Convexity of reachable sets of nonlinear ordinary differential equations

TL;DR

This paper establishes a precise condition under which the reachable set of a nonlinear ODE from a small initial ball is convex, facilitating easier approximation and analysis of such sets.

Contribution

It provides a necessary and sufficient convexity condition for reachable sets of nonlinear ODEs, including explicit bounds on initial set size based on the system's dynamics.

Findings

Convexity of reachable sets is guaranteed for sufficiently small initial balls.

An explicit upper bound on the initial ball radius is derived from the differential equation.

The results enable simple polyhedral approximations of reachable sets in practice.

Abstract

We present a necessary and sufficient condition for the reachable set, i.e., the set of states reachable from a ball of initial states at some time, of an ordinary differential equation to be convex. In particular, convexity is guaranteed if the ball of initial states is sufficiently small, and we provide an upper bound on the radius of that ball, which can be directly obtained from the right hand side of the differential equation. In finite dimensions, our results cover the case of ellipsoids of initial states. A potential application of our results is inner and outer polyhedral approximation of reachable sets, which becomes extremely simple and almost universally applicable if these sets are known to be convex. We demonstrate by means of an example that the balls of initial states for which the latter property follows from our results are large enough to be used in actual computations.