Ž i q ., where i and i are as defined in the proof of Theorem 2.1. Proceeding as in the proof of Theorem 2.1 we easily obtain ds
H Ž . Ž s . F L ¨ Ž i.
¨
0 i
x Ži .
for i G i0
Ž 20 .
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AKHMETOV AND ZAFER
and ds
H ŽŽq.. Ž s . F L 0 i
¨
¨
x Ži .
for i - i0 .
Ž 21 .
i
Using Živ. and Žvi. we also have
H Ž.q ¨Ž
¨
i
i
.
ds
Ž s.
F yŽ L q ␥ . x Ž i . .
Ž 22 .
It follows from Ž20., Ž21., and Ž22. that ¨ Ž i q . F ¨ Ž i . for all i g N. The remainder of the proof is similar to that of Theorem 2.1 and hence is omitted. Our last result in this paper is a Chetaev-type instability theorem, see page 216 in w12x, for the zero solution of Ž1.. THEOREM 2.3.
Assume that the following conditions are fulfilled.
Ži. For e¨ ery ⑀ ) 0 and for e¨ ery t G t 0 there exist points x g S⑀ such that V Ž t, x . ) 0. The set B of all points Ž t, x . such that x g S and such that ¨ Ž t, x . ) 0 is called the ‘‘domain ¨ ) 0.’’ The set B is bounded by the hypersurfaces 5 x 5 s and ¨ Ž t, x . s 0. We assume that ¨ is bounded from abo¨ e in B and 0 g ⭸ B for all t G t 0 . Žii. Dq V Ž t, x . is positi¨ e semidefinite on B_Di g N Ž Gi l B .. Žiii. Dq V Ž t, x . G y Ž V Ž t, x .. for some g A and for all Ž t, x . g Di g N Ž Gi l B .. Živ. V Ž i Ž x ., x q Ji Ž x .. G Ž V Ž i Ž x ., x . for some g A and for all Ž t, x . g Di g N Ž ⌫i l B ., where ⌫i s Ž t, x . : t s i Ž x .4 . Žv. There exists an L ) 0 such that < Ž x . y i0 < F L 5 x 5 for x g S and i g N. Žvi. There exists a ␥ ) 0 such that
HVŽVŽ x., xx. , x Ž Ž iŽ . i
..
ds
Ž s.
G Ž L q ␥ . 5 x5
for all x g S and i g N. Then the zero solution of Ž1. is unstable. Proof. Fix ⑀ ) 0 and t 0 , Ž t 0 , x 0 . g B, and let x Ž t . s x Ž t, t 0 , x 0 . be a solution of Ž1. having discontinuities at t s i for i g N. We shall show that x Ž t . must leave the ball S⑀ in finite time. In view of Žii. and Remark 2.1 we see that ¨ Ž t . is nondecreasing on each interval of its continuity in ⌳. We need to prove that ¨ Ž t . is nondecreasing for all t g ⌳. So we let
STABILITY OF THE ZERO SOLUTION
79
uŽ t . be the minimal solution of uX s y Ž u. on w i , i x such that uŽ i . s ¨ Ž i q ., where i and i are as defined in the proof of Theorem 2.1. By using Lemma 2 and Žvi. we see that
H Ž q. ¨Ž
i
0 i
¨
.
ds
Ž s.
G ␥ x Ži .
if i ) i0
Ž 23 .
if i - i0 .
Ž 24 .
and ds
H ŽŽ. . Ž s . G ␥ 0 i
¨
¨
x Ži .
i
From Ž23. and Ž24. we may deduce that ¨ Ž i . F ¨ Ž i q .. Therefore, ¨ Ž t . G ¨ Ž t 0 . for all t g ⌳, implying that Ž t, x Ž t .. g B_Di g N Gi for all t g ⌳. Let M ) 0 be a real number such that V Ž t, x . F M for all Ž t, x . g B, which is possible by Ži.. Since ¨ Ž t . G ¨ Ž t 0 ., there is a 1 ) 0 such that 5 x Ž t .5 G 1 for all t g ⌳. If we now define ms
min
¨ Ž t 0 .FsFM
Ž s.
then it follows from Ž23. and Ž24. that ¨ Ž i q. y ¨ Ž i . G ␥ m 1 .
Ž 25 .
