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Sparse optimal control of the Schlögl and Fitzhugh-Nagumo systems. (English) Zbl 1393.49019
Summary: 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 $$L^{1}$$-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.

##### MSC:
 49K20 Optimality conditions for problems involving partial differential equations 49M05 Numerical methods based on necessary conditions 35K57 Reaction-diffusion equations 37N25 Dynamical systems in biology
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