Stability criteria for impulsive differential equations in terms of two measures

In general, the stability notions of a given solution d(t) of an impulsive differential system cannot be transferred to the stability notions of the trivial solution by change of variables. This is because of the fact that moments of impulse effects of 4(t) need not be the same as that of a different solution x(t) and therefore demanding that the difference lx(t) d(t)1 be small for all future time seems unnatural (see [4, 73). As a result, the definitions of stability require suitable changes in the general situations and the corresponding theory has not yet been developed. However, in the case of impulsive differential systems with fixed moments of impulse effects such a problem does not arise and hence the standard definitions of stability will suffice. Employing suitable discontinuous Lyapunov functions, stability theory has recently been developed for such systems [ 1, 2, 3, 51. In this paper, we shall discuss stabiliy theory of impulsive differential systems with fixed moments of impulse effects in terms of two measures, since such a theory enables us to unify a variety of stability results found in the literature. We shall employ theory of impulsive differential inequalities as well as a direct approach to bring out the advantages of each approach.