On reduction of differential inclusions and Lyapunov stability

In this paper, locally Lipschitz regular functions are utilized to identify and remove infeasible directions from differential inclusions. The resulting reduced differential inclusion is point-wise smaller (in the sense of set containment) than the original differential inclusion. The reduced inclusion is utilized to develop a generalized notion of a time derivative for locally Lipschitz candidate Lyapunov functions. The developed generalized derivative yields less conservative statements of Lyapunov stability results and invariance-like results for differential inclusions. Illustrative examples are included to demonstrate the utility of the developed stability theorems.