On the Equivalence Between Dissipativity and Optimality of Discontinuous Nonlinear Regulators for Filippov Dynamical Systems

In this paper, we derive guaranteed gain, sector, and disk margins for nonlinear optimal and inverse optimal discontinuous feedback regulators that minimize a nonlinear-nonquadratic performance functional for Filippov dynamical systems. Specifically, we consider dynamical systems with Lebesgue measurable and locally essentially bounded vector fields characterized by differential inclusions involving Filippov set-valued maps. In addition, we extend classical dissipativity theory to address the problem of dissipative discontinuous dynamical systems. These results are then used to derive extended Kalman-Yakubovich-Popov conditions for characterizing necessary and sufficient conditions for dissipativity of discontinuous systems using Clarke gradients and locally Lipschitz continuous storage functions. Furthermore, using the newly developed dissipativity notions we develop a return difference inequality to provide connections between dissipativity and optimality of nonlinear discontinuous controllers for Filippov dynamical systems. Specifically, using the extended Kalman-Yakubovich-Popov conditions we show that our discontinuous feedback control law satisfies a return difference inequality if and only if the controller is dissipative with respect to a quadratic supply rate.