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Stabilization of second order evolution equations with unbounded feedback with delay – p.

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… (More)

We consider abstract second order evolution equations with unbounded feedback with time-varying delay. Existence results are obtained under some realistic assumptions. We prove the exponential decay under some conditions by introducing an abstract Lyapunov functional. Our abstract framework is applied to the wave, to the beam and to the plate equations with… (More)

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 time-varying with an a priori given upper bound on its derivative, which is less than 1. Sufficient and explicit conditions are derived that guarantee the… (More)

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… (More)

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 Lyapounov functionals. Such analysis is also extended to a nonlinear version of… (More)

- Ka¨ıs Ammari, Denis Mercier, Virginie Régnier, Julie Valein
- 2010

We consider N Euler-Bernoulli beams and N strings alternatively connected to one another and forming a particular network which is a chain beginning with a string. We study two stabilization problems on the same network and the spectrum of the corresponding conservative system: the characteristic equation as well as its asymptotic behavior are given. We… (More)

We consider an infinite dimensional system modeling a boost converter connected to a load via a transmission line. The governing equations form a system coupling the telegraph partial differential equation with the ordinary differential equations model-ing the converter. The coupling is given by the boundary conditions and the nonlinear controller we… (More)

— We consider a class of infinite dimensional systems involving a control function u taking values in [0, 1]. This class contains, in particular, the average models of some infinite dimensional switched systems. We prove that the system is well-posed and obtain some regularity properties. Moreover, when u is given in an appropriate feedback form and the… (More)