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Finite-time stabilization of a network of strings

Abstract : 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 $\alpha u_t$ with a coefficient $\alpha$ that may depend on the node is considered. We show that for a convenient choice of the sequence of coefficients $\alpha$, 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.
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https://hal.archives-ouvertes.fr/hal-01071150
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Submitted on : Friday, October 3, 2014 - 10:37:28 AM
Last modification on : Thursday, April 9, 2020 - 5:08:12 PM
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Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier. Finite-time stabilization of a network of strings. Mathematical Control and Related Fields, AIMS, 2015, 5 (4), pp.721-742. ⟨10.3934/mcrf.2015.5.721⟩. ⟨hal-01071150⟩

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