Computational Optimization and Applications

, Volume 51, Issue 2, pp 835–866 | Cite as

Multigrid second-order accurate solution of parabolic control-constrained problems

Article

Abstract

A mesh-independent and second-order accurate multigrid strategy to solve control-constrained parabolic optimal control problems is presented. The resulting algorithms appear to be robust with respect to change of values of the control parameters and have the ability to accommodate constraints on the control also in the limit case of bang-bang control. Central to the development of these multigrid schemes is the design of iterative smoothers which can be formulated as local semismooth Newton methods. The design of distributed controls is considered to drive nonlinear parabolic models to follow optimally a given trajectory or attain a final configuration. In both cases, results of numerical experiments and theoretical twogrid local Fourier analysis estimates demonstrate that the proposed schemes are able to solve parabolic optimality systems with textbook multigrid efficiency. Further results are presented to validate second-order accuracy and the possibility to track a trajectory over long time intervals by means of a receding-horizon approach.

Keywords

Multigrid methods Semismooth Newton method Parabolic partial differential equations Optimal control theory 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institut für Mathematik und Wissenschaftliches RechnenKarl-Franzens-Universität GrazGrazAustria
  2. 2.Research Group on Optimization, Departamento de MatemáticaEscuela Politécnica NacionalQuitoEcuador
  3. 3.Dipartimento e Facoltà di Ingegneria, Palazzo Dell’Aquila Bosco LucarelliUniversità degli Studi del SannioBeneventoItalia

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