Since ¨ Ž iq1 . G ¨ Ž i q . we get ¨ Ž iq1 . y ¨ Ž i . G ␥ m 1
for all i g N.
Ž 26 .
Summing Ž26. over i from 1 to k we see that ¨ Ž kq 1 . y ¨ Ž 1 . G Ž ␥ m 1 . k
for all k g N.
Ž 27 .
But Ž27. leads to a contradiction that ¨ Ž t . is unbounded in B. This completes the proof.
3. EXAMPLES EXAMPLE 3.1. Let i Ž x . s i y x 12 q x 22 so that Gi s Ž t, x . g G : i y x 12 q x 22 - t F i4 . We define S s D⬁is1 Gi and consider the impulsive
'
'
80
AKHMETOV AND ZAFER
system
½ ½
˙x 1 s ˙x 2 s
yx 2 ,
Ž t , x . f S, Ž t, x. g S
yx 1 ,
Ž t , x . f S, Ž t, x. g S
x1 , yx 2 ,
⌬ x 1 < ts i Ž x . s y␣ x 1 q  x 2 , ⌬ x 2 < ts i Ž x . s  x 1 y ␣ x 2 . We choose V Ž x . s x 12 q x 22 and make the following observations: Ža. V˙Ž x . s 0 if Ž t, x . f S and V˙Ž x . s y2V Ž x . if Ž t, x . g S. Since V Ž x q ⌬ x . s ŽŽ1 y ␣ . 2 q  2 .Ž x 12 q x 22 . y 4 Ž ␣ y 1. x 1 x 2 we have VŽ x q ⌬ x. F l Ž ␣ ,  .VŽ x. , where l Ž ␣ ,  . s Ž<  < q <1 y ␣ <. 2 . Žb. 5 x q ⌬ x 5 2 s ŽŽ1 y ␣ . 2 q  2 .Ž x 12 q x 22 . y 4 Ž ␣ y 1. x 1 x 2 and so 5 x q ⌬ x 5 2 G Ž<1 y ␣ < y <  <. 2 5 x 5 2 . It follows that if < <1 y ␣ < y <  < < G 1, then
i Ž x q ⌬ x . F i Ž x . . Žc. < i Ž x . y i0 < s x 12 q x 22 s 5 x 5. Žd. Let Ž s . s 2 s and Ž s . s l Ž ␣ . s, and fix a positive number ␥ - 1. If l Ž ␣ ,  . F 1, then ln l Ž ␣ ,  . F 2Ž1 y ␥ .5 x 5 and hence
'
lV
HV
ds 2s
F Ž 1 y ␥ . 5 x 5.
In view of Theorem 2.1 we deduce that the zero solution of Ž1. is asymptotically stable if < <1 y ␣ < y <  < < G 1 4
and
<1 y ␣ < q <  < F 1.
If we take i Ž x . s i y x 12 q x 22 , then it is easy to verify that condition Žv. is not satisfied and therefore Theorem 2.1 does not apply. However, since the additional conditions stated in Corollary 2.1 are true, we may conclude that the above conclusion is valid. We note that the beating is still absent in this case, since Ž drdt .5 x Ž t .5 s 0 for all Ž t, x . f S and Žb. holds.
