Results on relaxation theorems for hybrid systems

The Filippov-Wazewski relaxation theorem states that the solution set of a locally Lipschitz differential inclusion is dense in the solution set of its convexified differential inclusion in the uniform metric on compact time intervals. For a broad class of hybrid systems with logic variables, we propose conditions on system dynamics and on system data to establish finite-horizon relaxation theorems that rely on graphical distance of solutions. In turn these relaxation theorems are used to show continuous dependence on initial conditions. We also demonstrate by examples that the relaxation theorems could fail if any of the proposed conditions is removed