Finite-time stabilization of a network of strings

Fatiha Alabau-Boussouira (1, 2) , Vincent Perrollaz (3) , Lionel Rosier (4)

1 CORIDA - Robust control of infinite dimensional systems and applications
2 IECL - Institut Élie Cartan de Lorraine
3 LMPT - Laboratoire de Mathématiques et Physique Théorique
4 CAS - Centre Automatique et Systèmes

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.

Mots clés: transparent bound- ary condition; wave equation; Finite-time stabilization; network; Kirchhoff law; Riemann invariant

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques générales [math.GM]; Automatique / Robotique

Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier. Finite-time stabilization of a network of strings. Mathematical Control and Related Fields, 2015, 5 (4), pp.721-742. ⟨10.3934/mcrf.2015.5.721⟩. ⟨hal-01071150⟩