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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr


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1609-9389
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Volume 18, Issue 1

Issues

Sparse Optimal Control for Fractional Diffusion

Enrique Otárola / Abner J. Salgado
Published Online: 2017-09-02 | DOI: https://doi.org/10.1515/cmam-2017-0030

Abstract

We consider an optimal control problem that entails the minimization of a nondifferentiable cost functional, fractional diffusion as state equation and constraints on the control variable. We provide existence, uniqueness and regularity results together with first-order optimality conditions. In order to propose a solution technique, we realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator and consider an equivalent optimal control problem with a nonuniformly elliptic equation as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose a fully discrete scheme: piecewise constant functions for the control variable and first-degree tensor product finite elements for the state variable. We derive a priori error estimates for the control and state variables.

Keywords: Optimal Control Problem; Nondifferentiable Objective; Sparse Controls; Fractional Diffusion; Weighted Sobolev Spaces; Finite Elements; Stability; Anisotropic Estimates

MSC 2010: 26A33; 35J70; 49K20; 49M25; 65M12; 65M15; 65M60

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About the article

Received: 2017-04-04

Revised: 2017-08-03

Accepted: 2017-08-05

Published Online: 2017-09-02

Published in Print: 2018-01-01


Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1418784

Funding Source: Consejo Nacional de Innovación, Ciencia y Tecnología

Award identifier / Grant number: 3160201

The first author has been supported in part by CONICYT through FONDECYT project 3160201. The second author has been supported in part by NSF grant DMS-1418784.


Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 1, Pages 95–110, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0030.

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