Finite Element Analysis of a Constrained Dirichlet Boundary Control Problem Governed by a Linear Parabolic Equation

Abstract

This article considers finite element analysis of a Dirichlet boundary control problem governed by the linear parabolic equation. The Dirichlet control is considered in a closed and convex subset of the energy space H¹ (Ω ×(0,T)) . We discuss the well-posedness of the parabolic partial differential equation and derive some stability estimates. We prove the existence of a unique solution to the optimal control problem and derive the optimality system. The first-order necessary optimality condition results in a simplified Signorini-type problem for the control variable. The space discretization of the state variable is done using conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. To discretize the control we use the conforming prismatic Lagrange finite elements. We derive an optimal order of convergence of error in the control, state, and adjoint state under some regularity assumptions on the solutions. The theoretical results are corroborated by some numerical tests.