Dissipativity theory for discontinuous dynamical systems: Basic input, state, and output properties, and finite-time stability of feedback interconnections

In this paper we develop dissipativity theory for discontinuous dynamical systems. Specifically, using set-valued supply rate maps and set-valued connective supply rate maps consisting of locally Lebesgue integrable supply rates and connective supply rates, respectively, and set-valued storage maps consisting of piecewise continuous storage functions, dissipativity properties for discontinuous dynamical systems are presented. Furthermore, extended Kalman-Yakubovich-Popov set-valued conditions, in terms of the discontinuous system dynamics, characterizing dissipativity via generalized Clarke gradients and locally Lipschitz continuous storage functions are derived. Finally, these results are used to develop feedback interconnection stability results for discontinuous dynamical systems by appropriately combining the set-valued storage maps for the forward and feedback systems.