On the optimal control of viscous Cahn–Hilliard systems with hyperbolic relaxation of the chemical potential

In this paper, we study an optimal control problem for a viscous Cahn–Hilliard system with zero Neumann boundary conditions in which a hyperbolic relaxation term involving the second time derivative of the chemical potential has been added to the first equation of the system. For the initial-boundary value problem of this system, results concerning well-posedness, continuous dependence, and regularity are known. We show Fréchet differentiability of the associated control-to-state operator, study the associated adjoint state system, and derive first-order necessary optimality conditions. Concerning the nonlinearities driving the system, we can include the case of logarithmic potentials. In addition, we perform an asymptotic analysis of the optimal control problem as the relaxation coefficient approaches zero.