Reachable set of the Dubins car with an integral constraint on control

A three-dimensional reachable set for a nonlinear controlled object “Dubins car” is investigated. The control is the angular velocity of rotation of the linear velocity vector. The control action is constrained by an integral quadratic constraint. Based on the Pontryagin maximum principle, a description of the motions generating the boundary of the reachable set is given. The motions leading to the boundary are globally optimal Euler elastics. Simulation results are presented.