Optimization methods for Dirichlet control problems

Abstract

We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control is discretized explicitly using continuous piecewise linear approximations. Unconstrained, control-constrained, state-constrained and control-and-state constrained problems are analysed. A preconditioned conjugate method for a reduced problem in the control variable is proposed to solve the unconstrained problem, whereas semismooth Newton methods are discussed for the solution of constrained problems. State constraints are treated via a Moreau–Yosida penalization. Convergence is studied for both the continuous problems and the finite dimensional approximations. In the finite dimensional case, we are able to show convergence of the optimization procedures even in the absence of Tikhonov regularization parameter. Computational aspects are also treated and several numerical examples are included to illustrate the theoretical results.

Keywords:

• Dirichlet optimal control
• discretization
• constrained optimization
• preconditionate conjugate gradient
• semismooth Newton methods

Acknowledgements

The author would like the anonymous referees for their careful reading of the original manuscript and their suggestions and corrections.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was partially supported by the Spanish Ministerio de Ciencia e Innovación [project number MTM2014-57531-P] and [project number MTM2017-83185-P].