[Observabilité polynomiale uniforme des systèmes linéaires conservatifs semi-discrets en temps]
Dans cet article nous étudions la semi-discrétisation en temps des systèmes de dimension infinie qui sont polynomialement observables. En utilisant une méthode basée sur l’estimation de la résolvante, nous obtenons des inégalités d’observabilité polynomiale uniformes pour les solutions filtrées du problème semi-discret en temps. Nous présentons également des applications de notre résultat aux problèmes de stabilisation.
In this paper we study time semi-discrete approximations of a class of polynomially observable infinite dimensional systems. By using a method based on the resolvent estimate, we derive uniform polynomial observability inequalities within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. We also present an application of our result to stabilization problems.
Mots clés : Observability inequality, Time discretization, Filtering
@article{AMBP_2016__23_1_53_0,
author = {Hajjej, Zayd},
title = {Uniform polynomial observability of time-discrete conservative linear systems},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {53--73},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {23},
number = {1},
year = {2016},
doi = {10.5802/ambp.354},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/ambp.354/}
}
TY - JOUR AU - Hajjej, Zayd TI - Uniform polynomial observability of time-discrete conservative linear systems JO - Annales mathématiques Blaise Pascal PY - 2016 SP - 53 EP - 73 VL - 23 IS - 1 PB - Annales mathématiques Blaise Pascal UR - http://archive.numdam.org/articles/10.5802/ambp.354/ DO - 10.5802/ambp.354 LA - en ID - AMBP_2016__23_1_53_0 ER -
%0 Journal Article %A Hajjej, Zayd %T Uniform polynomial observability of time-discrete conservative linear systems %J Annales mathématiques Blaise Pascal %D 2016 %P 53-73 %V 23 %N 1 %I Annales mathématiques Blaise Pascal %U http://archive.numdam.org/articles/10.5802/ambp.354/ %R 10.5802/ambp.354 %G en %F AMBP_2016__23_1_53_0
Hajjej, Zayd. Uniform polynomial observability of time-discrete conservative linear systems. Annales mathématiques Blaise Pascal, Tome 23 (2016) no. 1, pp. 53-73. doi : 10.5802/ambp.354. http://archive.numdam.org/articles/10.5802/ambp.354/
[1] A sharp geometric condition for the boundary exponential stabilizability of a square plate by moment feedbacks only, Control of coupled partial differential equations (Internat. Ser. Numer. Math.), Volume 155, Birkhäuser, Basel, 2007, pp. 1-11 | DOI
[2] Dirichlet boundary stabilization of the wave equation, Asymptot. Anal., Volume 30 (2002) no. 2, pp. 117-130
[3] Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., Volume 6 (2001), p. 361-386 (electronic) | DOI
[4] Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992) no. 5, pp. 1024-1065 | DOI
[5] An approximation method for exact controls of vibrating systems, SIAM J. Control Optim., Volume 49 (2011) no. 3, pp. 1283-1305 | DOI
[6] A mixed formulation for the direct approximation of the control of minimal -norm for linear type wave equations, Calcolo, Volume 52 (2015) no. 3, pp. 245-288 | DOI
[7] On the observability of time-discrete conservative linear systems, J. Funct. Anal., Volume 254 (2008) no. 12, pp. 3037-3078 | DOI
[8] Uniformly exponentially stable approximations for a class of damped systems, J. Math. Pures Appl. (9), Volume 91 (2009) no. 1, pp. 20-48 | DOI
[9] A numerical approach to the exact boundary controllability of the wave equation. I. Dirichlet controls: description of the numerical methods, Japan J. Appl. Math., Volume 7 (1990) no. 1, pp. 1-76 | DOI
[10] Uniformly polynomially stable approximations for a class of second order evolution equations, Palest. J. Math., Volume 2 (2013) no. 2, pp. 312-329
[11] Boundary observability for the space semi-discretizations of the -D wave equation, M2AN Math. Model. Numer. Anal., Volume 33 (1999) no. 2, pp. 407-438 | DOI
[12] Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1 Contrôlabilité exacte, Recherches en Mathématiques Appliquées, 8, Masson, Paris, 1988, x+541 pages (With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch)
[13] Controllability cost of conservative systems: resolvent condition and transmutation, J. Funct. Anal., Volume 218 (2005) no. 2, pp. 425-444 | DOI
[14] A uniformly controllable and implicit scheme for the 1-D wave equation, M2AN Math. Model. Numer. Anal., Volume 39 (2005) no. 2, pp. 377-418 | DOI
[15] Uniform stabilization of a viscous numerical approximation for a locally damped wave equation, ESAIM Control Optim. Calc. Var., Volume 13 (2007) no. 2, p. 265-293 (electronic) | DOI
[16] Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations (1988) (available at http://people.maths.ox.ac.uk/trefethen/pdetext.html)
[17] Observation and control for operator semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009, xii+483 pages | DOI
[18] Admissible observation operators for linear semigroups, Israel J. Math., Volume 65 (1989) no. 1, pp. 17-43 | DOI
Cité par Sources :
