A Parametric Optimal Control Problem Involving a Diffusion Coefficient

Abstract

We investigate a parametric optimal control problem governed by a time dependent parabolic equation, where the control is a spatially varying diffusion coefficient. The objective is to minimize a tracking type cost functional under box constraints on the control. We establish the well posedness of the state equation, prove the existence of an optimal solution, and derive the first order optimality system consisting of the state equation, the adjoint equation, and a projection formula. A fully discrete numerical method combining finite element spatial discretization, implicit Euler time integration, and an adjoint based projected gradient algorithm is developed to solve the optimization problem efficiently. Numerical experiments demonstrate the accuracy, robustness, and effectiveness of the proposed approach. newline newline noindent textit