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Euler discretization for a class of nonlinear optimal control problems with control appearing linearly

Author

Listed:
  • Walter Alt

    (Friedrich-Schiller-Universität Jena)

  • Ursula Felgenhauer

    (Brandenburgische Technische Universität Cottbus-Senftenberg)

  • Martin Seydenschwanz

    (Siemens AG, Research in Digitalization and Automation)

Abstract

We investigate Euler discretization for a class of optimal control problems with a nonlinear cost functional of Mayer type, a nonlinear system equation with control appearing linearly and constraints defined by lower and upper bounds for the controls. Under the assumption that the cost functional satisfies a growth condition we prove for the discrete solutions Hölder type error estimates w.r.t. the mesh size of the discretization. If a stronger second-order optimality condition is satisfied the order of convergence can be improved. Numerical experiments confirm the theoretical findings.

Suggested Citation

  • Walter Alt & Ursula Felgenhauer & Martin Seydenschwanz, 2018. "Euler discretization for a class of nonlinear optimal control problems with control appearing linearly," Computational Optimization and Applications, Springer, vol. 69(3), pages 825-856, April.
    • Handle: RePEc:spr:coopap:v:69:y:2018:i:3:d:10.1007_s10589-017-9969-7
    DOI: 10.1007/s10589-017-9969-7
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    References listed on IDEAS

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    1. Klaus Deckelnick & Michael Hinze, 2012. "A note on the approximation of elliptic control problems with bang-bang controls," Computational Optimization and Applications, Springer, vol. 51(2), pages 931-939, March.
    2. Vili Dhamo & Fredi Tröltzsch, 2011. "Some aspects of reachability for parabolic boundary control problems with control constraints," Computational Optimization and Applications, Springer, vol. 50(1), pages 75-110, September.
    3. Stephen M. Robinson, 1976. "Regularity and Stability for Convex Multivalued Functions," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 130-143, May.
    4. Martin Seydenschwanz, 2015. "Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions," Computational Optimization and Applications, Springer, vol. 61(3), pages 731-760, July.
    5. Ursula Felgenhauer, 2016. "Discretization of semilinear bang-singular-bang control problems," Computational Optimization and Applications, Springer, vol. 64(1), pages 295-326, May.
    6. Alt, Walter & Schneider, Christopher & Seydenschwanz, Martin, 2016. "Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions," Applied Mathematics and Computation, Elsevier, vol. 287, pages 104-124.
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    Cited by:

    1. Gerardo Sánchez Licea, 2021. "Weak Measurable Optimal Controls for the Problems of Bolza," Mathematics, MDPI, vol. 9(2), pages 1-17, January.

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