Variational Stability For A Class Of Discontinuous System

In this paper,by using the Henstock integral and Lyapunov function, the variational stability for bounded variation solutions of a class of discontinuous systems is considered. For this kind of discontinuous systems ,we introduce some conceptions of the variational stability for bounded variation solutions,the Lyapunov type theorems for variational stability and variationally-asymptotically stability of bounded variation solutions are established. Finally,we give converse Lyapunov theorems and show that the variational stability and variationally-asymptotically stability imply the existence of Lyapunov functions with the properties described in Lyapunov type theorems.