Topological conjugacy of linear systems on Lie groups

In this paper we study a classification of linear systems on Lie groups with respect to the conjugacy of the corresponding flows. We also describe stability according to Lyapunov exponents.


Introduction
Consider the linear control system in R n given bẏ where A is a n × n matrix, (b 1 , . . . , b n ) is a vector in R n and u(t) = (u 1 (t), . . . , u n (t)) is locally integrable function. This kind of the linear control system has been extensively studied as it can be viewed in Agrachev and Sachkov [1], Jurdjevic [12] and Sontag [16], and the references therein. Commonly, three aspects of control system are studied: controllability, classification and optimality. Our first wish is to study the classification of the linear control systems on connected Lie groups, more specifically, the topological conjugacy.
Here we follow the work of Ayala and Tirao [2]. Inspired by a work due to Markus [13], Ayala and Tirao introduced the concept of linear control systems on connected Lie groups. They shown that the generalization of the linear control system (1) to a connected Lie group G is determined by the family of differential equations:ġ where X is an infinitesimal automorphism of G, X i are right invariant vector fields on G and u(t) = (u 1 (t), . . . , u n (t)) ∈ R n belongs to the class of unrestricted admissible control functions. It is worth note that there are several works about the control system (2) when the drift X is assumed to be a right invariant vector field. The reader interested can found more about this case, for instance, in the work of Biggs and Remsing [5] and the references therein.
To the linear control system (2) there exist a number of works concerning the controllability problem. Ayala and Tirao studied in [2] local controllability problems and the (ad)-rank conditions. After, Ayala and San Martin [3] establish controllability results for compact and semisimple Lie groups. In sequence, Jouan [9], [10] and Jouan and Dath [11] studied about equivalence and controllability of linear control systems. In [9], the importance of the linear control systems on Lie groups is once more highlighted. It is shown that the class of linear control systems classifies several classes of affine control systems on arbitrary connected manifolds.
Recently, Da Silva [7] and Ayala and Da Silva [4] shown that the dynamics of the flow associated with the drift are intrinsically connected with the behavior of the whole linear control system. They shown that, if the flow associated with the drift has trivial expanding or contracting subgroups, then the control system (2) is controllable.
Therefore, to know the dynamical properties of such flows is fundamental in order to understand the behavior of the linear control systems. Observing this fact, before studying the topological conjugacy between system of type (2) we view that it is necessary to study the topological conjugacy in a intermediary step, that is, we adopt the linear system on a connected Lie group G given bẏ where the drift X is an infinitesimal automorphism. Thus, the main purpose of this paper is to classify linear flows on Lie group via topological conjugacies and characterize asymptotic and exponential stability using Lyapunov exponents. Besides this purpose our work reveals the similarity between (3) and the linear system in R n given byẋ = Ax.
Then following a similar approach of the classical result we classify the linear flows according to the decomposition of the state space in stable, unstable and central Lie subgroups. Moreover, given a fixed point g ∈ G of the linear flow given by X , we define when g is stable, asymptotically stable and exponentially stable. From this we characterize these stabilities according to Lyapunov exponents.
The paper is organized as follows, in the second section we establish some results, prove that the stable and unstable subgroups are simply connected and characterize the stable and unstable elements as attractors and repellers. In the third section we prove an important result of the paper, that is, given two linear vector fields we give conditions on it and on their stable spaces in order to ensure that their respective flows are conjugated. Finally, in the last section we study the Lyapunov stability.

Stable and unstable Lie subgroups
In this section we present notations and basic tools to prove that the stable, unstable components of the state space are simply connected. Moreover in the main result of this section we characterize the stable and unstable elements as attractors and repellers respectively.

Definition:
Let {ϕ t } t∈R be a flow of automorphisms on a connected Lie group G. We say that ϕ t is contracting if there are constants c, µ > 0 such that ||(dϕ t ) e X|| ≤ ce µt ||X|| for any X ∈ g; We say that ϕ t is expanding if (ϕ t ) −1 = ϕ −t is contracting.
The next results states that the existence of contracting/expanding flow of automorphisms on a Lie group requires some topological properties.

