Relaxed optimal control for the stochastic Landau-Lifshitz-Bloch equation with jump noise

Abstract

We study the stochastic Landau-Lifshitz-Bloch equation perturbed by pure jump noise. In order to understand and also control the fluctuations and jumps observed in the hysteresis loop, we add noise and an external control to the effective field. We consider the control operator as one that can depend, even nonlinearly on both the control and the associated solution process. We reformulate the problem as a relaxed control problem by using random Young measures and show that the relaxed problem admits an optimal control. The proof employs the Aldous condition for establishing tightness of laws and a modified version of the Skorohod Theorem for obtaining subconvergence of laws.