Distributed Control and Observation

  • Claude Bardos
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 330)


Over the recent years, mostly under the impetus of the late JL Lions, important progress have been made for the control of distributed systems. This has contributed to the understanding of the duality which exist between the modal analysis and the need of very localized actuators. This duality leads to the phenomena of overspilling (excitation of higher order modes). On the other hand this type of research is closely related to the analysis of the exterior problem for the acoustic equation which was, under the influence of P. Lax, one of the main stimulus for the development of microlocal analysis. In this contribution only the acoustic equation is studied. However most of the idea carry on to other linear equations and to some non linear equations when considered in a perturbative regime where linearization techniques can be used; this includes perturbations of given solution of the Navier Stokes equation following the work of Imanuvilov and Fursikov (cf [12]) for a recent reference. Same ideas are used also in identification problems for instance in the migration method for oil recovery.


Principal Symbol Semi Group Unique Continuation Exact Controllability Carleman Estimate 
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  1. [1]
    S. Alinhac and M. S. Baouendi, A non uniqueness result for operators of principal type, Math. Zeit. 220 (1995), No. 4, 561–568.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    G. Bal and L. Ryzhnik: Time reversal for classicla waves in random media C. R. Acad. Sci. Paris t. 332 (2001).Google Scholar
  3. [3]
    C. Bardos and M. Fink: Mathematical foundations of the time reversal mirror to appear in Asymptotic Analysis.Google Scholar
  4. [4]
    C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, Control and stabilization of waves from the boundary, SIAM Journal on Control Theory and Applications, 30 (1992), 1024–1065.zbMATHMathSciNetGoogle Scholar
  5. [5]
    N. Burq: Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel, Acta Math., 180, (1988) 1–29.MathSciNetCrossRefGoogle Scholar
  6. [6]
    N. Burq: Mesures de default Séminaire Bourbaki 1996–97 n 826 March 1997.Google Scholar
  7. [7]
    Y. Colin de Verdière: Ergodicité et fonctions propres du Laplacien. Comm. Math. Phys, 102, (1985) 497–502.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    C. Draeger, M. Fink: One-channel time-reversal in chaotic cavities: Theoretical limits, Journal of Acoustical Society of America, 105, 2, (1999), 611–617.CrossRefGoogle Scholar
  9. [9]
    C. Draeger and M. Fink: One-Channel Time Reversal of Elastic Waves in a Chaotic 2D-Silicon Cavity. Phys. Rew. Letters, 79, 3, (1997) 407–410.CrossRefGoogle Scholar
  10. [10]
    E. Holmgren: Uber Systeme von linearen partiellen Differentialgleichnungen. Ofversigt af Kongl. Vetenskaps-Akad. Forh, 58, (1901) 91–105.Google Scholar
  11. [11]
    Hörmander L., The analysis of linear partial differential operators I, II, III, IV, Grundlehren der Math.Wiss. 256,257,274,275, Springer-Verlag (1985).Google Scholar
  12. [12]
    O. Y. Imanuvilov: Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var. 6 (2001), 39–72 (electronic).zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    F. John, On linear partial differential equation with analytic coefficients, Commun. Pure Appl. Math. 2 (1949), 209–253.zbMATHCrossRefGoogle Scholar
  14. [14]
    P. Lax and R. Phillips: Scattering Theory, Academic Press (1989).Google Scholar
  15. [15]
    G. Lebeau: Contrôle analytique 1: estimations à priori, Duke Math. J. 68 (1992), 1–30.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    G. Lebeau et L. Robbiano: Stabilisation de l’équation des ondes par le bord, Duke Math. J. 86, No. 3 (1997) 465–491.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    J. Leray: Uniformisation de la solution du problème linéaire analytique de Cauchy près de la variété qui porte les données de Cauchy (problème de Cauchy I), Bull. Soc. Math. France 85 (1957) 389–429.zbMATHMathSciNetGoogle Scholar
  18. [18]
    R. Melrose and J. Sjöstrand, Singularities of boundary value problems, Comm. on Pure and Appl. Math. 31, 593 (1978).zbMATHCrossRefGoogle Scholar
  19. [19]
    J.V. Ralston, Solutions of the wave equation with localized energy, Commun. on Pure and Appl. Math., 22 (1969), 807–823.zbMATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    L. Robbiano and C. Zuily Robbiano: Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients. Invent. Math. 131 (1998), no. 3, 493–539.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    A.I Schnirelman: Ergodic properties of eigenfunctions. Usp. Math. Nauk, 29, (1994) 181–182.Google Scholar
  22. [22]
    D. Tataru: Unique continuation for operators with partially analytic coefficients. J. Math. Pures Appl. (9) 78 (1999), no. 5, 505–521.zbMATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    S. Zelditch., Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, (1987) 919–941.MathSciNetCrossRefGoogle Scholar
  24. [24]
    S. Zelditch., Quantum transition amplitudes for ergodic and for completely integrable systems. J. Funct. Anal., 94, (1990) 415–436.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    S. Zelditch: Quantum mixing J. Funct. Anal., 140, (1996) 68–86.zbMATHMathSciNetCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Claude Bardos
    • 1
  1. 1.University Denis Diderot and LANN University Pierre et Marie Curie ParisParis

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