A Converse Lyapunov-Type Theorem for Control Systems with Regulated Cost

Given a nonlinear control system, a target set, a nonnegative integral cost, and a continuous function $W$, we say that the system is {\em globally asymptotically controllable to the target with $W$-regulated cost}, whenever, starting from any point $z$, among the strategies that achieve classical asymptotic controllability we can select one that also keeps the cost less than $W(z)$. In this paper, assuming mild regularity hypotheses on the data, we prove that a necessary and sufficient condition for global asymptotic controllability with regulated cost is the existence of a special, continuous Control Lyapunov Function, called a {\em Minimum Restraint Function}. The main novelty is the necessity implication, obtained here for the first time. Nevertheless, the sufficiency condition extends previous results based on semiconcavity of the Minimum Restraint Function, while we require mere continuity.