For the kinematic model of the Dubins car, a well-known description of the two-dimensional reachable set in the plane of geometric coordinates at the instant $t_f>0$ is provided if the scalar control on the interval $[0,t_f]$ is modulo bounded to a given number $\nu>0$. In this paper, an integral constraint on the control module's consumption is added to this geometric constraint. Under the condition $\nu{t_f} \leq 2\pi$, an effective description of the reachable set boundary is obtained. The structure of the boundary is investigated. Its characteristic regions are determined by three types of extremal controls satisfying Pontryagin maximum principle. The double constraint is expressed in the fact that for two of the three types of extremal motions specific to this problem, instead of a switching line, a stripe appears in the plane of geometric coordinates within which the control is zero. Using numerical simulation, a comparison of reachable sets with a double control constraint was made with reachable sets for the constraints considered separately.
Keywords: Dubins car, geometric constraint on control, impulse constraint, combined constraint, two-dimensional reachable set, Pontryagin maximum principle, parametric description of the boundary, numerical modeling
Received May 14, 2026
Revised June 17, 2026
Accepted June 22, 2026
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, 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 the Dubins car under double constraint on control: geometric and impulse. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 3, pp. 183–203.
