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

Gudi, T and Mallik, G and Sau, RCh (2022) Finite element analysis of the Dirichlet boundary control problem governed by linear parabolic equation. In: SIAM Journal on Control and Optimization, 60 (6). pp. 3262-3288.

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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 x (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. © 2022 Society for Industrial and Applied Mathematics.

Item Type:	Journal Article
Publication:	SIAM Journal on Control and Optimization
Publisher:	Society for Industrial and Applied Mathematics Publications
Additional Information:	The copyright for this article belongs to Society for Industrial and Applied Mathematics Publications.
Keywords:	Constrained optimization; Galerkin methods; Lagrange multipliers; Optimal control systems; Partial differential equations, Control constraint; Dirichlet boundary controls; Error estimates; Finite element analyse; Lagrange finite elements; Linear parabolic PDE; Linear-parabolic; Parabolic PDEs; PDE-constrained optimization; Prismatic lagrange finite element, Finite element method
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	31 Jan 2023 06:26
Last Modified:	31 Jan 2023 06:26
URI:	https://eprints.iisc.ac.in/id/eprint/79589

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