Optimal control of the stochastic Landau-Lifshitz-Bloch equation

TL;DR

This paper develops an optimal control framework for the stochastic Landau-Lifshitz-Bloch equation, establishing existence of solutions and optimal controls using Young measures, with results on pathwise uniqueness in lower dimensions.

Contribution

It introduces a relaxed control approach for the stochastic Landau-Lifshitz-Bloch equation, proving existence of weak solutions and optimal controls with Young measures, and demonstrates pathwise uniqueness in certain dimensions.

Findings

Existence of weak martingale solutions for the controlled equation.

Existence of weak relaxed optimal controls for general cost functionals.

Pathwise uniqueness established in dimensions 1 and 2.

Abstract

We consider the stochastic Landau-Lifshitz-Bloch equation in dimensions 1,2,3, perturbed by a real-valued Wiener process. We consider a Suslin space-valued control process with a general control operator, which can depend on both the control and the corresponding solution. We reduce the equation to a more general (relaxed) form, where the concept of Young measures is used. We then show the existence of a weak martingale solution to the controlled equation (relaxed). In the second part of the work, we show that for a general lower semicontinuous cost functional, the problem admits a weak relaxed optimal control. This is done using the theory of Young measures. Moreover, pathwise uniqueness is shown (for dimensions 1,2), which implies the existence of a strong solution.