On optimal control problem for the heat equation with integral boundary condition

In this paper we consider the optimal control problem for the heat equation with an integral boundary condition. Control functions are the free term and the coefficient of the equation of state and the free term of the integral boundary condition. The coefficients and the constant term of the equation of state are elements of a Lebesgue space and the free term of the integral condition is an element of Sobolev space. The functional goal is the final. The questions of correct setting of optimal control problem in the weak topology of controls space are studied. We prove that in this problem there exist at least one optimal control. The set of optimal controls is weakly compact in the space of controls and any minimizing sequence of controls of a functional of goal converges weakly to the set of optimal controls. There is proved Frechet differentiability of the functional of purpose on the set of admissible controls. The formulas for the differential of the gradient of the purpose functional are obtained. The necessary optimality condition is established in the form of variational inequality.