Some inverse and control problems for fluids  [ Quelques problèmes inverses et de contrôles pour les fluides ]
Annales Mathématiques Blaise Pascal, Tome 20 (2013) no. 1, pp. 101-138.

Ce papier discute quelques problèmes inverses et de contrôle pour des systèmes de type Navier-Stokes. On insiste sur quelques aspects de nature à la fois théorique et numérique ayant menés récemment à des résultats nouveaux : Problèmes inverses géométriques, Contrôlabilité Eulérienne et Lagrangienne, Réduction de tourbillons par optimisation de forme, etc.

This paper deals with some inverse and control problems for the Navier-Stokes and related systems. We will focus on some particular aspects that have recently led to interesting (theoretical and numerical) results: geometric inverse problems, Eulerian and Lagrangian controllability and vortex reduction oriented to shape optimization.

DOI : https://doi.org/10.5802/ambp.323
Classification : 35R30,  76B75,  76D55
Mots clés : Navier-Stokes equations, Euler equations, inverse problems, exact and approximate controllability, Lagrangian controllability, vortex reduction, shape optimization
@article{AMBP_2013__20_1_101_0,
     author = {Fern\'andez-Cara, Enrique and Horsin, Thierry and Kasumba, Henry},
     title = {Some inverse and control problems for fluids},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {101--138},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {20},
     number = {1},
     year = {2013},
     doi = {10.5802/ambp.323},
     mrnumber = {3112241},
     zbl = {1290.35325},
     language = {en},
     url = {www.numdam.org/item/AMBP_2013__20_1_101_0/}
}
Fernández-Cara, Enrique; Horsin, Thierry; Kasumba, Henry. Some inverse and control problems for fluids. Annales Mathématiques Blaise Pascal, Tome 20 (2013) no. 1, pp. 101-138. doi : 10.5802/ambp.323. http://www.numdam.org/item/AMBP_2013__20_1_101_0/

[1] Alessandrini, G.; Beretta, E.; Rosset, E.; Vessella, S. Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 29 (2000) no. 4, pp. 755-806 | Numdam | MR 1822407 | Zbl 1034.35148

[2] Alvarez, C.; Conca, C.; Friz, L.; Kavian, O.; Ortega, J. H. Identification of immersed obstacles via boundary measurements, Inverse Problems, Volume 21 (2005) no. 5, pp. 1531-1552 | Article | MR 2173409 | Zbl 1088.35080

[3] Andrieux, S.; Ben Abda, A.; Jaou, M. On some inverse geometrical problems, Partial differential equation methods in control and shape analysis (Pisa) (Lecture Notes in Pure and Appl. Math.) Volume 188, Dekker, New York, 1997, pp. 11-27 | MR 1452881 | Zbl 0881.35122

[4] Arrow, K. J.; Hurwicz, L.; Uzawa, H. Studies in linear and non-linear programming, With contributions by H. B. Chenery, S. M. Johnson, S. Karlin, T. Marschak, R. M. Solow. Stanford Mathematical Studies in the Social Sciences, vol. II, Stanford University Press, Stanford, Calif., 1958 | MR 108399 | Zbl 0091.16002

[5] Badra, M.; Caubet, F.; Dambrine, M. Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., Volume 21 (2011) no. 10, pp. 2069-2101 | Article | MR 2851707 | Zbl 1239.35182

[6] Bello, J. A.; Fernández-Cara, E.; Lemoine, J.; Simon, J. The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow, SIAM J. Control Optim., Volume 35 (1997) no. 2, pp. 626-640 | Article | MR 1436642 | Zbl 0873.76019

[7] Ben Belgacem, F.; Kaber, S. M. On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree, Inverse Problems, Volume 27 (2011) no. 5, pp. 055012, 19 | Article | MR 2793831 | Zbl 1217.35208

