# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020048

## The viability of switched nonlinear systems with piecewise smooth Lyapunov functions

 1 School of Management, University of Shanghai for Science and Technology Shanghai, 200093, China 2 School of Science, Inner Mongolia University of Science and Technology Baotou, 014010, China

* Corresponding author: Yan Gao

Received  May 2019 Revised  November 2019 Published  March 2020

In this paper, we focus on the viability and attraction for switched nonlinear systems with nonsmooth Lyapunov functions. We determine the viable set and region of attraction for switched systems in which Lyapunov functions are piecewise smooth. The switching law is constructed by using the directional derivatives of a piecewise smooth Lyapunov function along the trajectories of the subsystems. Sufficient conditions are derived to guarantee the viability and attraction of switched nonlinear systems on the level set of a piecewise smooth Lyapunov function. We further extend the method to switched systems involving possible sliding motions. The approach in the paper provides a unified framework for studying viability and attraction with a systematic consideration of sliding motions. Finally, considering two certain classes of piecewise smooth functions, the related conditions of the viability and attraction for the level set are developed.

Citation: Jianfeng Lv, Yan Gao, Na Zhao. The viability of switched nonlinear systems with piecewise smooth Lyapunov functions. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020048
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##### References:
Trajectories of switched systems excluding sliding motions
Trajectories of switched systems including sliding motions
The phase portraits of subsystems 1 and 2
State responses under the switching law
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