AUTHORS: Genqi Xu, A. Jalili Rahmati, F. Badpar

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ABSTRACT: In this paper, we consider the exponential stabilization problem of a Timoshenko beam with interior local controls with input delays. In the past, most of the stabilization results for the Timoshenko beam were on the boundary control with input delays. In the present paper we shall extend the method treating the boundary control with delays to the case of interior local control with delays. Essentially we design a new dynamic feedback control laws that stabilizes exponentially the system. Detail of the design procedure of the dynamic feedback controller and analysis of the exponential stability are given.

KEYWORDS: Timoshenko beam, input delay, dynamic feedback controllers, exponential stabilization

REFERENCES:

[1] J. U. Kim, Y. Renardy, Boundary Control of the Timoshenko Beam, SIAM Journal on Control and Optimization, (1987), 25(6): pp. 1417-13.

[2] G. Q. Xu, D. X. Feng, The Riesz basis property of a Timoshenko beam with boundary feedback and application, IMA Journal of Applied Mathematics, (2002), 67, pp.357-370.

[3] G. Q. Xu, S. P. Yung, Exponential decay rate for a Timoshenko beam with boundary damping, Journal of Optimization Theory and Applications, (2004), 123, pp.669-693.

[4] G. Q. Xu, Boundary feedback exponential stabilization of a Timoshenko beam with both ends free, International Journal of Control, (2005), 78, pp.286-297.

[5] R. Datko, J. Lagnese, M. P. Polis, An example of the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, (1986), 24, pp.152-156.

[6] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delay in their feedbacks, SIAM Journal on Control and Optimization, (1988), 26, pp.697-713.

[7] R. Datko, Two examples of ill-posedness with respect to small delays in stabilized elastic systems, IEEE Transactions on Automatic Control, (1993), 38, pp.163-166.

[8] G. Q. Xu, S. P. Yung, L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: Control, Optimization and Calculus of Variations, (2006), 12, pp.770-785.

[9] S. Nicaise, C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization, (2006), 45(5), pp.1561-1585.

[10] S. Nicaise, J. Valein, Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks, Networks and Heterogeneous Media, (2007), 2(3), pp.425-479.

[11] Y. F. Shang, G. Q. Xu,Y. L. Chen, Stability analysis of Euler-Bernoulli beam with input delay in the boundary control, Asian Journal of Control, (2012), 14, pp.186-196.

[12] Z. J. Han, G. Q. Xu, Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks. ESAIM:Control, Optimization and Calculus of Variations, (2010),17, pp.552-574.

[13] Y. F. Shang, G. Q. Xu, Stabilization of an EulerBernoulli beam with input delay in the boundary control, Systems and Control Letters, (2012), 61, pp.1069-1078.

[14] G. Q. Xu, H. X. Wang, Stabilisation of Timoshenko beam system with delay in the boundary control, International Journal of Control, (2013), 86(6): pp. 1165-1178.

[15] H. Wang and G. Q. Xu, Exponential stabilization of 1-d wave equation with input delay, WSEAS Trans. Math., 12, 2013, pp.1001-1013.

[16] Z. J. Han and G. Q. Xu, Output-based stabilization of EulerCBernoulli beam with time-delay in boundary input, IMA Journal of Mathematical Control and Information, (2014), 31 (4): pp.533-550. doi:10.1093/imamci/dnt030

[17] Y. F. Shang and G. Q. Xu, Dynamic feedback control and exponential stabilization of a compound system, Journal of Mathematical Analysis and Applications, 422, (2015), pp. 858-879.

[18] X. F. Liu and G. Q. Xu, Exponenntial stabilization for Timoshenko beam with distributed delay in the boundary control, Abstract and Applied Analysis, (2013), doi:10.1155/2013/726794.

[19] X. F. Liu and G. Q. Xu, Output-based stabilization of Timoshenko beam with the boundary control and input distributed delay, Journal of Dynamic and Control System, (2015), doi: 10.1007/s10883-015-9293-4

[20] Y. F. Shang, G. Q. Xu, X. Li, Output-based stabilization for a one-dimensional wave equation with distributed input delay in the boundary control, IMA Journal of Mathematical Control and Information, (2014), doi:10.1093/imamci/dnu030

[21] D. H. Shi, D. X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, IMA Journal of Mathematical Control and Information, (2001), 18(3): pp.395-403.

[22] A. Soufyane, A. Whebe, Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electronic Journal of Differential Equations, (2003).

[23] C. A. Raposo, J. A. D. Chuquipoma, J. A. J. Avila, M. L. Santos, Exponential decay and numerical solution for a Timoshenko system with delay term in the internal feedback, International Journal of Analysis and Applications, (2013), 3(1), pp.1-13.

[24] B. Z. Guo, Y. H. Luo, Controllability and stability of a second-order hyperbolic system with collocated sensor/ actuator, Systems and Control Letters, (2002), 46, pp.45-65.