Spectrum-determined Growth Condition Research Articles

Optimal Energy Decay for a Nonhomogeneous Flexible Beam with a Tip Mass

In this paper, a simple boundary feedback control moment is proposed to stabilize a nonhomogeneous flexible beam with a tip mass. By adopting the Riesz basis approach, it is shown that the close-loop system is a Riesz spectral system. Consequently, the exponential stability, spectrum-determined growth condition, and optimal decay rate are obtained.

On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type

In this paper, we study the Riesz basis property of the generalized eigenfunctions of a one-dimensional hyperbolic system in the energy state space. This characterizes the dynamic behavior of the system, particularly the stability, in terms of its eigenfrequencies. This system is derived from a thermoelastic equation with memory type. The asymptotic expansions for eigenvalues and eigenfunctions are developed. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. This deduces the spectrum-determined growth condition for the C 0 -semigroup associated with the system, and as a consequence, the exponential stability of the system is concluded.

Stability of a nonuniform Rayleigh beam with indefinite damping

This is a continuation of our earlier work [J.M. Wang, G.Q. Xu, S.P. Yung, Exponential stability for variable coefficients Rayleigh beams under boundary feedback control: a Riesz basis approach, Systems Control Lett. 51 (1) (2004) 33–50] on the study of a nonhomogeneous Rayleigh beam and this time the stabilization is achieved via an internal damping instead of the boundary feedbacks. We continue to address a conjecture of Guo [Basis property of a Rayleigh beam with boundary stabilization, J. Optim. Theory Appl. 112(3) (2002) 529–547] in this paper and demonstrate how the damping term can affect the decay rate asymptotically. By a detailed spectral analysis, we obtain a necessary condition for the stability and establish the Riesz basis property as well as the spectrum determined growth condition for the system. Furthermore, when the damping is indefinite, we provide a condition on how “negative” the damping can be without destroying the exponential stability.

Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach

In this paper, we consider the boundary stabilization of a sandwich beam which consists of two outer stiff layers and a compliant middle layer. Using Riesz basis approach, we show that there is a sequence of generalized eigenfunctions, which forms a Riesz basis in the state space. As a consequence, the spectrum-determined growth condition as well as the exponential stability of the closed-loop system are concluded. Finally, the well-posedness and regularity in the sense of Salamon-Weiss class as well as the exact controllability are also addressed.

Stabilization and optimal decay rate for a non-homogeneous rotating body-beam with dynamic boundary controls

In this paper, we consider the boundary stabilization of a flexible beam attached to the center of a rigid disk. The disk rotates with a non-uniform angular velocity while the beam has non-homogeneous spatial coefficients. To stabilize the system, we propose a feedback law which consists of a control torque applied on the disk and either a dynamic boundary control moment or a dynamic boundary control force or both of them applied at the free end of the beam. By the frequency multiplier method, we show that no matter how non-homogeneous the beam is, and no matter how the angular velocity is varying but not exceeding a certain bound, the nonlinear closed loop system is always exponential stable. Furthermore, by the spectral analysis method, it is shown that the closed loop system with uniform angular velocity has a sequence of generalized eigenfunctions, which form a Riesz basis for the state space, and hence the spectrum-determined growth condition as well as the optimal decay rate are obtained.

Exponential Stabilization ofLaminated 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 $C_0$-group due to the fact that the three asymptotes of the spectrum are parallel to the imaginary axis.

Differentiability of the C0-semigroup and Failure of Riesz Basis for a Mono-tubular Heat Exchanger Equation with Output Feedback: A Case Study

In this article, we show that the C0-semigroup associated with the closed-loop system of a mono-tubular heat exchanger equation with static output feedback studied in [System & Control Letters, 50 (2003), 363–369] is differentiable after some finite time period. This result strengthes the main conclusion of the spectrum-determined growth condition in aforementioned paper. Moreover, it is shown that the system is not a Riesz spectral system although the root subspace is complete in the energy Hilbert space. This is in sharp contrast to results for certain second order hyperbolic systems.

Basis property and stabilization of a translating tensioned beam through a pointwise control force

Introducing a recently developed powerful method, Riesz basis method as it is called, and applying it to the stabilization of a translating tensioned beam through a pointwise control force are the two purposes of the paper. An asymptotic analysis, which is different from the one in treating the same problem in literature, is used to get a complete asymptotic expansion of the eigenvalues and eigenfunctions for the beam. The controlled system is written as an evolution equation in a state Hilbert space whose energy norm (induced by inner product) is the total energy of the beam. The spectrum-determined growth condition is obtained by showing that there is a set of generalized eigenfunctions of the system, which forms a Riesz basis for the state space. As a consequence, the stability, particularly the exponential stability of the system, is developed. The analysis used here can be extended to other distributed systems and the results significantly improve that in [1].

