A dual dynamic programming for multidimensional parabolic optimal control problems

In the paper the optimal control problems governed by parabolic equations 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 anti 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 and some approximation of the problem considered which seems to be very useful from the practical point of view.