On the Stabilization of a Flexible Beam with a Tip Mass

We study the stability of a flexible beam that is clamped at one end and free at the other; a mass is also attached to the free end of the beam. To stabilize this system we apply a boundary control force at the free end of the beam. We prove that the closed-loop system is well-posed and is exponentially stable. We then analyze the spectrum of the system for a special case and prove that the spectrum determines the exponential decay rate for the considered case.

[1]  F. Smithies.  Linear Operators , 2019, Nature.

[2]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[3]  W. Rudin.  Real and complex analysis , 1968 .

[4]  I. Singer,et al.  Bases in Banach Spaces II , 1970 .

[5]  D. Russell.  Decay rates for weakly damped systems in Hilbert space obtained with control-theoretic methods☆ , 1975 .

[6]  Hiroki Tanabe,et al.  Equations of evolution , 1979 .

[7]  J. S. Gibson,et al.  A Note on Stabilization of Infinite Dimensional Linear Oscillators by Compact Linear Feedback , 1980 .

[8]  Steven G. Krantz,et al.  The Euler-Bernoulli beam equation with boundary energy dissipation , 1988 .

[9]  L. Markus,et al.  Stabilization of a hybrid system of elasticity by feedback boundary damping , 1988 .

[10]  R. Triggiani,et al.  Lack of uniform stabilization for noncontractive semigroups under compact perturbation , 1989 .

[11]  Jianxin Zhou,et al.  The wave propagation method for the analysis of boundary stabilization in vibrating structures , 1990 .

[12]  F. Conrad.  Stabilization of beams by pointwise feedback control , 1990 .

[13]  A. V. Balakrishnan.  Compensator design for stability enhancement with collocated controllers , 1991 .

[14]  B. Rao.  Decay estimates of solutions for a hybrid system of flexible structures , 1993, European Journal of Applied Mathematics.

[15]  B. Rao.  Stabilisation uniforme d'un système hybride en élasticité , 1993 .

[16]  Ö. Morgül,et al.  On the stabilization of a cable with a tip mass , 1994, IEEE Trans. Autom. Control..

[17]  Bopeng Rao,et al.  Uniform Stabilization of a Hybrid System of Elasticity , 1995 .