Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation

Optimal sparse control problems are considered for the FitzHugh–Nagumo system including the so-called Schlögl model. The nondifferentiable objective functional of tracking type includes a quadratic Tikhonov regularization term and the L1-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 ν > 0 and also for the case ν = 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 ν → 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.This author’s research was partially supported by the Spanish Ministerio de Econom´ıa y Competitividad under project MTM2011-22711

Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth

In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [12]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the $L^1-$norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions

State-constrained control-affine parabolic problems II: Second order sufficient optimality conditions

In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order sufficient conditions relying on the Goh transform

Second-order sufficient conditions in the sparse optimal control of a phase field tumor growth model with logarithmic potential

his paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn--Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term in order to enhance the occurrence of sparsity effects in the optimal controls, i.e., of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions have been established for the case without sparsity, the case with sparsity has not been treated before. The results obtained in this paper also improve the known results on this phase field model for the case without sparsity