Local exact controllability for the 1-d compressible Navier-Stokes equations
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 39, 14 p.

In this talk, I will present a recent result obtained in [6] with O. Glass, S. Guerrero and J.-P. Puel on the local exact controllability of the 1-d compressible Navier-Stokes equations. The goal of these notes is to give an informal presentation of this article and we refer the reader to it for extensive details.

DOI : 10.5802/slsedp.30
Ervedoza, Sylvain 1, 2

1 CNRS Institut de Mathématiques de Toulouse UMR 5219 F-31062 Toulouse France
2 Université de Toulouse UPS, INSA, INP, ISAE, UT1, UTM, IMT F-31062 Toulouse France
@article{SLSEDP_2011-2012____A39_0,
     author = {Ervedoza, Sylvain},
     title = {Local exact controllability for the $1$-d compressible {Navier-Stokes} equations},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:39},
     pages = {1--14},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2011-2012},
     doi = {10.5802/slsedp.30},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/slsedp.30/}
}
TY  - JOUR
AU  - Ervedoza, Sylvain
TI  - Local exact controllability for the $1$-d compressible Navier-Stokes equations
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:39
PY  - 2011-2012
SP  - 1
EP  - 14
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/articles/10.5802/slsedp.30/
DO  - 10.5802/slsedp.30
LA  - en
ID  - SLSEDP_2011-2012____A39_0
ER  - 
%0 Journal Article
%A Ervedoza, Sylvain
%T Local exact controllability for the $1$-d compressible Navier-Stokes equations
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:39
%D 2011-2012
%P 1-14
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://www.numdam.org/articles/10.5802/slsedp.30/
%R 10.5802/slsedp.30
%G en
%F SLSEDP_2011-2012____A39_0
Ervedoza, Sylvain. Local exact controllability for the $1$-d compressible Navier-Stokes equations. Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 39, 14 p. doi : 10.5802/slsedp.30. http://www.numdam.org/articles/10.5802/slsedp.30/

[1] E.V. Amosova. Exact local controllability for the equations of viscous gas dynamics. Differentsial’nye Uravneniya, 47(12):1754–1772, 2011. | MR | Zbl

[2] S. Chowdhury, M. Ramaswamy, and J.-P. Raymond. Controllability and stabilizability of the linearized compressible Navier-Stokes in one dimension. Submitted, 2012.

[3] J.-M. Coron. On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. (9), 75(2):155–188, 1996. | MR | Zbl

[4] J.-M. Coron. Control and nonlinearity, volume 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2007. | MR | Zbl

[5] J.-M. Coron and A. V. Fursikov. Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys., 4(4):429–448, 1996. | MR | Zbl

[6] S. Ervedoza, O. Glass, S. Guerrero, and J.-P. Puel. Local exact controllability for the 1-D compressible Navier Stokes equation. Arch. Ration. Mech. Anal., to appear. | MR

[7] E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov, and J.-P. Puel. Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. (9), 83(12):1501–1542, 2004. | MR

[8] A. V. Fursikov and O. Y. Imanuvilov. Controllability of evolution equations, volume 34 of Lecture Notes Series. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996. | MR | Zbl

[9] O. Glass. Exact boundary controllability of 3-D Euler equation. ESAIM Control Optim. Calc. Var., 5:1–44 (electronic), 2000. | Numdam | MR | Zbl

[10] O. Glass. On the controllability of the 1-D isentropic Euler equation. J. Eur. Math. Soc. (JEMS), 9(3):427–486, 2007. | MR | Zbl

[11] M. González-Burgos, S. Guerrero, and J.-P. Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Commun. Pure Appl. Anal., 8(1):311–333, 2009. | MR | Zbl

[12] S. Guerrero and O. Y. Imanuvilov. Remarks on global controllability for the Burgers equation with two control forces. Ann. Inst. H. Poincaré Anal. Non Linéaire, 24(6):897–906, 2007. | Numdam | MR | Zbl

[13] O. Y. Imanuvilov. Remarks on exact controllability for the Navier-Stokes equations. ESAIM Control Optim. Calc. Var., 6:39–72 (electronic), 2001. | Numdam | MR | Zbl

[14] O. Y. Imanuvilov and J.-P. Puel. On global controllability of 2-D Burgers equation. Discrete Contin. Dyn. Syst., 23(1-2):299–313, 2009. | MR | Zbl

[15] T.-T. Li and B.-P. Rao. Exact boundary controllability for quasi-linear hyperbolic systems. SIAM J. Control Optim., 41(6):1748–1755 (electronic), 2003. | MR | Zbl

[16] P. Martin, L. Rosier, and P. Rouchon. Null-controllability of a structurally damped wave equation with moving point control. Submitted, 2012.

[17] A. Matsumura and T. Nishida. The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ., 20(1):67–104, 1980. | MR | Zbl

[18] H. Nersisyan. Controllability of the 3D compressible Euler system. Comm. Partial Differential Equations, 36(9):1544–1564, 2011. | MR | Zbl

[19] L. Rosier and P. Rouchon. On the controllability of a wave equation with structural damping. Int. J. Tomogr. Stat., 5(W07):79–84, 2007. | MR

Cité par Sources :