Control systems on flag manifolds and their chain control sets

A right-invariant control system Σ on a connected Lie group G induce affine control systems Σo on every flag manifold Fe = G=Pe. In this paper we show that the chain control sets of the induced systems 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. Some relevant examples are included.