Exact Controllability for Stochastic First-Order Multi-Dimensional Hyperbolic Systems

TL;DR

This paper establishes the exact controllability of multi-dimensional stochastic hyperbolic systems using new Carleman estimates, identifying conditions under which control is feasible and demonstrating the necessity of both internal and boundary controls.

Contribution

It introduces a novel global Carleman estimate for backward stochastic systems, extending controllability results to stochastic hyperbolic systems with explicit geometric conditions.

Findings

Controllability holds if control time exceeds a sharp threshold T0.

Both internal and boundary controls are necessary for controllability.

The results extend deterministic controllability theory to stochastic systems.

Abstract

This paper investigates the exact controllability problem for multi-dimensional stochastic first-order symmetric hyperbolic systems with control inputs acting in two distinct ways: an internal control applied to the diffusion term and a boundary control applied to the drift term. By means of a classical duality argument, the controllability problem is reduced to an observability estimate for the corresponding backward stochastic system. The main technical contribution is the establishment of a new global Carleman estimate for such backward systems, combined with a weighted energy identity. This enables us to prove the desired observability inequality under a geometric structural condition (Condition \ref{cond1}), which ensures that all characteristic rays propagate toward the boundary within a finite time. As a result, we obtain exact controllability provided the control time $T$ exceeds a sharp threshold $T_0$ given explicitly in terms of the system geometry. Furthermore, we complement the positive result with several negative controllability theorems, which demonstrate that both controls are necessary and must act in a distributed manner. Our analysis not only extends controllability theory from deterministic to stochastic multi-dimensional hyperbolic systems but also provides, as a byproduct, new results for deterministic systems under a structural hypothesis. Applications to stochastic traffic flow, epidemiological models, and shallow-water equations are discussed.