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Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions

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  • Martin Seydenschwanz

Abstract

We analyze a combined regularization–discretization approach for a class of linear-quadratic optimal control problems. By choosing the regularization parameter $$\alpha $$ α with respect to the mesh size $$h$$ h of the discretization we approximate the optimal bang–bang control. Under weaker assumptions on the structure of the switching function we generalize existing convergence results and prove error estimates of order $${\mathcal {O}}(h^{1/(k+1)})$$ O ( h 1 / ( k + 1 ) ) with respect to the controllability index $$k$$ k . Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:61:y:2015:i:3:p:731-760
    DOI: 10.1007/s10589-015-9730-z
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    References listed on IDEAS

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    1. Walter Alt & Nils Bräutigam, 2009. "Finite-difference discretizations of quadratic control problems governed by ordinary elliptic differential equations," Computational Optimization and Applications, Springer, vol. 43(1), pages 133-150, May.
    2. M. Hinze & C. Meyer, 2010. "Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems," Computational Optimization and Applications, Springer, vol. 46(3), pages 487-510, July.
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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. 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.
    3. Nikolaus Daniels, 2018. "Tikhonov regularization of control-constrained optimal control problems," Computational Optimization and Applications, Springer, vol. 70(1), pages 295-320, May.
    4. 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.
    5. 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.
    6. Dang Hieu & Pham Kim Quy, 2023. "One-Step iterative method for bilevel equilibrium problem in Hilbert space," Journal of Global Optimization, Springer, vol. 85(2), pages 487-510, February.
    7. Dang Van Hieu & Jean Jacques Strodiot & Le Dung Muu, 2020. "An Explicit Extragradient Algorithm for Solving Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 476-503, May.
    8. Dang Hieu & Pham Ky Anh & Nguyen Hai Ha, 2021. "Regularization Proximal Method for Monotone Variational Inclusions," Networks and Spatial Economics, Springer, vol. 21(4), pages 905-932, December.
    9. 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.

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