Skip to main content
SpringerLink
Log in

Some Questions of Control in Fluid Mechanics

  • Chapter
  • First Online:
Control of Partial Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2048))

Abstract

The goal of these lecture notes is to present some techniques of non-linear control of PDEs, in the context of fluid mechanics. We will consider the problem of controllability of two different models, namely the Euler equation for perfect incompressible fluids, and the one-dimensional isentropic Euler equation for compressible fluids. The standard techniques used to deal with the Cauchy problem for these two models are of rather different nature, despite the fact that the models are close. As we will see, this difference will also appear when constructing solutions of the controllability problem; however a common technique (or point of view) will be used in both cases. This technique, introduced by J.-M. Coron as the return method, is a way to exploit the nonlinearity of the equation for control purposes. Hence we will see its application in two rather different types of PDEs. The plan of these notes is the following. In a first part, we recall in a very basic way some types of questions that can be raised in PDE control (in a non-exhaustive way). In a second part, we expose results concerning the controllability of the incompressible Euler equation. In a third part, we show how the techniques used to prove the controllability of the incompressible Euler equation can be used to prove some other controllability properties for this equation, namely the so-called Lagrangian controllability. In a fourth and last part, we consider the controllability of the isentropic Euler equation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. A.A. Agrachev, A.V. Sarychev, Navier–Stokes equations: Controllability by means of low modes forcing. J. Math. Fluid Mech. 7(1), 108–152 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. A.A. Agrachev, A.V. Sarychev, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing. Comm. Math. Phys. 265(3), 673–697 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. H.D. Alber, Local existence of weak solutions to the quasilinear wave equation for large initial values. Math. Z. 190(2), 249–276 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws. NoDEA Nonlinear Differ. Equat. Appl. 4(1), 1–42 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Amadori, R.M. Colombo, Continuous dependence for 2 ×2 conservation laws with boundary. J. Differ. Equat. 138(2), 229–266 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Ancona, G.M. Coclite, On the attainable set for Temple class systems with boundary controls. SIAM J. Contr. Optim. 43(6), 2166–2190 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Ancona, A. Marson, On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Contr. Optim. 36(1), 290–312 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. F. Ancona, A. Marson, Asymptotic stabilization of systems of conservation laws by controls acting at a single boundary point. Control methods in PDE-dynamical systems. Contemp. Math. 426, 1–43 (2007)

    Article  MathSciNet  Google Scholar 

  9. F. Ancona, A. Marson, Existence theory by front tracking for general nonlinear hyperbolic systems. Arch. Ration. Mech. Anal. 185, 287–340 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. A.L. Bertozzi, P. Constantin, Global regularity for vortex patches. Comm. Math. Phys. 152(1), 19–28 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Bianchini, A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161(1), 223–342 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Bressan, Global solutions of systems of conservation laws by wave front tracking. J. Math. Anal. Appl. 170, 414–432 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Bressan, Hyperbolic Systems of Conservation Laws, the One-Dimensional Problem, Oxford Lecture Series in Mathematics and its Applications 20, (Oxford University Press, Oxford, 2000)

    Google Scholar 

  14. A. Bressan, G.M. Coclite, On the boundary control of systems of conservation laws. SIAM J. Contr. Optim. 41(2), 607–622 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Bressan, R.M. Colombo, Decay of positive waves in nonlinear systems of conservation laws. Ann. Sc. Norm. Sup. Pisa IV-26, 133–160 (1998)

    Google Scholar 

  16. A. Bressan, R.M. Colombo, Unique solutions of 2 ×2 conservation laws with large data. Indiana Univ. Math. J. 44(3), 677–725 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Chapouly, Global controllability of nonviscous and viscous Burgers-type equations. SIAM J. Contr. Optim. 48(3), 1567–1599 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Chapouly, On the global null controllability of a Navier–Stokes system with Navier slip boundary conditions. J. Differ. Equat. 247(7), 2094–2123 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. Ann. Sci. École Norm. Sup. 26(4), 517–542 (1993)

