Optimal control of a nonlinear PDE governed by fractional Laplacian
type: Item
instance of: scholarly article
title: Optimal control of a nonlinear PDE governed by fractional Laplacian
zbMATH Open document ID: 1486.49005
author: Rafal Kamocki
published in: Applied Mathematics and Optimization
publication date: 2022-03-25T00:00:00Z
review text: The following optimal control problem is considered: $([(-\Delta)_{\omega}]^{\beta}z)(x) = f(x,z(x),u(x))$, $x\in\Omega$ a. e., $\min J (z, u) = \int_{\Omega}$ $f_0(x,z(x),u(x))dx$ where $u(x)\in M$ for $x\in\Omega$ a.e., $\Omega\subset\mathbb{R}^N$, $N\geq 1$, is a bounded set, $\beta >0$, $M\subset\mathbb{R}^m$ and $[(-\Delta)_{\omega}]^{\beta}$ denotes a weak fractional Laplace operator of order $\beta$ with zero Dirichlet boundary values on the boundary of $\Omega$. The aim of the paper is to investigate the existence of optimal solutions to the system. The first section is devoted to preliminaries where one mainly explains the notion of weak Dirichlet-Laplace operator of fractional order. The result on the existence of optimal solutions is presented in the second section. If $M$ is a compact set, then, under five quite restrictive assumptions on $f$, $f_0$ and $J$, one proves the existence of an optimal solution $(z^*,u^*)\in\mathrm{dom}([(-\Delta)_{\omega}]^{\beta}))\times \mathcal{U}_M$ to the stated optimal control problem, $\mathcal{U}_M$ denoting the set of controls. An example is shown in the third section.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 7498412
zbMATH Keywords: lower closer theorem
zbMATH Keywords: implicit measurable function theorem
zbMATH Keywords: fractional Dirichlet-Laplace operator
zbMATH Keywords: existence of optimal solutions
MaRDI profile type: Publication
full work available at URL: s00245-021-09802-7
OpenAlex ID: W3179754132
Wikidata QID: Q114230604
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DOI: 10.1007/S00245-021-09802-7
description: scientific article; zbMATH DE number 7498412