Boundedness control sets for linear systems on Lie groups

Let Σ be a linear system on a connected Lie group G and assume that the reachable set A from the identity element e ∈ G is open. In this paper, we give an algebraic condition to warrant the boundedness of the existent control set with a nonempty interior containing e. We concentrate the search for the class of decomposable groups which includes any solvable group and also every compact semisimple group.


Introduction
Let G be a connected Lie group with Lie algebra g. In [ ] the authors introduce the notion of a linear system Σ on G which is determined by a family of di erential equationṡ g(t) = X (g(t)) + m j= u j (t)X j (g(t)), where X is a linear vector eld, X j ∈ g considered as right invariant vector elds and u ∈ U ⊂ L ∞ (R, Ω ⊂ R m ) is the class of admissible controls. We deal with Ω as a subset of R m with ∈ int Ω. Furthermore, Σ is called restricted if Ω is compact and unrestricted if Ω = R m .
We denote by φ t,u (g) = φ(t, g, u) the solution of Σ with control u, initial condition g at time t. The controllability property of any system is a relevant issue in system theory. It gives you the possibility to connect any two arbitrary states of the manifold through a Σ-solution in positive time. For instance, when G is the Euclidean space R n an unrestricted linear system is controllable if and only if it satis es de Kalman rank condition [ ], which is nothing more than the ad-rank condition, see Remark 2.2 in chapter two. However, controllability is rare in the literature, especially for Σ. Assume G is nilpotent and the accessibility set A from the identity element e ∈ G provided by is an open set. It turns out that Σ is controllable on G ⇔ Spec Ly (D) ∩ R = { } .
Here, D ∈ ∂g is a g-derivation associated to X and the Lyapunov spectrum Spec Ly (D) consists of the real parts of the D-eigenvalues.
Recently, the authors in [ ] proved that the requirement Spec Ly (D) ∩ R = { } implies controllability for any Lie group with nite semisimple center, that is, for any Lie group which admits a maximal semisimple Lie subgroup with nite center. Certainly, the condition on the Lyapunov spectrum of D is very strong. Actually, each D-eigenvalue must live on the axis iR.
For restricted system there exists the notion of control set introduced in [ ], basically, a subset C where controllability holds on int(C). For a locally controllable system, it is shown in [ ] the shape of the control sets with nonempty interior. Under our assumptions, the control set containing the identity element e ∈ G reads as where A * is the reachable set of Σ − , i.e., when time in Σ is reversed.
In this paper, we are interested in research on algebraic condition to ensure the boundedness of C e . We concentrate the study on solvable Lie groups because in this case the space state is rstly decomposable. This means that G can be written as a direct product of the closed subgroups G + , G and G − with Lie algebras g + , g , and g − induced by the g-derivation D which determines the drift vector eld X . Secondly, any solvable Lie group has trivially the nite semisimple center property. Hence, we can apply any result about control sets from [ ]. In particular, denote by The authors show that for semisimple or nilpotent Lie groups the compactness of A G − , A * G + and G together is a su cient condition for the boundedness of C e . Furthermore, for the class of nilpotent simply connected Lie groups these conditions are also necessary. However, to compute e ectively these three sets is a very hard task. Hence, our main aims in this paper are to search for algebraic computable conditions to get the boundedness of C e . Next, we resume the chapters.
In Section 2, we review some of the standard facts on linear systems. In particular, we summarize without proof the primary relevant material on the dynamic structure, the reachable sets and the existence and uniqueness of control sets with a nonempty interior of Σ. We also mention the D-decomposition of the Lie algebra g and the corresponding Lie groups induced by X . In Section 3, our main result is stated and proved. A su cient algebraic condition for the boundedness of C e is given. In Section 4, we remark some possible extensions. And nally, Section 5 contains a couple of examples in low dimensional Lie groups.

