Nagumo's Theorem for Dubovitskij-Miljutin Tangent Cones

TL;DR

This paper extends Nagumo's Theorem to include Dubovitskij-Miljutin tangent cones, providing a new method to verify positive invariance in perturbed dynamical systems.

Contribution

It introduces an equivalence between positive invariance with perturbations and the Dubovitskij-Miljutin tangent cone, advancing the theoretical understanding of perturbed systems.

Findings

Established an equivalence for additive perturbations in dynamical systems.

Provided a method to check positive invariance with perturbations.

Linked positive invariance to Dubovitskij-Miljutin tangent cones.

Abstract

This paper focuses on a class of additive perturbations in dynamical systems. An equivalence statement for this construction is discovered, and consequently, a method of checking a notion of positive invariance with perturbation. The resulting conclusion is an equivalence between a more strict definition of positive invariance, based on a perturbation extension of the system and the Dubovitskij-Miljutin tangent cone.