Boundedness control sets for linear systems on Lie groups

Víctor Ayala; María Torreblanca Todco

doi:10.1515/math-2018-0035

Article Open Access

Boundedness control sets for linear systems on Lie groups

Víctor Ayala and María Torreblanca Todco

Published/Copyright: April 18, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

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.

Keywords: Decomposable group; Linear system; Control set; Boundedness

MSC 2010: 16W25; 93B05; 93C05

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

g˙(t)=X(g(t))+∑j=1muj(t)Xj(g(t)),

where 𝓧 is a linear vector field, X^j ∈ 𝔤 considered as right invariant vector fields 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 0 ∈ intΩ. 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 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

A={x∈G:∃u∈U and t≥0 withx=ϕt,u(e)}

is an open set. It turns out that

Σ is controllable on G⇔SpecLy(D)∩R={0}.

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 [2] proved that the requirement Spec_Ly(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

Ce=cl(A)∩A∗

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 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 𝓐_G⁻ = 𝓐 ∩ G⁻ and AG+∗ = 𝓐^∗ ∩ G⁺. The authors show that for semisimple or nilpotent Lie groups the compactness of 𝓐 _G⁻, AG+∗ and G⁰ together is a sufficient 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 effectively 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 sufficient algebraic condition for the boundedness of 𝓒_e 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

g˙(t)=X(g(t))+∑j=1muj(t)Xj(g(t)),u∈U.

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 (φ_t)_t∈ℝ the flow of 𝓧. By definition, 𝓧 is an infinitesimal automorphism, which means

{φt:t∈R} is a subgroup of Aut(G),

where Aut(G) is the Lie group of all G-automorphisms. Associated with 𝓧 there exists a derivation 𝓓 of 𝔤 supplied by

DY=−[X,Y](e), for all Y∈g.

The relationship between φ_t and 𝓓 is given by the formulas, [6],

(dφt)e=etDandφt(exp⁡Y)=exp⁡(etDY),t∈R,Y∈g.

On the other hand, the family of vector field Xu=∑j=1mujXj depends on m fixed right invariant vector fields X^j ∈ 𝔤 and the family of admissible control u = (u₁, …, u_m) ∈ 𝓤 which has the mission to redirect 𝓧 to reach the desired goal.

2.2 Reachable sets

For a state g ∈ G, the reachable set from g up to the time t is defined by

A≤t(g)={h∈G:∃u∈U and τ∈[0,t] withh=ϕτ,u(g)}.

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 𝓐_t₁ ⊂ 𝓐_t₂
for allg ∈ G, 𝓐_t(g) = 𝓐_tφ_t(g)
for allu ∈ 𝓤, g ∈ Gandt ≥ 0 it followsϕ_t,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

A≤t∗(g)={h∈M:∃u∈U and τ∈[0,t] andϕτ,u(h)=g}.

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.,

Span{Di(Yj):whereD0=Id,j=1,…,mandi=0,1,…}=g.

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

gα={Y∈g:(D−α)nY=0 for some n≥1}.

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 0 otherwise. Of course, 𝔤 decomposes as

g=g+⊕g0⊕g−, whereg+=⨁α:Re(α)>0gα,g0=⨁α:Re(α)=0gα and g−=⨁α:Re(α)<0gα.

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

C=cl(A(y))∩A∗(y).

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 𝓐_G⁻ = 𝓐 ∩ G⁻ and AG+∗ = 𝓐^∗ ∩ G⁺ then

A=AG−G+,0,A∗=AG+∗G−,0.

Furthermore, 𝓐_G⁻, AG+∗ and G⁰ 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

Ce=cl(A)∩A∗ bounded⇒cl(AG−),cl(AG+∗)andG0 bounded.

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 [5].

Theorem 3.1

Let us assume thatGis semisimple or nilpotent. If cl(𝓐_G⁻), cl(AG+∗) andG⁰are compact subsets ofGthen 𝓒_eis bounded.

Recall that a linear transformation L is said to be hyperbolic if L has just eigenvalues with nonzero real parts, in other words

SpecLy(D)∩R={0}.

