Control Systems on Flag Manifolds and their chain Control sets

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.


1.
Introduction. Let G be a noncompact connected semisimple Lie group. A right-invariant control system on a connected Lie group G is determined by a family of differential equations given bẏ g(t) = X 0 (g(t)) + m j=1 u i (t)X j (g(t)), u ∈ U (Σ) where X 0 , X 1 , . . . , X m are right invariant vector fields and U ⊂ L ∞ (R, U ⊂ R m ).
Here U ⊂ R m is a compact and convex set with 0 ∈ int U . This class of system induce an affine control system on any flag manifold F Θ = G/P Θ of G defined bẏ where f Θ i = (dπ Θ )X i for i = 0, . . . , m and π Θ : G → F Θ is the canonical projection. It is worth pointing out that to study systems on flag manifolds is desirable for at least two reasons: First, flag manifolds include, spheres, projective spaces and Grassmannian spaces. Secondly, right-invariant systems appears in many important applications in engineering and physics such as the orientation of rigid bodies and optimal problems (see [4], [8] and [9]).
In order to understand the dynamical behavior of Σ Θ one have to analyze control and chain control sets of the system. There is a way to analyze the chain control sets of Σ Θ through the induced control flow φ Θ on the fiber bundle In fact, since F Θ is a compact manifold, Theorem 4.3.11 of [5] implies that there exists a bijection between the Morse components of the control flow φ Θ and the chain control sets of the system Σ Θ . Also, by [3], all the Morse sets of the flow φ Θ are given fiberwise as fixed points in F Θ of split elements of G.
On the other hand, since the positive orbit S := O + (1) is a semigroup we can use the notion of control and chain control sets associated with S (see for instance [2], [12] and [14]). In particular, it is know that any effective chain control set of S contains a control set with nonempty interior.
Our viewpoint sheds some new light on invariant control systems and its relation with the Lie semigroup theory. In fact, in this paper we show that both notion actually agree, that is, the chain control sets of the induced systems Σ Θ and the effective chain control sets associated with S are actually the same. That implies, in particular, that any given chain control set of Σ Θ contains a control set with nonempty interior. Moreover, if the number of chain control sets coincides with the number of the control sets with nonempty interior then any chain control set is the closure of the control set that it contains.
The paper is structured as follows: In Section 2 we give some preliminaries on control and Lie theory and enunciate some results that will be needed. In Section 3 we state and prove our main result, showing that both notions of chain control sets coincides. As a consequence, we prove that if the number of chain control sets and of control sets with nonempty interior coincide, then any chain control set is the closure of the only control set with nonempty interior that it contains. Finally, we include a couple of relevant examples.

2.
Preliminaries. In this section we introduce all the concepts of control and Lie theory needed in order to present and prove our main result.
2.1. Control theory. Let M be a Riemannian differentiable manifold and consider an admissible class of control U ⊂ L ∞ (R, U ⊂ R m ). A control affine system on M is defined by the family of differential equationṡ (1) where f 0 , f 1 , . . . , f m are smooth vector fields on M . The set U is a compact metrizable space and the shift flow is a continuous chain transitive dynamical system (see [5]). For u ∈ U fixed, we denote by ϕ(·, x, u) the unique solution of (1) with ϕ(0, x, u) = x. If the vector fields f 0 , . . . , f m are elements of C ∞ , then ϕ has also the same class of differentiability with respect to M and the corresponding partial derivatives depend continuously on (t, x, u) ∈ R × M × U (see Thm. 1.1 of [10]).
If we assume that all the solutions determined by U are defined in the whole time axis 1 , we obtain the well defined map called the transition map of the system. It turns out that this map together with the shift flow determines a skew-product flow as follows which is called the control flow of the system (see Sec. 4.3 of [5]).
We denote by the set of reachable points from x ∈ M at time τ > 0, the set of reachable points from x up to time τ , and the positive orbit of x, respectively. Analogously we and O − (x) the set of the controllable points to x in time τ > 0, the set of controllable points to x up to time τ and the negative orbit of x, respectively. The affine control system (1) for all x ∈ D (iii) D is maximal with respect to the set inclusion and properties (i) and (ii).
In [5], Proposition 3.2.4, it is shown that a subset D with nonempty interior, that is maximal with the property (ii) above is also a control set.

