Input-to-State Stability of a Scalar Conservation Law with Nonlocal Velocity

Gediyon Weldegiyorgis,Michael Herty,Simone Göttlich

doi:10.3390/axioms10010012

Gediyon Weldegiyorgis, Michael Herty + Show 1 more

Open Access

https://doi.org/10.3390/axioms10010012

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Journal: Axioms	Publication Date: Jan 21, 2021
Citations: 1	License type: CC BY 4.0

Affiliation: University of Pretoria, University of Mannheim

Abstract

In this paper, we study input-to-state stability (ISS) of an equilibrium for a scalar conservation law with nonlocal velocity and measurement error arising in a highly re-entrant manufacturing system. By using a suitable Lyapunov function, we prove sufficient and necessary conditions on ISS. We propose a numerical discretization of the scalar conservation law with nonlocal velocity and measurement error. A suitable discrete Lyapunov function is analyzed to provide ISS of a discrete equilibrium for the proposed numerical approximation. Finally, we show computational results to validate the theoretical findings.

Highlights

The nature of modern high-volume production is characterized by a large number of items passing through many production steps
The paper is organized as follows: In Section 2, we present stabilization results of input-to-state stability (ISS) for a scalar conservation law with nonlocal velocity and measurement error
We study ISS of a closed-loop system of scalar conservation laws with nonlocal velocity and measurement error of the form:

Summary

Introduction

The nature of modern high-volume production is characterized by a large number of items passing through many production steps. The paper is organized as follows: In Section 2, we present stabilization results of ISS for a scalar conservation law with nonlocal velocity and measurement error. An equilibrium ρ∗ ≥ 0 of the closedloop system in Equation (8) is exponential ISS in L2 -norm with respect to any disturbance function d(·) ∈ L∞ (0, ∞) such that kdk L∞ (0,∞) ≤ D if there exist positive constants γ1 , γ2 , γ3 independent of d such that, for every initial condition ρ0 ( x ) ∈ L2 (0, 1), the L2 -solution to the closed-loop system in Equation (8) satisfies kρ(t, ·) − ρ∗ k L2 ≤ γ2 e−γ1 t kρ0 − ρ∗ k L2 + γ3 kd(s)k L∞ (0,t) , t ∈ [0, +∞). Therein, a detailed discussion of the case d ≡ 0 has been presented and we refer in particular to ([10], Theorem 3.1)

Numerical Study of Asymptotic Stability of a Scalar Conservation Law with

Numerical Simulations

Example 1

Example 2

Conclusions and Outlook