Asymptotic controllability and optimal control

We consider a control problem where the state must approach asymptotically a target C while paying an integral cost with a non-negative Lagrangian l. The dynamics f is just continuous, and no assumptions are made on the zero level set of the Lagrangian l. Through an inequality involving a positive number and a Minimum Restraint Function U=U(x) – a special type of Control Lyapunov Function – we provide a condition implying that (i) the system is asymptotically controllable, and (ii) the value function is bounded by . The result has significant consequences for the uniqueness issue of the corresponding Hamilton–Jacobi equation. Furthermore it may be regarded as a first step in the direction of a feedback construction.