Definition 2.2.
A set E ⊂ M is called a chain control set of (1) if it satisfies the following properties: For all x, y ∈ E and ε, τ > 0 there exists a controlled (ε, τ )-chain from x to y in M iii) E is maximal with respect to the set inclusion and the properties (i) and (ii) above.
A set E satisfying condition (ii) in the above definition is called controlled chain transitive. Remark 1. From the general theory, every chain control set of the control system (1) is closed, but this is not necessarely true for control sets. Moreover, every control set with nonempty interior is contained in a chain control set if local accessibility holds (see Sec. 4.3 of [5]).

2.2.
Lie theory and flag manifolds. In order to state and prove our main result, in this section we give some facts about semisimple Lie groups and their induced flag manifolds.
2.2.1. Semisimple Lie groups. Let G be a connected non-compact semisimple Lie group G with finite center and Lie algebra g. Fix a Cartan involution ζ : g → g with associated Cartan decomposition g = k ⊕ s and let a ⊂ s be a maximal Abelian subalgebra and a + ⊂ a a Weyl chamber. Let us denote by Π, Π + and Π − := −Π + the set of roots, the set of positive roots and set of negative roots, respectively, associated with a + . The Iwasawa decomposition of the Lie algebra g reads as Let K, A and N ± be the connected Lie subgroups of G with Lie algebras k, a and n ± , respectively. The Iwasawa decomposition of the Lie group G is given by G = KAN ± . The Weyl group of g is the finite group generated by the reflections on the hyperplanes ker α for α ∈ Π. Alternatively, the Weyl group can be obtained by the quotient M * /M , where M * and M are the normalizer and the centralizer of a in K, respectively.
We denote by Λ ⊂ Π + the set of the positive roots which cannot be written as linear combinations of other positive roots. The Weyl group coincides with the group generated by the reflections associated to α ∈ Λ. There is only one involutive Here m is the Lie algebra of M , n − (Θ) is the sum of the eigenspaces g α when α ∈ Θ ∩ Π − and Θ ⊂ Π is the set of roots given as linear combination of the simple roots in Θ. The corresponding parabolic subgroup P Θ is the normalizer of p Θ in G. The flag manifold F Θ is the orbit F Θ := Ad(G)p Θ on a Grasmannian manifold. It is also identified with the homogeneous space G/P Θ . When Θ = ∅, the sets p := p ∅ and F := F ∅ are called the minimal parabolic subalgebra and the maximal flag manifold, respectively.
An element of g of the form Y = Ad(g)H with g ∈ G and H ∈ cl a + is called a split element. When H ∈ a + we say that Y is a split regular element. The flow induced by a split element H ∈ cl a + on F Θ , is given by It turns out that the associated vector field is a gradient vector field with respect to an appropriate Riemannian metric on F Θ .
The connected components of the fixed point set of this flow are given by The sets fix Θ (H, w) are in bijection with the double coset space W H \W/W Θ where W H and W Θ are the subgroups of the Weyl group generated by the simple roots in Θ(H) := {α ∈ Λ, α(H) = 0} and in Θ, respectively. Each When H is a split regular element the set fix Θ (H, w) reduces to the point wb Θ and consequently fix Θ (Y, w) = gwb Θ .

