An obstruction to small-time  local null controllability  for a viscous Burgers' equation

Marbach, Frédéric

doi:10.24033/asens.2373

An obstruction to small-time local null controllability for a viscous Burgers' equation
[Obstruction à la contrôlabilité locale en temps petit pour une équation de Burgers visqueuse]

Marbach, Frédéric

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1129-1177.

Résumé
Abstract

Nous nous intéressons à la contrôlabilité locale en temps petit pour l'équation de Burgers visqueuse $y_{t} - y_{x x} + y y_{x} = u (t)$ , posée sur un segment, avec des conditions de Dirichlet nulles au bord. Le terme source au second membre est un contrôle scalaire qui joue un rôle similaire à celui d'une pression. Dans ce contexte, la condition de crochet de Lie nécessaire classique introduite par Sussmann ne permet pas de conclure. Cependant, en utilisant un développement à l'ordre deux du système étudié, nous mettons en lumière une obstruction de nature quadratique à la contrôlabilité locale en temps petit. Cette obstruction tient alors même que la vitesse de propagation de l'information dans cette équation de Burgers est infinie. Elle fait intervenir la norme $H^{- 5 / 4}$ du contrôle. La démonstration nécessite le calcul soigneux du noyau d'un opérateur intégral, ainsi que l'estimation d'opérateurs résiduels à l'aide de la théorie de régularité pour les opérateurs intégraux faiblement singuliers.

In this work, we are interested in the small-time local null controllability for the viscous Burgers' equation $y_{t} - y_{x x} + y y_{x} = u (t)$ on a line segment, with null boundary conditions. The second-hand side is a scalar control playing a role similar to that of a pressure. In this setting, the classical Lie bracket necessary condition introduced by Sussmann fails to conclude. However, using a quadratic expansion of our system, we exhibit a second order obstruction to small-time local null controllability. This obstruction holds although the information propagation speed is infinite for the Burgers equation. Our obstruction involves the $H^{- 5 / 4}$ norm of the control. The proof requires the careful derivation of an integral kernel operator and the estimation of residues by means of weakly singular integral operator estimates.

Publié le : 2018-11-12

DOI : 10.24033/asens.2373

Classification : 93B05, 93C20, 47G10.
Keywords: Burgers, controllability, quadratic obstruction, weakly singular integral operator.
Mot clés : Burgers, contrôlabilité, obstruction quadratique, opérateur intégral faiblement singulier.

@article{ASENS_2018__51_5_1129_0,
     author = {Marbach, Fr\'ed\'eric},
     title = {An obstruction to small-time  local null controllability  for a viscous {Burgers'} equation},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1129--1177},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {5},
     year = {2018},
     doi = {10.24033/asens.2373},
     mrnumber = {3942039},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2373/}
}

TY  - JOUR
AU  - Marbach, Frédéric
TI  - An obstruction to small-time  local null controllability  for a viscous Burgers' equation
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2018
SP  - 1129
EP  - 1177
VL  - 51
IS  - 5
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://archive.numdam.org/articles/10.24033/asens.2373/
DO  - 10.24033/asens.2373
LA  - en
ID  - ASENS_2018__51_5_1129_0
ER  -

%0 Journal Article
%A Marbach, Frédéric
%T An obstruction to small-time  local null controllability  for a viscous Burgers' equation
%J Annales scientifiques de l'École Normale Supérieure
%D 2018
%P 1129-1177
%V 51
%N 5
%I Société Mathématique de France. Tous droits réservés
%U http://archive.numdam.org/articles/10.24033/asens.2373/
%R 10.24033/asens.2373
%G en
%F ASENS_2018__51_5_1129_0

Marbach, Frédéric. An obstruction to small-time  local null controllability  for a viscous Burgers' equation. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1129-1177. doi : 10.24033/asens.2373. http://archive.numdam.org/articles/10.24033/asens.2373/

Bibliographie
Cité par

Adimurthi, A.; Ghoshal, S. S.; Gowda, G. Exact controllability of scalar conservation laws with strict convex flux, Mathematical Control & Related Fields, Volume 4 (2014), pp. 401-449 | DOI | MR | Zbl

Ancona, F.; Marson, A. On the attainable set for scalar nonlinear conservation laws with boundary control, SIAM J. Control Optim., Volume 36 (1998), pp. 290-312 (ISSN: 0363-0129) | DOI | MR | Zbl

Beauchard, K.; Coron, J.-M. Controllability of a quantum particle in a moving potential well, J. Funct. Anal., Volume 232 (2006), pp. 328-389 (ISSN: 0022-1236) | DOI | MR | Zbl

Bahouri, H.; Chemin, J.-Y.; Danchin, R., Grundl. math. Wiss., 343, Springer, Heidelberg, 2011, 523 pages (ISBN: 978-3-642-16829-1) | DOI | MR | Zbl

