Positive Invariance Condition for Continuous Dynamical Systems Based on Nagumo Theorem

TL;DR

This paper establishes conditions for positive invariance of convex sets in continuous dynamical systems using Nagumo's Theorem, with verification methods based on optimization theory.

Contribution

It provides a unified framework for positive invariance conditions of convex sets in continuous systems using tangent cones and optimization approaches.

Findings

Derived tangent cones for classical convex sets

Established necessary and sufficient invariance conditions

Proposed optimization-based verification methods

Abstract

In this paper, we obtain sufficient and necessary conditions of some classical convex sets as positively invariant sets for a continuous dynamical system, namely positive invariance conditions. The approach is based on Nagumo Theorem by deriving the tangent cones of these sets. We also propose approaches using optimization theory and models to verify the existence of these sufficient and necessary conditions.