On a shape control problem for the stationary Navier-Stokes equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 34 (2000) no. 6, p. 1233-1258
@article{M2AN_2000__34_6_1233_0,
     author = {Gunzburger, Max D. and Kim, Hongchul and Manservisi, Sandro},
     title = {On a shape control problem for the stationary Navier-Stokes equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {Dunod},
     volume = {34},
     number = {6},
     year = {2000},
     pages = {1233-1258},
     zbl = {0981.76027},
     mrnumber = {1812735},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2000__34_6_1233_0}
}
Gunzburger, Max D.; Kim, Hongchul; Manservisi, Sandro. On a shape control problem for the stationary Navier-Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 34 (2000) no. 6, pp. 1233-1258. http://www.numdam.org/item/M2AN_2000__34_6_1233_0/

[1] F. Abergel and R. Temam, On some control problems in fluid mechanics. Theor. Comp. Fluid Dyn. 1 (1990) 303-326. | Zbl 0708.76106

[2] R. Adams, Sobolev Spaces, Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[3] V. Alekseev, V. Tikhomirov and S. Fomin, Optimal Control. Consultants Bureau, New York (1987). | MR 924574 | Zbl 0689.49001

[4] G. Armugan and O. Pironneau, On the problem of riblets as a drag reduction device. Optimal Control Appl. Methods 10 (1989) 93-112. | MR 997236 | Zbl 0667.49002

[5] I. Babuska, The finite element method with Lagrangian multipliers. Numer. Math. 16 (1973) 179-192. | MR 359352 | Zbl 0258.65108

[6] D. Bedivan, Existence of a solution for complete least squares optimal shape problems. Numer. Funct. Anal. Optim. 18 (1997) 495-505. | MR 1467658 | Zbl 0895.35042

[7] D. Bedivan and G. Fix, An extension theorem for the space Hdiv. Appl. Math. Lett. (to appear). | Zbl 0915.46024

[8] D. Begis and R. Glowinski, Application de la méthode des éléments finis à l'approximation d'un problème de domaine optimal. Méthodes de résolution des problèmes approchés. Appl. Math. Optim. 2 (1975) 130-169. | MR 443372 | Zbl 0323.90063

[9] D. Chenais, On the existence of a solution in a domain identification problem. J. Math. Anal. Appl 52 (1975) 189-219. | MR 385666 | Zbl 0317.49005

[10] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[11] P. Ciarlet, Introduction to Numerical Linear Algebra and Optimization. Cambridge University, Cambridge (1989). | MR 1015713 | Zbl 0672.65001

[12] E. Dean, Q. Dinh, R. Glowinski, J. He, T. Pan and J. Periaux, Least squares domain embedding methods for Neumann problems: applications to fluid dynamics, in Domain Decomposition Methods for Partial Differential Equations, D. Keyes et al. Eds., SIAM, Philadelphia (1992). | MR 1189595 | Zbl 0768.76045

[13] N. Di Cesare, O. Pironneau and E. Polak, Consistent approximations for an optimal design problem. Report 98005, Labotatoire d'Analyse Numérique, Paris (1998).

[14] N. Fujii, Lower semi-continuity in domain optimization problems. J. Optim. Theory Appl. 57 (1988) 407-422. | MR 972916 | Zbl 0629.49006

[15] V. Girault and R. Raviart, The Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer, New York (1986). | MR 851383 | Zbl 0585.65077

[16] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984). | MR 737005 | Zbl 0536.65054

[17] R. Glowinski and O. Pironneau, Toward the computation of minimum drag profile in viscous laminar flow. Appl. Math. Model. 1 (1976) 58-66. | MR 455851 | Zbl 0361.76035

[18] M. Gunzburger, L. Hou and T. Svobodny, Analysis and finite element approximations of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls. RAIRO Modél. Math. Anal. Numér. 25 (1991) 711-748. | Numdam | MR 1135991 | Zbl 0737.76045

[19] M. Gunzburger, L. Hou and T. Svobodny, Optimal control and optimization of viscous, incompressible flow, in Incompressible Computational Fluid Dynamics, M. Gunzburger and R. Nicolaides Eds., Cambridge University, New York (1993) 109-150.

[20] M. Gunzburger and H. Kim, Existence of a shape control problem for the stationary Navier-Stokes equations. SIAM J. Control Optim. 36 (1998) 895-909. | MR 1613877 | Zbl 0917.49004

[21] M. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier-Stokes equations with distributed control. SIAM J. Numer. Anal. 37 (2000) 1481-1512. | MR 1759904 | Zbl 0963.35150

[22] M. Gunzburger and S. Manservisi, The velocity tracking problem for Navier-Stokes flows with bounded distributed control. SIAM J. Control Optim. 37 (1999) 1913-1945. | MR 1720145 | Zbl 0938.35118

[23] M. Gunzburger and S. Manservisi, A variational inequality formulation of an inverse elasticity problem. Comput. Methods Appl. Mech. Engrg. 189 (2000) 803-823. | MR 1755697 | Zbl 0969.76025

[24] M. Gunzburger and S. Manservisi, Some numerical computations of optimal shapes for Navier-Stokes flows (in préparation).

[25] J. Haslinger, K.H. Hoffmann and M. Kocvara, Control fictitious domain method for solving optimal shape design problems. RAIRO Modél Math. Anal. Numér. 27 (1993) 157-182. | Numdam | MR 1211614 | Zbl 0772.65043

[26] J. Haslinger and P. Neittaanmaki, Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd edn. Wiley, Chichester (1996). | MR 1419500 | Zbl 0845.73001

[27] K. Kunisch and G. Pensil, Shape optimization for mixed boundary value problems based on an embedding domain method (to appear). | Zbl 0914.49027

[28] O. Pironneau, Optimal Shape Design in Fluid Mechanics. Thesis, University of Paris, France (1976).

[29] O. Pironneau, On optimal design in fluid mechanics. J. Fluid. Mech. 64 (1974) 97-110. | MR 347229 | Zbl 0281.76020

[30] O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984). | MR 725856 | Zbl 0534.49001

[31] R. Showalter, Hilbert Space Methods for Partial Differential Equations. Electron. J. Differential Equations (1994) http://ejde.math.swt.edu/mono-toc.html | MR 1302484 | Zbl 0991.35001

[32] J. Simon, Domain variation for Stokes fiow, in Lecture Notes in Control and Inform. Sci. 159, X. Li and J. Yang Eds., Springer, Berlin (1990) 28-42. | MR 1129956 | Zbl 0801.76075

[33] J. Simon, Domain variation for drag Stokes flows, in Lecture notes in Control and Inform. Sci.114, A. Bermudez Ed., Springer, Berlin (1987) 277-283. | Zbl 0801.76075

[34] T. Slawig, Domain Optimization for the Stationary Stokes and Navier-Stokes Equations by Embedding Domain Technique. Thesis, TU Berlin, Berlin (1998).

[35] J. Sokolowski and J. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, Berlin (1992). | MR 1215733 | Zbl 0761.73003

[36] S. Stojanovic, Non-smooth analysis and shape optimization in flow problems. IMA Preprint Series 1046, IMA, Minneapolis (1992).

[37] R. Temam, Navier-Stokes equation. North-Holland, Amsterdam (1979).

[38] R. Temam, Navier-Stokes equations and Nonlinear Functional Analysis. SIAM, Philadelphia (1993). | MR 1318914 | Zbl 0833.35110

[39] V. Tikhomirov, Fundamental Principles of the Theory of Extremal Problems. Wiley, Chichester (1986). | MR 866483 | Zbl 0595.49001