Decay Rates for a Beam with Pointwise Force and Moment Feedback

We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0,π) with certain boundary conditions. The energy decay rate in both cases is investigated. In the case of the Rayleigh beam, we show that the decay rate is exponential if and only if ξ/π is a rational number with coprime factorization ξ/π=p/q, where q is odd. Moreover, for any other location of the actuator we give explicit polynomial decay estimates valid for regular initial data. In the case of the Euler–Bernoulli beam, even for a nonhomogeneous material, exponential decay of the energy is proved, independently of the position of the actuator.

Authors and Affiliations

1. Institut Elie Cartan, Département de Mathématiques, Université de Nancy I, F-54506 Vandoeuvre lès Nancy Cedex, France. {ammari,tucsnak}@iecn.u-nancy.fr., , , , , , FR

   Kais Ammari & Marius Tucsnak

2. Department of Mathematics and Statistics, University of Minnesota, Duluth, Minnesota 55812-2496, U.S.A. zliu@d.umn.edu., , , , , , US

   Zhuangyi Liu

Date received: October 30, 2000. Date revised: December 20, 2001.

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