Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T08:09:32.805Z Has data issue: false hasContentIssue false
  • English
  • Français

Globalization of SQP-Methods in Control of the Instationary Navier-Stokes Equations

Published online by Cambridge University Press:  15 September 2002

Michael Hintermüller  and
Michael Hinze
Michael Hintermüller
Affiliation:
Department of Mathematics, Karl-Franzens University of Graz, A-8010 Graz, Austria. [email protected].
Michael Hinze
Affiliation:
Fakultät für Mathematik und Naturwissenschaften, TU-Dresden, D-01069 Dresden, Germany. [email protected].
Get access

Abstract

A numerically inexpensive globalization strategy of sequential quadratic programming methods (SQP-methods) for control of the instationary Navier Stokes equations is investigated. Based on the proper functional analytic setting a convergence analysis for the globalized methodis given. It is argued that the a priori formidable SQP-step can be decomposed into linear primal and linear adjoint systems, which is amenable for existing CFL-software. A report on a numerical test demonstrates the feasibility of the approach.

Keywords

Globalized SQP-methodline searchNavier Stokes equationsoptimal control.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 4 , July 2002 , pp. 725 - 746
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abergel, F. and Temam, R., On some Control Problems in Fluid Mechanics. Theoret. Comput. Fluid Dyn. 1 (1990) 303-325. CrossRef
Bänsch, E., An adaptive Finite-Element-Strategy for the three-dimensional time-dependent Navier-Stokes Equations. J. Comput. Math. 36 (1991) 3-28. CrossRef
D. Bertsekas, Nonlinear Programming. Athena Scientific, Belmont, Massachusetts (1995).
J.F. Bonnans et al., Optimisation Numérique. Math. Appl. 27 , Springer-Verlag, Berlin (1997).
Ghattas, O. and Bark, J.J., Optimal control of two-and three-dimensional incompressible Navier-Stokes Flows. J. Comput. Physics 136 (1997) 231-244. CrossRef
P.E. Gill et al., Practical Optimization. Academic Press, San Diego, California (1981).
R. Glowinski, Finite element methods for the numerical simulation of incompressible viscous flow. Introduction to the Control of the Navier-Stokes Equations. Lect. Appl. Math. 28 (1991).
W.A. Gruver and E. Sachs, Algorithmic Methods in Optimal Control. Res. Notes Math. 47 , Pitman, London (1980).
M. Heinkenschloss, Formulation and analysis of a sequential quadratic programming method for the optimal Dirichlet boundary control of Navier-Stokes flow, in Optimal Control: Theory, Algorithms, and Applications, Kluwer Academic Publishers B.V. (1998) 178-203.
M. Hintermüller, On a globalized augmented Lagrangian-SQP algorithm for nonlinear optimal control problems with box constraints, in Fast solution methods for discretized optimization problems, K.-H. Hoffmann, R.H.W. Hoppe and V. Schulz Eds., Internat. Ser. Numer. Math. 138 (2001) 139-153.
M. Hinze, Optimal and instantaneous control of the instationary Navier-Stokes equations, Habilitationsschrift (1999). Fachbereich Mathematik, Technische Universität Berlin, download see http://www.math.tu-dresden.de/~hinze/publications.html.
Hinze, M. and Kunisch, K., Second order methods for optimal control of time-dependent fluid flow. SIAM J. Optim. Control 40 (2001) 925-946. CrossRef
Hood, P. and Taylor, C., A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. & Fluids 1 (1973) 73-100.
C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations. SIAM (1995).
F.S. Kupfer, An infinite-dimensional convergence theory for reduced SQP-methods in Hilbert space. SIAM J. Optim. 6 (1996).
E. Polak, Optimization. Appl. Math. Sci. 124 , Springer-Verlag, New York (1997).
M.J.D. Powell, Variable metric methods for constrained optimization, in Mathematical Programming, The State of the Art, Eds. Bachem, Grötschel, Korte, Bonn (1982).
W.C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations. CBMS-NSF Regional Conference Series in Applied Mathematics 70 , SIAM, Philadelphia (1998).
K. Schittkowski, On the convergence of a sequential quadratic programming method with an augmented Lagrangian line search function. Math. Operationsforschung u. Statist, Ser. Optim. 14 (1983) 197-216.
R. Temam, Navier-Stokes Equations. North-Holland (1979).