Asymptotic behavior of a delayed wave equation without displacement term

International audienceThis paper is dedicated to the investigation of the asymptotic behavior of a delayed wave equation without the presence of any displacement term. First, it is shown that the problem is well-posed in the sense of semigroups theory. Thereafter, LaSalle's invariance principle is invoked in order to establish the asymptotic convergence for the solutions of the system to a stationary position which depends on the initial data. More importantly, without any geometric condition such as BLR condition (Bardos et al. in SIAM J Control Optim 30 1024-1064, 1992; Lebeau and Robbiano in Duke Math J 86 465-491, 1997) in the control zone, the logarithmic convergence is proved by using an interpolation inequality combined with a resolvent method

Theme 4 | Simulation et optimisation de systemes complexes

This paper deals with nonlinear feedback stabilization problem of a #exible beam clamped at a rigid body and free at the other end. We assume that there is no damping. The feedbacklaw proposed here consists of a nonlinear control torque applied to the rigid body and either a nonlinear boundary control moment or a nonlinear boundary control force or both of them applied to the free end of the beam. This nonlinear feedback, which insures the exponential decay of the beam vibrations, extends thelinear case studied by Laousy et al. to a more general class of controls. This new class of controls is in particular of the interest to be robust

Asymptotic behavior of a 2D overhead crane with input delays in the boundary control

International audienceThe paper investigates the asymptotic behavior of a 2D overhead crane with input delays in the boundary control. A linear boundary control is proposed. The main feature of such a control lies in the fact that it solely depends on the velocity but under the presence of time-delays. We end-up with a closed-loop system where no displacement term is involved. It is shown that the problem is well-posed in the sense of semigroups theory. LaSalle's invariance principle is invoked in order to establish the asymptotic convergence for the solutions of the system to a stationary position which depends on the initial data. Using a resolvent method, it is proved that the convergence is indeed of polynomial type as long as the delay term satisfies a smallness condition. Lastly, non-convergence results are put forward in the case when such a condition on the delay term is not fulfilled

Stabilisation d'une equation de vibrations

Theme 4 - Simulation et optimisation de systemes complexes - Projet CongeAvailable at INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1997 n.3085 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueSIGLEFRFranc