Solutions of linear control systems on Lie groups

Victor Ayala; Alexandre J. Santana; Simão N. Stelmastchuk; Marcos A. Verdi

doi:10.1515/math-2024-0098

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Solutions of linear control systems on Lie groups

Victor Ayala , Alexandre J. Santana , Simão N. Stelmastchuk and Marcos A. Verdi

Published/Copyright: December 31, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

An explicit solution for linear control systems on Lie groups is presented using the variation of parameters method. This solution is formulated as a series of Lie brackets, and the convergence of these series is analyzed. Additionally, a concept of equivalence between two elements within this class is introduced.

Keywords: Linear systems on Lie groups; reachable sets; solution of linear systems

MSC 2010: 93C05; 34A05; 37C15

1 Introduction

The main purpose of this study is to explicitly establish the solution of linear control systems on Lie groups. While this class of control systems has been studied since the 1980s, the number of papers addressing this concept has recently increased significantly (see Ayala et al. [1], Da Silva et al. [2,3] and the references therein). In [4], Ayala and Kizil studied the solutions of linear control systems, leveraging that, in this context, the solution evaluated in a point g of a Lie group is given by the product of the solution at the identity by the drift flow at g . However, this work takes a different approach by presenting a computable solution using the method of variation of parameters. To the best knowledge available, this type of solution for linear systems has not yet been documented in the literature. To define linear control systems, consider G a real finite-dimensional and connected Lie group with Lie algebra g (set of right invariant vector fields on G ). Denoting by e the identity of G , a vector field X on G is called linear if for all Y ∈ g , [ X , Y ] ∈ g and X ( e ) = 0 . A linear control system or a linear system on G is given by

(1) g ′ = X ( g ) + ∑ i = 1 l u i Y i ( g ) ,

where the drift X is a linear vector field on G , the vector fields Y 1 , Y 2 , … , Y l are right invariant on G and u = ( u 1 , … , u l ) are the input maps such that u i : R → R are piecewise constant functions.

To describe the explicit solution of system (1), the work of Magnus [5] is considered, which provides the solution of a differential equation

g ′ = A ( t ) ⋅ g ,

on Lie groups G . Magnus presents the solution in the form of a series, applicable in a general context (see Theorem III in [5]). In the following decades, some authors established sufficient conditions for the convergence of this series (see Theorem 2.9 in the study by Iserles and Nooorsett [6] and Theorem 3 of Moan-Niesen [7]). In this context, to obtain the solution to (1), the variation-of-parameters method is used, combined with the results of [5]. Consequently, the solution is expressed as a series, and we proceed to investigate the convergence of this series.

This article is organized as follows: In Section 2, the solution to system (1) is established, along with a discussion of its convergence within nilpotent Lie groups and matrix Lie groups. In addition, an example in low dimension is provided. Section 3 examines the equivalence of systems using Magnus’s series, demonstrating that this equivalence preserves the reachable sets of control systems.

2 Solution of linear system on Lie group

The main outcome of this section is Theorem 1, which clearly establishes the solution for the system described in equation (1). In addition, a low-dimensional example is presented, along with an examination of the conditions for the convergence of this solution in the context of matrix Lie groups. To accomplish this, we first considered the following differential equation on G :

(2) g ′ = A ( t ) ⋅ g ,

where the real integrable matrix A ( t ) represents a curve in the Lie algebra g = T e G , A ( t ) ⋅ g = d ( R g ) e ( A ( t ) ) , and R g standing for the right translation. The solution of equation (2) (see Theorem III in [5]) reads as follows:

(3) g ( t ) = e x p Ω ( t ) ,

where

Ω ( t ) = ∫ 0 t A ( s ) d s − 1 2 ∫ 0 t ∫ 0 s A ( r ) d r , A ( s ) d s + 1 4 ∫ 0 t ∫ 0 s ∫ 0 r A ( σ ) d σ , A ( r ) d r , A ( s ) d s − ⋯

Denote by ϕ t the flow of the linear vector field X . Solutions of (1) are sought in the form g ( t ) = ϕ t ( c ( t , u ) ) , where c ( t , u ) ∈ G for fixed u . To enhance clarity in the discussion, using the notation c ( t ) is advisable. Therefore,

(4) g ′ ( t ) = X ( ϕ t ( c ( t ) ) ) + ( d ϕ t ) c ( t ) ( c ′ ( t ) ) .

