On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains

TL;DR

This paper investigates the regularity of solutions to Dirichlet optimal control problems in polygonal domains, showing that optimal controls remain continuous even in non-convex domains with constraints.

Contribution

It provides detailed Sobolev space regularity results and proves the continuity of optimal controls under constraints in non-convex polygonal domains.

Findings

Optimal control solutions are continuous despite non-convexity.

Regularity results are established in classical Sobolev spaces.

Control constraints do not impair the continuity of the optimal control.

Abstract

A linear quadratic Dirichlet control problem posed on a possibly non-convex polygonal domain is analyzed. Detailed regularity results are provided in classical Sobolev (Slobodetskii) spaces. In particular, it is proved that in the presence of control constraints, the optimal control is continuous despite the non-convexity of the domain.