Skip to main content
SpringerLink
Log in

A Semi-smooth Newton Method for Regularized State-constrained Optimal Control of the Navier-Stokes Equations

  • Published:
Computing Aims and scope Submit manuscript

Abstract

In this paper, we study semi-smooth Newton methods for the numerical solution of regularized pointwise state-constrained optimal control problems governed by the Navier-Stokes equations. After deriving an appropriate optimality system for the original problem, a class of Moreau-Yosida regularized problems is introduced and the convergence of their solutions to the original optimal one is proved. For each regularized problem a semi-smooth Newton method is applied and its local superlinear convergence verified. Finally, selected numerical results illustrate the behavior of the method and a comparison between the max-min and the Fischer-Burmeister as complementarity functionals is carried out.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • M. Bergounioux K. Kunisch (2002) ArticleTitlePrimal dual strategy for state-constrained optimal control problems Comput. Optim. Appl. 22 193–224 Occurrence Handle10.1023/A:1015489608037 Occurrence Handle1911062

    Article  MathSciNet  Google Scholar 

  • M. Bergounioux K. Kunisch (2002) ArticleTitleOn the structure of Lagrange multipliers for state-constrained optimal control problems Sys. Control Lett. 48 160–176 Occurrence Handle2020634

    MathSciNet  Google Scholar 

  • M. Bergounioux M. Hadou M. Hintermüller K. Kunisch (2000) ArticleTitleA comparison of interior point methods and a Moreau-Yosida based active set strategy for constrained optimal control problems SIAM J. Optim. 11 495–521 Occurrence Handle10.1137/S1052623498343131 Occurrence Handle1787272

    Article  MathSciNet  Google Scholar 

  • Brézis, H.: Análisis funcional. Alianza Editorial 1984.

  • E. Casas (1986) ArticleTitleControl of an elliptic problem with pointwise state constraints SIAM J. Control Optim. 24 1309–1318 Occurrence Handle10.1137/0324078 Occurrence Handle861100

    Article  MathSciNet  Google Scholar 

  • E. Casas (1993) ArticleTitleBoundary control of semilinear elliptic equations with pointwise state constraints SIAM J. Control Optim. 31 993–1006 Occurrence Handle10.1137/0331044 Occurrence Handle1227543

    Article  MathSciNet  Google Scholar 

  • Casas, E., Mateos, M., Raymond, J.-P.: Error estimates for the numerical approximation of a distributed control problem for the steady-state Navier-Stokes equations. Preprint.

  • B. Chen X. Chen C. Kanzow (2000) ArticleTitleA penalized Fischer-Burmeister NCP-function Math. Programming, Ser. A 88 211–216 Occurrence Handle10.1007/PL00011375 Occurrence Handle1765901

    Article  MathSciNet  Google Scholar 

  • J. C. Reyes Particlede los (2005) ArticleTitleA primal-dual active set method for bilaterally control constrained optimal control of the Navier-Stokes equations Numer. Funct. Anal. Optim. 25 657–683 Occurrence Handle10.1081/NFA-200045798

    Article  Google Scholar 

  • de los Reyes, J. C., Griesse, R.: State constrained optimal control of the Navier-Stokes equations. RICAM Report 2005–18, 2005.

  • J. C. Reyes Particlede los K. Kunisch (2005) ArticleTitleA semi-smooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations Nonlinear Anal. Theor. Meth. Appl. 62 1289–1316 Occurrence Handle10.1016/j.na.2005.04.035

    Article  Google Scholar 

  • M. Hintermüller K. Ito K. Kunisch (2003) ArticleTitleThe primal dual active set strategy as a semi-smooth Newton method SIAM J. Optim. 13 865–888 Occurrence Handle10.1137/S1052623401383558

    Article  Google Scholar 

  • K. Ito K. Kunisch (2003) ArticleTitleSemi-smooth Newton methods for state constrained optimal control problems Sys. Control Lett. 50 221–228 Occurrence Handle10.1016/S0167-6911(03)00156-7 Occurrence Handle2010814

    Article  MathSciNet  Google Scholar 

  • H. Jiang L. Qi (1997) ArticleTitleA new nonsmooth equations approach to nonlinear complementarity problems SIAM J. Control Optim. 35 178–193 Occurrence Handle10.1137/S0363012994276494 Occurrence Handle1430288

    Article  MathSciNet  Google Scholar 

  • C. Kanzow (2004) ArticleTitleInexact semi-smooth Newton methods for large scale complementarity problems Optim. Meth. Software 19 IssueID3–4 309–325 Occurrence Handle10.1080/10556780310001636369 Occurrence Handle2064249

    Article  MathSciNet  Google Scholar 

  • K. Kunisch X. Marduel (2001) ArticleTitleSuboptimal control of transient nonisothermal viscoelastic fluid flows Phys. Fluids 13 2478–2491 Occurrence Handle10.1063/1.1384869 Occurrence Handle1855137

    Article  MathSciNet  Google Scholar 

  • Rudin, W.: Real and complex analysis. McGraw-Hill 1966.

  • Temam, R.: Navier-Stokes equations: Theory and numerical analysis. Rhode Island AMS Chelsea Publishing 2001.

  • S. Turek (1999) Efficient solvers for incompressible flow problems Springer Berlin Occurrence Handle0930.76002

    MATH  Google Scholar 

  • M. Ulbrich (2003) ArticleTitleConstrained optimal control of Navier-Stokes flow by semi-smooth Newton methods Sys. Control Lett. 48 297–311 Occurrence Handle10.1016/S0167-6911(02)00274-8 Occurrence Handle2020646

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, EPN Quito, Ecuador and Institute for Mathematics, TU, Berlin, Germany

    J. C. de los Reyes

  2. Institute for Mathematics and Scientific Computing, Karl-Franzens-University of Graz, Austria

    K. Kunisch

Authors
  1. J. C. de los Reyes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Kunisch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. C. de los Reyes.

Rights and permissions

Reprints and permissions

About this article

Cite this article

de los Reyes, J.C., Kunisch, K. A Semi-smooth Newton Method for Regularized State-constrained Optimal Control of the Navier-Stokes Equations. Computing 78, 287–309 (2006). https://doi.org/10.1007/s00607-006-0183-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-006-0183-1

Mathematics Subject Classifications

Keywords

Search

Navigation