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

Jean-Michel Coron

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

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

Abstract

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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.

How to cite

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Coron, 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

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  1. J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems5 (1992) 295-312.  Zbl0760.93067
  2. J.-M. Coron, Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels. C. R. Acad. Sci. Paris317 (1993) 271-276.  
  3. J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl.75 (1996) 155-188.  Zbl0848.76013
  4. 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
  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).  Zbl0099.29504
  7. L. Debnath, Nonlinear water waves. Academic Press, San Diego (1994).  Zbl0793.76001
  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.  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. I325 (1997) 987-992.  
  11. O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV5 (2000) 1-44.  Zbl0940.93012
  12. L. Hörmander, Lectures on nonlinear hyperbolic differential equations. Springer-Verlag, Berlin Heidelberg, Math. Appl.26 (1997).  Zbl0881.35001
  13. Th. Horsin, On the controllability of the Burgers equation. ESAIM: COCV3 (1998) 83-95.  Zbl0897.93034
  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.  
  16. J.-L. Lions, On the controllability of distributed systems. Proc. Natl. Acad. Sci. USA94 (1997) 4828-4835.  Zbl0876.93044
  17. J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV1 (1995) 1-15.  Zbl0878.93034
  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.  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).  Zbl0537.76001
  21. N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems. Preprint, CIT-CDS 00-004.  Zbl0967.93073
  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. Paris53 (1871) 147-154.  Zbl03.0482.04
  23. D. Serre, Systèmes de lois de conservations, I et II. Diderot Éditeur, Arts et Sciences, Paris, New York, Amsterdam (1996).  
  24. E.D. Sontag, Control of systems without drift via generic loops. IEEE Trans. Automat. Control. 40 (1995) 1210-1219.  Zbl0837.93019

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