Semigroups.
In what follows we introduce the notion of control and chain control sets via semigroup action. Let S be a semigroup of a Lie group G and M a space provided with a G-transitive action.
For semigroups with nonempty interior of a semisimple Lie group G we have the following result from [14]. Theorem 2.3. For any w ∈ W there is an effective S-control set D Θ (w) ⊂ F Θ whose core is given by It turns out that there exists a unique Θ(S) ⊂ Λ such that the set {w ∈ W : and the number of effective control sets in F Θ is equal to the number of elements in the coset space W Θ(S) \ W/W Θ . The subset Θ(S) is called the flag type of the semigroup S.
There exists also a concept of chain control sets associated with a semigroup as follows: Let F be a family of subsets of S. For x, y ∈ M , ε > 0 and A ∈ F a (S, ε, A)chain from x to y is given by an integer n ∈ N and sequences x 0 , . . . , x n ∈ M , g 0 , . . . , g n−1 ∈ A such that x 0 = x, x n = y and (ii) For all x, y ∈ E F , ε > 0 and A ∈ F there exists a (S, ε, A)-chain from x to y.

set inclusion) with the properties (i) and (ii).
A F-chain control set is said to be effective if it contains an effective S-control set.
When G is a noncompact semisimple Lie group with finite center, it is possible to obtain a refined result (see [2], Proposition 4.7).
is called the F-flag type of the semigroup S. The number of chain control sets in F Θ is equal to the number of elements in the coset space W F (S) \ W/W Θ . If there is no possible confusion, we denote by E Θ (w) the above F-chain control sets on F Θ .
By the same Proposition 1 above, for any family of subsets F of S Remark 2. The above definitions of S-control sets and F-chain control sets differs from their analogous ones for control systems, mainly by the assumption that both, S-control sets and F-chain control sets, have nonempty interior which is not necessarily true for control systems.

2.2.3.
Flow on flag bundles. Let X be a compact metric space and φ : A Morse decomposition of φ is a finite collection {M 1 , . . . , M n } of nonempty pairwise disjoint isolated invariant compact sets satisfying (A) for all x ∈ X the ω * and ω-limit sets ω * (x) and ω(x) are contained in The elements in a Morse decomposition are called Morse sets. Let G be a connected semisimple noncompact Lie group with finite center and π : Q → X a G-principal bundle, where X is a compact metric space. The Lie group G acts continuously from the right on Q, this action preserves the fibers, and is free and transitive on each fiber. The flag bundle Now let φ t : Q → Q be a flow of automorphisms, i.e., φ t (q · g) = φ t (q) · g, and assume that the induced flow on X is chain transitive. For the induced flow φ Θ t : E Θ → E Θ we have the following result (see Thm. 9.11 of [3], Thm. 5.2 of [13]).
Theorem 2.4. There exist H φ ∈ cl(a + ) and a continuous φ-invariant map satisfying h(q · g) = Ad(g −1 )h(q), q ∈ Q, g ∈ G such that the induced flow on E Θ admits a finest Morse decomposition whose elements are given fiberwise by The subset Θ(φ) = Θ(H φ ) is called the flag type of the flow φ. The number of the Morse sets in E Θ is equal to the number of elements in the coset space 3. The main result. The aim of this paper is to bring together two areas in which we can analyze the behavior of an invariant control system both, from the control theory and from the semigroup theory point of views. In particular, the main theorem implies that any chain control set of a locally accessible induced system contains a control set with nonempty interior, which is in general not necessarily true (see Example 3.4.2 of [5]). Moreover, when Θ(S) = Θ(φ) every chain control set is the closure of the only control set with nonempty interior which it contains.
Let G be a noncompact connected semisimple Lie group with finite center and consider the right-invariant system Σ on G. Moreover, assume that Σ is a locally accessible right-invariant system on G, and therefore, that all the induced system Σ Θ are also locally accessible. From now S will stand for the semigroup given by O + (1), which by the local accessibility condition has nonempty interior in G.

