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Spectral Analysis of Thermo-elastic Plates with Rotational Forces

  • S. K. Chang
  • Roberto Triggiani
Part of the Applied Optimization book series (APOP, volume 15)

Abstract

We perform a spectral analysis of abstract thermo-elastic plate equations with ‘hinged’ B.C. in the presence of rotational forces, whereby the elastic equation is the (hyperbolic) Kirchoff equation. A precise description is given, which in particular shows that the resulting s.c. semi-group of contractions is neither compact nor differentiable for t > 0 (it contains an infinite-dimensional group invariant component). This is in sharp contrast with the case where rotational forces are neglected, whereby the elastic equation is the Euler-Bernoulli equation: in this latter case, the semigroup is, instead, analytic, under all canonical sets of B.C.

Keywords

Spectral Analysis Contraction Semigroup Bounded Perturbation Rotational Force Complex Conjugate Root 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Chang, S.K., Lasiecka, I. and Triggiani, R. (1997), “Lack of compactness and differentiability of the s.c. semigroup arising in thermo-elastic plate theory with rotational forces.”Google Scholar
  2. [2]
    Chen, S. and Triggiani, R. (1988), “Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems. The case a = 2,” in Proceedings of Seminar in Approximation and Optimization held at the University of Havana, Cuba, January 12–14, 1987, Lecture Notes in Mathematics #135.4, Springer-Verlag, Berlin.Google Scholar
  3. [3]
    Chen, S. and Triggiani, R. (1989), “Proof of extensions of two conjectures on structural damping for elastic systems. The case a>2,” Pacific J. of Mathematics, Vol. 136, 15–55.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Chen, S. and Triggiani, R. (1990), “Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0<a<1,” Proc. Amer. Math. Soc., Vol. 110, 401–415.MathSciNetMATHGoogle Scholar
  5. [5]
    Fattorini, H.O. (1983), “The Cauchy problem,” in Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, Massachusetts.Google Scholar
  6. [6]
    Hansen, S. (1992), “Exponential energy decay in a linear thermo-elastic rod,” J. Math. Anal e4 Appl., Vol. 167, 429–442.MATHCrossRefGoogle Scholar
  7. [7]
    Krein, S.G. (1971), “Linear differential equations in Banach space,” Trans. Amer. Math. Soc., Vol. 29.Google Scholar
  8. [8]
    Lagnese, J. (1989), Boundary Stabilization of Thin Plates, SIAM, Philadelphia.MATHCrossRefGoogle Scholar
  9. [9]
    Lagnese, J. and Lions, J.L. (1988), Modelling, Analysis and Control of Thin Places, Masson, Paris.Google Scholar
  10. [10]
    Lasiecka, I. (1997), “Control and stabilization of interactive structures,” in Systems and Control in the 21st Century, Birkhäuser Verlag, Basel, 245–262.Google Scholar
  11. [11]
    Lasiecka, I. and Triggiani, R., “Two direct proofs on the analyticity of the s.c. semigroup arising in abstract thermo-elastic equations,” Advances in Differential Equations, IFIP Workshop, University of Florida, February 1997, to appear.Google Scholar
  12. [12]
    Lasiecka, I. and Triggiani, R., “Lack of compactness and differentiability of the s.c. semigroup arising in thermo-elastic plate theory with rotational forces,” Advances in Differential Equations, IFIP Workshop, University of Florida, February 1997, submitted.Google Scholar
  13. [13] Lasiecka, I. and Triggiani, R., “Analyticity of thermo-elastic plate equations with coupled B.C.,” presented at: (i) Workshop on `Deterministic and Stochastic Evolutionary Systems,’ Scuola Normale Superiore, Pisa, Italy, July 1997; (ii)
    IFIP TC7 Conference on System Modelling and Optimization, Detroit, U.S., July 1997; (iii) MMAR Symposium, Miedzyzdroje, Poland, August 97, submitted.Google Scholar
  14. [14]
    Lasiecka, I. and Triggiani, R., “Control theory for partial differential equations: continuous and approximation theories,” in Encyclopedia of Mathematics and its Applications,Cambridge University Press, Cambridge, to appear.Google Scholar
  15. [15]
    Lasiecka, I. and Triggiani, R. (1991), “Exact controllability and uniform stabilization of Kirchoff plates with boundary control only on OwlE and homogeneous boundary displacement,” J. Diff. Eqn., Vol. 93, 62–101.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Liu, K. and Liu, Z. (1996), “Exponential stability and analyticity of abstract linear thermo-elastic systems,” preprint.Google Scholar
  17. [17]
    Liu, Z. and Renardy, M. (1995), “A note on the equations of a thermoelastic plate,” Appl. Math. Letters, Vol. 8, 1–6.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar
  19. [19]
    Triggiani, R. (1997), “Analyticity, and lack thereof, of semi-groups arising from thermo-elastic plates,” in Proceedings of Computational Science for the 21st Century, May 5–7, 1997, Wiley, New York.Google Scholar
  20. [20]
    Xia, D. (1983), Spectral Theory of Hyponormal Operators, Birkhäuser Verlag, Basel.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • S. K. Chang
    • 1
  • Roberto Triggiani
    • 2
  1. 1.Department of MathematicsYeungnam University KyungsanKyungpookKorea
  2. 2.Applied MathematicsUniversity of Virginia CharlottesvilleUSA

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