Variational Stability For A Class Of Impulsive Differential Systems At Fixed Times

In this paper, a kind of discontinuous systems discussed in paper [22] is generalized to containing impulsive effect by using conclusions of bounded variation solutions for discontinuous systems and the theory of impulsive systems. Existence and uniqueness of bounded variation solutions for first order impulsive differential systems at fixed times on a finite interval is discussed and the sufficient conditions of existence and uniqueness of bounded variation solutions for impulsive differential systems are established. At the same time, the variational stability of bounded variation solutions at the effect of perturbation and non-perturbation for the impulsive differential system are discussed, the Ljapunov type theorems for variational stability and asymptotically variational stability for impulsive differential systems are established. These results are a generalization of corresponding results of bounded variation solution and variational stability of bounded variation solutions to discontinuous systems in paper [22] and [23].