Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem

Gudi, Thirupathi; Sau, Ramesh Ch.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 78.

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We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L² cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.

Classification : 65N30, 65N15, 65N12, 65K10
Mots-clés : Diffusion equation, PDE-constrained optimization, control-constraints, finite element method, error bounds

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