Proposition:
Let G be a connected Lie group and (ϕ t ) a flow of automorphism on G. If ϕ t is contracting or expanding, the Lie group G is simply connected.
Proof: Let us assume that ϕ t is contracting, since the expanding case is analogous. Let G be the connected simply connected cover of G and consider D to be the central discrete subgroup of G such that G = G/D.
Since the canonical projection π : G → G is a covering map and G is simply connected, we can lift (ϕ t ) t∈R to a flow ( ϕ t ) t∈R on G such that In particular ϕ t (D) = D for all t ∈ R. Being that D is discrete and ϕ t is continuous, we must have that ϕ t (x) = x for all x ∈ D and t ∈ R. However, since ϕ t is contracting, ϕ t is also contracting and so, ϕ t (x) →ẽ when t → ∞, whereẽ ∈ G is the identity element. Being that D is discrete and ϕ t -invariant, we must have for x ∈ D and t > 0 large enough that ϕ t (x) = e which implies that x = e. Therefore, D = {e} and so G = G, which concludes the proof.

Corollary:
If G is a compact Lie group it admits no expanding or contracting flow of automorphisms.
Now we can prove that unstable and stable subgroups of the linear flow are simply connected. Let us assume that G is a connected Lie group and let X be a linear vector field on G with linear flow (ϕ t ) t∈R . Associated with X we have the connected ϕ-invariant Lie subgroups G + , G 0 and G − with Lie algebras given, respectively, by where α is an eigenvalue of the derivation D associated with X and g α its generalized eigenspace.
We consider also G +,0 and G −,0 as the connected ϕ-invariant Lie subgroups with Lie algebras g +,0 = g + ⊕ g 0 and g −,0 = g − ⊕ g 0 respectively. The next proposition states the main properties of the above subgroups, its proof can be found in [7] and [8].

Proposition:
It holds: 4. All the above subgroups are closed in G;

If G is solvable then
Moreover, the singularities of X are in G 0 ; 6. If D is inner and G 0 is compact then G = G 0 . Moreover, if G 0 is compact, then G has the decomposition (5) above.
The subgroups G + , G − are called, respectively, the unstable and stable subgroups of the linear flow ϕ t . We denote the restriction of ϕ t to G + and G − , respectively, by ϕ + t and ϕ − t . The next proposition gives us another topological property of such subgroups.

Proposition:
The Lie subgroups G + and G − are simply connected.
Proof: Since (dϕ t ) e = e D restricted to g + and to g − has only eigenvalues with positive and negative real parts, respectively, we have that ϕ + t t∈R is an expanding flow of automorphisms and ϕ − t t∈R a contracting flow of automorphisms. Proposition 2.2 implies that G + and G − are simply connected.
Next we will prove a technical lemma that will be needed in the next sections.

Lemma: Let us assume that G is a connected Lie group and let
we have by the continuity of the product of the Lie group G that x n = h 1,n h 2,n → h 1 h 2 = x in G. Let us assume then that x n → x. By consideringh 1,n = h −1 1 h 1,n andh 2,n = h 2,n h −1 2 we can assume that x n → e and we need to show that h i,n → e, i = 1, 2. Let U i be a neighborhood of e ∈ H i , i = 1, 2. By the conditions on H i , there are is an open set of G (see Lemma 6.14 of [?]) and being that x n → e there is N ∈ N such that for n ≥ N we have x n ∈ W which by the condition that showing that h i,n → e for i = 1, 2 as desired.
Next we characterize the stable and unstable elements as attractors and repellers respectively. First we introduce the concept of hyperbolic linear vector field.

Definition
: Let X be a linear vector field. We say that X is hyperbolic if its associated derivation D is hyperbolic, that is, D has no eigenvalues with zero real part.