Preliminaries
In what follows Σ will denote a linear system on a connected Lie group G. In this section, we establish the basic de nitions and the main results about the topological and dynamic structure of Σ. In particular, we list some properties of the reachable sets of Σ and we mention the Lie algebra decomposition induces by the drift vector eld X on g = g + ⊕ g ⊕ g − and, its dynamics consequences on the corresponding closed subgroups G + , G , and G − of G.
. The dynamic structure of Σ As we mention, a linear system Σ is furnished bẏ Essentially, its dynamic behavior is determined by two di erent classes of vector elds. First, the uncontrolled di erential equationġ(t) = X (g(t)). Denote by (ϕ t ) t∈R the ow of X . By de nition, X is an in nitesimal automorphism, which means where Aut(G) is the Lie group of all G-automorphisms. Associated with X there exists a derivation D of g supplied by , for all Y ∈ g. The relationship between ϕ t and D is given by the formulas, [ ], (dϕ t ) e = e tD and ϕ t (exp Y) = exp(e tD Y), t ∈ R, Y ∈ g.
On the other hand, the family of vector eld X u = ∑ m j= u j X j depends on m xed right invariant vector elds X j ∈ g and the family of admissible control u = (u , ..., u m ) ∈ U which has the mission to redirect X to reach the desired goal.

. Reachable sets
For a state g ∈ G, the reachable set from g up to the time t is de ned by and A(g) = ⋃ t > A ≤ t (g) is the reachable set from g. We denote A(e) by A.
Next, we collect the main properties of the reachable sets, see [ ] and [ ] .
Proposition 2.1. For a linear system Σ on the connected Lie group G it holds 1.

e ∈ int A if and only if A is open
The controllable set to g up to the time t is de ned by The controlled set to g is A * (g) = ⋃ t > A * ≤ t (g). We denote A * (e) by A * .

Remark 2.2. We assume from the start that
A is an open set and it happens for instance, when the system satis es the ad-rank condition, i.e., .., m and i = , , ... = g.
The system is said to be locally accessible at g if int(A ≤ t (g)) and int(A * ≤ t (g)) is nonempty for any t ≥ , and controllable from g if A(g) = G. .

D-Decomposable Lie groups
In this section, we look more closely at the Lie algebra decomposition induced by the derivation D associated with the drift vector eld X . We address the generalized eigenspaces of D provided by Here, α runs over the spectrum Spec(D). It turns out that [g α , g β ] ⊂ g α+β if α+β ∈ Spec(D) and otherwise. Of course, g decomposes as It follows that g + , g , g − are Lie subalgebras and g + , g − are nilpotent, see Proposition 3.1 in [6]. Let us denote by G + , G − , G , G +, , and G −, the connected Lie subgroups of G with Lie algebras g + , g − , g , g +, = g + ⊕ g and g −, = g − ⊕ g , respectively.

De nition 2.3. Let D be a g-derivation. The Lie algebra g is said to be
We collect some basic properties of these subgroups, Proposition 2.9 in [7].

Control sets
A more realistic approach to the controllability property of a system comes from the following notion.
In [ ] the authors prove general results about the shape of an existent control set, that we specialize in our particular class of linear systems, as follows: Lemma 2.5. Let C be a control set of Σ. If the system is locally accessible at any point of int(C) then for any In particular, the system is controllable on int C.
Instead to study the strong (global) controllability property of Σ we are looking for a weak conditions to obtain regions where controllability holds.

