Within the framework of the mathematical model ``Dubins car'', the reachable set on the plane is investigated. It is assumed that scalar control is constrained by a combined constraint. It includes a geometric constraint on the instantaneous control values and an integral quadratic constraint on the control as a whole. The construction of the reachable set is based on the Pontryagin maximum principle formulated for motions arriving at its boundary. The structure of emerging extreme motions is investigated. These motions consist of parts that are Euler elasticae and parts with constant control. Formulas for finding the constants of the conjugate system of the maximum principle are written out. On their basis, a method of one-parameter description of the reachable set boundary is introduced. Examples of numerical calculations of the reachable set boundary are given. The difference between the resulting set and the set that is the intersection of two reachable sets constructed only for the case of the geometric constraint and only the integral constraint is shown.
Keywords: Dubins car, geometric and integral constraints on control, Pontryagin maximum principle, two-dimensional reachable set, parametric description of the boundary, Euler elasticae, numerical modelling
Received February 15, 2025
Revised March 31, 2025
Accepted April 1, 2025
Published online April 7, 2025
Valerii S. Patsko, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: patsko@imm.uran.ru
Georgii I.Trubnikov, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, Ural Federal University named after the first President of Russia B.N.Yeltsin, Yekaterinburg, 620000 Russia, e-mail: jora_it@mail.ru
Andrey A. Fedotov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 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 both geometric and integral constraints on control. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol.31, no. 2, pp. 162–180
