Filippov's selection theorem and the existence of solutions for optimal control problems in time scales

Abstract

We obtain an extension of Filippov's Lemma to control systems in time scales, where the multifunction associated with the dynamics is delta measurable on time and Lipschitz on the state variable. Using a modified version of a recent result on compactness of the set of trajectories for inclusions on time scales, in combination with the Filippov's selection theorem obtained here, we prove the existence of solutions to optimal control problems in time scales. At the outset, we provide some measurability properties for functions and multifunctions defined in time scales.