Theorem 3.2

LetGbe a nilpotent simply connected Lie group. Then,

Cisbounded⇔cl(AG−),cl(AG+∗)arecompactsandDishyperbolic.

Remark 3.3

The main aim of the paper is to find algebraic conditions to decide wether cl(𝓐_G⁻), cl(AG+∗) are bounded sets. With that and the hyperbolic notion we can ensure the boundedness of the control set 𝓒_e.

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. IfSpec_Ly(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

ρ(φt(g),e)≤c−1e−λtρ(g,e), for any t≥0,(∗)

In fact, consider a curve γ : [0, 1] → G with γ(0) = e and γ(1) = g. Thus, φ_t ∘ γ is a curve connecting e to φ_t(g) and

ρ(φt(g),e)≤∫01‖(d(φt)γ(s)(γ˙(s))‖ds.

Now, any G-homomorphism ϕ satisfies the formula

ϕ∘Lg=Lϕ(g)∘ϕ.

Subsequently,

(dϕ)g=(dLϕ(g))e∘(dϕ)e∘(dLg−1)g.

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(φt)g‖=‖etD‖.

By hypothesis 𝓓 is a stable matrix, then (∗) follows.

Take t > 0 such that 𝓐_t ⊂ B(e, 1) the open ball with center e and radius 1. Just observe that for every positive number τ

φτ(B(e,1))⊂B(e,c−1e−λτ).

By using the same argument we obtain

A(n+1)t=Atφt(At) φ2t(At)…φnt(At)⊂B(e,1)B(e,c−1e−λt)B(e,c−1e−2λt)…B(e,c−1e−nλt).

Now, any g ∈ G can be decomposed as

g=g0g1g2…gn with gi∈B(e,c−1e−itλ),i=0,1,…,n.

Since the metric is left invariant, the following inequalities are true

ρ(g,e)=ρ(g0g1g2…gn,e)≤ρ(g0g1g2…gn,g0)+ρ(g0,e)≤ρ(g1g2…gn,e)+ρ(g0,e)≤ρ(g1g2…gn,g1)+ρ(g1,e)+ρ(g0,e)≤ρ(g2…gn,e)+ρ(g1,e)+ρ(g0,e)…≤Σi=0nρ(gi,e)<Σi=0nc−1e−iλt<c−1Σi=0∞(e−λt)i<∞.

Hence, there exists a radius R such that

R=c−1Σi=0∞(e−λt)i>0⇒Ant⊂B(e,R), for any n∈N.

This ends the proof, actually

A=∪n∈NAnt⊂B(e,R).

□

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(𝓐_G⁻) is bounded.

Proof

According to our hypothesis the group is decomposable, thus

G=G−G0G+.

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

πG−:G−→G/G+,0 is a homeomorphism.

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 π_G⁻ 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, π_G⁻(V) = π(V) = π(VW) is an open set in G/G^+,0. 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^+,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⁻ = (π_G⁻)⁻¹(K) ⊂ G⁻ such that 𝓐⁻ ⊂ π(K⁻). From that, we obtain

π(A)=A−⇒A⊂K−G+,0⇒AG−=A∩G−⊂K−.

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(AG+∗) 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 [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 𝓒_e = cl(𝓐) ∩ 𝓐^∗is bounded
G = G⁻ → 𝓐^∗ = G and 𝓒_e = cl(𝓐) is compact
G = G⁺ → 𝓐 = Gand 𝓒_e = 𝓐^∗is open

Proof

We have,
D hyperbolic ⇔g0={0}⇔g0 is compact⇔G0 is compact.
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(𝓐_G⁻) and cl(AG+∗) are bounded. Thus, Theorem 3.1 shows that 𝓒_e is bounded. On the other hand, if 𝓒_e is bounded it follows that G⁰ ⊂ 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 thatΣis controllable from the identity, i.e., for any arbitraryg ∈ Gthere exists a controluand a positive timetsuch thatϕ_t,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 (𝓐_G⁻) and cl(AG+∗) is to assume that both 𝔤^+,0 and 𝔤^−,0 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+,0andg−,0areidealsofg⇔[g0,g+]=0and[g+,g−]⊂g0.

