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Extremal Trajectories in a Time-Optimal Problem on the Group of Motions of a Plane with Admissible Control in a Circular Sector

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Abstract

We consider a time-optimal problem for a car model that can move forward on a plane and turn with a given minimum turning radius. Trajectories of this system are applicable in image processing for the detection of salient lines. We prove the controllability and existence of optimal trajectories. Applying the necessary optimality condition given by the Pontryagin maximum principle, we derive a Hamiltonian system for the extremals. We provide qualitative analysis of the Hamiltonian system and obtain explicit expressions for the extremal controls and trajectories.

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Acknowledgments

The authors thank the anonymous referee for valuable comments that helped to improve this paper.

Funding

The work of the first author (Sections 1, 4, 5) was supported by the Russian Science Foundation under grant no. 22-21-00877, https://rscf.ru/project/22-21-00877/.

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Authors and Affiliations

  1. Ailamazyan Program Systems Institute of Russian Academy of Sciences, Pereslavl-Zalessky, Yaroslavl Region, 152021, Russia

    Alexey P. Mashtakov & Yuri L. Sachkov

Authors
  1. Alexey P. Mashtakov
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  2. Yuri L. Sachkov
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Corresponding authors

Correspondence to Alexey P. Mashtakov or Yuri L. Sachkov.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Vol. 321, pp. 215–222 https://doi.org/10.4213/tm4341.

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Mashtakov, A.P., Sachkov, Y.L. Extremal Trajectories in a Time-Optimal Problem on the Group of Motions of a Plane with Admissible Control in a Circular Sector. Proc. Steklov Inst. Math. 321, 200–207 (2023). https://doi.org/10.1134/S0081543823020141

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  • DOI: https://doi.org/10.1134/S0081543823020141

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