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

TL;DR

The work extends exact controllability theory to stochastic multi-dimensional first-order symmetric hyperbolic systems by turning the problem into an observability estimate for a backward stochastic adjoint system via a new global Carleman estimate and a weighted energy identity. Under a geometric Condition 1 ensuring all rays reach the boundary in finite time, the authors derive a sharp time threshold $T_0$ and prove exact controllability for $T > T_0$ using duality and transposition solutions. They also establish negative results showing that both boundary and distributed diffusion controls are necessary, and they discuss extensions to deterministic systems and practical applications in stochastic traffic flow, epidemiology, and shallow-water dynamics.

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.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• proof
• proof : Proof of Proposition \ref{['hidden']}
• proof : Proof of Proposition \ref{['wellposednessfq']}
• proof
• proof : Proof of \ref{['thmObservability']}
• proof : Proof of Theorem \ref{['thmControllability']}