On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback
ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 3, p. 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.

DOI : https://doi.org/10.1051/cocv:2008001
Classification:  93C20,  93C25,  35J10,  47E05
Keywords: Rayleigh beam, collocated control, spectral analysis, exponential stability
@article{COCV_2008__14_3_632_0,
     author = {Guo, Bao-Zhu and Wang, Jun-Min and Zhou, Cui-Lian},
     title = {On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {3},
     year = {2008},
     pages = {632-656},
     doi = {10.1051/cocv:2008001},
     zbl = {1146.93026},
     mrnumber = {2434070},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2008__14_3_632_0}
}
Guo, Bao-Zhu; Wang, Jun-Min; Zhou, Cui-Lian. On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 3, pp. 632-656. doi : 10.1051/cocv:2008001. http://www.numdam.org/item/COCV_2008__14_3_632_0/

[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 1814271 | Zbl 0983.35021

[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 1923150 | Zbl 1042.93034

[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 1366650 | Zbl 0866.93001

[4] S.A. Avdonin and S.A. Ivanov, Riesz bases of exponentials and divided differences. St. Petersburg Math. J. 13 (2002) 339-351. | MR 1850184 | Zbl 0999.42018

[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 2007587 | Zbl 0986.93037

[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 1486069 | Zbl 0898.35055

[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 1626876 | Zbl 0909.35085

[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 1799611 | Zbl 1005.74041

[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 885183 | Zbl 0621.93053

[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 1025953 | Zbl 0685.73046

[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 1257004 | Zbl 0818.35072

[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 1355412 | Zbl 0847.35078

[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 2225306 | Zbl 1139.93026

[14] R. Dáger and E. Zuazua, Wave Propagation2006). | MR 2169126 | Zbl 1083.74002

[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 1851729 | Zbl 0985.35054

[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 2228960 | Zbl 1137.35344

[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 2062595 | Zbl 1157.93331

[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 2195332 | Zbl 1153.35368

[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 1900674 | Zbl 0999.35012

[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 2228174 | Zbl 1114.35023

[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 1745384

[23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | MR 710486 | Zbl 0516.47023

[24] R. Rebarber, Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33 (1995) 1-28. | MR 1311658 | Zbl 0819.93042

[25] M. Renardy, On the linear stability of hyperbolic PDEs and viscoelastic flows. Z. Angew. Math. Phys. 45 (1994) 854-865. | MR 1306936 | Zbl 0820.76008

[26] A.A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions. J. Soviet Math. 33 (1986) 1311-1342. | Zbl 0609.34019

[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 2246750 | Zbl 1100.93035

[28] G. Weiss and R.F. Curtain, Exponential stabilization of a Rayleigh beam using colocated control. IEEE Trans. Automatic Control (to appear). | MR 2401018

[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 2002142 | Zbl 1066.93028

[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 1998716 | Zbl 1068.93024

[31] R.M. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, Inc., London (1980). | MR 591684 | Zbl 0493.42001