A Corollary for Switched Nonsmooth Systems with Applications to Switching in Adaptive Control

This paper generalizes the Lasalle-Yoshizawa Theorem to switched nonsmooth systems. It is established that Filippov (Krasovskii) regularization of a switched system is contained within the convex hull of the Filippov (Krasovskii) regularizations of the subsystems. A common candidate Lyapunov function that has a negative semidefinite derivative along the trajectories of the subsystems is shown to be sufficient to establish LaSalleYoshizawa results for the switched system. Results for regular and non-regular candidate Lyapunov functions are presented using appropriate generalization of the time derivative. The developed generalization is motivated by adaptive control of switched systems where the derivative of the candidate Lyapunov function is typically negative semidefinite.