Optimal Control Problem for an Elliptic Equation with Periodicity Conditions and Control at Solution

Optimal control problems for elliptic equations with classical boundary conditions have been thoroughly studied. However, these problems with periodicity conditions are less well-researched. This paper focuses on the optimal control problem for an elliptic equation with periodicity conditions. The control function is the quotient at the solution to the elliptic equation and belongs to the Lebesgue space with a finite summability index. The solution to the boundary value problem for the elliptic equation is defined as a generalized solution from the Sobolev space. The paper examines the correctness of the considered optimal control problem, derives a formula for the gradient of the target functional, and determines a necessary condition for control optimality.