Local exact lagrangian controllability of the Burgers viscous equation

Horsin, Thierry

Annales de l'I.H.P. Analyse non linéaire, Tome 25 (2008) no. 2, pp. 219-230

[1] Ancona F., Marson A., On the attainable set for scalar nonlinear conservation laws with boundary control, SIAM J. Control Optim. 36 (1) (1998) 290-312. | Zbl | MR

[2] Boulakia M., Osses A., Two-dimensional local null controllability of a rigid structure in a Navier-Stokes fluid, C. R. Acad. Sci. Paris, Ser. I 343 (2) (2006) 105-109. | Zbl | MR

[3] H. Brezis, T. Cazenave, Semilinear Evolution Equations and Applications in Mechanics and Physics, Pitman Lecture Notes, Addison-Wesley, Reading MA, in press.

[4] Coddington E.A., Levinson N., Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York, 1955. | Zbl | MR

[5] Conca C., San Martin J., Tucsnak M., Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations 25 (5-6) (2000) 1019-1042. | Zbl | MR

[6] J.-M. Coron, Control and nonlinearity, in preparation. | Zbl

[7] Coron J.-M., Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels, C. R. Acad. Sci. Paris 317 (1993) 271-276. | Zbl | MR

[8] Coron J.-M., On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Control Optim. Calc. Var. 1 (1995/1996) 35-75, (electronic). | Zbl | MR | Numdam

[9] Coron J.-M., Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations, ESAIM Control Optim. Calc. Var. 8 (2002) 513-554, A tribute to J.L. Lions. | Zbl | MR | Numdam

[10] Coron J.-M., Trélat E., Global steady-state controllability of one-dimensional semilinear heat equations, SIAM J. Control Optim. 43 (2) (2004) 549-569, (electronic). | Zbl | MR

[11] Desjardins B., Esteban M.J., On weak solutions for fluid-rigid structure interaction: compressible and incompressible models, Comm. Partial Differential Equations 25 (7-8) (2000) 1399-1413. | Zbl | MR

[12] Di Perna R., Lions P.-L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989) 511-547. | Zbl | MR

[13] Doubova A., Fernandez-Cara E., Some control results for simplified one-dimensional models of fluid-solid interaction, Math. Models Methods Appl. Sci. 15 (5) (2005) 783-824. | Zbl | MR

[14] Fabre C., Puel J.-P., Zuazua E., Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A 125 (1) (1995) 31-61. | Zbl | MR

[15] Fattorini H.O., Russell D.L., Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal. 43 (1971) 272-292. | Zbl | MR

[16] Fernández-Cara E., Guerrero S., Remarks on the controllability of the Burgers equation, C. R. Acad. Sci. Paris, Ser. I 341 (2005) 229-232. | Zbl | MR

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

[18] Fursikov A.V., Imanuvilov O.Yu., Controllability of Evolution Equations, Lecture Notes Series, vol. 34, Seoul National University Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. | Zbl | MR

[19] Godlewski E., Raviart P.-A., Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. | Zbl | MR

[20] S. Guerrero, O.Yu. Imanuvilov, Remarks on global controllability for the Burgers equation with two control forces (2006), submitted for publication. | MR | Numdam

[21] Hörmander L., Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin), vol. 26, Springer-Verlag, Berlin, 1997. | Zbl | MR

[22] Horsin Molinaro T., On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var. 3 (1998) 83-95, (electronic). | Zbl | MR | Numdam

[23] Horsin Molinaro T., Application of the exact null controllability of the heat equation to moving sets, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 849-852. | Zbl | MR

[24] Imanuvilov O.Yu., On exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var. 3 (1998) 97-131, (electronic). | Zbl | MR | Numdam

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

[26] Ju L., Gunzburger M.D., Hou L.S., Approximation of exact boundary controllability problems for the 1-D wave equation by optimization-based methods, in: Recent Advances in Scientific Computing and Partial Differential Equations, Hong Kong, 2002, Contemp. Math., vol. 330, Amer. Math. Soc., Providence, RI, 2003, pp. 133-153. | Zbl | MR

[27] Lebeau G., Robbiano L., Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations 20 (1-2) (1995) 335-356. | Zbl | MR

[28] H. Maillot, E. Zuazua, Mouvement d'une particule dans un fluide, Prepublication, 1999.

[29] Mirrahimi M., Rouchon P., Controllability of quantum harmonic oscillators, IEEE Trans. Automat. Control 49 (5) (2004) 745-747. | MR

[30] Nodet M., Variational assimilation of Lagrangian data in oceanography, Inverse Problems 22 (1) (2006) 245-263. | Zbl | MR

[31] A. Osses, Quelques méthodes théoriques et numériques de contrôlabilité et problèmes d'interactions fluide-structure, PhD thesis, Ecole Polytechnique, Paris, 1998.

[32] Sontag E.D., Deterministic finite-dimensional systems, in: Mathematical Control Theory, second ed., Texts in Applied Mathematics, vol. 6, Springer-Verlag, New York, 1998, pp. xvi+531. | Zbl | MR

[33] Vázquez J.L., Zuazua E., Lack of collision in a simplified 1D model for fluid-solid interaction, Math. Models Methods Appl. Sci. 16 (5) (2006) 637-678. | MR