Stabilizability in impulsive optimization problems

We introduce the concepts of impulsive and of regular Sample and Euler stabilizability for impulsive control systems, where a cost is also considered. A condition guaranteeing the existence of a discontinuous stabilizing feedback such that the corresponding (impulsive or regular) sampling and Euler solutions have costs all bounded above by the same continuous, state-dependent function, is presented. This condition, based on the existence of a special Control Lyapunov Function, implies also that the infima of the cost over impulsive and over regular inputs and solutions, coincide. The proofs are constructive and we exhibit explicit control syntheses in feedback form.