Convergence of simultaneous distributed-boundary parabolic optimal control problems

TL;DR

This paper studies the convergence of optimal controls in heat conduction problems with mixed boundary conditions as the heat transfer coefficient increases, establishing existence, uniqueness, and optimality conditions.

Contribution

It introduces a framework for analyzing the convergence of simultaneous distributed-boundary optimal controls in parabolic problems as the boundary heat transfer coefficient tends to infinity.

Findings

Proves existence and uniqueness of optimal controls.

Derives first order optimality conditions using adjoint states.

Shows convergence of controls and states as the heat transfer coefficient increases.

Abstract

We consider a heat conduction problem $S$ with mixed boundary conditions in a n-dimensional domain $\Omega$ with regular boundary $\Gamma$ and a family of problems $S_{\alpha}$, where the parameter $\alpha>0$ is the heat transfer coefficient on the portion of the boundary $\Gamma_{1}$. In relation to these state systems, we formulate simultaneous \emph{distributed-boundary} optimal control problems on the internal energy $g$ and the heat flux $q$ on the complementary portion of the boundary $\Gamma_{2}$. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system and the adjoint states when the heat transfer coefficient $\alpha$ goes to infinity. Finally, we prove estimations between the simultaneous distributed-boundary optimal control and the distributed optimal control problem studied in a previous paper of the first author.