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

Boumediène Chentouf, Jun Min Wang

Research output: Contribution to journalArticle

25 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)667-691
Number of pages25
JournalJournal of Mathematical Analysis and Applications
Volume318
Issue number2
DOIs
Publication statusPublished - Jun 15 2006

Fingerprint

Dynamic Control
Boundary Control
Angular velocity
Decay Rate
Rotating
Stabilization
Closed loop systems
Closed-loop System
Boundary Stabilization
Frequency multiplying circuits
Flexible Beam
Multiplier Method
Feedback Law
Riesz Basis
Torque control
Force control
Growth Conditions
Spectral Analysis
Eigenvalues and eigenfunctions
Spectrum analysis

Keywords

  • Dynamic boundary control
  • Exponential stability
  • Frequency multiplier method
  • Non-homogeneous coefficients
  • Riesz basis
  • Rotating body-beam
  • Spectral analysis

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Stabilization and optimal decay rate for a non-homogeneous rotating body-beam with dynamic boundary controls. / Chentouf, Boumediène; Wang, Jun Min.

In: Journal of Mathematical Analysis and Applications, Vol. 318, No. 2, 15.06.2006, p. 667-691.

Research output: Contribution to journalArticle

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N2 - 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.

AB - 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.

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