On the convexity of the reachable set with respect to a part of coordinates at small time intervals

We investigate the convexity of the reachable sets for some of the coordinates of nonlinear systems with integral constraints on the control at small time intervals. We have proved sufficient convexity conditions in the form of constraints on the asymptotics of the eigenvalues of the Gramian of the controllability of a linearized system for some of the coordinates. There are two nonlinear third-order systems under study as examples. The system linearized along a trajectory generated by zero control is uncontrollable, and the system in the other example is completely controllable. We investigate the sufficient conditions for convexity of projection of reachable sets. Numerical modeling has been carried out, demonstrating the non-convexity of some projections even for small time intervals.