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

ESAIM: Control, Optimisation and Calculus of Variations (2002)

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

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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 (2002): 513-554. <http://eudml.org/doc/244897>.

@article{Coron2002,

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},

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/244897},

volume = {8},

year = {2002},

}

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

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/244897

ER -

## References

top- [1] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems 5 (1992) 295-312. Zbl0760.93067MR1164379
- [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. Zbl0781.76013
- [3] J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155-188. Zbl0848.76013MR1380673
- [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. Zbl0872.93040
- [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. Zbl0938.93030
- [6] R. Courant and D. Hilbert, Methods of mathematical physics, II. Interscience publishers, John Wiley & Sons, New York London Sydney (1962). Zbl0099.29504MR140802
- [7] L. Debnath, Nonlinear water waves. Academic Press, San Diego (1994). Zbl0793.76001MR1266390
- [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. Zbl0970.35116
- [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. Zbl0897.76014
- [11] O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV 5 (2000) 1-44. Zbl0940.93012MR1745685
- [12] L. Hörmander, Lectures on nonlinear hyperbolic differential equations. Springer-Verlag, Berlin Heidelberg, Math. Appl. 26 (1997). Zbl0881.35001MR1466700
- [13] Th. Horsin, On the controllability of the Burgers equation. ESAIM: COCV 3 (1998) 83-95. Zbl0897.93034MR1612027
- [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. Zbl0735.93006MR1147191
- [16] J.-L. Lions, On the controllability of distributed systems. Proc. Natl. Acad. Sci. USA 94 (1997) 4828-4835. Zbl0876.93044MR1453828
- [17] J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV 1 (1995) 1-15. Zbl0878.93034MR1382513
- [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. Zbl1053.93009
- [19] Li Ta Tsien and Yu Wen–Ci, Boundary value problems for quasilinear hyperbolic systems. Duke university, Durham, Math. Ser. V (1985). Zbl0627.35001
- [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). Zbl0537.76001MR748308
- [21] N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems. Preprint, CIT-CDS 00-004. Zbl0967.93073MR1893517
- [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. Zbl03.0482.04JFM03.0482.04
- [23] D. Serre, Systèmes de lois de conservations, I et II. Diderot Éditeur, Arts et Sciences, Paris, New York, Amsterdam (1996). MR1459988
- [24] E.D. Sontag, Control of systems without drift via generic loops. IEEE Trans. Automat. Control. 40 (1995) 1210-1219. Zbl0837.93019MR1344033

## Citations in EuDML Documents

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- Lionel Rosier, Control of the surface of a fluid by a wavemaker
- Olivier Glass, A controllability result for the $1$-D isentropic Euler equation
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- Eduardo Cerpa, Emmanuelle Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain
- Karine Beauchard, Controllablity of a quantum particle in a 1D variable domain
- Karine Beauchard, Controllability of a quantum particle in a 1D variable domain
- Thierry Horsin, Local exact lagrangian controllability of the Burgers viscous equation

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