IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v81y2022i3d10.1007_s10589-022-00350-6.html
   My bibliography  Save this article

Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation

Author

Listed:
  • Elisha R. Pager

    (University of Florida)

  • Anil V. Rao

    (University of Florida)

Abstract

A method is developed for solving bang-bang and singular optimal control problems using adaptive Legendre–Gauss–Radau collocation. The method is divided into several parts. First, a structure detection method is developed that identifies switch times in the control and analyzes the corresponding switching function for segments where the solution is either bang-bang or singular. Second, after the structure has been detected, the domain is decomposed into multiple domains such that the multiple-domain formulation includes additional decision variables that represent the switch times in the optimal control. In domains classified as bang-bang, the control is set to either its upper or lower limit. In domains identified as singular, the objective function is augmented with a regularization term to avoid the singular arc. An iterative procedure is then developed for singular domains to obtain a control that lies in close proximity to the singular control. The method is demonstrated on four examples, three of which have either a bang-bang and/or singular optimal control while the fourth has a smooth and nonsingular optimal control. The results demonstrate that the method of this paper provides accurate solutions to problems whose solutions are either bang-bang or singular when compared against previously developed mesh refinement methods that are not tailored for solving nonsmooth and/or singular optimal control problems, and produces results that are equivalent to those obtained using previously developed mesh refinement methods for optimal control problems whose solutions are smooth.

Suggested Citation

  • Elisha R. Pager & Anil V. Rao, 2022. "Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation," Computational Optimization and Applications, Springer, vol. 81(3), pages 857-887, April.
  • Handle: RePEc:spr:coopap:v:81:y:2022:i:3:d:10.1007_s10589-022-00350-6
    DOI: 10.1007/s10589-022-00350-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-022-00350-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10589-022-00350-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. William W. Hager & Hongyan Hou & Subhashree Mohapatra & Anil V. Rao & Xiang-Sheng Wang, 2019. "Correction to: Convergence rate for a Radau hp collocation method applied to constrained optimal control," Computational Optimization and Applications, Springer, vol. 74(1), pages 315-316, September.
    2. William W. Hager & Hongyan Hou & Subhashree Mohapatra & Anil V. Rao & Xiang-Sheng Wang, 2019. "Convergence rate for a Radau hp collocation method applied to constrained optimal control," Computational Optimization and Applications, Springer, vol. 74(1), pages 275-314, September.
    3. William W. Hager & Hongyan Hou & Anil V. Rao, 2016. "Convergence Rate for a Gauss Collocation Method Applied to Unconstrained Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 801-824, June.
    4. C.Y. Kaya & J.L. Noakes, 2003. "Computational Method for Time-Optimal Switching Control," Journal of Optimization Theory and Applications, Springer, vol. 117(1), pages 69-92, April.
    5. M. Soledad Aronna & J. Frédéric Bonnans & Pierre Martinon, 2013. "A Shooting Algorithm for Optimal Control Problems with Singular Arcs," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 419-459, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wanchun Chen & Wenhao Du & William W. Hager & Liang Yang, 2019. "Bounds for integration matrices that arise in Gauss and Radau collocation," Computational Optimization and Applications, Springer, vol. 74(1), pages 259-273, September.
    2. Joseph D. Eide & William W. Hager & Anil V. Rao, 2021. "Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions," Journal of Optimization Theory and Applications, Springer, vol. 191(2), pages 600-633, December.
    3. Zhang, Zemian & Chen, Xuesong, 2021. "A conjugate gradient method for distributed optimal control problems with nonhomogeneous Helmholtz equation," Applied Mathematics and Computation, Elsevier, vol. 402(C).
    4. C. Y. Kaya & J. M. Martínez, 2007. "Euler Discretization and Inexact Restoration for Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 134(2), pages 191-206, August.
    5. William W. Hager & Hongyan Hou & Subhashree Mohapatra & Anil V. Rao & Xiang-Sheng Wang, 2019. "Convergence rate for a Radau hp collocation method applied to constrained optimal control," Computational Optimization and Applications, Springer, vol. 74(1), pages 275-314, September.
    6. Michael McAsey & Libin Mou & Weimin Han, 2012. "Convergence of the forward-backward sweep method in optimal control," Computational Optimization and Applications, Springer, vol. 53(1), pages 207-226, September.
    7. Ursula Felgenhauer, 2016. "Discretization of semilinear bang-singular-bang control problems," Computational Optimization and Applications, Springer, vol. 64(1), pages 295-326, May.
    8. C. Kaya & Helmut Maurer, 2014. "A numerical method for nonconvex multi-objective optimal control problems," Computational Optimization and Applications, Springer, vol. 57(3), pages 685-702, April.
    9. Xiang Wu & Kanjian Zhang & Changyin Sun, 2013. "Parameter Tuning of Multi-Proportional-Integral-Derivative Controllers Based on Optimal Switching Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 454-472, November.
    10. Kaya, C. Yalcin, 2004. "Time-optimal switching control for the US cocaine epidemic," Socio-Economic Planning Sciences, Elsevier, vol. 38(1), pages 57-72, March.
    11. Yi Jiang & Yi He & Jie Sun, 2011. "Subdifferential properties of the minimal time function of linear control systems," Journal of Global Optimization, Springer, vol. 51(3), pages 395-412, November.
    12. Wenhui Luo & Xuewen Tan & Xiufen Zou & Qing Tan, 2023. "Optimal Treatment of Prostate Cancer Based on State Constraint," Mathematics, MDPI, vol. 11(19), pages 1-17, September.
    13. K. H. Wong & H. W. J. Lee & C. K. Chan & C. Myburgh, 2013. "Control Parametrization and Finite Element Method for Controlling Multi-species Reactive Transport in an Underground Channel," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 168-187, April.
    14. G. Vossen, 2010. "Switching Time Optimization for Bang-Bang and Singular Controls," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 409-429, February.
    15. Nahid Banihashemi & C. Yalçın Kaya, 2013. "Inexact Restoration for Euler Discretization of Box-Constrained Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 726-760, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:81:y:2022:i:3:d:10.1007_s10589-022-00350-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.