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

In this paper we investigate some properties of a class of C0 semigroups on Banach spaces. Suppose that the spectrum of the infinitesimal generator is discrete and there are only finitely many eigenvalues in each vertical strip, we show that such semigroups can be expanded by their generalized eigenvectors under certain conditions. As a consequence, we assert that the semigroups with these conditions are eventually differentiable provided that the system of generalized eigenvectors is complete. As an example, we apply our results to a one-dimensional wave control system. © 2006 Elsevier Inc. All rights reserved.postprin

Lyapunov stability of abstract nonlinear dynamic system in Banach space

The Lyapunov stability theory for nonlinear time-varying dynamic system in Banach space is given in this paper. The Lyapunov stable theorem and the Barbashin-Krasovskii-LaSalle invariant set principle in classical theory are extended to infinite-dimensional Banach space. Under the assumptions of the existence of solution and the additive property of motions, sufficient and necessary conditions for uniform stability and uniform asymptotic stability are obtained, and the Lyapunov functions are explicitly constructed. This extension can be used as a criterion of stability for continuous and discontinuous systems.postprin

Exponential stability of variable coefficients Rayleigh beams under boundary feedback controls: A Riesz basis approach

In this paper, we study the boundary stabilizing feedback control problem of Rayleigh beams that have non-homogeneous spatial parameters. We show that no matter how non-homogeneous the Rayleigh beam is, as long as it has positive mass density, stiffness and mass moment of inertia, it can always be exponentially stabilized when the control parameters are properly chosen. The main steps are a detail asymptotic analysis of the spectrum of the system and the proving of that the generalized eigenfunctions of the feedback control system form a Riesz basis in the state Hilbert space. As a by-product, a conjecture in Guo (J. Optim. Theory Appl. 112(3) (2002) 529) is answered. © 2003 Elsevier B.V. All rights reserved.postprin

Riesz basis property of the generalized eigenvector system of a Timoshenko beam

The Riesz basis property of the generalized eigenvector system of a Timoshenko beam with boundary feedback controls applied to two ends is studied in this paper. The spectral property of the operator A determined by the closed loop system is investigated. It is shown that operator A has compact resolvent and generates a C 0 semigroup, and its spectrum consists of two branches and has two asymptotes under some conditions. Furthermore it is proved that the sequence of all generalized eigenvectors of the system principal operator forms a Riesz basis for the state Hilbert space.postprin

A Design of Robust Feedback Control for Flexible Robotic Mechanisms

We study flexible robotic arms that are free to rotate and bend in the horizontal plane but are stiff in vertical bending and torsion. A motor connected to the hinged end drives the arm to a prescribed target position and it is the aim of this paper to design boundary feedback controller to stabilize the arm once it reaches the atarget position. A distinct difficulty is the non-dissipativity that arises from the requirement that the designed feedback should retain 0 as an eigenvalue in order not to change the rigid body mode shape of the arm. On preserving this zero eigenvalue, we have successfully constructed a boundary feedback that is robustly stable with respect to the target position by showing it is exponentially stable there.published_or_final_versio

Asymptotic Frequency Distributions for Variable Coefficients Rayleigh Beams under Boundary Feedback Control

The analysis of the boundary damping rate for eigenmodes of a Rayleigh Beam with variable coefficients is usually difficult because explicit solution formula are hard to come by. In this paper, by using the estimating devices of [9], we can carry out an asymptotic analysis and conclude that there is a uniform damping rate for the high frequency modes. As a result, Riesz basis property and exponential stability can be deduced and a conjecture in [3] is settled.published_or_final_versio

A unified 'bang-bang' principle with respect to R-invariant performance benchmarks

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Exponential stabilization of laminated beams with structural damping and boundary feedback controls

We study the boundary stabilization of laminated beams with structural damping which describes the slip occurring at the interface of two-layered objects. By using an invertible matrix function with an eigenvalue parameter and an asymptotic technique for the first order matrix differential equation, we find out an explicit asymptotic formula for the matrix fundamental solutions and then carry out the asymptotic analyses for the eigenpairs. Furthermore, we prove that there is a sequence of generalized eigenfunctions that forms a Riesz basis in the state Hilbert space, and hence the spectrum determined growth condition holds. Furthermore, exponential stability of the closed-loop system can be deduced from the eigenvalue expressions. In particular, the semigroup generated by the system operator is a Co-group due to the fact that the three asymptotes of the spectrum are parallel to the imaginary axis. © 2005 Society for Industrial and Applied Mathematics.published_or_final_versio

Time Consistent Portfolio Selection Under Short-Selling Prohibition: from Discrete to Continuous Setting

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Defect detection on patterned jacquard fabric

The techniques for defect detection on plain (unpatterned) fabrics have been well developed nowadays. This paper is on developing visual inspection methods for defect detection on patterned fabrics. A review on some defect detection methods on patterned fabrics is given. Then, a new method for patterned fabric inspection called Golden Image Subtraction (GIS) is introduced. GIS is an efficient and fast method, which can segment out the defective regions on patterned fabric effectively. An improved version of the GIS method using wavelet transform is also given. This research results contribute to the development of an automated fabric inspection machine for the textile industry.published_or_final_versio