Controllability properties for the Navier–Stokes system are closely related to observability properties for the adjoint Oseen–Stokes system; boundary observability inequalities are derived, for that adjoint system, that will be appropriate to deal with suitable constrained controls, like finite-dimensional controls supported in a given subset of the boundary. As an illustration, new boundary controllability results for the Oseen–Stokes system are derived. Finally, some further plausible consequences of the derived inequalities, concerning the Navier–Stokes system, are discussed.
DOI : 10.1051/cocv/2014045
Mots clés : Oseen–Stokes system, boundary observability inequalities, boundary control
@article{COCV_2015__21_3_723_0,
author = {Rodrigues, S\'ergio S.},
title = {Boundary observability inequalities for the {3D} {Oseen{\textendash}Stokes} system and applications},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {723--756},
publisher = {EDP-Sciences},
volume = {21},
number = {3},
year = {2015},
doi = {10.1051/cocv/2014045},
zbl = {1334.35252},
mrnumber = {3358628},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2014045/}
}
TY - JOUR AU - Rodrigues, Sérgio S. TI - Boundary observability inequalities for the 3D Oseen–Stokes system and applications JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 723 EP - 756 VL - 21 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014045/ DO - 10.1051/cocv/2014045 LA - en ID - COCV_2015__21_3_723_0 ER -
%0 Journal Article %A Rodrigues, Sérgio S. %T Boundary observability inequalities for the 3D Oseen–Stokes system and applications %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 723-756 %V 21 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014045/ %R 10.1051/cocv/2014045 %G en %F COCV_2015__21_3_723_0
Rodrigues, Sérgio S. Boundary observability inequalities for the 3D Oseen–Stokes system and applications. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 723-756. doi : 10.1051/cocv/2014045. http://www.numdam.org/articles/10.1051/cocv/2014045/
T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, number 252. Grundlehren der Mathematischen Wissenschaften. Springer (1982). | MR | Zbl
and , Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier–Stokes system. SIAM J. Control Optim. 49 (2011) 420–463. | DOI | MR | Zbl
, Stabilization of Navier–Stokes equations by oblique boundary feedback controllers. SIAM J. Control Optim. 50 (2012) 2288–2307. | DOI | MR | Zbl
, and , Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers. Nonlinear Anal. 64 (2006) 2704–2746. | DOI | MR | Zbl
, and , Internal exponential stabilization to a nonstationary solution for 3D Navier–Stokes equations. SIAM J. Control Optim. 49 (2011) 1454–1478. | DOI | MR | Zbl
and , Internal stabilization of Navier–Stokes equations with finite-dimensional controllers. Indiana Univ. Math. J. 53 (2004) 1443–1494. | DOI | MR | Zbl
H. Cartan, Formes Différentielles. Collection Méthodes. Hermann Paris (1967). | MR | Zbl
J.B. Conway, A Course in Functional Analysis, number 96. Grad. Texts Math. Springer, 2nd edition (1985). | MR | Zbl
M.P. do Carmo, Differential Forms and Applications. Universitext, Springer (1994). | MR | Zbl
, Sobolev spaces of differential forms and the Rham–Hodge isomorphism. J. Differ. Geom. 16 (1981) 63–73. | DOI | MR | Zbl
N. Dunford and J.T. Schwartz, Linear Operators. Part I: General Theory. Number VII, 4th printing. Pure Appl. Math. Interscience Publishers. John Wiley & Sons (1967). | Zbl
, , and , Local exact controllability of the Navier–Stokes system. J. Math. Pures Appl. 83 (2004) 1501–1542. | DOI | MR | Zbl
, Stabilization for the 3D Navier–Stokes system by feedback boundary control. Discrete Contin. Dyn. Syst. 10 (2004) 289–314. | DOI | MR | Zbl
, and , Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications. Trans. Amer. Math. Soc. 354 (2002) 1079–1116. | DOI | MR | Zbl
and , Local exact boundary controllability of the Boussinesq equation. SIAM J. Control Optim. 36 (1998) 391–421. | DOI | MR | Zbl
and , Exact controllability of the Navier–Stokes and Boussinesq equations. Russian Math. Surveys 54 (1999) 565–618. | DOI | MR | Zbl
, and . Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Commun. Pure Appl. Anal. 8 (2009) 311–333. | DOI | MR | Zbl
, Caractérisation de quelques espaces d’interpolation. Arch. Ration. Mech. Anal. 25 (1967) 40–63. | DOI | MR | Zbl
, Remarks on exact controllability for the Navier–Stokes equations. ESAIM: COCV 6 (2001) 39–72. | Numdam | MR | Zbl
J. Jost, Riemannian Geometry and Geometric Analysis. Univesitext, Springer, 4th edition (2005). | MR | Zbl
A. Kröner and S.S. Rodrigues, Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations. RICAM-Report No. 2014-02 (submitted) (2014). Available at http://www.ricam.oeaw.ac.at/publications/reports/. | MR
J.-L. Lions. Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod et Gauthier–Villars (1969). | MR | Zbl
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. I, number 181. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen. Springer-Verlag (1972). | Zbl
J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques. Masson Cie Éditeurs (1967). | MR | Zbl
and , Boundary feedback stabilization of the two-dimensional Navier–Stokes equations with finite-dimensional controllers. Discrete Contin. Dyn. Syst. 27 (2010) 1159–1187. | DOI | MR | Zbl
S.S. Rodrigues, Methods of Geometric Control Theory in Problems of Mathematical Physics. Ph.D. thesis, Universidade de Aveiro, Portugal (2008). Available at http://hdl.handle.net/10773/2931. | MR
, Local exact boundary controllability of 3D Navier–Stokes equations. Nonlinear Anal. 95 (2014) 175–190. | DOI | MR | Zbl
G. Schwarz, Hodge Decomposition – A Method for Solving Boundary Value Problems, Number 1607 in Lecture Notes in Mathematics. Springer (1995). | MR | Zbl
A. Shirikyan, Control and mixing for 2D Navier–Stokes equations with space-time localised noise (2011). Available at . | arXiv
K.T. Smith, Primer of Modern Analysis, Undergraduate Texts in Mathematics. Springer (1983). | MR | Zbl
M.E. Taylor, Partial Differential Equations I – Basic Theory, number 115. Appl. Math. Sci. Springer (1997). (corrected 2nd printing). | Zbl
R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, number 66. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 2nd edition (1995). | MR | Zbl
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, number 68. Appl. Math. Sci. Springer, 2nd edition (1997). | MR | Zbl
R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, reprint of the 1984 edition (2001). | MR | Zbl
A. Trautman, Differential Geometry for Physicists, Stony Brook Lectures. Monogr. Textbooks Phys. Sci. Bibliopolis (1984). | MR | Zbl
A.J. Weir, Lebesgue Integration and Measure. Cambridge University Press (1973). | MR | Zbl
Cité par Sources :
