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Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions

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  • Alt, Walter
  • Schneider, Christopher
  • Seydenschwanz, Martin

Abstract

We analyze the implicit Euler discretization for a class of convex linear-quadratic optimal control problems with control appearing linearly. Constraints are defined by lower and upper bounds for the controls, and the cost functional may depend on a regularization parameter ν. Without any structural assumption on the optimal control we prove convergence of order 1 w.r.t. the mesh size for the discrete optimal values. Under the additional assumption that the optimal control is of bang-bang type and the switching function satisfies a growth condition around their zeros we show that the solutions are calm functions of perturbation and regularization parameters. By applying this result to the implicit Euler discretization we improve existing error estimates for discretizations based on the explicit Euler method. Numerical experiments confirm the theoretical findings and demonstrate the usefulness of implicit methods and regularization in case of bang-bang controls.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:287-288:y:2016:i::p:104-124
    DOI: 10.1016/j.amc.2016.04.028
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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. 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.
    3. Walter Alt & C. Yalçın Kaya & Christopher Schneider, 2016. "Dualization and discretization of linear-quadratic control problems with bang–bang solutions," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 47-77, February.
    4. 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.
    5. Matthias Gerdts & Björn Hüpping, 2012. "Virtual control regularization of state constrained linear quadratic optimal control problems," Computational Optimization and Applications, Springer, vol. 51(2), pages 867-882, March.
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    Cited by:

    1. T. Scarinci & V. M. Veliov, 2018. "Higher-order numerical scheme for linear quadratic problems with bang–bang controls," Computational Optimization and Applications, Springer, vol. 69(2), pages 403-422, March.
    2. Liang, Yuling & Zhang, Huaguang & Zhang, Juan & Luo, Yanhong, 2021. "Integral reinforcement learning-based guaranteed cost control for unknown nonlinear systems subject to input constraints and uncertainties," Applied Mathematics and Computation, Elsevier, vol. 408(C).
    3. 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.
    4. J. Preininger & P. T. Vuong, 2018. "On the convergence of the gradient projection method for convex optimal control problems with bang–bang solutions," Computational Optimization and Applications, Springer, vol. 70(1), pages 221-238, May.

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