Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T15:20:55.914Z Has data issue: false hasContentIssue false

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

Abstract

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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