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.

中文翻译：

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具有分段光滑Lyapunov函数的非线性切换系统的生存性。

在本文中，我们集中于具有非光滑Lyapunov函数的切换非线性系统的生存力和吸引力。我们确定Lyapunov函数分段光滑的交换系统的可行集和吸引区域。通过使用沿着子系统轨迹的分段光滑Lyapunov函数的方向导数来构造开关定律。导出了充分的条件，以保证分段非线性Lyapunov函数的水平集上切换非线性系统的生存力和吸引力。我们进一步将该方法扩展到涉及可能的滑动运动的切换系统。本文中的方法为研究活力和吸引力提供了一个统一的框架，并系统地考虑了滑动运动。最后，