On the problem of optimal control in the coefficients of an elliptic equation

In this paper we consider the optimal control problem for linear elliptic equations of the second order. Control functions are included in the coefficients of the equation for the state, including the coefficients of the highest derivatives. Space management is a product of Lebesgue and Sobolev spaces. The functional purpose is the sum of the integrals over the region and part of its border. The problems of correct statement of the problem in the weak topology of the space of controls are studied. It is proved that a set of optimal control problems is not empty, it is weakly compact and every minimizing sequence of the functional goals converges weakly in the space of controls to the set of optimal controls. The examples show that the solution of the problem can be not unique and minimizing sequence for the functional purpose can not have a limit in the strong topology of space management. Differentiability of proved Frechet functional is proved and the expression for its gradient is found. A necessary condition for optimality in the form of variational inequalities.