[8] Boyer, F.; Hubert, F.; Le Rousseau, J. Uniform controllability properties for space/time-discretized parabolic equations, Numer. Math., Volume 118 (2011) no. 4, pp. 601-661 | Article | MR 2822494 | Zbl 1222.93029

[9] Canuto, B.; Kavian, O. Determining coefficients in a class of heat equations via boundary measurements, SIAM J. Math. Anal., Volume 32 (2001) no. 5, p. 963-986 (electronic) | Article | MR 1828313 | Zbl 0981.35096

[10] Carthel, C.; Glowinski, R.; Lions, J.-L. On exact and approximate boundary controllabilities for the heat equation: a numerical approach, J. Optim. Theory Appl., Volume 82 (1994) no. 3, pp. 429-484 | Article | MR 1290658 | Zbl 0825.93316

[11] Bermúdez de Castro, A. Continuum thermomechanics, Progress in Mathematical Physics, Volume 43, Birkhäuser Verlag, Basel, 2005 | MR 2145925 | Zbl 1070.74001

[12] Cindea, N.; Fernández-Cara, E.; Münch, A. Numerical null controllability of the wave equation through primal method and Carleman estimates, ESAIM: COCV, Volume (to appear, 2013) no. 3 | Zbl 1292.35162

[13] Cindea, N.; Fernández-Cara, E.; Münch, A.; De Souza, D. On the numerical null controllability of the Stokes and Navier-Stokes systems, In preparation (2013)

[14] Conca, C.; Schwindt, E. L.; Takahashi, T. On the identifiability of a rigid body moving in a stationary viscous fluid, Inverse Problems, Volume 28 (2012) no. 1, pp. 015005, 22 | Article | MR 2864506 | Zbl 1235.76021

[15] 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), p. 35-75 (electronic) | Article | EuDML 90500 | Numdam | MR 1393067 | Zbl 0872.93040

[16] Coron, J.-M. On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl. (9), Volume 75 (1996) no. 2, pp. 155-188 | MR 1380673 | Zbl 0848.76013

[17] Coron, J.-M.; Fursikov, A. V. Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary, Russian J. Math. Phys., Volume 4 (1996) no. 4, pp. 429-448 | MR 1470445 | Zbl 0938.93030

[18] Doubova, A.; Fernández-Cara, E.; González-Burgos, M.; Ortega, J. H. A geometric inverse problem for the Boussinesq system, Discrete Contin. Dyn. Syst. Ser. B, Volume 6 (2006) no. 6, pp. 1213-1238 | Article | MR 2240741 | Zbl 1116.35116

[19] Doubova, A.; Fernández-Cara, E.; Ortega, J. H. On the identification of a single body immersed in a Navier-Stokes fluid, European J. Appl. Math., Volume 18 (2007) no. 1, pp. 57-80 | Article | MR 2335200 | Zbl 1142.35101

[20] Ervedoza, S.; Valein, J. On the observability of abstract time-discrete linear parabolic equations, Rev. Mat. Complut., Volume 23 (2010) no. 1, pp. 163-190 | Article | MR 2578577 | Zbl 1191.35161

[21] Euler, L. General laws of the motion of fluids, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza (1999) no. 6, pp. 26-54 | MR 1754941 | Zbl 0955.76500

[22] Fabre, C. Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems, ESAIM Contrôle Optim. Calc. Var., Volume 1 (1995/96), p. 267-302 (electronic) | Article | EuDML 90499 | Numdam | MR 1418484 | Zbl 0872.93039

[23] Fernández-Cara, E.; Guerrero, S.; Imanuvilov, O. Yu.; Puel, J.-P. Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9), Volume 83 (2004) no. 12, pp. 1501-1542 | Article | MR 2103189 | Zbl 1267.93020

[24] Fernández-Cara, E.; Guerrero, S.; Imanuvilov, O. Yu.; Puel, J.-P. Some controllability results for the N-dimensional Navier-Stokes and Boussinesq systems with N-1 scalar controls, SIAM J. Control Optim., Volume 45 (2006) no. 1, p. 146-173 (electronic) | Article | MR 2225301 | Zbl 1109.93006

