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Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations

Published online by Cambridge University Press:  15 December 2005

Fredi Tröltzsch  and
Daniel Wachsmuth
Fredi Tröltzsch
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Str. d. 17. Juni 136, 10632 Berlin, Germany; troeltz@math.tu-berlin.de; wachsmut@math.tu-berlin.de
Daniel Wachsmuth
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Str. d. 17. Juni 136, 10632 Berlin, Germany; troeltz@math.tu-berlin.de; wachsmut@math.tu-berlin.de
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Abstract

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 Ls-neighborhood, whereby the underlying analysis allows to use weaker norms than L∞.

Keywords

Optimal controlNavier-Stokes equationscontrol constraintssecond-order optimality conditionsfirst-order necessary conditions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 1 , January 2006 , pp. 93 - 119
Copyright
© EDP Sciences, SMAI, 2006

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References

Abergel, F. and Temam, R., On some control problems in fluid mechanics. Theoret. Comput. Fluid Dynam. 1 (1990) 303–325. CrossRef
R.A. Adams, Sobolev spaces. Academic Press, San Diego (1978).
Arada, N., Raymond, J.-P. and Tröltzsch, F., On an augmented Lagrangian SQP method for a class of optimal control problems in Banach spaces. Comput. Optim. Appl. 22 (2002) 369–398. CrossRef
Bonnans, J.F., Second-order analysis for control constrained optimal control problems of semilinear elliptic equations. Appl. Math. Optim. 38 (1998) 303–325. CrossRef
Bonnans, J.F. and Zidani, H., Optimal control problems with partially polyhedric constraints. SIAM J. Control Optim. 37 (1999) 1726–1741. CrossRef
H. Brezis, Analyse fonctionelle. Masson, Paris (1983).
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).
Casas, E. and Mateos, M., Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431–1454. CrossRef
Casas, E. and Mateos, M., Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. 21 (2002) 67–100.
Casas, E., Tröltzsch, F. and Unger, A., Second-order sufficient optimality conditions for a nonlinear elliptic control problem. J. Anal. Appl. 15 (1996) 687–707.
Casas, E., Tröltzsch, F. and Unger, A., Second-order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369–1391. CrossRef
P. Constantin and C. Foias, Navier-Stokes equations. The University of Chicago Press, Chicago (1988).
R. Dautray and J.L. Lions, Evolution problems I, Mathematical analysis and numerical methods for science and technology 5. Springer, Berlin (1992).
Desai, M. and Ito, K., Optimal controls of Navier-Stokes equations. SIAM J. Control Optim. 32 (1994) 1428–1446. CrossRef
Dontchev, A.L., Hager, W.W., Poore, A.B. and Yang, B., Optimality, stability, and convergence in optimal control. Appl. Math. Optim. 31 (1995) 297–326. CrossRef
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.
Fattorini, H.O. and Sritharan, S., Necessary and sufficient for optimal controls in viscous flow problems. Proc. Roy. Soc. Edinburgh 124 (1994) 211–251. CrossRef
M.D. Gunzburger Ed., Flow control. Springer, New York (1995).
Gunzburger, M.D. and Manservisi, S., The velocity tracking problem for Navier-Stokes flows with bounded distributed controls. SIAM J. Control Optim. 37 (1999) 1913–1945. CrossRef
Gunzburger, M.D. and Manservisi, S., Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal. 37 (2000) 1481–1512. CrossRef
M. Hinze, Optimal and instantaneous control of the instationary Navier-Stokes equations. Habilitation, TU Berlin (2002).
Hinze, M. and Kunisch, K., Second-order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40 (2001) 925–946. CrossRef
Maurer, H. and Zowe, J., First- and second-order conditions in infinite-dimensional programming problems. Math. Programming 16 (1979) 98–110. CrossRef
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.
Raymond, J.-P. and Tröltzsch, F., Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dynam. Syst. 6 (2000) 431–450.
Roubíček, T. and Tröltzsch, F., Lipschitz stability of optimal controls for the steady-state Navier-Stokes equations. Control Cybernet. 32 (2002) 683–705.
Sritharan, S., Dynamic programming of the Navier-Stokes equations. Syst. Control Lett. 16 (1991) 299–307. CrossRef
R. Temam, Navier-Stokes equations. North Holland, Amsterdam (1979).
Tröltzsch, F., Lipschitz stability of solutions of linear-quadratic parabolic control problems with respect to perturbations. Dyn. Contin. Discrete Impulsive Syst. 7 (2000) 289–306.