Riesz basis generation, eigenvalues distribution, and exponential stability for a euler-bernoulli beam with joint feedback control

Abstract

Using an abstract result on Riesz basis generation for discrete operators in general Hilbert spaces, we show, in this article, that the generalized eigenfunctions of an Euler-Bernoulli beam equation with joint linear feedback control form a Riesz basis for the state space. The spectrum-determined growth condition is hence obtained. Meanwhile, the exponential stability as well as the asymptotic expansion of eigenvalues are also readily obtained by a straightforward computation.

On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback

“…Guo and Chan [15] established the Riesz basis property for Euler-Bernoulli beams with various boundary conditions. Xu and Yung [30] considered a Timoshenko beam with pointwise feedback control.…”

“…Pointwise stabilization of flexible structures has been studied extensively in the context of infinite-dimensional systems control over the past two decades due to wide applications in space technology and robotics [1,2,[5][6][7][8][9][10]14,15,[28][29][30]. Two fundamental issues, namely exponential stability and Riesz basis property, are investigated in these studies.…”

Properties of a class of C0 semigroups on Banach spaces and their applications

“…In [8,9,[11][12][13][14], the partial sum n i=1 E(λ i ; A) is shown to be uniformly bounded and unconditionally convergent in the strong topology. However, we observe that the case that sup n E(λ n ; A) = ∞ is not included there and we shall address this in this paper.…”

“…In fact, many researchers have worked on this problem, for instance see Lang and Locker [8,9], Verduyn Lunel [10], Guo [11], Rao [12], Shubov [13], Xu [14], etc. In [8,9,[11][12][13][14], the partial sum n i=1 E(λ i ; A) is shown to be uniformly bounded and unconditionally convergent in the strong topology.…”

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

“…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]. In this paper, we study the following general Petrovsky type system [15] in one space variable in normal form:…”