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, Tome 4 (2017) , pp. 613-632.

Nous considérons l’équation de Burgers visqueuse 1D avec un contrôle localisé dans un intervalle fini. Nous montrons que, pour tout ε>0, on peut trouver un temps T d’ordre logε -1 tel que tout état initial peut être amené dans un ε-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.

We consider the 1D viscous Burgers equation with a control localised in a finite interval. It is proved that, for any ε>0, one can find a time T of order logε -1 such that any initial state can be steered to the ε-neighbourhood of a given trajectory at time T. This property combined with an earlier result on local exact controllability shows that the Burgers equation is globally exactly controllable to trajectories in a finite time that does not depend on the initial conditions.

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DOI : https://doi.org/10.5802/jep.53
Classification : 35L65,  35Q93,  93C20
Mots clés : Équation de Burgers, stabilisation exponentielle, contrôle localisé, inégalité de Harnack
@article{JEP_2017__4__613_0,
     author = {Armen Shirikyan},
     title = {Global exponential stabilisation for {the~Burgers} equation with localised control},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {613--632},
     publisher = {\'Ecole polytechnique},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.53},
     mrnumber = {3665610},
     zbl = {1370.35058},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.53/}
}
Armen Shirikyan. Global exponential stabilisation for the Burgers equation with localised control. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017) , pp. 613-632. doi : 10.5802/jep.53. https://jep.centre-mersenne.org/articles/10.5802/jep.53/

[AL83] H. W. Alt & S. Luckhaus - “Quasilinear elliptic-parabolic differential equations”, Math. Z. 183 (1983) no. 3, p. 311-341 | Article | MR 706391 | Zbl 0497.35049

[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, vol. 25, North-Holland Publishing, Amsterdam, 1992 | MR 1156492 | Zbl 0778.58002

[Cha09] M. Chapouly - “Global controllability of nonviscous and viscous Burgers-type equations”, SIAM J. Control Optim. 48 (2009) no. 3, p. 1567-1599 | Article | MR 2516179 | Zbl 1282.93050

[Cor07a] J.-M. Coron - Control and nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007 | MR 2302744

[Cor07b] J.-M. Coron - “Some open problems on the control of nonlinear partial differential equations”, in Perspectives in nonlinear partial differential equations (H. Berestycki, ed.), Contemp. Math., vol. 446, American Mathematical Society, Providence, RI, 2007, p. 215-243 | Article | MR 2376661

[Cor10] J.-M. Coron - “On the controllability of nonlinear partial differential equations”, in Proceedings of the ICM, Vol. I, Hindustan Book Agency, New Delhi, 2010, p. 238-264

[Dia96] J. I. Diaz - “Obstruction and some approximate controllability results for the Burgers equation and related problems”, in Control of partial differential equations and applications (Laredo, 1994), Dekker, New York, 1996, p. 63-76 | Zbl 0853.93014

[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 | MR 2311198 | Zbl 1130.93015

[FI95] A. V. Fursikov & O. Y. Imanuvilov - “On controllability of certain systems simulating a fluid flow”, in Flow control (Minneapolis, MN, 1992) (M. D. Gunzburger, ed.), IMA Vol. Math. Appl., vol. 68, Springer, New York, 1995, p. 149-184 | MR 1348646

[FI96] A. V. Fursikov & O. Yu. Imanuvilov - Controllability of evolution equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996 | MR 1406566 | Zbl 0862.49004

[Fur00] A. V. Fursikov - Optimal control of distributed systems. Theory and applications, Translations of Mathematical Monographs, vol. 187, American Mathematical Society, Providence, RI, 2000 | MR 1726442 | Zbl 0998.93505

[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 | MR 2346380 | Zbl 1140.93013

[GI07] S. Guerrero & O. Y. 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 | Numdam | MR 2371111 | Zbl 1248.93024

[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 | Numdam | MR 2396520 | Zbl 1145.35330

[Hör97] L. Hörmander - Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications, vol. 26, Springer-Verlag, Berlin, 1997 | MR 1466700 | Zbl 0881.35001

[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 | MR 3337988 | Zbl 1312.93052

[Kry87] N. V. Krylov - Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987 | MR 901759 | Zbl 0619.35004

[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 | MR 563790

[Lan98] E. M. Landis - Second order equations of elliptic and parabolic type, Translations of Mathematical Monographs, vol. 171, American Mathematical Society, Providence, RI, 1998 | MR 1487894 | Zbl 0895.35001

[LSU68] O. A. Ladyženskaja, V. A. Solonnikov & N. N. Uralʼceva - Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1968 | Article | MR 2968071 | Zbl 1251.93033

[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 | MR 241822

[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 | MR 3227326 | Zbl 1291.93043

[Tay97] M. E. Taylor - Partial differential equations. I–III, Applied Mathematical Sciences, vol. 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 | Numdam | MR 2744156