Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients

Abstract

Abstract. This paper studies the basis property and the stability of a distributed system described by a nonuniform Euler-Bernoulli beam equation under linear boundary feedback control. It is shown that there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state Hilbert space. The asymptotic distribution of eigenvalues, the spectrumdetermined growth condition, and the exponential stability are concluded. The results are applied to a nonuniform beam equation with viscous damping of variable coefficient as a generalization of existing results for the uniform beam.

On Spectrum of a General Petrovsky Type Equation and Riesz Basis of N-Connected Beams with Linear Feedback at Joints

“…T have the asymptotic expansions (4.15) and (4.23), respectively, for sufficiently large positive integers k. The following result developed recently in [7] turns out to be very useful for the verification of Riesz basis generation for beam equations. …”

“…If successful, the growth condition can then be concluded as a consequence of existence of the Riesz basis. The Riesz basis for single beam equations was developed in [6][7][8]. The basis property for 2-connected beams was studied in [9,10].…”

“…) into (6.1) and comparing the order of both sides as in [7], we can obtain the following asymptotic expressions of eigenvalues:…”

Stabilization of Variable Coefficients Euler-Bernoulli Beam with Viscous Damping under a Force Control in Rotation and Velocity Rotation

“…In order to solve (21), spatial transformations as introduced in (Guo, 2002) are performed, which convert the first equation of (21) into a more convenient form. For this purpose, for 0 < x < 1, the system (21) is firstly rewritten as :…”

“…This approach was used by many authors for study the Euler-Bernoulli beams equations with variable coefficients (see e.g. Guo, 2002;Guo & Wang, 2006;Wang, 2004;Wang, Xu & Yung, 2005). In our case, we rely on idea of Wang et al (see e.g.…”