The problem of constructing a two-dimensional reachable set for the Dubins car on a plane of geometric coordinates at a fixed terminal time $t_f$ is considered. Scalar control specifies the instantaneous angular velocity of rotation and is assumed to be continuous. The integral of the absolute value of control is considered to be limited. It is known that in this case the reachable set is not closed. In the paper, we investigate the closure of the reachable set using “impulse” controls. It has been established that each point of the closure can be reached using impulse control. In this case, the number of impulses does not exceed two. If there are two impulses, then the first one is realized at the initial instant $t_0 = 0$, and the second one is performed at the instant $(2∕3)t_f$. The proof of this fact is based on a special property of the ellipse focal radii. A parametric description of the boundary of the reachable set is given. Simulation results are presented.
Keywords: Dubins car, impulse control, focal radii of an ellipse, two-dimensional reachable set, parametric description of the boundary, numerical modeling
Received January 26, 2026
Revised February 9, 2026
Accepted February 12, 2026
Funding Agency: The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2026-737).
Valerii Semenovich Patsko, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: patsko@imm.uran.ru
Georgii Igorevich Trubnikov, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, Ural Federal University named after the first President of Russia B. N. Yeltsin, Yekaterinburg, 620000 Russia, e-mail: jora_it@mail.ru
Andrey Anatol’evich Fedotov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: andreyfedotov@mail.ru
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Cite this article as: V.S. Patsko, G.I. Trubnikov, A.A. Fedotov. Two-dimensional reachable set of Dubins car with impulse control. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 164–182.
