Skip to main content
SpringerLink
Log in

Spectrum and analyticity of semigroups arising in elasticity theory and hydromechanics

  • Research article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

Cauchy problems for a second order linear differential operator equation

$$\ddot{z}(t)+A_{0}z(t)+D\dot{z}(t)=0$$

in a Hilbert space H are studied. Equations of this kind arise for example in elasticity and hydrodynamics. It is assumed that A 0 is a uniformly positive operator and that A −1/20 DA −1/20 is a bounded accretive operator in H. The location of the spectrum of the corresponding semigroup generator is described and sufficient conditions for analyticity are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Banks, H.T., Ito, K.: A unified framework for approximation in inverse problems for distributed parameter systems. Control Theory Adv. Technol. 4(1), 73–90 (1988)

    MathSciNet  Google Scholar 

  2. Banks, H.T., Ito, K., Wang, Y.: Well posedness for damped second-order systems with unbounded input operators. Differ. Integral Equ. 8(3), 587–606 (1995)

    MATH  MathSciNet  Google Scholar 

  3. Bátkai, A., Engel, K.: Exponential decay of 2×2 operator matrix semigroups. J. Comput. Anal. Appl. 6(2), 153–163 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Chen, G., Russell, D.L.: A mathematical model for linear elastic systems with structural damping. Q. Appl. Math. 39, 433–454 (1982)

    MATH  MathSciNet  Google Scholar 

  5. Chen, S., Triggiani, R.: Proof of extensions of two conjectures on structural damping for elastic systems. Pac. J. Math. 136(1), 15–55 (1989)

    MATH  MathSciNet  Google Scholar 

  6. Chen, S., Triggiani, R.: Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications. J. Differ. Equ. 88(2), 279–293 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, S., Liu, K., Liu, Z.: Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping. SIAM J. Appl. Math. 59(2), 651–668 (1999) (electronic)

    MATH  MathSciNet  Google Scholar 

  8. Engel, K., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York (2000)

    MATH  Google Scholar 

  9. Haase, M.: The Functional Calculus for Sectorial Operators. Birkhäuser, Basel (2006)

    MATH  Google Scholar 

  10. Hryniv, R.O., Shkalikov, A.A.: Operator models in elasticity theory and hydrodynamics and associated analytic semigroups. Mosc. Univ. Math. Bull. 54(5), 1–10 (1999)

    Google Scholar 

  11. Hryniv, R.O., Shkalikov, A.A.: Exponential stability of semigroups related to operator models in mechanics. Math. Notes 73(5), 618–624 (2003)

    Article  MathSciNet  Google Scholar 

  12. Hryniv, R.O., Shkalikov, A.A.: Exponential decay of solution energy for equations associated with some operator models of mechanics. Funct. Anal. Appl. 38(3), 163–172 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Huang, F.: On the mathematical model for linear elastic systems with analytic damping. SIAM J. Control Optim. 26(3), 714–724 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Huang, F.: Some problems for linear elastic systems with damping. Acta Math. Sci. 10(3), 319–326 (1990)

    MATH  MathSciNet  Google Scholar 

  15. Huang, S.-Z.: On energy decay rate of linear damped elastic systems. Tübinger Ber. Funktionalanal. 6, 65–91 (1997)

    Google Scholar 

  16. Jacob, B., Trunk, C.: Location of the spectrum of operator matrices which are associated to second order equations. Oper. Matrices 1, 45–60 (2007)

    MATH  MathSciNet  Google Scholar 

  17. Jacob, B., Morris, K., Trunk, C.: Minimum-phase infinite-dimensional second-order systems. IEEE Trans. Autom. Control 52, 1654–1665 (2007)

    Article  MathSciNet  Google Scholar 

  18. Jacob, B., Trunk, C., Winklmeier, M.: Analyticity and Riesz basis property of semigroups associated to damped vibrations. J. Evol. Equ. 8(2), 263–281 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976)

    MATH  Google Scholar 

  20. Kopachevsky, N.D., Krein, S.G.: Operator Approach to Linear Problems of Hydrodynamics. Volume 1: Self-adjoint Problems for an Ideal Fluid. Birkhäuser, Basel (2001)

    Google Scholar 

  21. Lancaster, P., Shkalikov, A.A.: Damped vibrations of beams and related spectral problems. Can. Appl. Math. Q. 2(1), 45–90 (1994)

    MATH  MathSciNet  Google Scholar 

  22. Shkalikov, A.A.: Operator pencils arising in elasticity and hydrodynamics: the instability index formula. In: Recent Developments in Operator Theory and Its Applications, Winnipeg, MB, 1994. Oper. Theory Adv. Appl., vol. 87, pp. 358–385. Birkhäuser, Basel (1996)

    Google Scholar 

  23. Tucsnak, M., Weiss, G.: How to get a conservative well-posed linear system out of thin air. II. Controllability and stability. SIAM J. Control Optim. 42(3), 907–935 (2003) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  24. Veselić, K.: Energy decay of damped systems. Z. Angew. Math. Mech. 84(12), 856–863 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Weiss, G., Tucsnak, M.: 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, 247–274 (2003) (electronic)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098, Paderborn, Germany

    Birgit Jacob

  2. Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, 98684, Ilmenau, Germany

    Carsten Trunk

Authors
  1. Birgit Jacob
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carsten Trunk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Birgit Jacob.

Additional information

Communicated by Rainer Nagel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jacob, B., Trunk, C. Spectrum and analyticity of semigroups arising in elasticity theory and hydromechanics. Semigroup Forum 79, 79–100 (2009). https://doi.org/10.1007/s00233-009-9148-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-009-9148-y

Keywords

Search

Navigation