Abstract
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.
1 Introduction
Let G be a connected Lie group with Lie algebra 𝔤. In [1] the authors introduce the notion of a linear system Σ on G which is determined by a family of differential equations
where 𝓧 is a linear vector field, Xj ∈ 𝔤 considered as right invariant vector fields and u ∈ 𝓤 ⊂ L∞(ℝ, Ω ⊂ ℝm) is the class of admissible controls. We deal with Ω as a subset of ℝm with 0 ∈ intΩ. Furthermore, Σ is called restricted if Ω is compact and unrestricted if Ω = ℝm.
We denote by ϕt,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 ℝn an unrestricted linear system is controllable if and only if it satisfies de Kalman rank condition [1], 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
Here, 𝓓 ∈ ∂𝔤 is a 𝔤-derivation associated to 𝓧 and the Lyapunov spectrum SpecLy(𝓓) consists of the real parts of the 𝓓-eigenvalues.
Recently, the authors in [2] proved that the requirement SpecLy(𝓓) ∩ ℝ = {0} implies controllability for any Lie group with finite semisimple center, that is, for any Lie group which admits a maximal semisimple Lie subgroup with finite 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 [3], basically, a subset 𝓒 where controllability holds on int(𝓒). For a locally controllable system, it is shown in [4] 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 𝓒e. We concentrate the study on solvable Lie groups because in this case the space state is firstly decomposable. This means that G can be written as a direct product of the closed subgroups G+, G0 and G− with Lie algebras 𝔤+, 𝔤0, and 𝔤− induced by the 𝔤-derivation 𝓓 which determines the drift vector field 𝓧. Secondly, any solvable Lie group has trivially the finite semisimple center property. Hence, we can apply any result about control sets from [5]. In particular, denote by 𝓐G− = 𝓐 ∩ G− and
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 sufficient algebraic condition for the boundedness of 𝓒e is given. In Section 4, we remark some possible extensions. And finally, Section 5 contains a couple of examples in low dimensional Lie groups.
2 Preliminaries
In what follows Σ will denote a linear system on a connected Lie group G. In this section, we establish the basic definitions 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 field 𝓧 on 𝔤 = 𝔤+ ⊕ 𝔤0 ⊕ 𝔤− and, its dynamics consequences on the corresponding closed subgroups G+, G0, and G− of G.
2.1 The dynamic structure of Σ
As we mention, a linear system Σ is furnished by
Essentially, its dynamic behavior is determined by two different classes of vector fields. First, the uncontrolled differential equation ġ(t) = 𝓧 (g(t)). Denote by (φt)t∈ℝ the flow of 𝓧. By definition, 𝓧 is an infinitesimal 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
The relationship between φt and 𝓓 is given by the formulas, [6],
On the other hand, the family of vector field
2.2 Reachable sets
For a state g ∈ G, the reachable set from g up to the time t is defined by
and 𝓐(g) = ⋃t>0 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 [7] and [8].
Proposition 2.1
For a linear systemΣon the connected Lie groupGit holds
0 ≤ t1 ≤ t2implies 𝓐t1 ⊂ 𝓐t2
for allg ∈ G, 𝓐t(g) = 𝓐tφt(g)
for allu ∈ 𝓤, g ∈ Gandt ≥ 0 it followsϕt,u(𝓐(g)) ⊂ 𝓐(g)
e ∈ int𝓐 if and only if 𝓐 is open
The controllable set to g up to the time t is defined by
The controlled set to g is 𝓐∗(g) = ⋃t>0𝓐≤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 satisfies the ad-rank condition, i.e.,
The system is said to be locally accessible atgif int(𝓐≤t(g)) and int(𝓐≤t∗(g)) is nonempty for anyt ≥ 0, and controllable fromgif 𝓐(g) = G.
2.3 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 field 𝓧. We address the generalized eigenspaces of 𝓓 provided by
Here, α runs over the spectrum Spec(𝓓). It turns out that [𝔤α,𝔤β] ⊂ 𝔤α+β if α+β ∈ Spec(𝓓) and 0 otherwise. Of course, 𝔤 decomposes as
It follows that 𝔤+, 𝔤0, 𝔤− are Lie subalgebras and 𝔤+, 𝔤− are nilpotent, see Proposition 3.1 in [6].
