Inverse control-type problem of determining highest coefficient for a one-dimensional parabolic equation

In this paper, we consider one inverse control-type problem of determining the leading coefficient of a one-dimensional parabolic equation. The problem under consideration is a variational statement of a coefficient inverse problem for a parabolic equation. The sought for coefficient of the parabolic equation depends on the spatial variable. An integral boundary condition is set for the parabolic equation. The desired highest coefficient of the parabolic equation plays role of the control function, which is an element of the Sobolev space. The set of admissible control functions belong to the Sobolev space. The objective functional for the control problem is compiled based on the integral overdetermination condition set in the inverse problem. This condition may be interpreted as tasks of a weighted mean of the solution of the equation under consideration as per time variable. The solution of the boundary value problem for a parabolic equation, for each given control function, is defined as a generalized solution from the Sobolev space. The existence of a solution of the considered inverse control-type problem is proved. An adjoint boundary value problem for the control problem under study is introduced. The Frechet differentiability of the objective functional in a set of admissible control functions is proved. In addition, an auxiliary boundary value problem is introduced; and using the solution of this problem, an expression for the gradient of the objective functional is found. The necessary optimality condition for the admissible control function is obtained.