Properties of solutions in the Dubins car control problem

This paper addresses the time-optimal control problem of the Dubins car, which is closely related to the problem of constructing the shortest curve with bounded curvature between two points in a plane. This connection allows researchers to apply both geometric methods and control theory techniques during their investigations. It is established that the time-optimal control for the Dubins car is a piecewise constant function with no more than two switchings. This characteristic enables the categorization of all such controls into several types, facilitating the examination of the solutions to the control problem for each type individually. The paper derives explicit formulas for determining the switching times of the control signal. In each case, necessary and sufficient conditions for the existence of solutions are obtained. For certain control types, the uniqueness of optimal solutions is established. Additionally, the dependence of the movement time on the initial and terminal conditions is studied.