Stability results involving time-averaged Lyapunov function derivatives

摘要

The stability results which comprise the Direct Method of Lyapunov involve the existenceof auxiliary functions (Lyapunov functions) endowed with certain definiteness properties.Although the Direct Method is very general and powerful, it has some limitations: there aredynamical systems with known stability properties for which there do not exist Lyapunovfunctions which satisfy the hypotheses of a Lyapunov stability theorem. In the present paper we identify a scalar switched dynamical system whose equilibrium(at the origin) has known stability properties (e.g., uniform asymptotic stability) and weprove that there does not exist a Lyapunov function which satisfies any one of the Lyapunovstability theorems (e.g., the Lyapunov theorem for uniform asymptotic stability). Usingthis example as motivation, we establish stability results which eliminated some of thelimitations of the Direct Method alluded to. These results involve time-averaged Lyapunovfunction derivatives (TALFD's). We show that these results are amenable to the analysis ofthe same dynamical systems for which the Direct Method fails. Furthermore, and moreimportantly, we prove that the stability results involving TALFD's are less conservative thanthe results which comprise the Direct Method (which henceforth, we refer to as the classicalLyapunov stability results). While we confine our presentation to continuous finite-dimensional dynamicalsystems, the results presented herein can readily be extended to arbitrary continuousdynamical systems defined on metric spaces. Furthermore, with appropriate modifications,stability results involving TALFD's can be generalized to discontinuous dynamical systems(DDS).