Second-order optimality conditions for the sparse optimal control of nonviscous Cahn–Hilliard systems

In this paper, we study the optimal control of an initial-boundary value problem for the classical nonviscous Cahn–Hilliard system with zero Neumann boundary conditions. Phase field systems of this type govern the evolution of diffusive phase transition processes with conserved order parameter. For such systems, optimal control problems have been studied in the past. We focus here on the situation when the cost functional of the optimal control problem contains a sparsity-enhancing nondifferentiable term like the L 1 -norm. For such cases, we establish first-order necessary and secondorder sufficient optimality conditions for locally optimal controls, where in the approach to second-order sufficient conditions we employ a technique introduced by Casas, Ryll and Tröltzsch in the paper [SIAM J. Control Optim. 53 (2015) 2168–2202]. The main novelty of this paper is that this method, which has recently been successfully applied to systems of viscous Cahn–Hilliard type, can be adapted also to the classical nonviscous case. Since in the case without viscosity the solutions to the state and adjoint systems turn out to be considerably less regular than in the viscous case, numerous additional technical difficulties have to be overcome, and additional conditions have to be imposed. In particular, we have to restrict ourselves to the case when the nonlinearity driving the phase separation is regular, while in the presence of a viscosity term also nonlinearities of logarithmic type turn could be admitted. In addition, the implicit function theorem, which was employed to establish the needed differentiability properties of the control-to-state operator in the viscous case, does not apply in our situation and has to be substituted by other arguments.