Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 1, pp. 93-119.

In this paper sufficient optimality conditions are established for optimal control of both steady-state and instationary Navier-Stokes equations. The second-order condition requires coercivity of the Lagrange function on a suitable subspace together with first-order necessary conditions. It ensures local optimality of a reference function in a ${L}^{s}$-neighborhood, whereby the underlying analysis allows to use weaker norms than ${L}^{\infty }$.

DOI : https://doi.org/10.1051/cocv:2005029
Classification : 49K20,  49K27
Mots clés : optimal control, Navier-Stokes equations, control constraints, second-order optimality conditions, first-order necessary conditions
@article{COCV_2006__12_1_93_0,
author = {Tr\"oltzsch, Fredi and Wachsmuth, Daniel},
title = {Second-order sufficient optimality conditions for the optimal control of {Navier-Stokes} equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {93--119},
publisher = {EDP-Sciences},
volume = {12},
number = {1},
year = {2006},
doi = {10.1051/cocv:2005029},
zbl = {1111.49017},
mrnumber = {2192070},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2005029/}
}
TY  - JOUR
AU  - Tröltzsch, Fredi
AU  - Wachsmuth, Daniel
TI  - Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2006
DA  - 2006///
SP  - 93
EP  - 119
VL  - 12
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2005029/
UR  - https://zbmath.org/?q=an%3A1111.49017
UR  - https://www.ams.org/mathscinet-getitem?mr=2192070
UR  - https://doi.org/10.1051/cocv:2005029
DO  - 10.1051/cocv:2005029
LA  - en
ID  - COCV_2006__12_1_93_0
ER  - 
Tröltzsch, Fredi; Wachsmuth, Daniel. Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 1, pp. 93-119. doi : 10.1051/cocv:2005029. http://www.numdam.org/articles/10.1051/cocv:2005029/

[1] F. Abergel and R. Temam, On some control problems in fluid mechanics. Theoret. Comput. Fluid Dynam. 1 (1990) 303-325. | Zbl 0708.76106

[2] R.A. Adams, Sobolev spaces. Academic Press, San Diego (1978). | Zbl 0314.46030

[3] N. Arada, J.-P. Raymond and F. Tröltzsch, On an augmented Lagrangian SQP method for a class of optimal control problems in Banach spaces. Comput. Optim. Appl. 22 (2002) 369-398. | Zbl 1039.90094

[4] J.F. Bonnans, Second-order analysis for control constrained optimal control problems of semilinear elliptic equations. Appl. Math. Optim. 38 (1998) 303-325. | Zbl 0917.49020

[5] J.F. Bonnans and H. Zidani, Optimal control problems with partially polyhedric constraints. SIAM J. Control Optim. 37 (1999) 1726-1741. | Zbl 0945.49020

[6] H. Brezis, Analyse fonctionelle. Masson, Paris (1983). | MR 697382 | Zbl 0511.46001

[7] E. Casas, An optimal control problem governed by the evolution Navier-Stokes equations, in Optimal control of viscous flows. Frontiers in applied mathematics, S.S. Sritharan Ed., SIAM, Philadelphia (1993). | MR 1632422

[8] E. Casas and M. Mateos, Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431-1454. | Zbl 1037.49024

[9] E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. 21 (2002) 67-100. | Zbl 1119.49309

[10] E. Casas, F. Tröltzsch and A. Unger, Second-order sufficient optimality conditions for a nonlinear elliptic control problem. J. Anal. Appl. 15 (1996) 687-707. | Zbl 0879.49020

[11] E. Casas, F. Tröltzsch and A. Unger, Second-order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369-1391. | Zbl 0962.49016

[12] P. Constantin and C. Foias, Navier-Stokes equations. The University of Chicago Press, Chicago (1988). | MR 972259 | Zbl 0687.35071

[13] R. Dautray and J.L. Lions, Evolution problems I, Mathematical analysis and numerical methods for science and technology 5. Springer, Berlin (1992). | MR 1156075

[14] M. Desai and K. Ito, Optimal controls of Navier-Stokes equations. SIAM J. Control Optim. 32 (1994) 1428-1446. | Zbl 0813.35078

[15] A.L. Dontchev, W.W. Hager, A.B. Poore and B. Yang, Optimality, stability, and convergence in optimal control. Appl. Math. Optim. 31 (1995) 297-326. | Zbl 0821.49022

[16] J.C. Dunn, On second-order sufficient conditions for structured nonlinear programs in infinite-dimensional function spaces, in Mathematical programming with data perturbations, A. Fiacco Ed., Marcel Dekker (1998) 83-107. | Zbl 0891.90147

[17] H.O. Fattorini and S. Sritharan, Necessary and sufficient for optimal controls in viscous flow problems. Proc. Roy. Soc. Edinburgh 124 (1994) 211-251. | Zbl 0800.49047

[18] M.D. Gunzburger Ed., Flow control. Springer, New York (1995). | MR 1348639 | Zbl 0816.00037

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

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

[21] M. Hinze, Optimal and instantaneous control of the instationary Navier-Stokes equations. Habilitation, TU Berlin (2002).

[22] M. Hinze and K. Kunisch, Second-order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40 (2001) 925-946. | Zbl 1012.49026

[23] H. Maurer and J. Zowe, First- and second-order conditions in infinite-dimensional programming problems. Math. Programming 16 (1979) 98-110. | Zbl 0398.90109

[24] H.D. Mittelmann and F. Tröltzsch, Sufficient optimality in a parabolic control problem, in Trends in Industrial and Applied Mathematics, A.H. Siddiqi and M. Kocvara Ed., Dordrecht, Kluwer (2002) 305-316.

[25] J.-P. Raymond and F. Tröltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dynam. Syst. 6 (2000) 431-450. | Zbl 1010.49015

[26] T. Roubíček and F. Tröltzsch, Lipschitz stability of optimal controls for the steady-state Navier-Stokes equations. Control Cybernet. 32 (2002) 683-705. | Zbl 1127.49021

[27] S. Sritharan, Dynamic programming of the Navier-Stokes equations. Syst. Control Lett. 16 (1991) 299-307. | Zbl 0737.49021

[28] R. Temam, Navier-Stokes equations. North Holland, Amsterdam (1979). | MR 603444 | Zbl 0426.35003

[29] F. Tröltzsch, Lipschitz stability of solutions of linear-quadratic parabolic control problems with respect to perturbations. Dyn. Contin. Discrete Impulsive Syst. 7 (2000) 289-306. | Zbl 0954.49017

Cité par Sources :