Remark:
If g α is the generalized eigenspace associated with the eigenvalue α of D, it is well known that when α+β is an eigenvalue of D and zero otherwise (see for instance Proposition 3.1 of [15]). This implies, in particular, that a necessary condition for the existence of a hyperbolic linear vector field on a Lie group G with dim G < ∞ is that G is a nilpotent Lie group.
To define the attractors and repellers we need on Lie Group G a metric space structure. Let ̺ stands for a left invariant Riemmanian distance on G. Using In particular, since (dϕ − t ) e = e tD| g − has only eigenvalues with negative real part, there are constants c, µ > 0 such that Analogously, we have that Now we have the main result of this section.

Theorem:
If X is a hyperbolic linear vector field on G then

It holds that
Proof: We will prove only the first assertion, since the proof of second assertion is analogous.
. Taking t → ∞ we get ̺(ϕ − t (g), e) → 0, that is, lim t→∞ ϕ t (g) = e. Conversely, let g ∈ G and assume that lim t→∞ ϕ t (g) = e. Being that X is hyperbolic, G is nilpotent and consequently, by Proposition 2.4, we have that g can be written uniquely as g = g − g + with g ± ∈ G ± . By the left invariance of the metric we have that Since g − ∈ G − we have by the first part of the proof and by our assumption that lim implying that lim t→∞ ϕ t (g + ) = e. This together with the inequality (7) implies that g + = e and consequently that g = g − ∈ G − as desired.

Remark:
It is well know that the solution to the linear system (4) in R n with initial condition x 0 is e At x 0 . Furthermore, if we consider the Euclidian metric, then a version of Theorem 2.9 is easily obtained. In fact, the definition of hyperbolic system in R n gives this result trivially (see for example [14]).

Conjugation between linear flows
In this section we classify the linear vector fields based on topological conjugacies between their associated flows. From now on we will consider X and Y to be linear vector fields on connected Lie groups G and H, respectively, and denote their linear flows by (ϕ t ) t∈R and (ψ t ) t∈R and their associated derivation by D and F , respectively. We say that X and Y are topological conjugated if there exists a homeomorphism π : G → H that commutates ϕ t and ψ t , that is, π(ϕ t (g)) = ψ t (π(g)), for any g ∈ G.
The next result establishes a first conjugation property of these restrictions.

Lemma: It holds that
where D + and F + are the restrictions of D and F to g + and h + , respectively. It follows that all eigenvalues of D have positive real part. Analogously, the same assertion is true for the eigenvalues of F and F .
Suppose now that n = m, then Theorem 7.1 in [?] assures that there exists a homeomorphism ζ : R n → R n such that ζ(e t D X) = e t F ζ(X), for any X ∈ R n .
Defining ξ : g + → h + by ξ = T −1 ζ S we see that which shows the topological conjugacy.
The map ζ : R n → R m given by ζ(v) = T ξ S −1 (v) is certainly a homeomorphism which by the Invariance of Domain Theorem implies that dim g + = n = m = dim h + concluding the proof. Proof: Let us do the unstable case, since the stable is analogous. By the above lemma, there exists ξ : g + → h + such that ξ(e tD + X) = e tF + ξ(X), for any X ∈ g + .
By Proposition 2.5 the subgroups G + and H + are simply connected, which implies that the map is well defined. Moreover, since both exp G + and exp H + are diffeomorphisms and ξ is a homeomorphism, we have that π is a homeomorphism. Let us show that π conjugates ϕ + t and ψ + t . Since ϕ + t • exp G + = exp G + • e tD + and ψ + t • exp H + = exp H + • e tF + we have, for any g ∈ G + , that Conversely, suppose that there exists a homeomorphism π : G + → H + such that π(ϕ t (g)) = ψ t (π(g)), g ∈ G + .

