GENERALIZED BELLMAN-HAMILTON-JACOBI OPTIMALITY CONDITIONS FOR A CONTROL PROBLEM WITH A BOUNDARY-CONDITION

In this paper we study conditions for optimality of a deterministic co ntrol problem where the state of the system is required to stop at the boundary. Using the Clarke generalized gradient, we refine the classi cal verification theorem and show that it is not only sufficient but a lso necessary for optimality. It is also shown that the solution to th e generalized Bellman-Jacobi-Hamilton equation involving the Clarke ge neralized gradient is unique among the class of regular functions.