Sparse Optimal Control of the Schlögl and FitzHugh–Nagumo Systems

Eduardo Casas 1 , Christopher Ryll 2 , and Fredi Tröltzsch 3
  • 1 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
  • 2 Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany
  • 3 Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany

Abstract.

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.

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