Main result
In this section, our main results are stated and proved. For that, we apply several results appearing in [ ].
From now we assume that G is decomposable. It turns out that A and A * are also decomposable. Denote by Furthermore, A G − , A * G + and G are contained in A ∩ A * . We assume that the reachable set A is open then the system is locally accessible in a neighborhood of e. From Lemma 2.5, Σ has a control set C e = cl(A) ∩ A * . On the other hand, by hypothesis g is D-decomposable hence C e is the only control sets with nonempty interior.
It is clear that In the sequel, we analyze a kind of converse. Actually, in some special cases, the boundedness of these three sets imply the boundedness of C e . The following two results were proved in [ ]. In fact, consider a curve γ ∶ [ , ] → G with γ( ) = e and γ( ) = g. Thus, ϕ t ○ γ is a curve connecting e to ϕ t (g) and Now, any G-homomorphism φ satis es the formula Subsequently, The homomorphism ϕ t belongs to Aut(G) for any t ∈ R and D = d(ϕ t ) e . Since the metric ρ is left invariant, we get d(ϕ t ) g = e tD .
By hypothesis D is a stable matrix, then ( * ) follows.
Take t > such that A t ⊂ B(e, ) the open ball with center e and radius . Just observe that for every positive number τ ϕ τ (B(e, )) ⊂ B(e, c − e −λτ ).
By using the same argument we obtain Now, any g ∈ G can be decomposed as g = g g g ...g n with g i ∈ B(e, c − e −itλ ), i = , , ..., n.
Hence, there exists a radius R such that This ends the proof, actually Now, we are able to prove our main result.
Theorem 3.5. Let Σ be a linear system on a decomposable connected Lie group G. Assume that g +, is an ideal of g then cl(A G − ) is bounded.
Proof. According to our hypothesis the group is decomposable, thus Since g +, is an ideal the Lie subgroup G +, is normal. In particular, the homogeneous space G G +, is a Lie group isomorphic to G − . Let us consider the canonical projection π ∶ G → G G +, . It turns out that π(A) = A − . Furthermore, on G G +, the derivation D associated to the drift vector eld X of Σ has just eigenvalues with negative real parts. In other words, D − is the corresponding derivation associated with the system Σ − in G − . In fact, the Lie algebra of G G +, is isomorphic to g +, = g + ⊕ g which is isomorphic to g − . Therefore, Proposition 3.4 implies that the reachable set A − of Σ − is bounded in G − . In the sequel, we prove that this condition is enough to show that the reachable set A is bounded in G − . However, we rst need to show that Actually, since any element in G has a unique decomposition in G − G G + the application is bijective. By the own de nition of the quotient topology on G G +, the projection π restricted to G − is continuous. Next, we prove that π G − is an open map. First, there are neigborhoods V ⊂ G − and W ⊂ G +, of the identity e ∈ G such that the product VW is also a neigborhood of e. In particular, π G − (V) = π(V) = π(VW) is an open set in G G +, . If g ∈ G we consider the translations L g (V) = gV and L g (W) = gW. Since Lg is a homeomorphism, the proof is done and π G − is a homeomorphism. Once again, the group G is decomposable, thus π(G − ) = G G +, and it is possible to cover A with the projection of a compact subset of A. In fact, for any compact K containing A − de ne the compact set From that, we obtain Since A − is bounded, it follows that cl(A G − ) is also bounded as we claim.
Corollary 3.6. Let Σ be a linear system on a decomposable connected Lie group G. Assume that g −, is an ideal of g then cl(A * G + ) is bounded.
Proof. The proof is completely analogous to that of Theorem 3.5.
Every nilpotent Lie group as a solvable group is decomposable, see [ ].
Theorem 3.7. Let Σ be a linear system on a nilpotent simply connected Lie group G. Assume that g +, and g −, are ideals of g. Then, Proof. . We have, The last equivalence depends strongly on the fact that in this particular case the exponential map is a global di eomorphism. Just observe that in general this is not true. For instance, exp(R) = S , however, the -dimensional sphere is not simply connected. Now, our hypothesis and Theorem 3.5 implies that cl(A G − ) and cl(A * G + ) are bounded. Thus, Theorem 3.1 shows that C e is bounded. On the other hand, if C e is bounded it follows that G ⊂ cl(A) is compact and ending the proof. . To prove the second item we observe that under the hypothesis G ⊂ A * . So, C e = cl(A) is trivially closed and bounded by Theorem 3.1. . If G = G + we get A = G. Thus, C e coincides with the open set A * . Remark 3.8. We observe that item third of Theorem 3.7 shows that Σ is controllable from the identity, i.e., for any arbitrary g ∈ G there exists a control u and a positive time t such that φ t,u (e) = g. For other results in the same spirit, we invite the readers to take a look at the following references, [ , , , , ]. Furthermore, in [ ] the author shows that the class of linear control systems is important in a theoretical way. He proves an equivalent theorem which involves a class of nonlinear control systems on general manifolds.
A su cient condition for the simultaneous boundedness of cl(A G − ) and cl(A * G + ) is to assume that both g +, and g −, are ideals of g. An equivalent condition is given by the next proposition. Proposition 3.9. Let g be a Lie algebra and D ∈ ∂g. It turns out g +, and g −, are ideals of g ⇔ g , g + = and g + , g − ⊂ g .

