Control Systems on Flag Manifolds and their chain Control sets

TL;DR

This paper establishes a correspondence between chain control sets and control sets with nonempty interior on flag manifolds, providing new insights into the structure of control systems on these geometric spaces.

Contribution

It proves that chain control sets on flag manifolds coincide with those defined via semigroup actions, linking control set properties to chain control sets.

Findings

Chain control sets coincide with semigroup-defined sets on flag manifolds.

Every chain control set contains a control set with nonempty interior.

Closure of control sets with nonempty interior equals chain control sets.

Abstract

In this paper we shown that the chain control sets of induced systems on flag manifolds coincides with their analogous one defined via semigroup actions. Consequently, any chain control set of the system contains a control set with nonempty interior and, if the number of the control sets with nonempty interior coincides with the number of the chain control sets, then the closure of any control set with nonempty interior is a chain control set.