Analysis and Optimal Boundary Control of a Nonstandard System of Phase Field Equations

Abstract

Abstract. We investigate a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced in [16], describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in [5], [6] for the case of homogeneous Neumann boundary conditions. In this paper, we investigate the case that the boundary condition for one of the unknowns of the system is of third kind and nonhomogeneous. For the resulting system, we show well-posedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the first-order necessary optimality conditions are derived. Owing to the strong nonlinear couplings in the PDE system, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional will be of standard type.

Optimal distributed control of a diffuse interface model of tumor growth

“…While there exist many contributions concerning the well-posedness of various types of Cahn-Hilliard systems, only a few deal with their optimal control. In this connection, we mention the papers [12,13,28,40], which deal with zero Neumann boundary conditions like (1.6), while in the recent papers [7,8,13,14] dynamic boundary conditions have been studied. A num-ber of papers also investigates optimal control problems for convective Cahn-Hilliard systems (cf.…”

Asymptotic limits and optimal control for the Cahn–Hilliard system with convection and dynamic boundary conditions

“…As far as optimal control problems are concerned, there exist numerous recent contributions to general viscous Cahn-Hilliard systems. In this connection, we refer to the papers [9,12,16,17,11] for the case of standard boundary conditions and to [6,7,13,14,15,22,27] for the case of dynamic boundary conditions. For the 'pure' case τ Ω = τ Γ = 0, there are the works of [52,37,58,59].…”

Boundary control problem and optimality conditions for the Cahn–Hilliard equation with dynamic boundary conditions

“…In this regard, let us address to [19], where both the viscous and the non-viscous Cahn-Hilliard equations, combined with these kinds of boundary conditions, have been investigated by assuming the boundary potential to be dominant on the bulk one. Furthermore, we have to mention [4,9,13,16,23,25,33,[36][37][38]42], where other problems related to the Cahn-Hilliard equation combined with dynamic boundary conditions have been analyzed, and [3,7,8,11,20,29,35] for the coupling of dynamic boundary conditions with different phase field models such as the Allen-Cahn or the Penrose-Fife model. So, according to [19] we supply the above system (1.1)-(1.2) with ∂ n w = 0 on Σ := Γ × (0, T ), (1.5) ∂ n y + ∂ t y Γ − ∆ Γ y Γ + f ′ Γ (y Γ ) = u Γ on Σ, (1.6) where Γ is the boundary of Ω, y Γ denotes the trace of y, ∆ Γ stands for the Laplace-Beltrami operator on the boundary, and ∂ n represents the outward normal derivative.…”