A constructive proof of Gibson's stability theorem

Fatiha Alabau-Boussouira; Piermarco Cannarsa

doi:10.3934/dcdss.2013.6.611

Article Contents

2013, Volume 6, Issue 3: 611-617. Doi: 10.3934/dcdss.2013.6.611

This issue Previous Article Preface Next Article Product structures and fractional integration along curves in the space

A constructive proof of Gibson's stability theorem

Fatiha Alabau-Boussouira^1, and
Piermarco Cannarsa^2,

1.
L.M.A.M., CNRS-UMR 7122, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01
2.
Dipartimento di Matematica, Università di Roma 'Tor Vergata', Via della Ricerca Scientica 1, 00133 Roma

Received: January 31, 2010

Revised: May 31, 2011

Published: May 2013

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Abstract

A useful stability result due to Gibson [SIAM J. Control Optim., 18 (1980), 311--316] ensures that, perturbing the generator of an exponentially stable semigroup by a compact operator, one obtains an exponentially stable semigroup again, provided the perturbed semigroup is strongly stable. In this paper we give a new proof of Gibson's theorem based on constructive reasoning, extend the analysis to Banach spaces, and relax the above compactness assumption. Moreover, we discuss some applications of such an abstract result to equations and systems of evolution.

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Mathematics Subject Classification: Primary: 47A50, 47A55; Secondary: 35B40, 93D20.

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References

[1]	F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I, 328 (1999), 1015-1020.doi: 10.1016/S0764-4442(99)80316-4.
[2]	F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541.doi: 10.1137/S0363012901385368.
[3]	F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems of evolution equations, J. Evol. Equ., 2 (2002), 127-150.doi: 10.1007/s00028-002-8083-0.
[4]	K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equation," Springer-Verlag, New York, 2000.
[5]	J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback, SIAM J. Control Optim., 18 (1980), 311-316.doi: 10.1137/0318022.
[6]	A. Haraux, "Semi-groupes Linéaires et Équations D'évolutions Linéaires Périodiques," Publications du Laboratoire d'Analyse Numérique 78011, Université Pierre et Marie Curie, Paris, 1978.
[7]	L. Hörmander, "Linear Partial Differential Operators," Springer-Verlag, Berlin, 1963.
[8]	V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method," in "Collection RMA,'' 36, Masson-John Wiley, Paris-Chicester, 1994.