Finite Time Stabilization of Nonautonomous First-Order Hyperbolic Systems

The paper concerns the phenomenon of finite time stabilization in initial boundary value problems for nonautonomous decoupled linear first-order one-dimensional hyperbolic systems. We establish sufficient and necessary conditions ensuring that solutions stabilize to zero in a finite time for any initial $L^2$-data. We give a combinatorial criterion stating that the stabilization occurs if and only if the matrix of reflection boundary coefficients corresponds to a directed acyclic graph. An equivalent algebraic criterion is that the adjacency matrix of this graph is nilpotent. In the case of autonomous hyperbolic systems we also provide a spectral stabilization criterion. Moreover, we analyse robustness properties of all these criteria.