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Time minimization problem on the group of motions of a plane with admissible control in a half-disk

We study a time minimization problem on the group of motions of a plane with admissible control in a half-disk. The considered control system describes a model of a car that can move forward on a plane and turn in place. Optimal trajectories of this system are used in image processing for the detection of salient lines. In particular, such trajectories are used in the analysis of medical images when searching for blood vessels in photos of the human retina. The problem is of interest in geometric control theory as a model example in which the set of admissible controls contains zero at the boundary. We prove complete controllability and the existence of optimal trajectories. By analyzing the Hamiltonian system of Pontryagin maximum principle we derive explicit formulas for extremal controls and trajectories. The optimality of extremals is partially investigated. The structure of optimal synthesis is described.