Comparing (1) and (4), we have that c ( t ) satisfies

(5) ∑ i = 1 l u i ( t ) Y i ( ϕ t ( c ( t ) ) ) = ( d ϕ t ) c ( t ) ( c ′ ( t ) ) .

Now, take b ( t ) = c ′ ( t ) ⋅ c ( t ) − 1 , where this product is ( d R c ( t ) − 1 ) c ( t ) ( c ′ ( t ) ) . Then

(6) ( ( d R c ( t ) − 1 ) c ( t ) ) − 1 b ( t ) = c ′ ( t ) ,

and thus,

( d R c ( t ) ) e ( b ( t ) ) = c ′ ( t ) .

Therefore, we obtain the following equation:

(7) c ′ ( t ) = b ( t ) ⋅ c ( t ) .

Now, by (3), the solution of (7) is

(8) c ( t ) = e x p C ( t ) ,

where

C ( t ) = ∫ 0 t b ( s ) d s − 1 2 ∫ 0 t ∫ 0 s b ( r ) d r , b ( s ) d s + 1 4 ∫ 0 t ∫ 0 s ∫ 0 r b ( σ ) d σ , b ( r ) d r , b ( s ) d s − ⋯

Let D be the derivation associated with X , specifically the derivation of g such that D = − a d ( X ) (see Section 3 of [8] for more details). Note that using (4) and (6), we have

(9) ( d ϕ t ) c ( t ) ( c ′ ( t ) ) = ( d ϕ t ) c ( t ) ( ( d R c ( t ) ) e ( b ( t ) ) ) = ( d ( ϕ t ∘ R c ( t ) ) ) e ( b ( t ) ) .

Because X is a linear field, its flow ϕ t : G → G is a homomorphism of G . Thus,

ϕ t ∘ R c ( t ) = R ϕ t ( c ( t ) ) ∘ ϕ t .

Consequently, recognizing that ϕ t ( c ( t ) ) = g ( t ) , we have that the last term of (9) becomes

(10) ( d ( R g ( t ) ∘ ϕ t ) ) e ( b ( t ) ) = ( d R g ( t ) ) e ∘ ( d ϕ t ) e ( b ( t ) ) .

By combining (5), (9), and (10), we obtain

∑ i = 1 l u i ( t ) Y i ( ϕ t ( c ( t ) ) ) = ( d R g ( t ) ) e ∘ ( d ϕ t ) e ( b ( t ) ) .

It follows that

( d ϕ t ) e ( b ( t ) ) = ( ( d R g ( t ) ) e ) − 1 ( ∑ i = 1 l u i ( t ) Y i ( g ( t ) ) ) = ∑ i = 1 l u i ( t ) ( ( d R g ( t ) ) e ) − 1 ( Y i ( g ( t ) ) ) .

Denoting Y i ( e ) by Y i and considering that these vector fields are right invariant, the last equality becomes

(11) ( d ϕ t ) e ( b ( t ) ) = ∑ i = 1 l u i ( t ) Y i .

Knowing that ( d ϕ t ) e = e t D and based on equation (11), it follows that

b ( t ) = e − t D ∑ i = 1 l u i ( t ) Y i = ∑ i = 1 l u i ( t ) e − t D Y i .

Then

(12) C ( t , u ) = ∫ 0 t ∑ i = 1 l u i ( s ) e − s D Y i d s + 1 2 ∫ 0 t ∫ 0 s ∑ i = 1 l u i ( r ) e − r D Y i d r , ∑ i = 1 l u i ( s ) e − s D Y i d s + 1 4 ∫ 0 t ∫ 0 s ∫ 0 r ∑ i = 1 l u i ( σ ) e − σ D Y i d σ , ∑ i = 1 l u i ( r ) e − r D Y i d r , ∑ i = 1 l u i ( s ) e − s D Y i d s − ⋯

Finally, the solution of system (1) is

g ( t , u , e ) = ϕ t ( c ( t ) ) = ϕ t ( e x p C ( t , u ) ) = e x p ( e t D C ( t , u ) ) .

To summarize, suppose that G satisfies the convergence condition given by Magnus, that is, ∫ 0 t ‖ A ( τ ) ‖ 2 d τ < π (see Theorem 3 in [7]). Then, considering the series C ( t , u ) , we can state the following theorem.

