Solution Curve for Linear Control Systems on Lie Groups

arXiv:1911.10294v1 [math.OC] 23 Nov 2019 Solution Curve for Linear Control Systems on Lie Groups João Paulo Lima de Oliveira Alexandre J. Santana Departamento de Matemática, Universidade Estadual de Maringá Maringá, Brazil Simão N. Stelmastchuk Universidade Federal do Paraná, Jandaia do Sul, Brazil December 3, 2019 Abstract The purpose of this paper is to describe explicitly the solution for linear control systems on Lie groups. In case of linear control systems with inner derivations, the solution is given basically by the product of the exponential of the associated invariant system and the exponential of the associated invariant drift field. We present the solutions in low dimensional cases and apply the results to obtain some controllability results. AMS 2010 subject classification: 93B05, 93C25, 34H05. Key words: linear control system, solutions, controllability. 1 Introduction Linear control system on Rn are control system given by differential equation ẋ = Ax + Bu, x ∈ Rn , (1) where A is a n × n-matrix, B is a m × n, and u = (u1 , u2 , . . . , um ) is an admissible control. Markus in [9] studied this systems in case of matrix groups and, later, Ayala and Tirao [2] extended for general Lie groups. Recall that a linear control system on connected Lie group G is a control system given by the differential equation m X ui (t)Xi (g), g ∈ G, (2) ġ = X (g) + i=1 where X is a linear vector field, namely, its flow ϕt is a family of automorphisms of G, X1 , . . . , Xm are invariant vector fields, and u = (u1 , . . . , un ) is an admissible control. 1 Despite linear control systems (1) have a good description of their solutions (see for instance ([1]), it is not true in case of systems on Lie groups (2). The first results in this way is found in [2]. Our purpose is to contribute in this direction describing the solutions of linear control flow φt (u, .). In fact, we construct the solution of the linear control system (2) using a technique of Cardetti and Mittenhuber [5], that is, considering an invariant system on a semi-direct product G ×ϕ R. Then the solution curve is obtained as the integral curve of a certain invariant vector field. We also show that if the derivation D, associated to the linear vector field X , is inner then the solution of linear control system has a simpler description (see Theorem 5.1). In current literature, we can note that there are not general results on controllability problems of linear control system on Lie groups (see e.g. [2], [5], [6], [7], [9], [11]). Then with the description of the solutions we can obtain some results on controllability. The paper is organized as follows, in the second section we establish some basic facts about linear controls system. In the third section we construct the solution of linear control system. In fourth section, we study controllability of linear control system on a subgroup H of G. Finally, in the last section we study solutions of linear control system (2) when the derivation is inner, and, as application of this section, we can construct solutions on matrix groups GL(n, R)+ and on all 3-dimensional, semisimple Lie groups: SL(2, R), SU (2), SO(3, R), and SO(2, 1). 