Exact Controllability for Stochastic First-Order Multi-Dimensional Hyperbolic Systems
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911399226114048 |
|---|---|
| author | Li, Zengyu Lü, Qi Wang, Yu Yang, Haitian |
| author_facet | Li, Zengyu Lü, Qi Wang, Yu Yang, Haitian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18270 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact Controllability for Stochastic First-Order Multi-Dimensional Hyperbolic Systems Li, Zengyu Lü, Qi Wang, Yu Yang, Haitian Optimization and Control 93B05, 93B07 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. |
| title | Exact Controllability for Stochastic First-Order Multi-Dimensional Hyperbolic Systems |
| topic | Optimization and Control 93B05, 93B07 |
| url | https://arxiv.org/abs/2601.18270 |