Theorem 1

Let G be a connected Lie group. Assume the aforementioned convergence condition. Then the solution g ( t , u , e ) of system (1), with the initial condition g ( 0 , u , e ) = e , is given by

g ( t , u , e ) = e x p ( e t D ⋅ C ( t , u ) ) .

Corollary 2

Under the assumption of Theorem 1, the solution g ( t , u , x ) of system (1), with initial condition g ( 0 , u , x 0 ) = x 0 , is given by

g ( t , u , x ) = e x p ( e t D ⋅ C ( t , u ) ) ⋅ ϕ t ( x ) .

Example 3

Take the Heisenberg group given by the set of matrices of the form

1 x z 0 1 y 0 1 1 ,

where x , y , z ∈ R . Its Lie algebra g can be identified with R 3 , where the bracket is given by

[ ( x 1 , y 1 , z 1 ) , ( x 2 , y 2 , z 2 ) ] = ( 0 , 0 , x 1 y 2 − x 2 y 1 ) .

Take a basis { X , Y , Z } of g such that [ X , Y ] = Z and the other brackets are zero. A straightfoward computation shows that the linear vector fields on G can be written as follows:

X = ( a x + d y ) ∂ ∂ x + ( b x + e y ) ∂ ∂ y + ( 1 2 ( b x 2 + d y 2 ) + c x + f y + ( a + e ) z ) ∂ ∂ z ,

and the associated derivation D = − a d ( X ) is

(13) D = a d 0 b e 0 c f a + e .

Since brackets involving three or more elements are null, it turns out that the solution g ( t , u ) of the system

g ′ = X ( g ) + u ( t ) Y ( g )

is given by

g ( t , u ) = exp e t D ∫ 0 t u ( s ) e − s D Y d s − 1 2 ∫ 0 t ∫ 0 s u ( r ) e − r D Y d r , u ( s ) e − s D Y d s .

In the remainder of this section, we discuss the convergence of series C ( t , u ) . In the case of nilpotent Lie groups, we have C ( t , u ) converges for every t > 0 .

Our next step is to show the condition of convergence in the case of matrix Lie groups. If G is a matrix group and ∫ 0 t ‖ A ( τ ) ‖ d τ < π , then the Magnus series converges, and its sum C ( t , u ) satisfies exp ( C ( t , u ) ) = c ( t ) [7].

In our context, we have that A ( t ) = b ( t ) = ∑ i = 0 m u i ( t ) e − t D Y i . Moreover, assume that G semisimple, we have that D = ad ( X ) for some X ∈ g , then

‖ A ( τ ) ‖ ≤ ∑ i = 1 l ∣ u i ( t ) ∣ ‖ e − t D Y i ‖ ≤ ∑ i = 1 l ∣ u i ( t ) ∣ ‖ e − t ad ( X ) Y i ‖ ≤ ∑ i = 1 l ∣ u i ( t ) ∣ ‖ ad ( e − t ) X Y i ‖ .

Suppose that the norm is given by the Cartan Killing metric. Then, by Ad -invariance, we obtain

‖ A ( τ ) ‖ ≤ ∑ i = 1 l ∣ u i ( t ) ∣ ‖ Y i ‖ .

Now considering that the control maps are bounded, one can take the norm given by ∣ ∣ u i ∣ ∣ = sup { ∣ u i ( t ) ∣ : t ∈ R } . Since

∫ 0 t ∣ ∣ A ( τ ) ∣ ∣ d τ ≤ ∑ i = 1 l ∫ 0 t ∣ u i ( t ) ∣ ∣ ∣ Y 1 ∣ ∣ d τ ≤ t ∑ i = 1 l ∣ ∣ u i ∣ ∣ ∣ ∣ Y i ∣ ∣ ,

we obtain the following condition: if

t < π ∑ i = 1 l ∣ ∣ u i ∣ ∣ ∣ ∣ Y i ∣ ∣ ,

then ∫ 0 t ∣ ∣ ∣ A ( τ ) ∣ ∣ d τ ≤ π , ending the proof.

Proposition 4

Let G be a matrix Lie group with an Ad-invariant metric. Then, for a bounded control u, the series C ( t , u ) converges if

t < π ∑ i = 1 l ∣ ∣ u i ∣ ∣ ∣ ∣ Y i ∣ ∣ .

As a consequence of Theorem 2.9 [6], we have the next proposition.

