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June  2013, 6(3): 611-617. doi: 10.3934/dcdss.2013.6.611

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, France
2. 

Dipartimento di Matematica, Università di Roma 'Tor Vergata', Via della Ricerca Scienti ca 1, 00133 Roma, Italy

Received  February 2010 Revised  June 2011 Published  December 2012

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.
Keywords: Poincaré recurrences, multifractal analysis., Dimension theory.
Mathematics Subject Classification: Primary: 47A50, 47A55; Secondary: 35B40, 93D2.
Citation: Fatiha Alabau-Boussouira, Piermarco Cannarsa. A constructive proof of Gibson's stability theorem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 611-617. doi: 10.3934/dcdss.2013.6.611
References:
[1]

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés,, C. R. Acad. Sci. Paris Sér. I, 328 (1999), 1015.  doi: 10.1016/S0764-4442(99)80316-4.  Google Scholar

[2]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems,, SIAM J. Control Optim., 41 (2002), 511.  doi: 10.1137/S0363012901385368.  Google Scholar

[3]

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems of evolution equations,, J. Evol. Equ., 2 (2002), 127.  doi: 10.1007/s00028-002-8083-0.  Google Scholar

[4]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equation,", Springer-Verlag, (2000).   Google Scholar

[5]

J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback,, SIAM J. Control Optim., 18 (1980), 311.  doi: 10.1137/0318022.  Google Scholar

[6]

A. Haraux, "Semi-groupes Linéaires et Équations D'évolutions Linéaires Périodiques,", Publications du Laboratoire d'Analyse Numérique 78011, (7801).   Google Scholar

[7]

L. Hörmander, "Linear Partial Differential Operators,", Springer-Verlag, (1963).   Google Scholar

[8]

V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method,", in, 36 (1994).   Google Scholar

show all references

References:
[1]

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés,, C. R. Acad. Sci. Paris Sér. I, 328 (1999), 1015.  doi: 10.1016/S0764-4442(99)80316-4.  Google Scholar

[2]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems,, SIAM J. Control Optim., 41 (2002), 511.  doi: 10.1137/S0363012901385368.  Google Scholar

[3]

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled systems of evolution equations,, J. Evol. Equ., 2 (2002), 127.  doi: 10.1007/s00028-002-8083-0.  Google Scholar

[4]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equation,", Springer-Verlag, (2000).   Google Scholar

[5]

J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback,, SIAM J. Control Optim., 18 (1980), 311.  doi: 10.1137/0318022.  Google Scholar

[6]

A. Haraux, "Semi-groupes Linéaires et Équations D'évolutions Linéaires Périodiques,", Publications du Laboratoire d'Analyse Numérique 78011, (7801).   Google Scholar

[7]

L. Hörmander, "Linear Partial Differential Operators,", Springer-Verlag, (1963).   Google Scholar

[8]

V. Komornik, "Exact Controllability and Stabilization. The Multiplier Method,", in, 36 (1994).   Google Scholar

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