Abstract
A right-invariant control system $Σ$ on a connected Lie group $G$ induce affine control systems $Σ_{Θ}$ on every flag manifold $\mathbb{F}_{Θ}=G/P_{Θ}$. 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.
Highlights
A right-invariant control system on a connected Lie group G is the family of differential equations given by m g(t) = X(g(t)) + ui(t)Y j(g(t)), u ∈ U
If Σ is a right-invariant system on G, we have induced affine control systems on every flag manifold FΘ = G/PΘ given by m x (t) = f0Θ (x(t)) + uj(t)fjΘ (x(t)), u ∈ U
We prove that if the number of chain control sets and of control sets with nonempty interior coincide, them any chain control set is the closure of the only control set with nonempty interior that it contains
Summary
If Σ is a right-invariant system on G, we have induced affine control systems on every flag manifold FΘ = G/PΘ given by m x (t) = f0Θ (x(t)) + uj(t)fjΘ (x(t)) , u ∈ U j=1. Any effective chain control set of S contains a control set with nonempty interior. In this paper we show that both notion agree, that is, the chain control sets of the induced systems ΣΘ and the effective chain control sets associated with S are the same. In particular, that any given chain control sets of ΣΘ contains a control set with nonempty interior. We prove that if the number of chain control sets and of control sets with nonempty interior coincide, them any chain control set is the closure of the only control set with nonempty interior that it contains
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