Proposition 5

With the same assumptions as in the previous proposition, suppose there exists a t ∗ > 0 such that the control u is bounded by t λ in [ 0 , t ∗ ] . Then C ( t , u ) converges if

0 ≤ t ≤ min t ∗ , λ + 1 8 ∑ i = 0 l ‖ Y i ‖ 1 ⁄ ( λ + 1 ) .

Furthermore, t does not depend on the derivation D.

A convergence condition can now be established based on the derivation, specifically the largest eigenvalue of that derivation. In fact, it is important to note that

‖ A ( τ ) ‖ ≤ ∑ i = 1 l ∣ u i ( t ) ∣ ‖ e − t D Y i ‖ ≤ ∑ i = 1 l ∣ u i ( t ) ∣ ‖ e − t D ‖ ‖ Y i ‖ ≤ ∑ i = 1 l ∣ u i ( t ) ∣ e ∣ t ∣ ‖ D ‖ ‖ Y i ‖ ≤ ∑ i = 1 l ∣ u i ( t ) ∣ e ∣ t ∣ α ‖ Y i ‖ ≤ ∑ i = 1 l ∣ u i ( t ) ∣ e ∣ t ∣ α max { ∣ Y i ‖ } ,

where α is the largest eigenvalue of D . Then,

∫ 0 t ∣ ∣ A ( τ ) d τ ≤ ∫ 0 t ∑ i = 1 l ∣ u i ( τ ) ∣ e τ α max { ∣ ∣ Y i ∣ ∣ } d τ ≤ ∫ 0 t ∑ i = 1 l ∣ ∣ u i ∣ ∣ e τ α max { ∣ ∣ Y i ∣ ∣ } d τ = ∑ i = 1 l ∣ ∣ u i ∣ ∣ max { ∣ ∣ Y i ∣ ∣ } ∫ 0 t e τ α d τ = ∑ i = 1 l ∣ ∣ u i ∣ ∣ max { ∣ ∣ Y i ∣ ∣ } e t α − 1 α = max { ∣ ∣ Y i ∣ ∣ } e t α − 1 α ∑ i = 1 l ∣ ∣ u i ∣ ∣ .

Moreover,

max { ∣ ∣ Y i ∣ ∣ } e t α − 1 α ∑ i = 1 l ∣ ∣ u i ∣ ∣ < π ⇔ e t α < α π max { ∣ ∣ Y i ∣ ∣ } ∑ i = 1 l ∣ ∣ u i ∣ ∣ + 1 ⇔ t < 1 α ln α π max { ∣ ∣ Y i ∣ ∣ } ∑ i = 1 l ∣ ∣ u i ∣ ∣ + 1 .

Thus, the following result has been established.

Proposition 6

Suppose u is a bounded admissible control. Then, for

t < 1 α ln α π max { ∣ ∣ Y i ∣ ∣ } ∑ i = 1 l ∣ ∣ u i ∣ ∣ + 1 ,

the series C ( t , u ) converges.

3 Equivalence of linear systems

This section discusses the equivalence of linear control systems using the previous series C ( t , u ) and demonstrates that this equivalence preserves the reachable sets. This equivalence is given by a conjugation of the systems flows, and hence, we first note that in general, the topological conjugation of two flows is given by a homeomorphism between the state spaces of the flows. However, because the continuous isomorphism of Lie groups is differentiable, we consider conjugation by differentiable isomorphisms. Consider the following two linear systems on G

(14) g ′ = X ( g ) + ∑ i = 1 l u i ( t ) X i ( g ) ,

(15) h ′ = Y ( h ) + ∑ i = 1 l u i ( t ) Y i ( h ) .

We denote by ϕ t and ψ t the flows of X and Y , and by D X and D Y their derivations, respectively. Suppose that there exists a differentiable homomorphism H : G → G such that

H ∘ ϕ t = ψ t ∘ H ;
( d H ) e X i ( e ) = Y i ( h ( e ) ) for every i = 1 , … , n .

The following equivalences can be established.

Proposition 7

Let H : G → G be a differentiable homomorphism.

The following sentences are equivalent:

(i) H ∘ ϕ t = ψ t ∘ H ;

(ii) ( d H ) ϕ t ( g ) ( X ( ϕ t ( g ) ) ) = Y ( ψ t ( H ( g ) ) ) ;

(iii) ( d H ) e ( e t D X X ) = e t D Y ( d H ) e ( X ) ;

(iv) ( d H ) e ( D X ) = D Y ( d H ) e ( X ) .

