A Neumann Boundary Control for Multidimensional Parabolic "Minmax" Control Problems

In this paper, the optimal control problems of "minmax" type, governed by parabolic equations with the second control on the boundary (Neumann condition), are considered. We apply a new dual dynamic programming approach to derive sufficient optimality conditions for such problems. The idea is to move all the notions from a state space to a dual space and to obtain a new verification theorem providing the conditions which should be satisfied by a solution of the dual partial differential equation of dynamic programming. We also give sufficient optimality conditions for the existence of an optimal dual feedback control.