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.

[1]  Gauthier Sallet,et al.  Boundary feedback stabilization of a rotating body-beam system , 1996, IEEE Trans. Autom. Control..

[2]  Genqi Xu,et al.  Riesz basis property, exponential stability of variable coefficient Euler-Bernoulli beams with indefinite damping , 2005 .

[3]  Kangsheng Liu.  Locally Distributed Control and Damping for the Conservative Systems , 1997 .

[4]  Bao-Zhu Guo,et al.  On the Boundary Control of a Hybrid System with Variable Coefficients , 2002 .

[5]  Omer Morgiil.  Dynamic Boundary Control of a Euler-Bernoulli Beam , 1992 .

[6]  Kangsheng Liu,et al.  Exponential Decay of Energy of the Euler--Bernoulli Beam with Locally Distributed Kelvin--Voigt Damping , 1998 .

[7]  J. Baillieul,et al.  Stabilizability and stabilization of a rotating body-beam system with torque control , 1993, IEEE Trans. Autom. Control..

[8]  A. Kolmogorov,et al.  Éléments de la théorie des fonctions et de l'analyse fonctionnelle , 1994 .

[9]  Hing-leung Li,et al.  Development and characterization of a rotary motor driven by anisotropic piezoelectric composite laminate , 1998 .

[10]  Bao-Zhu Guo,et al.  Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients , 2001, SIAM J. Control. Optim..

[11]  Bao-Zhu Guo,et al.  Riesz Basis Approach to the Stabilization of a Flexible Beam with a Tip Mass , 2000, SIAM J. Control. Optim..

[12]  Dynamic boundary controls of a rotating body-beam system with time-varying angular velocity , 2004 .

[13]  J. Baillieul,et al.  Rotational elastic dynamics , 1987 .

[14]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[15]  Kaïs Ammari,et al.  Decay Rates for a Beam with Pointwise Force and Moment Feedback , 2002, Math. Control. Signals Syst..

[16]  B. Guo,et al.  The Riesz basis property of discrete operators and application to a Euler–Bernoulli beam equation with boundary linear feedback control , 2001 .

[17]  P. Goldbart,et al.  Linear differential operators , 1967 .

[18]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[19]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.