An optimal control problem by parabolic equation in the class of smooth controls

We consider an optimal control problem by parabolic equation with differential constraint on the boundary. The problem is considered in the class of smooth controls. Functions of controls are satisfied by constraints in each point. Such problems describe the processes of mass transfer to the column of reverse mixing. We obtain a necessary optimality condition for the optimal control problem. We propose the method for improvement of admissible controls and carry out the numerical experiment.