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A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

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Abstract

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Würzburg, Am Hubland, 97074, Würzburg, Germany

    Matthias Gerdts

  2. Department of Mathematics, University of Hamburg, Bundesstr. 55, 20146, Hamburg, Germany

    Martin Kunkel

Authors
  1. Matthias Gerdts
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  2. Martin Kunkel
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Correspondence to Matthias Gerdts.

Additional information

M. Gerdts is supported by DFG grant GE 1163/5-1.

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Gerdts, M., Kunkel, M. A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems. Comput Optim Appl 48, 601–633 (2011). https://doi.org/10.1007/s10589-009-9275-0

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  • DOI: https://doi.org/10.1007/s10589-009-9275-0

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