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 derivative in the direction(s) of
a set-valued map for locally Lipschitz candidate Lyapunov functions. The
developed generalized derivative yields less conservative statements of
Lyapunov stability results, invariance-like results, and Matrosov results for
differential inclusions. Illustrative examples are included to demonstrate the
utility of the developed stability theorems.
