Optimal control of the viscous Cahn–Hilliard system with hyperbolic relaxation of the chemical potential

Jürgen Sprekels (WIAS Berlin)

Abstract: We consider the optimal sparse control of a system of viscous Cahn–Hilliard type in which the chemical potential contains a hyperbolic relaxation term. Recent results concerning the state system, in particular, the Fréchet differentiability of the control-to-state operator in suitable Banach spaces, are reported. For the optimal control problem, we derive first-order necessary optimality conditions for locally optimal controls and sparsity results. Finally, we study the asymptotic behavior of the optimal control problem as the relaxation parameter approaches zero, showing suitable convergence results for states, optimal controls, and adjoint states.