Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 513-554.

We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to the control which consists of horizontal moves. We assume that the motion of the fluid is well-described by the Saint-Venant equations (also called the shallow water equations). We prove the local controllability of this nonlinear control system around any steady state. As a corollary we get that one can move from any steady state to any other steady state.

DOI : 10.1051/cocv:2002050
Classification : 76B75, 93B05, 76B15, 35F30
Mots clés : controllability, hyperbolic systems, shallow water 
@article{COCV_2002__8__513_0,
     author = {Coron, Jean-Michel},
     title = {Local controllability of a {1-D} tank containing a fluid modeled by the shallow water equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {513--554},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002050},
     mrnumber = {1932962},
     zbl = {1071.76012},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2002050/}
}
TY  - JOUR
AU  - Coron, Jean-Michel
TI  - Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2002
SP  - 513
EP  - 554
VL  - 8
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2002050/
DO  - 10.1051/cocv:2002050
LA  - en
ID  - COCV_2002__8__513_0
ER  - 
%0 Journal Article
%A Coron, Jean-Michel
%T Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2002
%P 513-554
%V 8
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2002050/
%R 10.1051/cocv:2002050
%G en
%F COCV_2002__8__513_0
Coron, Jean-Michel. Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 513-554. doi : 10.1051/cocv:2002050. http://www.numdam.org/articles/10.1051/cocv:2002050/

[1] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems 5 (1992) 295-312. | MR | Zbl

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

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

[4] J.-M. Coron, On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1996) 35-75. | Numdam | Zbl

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

[6] R. Courant and D. Hilbert, Methods of mathematical physics, II. Interscience publishers, John Wiley & Sons, New York London Sydney (1962). | MR | Zbl

[7] L. Debnath, Nonlinear water waves. Academic Press, San Diego (1994). | MR | Zbl

[8] F. Dubois, N. Petit and P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid, ECC 99.

[9] A.V. Fursikov and O.Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations. Russian Math. Surveys. 54 (1999) 565-618. | Zbl

[10] O. Glass, Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles en dimension 3. C. R. Acad. Sci. Paris Sér. I 325 (1997) 987-992. | Zbl

[11] O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV 5 (2000) 1-44. | Numdam | MR | Zbl

[12] L. Hörmander, Lectures on nonlinear hyperbolic differential equations. Springer-Verlag, Berlin Heidelberg, Math. Appl. 26 (1997). | MR | Zbl

[13] Th. Horsin, On the controllability of the Burgers equation. ESAIM: COCV 3 (1998) 83-95. | Numdam | MR | Zbl

[14] J.-L. Lions, Are there connections between turbulence and controllability?, in 9th INRIA International Conference. Antibes (1990).

[15] J.-L. Lions, Exact controllability for distributed systems. Some trends and some problems, in Applied and industrial mathematics, Proc. Symp., Venice/Italy 1989. D. Reidel Publ. Co. Math. Appl. 56 (1991) 59-84. | MR | Zbl

[16] J.-L. Lions, On the controllability of distributed systems. Proc. Natl. Acad. Sci. USA 94 (1997) 4828-4835. | MR | Zbl

[17] J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV 1 (1995) 1-15. | Numdam | MR | Zbl

[18] J.-L. Lions and E. Zuazua, Exact boundary controllability of Galerkin's approximations of Navier-Stokes equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV 26 (1998) 605-621. | Numdam | Zbl

[19] Li Ta Tsien and Yu Wen-Ci, Boundary value problems for quasilinear hyperbolic systems. Duke university, Durham, Math. Ser. V (1985). | Zbl

[20] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables. Sringer-Verlag, New York Berlin Heidelberg Tokyo, Appl. Math. Sci. 53 (1984). | MR | Zbl

[21] N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems. Preprint, CIT-CDS 00-004. | MR

[22] A.J.C.B. De Saint-Venant, Théorie du mouvement non permanent des eaux, avec applications aux crues des rivières et à l'introduction des marées dans leur lit. C. R. Acad. Sci. Paris 53 (1871) 147-154. | JFM

[23] D. Serre, Systèmes de lois de conservations, I et II. Diderot Éditeur, Arts et Sciences, Paris, New York, Amsterdam (1996). | MR

[24] E.D. Sontag, Control of systems without drift via generic loops. IEEE Trans. Automat. Control. 40 (1995) 1210-1219. | MR | Zbl

Cité par Sources :