Spatial decay of perturbations in hyperbolic equations with optimal boundary control

TL;DR

This paper investigates the conditions under which boundary control can ensure exponential decay of localized perturbations in hyperbolic transport equations, providing necessary and sufficient criteria for domain-uniform stabilization.

Contribution

It establishes a precise criterion for boundary control domains that guarantee domain-uniform stabilization of hyperbolic transport equations with boundary and point controls.

Findings

Necessary and sufficient control domain conditions identified

Results apply to Dirichlet and Neumann boundary conditions

Numerical example demonstrates practical applicability

Abstract

Recently, domain-uniform stabilizability and detectability has been the central assumption %in order robustness results on the to ensure robustness in the sense of exponential decay of spatially localized perturbations in optimally controlled evolution equations. In the present paper we analyze a chain of transport equations with boundary and point controls with regard to this property. Both for Dirichlet and Neumann boundary and coupling conditions, we show a necessary and sufficient criterion on control domains which allow for the domain-uniform stabilization of this equation. We illustrate the results by means of a numerical example.