Proof

First, suppose H ∘ ϕ t = ψ t ∘ H , then

( d H ) ϕ t ( g ) d d t ( ϕ t ( g ) ) = d d t ( ψ t ∘ H ( g ) ) ,

hence it follows (ii) ( d H ) ϕ t ( g ) X ( g ) = Y ( H ( g ) ) . The uniqueness of solutions to differential equations is used to deduce (i) from (ii).

Now, consider (i), take the derivative at e on both sides of this equation and use the fact that ( d ϕ t ) e = e t D X to obtain (iii).

Now, assuming (iii), d ( H ∘ ϕ t ) e = d ( ψ t ∘ H ) e . Since G is connected, (i) holds.

It is clear that equality (iii) implies (iv). To obtain (iii) from (iv), it is sufficient to write e t D X as a power series and apply ( d H ) e .□

Lemma 8

Suppose that there exists a homomorphism H : G → G such that (A) and (B) hold. Denote by C 1 ( t , u ) and C 2 ( t , u ) the series given in (12) corresponding to the linear control systems (14) and (15), respectively. Then, ( d H ) e ( C 1 ( t ) ) = C 2 ( t ) .

Proof

Observe that C 1 ( t ) and C 2 ( t ) satisfy the differential equations

d C 1 ( t ) d t = ∑ k = 0 ∞ B k k ! a d C 1 k ( γ ( t ) ) and d C 2 ( t ) d t = ∑ k = 0 ∞ B k k ! a d C 2 k ( δ ( t ) ) ,

where B k denotes the Bernoulli numbers,

γ ( t ) = ∑ i = 1 l u i ( t ) e − t D X X i and δ ( t ) = ∑ i = 1 l u i ( t ) e − t D Y Y i .

From the aforementioned proposition, it follows that ( d H ) e ( γ ( t ) ) = δ ( t ) . Furthermore, because ( d H ) is an isomorphism of the Lie algebras, we have

d d t ( ( d H ) e ( C 1 ( t ) ) ) = ( d H ) e ∑ k = 0 ∞ B k k ! a d C 1 k ( γ ( t ) ) = ∑ k = 0 ∞ B k k ! a d ( d H ) e ( C 1 ) k ( δ ( t ) ) .

Then, ( d H ) e ( C 1 ( t ) ) and C 2 ( t ) satisfy the same differential equation with initial condition ( d H ) e ( C 1 ( 0 ) ) = 0 = C 2 ( 0 ) . Therefore, ( d H ) e ( C 1 ( t ) ) = C 2 ( t ) .□

Theorem 9

Let H : G → G be a homomorphism of Lie groups. The solutions of (14) and (15) are denoted by f ( t , u ) and g ( t , u ) , respectively. Therefore, the following conditions are equivalent:

H ( g ( t , u ) ) = f ( t , u ) ;
H ∘ ϕ t = ψ t ∘ H and ( d H ) e X i ( e ) = Y i ( H ( e ) ) for every i = 1 , … , n .

Proof

Suppose that the second condition holds. From Theorem 1, the solutions of linear systems (14) and (15) are as follows:

g ( t , u ) = exp ( e t D X C 1 ( t ) ) and f ( t , u ) = exp ( e t D Y C 2 ( t ) ) ,

where C 1 ( t ) and C 2 ( t ) are given in (12). Then

H ( g ( t , u ) ) = H ( exp ( e t D X C 1 ( t ) ) ) = exp ( ( d H ) e e t D X C 1 ( t ) ) .

From Proposition 7, we have H ( g ( t , u ) ) = exp ( e t D Y ( d H ) e C 1 ( t ) ) , then by Lemma 8, we obtain condition (1). Now suppose that H ( g ( t , u ) ) = f ( t , u ) . Then,

(16) ( d H ) g ( t , u ) X ( g ( t , u ) ) + ∑ i = 1 l u i ( t ) X i ( g ( t , u ) ) = Y ( f ( t , u ) ) + ∑ i = 1 l u i ( t ) Y i ( f ( t , u ) ) .

