Riesz basis property of serially connected Timoshenko beams

Xu, Genqi; Han, Zhong‐Jie; Yung, S. P.

doi:10.1080/00207170601100904

Cited by 50 publications

(54 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We need the following result, which comes from [18] and is an extension of the result in [29]. (1) The spectrum of A has a decomposition…”

Section: So It Holds Thatmentioning

confidence: 98%

“…We still continue our discussion under this condition. According to a result in [18], we only need to show that inf ξ ∈R detĤ (iξ) = 0. For this end, we shall choose a sequence ξ s → ∞, s → ∞ as follows such that there is a subsequence of it satisfying lim j →∞ detĤ (iξ n j ) = 0.…”

Section: Exponential Stability Analysis Of the Systemmentioning

confidence: 99%

“…Besides the methods mentioned above, some researchers used the frequency analysis method to study the control problems, for example, [16][17][18] for serially connected strings and Timoshenko beams, [15] for symmetric tree-shaped Euler-Bernoulli beam networks. We observe that for a controlled network, the operator determined by the closed loop system is non-normal, whose spectral analysis and basis generation of the root vectors are tough problems in operator theory.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Riesz Basis Property and Stability of Planar Networks of Controlled Strings

Han

Wang

2009

Acta Appl Math

View full text Add to dashboard Cite

In this paper we study the basis property of root vectors of a star-shaped networks of strings whose exterior ends are clamped and common node has a damping. The main operator determined by the networks is non-normal. By the asymptotical technique, we show that its spectra distribute in a strip parallel to the imaginary axis, and there is a sequence of root vectors (generalized eigenvectors and eigenvectors) that forms a Riesz basis with parentheses for the Hilbert state space. As a application of this result, we discuss the stability of the system.

show abstract

“…We need the following result, which comes from [18] and is an extension of the result in [29]. (1) The spectrum of A has a decomposition…”

Section: So It Holds Thatmentioning

confidence: 98%

Section: Exponential Stability Analysis Of the Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Riesz Basis Property and Stability of Planar Networks of Controlled Strings

Han

Wang

2009

Acta Appl Math

View full text Add to dashboard Cite

show abstract

“…The following proposition gives a sufficient condition for Riesz basis with parentheses, which is from [20] and is an extension result of [21]. (1) The spectrum of A has a decomposition σ (A) = σ 1 (A) σ 2 (A); (2) There is a real number α ∈ ‫ޒ‬ such that sup{ λ|λ ∈ σ 1 (A)} ≤ α ≤ inf{ λ|λ ∈ σ 2 (A)}; (3) σ 2 (A) = {λ k } k∈‫ގ‬ consists of isolated eigenvalues of A and is a union of finitely separated sets.…”

Section: Recall the Notations Hmentioning

confidence: 99%

“…The second difficulty comes from stability analysis of the system with variable coefficients, since the multiplier approach does not work for the networks. Here, we mainly use the Riesz basis approach as used in [19,20,22] to first obtain the spectrum-determined growth condition, and then to prove that the imaginary axis is not an asymptote of spectrum to obtain the stability. Although the proofs look like routine checks, the discussion of the Riesz basis property of eigenvectors and asymptote of spectrum have been difficult tasks in the spectral theory of linear operators.…”

mentioning

confidence: 99%

Exponential Stabilisation of a Tree-Shaped Network of Strings With Variable Coefficients

Guo¹,

Xu²

2011

Glasgow Math. J.

View full text Add to dashboard Cite

Abstract. In this paper, we deal with a tree-shaped network of strings with a fixed root node. By imposing velocity feedback controllers on all vertices except the root node, we show that the spectrum of the system operator consists of all isolated eigenvalues of finite multiplicity and is distributed in a strip parallel to the imaginary axis under certain conditions. Moreover, we prove that there exists a sequence of eigenvectors and generalised eigenvectors that forms a Riesz basis with parentheses, and that the imaginary axis is not an asymptote of the spectrum. Thereby, we deduce that the system is exponentially stable.2010 Mathematics Subject Classification. 93C20, 93D15, 35B35, 35P10, 47A70. Introduction.In past decades, there had been extensive literature devoted to research on controllability and stabilisation of one-dimensional (1D) networks. For instance, we refer to earlier works [5,8,9, 10], recent works [1, 2, 7, 18, 20, 22, 23] and the references therein. The main tools used in the literature are the Hilbert uniqueness method, the multiplier approach and the Riesz basis approach. We observe that the models discussed in the papers mentioned above are mainly of uniform structure; there are few results on non-homogeneous structure of 1D networks. Maybe part of the reason for the non-uniform networks is that there is no solution formula such as D'Alembert formula for 1D wave equation, and the investigation of controllability and observability of the networks tends to be very difficult. Therefore, Dager and Zuazua in [7] proposed that, as an open problem, the control problem of the network of 1D strings with variable coefficients is an important subject in future study. In this paper, we will study the stabilisation problem of a tree-shaped network of strings with variable coefficients using the spectral method. Since controllability and stabilisation of the system depend not only on the physical characteristics that support the graphs but also on the structures of the graphs, we suppose that the tree-shaped network is comprised of three strings with one fixed vertex called the root node, and three vertices

show abstract

Stabilization of a Timoshenko Beam With Disturbance Observer‐Based Time Varying Boundary Controls

Liu

Shang

2017

Asian Journal of Control

View full text Add to dashboard Cite

This paper is concerned with the boundary feedback stabilization for a Timoshenko beam with external disturbances in the boundary inputs. Based on the idea of active disturbance rejection controls, extended state observers with the time‐varying gains are designed to estimate disturbances and then a control strategy is presented by canceling the disturbances via the feedback channels. The well‐posedness of the resulting closed‐loop system is proved by the dual theory and admissibility theory, and the relationship between the stability and the disturbance is interpreted by Lyapunov's second method. At the end, the numerical experiment illustrate the effectiveness of the proposed control strategy.

show abstract

Riesz basis property of serially connected Timoshenko beams

Cited by 50 publications

References 20 publications

Riesz Basis Property and Stability of Planar Networks of Controlled Strings

Riesz Basis Property and Stability of Planar Networks of Controlled Strings

Exponential Stabilisation of a Tree-Shaped Network of Strings With Variable Coefficients

Stabilization of a Timoshenko Beam With Disturbance Observer‐Based Time Varying Boundary Controls

Contact Info

Product

Resources

About