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Finite-time stabilization of a network of strings

Authors: Fatiha Alabau-Boussouira, Vincent Perrollaz, and Lionel Rosier, Mathematical Control and Related Fields, Vol. 5 No 4, pp. 721-742, December 2015 DOI: 10.3934/mcrf.2015.5.721
We investigate the finite-time stabilization of a tree-shaped network of strings. Transparent boundary conditions are applied at all the external nodes. At any internal node, in addition to the usual continuity conditions, a modified Kirchhoff law incorporating a damping term αut with a coefficient α that may depend on the node is considered. We show that for a convenient choice of the sequence of coefficients α, any solution of the wave equation on the network becomes constant after a finite time. The condition on the coefficients proves to be sharp at least for a star-shaped tree. Similar results are derived when we replace the transparent boundary condition by the Dirichlet (resp. Neumann) boundary condition at one external node. Our results lead to the finite-time stabilization even though the systems may not be dissipative.
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author = {Vincent Perrollaz Fatiha Alabau-Boussouira, and Lionel Rosier},
title = {Finite-time stabilization of a network of strings},
journal = {Mathematical Control and Related Fields},
volume = {5},
number = {4},
pages = {721-742},
year = {2015},