Proposition 2. A subset D ⊂ F Θ is an effective S-control set if and only if it is a control set of Σ Θ with nonempty interior.
Proof. By Proposition 3.2.4 of [5] and the invariance of Σ we have that any effective S-control set in F Θ is in fact a control set of Σ Θ with nonempty interior.
Reciprocally, if D ⊂ F Θ is a control set with nonempty interior of Σ Θ , we obtain by invariance that D is a S-control set and we only have to show that it is effective.
By the local accessibility assumption, we have that int D is dense in D (see Lemma 3.2.13 of [5]). Moreover, since int S is dense in S and 1 ∈ S we get Therefore, by property (ii) in the definition of control set, which implies that D is in fact effective. We thus obtain the desired conclusion.
Remark 3. We should note that Proposition 2 does not use the fact that the Lie group G is semisimple. Thus the result remains true for any induced system on an arbitrary homogeneous space.
Consider the G-principal bundle π : U × G → U, where π is the projection on the first factor and G acts trivially from the right on U × G as (u, g) · h = (u, gh). The invariance of Σ shows that ϕ(t, gh, u) = ϕ(t, g, u)h which implies that the control flow φ t is a flow of automorphisms on U × G. Moreover, the induced flow on U coincides with the shift flow, which means that it is chain transitive. Therefore, Theorem 2.4 can be applied in order to obtain the Morse sets of the induced control flows φ Θ for any Θ ⊂ Λ.
Let F be the family of subset of S given by We are thus led to prove our main result. Proof. Our proof starts with the observation that for a subset E of F Θ the lift of E is given by x, u) ⊂ E}. By Theorem 4.3.11 of [5] there exists a bijection between the Morse sets of φ Θ and the chain control sets of the system Σ Θ given by: If M Θ is a Morse set of φ Θ , then π 2 (M Θ ) is a chain control set of Σ Θ , where π 2 : U × F Θ → F Θ is the projection on the second factor. Reciprocally, if E is a chain control set of Σ Θ its lift E is a Morse set of φ Θ .
The remainder of the proof falls naturally into three parts.
Step 1. If E F is a F-chain control set in F Θ then its lift E F is contained in a Morse set of φ Θ . By condition (ii) in the definition of F-chain control sets and by the rightinvariance of Σ we have that E F is controlled chain transitive. By Theorem 4.3.11 one gets in particular, that the lift E F is chain transitive for the flow φ Θ . Moreover, since E F is certainly φ Θ -invariant, Theorem B.2.26 of [5] applies and E F is contained in some Morse set. In fact the Morse sets are the maximal φ Θ -invariant chain transitive subsets of U × F Θ .
Step 2. Any effective F-chain control set in F Θ is a chain control set of Σ Θ .
Let E be an effective F-chain control in F Θ and D ⊂ E an effective control set. By Proposition 2, D is a control set of Σ Θ with nonempty interior. By Corollary 4.3.12 of [5] there exists a chain control set E of Σ Θ such that D ⊂ E . By the very definition of F-chain control sets and the fact that int D = ∅ we get that E is an effective F-chain control set in F Θ which by maximality implies that E ⊂ E. By the previous item the inclusion E ⊂ E must always happens and so E = E is a chain control set of Σ Θ .
Step 3. Any chain control set of Σ Θ is an effective F-chain control set.
Let E be a chain control set of Σ Θ . In order to show that E is a F-chain control set, it is enough to show that it contains an effective S-control set.
Let us define h : U → Ad(G)H φ by h(u) := h(u, 1), where h is the map defined by Theorem 2.4 applied to the control flow φ on the principal bundle U × G. The map h is continuous and Ad(ϕ(t, e, u))h(u) = h(θ t u).

Moreover, the relation between the Morse sets and the chain control sets together with Theorem 2.4 give us that
Let g ∈ int S and consider u ∈ U and τ > 0 such that g = ϕ(τ, e, u). By extending periodically u to a τ -periodic control function u * ∈ U it turns out that g n = ϕ(τ, 1, u) n = ϕ(nτ, 1, u * ) ∈ int S. Therefore, consequently fix Θ (h(u * ), w) is a g-invariant compact set. Hence, there exists a nonempty subset Ω ⊂ fix Θ (h(u * ), w), which is minimal for the g-action in F Θ . By Proposition 2.3 of [14], the set Ω has to be contained in the interior of a S-control set D in F Θ . Since Ω ⊂ E we must have that D ⊂ E showing that E is in fact an effective F-chain control set which concludes the proof.
As a direct consequence of our main result we get. Corollary 1. Any chain control set of the system Σ Θ contains a control set with nonempty interior.
We have also that the flag type of φ and the F-flag type of S coincides.

