Optimal control of the stochastic Landau-Lifshitz-Bloch equation

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.