SECOND METHOD OF LYAPUNOV FOR STABILITY OF LINEAR IMPULSIVE DIFFERENTIAL-DIFFERENCE EQUATIONS WITH VARIABLE IMPULSIVE PERTURBATIONS

The present work is devoted to the study of stability of the zero solution to linear impulsive differential-difference equations with variable impulsive perturbations. With the aid of piecewise continuous auxiliary functions, which are generalizations of the classical Lyapunov's functions, sufficient conditions are found for the uniform stability and uniform asymptotical stability of the zero solution to equations under consideration.


I. Introduction
The impulsive differential-difference equations are adequate mathematical models of various real processes and phenomena that are characterized by rapid change of their state and dependence on the pre-history at each moment.In spite of great possibili- ties of applications, the theory of these equations is developing rather slowly due to difficulties of technical and theoretical character.
If the impulses are realized at fixed moments of time, the results can be easily de- rived by virtue of the corresponding result in the continuous case.Studies of impul- sive differential-difference equations with variable impulsive perturbations carry lots of difficulties due to the presence of phenomena such as "beating" of the solutions, bi- furcation, loss of property of autonomy, etc.The importance of these equations in mathematical modeling necessitates to prove criteria for stability of their solutions.
The investigations of the present work are carried out with the aid of piecewise continuous Lyapunov's functions [3] and a technique that uses minimal subsets of suit- able spaces of piecewise continuous functions.The elements of these subsets help us estimate the derivatives of piecewise continuous auxiliary functions [1, 2].
Let 9o E C[[toh, to], n].We denote by z(t)= x(t;to,o the solution of system (1), (2) that satisfies the initial condition The symbol J + (t0,90) stands for the maximal interval of the type [t0,) at which the solution x(t;to,o is defined; co C[[ to h, to],n], and 11911-- maXs e [t o -h, t0] (s) is the norm of the function p C 0. We will specify the solution x(t)= x(t;to, Po of the initial problem (1), ( 2), (3) as follows: 1.For t [t o -h, to] the solution x(t) coincides with the initial function T0(t) For t E J + (to, o), t =/= rk(x(t)) k-1,2,..., the function x(t) is differentiable and c(t) A(t)x(t) + B(t)x(t-h).
We make the following assumptions: 114.The matrix-valued n xn-functions A(t) and B(t) are continuous for t E (to, OO).1t5.B(t) is a diagonal and A(t)is antisymmetric matrix function.
The integral curves of system (1), (2) meet successively each of the hypersur- faces o1, 0"2,'" exactly once.Condition H7 stipulate the absence of the "beating" phenomenon of the solutions to the system (1), (2), i.e., when a given integral curve meets more than once (or even infinitely many times) one and the same hypersurface.The "beating" pheno- mena is not present in the case when vk(x --tk, k 1,2,...,x Rn, i.e., when the impulses are realized at fixed moments. Definition 1: The zero solution of system (1), ( 2) is said to be a) uniformly stable, if b) (Vo e Co: I I o I I < ) (vte z + (to, So)): (t; to, o) < ; (vt _> t o + ,t e J + (to, o))" z(t; t 0, 0) < ; c) uniformly asymptotically stable if it is uniformly stable and uniformly attractive.
The function V is Lipschitzian with respect to its second argument on each of the sets Gk, k 1,2, 3.
In the sequel, we shall use the following functional classes, assuming that condi- tions H1, H2 and H3 a:e met: PC[[to, c), Rn] = x: [to, c)-Rn: x(t) is piecewise continuous with points of dis- continuity of the first'kind (i.e., the left and right limits eist there, and they are bounded) on the interval (to, C at which it is left continuous; fl {z E PC [[to, OO),n]'V(s,x(s)) <_ V(t,z(t)),t-h < s < t,t > to, V o}" Let V V0, z PC [[to, CX),gn], and t 7 rk(x)) + [A(t)z(t) + h)].
In proving the main results of the paper we shall use the following statements: Theorem 1: Let the following assumptions hold: 1.

Main Results
Theorem 2: Let the following conditions hold: 1.

2.
The elements of the matrix-valued n x n-function B(t) are nonpositive for all t E (t o, c).
Let 9o e C0: I I o I I < and let x(t) = x(t;to,o be a solution to problem (1-3).
Theorem 3: Let the following conditions hold: 1.
Conditions H l-H7 are met.