Finite element analysis of the Dirichlet boundary control problem governed by linear parabolic equation

TL;DR

This paper presents a finite element method for solving a Dirichlet boundary control problem governed by a linear parabolic PDE, including analysis of well-posedness, regularity, optimality conditions, and convergence with numerical validation.

Contribution

It introduces a finite element framework for Dirichlet boundary control of parabolic equations, deriving optimality conditions and establishing convergence rates.

Findings

Proved well-posedness and regularity of the control problem.

Derived optimal order of convergence for control, state, and adjoint variables.

Validated theoretical results with numerical experiments.

Abstract

A finite element analysis of a Dirichlet boundary control problem governed by the linear parabolic equation is presented in this article. The Dirichlet control is considered in a closed and convex subset of the energy space $H^1(\Omega \times(0,T)).$ We prove well-posedness and discuss some regularity results for the control problem. We derive the optimality system for the optimal control problem. The first order necessary optimality condition results in a simplified Signorini type problem for 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 control, state, and adjoint state. The theoretical results are corroborated by some numerical tests.