Analytical Description of the Two-Dimensional Reachable Set of the Dubins Car with an Integral Constraint on Control

The kinematic model of the “Dubins car” is investigated. Unlike the canonical case, the control (angular turning velocity) is constrained not by a geometric constraint on instantaneous values, but by an integral quadratic constraint, characterizing energy costs. The problem of constructing the two-dimensional reachable set in the plane of motion is posed. The solution uses Pontryagin maximum principle and the theory of elliptic integrals, as well as Jacobi elliptic functions. It is proved that controls leading to the boundary of the reachable set change their sign no more than once. A parametric description of the curves constituting the boundary of the set under study is given. The results of numerical modeling are presented and a comparison is made with the known results of constructing the reachable set under a geometric constraint on control.