Let us denote by G+, G−, G0, G+,0, and G−,0 the connected Lie subgroups of G with Lie algebras 𝔤+, 𝔤−, 𝔤0, 𝔤+,0 = 𝔤+ ⊕ 𝔤0 and 𝔤−,0 = 𝔤− ⊕ 𝔤0, respectively.
Definition 2.3
Let 𝓓 be a 𝔤-derivation. The Lie algebra 𝔤 is said to be 𝓓-decomposable if 𝔤 = 𝔤+ ⊕ 𝔤0 ⊕ 𝔤−.
We collect some basic properties of these subgroups, Proposition 2.9 in [7].
Proposition 2.4
Let 𝓓 be a 𝔤-derivation. It holds,
G+,0 = G+G0 = G0G+andG−,0 = G−G0 = G0G−
G+ ∩ G− = G+,0 ∩ G− = G−,0 ∩ G+ = {e}
G+,0 ∩ G−,0 = G0
G+,G0,G−,G+,0andG−,0are closed
IfGis solvable thenGis decomposable
2.4 Control sets
A more realistic approach to the controllability property of a system comes from the following notion. A nonempty set 𝓒 ⊂ G is called a control set, [3] if
for every g ∈ G there exists u ∈ 𝓤 such that ϕ(t, g, u) ⊂ 𝓒, t ≥ 0
𝓒 ⊂ cl(𝓐(g)) for every g ∈ 𝓒
𝓒 is maximal with properties (i) and (ii).
In [4] 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 ofΣ. If the system is locally accessible at any point ofint(𝓒) then for anyy ∈ int 𝓒
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.
3 Main result
In this section, our main results are stated and proved. For that, we apply several results appearing in [5]. From now we assume that G is decomposable. It turns out that 𝓐 and 𝓐∗ are also decomposable. Denote by 𝓐G− = 𝓐 ∩ G− and
Furthermore, 𝓐G−,
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 𝓒e = cl(𝓐) ∩ 𝓐∗. On the other hand, by hypothesis 𝔤 is 𝓓-decomposable hence 𝓒e 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 𝓒e. The following two results were proved in [5].
Theorem 3.1
Let us assume thatGis semisimple or nilpotent. If cl(𝓐G−), cl(
Recall that a linear transformation L is said to be hyperbolic if L has just eigenvalues with nonzero real parts, in other words
Theorem 3.2
LetGbe a nilpotent simply connected Lie group. Then,
Remark 3.3
The main aim of the paper is to find algebraic conditions to decide wether cl(𝓐G−), cl(
Let Σ be a linear system on a connected Lie group G. If 𝓓 is a stable matrix the reachable set on G− is bounded. More precisely:
Proposition 3.4
Let 𝓓 be the derivation associated with 𝓧. IfSpecLy(𝓓) ⊂ ℝ−then the reachable set 𝓐 = 𝓐−is bounded
Proof
Let us denote by ρ the left invariant metric of G, [9]. There exist c > 1 and λ > 0 such that
In fact, consider a curve γ : [0, 1] → G with γ(0) = e and γ(1) = g. Thus, φt ∘ γ is a curve connecting e to φt(g) and
Now, any G-homomorphism ϕ satisfies the formula
Subsequently,
The homomorphism φt belongs to Aut(G) for any t ∈ ℝ and 𝓓 = d(φt)e. Since the metric ρ is left invariant, we get
By hypothesis 𝓓 is a stable matrix, then (∗) follows.
Take t > 0 such that 𝓐t ⊂ B(e, 1) the open ball with center e and radius 1. Just observe that for every positive number τ
By using the same argument we obtain
Now, any g ∈ G can be decomposed as
Since the metric is left invariant, the following inequalities are true
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
LetΣbe a linear system on a decomposable connected Lie groupG. Assume that 𝔤+,0is an ideal of 𝔤 then cl(𝓐G−) is bounded.
