Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. (English) Zbl 1180.35095

Nicaise, Serge; Pignotti, Cristina

SIAM J. Control Optim. 45, No. 5, 1561-1585 (2006).

Summary: We consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using some observability inequalities. If one of the above assumptions is not satisfied, some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero.

MSC:

35B35	Stability in context of PDEs
35L05	Wave equation
93D15	Stabilization of systems by feedback
35L20	Initial-boundary value problems for second-order hyperbolic equations
35R10	Partial functional-differential equations

Keywords:

mixed Dirichlet-Neumann boundary condition; observability inequalities