Optimal Feedback Controls

Optimal control problems governed by ordinary differential equations with control constraints that are not necessarily compact are considered. Conditions imposed on the data and on the structure of the terminal sets imply that the minimum is attained and that the value function is locally Lipschitz. A necessary condition in terms of lower directional Dini derivates of the value function is given. The condition reduces to the Bellman–Hamilton–Jacobi (BHJ) condition at points of differentiability of the value, and for a subclass of the problems considered implies that the value is a viscosity solution of the BHJ equation. A strengthened version of the necessary condition gives an optimal feedback control and a procedure for approximating optimal controls.