Numerical analysis of stabilisation for random hyperbolic systems of conservation laws

Abstract

This paper extends the deterministic Lyapunov-based stabilisation framework to random hyperbolic systems of conservation laws, where uncertainties arise in boundary controls and initial data. Building on the finite-volume discretisation method from [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 stabilisation of the linear wave equations and linearised shallow-water flows with random perturbations. We also present the decay rates for a nonlinear example and for the linearised Saint-Venant system with source terms.