Extensions
In this paper, we concentrate the study on decomposable Lie groups. However, one might be tempted to try to extend the result to semisimple groups. Let us consider an unrestricted linear system Σ on a connected semisimple Lie group G. In this case, we have two possibilities . The compact case In [ ] the authors prove the following result: The LARC condition is weaker than the ad-rank condition. Actually, in the rst case you are allowed to compute the Lie brackets , which is forbidden in the other case. Therefore, C e = G for any transitive linear system on G.
. The noncompact case Here, we just comment that except the case G = G , the space state cannot be decomposable. In fact, in [ ] we show that the set G − G G + ⊊ G is just an open Bruhat cell which is dense in the group all. In particular, our results can not be extended in this direction.

Examples
In this section, we give some examples of boundedness and unboundedness control sets on some decomposable Lie groups. But rst, we explain how to nd the face of the drift X when it is induced by an inner derivation.
Remark 5.1. A particular class of linear vector elds is easy computed through a -parameter of inner Gautomorphisms. Take X ∈ g a right invariant vector eld and consider the solution X t (g) with initial condition g ∈ G. By the right invariance, the solution through the initial condition g is provided by the right translation by g of the solution X t (e) = exp G (tX) through the identity element. In order words X t (g) = exp G (tX)g.
Here, exp G ∶ g → G is the usual exponential map. Hence, X de nes by conjugation a -parameter group of inner automorphism as follows ϕ t (g) = X t (e) g X −t (e), g ∈ G, and ϕ t ∈ Aut(G) for any t ∈ R.
Therefore, it is possible to compute the linear vector eld as Recall that any derivation on a semisimple Lie group is inner. This property has interest for us in the compact case. On the other hand, in [ ] we built an algorithm which provides an e ective means to compute the Lie algebra ∂g that we use in this section.
From Remark 5.1 the linear vector eld X associated to D is given by Let Σ be the transitive linear system on G de ned bẏ where, D = ad(Y) comes from a = − and b = . Since ad(Y)X = −Y then Span {X, DX} = g. So, Σ satis es the ad-rank condition, A is open and of course, Σ satis es also LARC. Moreover, G is solvable thus the control set C e is the only one with nonempty interior. It turns out that, Thus, To conclude, the system is controllable from the identity. This fact is completely concordant with Theorem 3 in [11]. Actually, it is shown there that a transitive system in a canonical form, like Σ, is controllable if and only if b = .
Example 5.3. Let g = RX + RY + RZ be the Lie algebra of the connected and simply connected Heisenberg Lie group G of dimension three. The generators of g are provided by The only one non-vanishing Lie bracket is [X, Y] = Z. Any derivation D is represented by a real parameters matrix in the basis {X, Y , Z} as follow Consider the linear system Σ with derivation D determined by its coe cients a = d = − , b = , c = − , e = f = and the control vectors X and Z, We have, Spec Ly (D) = {− , − } . So, g −, = g − = g and g +, = are both ideals of g. On the other hand, Since D is a hyperbolic derivation, Theorem 3.7 shows that the existent control set C e is bounded. (t) = X (g(t)) + u (t)X(g(t)) + u (t)Y(g(t)), u ∈ U , with Ω = R, where X = ad(X). Since Σ satis es LARC, the control set is bounded and coincides with the group. The system is controllable from the identity.
Example 5.5. Take the linear system Σ on the Heisenberg group G like in Example , but with di erent dynamic behaviorġ (t) = X (g(t)) + u (t)(X − Y)(g(t)) + u (t)(X + Y + Z)(g(t)), u ∈ U , with Ω Let Σ be a linear system with an arbitrary derivation D ∈ ∂g such that the reachable set A of Σ is open. Hence, the control set C e is unbounded. In fact, a straightforward computation shows that the Lie algebra of g-derivations is ve dimensional and reads as Since the underlying topological space of G is the connected and simply connected manifold R , Theorem 3.5 applies. However, ∈ Spec Ly (D) for any D ∈ ∂g. Thus, no hyperbolic derivation exits, ending the proof.