We investigate the finite-time boundary stabilization of a one-dimensional first-order quasilinear system of diagonal form on [0,1].The dynamics of the boundary controls are governed by a finite-time stable ODE. We derive a (local) finite-time stability for the full system. The above feedback strategy is applied to the regulation of water flows in a network of canals. When zero-order terms are incorporated in the system, we show that the system remains exponentially stable with a decay rate related to the magnitude of the 

perturbative terms. Finally, we consider the finite-time stabilization of the wave equation on a tree. Transparent boundary conditions are applied at all the external nodes, and at the internal nodes, in addition to the usual continuity conditions, the Kirchhoff law is modified by incorporating a damping term.  We show that a finite-time stability still occurs for a certain choice of the damping coefficients at the internal nodes. This is a joint work with Vincent Perrollaz and Fatiha Alabau-Boussouira.

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