Proof
According to our hypothesis the group is decomposable, thus
Since 𝔤+,0 is an ideal the Lie subgroup G+,0 is normal. In particular, the homogeneous space G/G+,0 is a Lie group isomorphic to G−. Let us consider the canonical projection π : G → G/G+,0. It turns out that π(𝓐) = 𝓐−. Furthermore, on G/G+,0 the derivation 𝓓 associated to the drift vector field 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+,0 is isomorphic to 𝔤+,0 = 𝔤+ ⊕ 𝔤0 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 first need to show that
Actually, since any element in G has a unique decomposition in G−G0G+ the application is bijective. By the own definition of the quotient topology on G/G+,0 the projection π restricted to G− is continuous. Next, we prove that πG− is an open map. First, there are neigborhoods V ⊂ G− and W ⊂ G+,0 of the identity e ∈ G such that the product VW is also a neigborhood of e. In particular, πG−(V) = π(V) = π(VW) is an open set in G/G+,0. If g ∈ G we consider the translations Lg(V) = gV and Lg(W) = gW. Since Lg is a homeomorphism, the proof is done and πG− is a homeomorphism.
Once again, the group G is decomposable, thus π(G−) = G/G+,0 and it is possible to cover 𝓐 with the projection of a compact subset of 𝓐. In fact, for any compact K containing 𝓐− define the compact set K− = (πG−)−1(K) ⊂ G− such that 𝓐− ⊂ π(K−). From that, we obtain
Since 𝓐− is bounded, it follows that cl(𝓐 G−) is also bounded as we claim. □
Corollary 3.6
LetΣbe a linear system on a decomposable connected Lie groupG. Assume that 𝔤−,0is an ideal of 𝔤 then cl(
Proof
The proof is completely analogous to that of Theorem 3.5. □
Every nilpotent Lie group as a solvable group is decomposable, see [5].
Theorem 3.7
LetΣbe a linear system on a nilpotent simply connected Lie groupG. Assume that 𝔤+,0and 𝔤−,0are ideals of 𝔤. Then,
𝓓 hyperbolic ⇔ the control set 𝓒e = cl(𝓐) ∩ 𝓐∗is bounded
G = G− → 𝓐∗ = G and 𝓒e = cl(𝓐) is compact
G = G+ → 𝓐 = Gand 𝓒e = 𝓐∗is open
Proof
We have,
The last equivalence depends strongly on the fact that in this particular case the exponential map is a global diffeomorphism. Just observe that in general this is not true. For instance, exp(ℝ) = S1, however, the 1-dimensional sphere is not simply connected. Now, our hypothesis and Theorem 3.5 implies that cl(𝓐G−) and cl(
To prove the second item we observe that under the hypothesis G ⊂ 𝓐∗. So, 𝓒e = cl(𝓐) is trivially closed and bounded by Theorem 3.1.
If G = G+ we get 𝓐 = G. Thus, 𝓒e coincides with the open set 𝓐∗.
Remark 3.8
We observe that item third of Theorem 3.7 shows thatΣis controllable from the identity, i.e., for any arbitraryg ∈ Gthere exists a controluand a positive timetsuch thatϕt,u(e) = g. For other results in the same spirit, we invite the readers to take a look at the following references, [2, 7, 8, 10, 11]. Furthermore, in [12] 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 sufficient condition for the simultaneous boundedness of cl (𝓐G−) and cl(
Proposition 3.9
Let 𝔤 be a Lie algebra and 𝓓 ∈ ∂ 𝔤. It turns out
4 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 [10] the authors prove the following result:
Theorem 4.1
IfGis a connected and compact semisimple Lie group, a linear systemΣis controllable onGif and only if the system is transitive, i.e., satisfies the Lie algebra rank condition, (LARC), provided by
The LARC condition is weaker than the ad-rank condition. Actually, in the first case you are allowed to compute the Lie brackets [𝓓i1(Yj1),𝓓i2(Yj2)], which is forbidden in the other case. Therefore,
The noncompact case
Here, we just comment that except the case G = G0, the space state cannot be decomposable. In fact, in [5] we show that the set G−G0G+ ⊊ 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.
