Numerical Feedback Stabilization with Applications to Networks

Simone Göttlich,Peter Schillen

doi:10.1155/2017/6896153

Simone Göttlich, Peter Schillen

Open Access

https://doi.org/10.1155/2017/6896153

Copy DOI

Abstract

The focus is on the numerical consideration of feedback boundary control problems for linear systems of conservation laws including source terms. We explain under which conditions the numerical discretization can be used to design feedback boundary values for network applications such as electric transmission lines or traffic flow systems. Several numerical examples illustrate the properties of the results for different types of networks.

Highlights

During the last few years, a huge amount of literature has emerged that deals with theoretical stability results for boundary control of conservation laws; see, for example, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]
For the analytical case it can be proven that feedback boundary values, designed under certain conditions, yield an exponential decay of a continuous Lyapunov function [3, 9, 31] in the context of networks
Linear systems of balance laws can be expressed in a characteristic form in a straightforward way while nonlinear systems can be transformed into a characteristic form by linearization with respect to a steady-state solution; see [3, 26] for examples and more details

Summary

Introduction

During the last few years, a huge amount of literature has emerged that deals with theoretical stability results for boundary control of conservation laws; see, for example, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. For the analytical case it can be proven that feedback boundary values, designed under certain conditions, yield an exponential decay of a continuous Lyapunov function [3, 9, 31] in the context of networks. Inspired by [24, 25], we restrict ourselves to the numerical approximation of such systems and study the asymptotic behavior of that solution or to be more precise the conditions under which an exponential decay of the discrete solution to hyperbolic balance law can be attained. One can measure the outflow, modify it, and let it flow back again as inflow. This kind of control is called feedback boundary control

Objectives

Results

Conclusion