Stabilizability in optimal control

We extend the well known concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also associated costs; in particular, we introduce the notions of Sample and Euler stabilizability to a closed target set C with W-regulated cost, which roughly means that we require the existence of a stabilizing feedback such that all the corresponding sampling and Euler solutions have finite costs, bounded above by a continuous, state-dependent function W, divided by some positive constant c. We prove that the existence of a special Control Lyapunov Function W, called c-Minimum Restraint function, c-MRF, implies Sample and Euler stabilizability to C with W-regulated cost, so extending [Motta, Rampazzo 2013], [Lai, Motta, Rampazzo, 2016], where the existence of a c-MRF was only shown to yield global asymptotic controllability to C with W-regulated cost.