    MathSciNet  MATH  Google Scholar 

  20. J.-Y. Chemin, Fluides Parfaits Incompressibles, (Astérisque 230, 1995).

    Google Scholar 

  21. I-L. Chern, Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities. Comm. Pure Appl. Math. 42, 815–844 (1989)

    Google Scholar 

  22. A. Corli, M. Sablé-Tougeron, Perturbations of bounded variation of a strong shock wave. J. Differ. Equat. 138(2), 195–228 (1997)

    Article  MATH  Google Scholar 

  23. J.-M. Coron, Global Asymptotic Stabilization for controllable systems without drift. Math. Contr. Signal Syst. 5, 295–312 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. J.-M. Coron, Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels. C. R. Acad. Sci. Paris Sér. I Math. 317(3), 271–276 (1993)

    MathSciNet  MATH  Google Scholar 

  25. J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. 75, 155–188 (1996)

    MathSciNet  MATH  Google Scholar 

  26. J.-M. Coron, Sur la stabilization des fluides parfaits incompressibles bidimensionnels, séminaire : équations aux dérivées partielles, 1998–1999, Exp. No. VII, (École Polytechnique, Palaiseau, 1999)

    Google Scholar 

  27. J.-M. Coron, On the null asymptotic stabilization of 2-D incompressible Euler equation in a simply connected domain. SIAM J. Control Optim. 37(6), 1874–1896 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. J.-M. Coron, On the controllability of the 2-D incompressible Navier–Stokes equations with the Navier slip boundary conditions. ESAIM Contrôle Optim. Calc. Var. 1, 35–75 (1995/1996)

    Google Scholar 

  29. J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs 136, (American Mathematical Society, Providence, RI, 2007)

    Google Scholar 

  30. J.-M. Coron, A.V. Fursikov, Global exact controllability of the 2D Navier–Stokes equations on a manifold without boundary. Russ. J. Math. Phys. 4(4), 429–448 (1996)

    MathSciNet  MATH  Google Scholar 

  31. C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38, 33–41 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  32. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol 325 Grundlehren Math. Wissenschaften Series, (Springer, Berlin, 2000)

    Google Scholar 

  33. R. Danchin, Évolution d’une singularité de type cusp dans une poche de tourbillon. Rev. Mat. Iberoamericana 16(2), 281–329 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. N. Depauw, Poche de tourbillon pour Euler 2D dans un ouvert à bord. J. Math. Pures Appl. 78(3), 313–351 (1999)

    MathSciNet  MATH  Google Scholar 

  35. R.J. DiPerna, Global solutions to a class of nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 26, 1–28 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  36. R.J. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws. J. Differ. Equat. 20(1), 187–212 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  37. F. Dubois, P.G. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differ. Equat. 71(1), 93–122 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Dutrifoy, On 3-D vortex patches in bounded domains. Comm. Part. Differ. Equat. 28(7–8), 1237–1263 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Ebin, J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92, 102–163 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  40. E. Fernàndez-Cara, S. Guerrero, O.Y. Imanuvilov, J.-P. Puel, Some controllability results for the N-dimensional Navier–Stokes and Boussinesq systems with N − 1 scalar controls. SIAM J. Contr. Optim. 45(1), 146–173 (2006)

    Article  MATH  Google Scholar 

  41. A.V. Fursikov, O.Y. Imanuvilov, Exact local controllability of two-dimensional Navier–Stokes equations. Russ. Mat. Sb. 187(9), 103–138 (1996); translation in Sb. Math. 187(9), 1355–1390 (1996)

    Google Scholar 

  42. P. Gamblin, X. Saint Raymond, On three-dimensional vortex patches. Bull. Soc. Math. France 123(3), 375–424 (1995)

    MathSciNet  MATH  Google Scholar 

  43. S.J. Gardiner, Harmonic approximation, London Mathematical Society Lecture Note Series, 221 (Cambridge University Press, Cambridge, 1995)