2 Linear Vector Fields This section is intended to recall some necessary facts about linear vector fields and linear systems on Lie groups that will be useful along the paper (see [7] for more details). Let G be a connected Lie group with Lie algebra g. Throughout this paper, g is the set of right invariant vector fields. For every g ∈ G, the maps Rg , Lg : G → G are, respectively, the right and left translations on G. A vector field X on G is said to be linear if its flow, which is denoted by ϕt , is an one-parameter group of automorphisms. In [7] it is showed that a linear vector field X can be characterized by one of the following conditions: (i) for all t ∈ R, ϕt is an automorphism of G; (ii) for all Y ∈ g, [X , Y ] ∈ g and X (e) = 0, where e is identity of G. (iii) for all g, h ∈ G, X (gh) = d(Rh )g X (g) + d(Lg )h X (h). For any linear vector field X it is possible to associate a derivation D : g → g defined by D(Y ) = −[X , Y ]. From a derivation D and a liner flow ϕ given by X we can see that d(ϕt )e = etD ϕt (exp(Y )) = exp(etD Y ). and A particular case of derivations is inner derivation, that is, D = −ad(X) for some X ∈ g. In this case, the linear vector field X can be decomposed as X = X + dIX, where dIX is the left-invariant vector field induced by inverse map I(g) = g −1 . 2 A linear system on G is a control system of the form ΣL : m X dg uj Yj (g), = X (g) + dt j=1 where X is a linear vector field, Y1 , . . . , Ym are right-invariant vector fields and m the control functions u : R → U ⊂ Rm belong to a subset U ⊂ L∞ loc (R; R ) of the space of the locally integrable functions. Without loss of generality we can assume, throughout this paper, that U is the set of piece-wise constant functions. For u ∈ U and g ∈ G, we denote the solution of ΣL starting at g associated to u by φt (u, g) with t ∈ R. For g, h ∈ G, we say that h is reachable from g in time t if there is u such that φt (u, g) = h. Let us denote by At (g) the set of all points in G reachable from g in time t. The reachable set from g is defined as follows [ A(g) = At (g). t≥0 The system ΣL is said to be controllable if any h is reachable for any g. Equivalently, A(g) = G for any g ∈ G. 3 Solution for Linear Control Systems In this section we come up with a construction presented in [5] which associates to a linear control system ΣL an invariant one, denoted by ΣI . We begin by remembering that the flow ϕt of the linear vector field X yields a representation ϕ : R → Aut(G). This allows us to define the semi-direct product G ×ϕ R, that is, the set G × R endowed with the product (g, t)(h, s) = (gϕt (h), t + s) (see e.g. [10]). It is well-know that G ×ϕ R is a Lie group. Furthermore, its correspondent Lie algebra is the semi-direct product of Lie algebras g ×σ R, where σ : R → Der(g) is defined as σ(t)(Y ) = adtX (Y ) = [tX , Y ]. The relation between ϕ and σ is given by dϕ0 = σ. For any vector fields (Y, t), (W, s) ∈ g ×σ R, the Lie bracket is given by the formula [(Y, t)(W, s)] = ([Y + tX , W + sX ], 0) . Throughout this paper, for (g, r) ∈ G ×ϕ R, R(g,r) we denote the right translation and, when not specified, the differential dR(g,r) is evaluated at the group identity. Now we determine the value of a vector field (W, s) on an arbitrary point (g, r). Proposition 3.1 If (W, s) ∈ g ×σ R and (g, r) ∈ G ×ϕ R, then (W, s)(g, r) = (W (g) + sX (g), s) . Proof: We first write (W, s)(g, r) = dR(g,r) (W, s). Thus, in matrix notation, the differential dR(g,r) gives    dRg X (g) W dR(g,r) (W, s) = = (W (g) + sX (g), s) . 0 1 s 3  This Proposition allow us to describe exponentials of invariant vector fields of g ×σ R. Lemma 3.2 If (W, 0), (0, s) ∈ g ×σ R, then their exponentials are smooth curves (exp(tW ), 0) and (e, st), respectively, with t ∈ R. Proof: We first compute the exponential for (W, 0). To this purpose, we note that (W, 0)(g, r) = (W (g), 0) for all (g, r) ∈ G ×ϕ R in view of Proposition 3.1. By definition of exponential,   d d (exp(tW ), 0) = exp(tW ), 0 = (W (exp(tW )), 0) = (W, 0)(exp(tW ), 0). dt dt The result follows by uniqueness of solution. Analogously we can see that the curve (e, st) is the exponential of (0, s).  In the following, the previous Lemma will be used to determine the exponential of an invariant vector field (W, s) ∈ g ×σ R. Proposition 3.3 Let (W, s) be an invariant vector field on G ×σ R. It follows that ! n−1 Y ϕist/n ◦ exp(t/n · W ), st . (3) exp(t(W, s)) = lim n→∞ i=0 Proof: We first write (W, s) = (W, 0) + (0, s). Applying the Lie product formula gives exp(t(W, s)) = n lim (exp(t/n(W, 0)) · exp(t/n(0, s))) . n→∞ Using the above Lemma and the semidirect product we see that exp(t(W, s)) = = = n lim ((exp(t/n · W ), 0)(e, st/n)) n→∞ lim (exp(t/n · W ), st/n)n n→∞ lim n→∞ n−1 Y ϕist/n ◦ exp(t/n · W ), st , i=0 and the proof is complete. Using the relation d(ϕt )e = etD we can rewrite formula (3) as ! n−1 Y
exp t/n · eDt W , st , exp(t(W, s)) = lim n→∞ !  i=0 ist D. n ¯ Consider the vector fields X̄ = (0, 1), Yj = (Yj , 0) ∈ g ×σ R, for each j = 1, . . . , m. It follows from Proposition 3.1 that these vector fields can be expressed where, for simplification, we denote Dt = 4 as X¯(g, r) = (X (g), 0) and Ȳj (g, r) = (Yj (g), 0). We define the following invariant control system on G ×ϕ R: ΣI : m X d(g, r) uj Y¯j (g, r). = X¯ (g, r) + dt j=1 Equivalently, we have  dg/dt dr/dt  m X   uj Yj (g)   X (g) + = . j=1 1 This system was built in such a way that π(ΣI ) = ΣL , where π : G ×ϕ R → G is the projection on the first coordinate. If we denote AI (g, r)) the reachable set of ΣI at point (g, r) and AL (g) the reachable set of ΣL at point g, then π(AI (g, r)) = AL (g). Furthermore, the invariance of the system allows us to write AI (g, r) = AI (e, 0) · (g, r). And now, we move on to the main result. Theorem 3.4 For u = (u1 , . . . , um ) ∈ Rm the curve   n−1 m Y X t ϕit/n ◦ exp  φt (u, e) = lim uj Yj  n→∞ n i=0 j=1 (4) is the solution, with initial condition φ0 (u, e) = e, of the dynamical system ΣL : Proof: Let us denote W = m X dg uj Yj (g). = X (g) + dt j=1 m X uj Yj and exp(t(W, 1)) = (φt (u, e), t). From Propo- j=1 sition 3.1 we have that  (W, 1)(φt (u, e), t) = (W (φt (u, e))+X (φt (u, e)), 1) = X (φt (u, e)) + m X j=1 uj Yj (φt (u, e)), 1 . On the other side, the curve (φt (u, e), t) is the integral curve of W . Therefore (W, 1) exp(t(W, 1)) = (W, 1)(φt (u, e), t) = ( dφt (u, e) , 1). dt We thus get  dφt dt (u, e) 1  m X   uj Yj (φt (u, e))   X (φt (u, e)) + = . j=1 1 5  Taking the projection on the first coordinate we conclude that the curve φt (u, e) satisfies the differential equation of the dynamical system. As φ0 (u, e) = e we have that this is the solution of the system at the identity. On the other side, Proposition 3.3 give us a description of exp(t(W, 1)). Since exp(t(W, 1)) = (φt (u, e), t), we conclude that   m n−1 X Y t ϕit/n ◦ exp  φt (u, e) = lim uj Yj  . n→∞ n j=1 i=0  The above theorem allow us to recover a well-know result about linear systems. Corollary 3.5 Let g ∈ G be an arbitrary point and consider the linear dynamical system as in the previous theorem. The solution φt (u, g) of the system starting at g is given by the formula φt (u, g) = φt (u, e)ϕt (g). Proof: Consider an arbitrary point (g, r) ∈ G ×ϕ R. Denote φI ((g, r), t) the solution of control system ΣI at point (g, r). We have already seen that φL (g, t) = π(φI ((g, r), t)). Now, using the right invariance property and the semidirect product we get φt (u, g) = π (φI ((e, 0), t)(g, r)) . By the proof of the previous theorem, it follows that φI ((e, 0), t) = (φt (u, e), t). Therefore φt (u, g) = π(φt (u, e)ϕt (g), t + r) = φt (u, e)ϕt (g).  Remark 1 It is possible to apply the previous theorem (and its corollary) to obtain the solution curve for linear control systems by means of the cocycle property. In fact, without loss of generality, consider a piece-wise constant control u : [0, t + s] → U ⊂ Rm , with t, s ∈ R, given by concatenation  u1 , if r ∈ [0, t] u(r) = u2 , if r ∈ (t, t + s], where u1 and u2 are constants. Now, using Corollary 3.5 and cocycle property we get φt+s (u, e) = φs (u2 , e)ϕs (φt (u1 , e)). Example 3.1 (Linear Control Systems on Heisenberg Group) Let G be the Heisenberg group, that is, the set of all real matrix of the form   1 x z 0 1 y  . 