The Dichotomy Property in Stabilizability of $2\times2$ Linear Hyperbolic Systems

TL;DR

This paper investigates how the length of a 2x2 hyperbolic system affects its stabilizability, revealing a dichotomy where systems are either stabilizable for all lengths or only below a critical length, with control strategies for the latter.

Contribution

It establishes a dichotomy property in stabilizability related to system length and provides analytical criteria and control methods for systems exceeding the critical length.

Findings

System is stabilizable for all L>0 or not at all, depending on L.

Existence of a critical length L_c dividing stabilizability.

Finite-time stabilization achieved via backstepping control for L ≥ L_c.

Abstract

This paper is devoted to discuss the stabilizability of a class of $ 2 \times2 $ non-homogeneous hyperbolic systems. Motivated by the example in \cite[Page 197]{CB2016}, we analyze the influence of the interval length $L$ on stabilizability of the system. By spectral analysis, we prove that either the system is stabilizable for all $L>0$ or it possesses the dichotomy property: there exists a critical length $L_c>0$ such that the system is stabilizable for $L\in (0,L_c)$ but unstabilizable for $L\in [L_c,+\infty)$. In addition, for $L\in [L_c,+\infty)$, we obtain that the system can reach equilibrium state in finite time by backstepping control combined with observer. Finally, we also provide some numerical simulations to confirm our developed analytical criteria.