[25] Fernández-Cara, E.; Münch, A. Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods, Math. Control Relat. Fields, Volume 2 (2012) no. 3, pp. 217-246 | Article | MR 2991568 | Zbl 1264.35260

[26] Fernández-Cara, E.; Münch, A. Strong convergent approximations of null controls for the 1D heat equation, SeMA Journal, Volume 61 (2013) no. 1, pp. 49-78 | Article | Zbl 1263.35121

[27] Fowler, A. Mathematical geoscience, Interdisciplinary Applied Mathematics, Volume 36, Springer, London, 2011 | Article | MR 2760029 | Zbl 1219.86001

[28] Fursikov, A. V. Exact controllability and feedback stabilization from a boundary for the Navier-Stokes equations, Control of fluid flow (Lecture Notes in Control and Inform. Sci.) Volume 330, Springer, Berlin, 2006, pp. 173-188 | Article | MR 2243525 | Zbl 1161.76467

[29] Fursikov, A. V.; Gunzburger, M.; Hou, L. S.; Manservisi, S. Optimal control problems for the Navier-Stokes equations, Lectures on applied mathematics (Munich, 1999), Springer, Berlin, 2000, pp. 143-155 | MR 1767769 | Zbl 0962.49003

[30] Fursikov, A. V.; Imanuilov, O. Yu. Exact controllability of the Navier-Stokes and Boussinesq equations, Uspekhi Mat. Nauk, Volume 54 (1999) no. 3(327), pp. 93-146 | Article | MR 1728643 | Zbl 0970.35116

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

[32] Glass, O.; Horsin, T. Approximate Lagrangian controllability for the 2-D Euler equation. Application to the control of the shape of vortex patches, J. Math. Pures Appl. (9), Volume 93 (2010) no. 1, pp. 61-90 | Article | MR 2579376 | Zbl 1180.35531

[33] Glass, O.; Horsin, T. Prescribing the Motion of a Set of Particles in a Three-Dimensional Perfect Fluid, SIAM J. Control Optim., Volume 50 (2012) no. 5, pp. 2726-2742 | Article | MR 3022084 | Zbl 1263.76018

[34] Glowinski, R. Numerical methods for nonlinear variational problems, Scientific Computation, Springer-Verlag, Berlin, 2008 (Reprint of the 1984 original) | MR 2423313 | Zbl 1139.65050

[35] Glowinski, R.; Lions, J.-L.; He, J. Exact and approximate controllability for distributed parameter systems, Encyclopedia of Mathematics and its Applications, Volume 117, Cambridge University Press, Cambridge, 2008 (A numerical approach) | Article | MR 2404764 | Zbl 1142.93002

[36] González-Burgos, M.; Guerrero, S.; Puel, J.-P. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal., Volume 8 (2009) no. 1, pp. 311-333 | Article | MR 2449112 | Zbl 1152.93005

[37] Gunzburger, M. D. Perspectives in flow control and optimization, Advances in Design and Control, Volume 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003 | MR 1946726 | Zbl 1088.93001

[38] Hinze, M.; Kunisch, K. Second order methods for optimal control of time-dependent fluid flow, SIAM J. Control Optim., Volume 40 (2001) no. 3, p. 925-946 (electronic) | Article | MR 1871460 | Zbl 1012.49026

[39] Horsin, Th. Application of the exact null controllability of the heat equation to moving sets, C. R. Math. Acad. Sci. Paris, Volume 342 (2006) no. 11, pp. 849-852 | Article | MR 2224634 | Zbl 1138.93013

[40] Horsin, Th. Local exact Lagrangian controllability of the Burgers viscous equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 25 (2008) no. 2, pp. 219-230 | Article | EuDML 78786 | Numdam | MR 2396520 | Zbl 1145.35330