Taking t = 0 , it follows that

( d H ) e X ( e ) + ∑ i = 1 l u i ( t ) X i ( e ) = Y ( e ) + ∑ i = 1 l u i ( t ) Y j ( e ) ,

since g ( 0 , u ) = f ( 0 , u ) = e . As X and Y are linear vector fields, we have

∑ i = 1 l u i ( 0 ) [ ( d H ) e ( X i ( e ) ) − Y i ( e ) ] = 0 .

Knowing that the aforementioned equality holds for all control u ( t ) = ( u 1 ( t ) , … , u n ( t ) ) , we conclude that

(17) ( d H ) e ( X i ( e ) ) − Y i ( e ) = 0 .

It means that ( d H ) e ( X i ( e ) ) = Y i ( e ) , for every i = 1 , … , n .

On other hand, we can write equation (16) as follows:

( d H ) g ( t , u ) X ( g ( t , u ) ) − Y ( f ( t , u ) ) = ∑ i = 1 l u i ( t ) [ ( d H ) e ( X i ( g ( t , u ) ) ) − Y j ( f ( t , u ) ) ] .

By utilizing the right invariance of vector fields and knowing that H is a homomorphism, we obtain the results using equation (17):

( d H ) g ( t , u ) X ( g ( t , u ) ) − Y ( f ( t , u ) ) = 0 ,

that is, ( d H ) g ( t , u ) X ( g ( t , u ) ) = Y ( H ( g ( t , u ) ) ) . As it is not difficult to show H ∘ ϕ t = ψ t ∘ H , it follows the second condition.□

Theorem 9 provides an equivalence of linear control systems and it is not difficult to show that this equivalence preserves the reachable sets from the identity. We denote these sets by

( ℛ X ) t = { g ( t , u ) : u ∈ L ∞ [ 0 , t ] and g ( 0 , u ) = e } ,

for t > 0 . Then we have the following proposition.

Proposition 10

Suppose that there exists a homomorphism H : G → G such that (A) and (B) hold. Then H ( ( ℛ X ) t ) = ( ℛ Y ) t .

Proof

Let g ( t , u ) ∈ ( ℛ X ) t . From Theorem 9, we observe that H ( g ( t , u ) ) = H ( t , u ) , where H ( t , u ) ∈ ( ℛ Y ) t . Thus, h ( ( ℛ X ) t ) = ( ℛ Y ) t .□

We finish this section with the following remark.

Remark 11

With these results, it make sense to establish the following equivalence relation. Two systems are equivalent if there exists a group isomorphism H : G → G such that the corresponding Lie algebra isomorphism ( d H ) e satisfies ( d H ) e ( D X ) = D Y ( d H ) e and ( d H ) e X i ( e ) = Y i ( H ( e ) ) , for every i = 1 , … , n .

4 Conclusion

The main result, Theorem 1, provides a computable description of the solution of the linear system. This description is being applied to investigate the properties of this class of control systems. Recall that a classical topic of control theory is controllability. In our setup, it signifies the ability to navigate a system through its entire state space G using only permissible controls u . That is, consider a linear system on G as presented in the introduction, denote by g(t, u, h) its solution in h, then this system is called controllable if for all h 1 , h 2 ∈ G , there exist T > 0 and admissible control u such that g ( t , u , h 1 ) = h 2 for t ∈ ( o , T ) . Then it is clear the importance of a good description of the solution to study controllability. Another important concepts in control theory are the observability concept, which is a dual concept of controllability and stability. All these concepts needs of the solution of the system to be study [9]. We therefore hope that we will soon be able to apply the results of this article to study controllability. Finally, recall that the Jouan equivalence theorem allows us to extend any result of a linear control system on Lie groups to more general affine systems on manifolds which dynamics satisfy a finiteness condition, see reference [8].

Acknowledgements

The authors are grateful for the reviewer’s valuable comments, which helped to improve the manuscript.

Funding information: A. J. Santana was partially supported by CNPq grant no. 309409/2023-3.
Author contributions: VA: conceptualization, formal analysis, funding acquisition, reviewing and editing. AJS: supervision, investigation, funding acquisition and project administration. SNS: validation, formal analysis, methodology and visualization. MAV: software, resources, data curation and administration writing – original draft preparation. All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.
Conflict of interest: The authors state no conflicts of interest.

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Received: 2024-08-10

Revised: 2024-10-25

Accepted: 2024-11-08

Published Online: 2024-12-31

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Keywords for this article

Linear systems on Lie groups; reachable sets; solution of linear systems

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