Optimal distributed control of a generalized fractional Cahn-Hilliard system

TL;DR

This paper develops optimal distributed control strategies for a generalized fractional Cahn-Hilliard system, extending previous well-posedness results and addressing the challenges of control-to-state differentiability under boundedness assumptions.

Contribution

It introduces a rigorous control framework for fractional Cahn-Hilliard systems, establishing differentiability and optimality conditions under boundedness assumptions.

Findings

Existence of optimal controls under boundedness conditions

First-order necessary optimality conditions derived

Applicability to systems with polynomial and logarithmic nonlinearities

Abstract

In the recent paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (arXiv:1804.11290) by the same authors, general well-posedness results have been established for a a class of evolutionary systems of two equations having the structure of a viscous Cahn-Hilliard system, in which nonlinearities of double-well type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A,B having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in arXiv:1804.11290 by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Frechet differentiability result for the associated control-to-state mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and first-order necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system.