On convexity of small-time reachable sets of nonlinear control systems

The convexity of reachable sets plays an essential role in the development of algorithms for solving optimal control problems and problems of feedback control. For nonlinear control systems the reachable sets are generally not convex and may have a rather complicated structure. However, for systems with integral quadratic constraints on the control B. Polyak showed that the reachable sets are convex if the linearization of the system is controllable and control inputs are restricted from above in L2 norm by a sufficiently small number. In the present paper we use this result to prove sufficient conditions for the convexity of reachable sets of a nonlinear control-affine system on small time intervals, assuming that control resources are limited by a given (not necessarily small) value. These conditions are based on the asymptotics for the minimal eigenvalue of the controllability Gramian of system linearization as a function of the length of the time interval. We prove the asymptotics for a linear time-invariant system containing a small parameter that implies the convexity of small-time reachable sets for some classes of two-dimensional nonlinear control systems. The results of numerical simulations for illustrative examples are discussed.