Sparse Optimal Control of the Schlögl and FitzHugh-Nagumo Systems

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-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. 2010 Mathematical subject classification: 49K20, 49M05, 35K57, 37N25.