Gevrey class semigroups arising from elastic systems with gentle dissipation: the case

Authors:
Shu Ping Chen and Roberto Triggiani

Journal:
Proc. Amer. Math. Soc. **110** (1990), 401-415

MSC:
Primary 47D05; Secondary 34G10, 73D30

DOI:
https://doi.org/10.1090/S0002-9939-1990-1021208-4

MathSciNet review:
1021208

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Abstract: Let (the elastic operator) be a positive, self-adjoint operator with domain in the Hilbert space , and let (the dissipation operator) be another positive, self-adjoint operator satisfying for some constants and . Consider the operator

**[B.1 ]**A. V. Balakrishnan,*Damping operators in continuum models of flexible structures*, preprint.**[C-R.1]**G. Chen and D. L. Russell,*A mathematical model for linear elastic systems with structural damping*, Quart. Appl. Math.**39**(1981/82), no. 4, 433–454. MR**644099**, https://doi.org/10.1090/S0033-569X-1982-0644099-3**[C-T.1]**Shu Ping Chen and Roberto Triggiani,*Proof of two conjectures by G. Chen and D. L. Russell on structural damping for elastic systems*, Approximation and optimization (Havana, 1987) Lecture Notes in Math., vol. 1354, Springer, Berlin, 1988, pp. 234–256. MR**996678**, https://doi.org/10.1007/BFb0089601**[C-T.2]**Shu Ping Chen and Roberto Triggiani,*Proof of extensions of two conjectures on structural damping for elastic systems*, Pacific J. Math.**136**(1989), no. 1, 15–55. MR**971932****[C-T.3]**Shu Ping Chen and Roberto Triggiani,*Proof of extensions of two conjectures on structural damping for elastic systems*, Pacific J. Math.**136**(1989), no. 1, 15–55. MR**971932****[K.1]**S. G. Kreĭn,*Linear differential equations in Banach space*, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by J. M. Danskin; Translations of Mathematical Monographs, Vol. 29. MR**0342804****[K.2]**Tosio Kato,*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473****[L.1]**Walter Littman,*A generalization of a theorem of Datko and Pazy*, Advances in computing and control (Baton Rouge, LA, 1988) Lect. Notes Control Inf. Sci., vol. 130, Springer, Berlin, 1989, pp. 318–323. MR**1029070**, https://doi.org/10.1007/BFb0043280**[L.2]**J. L. Lions,*Controlabilite exacte des systemes distribues*, Masson, Paris, 1989.**[L-M.1]**J. L. Lions and E. Magenes,*Non homogeneous boundary value problems and applications*I, Springer-Verlag, New York, 1972.**[P.1]**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486****[T.l]**Roberto Triggiani,*On the stabilizability problem in Banach space*, J. Math. Anal. Appl.**52**(1975), no. 3, 383–403. MR**0445388**, https://doi.org/10.1016/0022-247X(75)90067-0**[T.2]**-,*Improving stability properties of hyperbolic damped equations by boundary feedback*, Springer-Verlag Lecture Notes (LNCIS), 1985, 400-409.**[T.3]**R. Triggiani,*Finite rank, relatively bounded perturbations of semi-groups generators. III. A sharp result on the lack of uniform stabilization*, Differential Integral Equations**3**(1990), no. 3, 503–522. MR**1047750****[T.4]**S. Taylor, Ph.D. thesis, Chapter 'Gevrey semigroups', School of Mathematics, University of Minnesota, 1989.

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DOI:
https://doi.org/10.1090/S0002-9939-1990-1021208-4

Article copyright:
© Copyright 1990
American Mathematical Society