Communications in Applied Analysis 18 (2014) 455–522 NONLINEAR DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDES: FILIPPOV SOLUTIONS, NONSMOOTH STABILITY AND DISSIPATIVITY THEORY, AND OPTIMAL DISCONTINUOUS FEEDBACK CONTROL

In this paper, we develop stability, dissipativity, and optimality notions for dynamical systems with discontinuous vector fields. Specifically, we consider dynamical systems with Lebesgue measurable and locally essentially bounded vector fields characterized by differential inclusions involving Filippov set-valued maps specifying a set of directions for the system velocity and admitting Filippov solutions with absolutely continuous curves. 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. In addition, feedback interconnection stability results for discontinuous systems are developed thereby providing a generalization of the small gain and positivity theorems to systems with discontinuous vector fields. Moreover, we consider a discontinuous control problem involving a notion of optimality that is directly related to a specified nonsmooth Lyapunov function to obtain a characterization of optimal discontinuous feedback controllers. 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-YakubovichPopov 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.