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Identification de la Non-Linearité D'Une équation Parabolique Quasilineaire

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On étudie dans cet article le problème de la détermination de la nonlinéarité dans une équation parabolique à partir de mesures ponctuelles sur la solution. Ceci correspond à la situation courante en physique, dans laquelle un paramètre dépend de l'état du système (par exemple un coefficient de conductibilité thermique dépend de la température).

Après avoir posé le problème comme un problème de contrôle dont on montre l'éxistence d'une solution, on calcule la dérivée du critère par rapport à la non-linéarité sans hypothèse sur la forme analytique de cette dernière.

Enfin on donne une application numérique de l'algorithme obtenu, qui montre que la situation considérée ici (estimation d'un paramètre dépendant de l'état) est plus favorable, du point de vue identification, que la situation où l'on se place habituellement (estimation d'un paramètre dépendant des variables d'espace et/ou de temps).

Abstract

In this work, we study the problem of the determination of the non-linearity in a parabolic equation from measurements over its solution. This corresponds to an usual physical situation, in which the parameter depends on the state of the system (for instance, the heat conduction coefficient depends on the temperature).

We first state the problem as a control problem, for which we show the existence of a solution, and we calculate the derivative of the criterion with respect to the nonlinearity without any assumption on the algebraic form of this latter.

We finally give a numerical application of the algorithm, which shows that the situation which is here taken in account (estimation of a parameter function depending on the state) is much better, from the identification point of view, than the usually studied situation for distributed system (estimation of a parameter function depending on space and/or time variables).

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References

  1. J. C. Bruch, Non linear equation of unsteady groundwater flow,Jour. of the Hydraulics Division 9589, March 1973, HY3.

  2. J. R. Cannon andP. Duchateau, Determining unknown coefficients in a nonlinear heat conduction problem,SIAM J. Appl. Math. 24, No° 3, May 1973.

  3. J. Cea,Optimization, Dunod, Paris, 1971.

    Google Scholar 

  4. G. Chavent,Sur une méthode de résolution du problème inverse dans les équations aux dérivées partielles paraboliques, Note CRAS, Paris, t. 260, Décembre 1969.

  5. G. Chavent,Analyse fonctionnelle et identification de coefficients répartis dans les équations aux dérivées partielles, Thèse, Paris, 1971.

    Google Scholar 

  6. G. Chavent, Identification of distributed parameters,Third IFAC Symposium, The Hague/Delft, The Netherlands, Juin 1973.

  7. G. Chavent, M. Dupuy andP. Lemonnier,History matching by use of optimal control theory, SPE 4627 presented at the Annual Fall Meeting of the SPE of AIME, Las Vegas, October 1973.

  8. W. H. Chen andJ. H. Seinfeld, Estimation of spatially varying parameters in partial differential equations,Int. J. Control, 1972,15, No° 3, 487–495.

    Google Scholar 

  9. M. I. Darby andN. Morton, Numerical solution for the nonlinear penetration of magnetic flux into type II superconductors,J. Comp. Physics 13 (1973), 59–69.

    Google Scholar 

  10. J. R. Freeman, A method for computing the transition from ambipolar to free diffusion in decaying plasma,J. Comp. Physics 13 (1973), 59–69.

    Google Scholar 

  11. W. Giesel andM. Renger, Numerical treatment of the unsaturated water flow equation: Comparison of experimental and computed results,Water Resources Research 9, No° 1, February 1973.

  12. D. D. Joseph andT. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,Arch. Rational Mech. Anal. 49.

  13. J. L. Lions,Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod, 1968.

  14. J. L. Lions,Quelques méthodes de résolution de problèmes aux limites non linéaires, Dunod-Gauthier Villars, Paris 1969.

    Google Scholar 

  15. H. D. Meyer, The numerical solution of non-linear parabolic problems by variational methods,SIAM J. Numer. Anal. 10, No° 4, September 1973.

  16. J. Salin andJ. G. Salin, Constant line sources of heat in infinite media, whose thermal resistivity are linear functions of the temperature,Int. J. Heat Mass Transfer 16, 1193–1198.

  17. M. F. Wheeler, A prioriL 2 error estimates for Galerkin's approximations to parabolic partial differential equations,SIAM J. Numer. Anal. 10, No° 4, September 1973.

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Authors and Affiliations

  1. Domaine de Voluceau, IRIA-Laboria, Rocquencourt, France

    G. Chavent & P. Lemonnier

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  1. G. Chavent
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  2. P. Lemonnier
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Chavent, G., Lemonnier, P. Identification de la Non-Linearité D'Une équation Parabolique Quasilineaire. Appl Math Optim 1, 121–162 (1974). https://doi.org/10.1007/BF01449027

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  • DOI: https://doi.org/10.1007/BF01449027

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