Optimal control for pointwise asymptotic stability in a hybrid control system

Pointwise asymptotic stability, or semistability, is a property of the set of equilibria of a dynamical system, where every equilibrium is Lyapunov stable and every solution is convergent to some equilibrium. Under an appropriate version of asymptotic controllability assumption, it is shown that the property can be achieved in a hybrid control system by open-loop optimal solutions of an infinite-horizon optimal control problem. For discrete-time systems, the optimal solutions can be generated by feedback. Regularity of the optimal value function and the existence of hybrid optimal controls are also studied.