Exact Controllability for a Refined Stochastic Hyperbolic Equation with Internal Controls

TL;DR

The paper proves internal exact controllability for a refined stochastic hyperbolic equation with internal controls by deriving an $L^2$ Carleman observability inequality for the associated backward stochastic hyperbolic equation. The authors construct an $L^2$ Carleman framework via an auxiliary time-discretized optimal control problem, obtain deterministic and stochastic Carleman estimates for the hyperbolic operator ${\cal A}$, and develop three energy estimates for the backward problem. A duality argument connects these observability results to exact controllability of the forward system, showing that exact control is possible for any $T>T_0$ and that the waiting time matches the deterministic case. The work extends controllability theory for stochastic hyperbolic equations to internal controls and provides a rigorous pathway combining Carleman estimates, energy methods, and duality to achieve interior controllability without additional waiting-time penalties.

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.

Theorems & Definitions (16)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.1