Some inverse and control problems for fluids

 Access to full text

 Full (PDF)

1. G. Alessandrini, E. Beretta, E. Rosset, S. Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), 755-806 Zbl1034.35148MR1822407
2. C. Alvarez, C. Conca, L. Friz, O. Kavian, J. H. Ortega, Identification of immersed obstacles via boundary measurements, Inverse Problems 21 (2005), 1531-1552 Zbl1088.35080MR2173409
3. S. Andrieux, A. Ben Abda, M. Jaou, On some inverse geometrical problems, Partial differential equation methods in control and shape analysis (Pisa) 188 (1997), 11-27, Dekker, New York Zbl0881.35122MR1452881
4. K. J. Arrow, L. Hurwicz, H. Uzawa, Studies in linear and non-linear programming, (1958), Stanford University Press, Stanford, Calif. Zbl0091.16002MR108399
5. M. Badra, F. Caubet, M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci. 21 (2011), 2069-2101 Zbl1239.35182MR2851707
6. J. A. Bello, E. Fernández-Cara, J. Lemoine, J. Simon, The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow, SIAM J. Control Optim. 35 (1997), 626-640 Zbl0873.76019MR1436642
7. F. Ben Belgacem, S. M. Kaber, On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree, Inverse Problems 27 (2011) Zbl1217.35208MR2793831
8. F. Boyer, F. Hubert, J. Le Rousseau, Uniform controllability properties for space/time-discretized parabolic equations, Numer. Math. 118 (2011), 601-661 Zbl1222.93029MR2822494
9. B. Canuto, O. Kavian, Determining coefficients in a class of heat equations via boundary measurements, SIAM J. Math. Anal. 32 (2001), 963-986 (electronic) Zbl0981.35096MR1828313
10. C. Carthel, R. Glowinski, J.-L. Lions, On exact and approximate boundary controllabilities for the heat equation: a numerical approach, J. Optim. Theory Appl. 82 (1994), 429-484 Zbl0825.93316MR1290658
11. A. Bermúdez de Castro, Continuum thermomechanics, 43 (2005), Birkhäuser Verlag, Basel Zbl1070.74001MR2145925
12. N. Cindea, E. Fernández-Cara, A. Münch, Numerical null controllability of the wave equation through primal method and Carleman estimates, ESAIM: COCV (to appear, 2013) Zbl1292.35162
13. N. Cindea, E. Fernández-Cara, A. Münch, D. De Souza, On the numerical null controllability of the Stokes and Navier-Stokes systems, In preparation (2013) Zbl1292.35162
14. C. Conca, E. L. Schwindt, T. Takahashi, On the identifiability of a rigid body moving in a stationary viscous fluid, Inverse Problems 28 (2012) Zbl1235.76021MR2864506
15. J.-M. Coron, On the controllability of the $2$-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Contrôle Optim. Calc. Var. 1 (1995/96), 35-75 (electronic) Zbl0872.93040MR1393067
16. J.-M. Coron, On the controllability of $2$-D incompressible perfect fluids, J. Math. Pures Appl. (9) 75 (1996), 155-188 Zbl0848.76013MR1380673
17. J.-M. Coron, A. V. Fursikov, Global exact controllability of the $2$D Navier-Stokes equations on a manifold without boundary, Russian J. Math. Phys. 4 (1996), 429-448 Zbl0938.93030MR1470445
18. A. Doubova, E. Fernández-Cara, M. González-Burgos, J. H. Ortega, A geometric inverse problem for the Boussinesq system, Discrete Contin. Dyn. Syst. Ser. B 6 (2006), 1213-1238 Zbl1116.35116MR2240741
19. A. Doubova, E. Fernández-Cara, J. H. Ortega, On the identification of a single body immersed in a Navier-Stokes fluid, European J. Appl. Math. 18 (2007), 57-80 Zbl1142.35101MR2335200
20. S. Ervedoza, J. Valein, On the observability of abstract time-discrete linear parabolic equations, Rev. Mat. Complut. 23 (2010), 163-190 Zbl1191.35161MR2578577
21. L. Euler, General laws of the motion of fluids, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza (1999), 26-54 Zbl0955.76500MR1754941
22. C. Fabre, Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems, ESAIM Contrôle Optim. Calc. Var. 1 (1995/96), 267-302 (electronic) Zbl0872.93039MR1418484
23. E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov, J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9) 83 (2004), 1501-1542 Zbl1267.93020MR2103189
24. E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov, J.-P. Puel, Some controllability results for the $N$-dimensional Navier-Stokes and Boussinesq systems with $N-1$ scalar controls, SIAM J. Control Optim. 45 (2006), 146-173 (electronic) Zbl1109.93006MR2225301
25. E. Fernández-Cara, A. Münch, Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods, Math. Control Relat. Fields 2 (2012), 217-246 Zbl1264.35260MR2991568
26. E. Fernández-Cara, A. Münch, Strong convergent approximations of null controls for the 1D heat equation, SeMA Journal 61 (2013), 49-78 Zbl1263.35121
27. A. Fowler, Mathematical geoscience, 36 (2011), Springer, London Zbl1219.86001MR2760029
