Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, p. 552-574

In this paper, the stability of a Timoshenko beam with time delays in the boundary input is studied. The system is fixed at the left end, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system is well-posed. By the complete spectral analysis, we show that there is a sequence of eigenvectors and generalized eigenvectors of the system operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum determined growth condition. Then we conclude the exponential stability of the system under certain conditions. Finally, we give some simulations to support our results.

DOI : https://doi.org/10.1051/cocv/2010009
Classification:  93D15,  93C20,  49K25,  34H05
Keywords: Timoshenko beam, exponential stability, time delay, Riesz basis, feedback control
@article{COCV_2011__17_2_552_0,
     author = {Han, Zhong-Jie and Xu, Gen-Qi},
     title = {Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {2},
     year = {2011},
     pages = {552-574},
     doi = {10.1051/cocv/2010009},
     zbl = {1251.93106},
     mrnumber = {2801331},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_2_552_0}
}
Han, Zhong-Jie; Xu, Gen-Qi. Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, pp. 552-574. doi : 10.1051/cocv/2010009. http://www.numdam.org/item/COCV_2011__17_2_552_0/

[1] J.W. Brown and R.V. Churchill, Complex variables and applications. Seventh Edition, China Machine Press, Beijing (2004). | Zbl 0299.30003

[2] R. Datko, Two examples of ill-posedness with respect to small time delays in stabilized elastic systems. IEEE Trans. Automat. Contr. 38 (1993) 163-166. | MR 1201514 | Zbl 0775.93184

[3] J.U. Kim and Y. Renardy, Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25 (1987) 1417-1429. | MR 912448 | Zbl 0632.93057

[4] W.H. Kwon, G.W. Lee and S.W. Kim, Performance improvement, using time delays in multi-variable controller design. Int. J. Control 52 (1990) 1455-1473. | Zbl 0708.93024

[5] J.S. Liang, Y.Q. Chen and B.Z. Guo, A new boundary control method for beam equation with delayed boundary measurement using modified smith predictors, in Proceedings of the 42nd IEEE Conference on Decision and Control, Hawaii (2003) 809-814.

[6] Yu.I. Lyubich and V.Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces. Studia Math. 88 (1988) 34-37. | MR 932004 | Zbl 0639.34050

[7] R. Mennicken and M. Möller, Non-self-adjoint boundary eigenvalue problem, North-Holland Mathematics Studies 192. North-Holland, Amsterdam (2003). | MR 1995773 | Zbl 1033.34001

[8] W. Michiels and S.I. Niculescu, Stability and stabilization of time-delay systems: An Eigenvalue-based approach. Society for Industrial and Applied Mathematics, Philadelphia (2007). | MR 2384531 | Zbl 1140.93026

[9] O. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equation. IEEE Trans. Automat. Contr. 40 (1995) 1626-1630. | MR 1347843 | Zbl 0831.93039

[10] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45 (2006) 1561-1585. | MR 2272156 | Zbl 1180.35095

[11] S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay. Differential and Integral Equations 21 (2008) 935-958. | MR 2483342 | Zbl 1224.35247

[12] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. NHM 2 (2007) 425-479. | MR 2318841 | Zbl 1211.35050

[13] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, Berlin (1983). | MR 710486 | Zbl 0516.47023

[14] K. Sriram and M.S. Gopinathan, A two variable delay model for the circadian rhythm of Neurospora crassa. J. Theor. Biol. 231 (2004) 23-38. | MR 2107548

[15] J. Srividhya and M.S. Gopinathan, A simple time delay model for eukaryotic cell cycle. J. Theor. Biol. 241 (2006) 617-627. | MR 2254911

[16] H. Suh and Z. Bien, Use of time-delay actions in the controller design. IEEE Trans. Automat. Contr. 25 (1980) 600-603. | Zbl 0432.93044

[17] S. Timoshenko, Vibration Problems in Engineering. Van Norstrand, New York (1955). | JFM 54.0845.02

[18] Q.P. Vu, J.M. Wang, G.Q. Xu and S.P. Yung, Spectral analysis and system of fundamental solutions for Timoshenko beams. Appl. Math. Lett. 18 (2005) 127-134. | MR 2121270 | Zbl 1070.74020

[19] G.Q. Xu and D.X. Feng, The Riesz basis property of a Timoshenko beam with boundary feedback and application. IMA J. Appl. Math. 67 (2002) 357-370. | MR 1926568 | Zbl 1136.74343

[20] G.Q. Xu and B.Z. Guo, Riesz basis property of evolution equations in Hilbert space and application to a coupled string equation. SIAM J. Control Optim. 42 (2003) 966-984. | MR 2002142 | Zbl 1066.93028

[21] G.Q. Xu and J.G. Jia, The group and Riesz basis properties of string systems with time delay and exact controllability with boundary control. IMA J. Math. Control Inf. 23 (2006) 85-96. | MR 2212262 | Zbl 1106.93013

[22] G.Q. Xu and S.P. Yung, The expansion of semigroup and criterion of Riesz basis J. Differ. Equ. 210 (2005) 1-24. | MR 2114122 | Zbl 1131.47042

[23] G.Q. Xu, Z.J. Han and S.P. Yung, Riesz basis property of serially connected Timoshenko beams. Int. J. Control 80 (2007) 470-485. | MR 2294162 | Zbl 1120.93026

[24] G.Q. Xu, S.P. Yung and L.K. Li, Stabilization of wave systems with input delay in the boundary control. ESAIM: COCV 12 (2006) 770-785. | Numdam | MR 2266817 | Zbl 1105.35016

[25] R.M. Young, An introduction to nonharmonic Fourier series. Academic Press, London (1980) 80-84. | MR 591684 | Zbl 0981.42001