Two-Dimensional Reachable Set of Dubins Car with Both Geometric and Integral Constraints on Control

Within the framework of the mathematical model “Dubins car,” the reachable set on the plane is investigated. We assume that a 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 Pontryagin’s maximum principle formulated for motions leading to its boundary. We study the structure of emerging extreme motions. These motions consist of segments that are Euler elasticae, and segments with constant control. Equations for the constants of the conjugate system of the maximum principle are written out. On their basis, we introduce a method for a one-parameter description of the boundary of the reachable set. We provide examples of numerical calculation of the boundary of the reachable set. The difference between the resulting set and the set that is the intersection of the reachable set only for the case of the geometric constraint, and the reachable set only for the integral constraint is demonstrated.