28. A. V. Fursikov, Exact controllability and feedback stabilization from a boundary for the Navier-Stokes equations, Control of fluid flow 330 (2006), 173-188, Springer, Berlin Zbl1161.76467MR2243525
29. A. V. Fursikov, M. Gunzburger, L. S. Hou, S. Manservisi, Optimal control problems for the Navier-Stokes equations, Lectures on applied mathematics (Munich, 1999) (2000), 143-155, Springer, Berlin Zbl0962.49003MR1767769
30. A. V. Fursikov, O. Yu. Imanuilov, Exact controllability of the Navier-Stokes and Boussinesq equations, Uspekhi Mat. Nauk 54 (1999), 93-146 Zbl0970.35116MR1728643
31. A. V. Fursikov, O. Yu. Imanuvilov, Controllability of evolution equations, 34 (1996), Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul Zbl0862.49004MR1406566
32. O. Glass, T. Horsin, Approximate Lagrangian controllability for the 2-D Euler equation. Application to the control of the shape of vortex patches, J. Math. Pures Appl. (9) 93 (2010), 61-90 Zbl1180.35531MR2579376
33. O. Glass, T. Horsin, Prescribing the Motion of a Set of Particles in a Three-Dimensional Perfect Fluid, SIAM J. Control Optim. 50 (2012), 2726-2742 Zbl1263.76018MR3022084
34. R. Glowinski, Numerical methods for nonlinear variational problems, (2008), Springer-Verlag, Berlin Zbl1139.65050MR2423313
35. R. Glowinski, J.-L. Lions, J. He, Exact and approximate controllability for distributed parameter systems, 117 (2008), Cambridge University Press, Cambridge Zbl1142.93002MR2404764
36. M. González-Burgos, S. Guerrero, J.-P. Puel, Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation, Commun. Pure Appl. Anal. 8 (2009), 311-333 Zbl1152.93005MR2449112
37. M. D. Gunzburger, Perspectives in flow control and optimization, 5 (2003), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA Zbl1088.93001MR1946726
38. M. Hinze, K. Kunisch, Second order methods for optimal control of time-dependent fluid flow, SIAM J. Control Optim. 40 (2001), 925-946 (electronic) Zbl1012.49026MR1871460
39. Th. Horsin, Application of the exact null controllability of the heat equation to moving sets, C. R. Math. Acad. Sci. Paris 342 (2006), 849-852 Zbl1138.93013MR2224634
40. Th. Horsin, Local exact Lagrangian controllability of the Burgers viscous equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), 219-230 Zbl1145.35330MR2396520
41. O. Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var. 6 (2001), 39-72 (electronic) Zbl0961.35104MR1804497
42. V. Isakov, Inverse problems for partial differential equations, 127 (2006), Springer, New York Zbl1092.35001MR2193218
43. H. Kasumba, K. Kunisch, On free surface PDE constrained shape optimization problems, Appl. Math. Comput. 218 (2012), 11429-11450 Zbl1278.49051MR2943988
44. H. Kasumba, K. Kunisch, Vortex control in channel flows using translational invariant cost functionals, Comput. Optim. Appl. 52 (2012), 691-717 Zbl1258.49070MR2950502
45. T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188-200 Zbl0166.45302MR211057
46. S. Kindermann, Convergence rates of the Hilbert uniqueness method via Tikhonov regularization, J. Optim. Theory Appl. 103 (1999), 657-673 Zbl0943.65109MR1727248
47. M. V. Klibanov, A. Timonov, Carleman estimates for coefficient inverse problems and numerical applications, (2004), VSP, Utrecht Zbl1069.65106MR2126149
48. A. B. Krygin, Extension of diffeomorphisms that preserve volume, Funkcional. Anal. i Priložen. 5 (1971), 72-76 Zbl0236.57016MR368067
49. K. Kunisch, B. Vexler, Optimal vortex reduction for instationary flows based on translation invariant cost functionals, SIAM J. Control Optim. 46 (2007), 1368-1397 Zbl1159.35398MR2346385
50. S. Labbé, E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems, Systems Control Lett. 55 (2006), 597-609 Zbl1129.93324MR2225370
51. J. L. Lagrange, Oeuvres. Tome 14, (1967–1892), Gauthier-Villars (Paris), Hildesheim 
52. S. Micu, E. Zuazua, Regularity issues for the null-controllability of the linear 1-d heat equation, Systems Control Lett. 60 (2011), 406-413 Zbl1225.93027MR2841484
53. A. Münch, E. Zuazua, Numerical approximation of null controls for the heat equation: ill-posedness and remedies, Inverse Problems 26 (2010) Zbl1203.35015MR2661697
54. A. A. Samarskii, P. N. Vabishchevich, Numerical methods for solving inverse problems of mathematical physics, (2007), Walter de Gruyter GmbH & Co. KG, Berlin Zbl1136.65105MR2381619
55. J. San Martín, T. Takahashi, M. Tucsnak, A control theoretic approach to the swimming of microscopic organisms, Quart. Appl. Math. 65 (2007), 405-424 Zbl1135.76058MR2354880
56. W. Yan, Y. He, Y. Ma, Shape reconstruction of an inverse boundary value problem of two-dimensional Navier-Stokes equations, Internat. J. Numer. Methods Fluids 62 (2010), 632-646 Zbl05668336MR2605011