Finite Time Stabilization of Nonautonomous First Order Hyperbolic Systems

TL;DR

This paper investigates conditions under which solutions to nonautonomous first-order hyperbolic systems stabilize in finite time, providing combinatorial and algebraic criteria for robust stabilization.

Contribution

It introduces new combinatorial and algebraic criteria for finite time stabilization of nonautonomous hyperbolic systems, including conditions for robustness and spectral criteria in autonomous cases.

Findings

Stabilization occurs if and only if the reflection boundary coefficient matrix is associated with an acyclic graph.

Robust stabilization is characterized by the nilpotency of the adjacency matrix.

Spectral stabilization criterion is provided but is nonrobust to coefficient perturbations.

Abstract

We address nonautonomous initial boundary value problems for decoupled linear first-order one-dimensional hyperbolic systems, investigating the phenomenon of finite time stabilization. We establish sufficient and necessary conditions ensuring that solutions stabilize to zero in a finite time for any initial $L^2$-data. In the nonautonomous case we give a combinatorial criterion stating that the robust stabilization occurs if and only if the matrix of reflection boundary coefficients corresponds to a directed acyclic graph. An equivalent robust algebraic criterion is that the adjacency matrix of this graph is nilpotent. In the autonomous case we also provide a spectral stabilization criterion, which is nonrobust with respect to perturbations of the coefficients of the hyperbolic system.