On a variant of Tykhonov regularization in optimal control under PDEs

TL;DR

This paper explores a flexible variant of Tikhonov regularization for PDE-constrained optimal control, allowing control-state independence and analyzing existence, optimality, numerical methods, and convergence of solutions.

Contribution

It introduces a new regularization approach that relaxes control-state differential constraints and studies its theoretical and numerical properties.

Findings

Existence and optimality of solutions established.

Numerical procedure developed and tested on academic examples.

Convergence of approximate solutions to original problem demonstrated.

Abstract

We make some remarks on a variant of the classical Tikhonov regularization in optimal control under PDEs which allows for a certain flexibility in dealing with non-linearities and state restrictions, in the sense that differential constraints between control and state are eliminated and pairs can run freely in their respective sets of feasibility, at the expense of introducing an additional variable in a collection of approximated problems. In addition to exploring basic issues like existence and optimality, we also discuss a numerical procedure and apply it to some academic, illustrative numerical tests, as well as examine the convergence of solutions of this new family of approximated problems to the solutions of the underlying optimal control problem.