    Google Scholar 

  44. O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM Contr. Optim. Calc. Var. 5, 1–44 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. O. Glass, An addendum to a J.-M. Coron theorem concerning the controllability of the Euler system for 2D incompressible inviscid fluids. J. Math. Pures Appl. 80(8), 845–877 (2001)

    Google Scholar 

  46. O. Glass, Asymptotic stabilizability by stationary feedback of the two-dimensional Euler equation: The multiconnected case. SIAM J. Contr. Optim. 44(3), 1105–1147 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  47. O. Glass, On the controllability of the 1-D isentropic Euler equation. J. Eur. Math. Soc. 9(3), 427–486 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  48. O. Glass, T. Horsin, Approximate Lagrangian controllability for the 2-D Euler equation. Application to the control of the shape of vortex patches. J. Math. Pures Appl. 93(1), 61–90 (2010)

    MathSciNet  MATH  Google Scholar 

  49. O. Glass, P.G. LeFloch, Nonlinear Hyperbolic Systems: Nondegenerate Flux, Inner Speed Variation, and Graph Solutions. Arch. Ration. Mech. Anal. 185, 409–480 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18, 697–715 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Glimm, P.D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws. Amer. Math. Soc., Providence, RI. 101, (1970)

    Google Scholar 

  52. H. Holden, N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Applied Mathematical Sciences, 152, (Springer, New York, 2002)

    Google Scholar 

  53. T. Horsin, On the controllability of the Burgers equation. ESAIM: Control Opt. Calc. Var. 3, 83–95 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  54. T. Horsin, Local exact Lagrangian controllability of the Burgers viscous equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(2), 219–230 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  55. O.Y. Imanuvilov, Remarks on exact controllability for the Navier–Stokes equations. ESAIM Control Optim. Calc. Var. 6, 39–72 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  56. T. Kato, Nonstationary flows of viscous and ideal fluids in R 3. J. Funct. Anal. 9, 296–305 (1972)

    Article  MATH  Google Scholar 

  57. T. Kato, On the Smoothness of Trajectories in Incompressible Perfect Fluids, Nonlinear wave equations, (Providence, RI, 1998); Contemp. Math. 263, 109–130. (2000), Amer. Math. Soc., Providence, RI.

    Google Scholar 

  58. A.V. Kazhikov, Note on the formulation of the problem of flow through a bounded region using equations of perfect fluid. PMM USSR, 44, 672–674 (1981)

    Google Scholar 

  59. S.G. Krantz, H.R. Parks, A primer of Real Analytic Functions, 2nd edn. Basler Lehrbücher, (Birkhäuser, Basel, 1992)

    Google Scholar 

  60. P.D. Lax, Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10, 537–566 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  61. P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Lectures in Mathematics, (ETH Zürich, Birkäuser, 2002)

    Book  MATH  Google Scholar 

  62. M. Lewicka, K. Trivisa, On the L 1 well posedness of systems of conservation laws near solutions containing two large shocks. J. Differ. Equat. 179(1), 133–177 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  63. T.-T. Li, B.-P. Rao, Exact boundary controllability for quasi-linear hyperbolic systems. SIAM J. Contr. Optim. 41(6), 1748–1755 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  64. T.-T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Series on Applied Mathematics, 3. American Institute of Mathematical Sciences (AIMS), (Springfield, MO, Higher Education Press, Beijing, 2010)

    Google Scholar 

  65. L. Lichtenstein, Über einige existenzprobleme der hydrodynamic. Math. Z. 28(1), 387–415 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  66. J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tomes 1 & 2. (Masson, RMA 8 & 9, Paris, 1988)

    Google Scholar 

  67. J.-L. Lions, in Are There Connections Between Turbulence and Controllability?, Analysis and Optimization of Systems, ed. by A. Bensoussan, J.-L. Lions (Springer, Berlin, 1990); Lecture Notes Control and Inform. Sci. 144

