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Armen Shirikyan
Global exponential stabilisation for the Burgers equation with localised control
(Stabilisation exponentielle globale pour l’équation de Burgers avec contrôle localisé)
Journal de l'École polytechnique — Mathématiques, 4 (2017), p. 613-632, doi: 10.5802/jep.53
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Class. Math.: 35L65, 35Q93, 93C20
Mots clés: Équation de Burgers, stabilisation exponentielle, contrôle localisé, inégalité de Harnack

Résumé - Abstract

Nous considérons l’équation de Burgers visqueuse 1D avec un contrôle localisé dans un intervalle fini. Nous montrons que, pour tout $\varepsilon >0$, on peut trouver un temps $T$ d’ordre $\log \varepsilon ^{-1}$ tel que tout état initial peut être amené dans un $\varepsilon $-voisinage d’une trajectoire donnée au temps $T$. Cette propriété, jointe à un résultat précédent de contrôle local exact, montre que l’équation de Burgers est globalement exactement contrôlable vers les trajectoires en un temps fini qui ne dépend pas des conditions initiales.

Bibliographie

[AL83] H. W. Alt & S. Luckhaus, “Quasilinear elliptic-parabolic differential equations”, Math. Z. 183 (1983) no. 3, p. 311-341 Article
[BIN79] O. V. Besov, V. P. Ilʼin & S. M. Nikolʼskiĭ, Integral representations of functions and imbedding theorems, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C., 1979
[BV92] A. V. Babin & M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications 25, North-Holland Publishing, Amsterdam, 1992  MR 1156492
[Cha09] M. Chapouly, “Global controllability of nonviscous and viscous Burgers-type equations”, SIAM J. Control Optim. 48 (2009) no. 3, p. 1567-1599 Article
[Cor07a] J.-M. Coron, Control and nonlinearity, Mathematical Surveys and Monographs 136, American Mathematical Society, Providence, RI, 2007
[Cor07b] J.-M. Coron, Some open problems on the control of nonlinear partial differential equations, in H. Berestycki, éd., Perspectives in nonlinear partial differential equations, Contemp. Math. 446, American Mathematical Society, 2007, p. 215–243
[Cor10] J.-M. Coron, On the controllability of nonlinear partial differential equations, Proceedings of the ICM, Vol. I, Hindustan Book Agency, 2010, p. 238–264
[Dia96] J. I. Diaz, Obstruction and some approximate controllability results for the Burgers equation and related problems, Control of partial differential equations and applications (Laredo, 1994), Dekker, New York, 1996, p. 63–76
[FCG07] E. Fernández-Cara & S. Guerrero, “Null controllability of the Burgers system with distributed controls”, Systems Control Lett. 56 (2007) no. 5, p. 366-372 Article
[FI95] A. V. Fursikov & O. Yu. Imanuvilov, On controllability of certain systems simulating a fluid flow, in M. D. Gunzburger, éd., Flow control (Minneapolis, MN, 1992), IMA Vol. Math. Appl. 68, Springer, 1995, p. 149–184
[FI96] A. V. 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
[Fur00] A. V. Fursikov, Optimal control of distributed systems. Theory and applications, Translations of Mathematical Monographs 187, American Mathematical Society, Providence, RI, 2000
[GG07] O. Glass & S. Guerrero, “On the uniform controllability of the Burgers equation”, SIAM J. Control Optim. 46 (2007) no. 4, p. 1211-1238 Article
[GI07] S. Guerrero & O. Yu. Imanuvilov, “Remarks on global controllability for the Burgers equation with two control forces”, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007) no. 6, p. 897-906 Article
[Hor08] T. Horsin, “Local exact Lagrangian controllability of the Burgers viscous equation”, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) no. 2, p. 219-230 Article
[Hör97] L. Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications 26, Springer-Verlag, Berlin, 1997
[KR15] A. Kröner & S. S. Rodrigues, “Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations”, SIAM J. Control Optim. 53 (2015) no. 2, p. 1020-1055 Article
[Kry87] N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series) 7, D. Reidel Publishing Co., Dordrecht, 1987
[KS80] N. V. Krylov & M. V. Safonov, “A property of the solutions of parabolic equations with measurable coefficients”, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980) no. 1, p. 161-175, 239
[Lan98] E. M. Landis, Second order equations of elliptic and parabolic type, Translations of Mathematical Monographs 171, American Mathematical Society, Providence, RI, 1998
[LSU68] O. A. Ladyženskaja, V. A. Solonnikov & N. N. Uralʼceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs 23, American Mathematical Society, Providence, RI, 1968
[Léa12] M. Léautaud, “Uniform controllability of scalar conservation laws in the vanishing viscosity limit”, SIAM J. Control Optim. 50 (2012) no. 3, p. 1661-1699 Article
[Mar14] F. Marbach, “Small time global null controllability for a viscous Burgers’ equation despite the presence of a boundary layer”, J. Math. Pures Appl. (9) 102 (2014) no. 2, p. 364-384 Article
[Tay97] M. E. Taylor, Partial differential equations. I–III, Applied Mathematical Sciences 115–117, Springer-Verlag, New York, 1996-97
[TBR10] L. Thevenet, J.-M. Buchot & J.-P. Raymond, “Nonlinear feedback stabilization of a two-dimensional Burgers equation”, ESAIM Contrôle Optim. Calc. Var. 16 (2010) no. 4, p. 929-955 Article |  MR 2744156