Berg, C.; Christensen, J. P. R.; Ressel, P., Graduate Texts in Math., 100, Springer, New York, 1984, 289 pages (ISBN: 0-387-90925-7) | DOI | MR | Zbl

Beauchard, K.; Coron, J.-M.; Teismann, H. Minimal time for the bilinear control of Schrödinger equations, Systems Control Lett., Volume 71 (2014), pp. 1-6 (ISSN: 0167-6911) | DOI | MR | Zbl

Beauchard, K. Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl., Volume 84 (2005), pp. 851-956 (ISSN: 0021-7824) | DOI | MR | Zbl

Beauchard, K. Null controllability of Kolmogorov-type equations, Math. Control Signals Systems, Volume 26 (2014), pp. 145-176 (ISSN: 0932-4194) | DOI | MR | Zbl

Beauchard, K.; Morancey, M. Local controllability of 1D Schrödinger equations with bilinear control and minimal time, Math. Control Relat. Fields, Volume 4 (2014), pp. 125-160 (ISSN: 2156-8472) | DOI | MR | Zbl

Beauchard, K.; Miller, L.; Morancey, M. 2D Grushin-type equations: minimal time and null controllable data, J. Differential Equations, Volume 259 (2015), pp. 5813-5845 (ISSN: 0022-0396) | DOI | MR

Bényi, Á.; Maldonado, D.; Naibo, V. What is $...$ a paraproduct?, Notices Amer. Math. Soc., Volume 57 (2010), pp. 858-860 (ISSN: 0002-9920) | MR | Zbl

Bony, J.-M. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., Volume 14 (1981), pp. 209-246 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Brockett, R. Controllability with quadratic drift, Math. Control Relat. Fields, Volume 3 (2013), pp. 433-446 (ISSN: 2156-8472) | DOI | MR | Zbl

Brunovský, P. A classification of linear controllable systems, Kybernetika (Prague), Volume 6 (1970), pp. 173-188 (ISSN: 0023-5954) | MR | Zbl

Coron, J.-M.; Crépeau, E. Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc. (JEMS), Volume 6 (2004), pp. 367-398 http://link.springer.de/cgi/linkref?issn=1435-9855&year=04&volume=6&page=367 (ISSN: 1435-9855) | DOI | MR | Zbl

Cerpa, E.; Crépeau, E. Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 26 (2009), pp. 457-475 (ISSN: 0294-1449) | DOI | Numdam | MR | Zbl

Cerpa, E. Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain, SIAM J. Control Optim., Volume 46 (2007), pp. 877-899 (ISSN: 0363-0129) | DOI | MR | Zbl

Coron, J.-M.; Guerrero, S. Singular optimal control: a linear 1-D parabolic-hyperbolic example, Asymptot. Anal., Volume 44 (2005), pp. 237-257 (ISSN: 0921-7134) | MR | Zbl

Chapouly, M. Global controllability of nonviscous and viscous Burgers-type equations, SIAM J. Control Optim., Volume 48 (2009), pp. 1567-1599 (ISSN: 0363-0129) | DOI | MR | Zbl

Coron, J.-M. On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well, C. R. Math. Acad. Sci. Paris, Volume 342 (2006), pp. 103-108 (ISSN: 1631-073X) | DOI | MR | Zbl

Coron, J.-M., Mathematical Surveys and Monographs, 136, Amer. Math. Soc., 2007, 426 pages (ISBN: 978-0-8218-3668-2; 0-8218-3668-4) | MR | Zbl

Coron, J.-M., Perspectives in nonlinear partial differential equations (Contemp. Math.), Volume 446, Amer. Math. Soc., 2007, pp. 215-243 | DOI | MR | Zbl

Coron, J.-M. On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Contrôle Optim. Calc. Var., Volume 1 (1995/96), pp. 35-75 (ISSN: 1292-8119) | DOI | Numdam | MR | Zbl

Coron, J.-M. On the controllability of 2-D incompressible perfect fluids, J. Math. pures appl., Volume 75 (1996), pp. 155-188 (ISSN: 0021-7824) | MR | Zbl

Calderón, A. P.; Zygmund, A. On the existence of certain singular integrals, Acta Math., Volume 88 (1952), pp. 85-139 (ISSN: 0001-5962) | DOI | MR | Zbl

Calderón, A. P.; Zygmund, A. On singular integrals, Amer. J. Math., Volume 78 (1956), pp. 289-309 (ISSN: 0002-9327) | DOI | MR | Zbl

Dardé, J.; Ervedoza, S. On the reachable set for the one-dimensional heat equation (preprint arXiv:1609.02692 ) | MR

Diaz, J. I., Control of partial differential equations and applications (Laredo, 1994) (Lecture Notes in Pure and Appl. Math.), Volume 174, Dekker, 1996, pp. 63-76 | MR | Zbl

David, G.; Journé, J.-L. A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math., Volume 120 (1984), pp. 371-397 (ISSN: 0003-486X) | DOI | MR | Zbl

