Optimal sparse control problems are considered for the FitzHugh--Nagumo system including the so-called Schlogl model. The nondifferentiable objective functional of tracking type includes a quadratic Tikhonov regularization term and the $L^1$-norm of the control that accounts for the sparsity. Though the objective functional is not differentiable, a theory of second order sufficient optimality conditions is established for Tikhonov regularization parameter $\nu>0$ and also for the case $\nu = 0$. In this context, also local minima are discussed that are strong in the sense of the calculus of variations. The second order conditions are used as the main assumption for proving the stability of locally optimal solutions with respect to $\nu \to 0$ and with respect to perturbations of the desired state functions. The theory is confirmed by numerical examples that are resolved with high precision to confirm that the optimal solution obeys the system of necessary optimality conditions.
['Eduardo Casas', 'Christopher Ryll', 'Fredi Tröltzsch']
Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation