Spectral operators generated by Timoshenko beam model. (English) Zbl 0985.93012

Summary: We announce a series of results on the spectral analysis for a class of nonselfadjoint operators which are the dynamics generators for systems governed by the equations of the Timoshenko beam model with a 2-parameter family of dissipative boundary conditions. Our results split into three groups.
(1) We present asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues) and for their generalized eigenvectors.
(2) We show that these operators are Riesz spectral. This result follows from the fact that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces.
(3) We give the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators.
Our results, on the one hand, provide a class of nontrivial examples of spectral operators (nonselfadjoint operators that admit an analog of spectral decomposition). On the other hand, these results give a key to the solutions of various control and stabilization problems for the Timoshenko beam model using the spectral decomposition method.


93C20 Control/observation systems governed by partial differential equations
47F05 General theory of partial differential operators
35L05 Wave equation
35P20 Asymptotic distributions of eigenvalues in context of PDEs
93B28 Operator-theoretic methods
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
74H45 Vibrations in dynamical problems in solid mechanics
74M05 Control, switches and devices (“smart materials”) in solid mechanics
Full Text: DOI


[1] A.V. Balakrishnan, Vibrating systems with singular mass-inertia matrices, in: S. Sivasundaram (Ed.), First International Conference on Nonlinear Problems in Aviation & Aerospace, 1996, pp. 23-32.
[2] A.V. Balakrishnan, Smart structures and super stability, in: Computational Science for 21st Century, Wiley, Chichester, 1997.
[3] A.V. Balakrishnan, Dynamics and control of articulated anisotropic Timoshenko beams, in: Dynamics and Control of Distributed Systems, Cambridge University Press, Cambridge, 1998, pp. 121-202.
[4] Chen, G.; Krantz, S.G.; Russell, D.L.; Wayne, C.E.; West, H.H.; Coleman, M.P., Analysis, … for coupled beams, SIAM J. appl. math., 49, 1665-1693, (1989) · Zbl 0685.73046
[5] Chen, G.; Wang, H., Asymptotic locations of eigenfrequencies of euler – bernoulli beam with … damping coefficients, SIAM J. control optim., 29, 2, 347-367, (1991) · Zbl 0759.93068
[6] Coleman, M.P.; Wang, H., Analysis of vibration spectrum of a Timoshenko beam with boundary damping by the wave method, Wave motion, 17, 223-239, (1993) · Zbl 0776.73036
[7] Cox, S.; Zuazua, E., The rate at which the energy decays in a damped string, Comm. P.D.E., 19, 1 & 2, 213-243, (1994) · Zbl 0818.35072
[8] Cox, S.; Zuazua, E., The rate at which the energy decays in the string damped at one end, Indiana univ. math. J., 44, 545-573, (1995) · Zbl 0847.35078
[9] M.V. Fedoryuk, Asymptotic Analysis, Springer, Berlin, 1991. · Zbl 0742.34004
[10] Geist, B.; McLaughlin, J.R., Eigenvalue formulas for the uniform Timoshenko beam: free-free problem, Electron. res. ann., AMS 4, 12-17, (1998) · Zbl 0897.34071
[11] I.Ts. Gohberg, M.G. Krein, Introduction to Theory of Linear Nonselfadjoint Operators, Trans. of Math. Monogr., vol. 18, AMS, Providence, RI, 1969. · Zbl 0181.13503
[12] Kim, J.U.; Renardy, Y., Boundary control of the Timoshenko beam, SIAM J. control optim., 25, 1417-1429, (1987) · Zbl 0632.93057
[13] V. Komornik, Exact Controllability and Stabilization, Wiley, New York, 1994. · Zbl 0937.93003
[14] J.E. Lagnese, Recent progress in stabilizability of thin beams and plates, in: Lecture Notes in Pure and Applied Mathematics, vol. 128, 1991, pp. 61-112. · Zbl 0764.93014
[15] J.E. Lagnese, G. Leugering, E.J.P.G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multilink Structures, Birkhauser, Basel, 1994. · Zbl 0810.73004
[16] N.K. Nikol’skii, Treatise on the Shift Operator, Springer, Berlin, 1986.
[17] Olver, F.W.J., Asymptotics and special functions, (1979), Academic Press New York · Zbl 0303.41035
[18] Rao, B.P., Optimal energy decay rate in a damped Raleigh beam, Discrete control dyn. systems, 4, 721-734, (1998) · Zbl 0952.35145
[19] Shubov, M.A., Asymptotics of resonances and eigenvalues for nonhomogeneous damped string, Asymptotic anal., 13, 31-78, (1996) · Zbl 0864.35108
[20] Shubov, M.A., Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogenous damped string, Integral equations oper. theory, 25, 289-328, (1996) · Zbl 0855.47010
[21] Shubov, M.A., Nonselfadjoint operators generated by the equation of nonhomogenous damped string, Trans. of amer. math. soc., 349, 11, 4481-4499, (1997) · Zbl 0889.47004
[22] Shubov, M.A., Spectral operators generated by damped hyperbolic equations, Integral equations oper. theory, 28, 358-372, (1997) · Zbl 0912.47016
[23] Shubov, M.A., Exact boundary and distributed controllability of radial damped wave equation, J. math. pure appl., 77, 415-437, (1998) · Zbl 0910.93013
[24] Shubov, M.A., Spectral decomposition method for controlled damped string. reduction of control time, Appl. anal., 68, 3-4, 241-259, (1998) · Zbl 0914.35018
[25] M.A. Shubov, Spectral operators generated by 3-dimensional damped wave equation and application to control theory, in: A.G. Ramm (Ed.), Spectral and Scattering Theory, Plenum Press, New York, 1998, pp. 177-188. · Zbl 0901.35065
[26] Shubov, M.A., Asymptotics of spectrum and eigenfunctions … nonhomogeneous radial damped wave equations, Asymptotic anal., 16, 245-272, (1998) · Zbl 0938.35113
[27] M.A. Shubov, Exact controllability of damped Timoshenko beam, IMA J. Math. Control Inform., to appear. · Zbl 0991.93016
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