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.