The dichotomy property in stabilizability of 2 × 2 linear hyperbolic systems

¹ School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China
² School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, PR China
³ Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, PR China

^* Corresponding author: wzq@fudan.edu.cn

Received: 7 May 2023
Accepted: 25 January 2024

Abstract

This paper is devoted to discuss the stabilizability of a class of 2 × 2 non-homogeneous hyperbolic systems. Motivated by the example in the Section 5.6 of Bastin and Coron’s book in 2016, 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 ∈ (0, L_c) but unstabilizable for L ∈ [L_c,+∞). In addition, for L ∈ [L_c, +∞), 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.