Spectral Theory for Operator Matrices Related to Models in Mechanics

C. Trunk

Matematičeskie zametki, Tome 83 (2008) no. 6, pp. 923-932.

Résumé

We derive various properties of the operator matrix $$ \mathscr A=\begin{vmatrix} 0 \\ -A_0-D \end{vmatrix}, $$ where $A_0$ is a uniformly positive operator and $A_0^{-1/2}DA_0^{-1/2}$ is a bounded nonnegative operator in a Hilbert space $H$. Such operator matrices are associated with second-order problems of the form $\ddot z(t)+A_0z(t)+D\dot z(t)=0$, which are used as models for transverse motions of thin beams in the presence of damping.

Détail

Keywords: operator matrices, second-order partial differential equations, spectrum, Riesz basis, definitizable operator, Krein space, analytic semigroup.

Bibliographie

[1] C. D. Benchimol, “A note on weak stabilizability of contraction semigroups”, SIAM J. Control Optimization, 16:3 (1978), 373–379 | DOI | MR | Zbl

[2] I. Lasiecka, R. Triggiani, “Uniform exponential energy decay of wave equations in a bounded region with $L^2(0,\infty;L^2(\Gamma))$ – feedback control in the Dirichlet boundary condition”, J. Differential Equations, 66:3 (1987), 340–390 | DOI | MR | Zbl

[3] N. Levan, “The stabilization problem: A Hilbert space operator decomposition approach”, IEEE Trans. Circuits and Systems, 25:9 (1978), 721–727 | DOI | MR | Zbl

[4] M. Slemrod, “Stabilization of boundary control systems”, J. Differential Equation, 22:2 (1976), 402–415 | DOI | MR | Zbl

[5] E. Hendrickson, I. Lasiecka, “Numerical approximations and regularizations of Riccati equations arising in hyperbolic dynamics with unbounded control operators”, Comput. Optim. Appl., 2:4 (1993), 343–390 | DOI | MR | Zbl

[6] B. Jacob, K. Morris, C. Trunk, “Minimum-phase infinite-dimensional second-order systems”, IEEE Trans. Automat. Control (to appear)

[7] E. Hendrickson, I. Lasiecka, “Finite-dimensional approximations of boundary control problems arising in partially observed hyperbolic systems”, Dynam. Contin. Discrete Impuls. Systems, 1:1 (1995), 101–142 | MR | Zbl

[8] H. T. Banks, K. Ito, “A unified framework for approximation in inverse problems for distributed parameter systems”, Control Theory Adv. Tech., 4:1 (1988), 73–90 | MR

[9] I. Lasiecka, “Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary”, J. Differential Equations, 79:2 (1989), 340–381 | DOI | MR | Zbl

[10] H. T. Banks, K. Ito, Y. Wang, “Well posedness for damped second-order systems with unbounded input operators”, Differential Integral Equations, 8:3 (1995), 587–606 | MR | Zbl

[11] S. Chen, K. Liu, Z. Liu, “Spectrum and stability for elastic systems with global or local Kelvin–Voigt damping”, SIAM J. Appl. Math., 59:2 (1999), 651–668 | DOI | MR | Zbl

[12] A. Bátkai, K.-J. Engel, “Exponential decay of $2\times2$ operator matrix semigroups”, J. Comput. Anal. Appl., 6:2 (2004), 153–163 | MR | Zbl

[13] R. O. Griniv, A. A. Shkalikov, “Eksponentsialnaya ustoichivost polugrupp, svyazannykh s nekotorymi operatornymi modelyami v mekhanike”, Matem. zametki, 73:5 (2003), 657–664 | MR | Zbl

[14] R. O. Griniv, A. A. Shkalikov, “Eksponentsialnoe ubyvanie energii reshenii uravnenii, otvechayuschikh nekotorym operatornym modelyam mekhaniki”, Funkts. analiz i ego pril., 38:3, 3–14 | MR | Zbl

