On reduction of differential inclusions and Lyapunov stability
For researchers in control theory and dynamical systems, this provides a theoretical framework to obtain less conservative stability conditions for differential inclusions.
This paper develops a method to reduce set-valued maps in differential inclusions by removing infeasible directions using locally Lipschitz, regular functions, leading to less conservative Lyapunov stability theorems. Examples demonstrate the utility of the approach.
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.