Some inverse and control problems for fluids

Fernández-Cara, Enrique; Horsin, Thierry; Kasumba, Henry

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.

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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) | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | Numdam | Zbl

[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 | DOI | MR | Zbl

[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) | DOI | EuDML | Numdam | MR | Zbl

[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 | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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) | DOI | EuDML | Numdam | MR | Zbl

[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 | DOI | MR | Zbl

[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) | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | Zbl

[27] Fowler, A. Mathematical geoscience, Interdisciplinary Applied Mathematics, 36, Springer, London, 2011 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

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

[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) | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | EuDML | Numdam | MR | Zbl

[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) | DOI | EuDML | Numdam | MR | Zbl

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

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl