On Applicability of Linearization Approach in Calculating Small-Time Reachable Sets of Nonlinear Control Systems

The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear systems affine in control variables at small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the L2 norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of a time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is determined using the Banach-Mazur metric in the space of sets.