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

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.