[15] M. Tucsnak, G. Weiss, “How to get a conservative well-posed linear system out of thin air. II: Controllability and stability”, SIAM J. Control Optim., 42:3 (2003), 907–935 | DOI | MR | Zbl

[16] K. Veselić, “Energy decay of damped systems”, ZAMM Z. Angew. Math. Mech., 84:12 (2004), 856–863 | DOI | MR | Zbl

[17] G. Chen, D. Russell, “A mathematical model for linear elastic systems with structural damping”, Quart. Appl. Math., 39:4 (1982), 433–454 | MR | Zbl

[18] S. Chen, R. Triggiani, “Proof of extensions of two conjectures on structural damping for elastic systems”, Pacific J. Math., 136:1 (1989), 15–55 | MR | Zbl

[19] S. Chen, R. Triggiani, “Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications”, J. Differential Equations, 88:2 (1990), 279–293 | DOI | MR | Zbl

[20] R. O. Griniv, A. A. Shkalikov, “Operatornye modeli v teorii uprugosti i gidromekhanike i sootvetstvuyuschie analiticheskie polugruppy”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1999, no. 5, 5–14 | MR | Zbl

[21] F. Huang, “On the mathematical model for linear elastic systems with analytic damping”, SIAM J. Control Optim., 26:3 (1988), 714–724 | DOI | MR | Zbl

[22] F. Huang, “Some problems for linear elastic systems with damping”, Acta Math. Sci., 10:3 (1990), 319–326 | MR | Zbl

[23] P. Lancaster, A. Shkalikov, “Damped vibrations of beams and related spectral problems”, Canad. Appl. Math. Quart., 2:1 (1994), 45–90 | MR | Zbl

[24] B. Jacob, C. Trunk, M. Winklmeier, Analyticity and Riesz basis property of semigroups associated to damped vibrations (to appear)

[25] J. Bognár, Indefinite Inner Product Spaces, Ergeb. Math. Grenzgeb., 78, Springer-Verlag, New York–Heidelberg, 1974 | MR | Zbl

[26] T. Ya. Azizov, I. S. Iokhvidov, Lineinye operatory v prostranstvakh s indefinitnoi metrikoi, Nauka, M., 1986 | MR | Zbl

[27] P. Lancaster, A. Markus, V. Matsaev, “Definitizable operators and quasihyperbolic operator polynomials”, J. Funct. Anal., 131:1 (1995), 1–28 | DOI | MR | Zbl

[28] H. Langer, A. Markus, V. Matsaev, “Locally definite operators in indefinite inner product spaces”, Math. Ann., 308:3 (1997), 405–424 | DOI | MR | Zbl

[29] T. Ya. Azizov, P. Jonas, C. Trunk, “Spectral points of type $\pi_+$ and $\pi_-$ of self-adjoint operators in Krein spaces”, J. Funct. Anal., 226:1 (2005), 114–137 | DOI | MR | Zbl

[30] H. Langer, “Spectral functions of definitizable operators in Krein spaces”, Functional Analysis (Dubrovnik, 1981), Lecture Notes in Math., 948, Springer, Berlin, 1982, 1–46 | DOI | MR | Zbl

[31] P. Jonas, “On locally definite operators in Krein spaces”, Spectral analysis and its applications, Theta Ser. Adv. Math., 2, Theta, Bucharest, 2003, 95–127 | MR | Zbl

[32] T. Ya. Azizov, J. Behrndt, P. Jonas, C. Trunk, Spectral points of type $\pi_+$ and $\pi_-$ for closed linear relations in Krein spaces (to appear)

[33] G. Weiss, M. Tucsnak, “How to get a conservative well-posed linear system out of thin air. I: Well-posedness and energy balance”, ESAIM Control Optim. Calc. Var., 9 (2003), 247–274 | MR | Zbl

[34] B. Jacob, C. Trunk, “Location of the spectrum of operator matrices which are associated to second order equations”, Oper. Matrices, 1:1 (2007), 45–60 | MR | Zbl

[35] N. K. Nikolskii, Lektsii ob operatore sdviga, Nauka, M., 1980 | MR | Zbl