Mathematical Control and Related Fields (MCRF)

Finite-time stabilization of a network of strings

Pages: 721 - 742, Volume 5, Issue 4, December 2015      doi:10.3934/mcrf.2015.5.721

       Abstract        References        Full Text (580.7K)       Related Articles       

Fatiha Alabau-Boussouira - Institut Elie Cartan de Lorraine, UMR-CNRS 7502, Université de Lorraine, Ile du Saulcy, 57045 Metz Cedex 1, France (email)
Vincent Perrollaz - Laboratoire de Mathématiques et Physique Théorique, Université de Tours, UFR Sciences et Techniques, Parc de Grandmont, 37200 Tours, France (email)
Lionel Rosier - Centre Automatique et Systèmes, MINES ParisTech, PSL Research University, 60 Boulevard Saint-Michel, 75272 Paris Cedex 06, France (email)

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. Our results lead to the finite-time stabilization even though the systems may not be dissipative.

Keywords:  Finite-time stabilization, network, wave equation, transparent boundary condition, Kirchhoff law, Riemann invariant.
Mathematics Subject Classification:  Primary: 93D15; Secondary: 34B45, 35L05.

Received: October 2014;      Revised: January 2015;      Available Online: October 2015.