Abstract
This paper addresses the time-optimal control problem for a kinematic model known as the Dubins car. The main goal of the study is to determine control inputs that enable the system to reach a given final state in minimum time. Such problems play an important role in optimal control theory and have applications in robotics, autonomous vehicle navigation, and trajectory planning tasks. Particular attention is given to analyzing the optimal trajectories of the Dubins car. It is known that, in the absence of state constraints, optimal trajectories have a specific structure and consist of segments of circular arcs and straight lines. These trajectories allow the system to move most efficiently while respecting the vehicle’s turning radius limitations. The study also considers the problem under state constraints, which may be represented as forbidden regions in the workspace. The presence of such constraints significantly affects the shape of optimal trajectories and complicates the control problem. Analyzing these conditions allows a more complete investigation of the properties of optimal solutions and helps identify feasible modes of system motion.