The Value Function in Optimal Control: Sensitivity, Controllability, and Time-Optimality

Abstract

We consider a general optimal control problem in which the constraints depend on a parameter $\alpha $, and the resulting value function $V(\alpha )$. A formula for the generalized gradient of V is proven and then used to obtain results on stability and controllability of the problem. A special study is made of the time-optimal control problem, one consequence of which is a new criterion assuring local null-controllability of the system and continuity of the minimal time function at the origin.