Variational formulation of an inverse problem for a parabolic equation with integral conditions

Coefficient inverse problems for partial differential equations can be posed as optimal control problems, i. e. in variation form. In such formulations, the sought-for coefficients of the state equations play the role of control functions, and the objective functionals are compiled on the basis of additional conditions. The paper discusses a variational formulation of the inverse problem of determining the lower coefficient of a multidimensional parabolic equation with an integral boundary condition and an additional integral condition. In this case, the role of the control function is played by the lower coefficient of the parabolic equation and is an element of the space of Lebesgue integrable functions with a finite summability index. The solution to the boundary value problem for a parabolic equation, for each given control function, is defined as a generalized solution from the Sobolev space. The objective functional is based on an additional integral condition. The existence of a solution to the problem is proved and the necessary optimality condition is obtained.