Journal of Evolution Equations

, Volume 9, Issue 2, pp 267–291 | Cite as

A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems

  • Farid Ammar-Khodja
  • Assia Benabdallah
  • Cédric Dupaix
  • Manuel González-Burgos


We present a generalization of the Kalman rank condition to the case of n × n linear parabolic systems with constant coefficients and diagonalizable diffusion matrix. To reach the result, we are led to prove a global Carleman estimate for the solutions of a scalar 2n-order parabolic equation and deduce from it an observability inequality for our adjoint system.


Kalman Condition Control Observability Carleman estimates Reaction diffusion systems 

Mathematical Subject Classification (2000).

3B07 35K05 35K55 35R30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Ammar-Khodja, A. Benabdallah and C. Dupaix, Null controllability of some reaction-diffusion systems with one control force J. Math.Anal.Appl.320 (2006), no. 2, 928–943Google Scholar
  2. 2.
    F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. Gonzá lez-Burgos, Controllability for a class of reaction-diffusion systems: the generalized Kalman’s condition, C. R.Acad.Sci. Paris, Ser. I 345 (2007), no. 10, 543–548Google Scholar
  3. 3.
    F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. Gonzá lez-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems in preparation.Google Scholar
  4. 4.
    F. Ammar-Khodja, A. Benabdallah, C. Dupaix and I. Kostine, Null controllability of some systems of parabolic type by one control force ESAIM Control Optim.Calc.Var.11 (2005), no. 3, 426–448.Google Scholar
  5. 5.
    A. Benabdallah, Y. Dermenjian, J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math.Anal.Appl. 336 (2007), no. 2, 865–887.Google Scholar
  6. 6.
    A. Doubova, A. Osses, and J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim.Calc.Var. 8 (2002), 621–661.Google Scholar
  7. 7.
    E. Fernández-Cara, M. Gonzá lez-Burgos, L. de Teresa, Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations, Commun.Pure Appl.Anal. 5 (2006), no. 3, 637–656.Google Scholar
  8. 8.
    E. Fernández-Cara, M. Gonzá lez-Burgos, L. de Teresa, About boundary controllability of cascade heat equations, In preparation.Google Scholar
  9. 9.
    E. Fernández-Cara, L. de Teresa, Null controllability of a cascade system of parabolic- hyperbolic equations, Discrete Contin.Dyn. Syst. 11 (2004), no. 2–3, 699–714.Google Scholar
  10. 10.
    E. Fernández-Cara, E. Zuazua, The cost of approximate controllability for heat equations: the linear case, Adv. Differential Equations 5 (2000), no. 4–6, 465–514.Google Scholar
  11. 11.
    A. Fursikov, O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.Google Scholar
  12. 12.
    M. González-Burgos, R. Pérez-García, M. González-Burgos, R. Pérez-Garcia, Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot.Anal.46 (2006), no. 2, 123–162.Google Scholar
  13. 13.
    M. González-Burgos, L. de Teresa Controllability results for cascade systems of m coupled parabolic PDEs by one control force, submitted to Systems & Control Letters.Google Scholar
  14. 14.
    S. Guerrero, Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim. 46, (2007), no. 2, 379–394.Google Scholar
  15. 15.
    E. L. Ince., Ordinary Differential Equations, Dover Publications, New York, 1944.Google Scholar
  16. 16.
    O. Yu. Imanuvilov, M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ.Res.Inst.Math.Sci. 39 (2003), no. 2, 227–274.Google Scholar
  17. 17.
    R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical Control Theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London 1969.Google Scholar
  18. 18.
    G. Lebeau, L. Robbiano, Contrôle exact de l’équation de la chaleur, Comm.Partial Differential Equations 20 (1995), no. 1–2, 335–356.Google Scholar
  19. 19.
    H. Leiva, Controllability of a system of parabolic equations with non-diagonal diffusion matrix, IMA J. Math.Control Inform. 22 (2005), no. 2, 187–199.Google Scholar
  20. 20.
    J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients, J. Differential Equations 233 (2007), no. 2, 417–447.Google Scholar
  21. 21.
    A. Pazy, Semigroups of linear operators and application to partial differential equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.Google Scholar
  22. 22.
    L. de Teresa, Insensitizing controls for a semilinear heat equation, Comm.Partial Differential Equations 25 (2000), no. 1–2, 39–72.Google Scholar
  23. 23.
    R. Triggiani, Constructive steering control functions for linear systems and abstract rank conditions, J. Optim.Theory Appl. 74 (1992), no. 2, 347–367.Google Scholar
  24. 24.
    J. Zabczyk, Mathematical Control Theory: An Introduction, Systems & Control: Foundations & Applications, Birkhäuser, Boston, 1992.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Farid Ammar-Khodja
    • 1
  • Assia Benabdallah
    • 2
  • Cédric Dupaix
    • 3
  • Manuel González-Burgos
    • 4
  1. 1.Laboratoire de Mathématiques UMR 6623Université de Franche-ComtéBesançon cedexFrance
  2. 2.CMI-LATP, UMR 6632Université de Provence, Technopôle Château-GombertMarseille cedex 13France
  3. 3.Laboratoire de Mathématiques UMR 6623Université de Franche-ComtéBesançon cedexFrance
  4. 4.Dpto, E.D.A.NUniversidad de Sevilla, Aptdo. 1160SevillaSpain

Personalised recommendations