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 .
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}, mrnumber = {2192070}, zbl = {1111.49017}, 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 SP - 93 EP - 119 VL - 12 IS - 1 PB - EDP-Sciences UR - http://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 http://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. http://www.numdam.org/articles/10.1051/cocv:2005029/
[1] On some control problems in fluid mechanics. Theoret. Comput. Fluid Dynam. 1 (1990) 303-325. | Zbl
and ,[2] Sobolev spaces. Academic Press, San Diego (1978). | Zbl
,[3] On an augmented Lagrangian SQP method for a class of optimal control problems in Banach spaces. Comput. Optim. Appl. 22 (2002) 369-398. | Zbl
, and ,[4] Second-order analysis for control constrained optimal control problems of semilinear elliptic equations. Appl. Math. Optim. 38 (1998) 303-325. | Zbl
,[5] Optimal control problems with partially polyhedric constraints. SIAM J. Control Optim. 37 (1999) 1726-1741. | Zbl
and ,[6] Analyse fonctionelle. Masson, Paris (1983). | MR | Zbl
,[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] Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431-1454. | Zbl
and ,[9] Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. 21 (2002) 67-100. | Zbl
and ,[10] Second-order sufficient optimality conditions for a nonlinear elliptic control problem. J. Anal. Appl. 15 (1996) 687-707. | Zbl
, and ,[11] Second-order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369-1391. | Zbl
, and ,[12] Navier-Stokes equations. The University of Chicago Press, Chicago (1988). | MR | Zbl
and ,[13] Evolution problems I, Mathematical analysis and numerical methods for science and technology 5. Springer, Berlin (1992). | MR
and ,[14] Optimal controls of Navier-Stokes equations. SIAM J. Control Optim. 32 (1994) 1428-1446. | Zbl
and ,[15] Optimality, stability, and convergence in optimal control. Appl. Math. Optim. 31 (1995) 297-326. | Zbl
, , and ,[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] Necessary and sufficient for optimal controls in viscous flow problems. Proc. Roy. Soc. Edinburgh 124 (1994) 211-251. | Zbl
and , ., Flow control. Springer, New York (1995). |[19] The velocity tracking problem for Navier-Stokes flows with bounded distributed controls. SIAM J. Control Optim. 37 (1999) 1913-1945. | Zbl
and ,[20] Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal. 37 (2000) 1481-1512. | Zbl
and ,[21] Optimal and instantaneous control of the instationary Navier-Stokes equations. Habilitation, TU Berlin (2002).
,[22] Second-order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40 (2001) 925-946. | Zbl
and ,[23] First- and second-order conditions in infinite-dimensional programming problems. Math. Programming 16 (1979) 98-110. | Zbl
and ,[24] 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.
and ,[25] Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dynam. Syst. 6 (2000) 431-450. | Zbl
and ,[26] Lipschitz stability of optimal controls for the steady-state Navier-Stokes equations. Control Cybernet. 32 (2002) 683-705. | Zbl
and ,[27] Dynamic programming of the Navier-Stokes equations. Syst. Control Lett. 16 (1991) 299-307. | Zbl
,[28] Navier-Stokes equations. North Holland, Amsterdam (1979). | MR | Zbl
,[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 :