High order Lyapunov-like functions for optimal control

We consider an optimal control problem where the state has to approach asymp -totically a closed target, while paying an integral cost with a non-negative Lagrangian £. We generalize the dissipative relation that usually defines a Control Lvapunov Function bv introducing a weaker differential inequality, which involves both the Lagrangian £ and higher order dynamics' directions expressed in form of iterated Lie brackets up to a certain degree k. The existence of a solution U of the resulting extended relation turns out to be sufficient for a twofold goal: on the one hand, it ensures that the system is globally asymptotically controllable to the target, and, 011 the other hand, it implies that the value function associated to the minimization problem is bounded above by a [/-dependent function. We call such a solution U a degree-k Minimum Restraint Function (k > 1). A11 example is provided where a smooth degree-1 Minimum Restraint Function fails to exist, while the distance from the target happens to be a C°° degree-2 Minimum Restraint Function