Glasnik Matematicki, Vol. 48, No. 2 (2013), 373-390.

OPTIMAL DAMPING OF THE INFINITE-DIMENSIONAL VIBRATIONAL SYSTEMS: COMMUTATIVE CASE

Ivica Nakić

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: nakic@math.hr

Abstract.   In this paper we treat the case of an abstract vibrational system of the form Mx″+Cx′+x=0, where the positive semi-definite selfadjoint operators M and C commute. We explicitly calculate the solution of the corresponding Lyapunov equation which enables us to obtain the set of optimal damping operators, thus extending already known results in the matrix case.

2010 Mathematics Subject Classification.   34G10, 70J99, 47D06.

Key words and phrases.   Vibrational systems, damping, Lyapunov equation.

Full text (PDF) (free access)

DOI: 10.3336/gm.48.2.10

References:

  1. N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Dover Publications Inc., New York, 1993.
    MathSciNet    

  2. P. Benner, Z. Tomljanović and N. Truhar, Dimension reduction for damping optimization in linear vibrating systems, Z. Angew. Math. Mech 91 (2011), 179-191.
    MathSciNet     CrossRef

  3. G. Chen and J. Zhou, Vibration and damping in distributed systems, Vol. I, CRC Press, Boca Raton, 1993.
    MathSciNet    

  4. S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math. 136 (1989), 15-55.
    MathSciNet     CrossRef

  5. S. J. Cox, I. Nakić, A. Rittmann and K. Veselić, Lyapunov optimization of a damped system, Systems Control Lett. 53 (2004), 187-194.
    MathSciNet     CrossRef

  6. S. J. Cox, Designing for optimal energy absorption. II. The damped wave equation, in Control and estimation of distributed parameter systems (Vorau, 1996), Birkhäuser, Basel, 1998, 103-109.
    MathSciNet    

  7. S. J. Cox, Designing for optimal energy absorption, I: Lumped parameter systems, ASME J. Vibration and Acoustics 120 (1998), 339-345.
    CrossRef

  8. R. Datko, Extending a theorem of A. M. Liapunov to Hilbert space, J. Math. Anal. Appl. 32 (1970), 610-616.
    MathSciNet     CrossRef

  9. K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, New York, 2000.
    MathSciNet    

  10. M. O. González, Classical complex analysis, Marcel Dekker Inc., New York, 1992.
    MathSciNet    

  11. E. Hille and R. S. Phillips, Functional analysis and semi-groups, American Mathematical Society, Providence, 1957.
    MathSciNet    

  12. F. L. Huang, On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control Optim. 26 (1988), 714-724.
    MathSciNet     CrossRef

  13. T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin, 1995.
    MathSciNet    

  14. A. N. Kolmogorov and S. V. Fomin, Introductory real analysis, Dover Publications Inc., New York, 1975.
    MathSciNet    

  15. P. Lancaster, Lambda-matrices and vibrating systems, %International series of monographs in pure and applied mathematics. Pergamon Press, 1966.
    MathSciNet    

  16. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
    MathSciNet     CrossRef

  17. V. Q. Phóng, The operator equation AX-XB=C with unbounded operators A and B and related abstract Cauchy problems, Math. Z. 208 (1991), 567-588.
    MathSciNet     CrossRef

  18. N. Truhar and K. Veselić, An efficient method for estimating the optimal dampers' viscosity for linear vibrating systems using Lyapunov equation, SIAM J. Matrix Anal. Appl. 31 (2009), 18-39.
    MathSciNet     CrossRef

  19. N. Truhar, An efficient algorithm for damper optimization for linear vibrating systems using lyapunov equation, J. Comput. Appl. Math. 172 (2004), 169-182.
    MathSciNet     CrossRef

  20. T. Veijola, Analytic damping model for an mem perforation cell, Microfluidics and Nanofluidics 2 (2006), 249-260.
    CrossRef

  21. K. Veselić, Damped oscillations of linear systems: a mathematical introduction, Springer, Heidelberg, 2011.
    MathSciNet     CrossRef

  22. K. Veselić, K. Brabender and K. Delinić, Passive control of linear systems, in Applied mathematics and computation (eds. M. Rogina et al.), Dept. of Math, University of Zagreb, Zagreb, 2001, 39-68.
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