Exponential stability of the wave equation with boundary time-varying delay

Serge Nicaise (1) , Cristina Pignotti (2) , Julie Valein (3, 4)

1 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015
2 Dipartimento di Matematica Pura e Applicata [Padova]
3 IECN - Institut Élie Cartan de Nancy
4 CORIDA - Robust control of infinite dimensional systems and applications

We consider the wave equation with a time-varying delay term in the boundary condition in a bounded and smooth domain [\Omega\subset\RR^n.] Under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and suitable Lyapunov functionals. Such analysis is also extended to a nonlinear version of the model.

Mots clés: delay feedbacks; Wave equation; stabilization

Domaines: Equations aux dérivées partielles [math.AP]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - Series S, 2011, 4 (3), pp.693-722. ⟨10.3934/dcdss.2011.4.693⟩. ⟨hal-00599629⟩