# On reduction of differential inclusions and Lyapunov stability

## TL;DR

This paper introduces a method to reduce differential inclusions using Lipschitz functions, leading to less conservative Lyapunov stability results and broader applicability in stability analysis.

## Contribution

It proposes a novel reduction technique for set-valued maps in differential inclusions and develops a generalized derivative for Lyapunov functions, improving stability analysis.

## Key findings

- Reduced set-valued maps are smaller and more precise.
- Generalized derivatives lead to less conservative stability theorems.
- Illustrative examples demonstrate practical utility.

## 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 point-wise 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.