Fernández-Cara, E.; Guerrero, S. Null controllability of the Burgers system with distributed controls, Systems Control Lett., Volume 56 (2007), pp. 366-372 (ISSN: 0167-6911) | DOI | MR | Zbl

Fursikov, A.; Imanuvilov, O., Flow control (Minneapolis, MN, 1992) (IMA Vol. Math. Appl.), Volume 68, Springer, 1995, pp. 149-184 | DOI | MR | Zbl

Fursikov, A.; Imanuvilov, O., Lecture Notes Series, 34, Seoul National University Research Institute of Mathematics Global Analysis Research Center, 1996, 163 pages | MR | Zbl

Frazier, M.; Jawerth, B. Decomposition of Besov spaces, Indiana Univ. Math. J., Volume 34 (1985), pp. 777-799 (ISSN: 0022-2518) | DOI | MR | Zbl

Frazier, M.; Jawerth, B., Function spaces and applications (Lund, 1986) (Lecture Notes in Math.), Volume 1302, Springer, Berlin, 1988, pp. 223-246 | DOI | MR | Zbl

Frazier, M.; Jawerth, B. A discrete transform and decompositions of distribution spaces, J. Funct. Anal., Volume 93 (1990), pp. 34-170 (ISSN: 0022-1236) | DOI | MR | Zbl

Frazier, M.; Jawerth, B.; Weiss, G., CBMS Regional Conference Series in Mathematics, 79, Amer. Math. Soc., Providence, RI, 1991, 132 pages (ISBN: 0-8218-0731-5) | MR | Zbl

Guerrero, S.; Imanuvilov, O. Remarks on global controllability for the Burgers equation with two control forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 24 (2007), pp. 897-906 (ISSN: 0294-1449) | DOI | Numdam | MR | Zbl

Horsin, T. On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var., Volume 3 (1998), pp. 83-95 (ISSN: 1292-8119) | DOI | Numdam | MR | Zbl

Lions, J.-L., Dunod, 1969, 554 pages | MR | Zbl

Lissy, P. The cost of the control in the case of a minimal time of control: the example of the one-dimensional heat equation (2017) | MR

Marbach, F. Small time global null controllability for a viscous Burgers' equation despite the presence of a boundary layer, J. Math. Pures Appl., Volume 102 (2014), pp. 364-384 (ISSN: 0021-7824) | DOI | MR | Zbl

Mercer, J. Functions of Positive and Negative Type, and their Connection with the Theory of Integral Equations, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, Volume 209 (1909), pp. pp. 415-446 http://www.jstor.org/stable/91043 | JFM

Martin, P.; Rosier, L.; Rouchon, P. On the reachable states for the boundary control of the heat equation, Appl. Math. Res. Express. AMRX, Volume 2 (2016), pp. 181-216 (ISSN: 1687-1200) | DOI | MR

Palʼcev, B. V. Asymptotic behavior of the spectrum and eigenfunctions of convolution operators on a finite interval with the kernel having a homogeneous Fourier transform, Dokl. Akad. Nauk SSSR, Volume 218 (1974), pp. 28-31 (ISSN: 0002-3264) | MR

Pazy, A., Applied Mathematical Sciences, 44, Springer, 1983, 279 pages (ISBN: 0-387-90845-5) | DOI | MR | Zbl

Perrollaz, V. Exact controllability of scalar conservation laws with an additional control in the context of entropy solutions, SIAM J. Control Optim., Volume 50 (2012), pp. 2025-2045 | DOI | MR | Zbl

Rosenblatt, M. Some results on the asymptotic behavior of eigenvalues for a class of integral equations with translation kernels, J. Math. Mech., Volume 12 (1963), pp. 619-628 | MR | Zbl

Rosier, L. Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., Volume 2 (1997), pp. 33-55 (ISSN: 1292-8119) | DOI | Numdam | MR | Zbl

Stein, E. M., Princeton Mathematical Series, No. 30, Princeton Univ. Press, Princeton, N.J., 1970, 290 pages | MR | Zbl

Sussmann, H. Lie brackets and local controllability: a sufficient condition for scalar-input systems, SIAM J. Control Optim., Volume 21 (1983), pp. 686-713 (ISSN: 0363-0129) | DOI | MR | Zbl

Torres, R. Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc., Volume 90 (1991) (ISSN: 0065-9266) | DOI | MR | Zbl

Trélat, E., Mathématiques concrètes, Vuibert, Paris, 2005, 246 pages (ISBN: 2-7117-7175-X) | MR

Widom, H. Asymptotic behavior of the eigenvalues of certain integral equations, Trans. Amer. Math. Soc., Volume 109 (1963), pp. 278-295 (ISSN: 0002-9947) | DOI | MR | Zbl

Youssfi, A. Regularity properties of singular integral operators, Studia Math., Volume 119 (1996), pp. 199-217 (ISSN: 0039-3223) | DOI | MR | Zbl

Cité par Sources :