Optimal control of mixed local-nonlocal parabolic PDE with singular boundary-exterior data

Jean-Daniel Djida; Gisèle Mophou; Mahamadi Warma

doi:10.3934/eect.2022015

Article Contents

2022, Volume 11, Issue 6: 2129-2163. Doi: 10.3934/eect.2022015

This issue Previous Article On analytic semigroup generators involving Caputo fractional derivative Next Article Telegraph systems on networks and port-Hamiltonians. Ⅲ. Explicit representation and long-term behaviour

Optimal control of mixed local-nonlocal parabolic PDE with singular boundary-exterior data

1.
African Institute for Mathematical Sciences (AIMS), P.O. Box 608, Limbe Crystal Gardens, South West Region, Cameroon, Institut für Analysis, Fakultät Mathematik, Technische Universität Dresden D-01062 Dresden, Germany
2.
Laboratoire L.A.M.I.A., Département de Mathématiques et Informatique, Université des Antilles, Campus Fouillole, 97159 Pointe-à-Pitre, (FWI), Guadeloupe, France
3.
Department of Mathematical Sciences, Center for Mathematics and Artificial Intelligence (CMAI), George Mason University, 4400 University Dr, Fairfax, VA 22030, USA

^* Corresponding author: Mahamadi Warma
^* Corresponding author: Mahamadi Warma

Received: March 2021

Revised: December 2022

Early access: March 2022

Published: December 2022

The first author is supported by the Deutscher Akademischer Austausch Dienst/German Academic Exchange Service (DAAD). The third author is partially supported by the AFOSR under Award NO: FA9550-18-1-0242 and by the US Army Research Office (ARO) under Award NO: W911NF-20-1-0115.

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Abstract

We consider parabolic equations on bounded smooth open sets $ {\Omega}\subset \mathbb{R}^N $ ($ N\ge 1 $) with mixed Dirichlet type boundary-exterior conditions associated with the elliptic operator $ \mathscr{L} : = - \Delta + (-\Delta)^{s} $ ($ 0<s<1 $). Firstly, we prove several well-posedness and regularity results of the associated elliptic and parabolic problems with smooth, and then with singular boundary-exterior data. Secondly, we show the existence of optimal solutions of associated optimal control problems, and we characterize the optimality conditions. This is the first time that such topics have been presented and studied in a unified fashion for mixed local-nonlocal PDEs with singular data.

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Mathematics Subject Classification: 49J20, 49K20, 35S15, 49N60.

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