5 Examples
In this section, we give some examples of boundedness and unboundedness control sets on some decomposable Lie groups. But first, we explain how to find the face of the drift 𝓧 when it is induced by an inner derivation.
Remark 5.1
A particular class of linear vector fields is easy computed through a 1-parameter of innerG-automorphisms. TakeX ∈ 𝔤 a right invariant vector field and consider the solutionXt(g) with initial conditiong ∈ G. By the right invariance, the solution through the initial conditiongis provided by the right translation bygof the solutionXt(e) = expG(tX) through the identity element. In order words
Here, expG:𝔤 → G is the usual exponential map. Hence, X defines by conjugation a 1-parameter group of inner automorphism as follows
Therefore, it is possible to compute the linear vector field as
The associated derivation 𝓓 : 𝔤 → 𝔤 is 𝓓(Y) = −[X, Y], Y ∈ 𝔤.
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 [13] we built an algorithm which provides an effective means to compute the Lie algebra ∂ 𝔤 that we use in this section.
Example 5.2
Consider the solvable affine group
with Lie algebra 𝔤 = Span {X, Y} and [X, Y] = Y. An easy computation shows that∂ 𝔤 is given just by inner derivation with the shape
From Remark 5.1 the linear vector fiel 𝓧 associated to 𝓓 is given by
Let Σ be the transitive linear system on G defined by
where, 𝓓 = ad(Y) comes from a = −1 and b = 0. Since ad(Y)X = −Y then Span {X,𝓓X} = 𝔤. So, Σ satisfies the ad-rank condition, 𝓐 is open and of course, Σ satisfies also LARC. Moreover, G is solvable thus the control set 𝓒e 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 = 0.
Example 5.3
Let 𝔤 = ℝX + ℝY + ℝZbe the Lie algebra of the connected and simply connected Heisenberg Lie groupG
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 6 real parameters matrix in the basis {X, Y,Z} as follow
Consider the linear system Σ with derivation 𝓓 determined by its coefficients a = d = −1, b = 1, c = −1, e = f = 0 and the control vectors X and Z,
We have, SpecLy(𝓓) = {−1, −2}. So, 𝔤−,0 = 𝔤− = 𝔤 and 𝔤+,0 = 0 are both ideals of 𝔤. On the other hand,
Since 𝓓 is a hyperbolic derivation, Theorem 3.7 shows that the existent control set 𝓒e is bounded.
Example 5.4
On the rotational group S0(3, ℝ) with Lie algebra 𝔰𝔬 (3, ℝ) the skew-symmetric real matrix of order three
consider the system
where 𝓧 = ad(X). SinceΣsatisfies LARC, the control set is bounded and coincides with the group. The system is controllable from the identity.
Example 5.5
Take the linear systemΣon the Heisenberg groupGlike in Example 2, but with different dynamic behavior
where the derivation 𝓓 is furnished bya = 1, d = −1 andb = c = e = f = 0. Hence, 𝓓(X − Y) = X + Y. Thus,
an 𝓐 is an open set.
If we restrict Σ to the plane ℝ2 = Span {X, Y} we get a classical linear system on the vector space Σℝ2
which satisfies Lemma 2.5. Moreover,
Thus, the control set 𝓒e restricted to the plane is bounded and reads
However, 𝓒e can not be bounded. Despite the fact that 𝔤+,0 = Span {X, Z} and 𝔤−,0 = Span {Y, Z} are ideals, the derivation 𝓓 = diag(1, −1, 0) is just hyperbolic on the plane not on G. Actually,
Example 5.6
Let us consider the nilpotent Lie groupGwith Lie algebra
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 𝓒e is unbounded. In fact, a straightforward computation shows that the Lie algebra of 𝔤-derivations is five dimensional and reads as
Since the underlying topological space of G is the connected and simply connected manifold ℝ4, Theorem 3.5 applies. However, 0 ∈ SpecLy(𝓓) for any 𝓓 ∈ ∂ 𝔤. Thus, no hyperbolic derivation exits, ending the proof.
Acknowledgement
Proyecto de Investigación Básica e Investigación Aplicada, n∘ 24, 2017, Concytec-Fondecyt, UNSA, Perú.
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