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

SpanLA{Di(Yj):whereD0=Id,j=1,…,mandi=0,1,…}=g.

The LARC condition is weaker than the ad-rank condition. Actually, in the first case you are allowed to compute the Lie brackets [𝓓^i₁(Y^j₁),𝓓^i₂(Y^j₂)], which is forbidden in the other case. Therefore,

Ce=GforanytransitivelinearsystemonG.

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 solutionX_t(g) with initial conditiong ∈ G. By the right invariance, the solution through the initial conditiongis provided by the right translation bygof the solutionX_t(e) = exp_G(tX) through the identity element. In order words

Xt(g)=expG⁡(tX)g.

Here, exp_G:𝔤 → G is the usual exponential map. Hence, X defines by conjugation a 1-parameter group of inner automorphism as follows

φt(g)=Xt(e)gX−t(e),g∈G, and φt∈Aut(G) for any t∈R.

Therefore, it is possible to compute the linear vector field as

X(g)=(ddt)t=0φt(g).

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

{G=(xy01):x>0andy∈R}

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

∂g={D=(00ab):a,b∈R}.

From Remark 5.1 the linear vector fiel 𝓧 associated to 𝓓 is given by

X(x,y)=(0a(x−1)+by00).

Let Σ be the transitive linear system on G defined by

g˙(t)=X(g(t))+u(t)X(g(t)),u∈U

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 𝓒_e is the only one with nonempty interior. It turns out that,

g+=g−=0⇒g0=g.

Thus,

G+,0=G⊂A and G−,0=G⊂A∗⇒Ce=G.

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

G={g=(1xz01y001):(x,y,z)∈R3}

of dimension three. The generators of 𝔤 are provided by

X=∂∂x,Y=∂∂y+x∂∂z and Z=∂∂z.

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

∂g={(ab0cd0efa+d):a,b,c,d,e,f∈R}.

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,

g˙(t)=X(g(t))+u1(t)X(g(t))+u2(t)Z(g(t)),u∈U, with Ω=[−1,1].

We have, Spec_Ly(𝓓) = {−1, −2}. So, 𝔤^−,0 = 𝔤⁻ = 𝔤 and 𝔤^+,0 = 0 are both ideals of 𝔤. On the other hand,

Span{X,Z,D(X)=X−Y}=g.

Since 𝓓 is a hyperbolic derivation, Theorem 3.7 shows that the existent control set 𝓒_e 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

g=Span{X,Y,Z}

consider the system

g˙(t)=X(g(t))+u1(t)X(g(t))+u2(t)Y(g(t)),u∈U,withΩ=R,

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

g˙(t)=X(g(t))+u1(t)(X−Y)(g(t))+u2(t)(X+Y+Z)(g(t)),u∈U,withΩ=[−1,1]

where the derivation 𝓓 is furnished bya = 1, d = −1 andb = c = e = f = 0. Hence, 𝓓(X − Y) = X + Y. Thus,

Span{X−Y,Z,X+Y}=g

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 Σ_ℝ²

(x˙y˙)=(100−1)(xy)+(11)u:u∈U with Ω=[−1,1]

which satisfies Lemma 2.5. Moreover,

A=R×(−1,1) and A∗=(−1,1)×R.

Thus, the control set 𝓒_e restricted to the plane is bounded and reads

(Ce)R2=(−1,1)×[−1,1], see [4].

However, 𝓒_e 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,

Ce=(−1,1)×[−1,1]×(1,1,1)R.

Example 5.6

Let us consider the nilpotent Lie groupGwith Lie algebra

g=RX1+RX2+RX3+RX4,andtherules[X4,X2]=X4,[X3,X2]=X4+X2,[X1,X2]=X3and[ X1,X3]=X4.

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 five dimensional and reads as

∂g={(a−a000000bca+b0deb+c+d2a+b):a,b,c,d,e∈R}.

Since the underlying topological space of G is the connected and simply connected manifold ℝ⁴, Theorem 3.5 applies. However, 0 ∈ Spec_Ly(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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Received: 2017-10-06

Accepted: 2018-01-25

Published Online: 2018-04-18

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0035

Keywords for this article

Decomposable group; Linear system; Control set; Boundedness

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