×

Stability of a nonuniform Rayleigh beam with indefinite damping. (English) Zbl 1100.93035

Summary: This is a continuation of our earlier work [J.M. Wang, G.Q. Xu, S.P. Yung, Exponential stability for variable coefficients Rayleigh beams under boundary feedback control: a Riesz basis approach, Systems Control Lett. 51 (1) (2004) 33–50 (2004; Zbl 1157.93498)] on the study of a nonhomogeneous Rayleigh beam and this time the stabilization is achieved via an internal damping instead of the boundary feedbacks. We continue to address a conjecture of Guo [Basis property of a Rayleigh beam with boundary stabilization, J. Optim. Theory Appl. 112(3) (2002) 529–547 (2002; Zbl 1147.93387)] in this paper and demonstrate how the damping term can affect the decay rate asymptotically. By a detailed spectral analysis, we obtain a necessary condition for the stability and establish the Riesz basis property as well as the spectrum determined growth condition for the system. Furthermore, when the damping is indefinite, we provide a condition on how “negative” the damping can be without destroying the exponential stability.

MSC:

93D09 Robust stability
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Birkhoff, G. D., On the asymptotic character of the solution of certain linear differential equations containing a parameter, Trans. Amer. Math. Soc., 9, 219-231 (1908) · JFM 39.0386.01
[2] Cox, S.; Zuazua, E., The rate at which energy decays in a damping string, Comm. Partial Differential Equations, 19, 213-243 (1994) · Zbl 0818.35072
[3] Freitas, P.; Zuazua, E., Stability results for the wave equation with indefinite damping, J. Differential Equations, 132, 338-352 (1996) · Zbl 0878.35067
[4] Gohberg, I., Classes of Linear Operators, vol. I (1990), Birkhäuser Verlag: Birkhäuser Verlag Basel-Boston-Berlin · Zbl 0722.00021
[5] Guo, B. Z., Basis property of a Rayleigh beam with boundary stabilization, J. Optim. Theory Appl., 112, 3, 529-547 (2002) · Zbl 1147.93387
[6] Guo, B. Z., Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients, SIAM J. Control Optim., 40, 6, 1905-1923 (2002) · Zbl 1015.93025
[7] Guo, B. Z.; Wang, J. M.; Yung, S. P., Boundary stabilization of a flexible manipulator with rotational inertia, Differential Integral Equations, 18, 9, 1013-1038 (2005) · Zbl 1212.93254
[8] Hansen, S. W.; Lasiecka, I., Analyticity, hyperbolicity and uniform stability of semigroups arising in models of composite beams, Math. Models Methods Appl. Sci., 10, 555-580 (2000) · Zbl 1036.74031
[9] Huang, F. L., Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1, 43-56 (1985) · Zbl 0593.34048
[10] Liu, K.; Liu, Z.; Rao, B., Exponential stability of an abstract non-dissipative linear system, SIAM J. Control Optim., 40, 149-165 (2001) · Zbl 0997.93085
[11] Luo, Z. H.; Guo, B. Z.; Morgül, O., Stability and Stabilization of Infinite Dimensional Systems with Applications (1999), Spring-Verlag: Spring-Verlag London
[12] Naimark, M. A., Linear Differential Operators, vol. I (1967), Frederick Ungar Publishing Company: Frederick Ungar Publishing Company New York · Zbl 0219.34001
[13] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer: Springer New York · Zbl 0516.47023
[14] Rao, B. P., Optimal energy decay rate in a damping Rayleigh beam, optimization methods in partial differential equations, Contemp. Math., 209, 211-229 (1997)
[15] Renardy, M., On the type of certain \(C_0\)-semigroups, Comm. Partial Differential Equations, 18, 1299-1307 (1993) · Zbl 0801.47029
[16] D.L. Russell, Mathematical models for the elastic beam and their control-theoretic implications, in: H. Brezis, M.G. Crandall F. Kapell (Eds.), Semigroups, Theory and Applications, vol. II, Longman Scientific and Technical, Harlow, 1986, pp. 177-216.; D.L. Russell, Mathematical models for the elastic beam and their control-theoretic implications, in: H. Brezis, M.G. Crandall F. Kapell (Eds.), Semigroups, Theory and Applications, vol. II, Longman Scientific and Technical, Harlow, 1986, pp. 177-216.
[17] Tretter, C., On λ-nonlinear boundary eigenvalue problem, (Mathematical Research, vol. 71 (1993), Akademie Verlag: Akademie Verlag Berlin) · Zbl 0974.34079
[18] Tretter, C., Kamke problems. Properties of the eigenfunctions, Math. Nachr., 170, 251-275 (1994) · Zbl 0813.34071
[19] Tretter, C., On fundamental systems for differential equations of Kamke type, Math. Z., 219, 609-629 (1995) · Zbl 0827.34004
[20] J.M. Wang, Riesz basis property of some infinite-dimensional control problems and its applications, Ph.D. Thesis, Hong Kong, 2004, \( \langle;\) http://sunzi1.lib.hku.hk/hkuto/record/B3016350X \(\rangle;\); J.M. Wang, Riesz basis property of some infinite-dimensional control problems and its applications, Ph.D. Thesis, Hong Kong, 2004, \( \langle;\) http://sunzi1.lib.hku.hk/hkuto/record/B3016350X \(\rangle;\)
[21] Wang, J. M.; Xu, G. Q.; Yung, S. P., Exponential stability for variable coefficients Rayleigh beams under boundary feedback control: a Riesz basis approach, Systems Control Lett., 51, 1, 33-50 (2004) · Zbl 1157.93498
[22] Wang, J. M.; Xu, G. Q.; Yung, S. P., Riesz basis property, exponential stability of variable coefficient Euler-Bernoulli beams with indefinite damping, IMA J. Appl. Math., 70, 3, 459-477 (2005) · Zbl 1157.34062
[23] Wang, J. M.; Xu, G. Q.; Yung, S. P., Exponential stabilization of laminated beams with structural damping and boundary feedback controls, SIAM J. Control Optim., 44, 5, 1575-1597 (2005) · Zbl 1132.93021
[24] Xu, G. Q.; Feng, D. X., On the spectrum determined growth assumption and the perturbation of \(C_0\)-semigroups, Integral Equations Operator Theory, 39, 3, 363-376 (2001) · Zbl 0986.47037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.