This paper extends the deterministic Lyapunov-based stabilization framework to random hyperbolic systems of conservation laws, where uncertainties arise in boundary controls and initial data. Building on the finite volume discretization method from [{\sc M. Banda and M. Herty}, Math. Control Relat. Fields., 3 (2013), pp. 121--142], we introduce a stochastic discrete Lyapunov function to prove the exponential decay of numerical solutions for systems with random perturbations. For linear systems, we derive explicit decay rates, which depend on boundary control parameters, grid resolutions, and the statistical properties of the random inputs. Theoretical decay rates are verified through numerical examples, including boundary stabilization of the linear wave equations and linearized shallow-water flows with random perturbations. We also present the decay rates for a nonlinear example and for the linearized Saint-Venant system with source terms.

翻译：本文将确定性Lyapunov镇定框架推广至随机双曲守恒律系统，其中不确定性存在于边界控制与初始数据中。基于[{\sc M. Banda and M. Herty}, Math. Control Relat. Fields., 3 (2013), pp. 121--142]中的有限体积离散化方法，我们引入随机离散Lyapunov函数，以证明随机扰动系统数值解的指数衰减性。对于线性系统，我们推导出显式衰减率，其取决于边界控制参数、网格分辨率以及随机输入的统计特性。理论衰减率通过数值算例得到验证，包括随机扰动下线性波动方程的边界镇定以及线性化浅水流动的边界镇定。我们还给出了非线性算例以及含源项线性化Saint-Venant系统的衰减率。