Theorem:
Let us assume that X and Y are hyperbolic. If the stable and unstable subgroups of ϕ t and ψ t have the same dimension, then ϕ t and ψ t are conjugated.
Proof: By the above theorem, there are homeomorphisms π u : G + → H + , that conjugates ϕ + t and ψ t , and π s : G − → H − that conjugates ϕ − t and ψ − t . By Proposition 2.4 we have, since G is nilpotent, that G = G + G − with G + ∩ G − = e G . Consequently, any g ∈ G has a unique decomposition g = g + g − with g + ∈ G + and g − ∈ G − . Moreover, the same statement is true for H. Therefore, the map π : G → H given by We will divide the rest of our proof in two steps: Step 1: π and π −1 are continuous. Let us show the continuity of π since the proof for π −1 is analogous. Let then (x n ) a sequence in G and assume that x n → x. By Proposition 2.4 there are unique sequences (g + n ) in G + and (g − n ) in G − such that x n = g + n g − n . If x = g + g − we have by Lemma 2.6 that x n → x if and only if g ± n → g ± in G ± . Since π u and π s are homeomorphism we have that π u (g + n ) → π u (g + ) and π s (g − n ) → π u (g − ) which again by Lemma 2.6 now applied to H, implies that showing that π is continuous.

Corollary:
Let us assume that X and Y are hyperbolic and that G = H. If the stable or the unstable subgroup of ϕ t and ψ t have the same dimension, then ϕ t and ψ t are conjugated.
Proof: In fact, if G + 1 and G + 2 are, respectively, the unstable subgroups of ϕ t and ψ t and, G − 1 and G − 2 , respectively, their stable subgroups, then We are now interested to prove the converse of Theorem 3.3. Then we need the next result that shows that any conjugation between hyperbolic linear vector fields has to take the neutral element of G to the neutral element of H.

Lemma: Let X and Y be hyperbolic linear vector fields on G and H,
respectively. If the homeomorphism π : G → H conjugate ϕ t and ψ t , then π(e G ) = e H .
Proof: We observe that e G ∈ G and e H ∈ H are, by item 5. of Proposition 2.4, the unique fixed points of flows ϕ t and ψ t , respectively. Then, the following equality ψ t (π(e G )) = π(ϕ t (e G )) = π(e G ).
shows the Lemma.
Now we are in conditions to prove the converse of Theorem 3.3.

Theorem:
Let us assume that X and Y are hyperbolic. If ϕ t and ψ t are conjugated, then their stable and unstable subgroups have the same dimension.
From Theorem 3.2 it is sufficient to show that ϕ ± t and ψ ± t are conjugated. We will only show that ϕ − t and ψ − t are conjugated, since the unstable case is analogous. We begin by showing that π(G − ) = H − . Take g ∈ G − . From Lemma 3.5 and Theorem 2.9 it follows that e H = π(e G ) = π lim t→∞ ϕ t (g) = lim t→∞ π(ϕ t (g)) = lim t→∞ ψ t (π(g)).
Again by Theorem 2.9 we get that π(g) ∈ H − showing that π(G − ) ⊂ H − . Analogously we show that π −1 (H − ) ⊂ G − and consequently that π(G − ) = H − . If we consider the restriction π s := π| G − we have that π s is a homeomorphism between G − and H − and it certainly conjugates ϕ − t and ψ − t which from Theorem 3.2 implies that the stables subgroups of ϕ t and ψ t have the same dimension.

Remark:
Someone can easily observe that Theorems 3.3 and 3.6 are versions to Lie group G of well known Theorems of topological conjugacy in R n ( see for example section 4.7 in [14]).

Lyapunov stability
In this section we will show that the stability properties of a linear flow on a Lie group G behaves in the same way as the one of the linear flow on the Lie algebra g induced by the derivation D.
In order to characterize the stability, let us define the Lyapunov exponent at g ∈ G in direction to v ∈ T g G by where the norm · is given by the left invariant metric.
Our next step is to show the invariance of the Lyapunov exponent. In fact, since that ϕ t • L g = L ϕt(g) • ϕ t , it follows that As · is a left invariant norm we have that It is clear that for v ∈ g we obtain λ(g, v(g)) = λ(e, v(e)). In other words, taking v ∈ g we obtain λ(e, v) = lim sup Our next Lemma is similar to well know result for Lyapunov exponent on R n .
This lemma will be used in the proof of forthcoming characterization of Lyapunov exponents. Before that we need to consider another decomposition of the Lie algebra g. Let us denote by λ 1 , . . . , λ k the k distinct values in of the real parts of the derivation D. We have then that g λi , where g λi := α;Re(α)=λi g α
We follow by introducing the version of stability of the system (3) on a Lie group for (see Definition 1.4.6 in [6]).

