Optimal control of a class of semilinear fractional elliptic equations

Cyrille Kenne; Gisèle Mophou; Mahamadi Warma; Cyrille Kenne; Gisèle Mophou; Mahamadi Warma

doi:10.3934/math.2026177

AIMS Mathematics

2026, Volume 11, Issue 2: 4415-4444. doi: 10.3934/math.2026177

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Optimal control of a class of semilinear fractional elliptic equations

1.
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
2.
Département de Mathématiques et Informatique, Université des Antilles, Campus Fouillole, 97159 Pointe-à-Pitre (FWI), Guadeloupe
3.
Laboratoire MAINEGE, Université Ouaga 3S, 06 BP 10347 Ouagadougou 06, Burkina Faso
4.
Department of Mathematical Sciences and the Center for Mathematics and Artificial Intelligence (CMAI), George Mason University, Fairfax, VA 22030, USA

Received: 04 December 2025 Revised: 25 January 2026 Accepted: 02 February 2026 Published: 12 February 2026
MSC : 49J20, 35J61, 35R11

In this paper, a class of semilinear fractional elliptic equations associated with the spectral or integral fractional Dirichlet Laplace operator was considered. Unlike most contributions to fractional optimal control, which assume that the control enters the state equation linearly, we addressed space-fractional equations in which the control appears in a nonlinear form. We proved existence and regularity results for the state equation and, using a measurable selection argument, established the existence of optimal controls. We then derived a pointwise Pontryagin-type minimum principle and first-order necessary optimality conditions. Second-order conditions for optimality are also obtained for $ L^{\infty} $, and $ L^2 $-local solutions under some structural assumptions.
- nonlinear fractional equation,
- optimal solutions,
- Pontryagin principle,
- first- and second-order optimality conditions
Citation: Cyrille Kenne, Gisèle Mophou, Mahamadi Warma. Optimal control of a class of semilinear fractional elliptic equations[J]. AIMS Mathematics, 2026, 11(2): 4415-4444. doi: 10.3934/math.2026177

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Abstract

In this paper, a class of semilinear fractional elliptic equations associated with the spectral or integral fractional Dirichlet Laplace operator was considered. Unlike most contributions to fractional optimal control, which assume that the control enters the state equation linearly, we addressed space-fractional equations in which the control appears in a nonlinear form. We proved existence and regularity results for the state equation and, using a measurable selection argument, established the existence of optimal controls. We then derived a pointwise Pontryagin-type minimum principle and first-order necessary optimality conditions. Second-order conditions for optimality are also obtained for $ L^{\infty} $, and $ L^2 $-local solutions under some structural assumptions.

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