Corollary 2. With the previous assumptions, it holds that
Proof. For any pair of subsets Θ 1 , Θ 2 ⊂ Λ we know Therefore, it is enough for us to show that W F (S) = W Θ(φ) . By Theorem 3.1 the family of the chain control sets of the induced systems in F and family of F-chain control sets in F coincides. Consequently Since π 2 (M(1)) is S-invariant, it contains the only S-invariant control set in F, and so π 2 (M(1)) = E(1). Let us take an element w ∈ W F (S). By definition E(w) is the only F-chain control set containing D(w). Moreover, by the choice of w, E(w) = E(1) = π 2 (M(1)).
Since any other choice of positive Weyl chamber just conjugates the flag types, w.l.o.g. we can assume that int S ∩ exp a + = ∅.
It follows that wb 0 ∈ D(w) 0 and there exists u ∈ U with wb 0 ∈ fix(h(u), 1). One can check that wb 0 = kb 0 where k ∈ K is such that Ad(k)H φ = h(u). The equality Hence w ∈ W Θ(φ) implying that W F (S) ⊂ W Θ(φ) and concluding the proof.
The above corollary implies in particular that Θ(S) ⊂ Θ(φ). We are now interested to see what happens when the equality holds, that is, when any chain control set of Σ Θ contains exactly one control set with nonempty interior. Definition 3.2. Let Θ ⊂ Λ and w ∈ W. We say that the chain control set E Θ (w) is (uniformly) hyperbolic if for each (u, x) ∈ M Θ (w) there exists a decomposition such that the following properties hold: (H1) (dϕ t,u ) x E ± u,x = E ± φt(u,x) for all t ∈ R and (u, x) ∈ M Θ (w).
u,x depend continuously on (u, x), i.e., the projections π ± u,x : In [6] it is shown that any chain control set E Θ (w) is partially hyperbolic, that is, the above decomposition of the tangent spaces have a third subspace that corresponds to the tangent space of the submanifold fix Θ (h(u), w). By Theorem 5.6 of [6], hyperbolicity happens if and only if Θ(φ) ⊂ w Θ if and only if for any u ∈ U the set fix Θ (h(u), w) consists of only one point. Consequently, if E Θ (w) is hyperbolic, Theorem 6.1 of [6] implies that cl(D Θ (w)) = E Θ (w).
The concept of uniformly hyperbocity together with the assumption that the flag types agree give us the following strong result about control and chain control sets on flag manifolds. Proof. Let us first assume that D(1) = E(1). Since Θ(S) ⊂ Θ(φ) always holds, we only have to show the opposite inclusion. For any w ∈ W Θ(φ) it follows that D(w) ⊂ E(w) = E(1) = D(1) and since two control sets are either disjoints or coincides, we must have D(w) = D (1) showing that W Θ(φ) ⊂ W Θ(S) and consequently Θ(φ) ⊂ Θ(S).
For any w ∈ W, the domain of attraction of D(w) is the subset of F given by A(D(w)) = {x ∈ F, gx ∈ D(w) for some g ∈ S} .
The following lemma relates the domain of attraction and the core of an effective control set.
Lemma 3.4. For any w ∈ W it holds that where D * (w) is the unique S −1 -control set in F whose core is D * (w) 0 = D(w) 0 and A * (D * (w)) its domain of attraction.
Proof. Since D(w) 0 = D * (w) 0 we already have that For any x ∈ A(D(w)) there exists g ∈ S such that gx ∈ D(w). By using condition (ii) in the definition of S-control sets we get that D(w) 0 ⊂ D(w) ⊂ cl(Sgx). Since D(w) 0 is an open set and int S is dense in S, we obtain Analogously, if x ∈ A * (D * (w)) we get that D * (w) 0 ∩ int S −1 (g x) = ∅ for some g ∈ S −1 . Therefore, if x ∈ A(D(w)) ∩ A * (D * (w)) there exists h ∈ S −1 and h ∈ S such that h g x, hgx ∈ D(w) 0 , since D(w) 0 = D * (w) 0 . Being that in D(w) 0 we have controllability, there exists k ∈ S such