Convexity of reachable sets of nonlinear ordinary differential equations

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.

💡 Research Summary

This paper addresses a fundamental geometric property of reachable sets for nonlinear ordinary differential equations (ODEs). Given a dynamical system (\dot x = f(x)) with a sufficiently smooth vector field, the authors consider an initial set that is a ball (or, more generally, an ellipsoid) of radius (r) centered at a point (x_0). The reachable set (\mathcal R(t;B_r(x_0))) consists of all states that can be reached at a fixed time (t>0) from any initial condition inside that ball. While reachable sets for linear systems are always convex, the situation for nonlinear systems is far more complicated; non‑convexity hampers many verification and control synthesis techniques that rely on convex approximations.

The core contribution is a necessary and sufficient condition guaranteeing convexity of (\mathcal R(t;B_r(x_0))). The condition is expressed in terms of two readily computable quantities derived from the right‑hand side (f):

1. A bound on the symmetric part of the Jacobian – there must exist a non‑negative constant (\lambda) such that for all (z\in\mathbb R^n) and all vectors (v) we have
   \