Globalization of SQP-methods in control of the instationary Navier-Stokes equations

Hintermüller, Michael; Hinze, Michael

ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 4, pp. 725-746.

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 method is 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.

[1] F. Abergel and R. Temam, On some Control Problems in Fluid Mechanics. Theoret. Comput. Fluid Dyn. 1 (1990) 303-325. | Zbl

[2] E. Bänsch, An adaptive Finite-Element-Strategy for the three-dimensional time-dependent Navier-Stokes Equations. J. Comput. Math. 36 (1991) 3-28. | Zbl

[3] D. Bertsekas, Nonlinear Programming. Athena Scientific, Belmont, Massachusetts (1995). | Zbl

[4] J.F. Bonnans et al., Optimisation Numérique. Math. Appl. 27, Springer-Verlag, Berlin (1997). | MR | Zbl

[5] O. Ghattas and J.J. Bark, Optimal control of two-and three-dimensional incompressible Navier-Stokes Flows. J. Comput. Physics 136 (1997) 231-244. | Zbl

[6] P.E. Gill et al., Practical Optimization. Academic Press, San Diego, California (1981). | MR | Zbl

[7] 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). | MR | Zbl

[8] W.A. Gruver and E. Sachs, Algorithmic Methods in Optimal Control. Res. Notes Math. 47, Pitman, London (1980). | MR | Zbl

[9] 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. | Zbl

[10] 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. | Zbl

[11] 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/$\tilde{}$hinze/publications.html.

[12] M. Hinze and K. Kunisch, Second order methods for optimal control of time-dependent fluid flow. SIAM J. Optim. Control 40 (2001) 925-946. | Zbl

[13] P. Hood and C. Taylor, A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. & Fluids 1 (1973) 73-100. | Zbl

[14] C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations. SIAM (1995). | MR | Zbl

[15] F.S. Kupfer, An infinite-dimensional convergence theory for reduced SQP-methods in Hilbert space. SIAM J. Optim. 6 (1996). | MR | Zbl

[16] E. Polak, Optimization. Appl. Math. Sci. 124, Springer-Verlag, New York (1997). | Zbl

[17] M.J.D. Powell, Variable metric methods for constrained optimization, in Mathematical Programming, The State of the Art, Eds. Bachem, Grötschel, Korte, Bonn (1982). | MR | Zbl

[18] W.C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations. CBMS-NSF Regional Conference Series in Applied Mathematics 70, SIAM, Philadelphia (1998). | MR | Zbl

[19] 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. | Zbl

[20] R. Temam, Navier-Stokes Equations. North-Holland (1979). | MR | Zbl