Asymptotic Behaviour of Controllability Gramians and Convexity of Small-Time Reachable Sets

We study the convexity of reachable sets for a nonlinear control-affine system on a small time interval under integral constraints on control variables. The convexity of a reachable set for a nonlinear control system under integral constraints was proved by B. Polyak under assumption that linearization of the system is controllable and L-2 norms of controls are bounded from above by a sufficiently small number. Using this result we propose sufficient conditions for the convexity of reachable sets of a control-affine system on a small time interval. These conditions are based on the estimates for the asymptotics of the minimal eigenvalue of controllability Gramian of the system linearization, which depends on a small parameter (a length of the interval). A procedure for calculating estimates using the expansion of the Gramian into a series with respect to the small parameter degrees is described and some illustrative examples are presented.