Convergence of simultaneous distributed-boundary parabolic optimal control problems

Domingo Tarzia; Carolina Bollo; Claudia Gariboldi

doi:10.3934/eect.2020045

Article Contents

2020, Volume 9, Issue 4: 1187-1201. Doi: 10.3934/eect.2020045

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Convergence of simultaneous distributed-boundary parabolic optimal control problems

1.
Depto. Matemática-CONICET, FCE, Univ. Austral, Paraguay 1950, S2000FZF Rosario, Argentina
2.
Depto. Matemática, FCEFQyN, Univ. Nac. de Río Cuarto, Ruta 36 Km 601, 5800 Río Cuarto, Argentina

^* Corresponding author: DTarzia@austral.edu.ar
^* Corresponding author: DTarzia@austral.edu.ar

Received: September 30, 2019

Early access: March 2020

Published: December 2020

The first and third authors is supported by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement 823731 CONMECH

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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 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.

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Mathematics Subject Classification: Primary: 49J20; Secondary: 35K05, 49K20.

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