Numerical Analysis of Stabilization for Random Hyperbolic Systems of Conservation Laws

TL;DR

This paper develops a stochastic Lyapunov-based stabilization method for random hyperbolic conservation laws, providing explicit decay rates and validating them through numerical examples involving linear and nonlinear systems.

Contribution

It extends deterministic stabilization techniques to stochastic systems using a novel stochastic Lyapunov function, deriving explicit decay rates dependent on system parameters.

Findings

Explicit decay rates for linear systems with random perturbations

Validation through numerical examples including wave equations and shallow-water flows

Decay rates provided for nonlinear and source-term systems

Abstract

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.