FINITE ELEMENT ANALYSIS OF A CONSTRAINED DIRICHLET BOUNDARY CONTROL PROBLEM GOVERNED BY A LINEAR PARABOLIC EQUATION

摘要

This article considers finit eelemen tanalysi so f aDirichle tboundar ycontro lproblem governed by the linear parabolic equation. The Dirichlet control is considered in a closed and convex subset of the energy space H~1(? × (0, T )). We discuss the well-posedness of the parabolic partial differentia lequatio nan dderiv esom estabilit yestimates .W eprov eth eexistenc eo f aunique solution to the optimal control problem and derive the optimality system. The first-orde rnecessary optimality condition results in a simplifie dSignorini-typ eproble mfo rth econtro lvariable .The space discretization of the state variable is done using conforming finit eelements ,wherea sth etime discretization is based on discontinuous Galerkin methods. To discretize the control we use the conforming prismatic Lagrange finit eelements .W ederiv ea noptima lorde ro fconvergenc eo ferror in the control, state, and adjoint state under some regularity assumptions on the solutions. The theoretical results are corroborated by some numerical tests.