In this paper we consider the wave equation on 1-d networks with a delay term in the boundary and/or transmission conditions. We first show the well posedness of the problem and the decay of an appropriate energy. We give a necessary and sufficient condition that guarantees the decay to zero of the energy. We further give sufficient conditions that lead to exponential or polynomial stability of the solution. Some examples are also given.

Exponential stability analysis via Lyapunov method is extended to the one-dimensional heat and wave equations with time-varying delay in the boundary conditions. The delay function is admitted to be timevarying with an a priori given upper bound on its derivative, which is less than 1. Sufficient and explicit conditions are derived that guarantee the exponential stability. Moreover the decay rate can be explicitly computed if the data are given.

We consider the wave equation with a time-varying delay term in the boundary condition in a bounded and smooth domain Ω ⊂ IR 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.

In this paper we study the stabilization of the wave equation on general 1-d networks. For that, we transfer known observability results in the context of control of conservative systems (see [14]) into a weighted observability estimate for the dissipative one. Then we use an interpolation inequality similar to the one proved in [7] to obtain the explicit decay estimate of the energy for smooth initial data. The obtained decay rate depends on the geometric and topological properties of the network. We give also some examples of particular networks in which our results apply yielding different decay rates.

Abstract.We consider abstract second order evolution equations with unbounded feedback with delay. Existence results are obtained under some realistic assumptions. Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability. Some new examples that enter into our abstract framework are presented.Mathematics Subject Classification. 93D15, 93C20.