Existence of optimal controls on hybrid time domains

Hybrid control systems are considered, combining continuous-time dynamics and discrete-time dynamics, and modeled by differential equations or inclusions, by difference equations or inclusions, and by constraints on the resulting dynamics. Solutions are defined on hybrid time domains. Finite-horizon and infinite-horizon optimal control problems for such control systems are considered. Existence of optimal open-loop controls is shown. The assumptions used include, essentially, the existence for the (non-hybrid) continuous-time case; the existence for the (non-hybrid) discrete-time case; mild conditions on the endpoint penalties; and closedness and boundedness, in the finite-horizon case, of the set of admissible hybrid time domains. Examples involving switching systems and hybrid automata are included.