that However, since int S is S-invariant we get that (h g ) −1 khg ∈ int S showing that x ∈ D(w) 0 and concluding the proof. By Theorem 6.3 of [12], the above domain of attraction can be characterized as follows: For a finite sequence α 1 , . . . , α n in Λ let us denote by s 1 , . . . , s n the reflections with respect these roots and consider P i := P {αi} . The corresponding flag manifold is F i = G/P i and we denote by π i the canonical projection of F onto F i . Moreover, for a given subset X of F we denote by γ i (X) := π −1 i π i (X) the operation of exhausting the subset X with the fibers of π i . Now, take w ∈ W and consider the reduced expressions w = r m · · · r 1 and w 0 w = s n · · · s 1 with exhausting maps γ i and γ j , respectively. It holds that A(D(w)) = γ 1 · · · γ n (D(w 0 )) and A * (D * (w)) = γ 1 · · · γ m (D(1)) Concerning chain control sets, from Propositions 9.9 and 9.10 of [3] and the fact that π 2 (M(w)) = E(w) it is straightforward to see that E(w) ⊂ γ 1 · · · γ m (E(1)) ∩ γ 1 · · · γ n (E(w 0 )) .
We can now prove the following result.
Using (3) we obtain E(w) ⊂ A * (D * (w)) ∩ cl (A(D(w))) . Bilinear systems on sl(2): In [1] the authors described all the possibilities of the control sets of a bilinear system on the projective space P. In what follows we will use their description to show a case where we always have Θ(φ) = Θ(S).
Let us consider the family of bilinear control systems on R 2 defined by where X, Y ∈ sl(2). Following [1], if det[X, Y ] = 0 it turns out that span{X, [X, Y ]} = sl(2) and the systems Σ ρ satisfies in particular the Lie algebra rank condition.
Let us still denote by Σ ρ the induced system on P. Under the assumptions that det X < 0 and det[X, Y ] > 0, Theorem 5.2 of [1] assures that the map ρ → D ρ is continuous in the Hausdorff metric. Here, D ρ denote the positive invariant control set on the induced system Σ ρ on P. Moreover, Lemma 4.7.4 (ii) of [5] implies that the continuity points of the map ρ → D ρ are exactly the points where D ρ = E ρ is a chain control set.
On the other hand, sl(2) acts transitively on P so P = sl(2)/P is a homogeneous space. A simple calculation shows that, in fact P is a minimal parabolic subgroup of G, implying that P is the maximal flag manifold F of sl (2). Therefore, D ρ = cl D ρ = E ρ which means by Theorem 3.3 that Θ(φ ρ ) = Θ(S ρ ) for any ρ > 0.
Control systems with small control range: In this example we show that, for invariant control systems with small control range we also have the equality of the flag types.
Let G be a noncompact connected semisimple Lie group and consider the invariant system (Σ)ġ(t) = X 0 (g(t)) + m i=1 u i (t)X i (g(t)), u ∈ U.
and consider the induced system on F. By Corollary 4.7.6 (ii) of [5], there exists ρ * > 0 such that the induced system Σ ρ on F satisfies D ρ (1) = E ρ (1) for all up to at most countably many values of ρ ∈ (0, ρ * ). Here Σ ρ is the induced system on F whose set of control functions is U ρ = L ∞ (R, ρΩ). Therefore, by Theorem 3.3 we get that Θ(φ ρ ) = Θ(S ρ ) for all up to at most countably many values of ρ ∈ (0, ρ * ).
In particular, if X 0 is an element of a positive Weyl chamber a + for some Iwasawa decomposition, the origin b 0 = 1 · P ∈ F is an hyperbolic fixed point of the uncontrolled system. Therefore, for ρ > 0 small enough we have that E ρ is a hyperbolic chain control set. By Theorem 4.6 of [6] we get that Θ(φ) = ∅ and so Θ(S) ⊂ Θ(φ) = ∅ implying the equality.