Exact Controllability for a Refined Stochastic Hyperbolic Equation with Internal Controls

Abstract

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.