Exact Controllability for a Refined Stochastic Hyperbolic Equation with Internal Controls

We establish the internal exact controllability of a refined stochastic hyperbolic equation by deriving a suitable observability inequality via Carleman estimates for the associated backward stochastic hyperbolic equation. In contrast to existing results on boundary exact controllability–which require longer waiting times, we demonstrate that the required waiting time for internal exact controllability in stochastic hyperbolic equations coincides exactly with that of their deterministic counterparts.