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Local controllability of a 1-D tank containing a fluid modeledby the shallow water equations

Published online by Cambridge University Press:  15 August 2002

Jean-Michel Coron
Jean-Michel Coron*
Affiliation:
Université Paris-Sud, Département de Mathématique, bâtiment 425, 91405 Orsay, France; Jean-Michel.Coron@math.u-psud.fr.
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Abstract

We consider a 1-D tank containing an inviscid incompressibleirrotational fluid. The tank is subject to the control which consistsof horizontal moves. We assume that the motion of the fluid is well-described by the Saint–Venant equations (alsocalled theshallow water equations). We prove the localcontrollability 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.

Keywords

Controllabilityhyperbolic systemsshallow water.

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Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 513 - 554
Copyright
© EDP Sciences, SMAI, 2002

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References

Coron, J.-M., Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems 5 (1992) 295-312. CrossRef
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.
Coron, J.-M., On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155-188.
Coron, J.-M., On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1996) 35-75. CrossRef
Coron, J.-M. and Fursikov, A., Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4 (1996) 429-448.
R. Courant and D. Hilbert, Methods of mathematical physics, II. Interscience publishers, John Wiley & Sons, New York London Sydney (1962).
L. Debnath, Nonlinear water waves. Academic Press, San Diego (1994).
F. Dubois, N. Petit and P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid, ECC 99.
Fursikov, A.V. and Imanuvilov, O.Yu., Exact controllability of the Navier-Stokes and Boussinesq equations. Russian Math. Surveys . 54 (1999) 565-618. CrossRef
Glass, O., 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. CrossRef
Glass, O., Exact boundary controllability of 3-D Euler equation. ESAIM: COCV 5 (2000) 1-44. CrossRef
L. Hörmander, Lectures on nonlinear hyperbolic differential equations. Springer-Verlag, Berlin Heidelberg, Math. Appl. 26 (1997).
Th. Horsin, On the controllability of the Burgers equation. ESAIM: COCV 3 (1998) 83-95. CrossRef
J.-L. Lions, Are there connections between turbulence and controllability?, in 9th INRIA International Conference. Antibes (1990).
Lions, J.-L., 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.
J.-L. Lions, On the controllability of distributed systems. Proc. Natl. Acad. Sci. USA 94 (1997) 4828-4835.
Lions, J.-L. and Zuazua, E., Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV 1 (1995) 1-15.
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.
Li Ta Tsien and Yu Wen-Ci, Boundary value problems for quasilinear hyperbolic systems. Duke university, Durham, Math. Ser. V (1985).
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).
N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems. Preprint, CIT-CDS 00-004.
de Saint-Venant, A.J.C.B., 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.
D. Serre, Systèmes de lois de conservations, I et II. Diderot Éditeur, Arts et Sciences, Paris, New York, Amsterdam (1996).
Sontag, E.D., Control of systems without drift via generic loops. IEEE Trans. Automat. Control . 40 (1995) 1210-1219. CrossRef