The Dichotomy Property in Stabilizability of 2×22\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

Null controllability and finite-time stabilization in minimal time of one-dimensional first-order 2 × 2 linear hyperbolic systems

The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order 2 ×2 linear hyperbolic systems. The main technical point is to show that we cannot obtain a better time. The proof combines the backstepping method with the Titchmarsh convolution theorem

Boundary Exponential Stabilization of 1-Dimensional Inhomogeneous Quasi-Linear Hyperbolic Systems

International audienceThis paper deals with the problem of boundary stabilization of first-order $n\times n$ inhomogeneous quasi-linear hyperbolic systems. A backstepping method is developed. The main result supplements the previous works on how to design multiboundary feedback controllers to achieve exponential stability with arbitrary decay rate of the original nonlinear system in the spatial $H^2$ sense