The Convex Geometry of Integrator Reach Sets

TL;DR

This paper analyzes the convex geometric properties of integrator reach sets, providing explicit formulas for their volume and diameter, which enhances understanding and computation of reachable states in control systems.

Contribution

It introduces novel closed-form expressions for the volume and diameter of integrator reach sets using convex analysis, advancing theoretical understanding and computational methods.

Findings

Explicit formulas for volume and diameter of reach sets

Illustrative examples demonstrating the formulas

Potential for improved reach set computation algorithms

Abstract

We study the convex geometry of the forward reach sets for integrator dynamics in finite dimensions with bounded control. We derive closed-form expressions for the volume and the diameter (i.e., maximal width) of these sets in terms of the state space dimension, control bound, and time. These results are novel, and use convex analysis to give an analytical handle on the "size" of the integrator reach set. Several concrete examples are provided to illustrate our results. We envision that the ideas presented here will motivate further theoretical and algorithmic development in reach set computation.