Exact controllability of the stochastic Maxwell equation: theory and numerical simulation

This article investigates the exact controllability of three-dimensional stochastic Maxwell equations, a coupled system comprising two stochastic partial differential equations. The research establishes the observability inequality for the backward stochastic Maxwell equations using the multiplier method, and subsequently, proves the exact controllability of the forward equations. The control acting on the diffusion term is found to be indispensable, since exact controllability is destroyed when this control is removed; it is further proved that the controllability result obtained in this paper is achieved with a minimal number of controls. Finally, a numerical algorithm combining a central difference for spatial discretization, a midpoint scheme for temporal discretization and Lagrange multiplier method is proposed, yielding numerical results that offer the control value and lead to deeper insights into the underlying theoretical framework.