Variational method of solving a coefficient inverse problem for an elliptic equation

One of the main types of inverse problems for equations with partial derivatives are the problems in which coefficients of equations or included values have to be determined based on some additional information. Such problems are called coefficient inverse problems for equations with partial derivatives. Inverse problems for equations with partial derivatives can be set in a variational form, i. e. like problems of optimal control by corresponding systems. Variational setting of one coefficient inverse problem for a two-dimensional elliptic equation with additional integral condition is considered. At that, the control function gets included in the coefficient when solving the equation of state, and is an element of a space of quadric totalized functions in the sense of Lebeg. Objective functional is set on the basis of an additional integral condition. Boundary conditions for equation of the state are mixed, i.e. the second boundary condition is given in one part of the boundary, and the first boundary condition is given in another part. Solving the boundary problem at each fixed control coefficient intends a generalized solution from the Sobolev space. The questions of correctness of the considered coefficient inverse problem in variational setting are studied. It is proved that the considered problem is correctly set in the weak topology of control functions’ space. I. e. the multitude of optimal controls is nonvacuous and weakly compact; and any minimizing sequence of the problem weakly converges to the multitude of optimal controls. Besides, differentiability of objective functional in the sense of Frechet is proved, and a formula for its gradient is obtained. The necessary optimum condition in the form of variational inequality is determined.