Translation of Nagumo's Foundational Work on Barrier Functions: On the Location of Integral Curves of Ordinary Differential Equations

TL;DR

This paper translates Nagumo's 1942 foundational work on barrier functions and set invariance in ordinary differential equations, making it accessible for modern researchers and highlighting its relevance to safety-critical control systems.

Contribution

It provides an accessible English translation of Nagumo's original paper, clarifying its foundational role in barrier functions and set invariance for contemporary control theory.

Findings

Establishes the foundational role of Nagumo's work in barrier functions

Highlights the relevance of set invariance in safety-critical systems

Provides historical context for modern control methods

Abstract

In 1942, Prof. Mitio Nagumo published his seminal paper on the location of integral curves of ordinary differential equations. Nagumo's paper provides the foundation of the set invariance of ordinary differential equations and barrier functions, which have recently gained popularity for the control design of safety critical dynamical systems. This translation shall serve the community with an easily accessible version of the original 1942 paper in English. A copy of Nagumo's paper in German is also attached as a reference. That copy was created by the Boeing Company, Germany, in an attempt to improve pdf format readability of the original paper.