Optimal Control pp 84-115 | Cite as

# Spectral Analysis of Thermo-elastic Plates with Rotational Forces

## 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## Preview

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## References

- [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]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]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.MathSciNetzbMATHCrossRefGoogle Scholar - [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.MathSciNetzbMATHGoogle Scholar - [5]Fattorini, H.O. (1983), “The Cauchy problem,” in
*Encyclopedia of Mathematics and its Applications*, Addison-Wesley, Reading, Massachusetts.Google Scholar - [6]Hansen, S. (1992), “Exponential energy decay in a linear thermo-elastic rod,”
*J. Math. Anal e4 Appl*.,*Vol*.*167*, 429–442.zbMATHCrossRefGoogle Scholar - [7]Krein, S.G. (1971), “Linear differential equations in Banach space,”
*Trans. Amer. Math. Soc*.,*Vol*.*29*.Google Scholar - [8]Lagnese, J. (1989),
*Boundary Stabilization of Thin Plates*, SIAM, Philadelphia.zbMATHCrossRefGoogle Scholar - [9]Lagnese, J. and Lions, J.L. (1988),
*Modelling*,*Analysis and Control of Thin Places*, Masson, Paris.Google Scholar - [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]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]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] 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]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]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.MathSciNetzbMATHCrossRefGoogle Scholar - [16]Liu, K. and Liu, Z. (1996), “Exponential stability and analyticity of abstract linear thermo-elastic systems,” preprint.Google Scholar
- [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]Pazy, A. (1983),
*Semigroups of Linear Operators and Applications to Partial Differential Equations*, Springer-Verlag, Berlin.zbMATHCrossRefGoogle Scholar - [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]Xia, D. (1983),
*Spectral Theory of Hyponormal Operators*, Birkhäuser Verlag, Basel.zbMATHGoogle Scholar