Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

Günter Leugering; Gisèle Mophou; Maryse Moutamal; Mahamadi Warma

doi:10.3934/mcrf.2022015

Article Contents

2023, Volume 13, Issue 2: 771-807. Doi: 10.3934/mcrf.2022015

This issue Previous Article Boundary control for transport equations Next Article Averaged turnpike property for differential equations with random constant coefficients

Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

1.
Department of Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11 (03.322), 91058 Erlangen, Germany
2.
Laboratoire L.A.M.I.A., Département de Mathématiques et Informatique, Université des Antilles, Campus Fouillole, 97159 Pointe-à-Pitre, (FWI), Guadeloupe, Laboratoire MAINEGE, Université Ouaga 3S, 06 BP 10347 Ouagadougou 06, Burkina Faso
3.
Department of Mathematics, University of Buea, Camerron, African Institute for Mathematical Sciences (AIMS), P.O. Box 608, Limbe Crystal Gardens, South West Region, Cameroon
4.
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: May 2021

Revised: February 2022

Early access: March 2022

Published: June 2023

The third author is supported by the Deutscher Akademischer Austausch Dienst/German Academic Exchange Service (DAAD). The fourth 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

In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.

Keywords:

Riemann-Liouville fractional derivative,
Caputo fractional derivative,
fractional integral,
Sturm-Liouville equations,
initial-boundary value problems,
optimal control,
optimality system,
optimaltiy conditions.

Mathematics Subject Classification: 35J20,49J45,49J20.

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Figure 1. A sketch of a star graph with n edges

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References

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