'
STABILITY OF THE ZERO SOLUTION
81
EXAMPLE 3.2. Let i Ž x . s i q x 12 q x 22 . Clearly Gi s Ž t, x . : i - t F i q x 12 q x 22 4 . Define S s D⬁is1 Gi and consider the impulsive system
˙x 1 s
½
˙x 2 s
yx 2 , yx 2 q
½
x 13 ,
x1 , x1 q
x 23 ,
Ž t , x . f S, Ž t, x. g S Ž t , x . f S, Ž t , x . g S,
⌬ x 1 < ts i Ž x . s y␣ x 1 q  x 2 , ⌬ x 2 < ts i Ž x . s  x 1 y ␣ x 2 . We choose V Ž x . s x 12 q x 22 and make the following observations: Ža. V˙Ž x . s 0 if Ž t, x . f S and V˙Ž x . F 2V 2 Ž x . if Ž t, x . g S. Since V Ž x q ⌬ x . s ŽŽ1 y ␣ . 2 q  2 .Ž x 12 q x 22 . y 4 Ž ␣ y 1. x 1 x 2 , we have VŽ x q ⌬ x. F l Ž ␣ ,  .VŽ x. , where l Ž ␣ ,  . s Ž<  < q <1 y ␣ <. 2 . Žb. Let x, y g R n such that 5 x 5 F h and 5 y 5 F h, where h ) 0 is some real number. It follows that
i Ž x . y i Ž y . F 2 h5 x y y 5. Žc. If l Ž ␣ ,  . - 1 then i Ž x q ⌬ x . F i q l Ž ␣ ,  .5 x 5 - i Ž x .. Žd. Let g s Žyx 2 q x 13, x 1 q x 23 . and mŽ h. s max 5 x 5 F h 5 g 5. Clearly, mŽ h. ª 0 as h ª 0 and so there exists h 0 such that 2 hmŽ h. - 1 for all h F h0 . Že. Let Ž s . s 2 s 2 and Ž s . s l Ž ␣ ,  . s, and fix a positive real 3
number ␥ . Choose 5 x 5 F min h 0 , Ž 1 y l . r Ž 2 l Ž 2 h 0 q ␥ . . 4 . It follows that 1 y l G 2 l Ž2 h q ␥ .5 x 5 3 and so
'
V
ds
Hl V 2 s
2
G Ž 2 h q ␥ . 5 x 5.
By Theorem 2.2, the zero solution of Ž1. is asymptotically stable if <  < q < ␣ y 1 < - 1. In this case, 2 hM - 1 is sufficient for the absence of beating.
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AKHMETOV AND ZAFER
We end this paper by pointing out that the results of this paper can be generalized easily if Gi s Ž t , x . g G : t is between t s i Ž x . and i Ž x . 4 , where t s i Ž x . is any given surface like i Ž x . such that i Ž0. s i Ž0.. In this article, i Ž x . s i Ž0.. REFERENCES 1. M. U. Akhmetov, On motion with impulse actions on a surface, Iz¨ .-Acad. Nauk Kaz. SSR Ser. Fiz.-Mat. 1 Ž1988., 11᎐14. 2. M. U. Akhmetov and N. A. Perestyuk, A comparison method for differential equations with impulse actions, Differentsial’naya Ura¨ neniya 26 Ž1990., 1475᎐1483. 3. A. F. Filippov, ‘‘Differential Equations with Discontinuous Right-Hand Sides,’’ Kluwer Academic, Dordrecht, 1988. 4. S. I. Gurgula, Investigation of the stability of solutions of impulsive systems by Lyapunov’s second method, Ukrainian Math. J. 1 Ž1982., 100᎐103. 5. S. I. Gurgula and N. A. Perestyuk, On the second Lyapunov’s method in impulsive systems, Dokl. Acad. Nauk. Ukrainian. SSR Ser. A 10 Ž1982., 11᎐14. 6. P. Hartman, ‘‘Ordinary Differential Equations,’’ Wiley, New York, 1964. 7. H. Kwakernaak and R. Sivan, ‘‘Linear Optimal Control Systems,’’ Wiley, New York, 1972. 8. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, ‘‘Theory of Impulsive Differential Equations,’’ World Scientific, SingaporerNew York, 1989. 9. V. Lakshmikantham and S. Leela, ‘‘Differential and Integral Inequalities,’’ Vol. 1, Academic Press, New York, 1969. 10. V. Lakshmikantham and X. Liu, On quasi stability for impulsive differential systems, Nonlinear Anal. 13 Ž1989., 819᎐828. 11. X. Liu and R. Pirapakaran, Global stability results for impulsive differential equations, Appl. Anal. 33 Ž1989., 87᎐102. 12. R. K. Miller and A. N. Michel, ‘‘Ordinary Differential Equations,’’ Academic Press, New York, 1982. 13. V. D. Milman and A. D. Myshkis, On motion stability with shoks, Sibirsk. Mat. Zh. 1 Ž1960., 233᎐237. 14. A. M. Samoilenko and N. A. Perestyuk, ‘‘Impulsive Differential Equations,’’ World Scientific, Singapore, 1995. 15. A. M. Samoilenko and N. A. Perestyuk, Stability of solutions of impulsive differential equations, Differentsial’naya Ura¨ neniya 13 Ž1977., 1981᎐1992.