Definition:
Let g ∈ G be a fixed point of X . We say that g is 2) asymptotically stable if it is stable and there exists a g-neighborhood W such that lim t→∞ ϕ t (x) = g whenever x ∈ W ; 3) exponentially stable if there exist c, µ and a g-neighborhood W such that for all x ∈ W it holds that ̺(ϕ t (x), g) ≤ ce −µt ̺(x, g), for all t ≥ 0;

4) unstable if it is not stable.
We should notice that, since property 3) is local, it does not depend on the metric that we choose on G.
Next we prove a technical lemma that will be needed for the main results of this section.

Lemma
: Let X and Y be linear vector fields on the Lie groups G and H, respectively, and π : G → H be a continuous map that commutates the linear flows of X and Y. If the fixed point g of X is stable (asymptotically stable) and there is a g-neighborhood U such that V = π(U ) is open in H and the restriction π| U is a homeomorphism, then the fixed point π(g) of Y is stable (asymptotically stable). Moreover, if π is a covering map the converse also holds.
Proof: Let us assume that g is stable for X and let U ′ be a π(g)-neighborhood.
By the property of π around g, there exists a g-neighborhood U such that π restricted to U is a homeomorphism and π(U ) ⊂ U ′ . By the stability, there exists a g-neighborhood V such that ϕ t (V ) ⊂ U for all t ≥ 0. Consequently V ′ = π(V ) is a π(g)-neighborhood and it holds that showing that π(g) is stable for Y.
If g is asymptotically stable, there is a g-neighborhood W such that lim t→∞ ϕ t (x) = g for any x ∈ W . We can assume w.l.o.g. that W is small enough such that π restricted to W is a homeomorphism. Then W ′ = π(W ) is a π(g)-neighborhood and showing that π(g) is asymptotically stable for Y.
Let us assume now that π is a covering map and that π(g) is stable for Y. Since π is a covering map, there is a distinguished π(g)-neighborhood U ′ , that is, π −1 (U ′ ) = α U α is a disjoint union in G such that π restricted to each U α is a homeomorphism onto U ′ . Let U be a given g-neighborhood and assume w.l.o.g. that U is the component of π −1 (U ′ ) that contains g. By stability, there exists a π(g)- and consequently ϕ t (x) ∈ π −1 (U ′ ) for all t ≥ 0. Since π −1 (U ′ ) is a disjoint union and x ∈ V ⊂ U we must have ϕ t (x) ∈ U for all t ≥ 0. Being that x ∈ V was arbitrary, we get that ϕ t (V ) ⊂ U for all t ≥ 0, showing that g is stable for the linear vector field X .
The asymptotically stability follows, as above, from the fact that π has a continuous local inverse.
The following theorem characterizes, as for the Euclidian case, asymptotic and exponential stability at the identity e ∈ G for a linear vector field in terms of the eigenvalues of D( see for instance Theorem 1.4.8 in [6]).