[41] Imanuvilov, O. Yu. Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., Volume 6 (2001), p. 39-72 (electronic) | Article | EuDML 90600 | Numdam | MR 1804497 | Zbl 0961.35104

[42] Isakov, V. Inverse problems for partial differential equations, Applied Mathematical Sciences, Volume 127, Springer, New York, 2006 | MR 2193218 | Zbl 1092.35001

[43] Kasumba, H.; Kunisch, K. On free surface PDE constrained shape optimization problems, Appl. Math. Comput., Volume 218 (2012) no. 23, pp. 11429-11450 | Article | MR 2943988 | Zbl 1278.49051

[44] Kasumba, H.; Kunisch, K. Vortex control in channel flows using translational invariant cost functionals, Comput. Optim. Appl., Volume 52 (2012) no. 3, pp. 691-717 | Article | MR 2950502 | Zbl 1258.49070

[45] Kato, T. On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal., Volume 25 (1967), pp. 188-200 | Article | MR 211057 | Zbl 0166.45302

[46] Kindermann, S. Convergence rates of the Hilbert uniqueness method via Tikhonov regularization, J. Optim. Theory Appl., Volume 103 (1999) no. 3, pp. 657-673 | Article | MR 1727248 | Zbl 0943.65109

[47] Klibanov, M. V.; Timonov, A. Carleman estimates for coefficient inverse problems and numerical applications, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2004 | Article | MR 2126149 | Zbl 1069.65106

[48] Krygin, A. B. Extension of diffeomorphisms that preserve volume, Funkcional. Anal. i Priložen., Volume 5 (1971) no. 2, pp. 72-76 | MR 368067 | Zbl 0236.57016

[49] Kunisch, K.; Vexler, B. Optimal vortex reduction for instationary flows based on translation invariant cost functionals, SIAM J. Control Optim., Volume 46 (2007) no. 4, pp. 1368-1397 | Article | MR 2346385 | Zbl 1159.35398

[50] Labbé, S.; Trélat, E. Uniform controllability of semidiscrete approximations of parabolic control systems, Systems Control Lett., Volume 55 (2006) no. 7, pp. 597-609 | Article | MR 2225370 | Zbl 1129.93324

[51] Lagrange, J. L. Oeuvres. Tome 14, Gauthier-Villars (Paris), Hildesheim, 1967–1892 (Publiées par les soins de J.-A. Serret [et G. Darboux] ; [Précédé d’une notice sur la vie et les ouvrages de J.-L. Lagrange, par M. Delambre])

[52] Micu, S.; Zuazua, E. Regularity issues for the null-controllability of the linear 1-d heat equation, Systems Control Lett., Volume 60 (2011) no. 6, pp. 406-413 | Article | MR 2841484 | Zbl 1225.93027

[53] Münch, A.; Zuazua, E. Numerical approximation of null controls for the heat equation: ill-posedness and remedies, Inverse Problems, Volume 26 (2010) no. 8, pp. 085018, 39 | Article | MR 2661697 | Zbl 1203.35015

[54] Samarskii, A. A.; Vabishchevich, P. N. Numerical methods for solving inverse problems of mathematical physics, Inverse and Ill-posed Problems Series, Walter de Gruyter GmbH & Co. KG, Berlin, 2007 | Article | MR 2381619 | Zbl 1136.65105

[55] San Martín, J.; Takahashi, T.; Tucsnak, M. A control theoretic approach to the swimming of microscopic organisms, Quart. Appl. Math., Volume 65 (2007) no. 3, pp. 405-424 | MR 2354880 | Zbl 1135.76058

[56] Yan, W.; He, Y.; Ma, Y. Shape reconstruction of an inverse boundary value problem of two-dimensional Navier-Stokes equations, Internat. J. Numer. Methods Fluids, Volume 62 (2010) no. 6, pp. 632-646 | Article | MR 2605011 | Zbl pre05668336