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Coron, Jean-Michel

Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations. (English) Zbl 1071.76012

ESAIM, Control Optim. Calc. Var. 8, 513-554 (2002).
Summary: We consider a one-dimensional tank containing an inviscid incompressible irrotational fluid. The tank is subject to the control which consists of horizontal accelerations. We assume that the motion of the fluid is 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.

Cited in 31 Documents

MSC:

76B75 Flow control and optimization for incompressible inviscid fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
93C20 Control/observation systems governed by partial differential equations
35Q35 PDEs in connection with fluid mechanics
93B05 Controllability

Keywords:

hyperbolic systems; Saint-Venant equations
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\textit{J.-M. Coron}, ESAIM, Control Optim. Calc. Var. 8, 513--554 (2002; Zbl 1071.76012)
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References:

[1] J.-M. Coron , Global asymptotic stabilization for controllable systems without drift . Math. Control Signals Systems 5 ( 1992 ) 295 - 312 . MR 1164379 | Zbl 0760.93067 · Zbl 0760.93067
[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 0781.76013 · Zbl 0781.76013
[3] J.-M. Coron , On the controllability of 2-D incompressible perfect fluids . J. Math. Pures Appl. 75 ( 1996 ) 155 - 188 . MR 1380673 | Zbl 0848.76013 · Zbl 0848.76013
[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 0872.93040 · Zbl 0872.93040
[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 . Zbl 0938.93030 · Zbl 0938.93030
[6] R. Courant and D. Hilbert , Methods of mathematical physics, II . Interscience publishers, John Wiley & Sons, New York London Sydney ( 1962 ). MR 140802 | Zbl 0099.29504 · Zbl 0099.29504
[7] L. Debnath , Nonlinear water waves . Academic Press, San Diego ( 1994 ). MR 1266390 | Zbl 0793.76001 · Zbl 0793.76001
[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 0970.35116 · Zbl 0970.35116
[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 . Zbl 0897.76014 · Zbl 0897.76014
[11] O. Glass , Exact boundary controllability of 3-D Euler equation . ESAIM: COCV 5 ( 2000 ) 1 - 44 . Numdam | MR 1745685 | Zbl 0940.93012 · Zbl 0940.93012
[12] L. Hörmander , Lectures on nonlinear hyperbolic differential equations . Springer-Verlag, Berlin Heidelberg, Math. Appl. 26 ( 1997 ). MR 1466700 | Zbl 0881.35001 · Zbl 0881.35001
[13] Th. Horsin , On the controllability of the Burgers equation . ESAIM: COCV 3 ( 1998 ) 83 - 95 . Numdam | MR 1612027 | Zbl 0897.93034 · Zbl 0897.93034
[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 1147191 | Zbl 0735.93006 · Zbl 0735.93006
[16] J.-L. Lions , On the controllability of distributed systems . Proc. Natl. Acad. Sci. USA 94 ( 1997 ) 4828 - 4835 . MR 1453828 | Zbl 0876.93044 · Zbl 0876.93044
[17] J.-L. Lions and E. Zuazua , Approximate controllability of a hydro-elastic coupled system . ESAIM: COCV 1 ( 1995 ) 1 - 15 . Numdam | MR 1382513 | Zbl 0878.93034 · Zbl 0878.93034
[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 1053.93009 · Zbl 1053.93009
[19] Li Ta Tsien and Yu Wen-Ci, Boundary value problems for quasilinear hyperbolic systems. Duke university, Durham, Math. Ser. V (1985). Zbl 0627.35001 · Zbl 0627.35001
[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 ). MR 748308 | Zbl 0537.76001 · Zbl 0537.76001
[21] N. Petit and P. Rouchon , Dynamics and solutions to some control problems for water-tank systems . Preprint, CIT-CDS 00 - 004 . MR 1893517 · Zbl 0967.93073
[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 03.0482.04 · JFM 03.0482.04
[23] D. Serre , Systèmes de lois de conservations, I et II. Diderot Éditeur, Arts et Sciences, Paris, New York, Amsterdam ( 1996 ). MR 1459988
[24] E.D. Sontag , Control of systems without drift via generic loops . IEEE Trans. Automat. Control. 40 ( 1995 ) 1210 - 1219 . MR 1344033 | Zbl 0837.93019 · Zbl 0837.93019
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