Boundary of the Reachable Set of a Control System on the Plane under the Assumption of Linear Independence of Control Vector Fields

We consider the control system ẋ=u f(x)+v g(x) on the plane, where the smooth vector fields f and g are linearly independent at every point and the controls u and v are nonnegative. We provide a complete description of the reachable set from a fixed point of the plane. In particular, we show that its boundary consists of orbits of the vector fields and may be disconnected. In addition, several equivalent sufficient conditions for the connectedness of the boundary are given.