Optimal distributed control of a stochastic Cahn-Hilliard equation

TL;DR

This paper develops a mathematical framework for optimally controlling a stochastic Cahn-Hilliard equation, ensuring existence of solutions and deriving necessary conditions for optimality using advanced probabilistic and analytical methods.

Contribution

It introduces a novel optimal control formulation for the stochastic Cahn-Hilliard equation and establishes existence and first-order optimality conditions.

Findings

Existence of an optimal control is proved.

First-order necessary conditions for optimality are derived.

Analytical techniques include compactness and monotonicity arguments.

Abstract

We study an optimal distributed control problem associated to a stochastic Cahn-Hilliard equation with a classical double-well potential and Wiener multiplicative noise, where the control is represented by a source-term in the definition of the chemical potential. By means of probabilistic and analytical compactness arguments, existence of an optimal control is proved. Then the linearized system and the corresponding backward adjoint system are analysed through monotonicity and compactness arguments, and first-order necessary conditions for optimality are proved.