Theorem:
For a linear vector field X the following statements are equivalents: (i) The identity e ∈ G is asymptotically stable; (ii) The identity e ∈ G is exponentially stable; (iii) All Lyapunov exponents of ϕ t are negative; (iv) The stable subgroup G − satisfies G = G − .
Proof: Since G = G − if and only if g = g − we have that (iii) and (iv) are equivalent, by Theorem 4.2. Moreover, by equation (6) we have that (iii) and (iv) implies (ii) and (ii) certainly implies (i). We just need to show, for instance, that (i) implies (iv), which we will do in two steps: Step 1: If e ∈ G is asymptotically stable, G is nilpotent; In fact, let U be a neighborhood of 0 ∈ g such that exp restricted to U is a diffeomorphism and such that exp(U ) ⊂ W . For any X ∈ ker D let δ > 0 small enough such that g = exp(δX) ∈ W . Since ϕ t (g) = g for any t ∈ R the asymptotic assumption implies that we must have g = e and consequently that X = 0 showing that ker D = {0}. The derivation D is then invertible which implies that g is a nilpotent Lie algebra and so G is a nilpotent Lie group.
Step 2: If e ∈ G is asymptotically stable, G = G − .
The derivation D on the Lie algebra g can be identified with the linear vector field on g given by X → D(X). Its associated linear flow is given by e tD . By the above step, G is a nilpotent Lie group which implies that exp : g → G is a covering map. Moreover, since ϕ t • exp = exp • e tD we have that e ∈ G is asymptotically stable if and only if 0 ∈ g is asymptotically stable for the linear vector field induced by D.
By the results in [6] for linear Euclidian systems we have that 0 ∈ g is asymptotically stable if and only if D has only eigenvalues with negative real part, that is, g = g − implying that G = G − and concluding the proof.

Remark:
We should notice that the above result shows us that, as for linear Euclidian systems, local stability is equal to global stability. Moreover, in order for e ∈ G be asymptotically stable for a linear vector field X is necessary that for G to be a simply connected nilpotent Lie group.
The next result concerns the stability of a linear vector field.

Theorem:
The identity e ∈ G is stable for the linear vector field X if G = G −,0 and D restricted to g 0 is semisimple.
Proof: First we note that G = G −,0 if and only if g = g −,0 . By Theorem 4.7 in [6] for linear Euclidian systems, the conditions that g = g −,0 and that D| g 0 is semisimple is equivalent to 0 ∈ g be stable for the linear vector field induced by D. Since exp is a local diffeomorphism around 0 ∈ g we have by Lemma 4.4 that 0 ∈ g stable for D implies that e ∈ G is stable for X concluding the proof.
The next result gives us a partial converse of the above theorem.

Theorem:
If e ∈ G is stable for the linear vector field X then G = G −,0 . Moreover, if exp G 0 : g 0 → G 0 is a covering map then e ∈ G stable for X implies also that D| g 0 is semisimple.
Proof: By equation (7) the only element in G + that have bounded positive X -orbit is the identity. Therefore, if e ∈ G is stable then G + = {e} and consequently G = G −,0 .
Since G 0 is ϕ-invariant, the linear flow X induces a linear vector field X G 0 on G 0 such that the associated linear flow is the restriction (ϕ t )| G 0 . Moreover, being that G 0 is a closed subgroup, it is not hard to prove that e ∈ G stable for X implies e ∈ G 0 stable for the restriction X G 0 .
If we assume that exp G 0 is a covering map, we have by Lemma 4.4 that e ∈ G 0 stable for X G 0 if and only if 0 ∈ g 0 stable for D| g 0 which by Theorem 4.7 of [6] implies that D| g 0 is semisimple.
When G is a nilpotent Lie group the subgroup G 0 is also nilpotent and so the map exp G 0 : g 0 → G 0 is a covering map. We have then the following.

Corollary:
If G is a nilpotent Lie group then e ∈ G is stable if and only if G = G −,0 and D| g 0 is semisimple.
4.10 Remark: Another example where we have that the stability of the linear vector field X on the neutral element implies that D| g 0 is semisimple is when G is a solvable Lie group and exp : g → G is a diffeomorphism, where G is the simply connected covering of G.

Conclusion
We conclude by observing that our work is an initial step for several studies. We explain this assertion with two important problems. First, since now we understand the topological conjugacy of linear systems of type (3), the next natural step is to study the topological conjugacy of linear control systems (2). Second, to study the concept of Morse index of the linear system (3). However, here, it is necessary to observe that by Corollary 2.3, the compacity necessary on G to this study implies that the flow of (3) has no expanding or contracting subgroups. Consequently a suitable homogenous space has to be considered in order to study the Morse index. To conclude, the similarity of results make us believe that one can show that several results, founded in classical literature of linear system in R n , are still true for the linear systems (3) on Lie groups.