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. 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.