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 -neighborhood, whereby the underlying analysis allows to use weaker norms than .
Keywords: 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},
year = {2006},
publisher = {EDP Sciences},
volume = {12},
number = {1},
doi = {10.1051/cocv:2005029},
mrnumber = {2192070},
zbl = {1111.49017},
language = {en},
url = {https://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 SP - 93 EP - 119 VL - 12 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv:2005029/ DO - 10.1051/cocv:2005029 LA - en ID - COCV_2006__12_1_93_0 ER -
%0 Journal Article %A Tröltzsch, Fredi %A Wachsmuth, Daniel %T Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 93-119 %V 12 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv:2005029/ %R 10.1051/cocv:2005029 %G en %F COCV_2006__12_1_93_0
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
[1] and, On some control problems in fluid mechanics. Theoret. Comput. Fluid Dynam. 1 (1990) 303-325. | Zbl
[2] , Sobolev spaces. Academic Press, San Diego (1978). | Zbl
[3] , and, On an augmented Lagrangian SQP method for a class of optimal control problems in Banach spaces. Comput. Optim. Appl. 22 (2002) 369-398. | Zbl
[4] , Second-order analysis for control constrained optimal control problems of semilinear elliptic equations. Appl. Math. Optim. 38 (1998) 303-325. | Zbl
[5] and, Optimal control problems with partially polyhedric constraints. SIAM J. Control Optim. 37 (1999) 1726-1741. | Zbl
[6] , Analyse fonctionelle. Masson, Paris (1983). | Zbl | MR
[7] , 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
[8] and, Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431-1454. | Zbl
[9] and, Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. 21 (2002) 67-100. | Zbl
[10] , and, Second-order sufficient optimality conditions for a nonlinear elliptic control problem. J. Anal. Appl. 15 (1996) 687-707. | Zbl
[11] , and, Second-order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369-1391. | Zbl
[12] and, Navier-Stokes equations. The University of Chicago Press, Chicago (1988). | Zbl | MR
[13] and, Evolution problems I, Mathematical analysis and numerical methods for science and technology 5. Springer, Berlin (1992). | MR
[14] and, Optimal controls of Navier-Stokes equations. SIAM J. Control Optim. 32 (1994) 1428-1446. | Zbl
[15] ,, and, Optimality, stability, and convergence in optimal control. Appl. Math. Optim. 31 (1995) 297-326. | Zbl
[16] , 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
[17] and, Necessary and sufficient for optimal controls in viscous flow problems. Proc. Roy. Soc. Edinburgh 124 (1994) 211-251. | Zbl
[18] ., Flow control. Springer, New York (1995). | Zbl | MR
[19] and, The velocity tracking problem for Navier-Stokes flows with bounded distributed controls. SIAM J. Control Optim. 37 (1999) 1913-1945. | Zbl
[20] and, Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal. 37 (2000) 1481-1512. | Zbl
[21] , Optimal and instantaneous control of the instationary Navier-Stokes equations. Habilitation, TU Berlin (2002).
[22] and, Second-order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40 (2001) 925-946. | Zbl
[23] and, First- and second-order conditions in infinite-dimensional programming problems. Math. Programming 16 (1979) 98-110. | Zbl
[24] and, 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] and, Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dynam. Syst. 6 (2000) 431-450. | Zbl
[26] and, Lipschitz stability of optimal controls for the steady-state Navier-Stokes equations. Control Cybernet. 32 (2002) 683-705. | Zbl
[27] , Dynamic programming of the Navier-Stokes equations. Syst. Control Lett. 16 (1991) 299-307. | Zbl
[28] , Navier-Stokes equations. North Holland, Amsterdam (1979). | Zbl | MR
[29] , Lipschitz stability of solutions of linear-quadratic parabolic control problems with respect to perturbations. Dyn. Contin. Discrete Impulsive Syst. 7 (2000) 289-306. | Zbl
Cité par Sources :
