# Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 8, page 513-554
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topCoron, 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 8 (2010): 513-554. <http://eudml.org/doc/90659>.

@article{Coron2010,

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.
},

author = {Coron, Jean-Michel},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Controllability; hyperbolic systems; shallow water.; Saint-Venant equations},

language = {eng},

month = {3},

pages = {513-554},

publisher = {EDP Sciences},

title = {Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations},

url = {http://eudml.org/doc/90659},

volume = {8},

year = {2010},

}

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

DA - 2010/3//

PB - EDP Sciences

VL - 8

SP - 513

EP - 554

AB -
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.

LA - eng

KW - Controllability; hyperbolic systems; shallow water.; Saint-Venant equations

UR - http://eudml.org/doc/90659

ER -

## References

top- J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems5 (1992) 295-312. Zbl0760.93067
- J.-M. Coron, Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels. C. R. Acad. Sci. Paris317 (1993) 271-276.
- J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl.75 (1996) 155-188. Zbl0848.76013
- J.-M. Coron, On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM: COCV1 (1996) 35-75. Zbl0872.93040
- 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. Zbl0938.93030
- R. Courant and D. Hilbert, Methods of mathematical physics, II. Interscience publishers, John Wiley & Sons, New York London Sydney (1962). Zbl0099.29504
- L. Debnath, Nonlinear water waves. Academic Press, San Diego (1994). Zbl0793.76001
- F. Dubois, N. Petit and P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid, ECC 99.
- A.V. Fursikov and O.Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations. Russian Math. Surveys. 54 (1999) 565-618. Zbl0970.35116
- O. Glass, Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles en dimension 3. C. R. Acad. Sci. Paris Sér. I325 (1997) 987-992.
- O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV5 (2000) 1-44. Zbl0940.93012
- L. Hörmander, Lectures on nonlinear hyperbolic differential equations. Springer-Verlag, Berlin Heidelberg, Math. Appl.26 (1997). Zbl0881.35001
- Th. Horsin, On the controllability of the Burgers equation. ESAIM: COCV3 (1998) 83-95. Zbl0897.93034
- J.-L. Lions, Are there connections between turbulence and controllability?, in 9th INRIA International Conference. Antibes (1990).
- 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.
- J.-L. Lions, On the controllability of distributed systems. Proc. Natl. Acad. Sci. USA94 (1997) 4828-4835. Zbl0876.93044
- J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV1 (1995) 1-15. Zbl0878.93034
- 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. Zbl1053.93009
- Li Ta Tsien and Yu Wen-Ci, Boundary value problems for quasilinear hyperbolic systems. Duke university, Durham, Math. Ser.V (1985). Zbl0627.35001
- 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). Zbl0537.76001
- N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems. Preprint, CIT-CDS 00-004. Zbl0967.93073
- 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. Paris53 (1871) 147-154. Zbl03.0482.04
- D. Serre, Systèmes de lois de conservations, I et II. Diderot Éditeur, Arts et Sciences, Paris, New York, Amsterdam (1996).
- E.D. Sontag, Control of systems without drift via generic loops. IEEE Trans. Automat. Control. 40 (1995) 1210-1219. Zbl0837.93019

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.