On reduction of differential inclusions and Lyapunov stability^*

¹ School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK, USA.
² Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, USA.
³ Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, USA.

^** Corresponding author: rushikesh.kamalapurkar@okstate.edu

Received: 10 May 2019
Accepted: 25 November 2019

Abstract

In this paper, locally Lipschitz, regular functions are utilized to identify and remove infeasible directions from set-valued maps that define differential inclusions. The resulting reduced set-valued map is pointwise smaller (in the sense of set containment) than the original set-valued map. The corresponding reduced differential inclusion, defined by the reduced set-valued map, is utilized to develop a generalized notion of a derivative for locally Lipschitz candidate Lyapunov functions in the direction(s) of a set-valued map. The developed generalized derivative yields less conservative statements of Lyapunov stability theorems, invariance theorems, invariance-like results, and Matrosov theorems for differential inclusions. Included illustrative examples demonstrate the utility of the developed theory.