A Sufficient Condition on Riesz Basis with Parentheses of Non--Self-Adjoint Operator and Application to a Serially Connected String System under Joint Feedbacks

This paper gives an abstract sufficient condition on Riesz basis with parentheses property for the generators of C0-groups in Hilbert spaces whose eigenvalues are comprised of some finite unification of separable sets after taking the algebraic multiplicities into account. The condition is then applied to the closed-loop system of a serially connected string system under joint damping feedbacks to show that there is a family of generalized eigenfunctions that form a Riesz basis with parentheses in the state space. The spectrum-determined growth condition is concluded as a consequence.

On the Boundary Control of a Hybrid System with Variable Coefficients

A model of the spacecraft control laboratory experiment with variable coefficients is considered. It is shown that the closed-loop system under boundary feedback damping has a sequence of generalized eigenfunctions, which form a Riesz basis for the state Hilbert space. The spectrum-determined growth condition, the exponential stability, and an asymptotic expression of the spectrum are obtained. Moreover, the exact controllability and exact observability of the system are also presented.

Basis Property of a Rayleigh Beam with Boundary Stabilization

A Rayleigh beam equation with boundary stabilization control is considered. Using an abstract result on the Riesz basis generation of discrete operators in Hilbert spaces, we show that the closed-loop system is a Riesz spectral system; that is, there is a sequence of generalized eigenfunctions of the system, which forms a Riesz basis in the state Hilbert space. The spectrum-determined growth condition, distribution of eigenvalues, as well as stability of the system are developed. This paper generalizes the results in Ref. 1.

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

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 spectrum-determined 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.

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

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 spectrum-determined growth condition of a vibration cable with a tip mass

We show that the spectrum-determined growth condition holds for the closed-loop system of a vibration cable with a tip mass and linear boundary feedback control. The optimal decay rate of the energy for a case left unsolved previously is determined, and the asymptotic expansion of the associated semigroup is obtained. As a consequence of the approach, we show, in a different point of view, the lack of uniform exponential stability of the system with only boundary velocity feedback control.

The first real eigenvalue of a one-dimensional linear thermoelastic system

In this note, we show, for a one-dimensional linear thermoelastic equation with Dirichlet-Dirichlet boundary conditions, that there is at least one real eigenvalue which is greater than the dominant eigenvalue of the “pure” heat equation with the same boundary conditions. The result concludes the spectrum-determined growth condition for the system by virtue of a result of Renardy [1]. Moreover, this property is shown to be preserved for the same system with boundary vibration control.

Root-Locus and Boundary Feedback Design for a Class of Distributed Parameter Systems

In this paper, a fairly complete parallel of the finite-dimensional root locus theory is presented for quite general, nonconstant coefficient, even order ordinary differential operators on a finite interval with control and output boundary conditions representative of a choice of colocated point actuators and sensors. Root-locus design methods for linear distributed parameter systems have also been studied for some time and the primary difficulties in rigorously interpreting root locus conclusions for distributed parameter systems are well known. First, the transfer function of a distributed parameter system may not be meromorphic at infinity so that many of the standard Rouche arguments, required even in the lumped case to determine the asymptotic behavior of the root loci, are not generally valid. Another difficulty is that the infinitesimal generator in the state-space model for a closed-loop system may not be selfadjoint, accretive or even satisfy the spectrum determined growth condition. Thus, regardless of whether the root loci---interpreted as closed-loop eigenvalues---lie in the open left-half plane, additional analysis would be required to conclude that the closed-loop system would be asymptotically stable. Formulating the systems in the classical format of a boundary control problem, the asymptotic analysis of the root loci can be based on the pioneering work by Birkhoff on eigenfunction expansions for boundary value problems, work that predated and indeed motivated the development of spectral theory in Hilbert space. Birkhoff's work also contains an asymptotic expansion of eigenfunctions in the spatial variable, generalizing the earlier Sturm--Liouville theory for second-order operators. By further extending this general asymptotic analysis to also include expansions in the gain parameter, a rigorous treatment of the open and closed loop transfer functions and of the corresponding return difference equation can be presented. The asymptotic analysis of the return difference equation forms the basis for both our rigorous formulation of the basic problem and its solution.