On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback

Guo, Bao-Zhu; Wang, Jun-Min; Zhou, Cui-Lian

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 632-656.

We study the dynamic behavior and stability of two connected Rayleigh beams that are subject to, in addition to two sensors and two actuators applied at the joint point, one of the actuators also specially distributed along the beams. We show that with the distributed control employed, there is a set of generalized eigenfunctions of the closed-loop system, which forms a Riesz basis with parenthesis for the state space. Then both the spectrum-determined growth condition and exponential stability are concluded for the system. Moreover, we show that the exponential stability is independent of the location of the joint. The range of the feedback gains that guarantee the system to be exponentially stable is identified.

[1] K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM J. Control Optim. 39 (2000) 1160-1181. | MR | Zbl

[2] K. Ammari, Z. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback. Math. Control Signals Systems 15 (2002) 229-255. | MR | Zbl

[3] S.A. Avdonin and S.A. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, Cambridge, UK (1995). | MR | Zbl

[4] S.A. Avdonin and S.A. Ivanov, Riesz bases of exponentials and divided differences. St. Petersburg Math. J. 13 (2002) 339-351. | MR | Zbl

[5] S.A. Avdonin and W. Moran, Simultaneous control problems for systems of elastic strings and beams. Syst. Control Lett. 44 (2001) 147-155. | MR | Zbl

[6] C. Castro and E. Zuazua, A hybrid system consisting of two flexible beams connected by a point mass: spectral analysis and well-posedness in asymmetric spaces. ESAIM: Proc. 2 (1997) 17-53. | MR | Zbl

[7] C. Castro and E. Zuazua, Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass. SIAM J. Control Optim. 36 (1998) 1576-1595. | MR | Zbl

[8] C. Castro and E. Zuazua, Exact boundary controllability of two Euler-Bernoulli beams connected by a point mass. Math. Comput. Modelling 32 (2000) 955-969. | MR | Zbl

[9] G. Chen, M.C. Delfour, A.M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams. SIAM J. Control Optim. 25 (1987) 526-546. | MR | Zbl

[10] G. Chen, S.G. Krantz, D.L. Russell, C.E. Wayne, H.H. West and M.P. Coleman, Analysis, designs, and behavior of dissipative joints for coupled beams. SIAM J. Appl. Math. 49 (1989) 1665-1693. | MR | Zbl

[11] S. Cox and E. Zuazua, The rate at which energy decays in a damped string. Comm. Partial Diff. Eq. 19 (1994) 213-243. | MR | Zbl

[12] S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44 (1995) 545-573. | MR | Zbl

[13] R.F. Curtain and G. Weiss, Exponential stabilization of well-posed systems by colocated feedback. SIAM J. Control Optim. 45 (2006) 273-297. | MR | Zbl

[14] R. Dáger and E. Zuazua, Wave Propagation2006). | MR | Zbl

[15] B.Z. Guo and K.Y. Chan, Riesz basis generation, eigenvalues distribution, and exponential stability for a Euler-Bernoulli beam with joint feedback control. Rev. Mat. Complut. 14 (2001) 205-229. | MR | Zbl

[16] B.Z. Guo and J.M. Wang, Riesz basis generation of an abstract second-order partial differential equation system with general non-separated boundary conditions. Numer. Funct. Anal. Optim. 27 (2006) 291-328. | MR | Zbl

[17] B.Z. Guo and G.Q. Xu, Riesz basis and exact controllability of $C_0$-groups with one-dimensional input operators. Syst. Control Lett. 52 (2004) 221-232. | MR | Zbl

[18] B.Z. Guo and G.Q. Xu, Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition. J. Funct. Anal. 231 (2006) 245-268. | MR | Zbl

[19] B.Ya. Levin, On bases of exponential functions in $L_2$. Zapiski Math. Otd. Phys. Math. Facul. Khark. Univ. 27 (1961) 39-48 (in Russian).

[20] K.S. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity. Z. Angew. Math. Phys. 53 (2002) 265-280. | MR | Zbl

[21] K.S. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping. Z. Angew. Math. Phys. 57 (2006) 419-432. | MR | Zbl

[22] Z.H. Luo, B.Z. Guo and Ö. Morgül, Stability and Stabilization of Linear Infinite Dimensional Systems with Applications. Springer-Verlag, London (1999). | MR

[23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | MR | Zbl

[24] R. Rebarber, Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33 (1995) 1-28. | MR | Zbl

[25] M. Renardy, On the linear stability of hyperbolic PDEs and viscoelastic flows. Z. Angew. Math. Phys. 45 (1994) 854-865. | MR | Zbl

[26] A.A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions. J. Soviet Math. 33 (1986) 1311-1342. | Zbl

[27] J.M. Wang and S.P. Yung, Stability of a nonuniform Rayleigh beam with internal dampings. Syst. Control Lett. 55 (2006) 863-870. | MR | Zbl

[28] G. Weiss and R.F. Curtain, Exponential stabilization of a Rayleigh beam using colocated control. IEEE Trans. Automatic Control (to appear). | MR

[29] G.Q. Xu and B.Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation. SIAM J. Control Optim. 42 (2003) 966-984. | MR | Zbl

[30] G.Q. Xu and S.P. Yung, Stabilization of Timoshenko beam by means of pointwise controls. ESAIM: COCV 9 (2003) 579-600. | Numdam | MR | Zbl

[31] R.M. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, Inc., London (1980). | MR | Zbl