×

Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations. (English) Zbl 1111.49017

The authors establish sufficient optimality conditions for a class of optimal control problems governed by Navier-Stokes equations. In the steady state case the problem is: minimize the quadratic cost functional \(J(u):=\int_\Omega | y(u,x)-y_d(x)| ^2\,dx+\gamma/2 \int_\Omega | u(x)| ^2\,dx\) subject to the Navier-Stokes equations \(-\nu\triangle y+(y\cdot \nabla)y+\nabla p=u\) in \(\Omega\), \(\operatorname{div}y=0\) in \(\Omega\), \(y=0\) on \(\partial\Omega\), where \(\Omega\subset {\mathbb R}^n\) is bounded. The controls satisfy the unilateral conditions \(u_a(x)\leq u(x)\leq u_b(x)\) on \(\Omega\). Under a smallness assumption the authors show that an optimal control which satisfies the necessary condition of first order defines a (local) minimizer providing the second derivative of the associated Lagrange function is positive for all nonzero functions from \(L^2(\Omega)\) which are zero on the active sets of the control in consideration. The proof relies on the smallness assumption (A2), the quadratic structure of the cost functional and on the definition of the admissible set of controls.

MSC:

49K20 Optimality conditions for problems involving partial differential equations
35Q30 Navier-Stokes equations
76D55 Flow control and optimization for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] F. Abergel and R. Temam , On some control problems in fluid mechanics . Theoret. Comput. Fluid Dynam. 1 ( 1990 ) 303 - 325 . Zbl 0708.76106 · Zbl 0708.76106
[2] R.A. Adams , Sobolev spaces . Academic Press, San Diego ( 1978 ). Zbl 0314.46030 · 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 · 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 · 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 · Zbl 0945.49020
[6] H. Brezis , Analyse fonctionelle . Masson, Paris ( 1983 ). MR 697382 | Zbl 0511.46001 · 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 · 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 · 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 · 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 · Zbl 0962.49016
[12] P. Constantin and C. Foias , Navier-Stokes equations . The University of Chicago Press, Chicago ( 1988 ). MR 972259 | Zbl 0687.35071 · 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 · Zbl 0755.35001
[14] M. Desai and K. Ito , Optimal controls of Navier-Stokes equations . SIAM J. Control Optim. 32 ( 1994 ) 1428 - 1446 . Zbl 0813.35078 · 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 · 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 · 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 · Zbl 0800.49047
[18] M.D. Gunzburger Ed ., Flow control. Springer, New York (1995). MR 1348639 | Zbl 0816.00037 · 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 · 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 · Zbl 0963.35150
[21] M. Hinze , Optimal and instantaneous control of the instationary Navier-Stokes equations . Habilitation, TU Berlin ( 2002 ). · Zbl 1073.49025
[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 · 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 · 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 · 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 · Zbl 1127.49021
[27] S. Sritharan , Dynamic programming of the Navier-Stokes equations . Syst. Control Lett. 16 ( 1991 ) 299 - 307 . Zbl 0737.49021 · Zbl 0737.49021
[28] R. Temam , Navier-Stokes equations . North Holland, Amsterdam ( 1979 ). MR 603444 | Zbl 0426.35003 · 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 · Zbl 0954.49017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.