American Institute of Mathematical Sciences

Advanced
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

[1]

V. Afraimovich, J. Schmeling, Edgardo Ugalde, Jesús Urías. Spectra of dimensions for Poincaré recurrences. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 901-914. doi: 10.3934/dcds.2000.6.901
[2]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835
[3]

Juan Wang, Xiaodan Zhang, Yun Zhao. Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2315-2332. doi: 10.3934/dcds.2014.34.2315
[4]

Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857
[5]

V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263
[6]

Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977
[7]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627
[8]

Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234
[9]

Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
[10]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469
[11]

Valentin Afraimovich, Jean-Rene Chazottes and Benoit Saussol. Local dimensions for Poincare recurrences. Electronic Research Announcements, 2000, 6: 64-74.

[12]

Zied Douzi, Bilel Selmi. On the mutual singularity of multifractal measures. Electronic Research Archive, 2020, 28 (1) : 423-432. doi: 10.3934/era.2020024
[13]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295
[14]

Jean-Pierre Francoise, Claude Piquet. Global recurrences of multi-time scaled systems. Conference Publications, 2011, 2011 (Special) : 430-436. doi: 10.3934/proc.2011.2011.430
[15]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425
[16]

Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635
[17]

Imen Bhouri, Houssem Tlili. On the multifractal formalism for Bernoulli products of invertible matrices. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1129-1145. doi: 10.3934/dcds.2009.24.1129
[18]

Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473
[19]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020173
[20]

Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences