On robust stability of switched systems in the context of Filippov solutions

We consider the robust stability problem of a class of nonlinear switched systems defined on compact sets with state-dependent switching. Instead of the Carathéodory solutions, we study the more general Filippov solutions. This encapsulates solutions with infinite switching in finite time and sliding modes in the neighborhood of the switching surfaces. In this regard, we formulate a Lyapunov-like stability theorem, based on the theory of differential inclusions. Additionally, we extend the results to switched systems with simplical uncertainty. We also demonstrate that, for the special case of polynomial switched systems defined on semi-algebraic sets, stability analysis can be checked based on sum of squares programming techniques.