Geodesic
Extremal Trajectories in a Time-Optimal Problem on the Group of Motions of a Plane with Admissible Control in a Circular Sector
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 215-222.

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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.
Keywords: geometric control, model of a car, extremal trajectories, Pontryagin maximum principle, group of motions of a plane. 
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Alexey P. Mashtakov; Yuri L. Sachkov. Extremal Trajectories in a Time-Optimal Problem on the Group of Motions of a Plane with Admissible Control in a Circular Sector. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 215-222. http://geodesic.mathdoc.fr/item/TM_2023_321_a13/

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