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EURO Journal on Computational Optimization

, Volume 4, Issue 1, pp 47–77 | Cite as

Dualization and discretization of linear-quadratic control problems with bang–bang solutions

  • Walter Alt
  • C. Yalçın Kaya
  • Christopher SchneiderEmail author
Original Paper

Abstract

We consider linear-quadratic (LQ) control problems, where the control variable appears linearly and is box-constrained. It is well-known that these problems exhibit bang–bang and singular solutions. We assume that the solution is of bang–bang type, which is computationally challenging to obtain. We employ a quadratic regularization of the LQ control problem by embedding the \(L^2\)-norm of the control variable into the cost functional. First, we find a dual problem guided by the methodology of Fenchel duality. Then we prove strong duality and the saddle point property, which together ensure that the primal solution can be recovered from the dual solution. We propose a discretization scheme for the dual problem, under which a diagram depicting the relations between the primal and dual problems and their discretization commutes. The commuting diagram ensures that, given convergence results for the discrete primal variables, discrete dual variables also converge to a solution of the dual problem with a similar error bound. We demonstrate via a simple but illustrative example that significant computational savings can be achieved by solving the dual, rather than the primal, problem.

Keywords

Linear-quadratic control Bang–bang control Duality Regularization Discretization 

Mathematics Subject Classification

49N10 49N15 49M25 49J30 49J15 

Notes

Acknowledgments

We would like to thank Radu Ioan Boţ und Regina S. Burachik for helpful discussions and also express our gratitude to the anonymous referees for valuable suggestions and comments.

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Copyright information

© EURO - The Association of European Operational Research Societies 2015

Authors and Affiliations

  • Walter Alt
    • 1
  • C. Yalçın Kaya
    • 2
  • Christopher Schneider
    • 1
    Email author
  1. 1.Fakultät für Mathematik und InformatikFriedrich-Schiller-UniversitätJenaGermany
  2. 2.School of Information Technology and Mathematical SciencesUniversity of South AustraliaMawson LakesAustralia

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