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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 13, Issue 4


Sparse Optimal Control of the Schlögl and FitzHugh–Nagumo Systems

Eduardo Casas
  • Departmento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, Av. Los Castros s/n, 39005 Santander, Spain
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Published Online: 2013-08-15 | DOI: https://doi.org/10.1515/cmam-2013-0016


We investigate the problem of sparse optimal controls for the so-called Schlögl model and the FitzHugh–Nagumo system. In these reaction–diffusion equations, traveling wave fronts occur that can be controlled in different ways. The L1-norm of the distributed control is included in the objective functional so that optimal controls exhibit effects of sparsity. We prove the differentiability of the control-to-state mapping for both dynamical systems, show the well-posedness of the optimal control problems and derive first-order necessary optimality conditions. Based on them, the sparsity of optimal controls is shown. The theory is illustrated by various numerical examples, where wave fronts or spiral waves are controlled in a desired way.

Keywords: Optimal Control; Reaction–Diffusion System; Schlögl Model; FitzHugh–Nagumo System; Sparse Control; Traveling Wave Fronts; Spiral Waves

About the article

Published Online: 2013-08-15

Published in Print: 2013-10-01

Citation Information: Computational Methods in Applied Mathematics, Volume 13, Issue 4, Pages 415–442, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2013-0016.

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