    Google Scholar 

  68. P.-L. Lions, B. Perthame, P.E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math. 49(6), 599–638 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  69. P.-L. Lions, B. Perthame, E. Tadmor, Existence and stability of entropy solutions to isentropic gas dynamics in Eulerian and Lagrangian variables. Comm. Math. Phys. 163, 415–431 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  70. A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, (Springer, New York, 1984)

    Book  MATH  Google Scholar 

  71. C.B. Morrey, Multiple Integrals in the Calculus of Variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, (Springer, New York, 1966)

    Google Scholar 

  72. H. Nersisyan, Controllability of the 3D compressible Euler system. Comm. Partial Differential Equations 36(9), 1544–1564 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  73. V. Perrollaz, Exact controllability of scalar conservation laws with an additional control in the context of entropy solutions, Preprint (2010)

    Google Scholar 

  74. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften, 299. (Springer, Berlin, 1992)

    Google Scholar 

  75. N.H. Risebro, A front-tracking alternative to the random choice method. Proc. Amer. Math. Soc. 117(4), 1125–1139 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  76. D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions. SIAM Rev. 20, 639–739 (1978)

    MATH  Google Scholar 

  77. S. Schochet, Sufficient conditions for local existence via Glimm’s scheme for large BV data.J. Differ. Equat. 89(2), 317–354 (1991)

    Google Scholar 

  78. P. Serfati, Une preuve directe d’existence globale des vortex patches 2D. C. R. Acad. Sci. Paris Sér. I Math. 318(6), 515–518 (1994)

    MathSciNet  MATH  Google Scholar 

  79. D. Serre, Systèmes de lois de conservation. I. Hyperbolicité, entropies, ondes de choc, & II. Structures géométriques, oscillation et problèmes mixtes. Fondations. (Diderot Editeur, Paris, 1996)

    Google Scholar 

  80. A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations. Comm. Math. Phys. 266(1), 123–151 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  81. F. Sueur, Vorticity internal transition layers for the Navier–Stokes equations, preprint, arXiv:0812.2145 (2008)

    Google Scholar 

  82. J. L. Walsh, The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions. Bull. Amer. Math. Soc. 35 (1929), no. 4, 499-544

    Article  MathSciNet  MATH  Google Scholar 

  83. H. Whitney, Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36(1), 63–89 (1934)

    Article  MathSciNet  Google Scholar 

  84. H. Whitney, The imbedding of manifolds in families of analytic manifolds. Ann. of Math. (2) 37(4), 865–878 (1936)

    Google Scholar 

  85. W. Wolibner, Un théorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long. Math. Z. 37, 698–726 (1933)

    Article  MathSciNet  Google Scholar 

  86. V.I. Yudovich, Non-stationary flows of an ideal incompressible fluid. Z̆. Vy čisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963); (Russian). English translation in USSR Comput. Math. Math. Physics 3, 1407–1456 (1963)

    Google Scholar 

  87. V.I. Yudovich, The flow of a perfect, incompressible liquid through a given region. Dokl. Akad. Nauk SSSR 146, 561–564 (1962); (Russian). English translation in Soviet Physics Dokl. 7, 789–791 (1962)

    Google Scholar 

  88. E. Zuazua, Exact controllability for the semilinear wave equation. J. Math. Pures Appl. (9) 69(1), 1–31 (1990)

    Google Scholar 

Download references

Acknowledgements

The author is very thankful to Professors P. Cannarsa and J.-M. Coron, as well as to the CIME staff, for organizing this wonderful summer school in Cetraro.

Author information

Authors and Affiliations

  1. Ceremade, CNRS UMR 7534, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Olivier Glass

Authors
  1. Olivier Glass
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olivier Glass .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Glass, O. (2012). Some Questions of Control in Fluid Mechanics. In: Control of Partial Differential Equations. Lecture Notes in Mathematics(), vol 2048. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27893-8_3

Download citation

Publish with us

Policies and ethics

Search

Navigation