Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks

Abstract

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.

Stability of the heat and of the wave equations with boundary time-varying delays

“…Our main novel contribution is an extension of previous results from [10,12] to time-varying delays. This extension is not straightforward due to the loss of translation-invariance.…”

“…Note further that to the best of our knowledge the heat equation with boundary delay has not been treated in the literature. Contrary to [10,12], the existence results do not follow from standard semi-group theory because the spatial operator depends on time due to the time-varying delay. Therefore we use the variable norm technique of Kato [7,8].…”

“…Thus, stability conditions and exponential bounds were derived for some scalar heat and wave equations with constant delays and with Dirichlet boundary conditions without delay in [16,17]. Stability and instability conditions for the wave equations with constant delay can be found in [10,12].…”

“…This extension is not straightforward due to the loss of translation-invariance. In the constant delay case the exponential stability was proved in [10,12] by using the observability inequality which can not be applicable in the time-varying case (since the system is not invariant by translation). Hence we introduce new Lyapunov functionals with exponential terms and an additional term for the wave equation, which take into account the dependence of the delay with respect to time.…”

Boundary controllability of a chain of serially connected Euler-Bernoulli beams with interior masses

“…Let us finally quote the paper by Nicaise and Valein ( [38]) on stabilization of the one-dimensional wave equation with a delay term in the feedbacks. They use the same method as we did in a previous paper [34] (technique developed by von Below in [7]) to get the characteristic equation associated to the eigenvalues and apply this spectral analysis to stabilization.…”