On Asymptotics of Small-Time Reachable Sets for Nonlinear Systems with Isoperimetric Constraints

We study the problem of approximate representation of reachable sets at small time intervals for nonlinear control-affine systems with isoperimetric control constraints. By isoperimetric constraints we mean integral constraints on the control input and state of the system. It is shown that under certain conditions imposed on the controllability Gramian of the linearized system the reachable set turns out to be convex and asymptotically close in shape to an ellipsoid in the state space for a sufficiently small time interval. This ellipsoid is a reachable set of the system linearized along the trajectory generated by zero control input. Copyright (C) 2021 The Authors.