Converse stability theorems for discontinuous dynamical systems: Improved results

In (Ye, et.al., 1998) we established, among other results, a set of sufficient conditions for the uniform asymptotic stability of invariant sets for discontinuous dynamical systems (DDS) defined on metric space, and under some additional minor assumptions, we also established a set of necessary conditions (a converse theorem). This converse theorem involves Lyapunov functions which need not necessarily be continuous. In the present paper, we show that under some additional very mild assumptions, the Lyapunov functions for the converse theorem need actually be continuous. This improvement in the regularity properties of the Lyapunov functions shows that the stability results in (Ye, et.al., 1998) (under the additional mild assumptions) are rather robust. To keep our presentation as simple as possible, we confine ourselves to discontinuous dynamical systems determined by ordinary differential equations. However, the methodology employed herein can be used to establish converse theorems for DDS involving continuous Lyapunov functions for dynamical systems defined on metric spaces concerning a variety of stability and boundedness types.