[Paper Review] An analysis of the input-to-state-stabilisation of linear hyperbolic systems of balance laws with boundary disturbances

Gediyon Weldegiyorgis, Mapundi K. Banda|arXiv (Cornell University)|Jun 3, 2020

Stability and Controllability of Differential Equations21 references5 citations

TL;DR

This paper proposes a discrete Input-to-State Stability (ISS) analysis for linear hyperbolic balance laws with boundary disturbances using an $L^2$-norm ISS-Lyapunov function. By discretizing the continuous ISS-Lyapunov function via a finite volume upwind scheme and time splitting, the authors establish conditions under which the discrete system remains ISS, demonstrating asymptotic decay of the Lyapunov function in numerical experiments on Saint-Venant and isothermal Euler equations.

ABSTRACT

In this paper, a linear hyperbolic system of balance laws with boundary disturbances in one dimension is considered. An explicit candidate Input-to-State Stability (ISS)-Lyapunov function in $ L^2- $norm is considered and discretised to investigate conditions for ISS of the discrete system as well. Finally, experimental results on test examples including the Saint-Venant equations with boundary disturbances are presented. The numerical results demonstrate the expected theoretical decay of the Lyapunov function.

Motivation & Objective

• To analyze input-to-state stability (ISS) of linear hyperbolic systems of balance laws under boundary disturbances.
• To develop a discrete ISS-Lyapunov function for numerical verification of stability in semi-discretized systems.
• To validate the theoretical ISS conditions through numerical simulations on physical models such as the Saint-Venant and isothermal Euler equations.
• To investigate the impact of numerical discretization, including numerical viscosity, on ISS properties.

Proposed method

• A continuous $L^2$-norm ISS-Lyapunov function is constructed for the linear hyperbolic system of balance laws with variable coefficients.
• The Lyapunov function is discretized using a finite volume method with an upwind scheme for spatial approximation.
• Time integration is performed using a time splitting method to decouple the hyperbolic dynamics from the boundary feedback.
• Stability conditions are derived by ensuring the discrete Lyapunov function decays over time under the given boundary feedback and disturbance structure.
• Theoretical conditions for discrete ISS are checked numerically by simulating the decay of the discrete Lyapunov function under various boundary disturbance profiles.
• The method is applied to two test cases: the Saint-Venant equations and the isothermal Euler equations with specific initial and boundary conditions.

Experimental results

Research questions

• RQ1 Under what conditions does the discrete $L^2$-norm Lyapunov function decay for a linear hyperbolic system with boundary disturbances?
• RQ2 Can the discretization of an ISS-Lyapunov function preserve the input-to-state stability property of the continuous system?
• RQ3 How do numerical artifacts such as numerical viscosity affect the decay rate of the discrete Lyapunov function?
• RQ4 Does the ISS property hold for the isothermal Euler equations when the matrix $M_j$ fails to be positive semi-definite?
• RQ5 To what extent do numerical simulations match the theoretical decay behavior of the Lyapunov function under boundary disturbances?

Key findings

• The numerical simulations show that the discrete $L^2$-norm Lyapunov function decays asymptotically over time, confirming discrete ISS for the Saint-Venant equations under boundary disturbances.
• For the Saint-Venant equations, the Lyapunov function decay is observed across multiple values of $\mu > 0$, with curves converging to zero, indicating robust stability.
• In the isothermal Euler equations case, the condition for ISS fails due to $\gamma_{22}(x) < 0$, preventing the matrix $M_j$ from being positive semi-definite, thus the ISS result does not hold.
• The decay of the Lyapunov function is explicitly observed in simulations with $T=10$, $J=1600$, and CFL=0.75, validating the theoretical decay rate.
• The numerical results confirm that the theoretical ISS conditions derived analytically can be computationally verified using the discretized Lyapunov function framework.
• The study highlights the need to carefully analyze numerical artifacts like numerical viscosity, which may influence the convergence rate of discrete stability results.