0 0 1 6 As usual, we identify this group with R3 in such a way that the group product is defined as (x1 , y1 , z1 ) · (x2 , y2 , z2 ) = (x1 + x2 , y1 + y2 , z1 + z2 + x1 y2 ). In this case, the Lie algebra of the Heisenberg group is the vector space R3 with the Lie bracket defined as [(x1 , y1 , z1 )(x2 , y2 , z2 )] = (0, 0, x1 y2 − x2 y1 ) and the exponential map exp : R3 → R3 as   xy exp(x, y, z) = x, y, +z . 2 The right invariant vector fields Y = (m, n, p) in G have the form Y (x, y, z) = m ∂ ∂ ∂ +n + (my + p) , ∂x ∂y ∂z while the matrix of a derivation D associated to a linear vector field X is written as   a11 a12 0 . 0 D = a21 a22 a31 a32 a11 + a22 We consider the following linear control system on G dg = X (g) + uY (g), dt where Y = (0, 0, p) and X is linear vector field associated to derivation D. In view of Remark 1, it is enough to express the solution of the system on an interval [0, T ] in which the control function u is constant. In formula (4), we note, by a direct calculation, that   t it/n·D t . ue Y = 0, 0, up n n Taking the exponential we obtain φt (u, e) = n−1 X t up 0, 0, lim n→∞ n i=0 ! =   Z t upds = (0, 0, upt), 0, 0, (5) 0 where u(t) = u, for 0 ≤ t ≤ T . From the solution (5) it is easy to see that the linear control system is not controllable on G. 4 Controlability In this section, our intention is to study controllability of a linear system in a Lie subgroup of G. We begin by establishing some current notation: L = a = Da = Lie{X , Y1 , . . . , Ym } Lie{Y1 , . . . , Ym } span{Di (Y ) : Y ∈ h, i ∈ N}. 7 In general the subalgebra a is not D-invariant. Consider, for example, a linear system dx = Ax + ub dt defined on R2 where A is a rotation matrix. It is clear that [A, b] = −Ab ∈ / a. For this reason, we consider h = LA(Da) the g subalgebra generated by Da and recall Proposition 3 from [7]: Proposition 4.1 The subalgebra h is D-invariant. Therefore, the Lie algebra L satisfies L = RX ⊕ h. Moreover, the linear system satisfies the rank condition if and only if h = g. We denote by H the closed subgroup generated by h = LA(Dh). It follows that ϕt (H) ⊂ H. Furthermore, X is tangent to H, in other words, X (h) ∈ Th H if h ∈ H. We now establish a first result about control system on H. Proposition 4.2 If h ∈ H, then for all control u and for all t we have φt (u, h) ∈ H. Proof: Corollary 3.5 gives that φt (u, h) = φt (u, e)ϕt (h). We know that ϕt (h) ∈ H for h ∈ H, since H is ϕt -invariant. From solution (4) we can see that ϕt (u, e) ∈ H. Therefore, φt (u, h) ∈ H.  Here, the key of proof is the fact φt (u, e) ∈ H. This fact, given by Theorem 3.4, allow us to study controllability in a new perspective. For example, a direct result about controllability is obtained. Theorem 4.3 If H 6⊆ G, then ΣL is not controllable on G. Equivalently, if h 6⊆ g, then ΣL is not controllable on G. Proof: We begin by observing that ϕt (u, h) ∈ H when h ∈ H by Proposition above. Thus, for any g ∈ G\H, it is not possible to find some control and time t > 0 such that ϕt (u, h) = g. It means that Σ is not controllable on G.  Theorem above shows a restriction to controllability of Σ in G. However, it does not show any information about controllability of Σ on H. Our idea is to give controllability conditions on H. Our first step is to point out that the rank condition occur naturally. Corollary 4.4 The linear system on H satisfies the rank condition. Proof: It is a consequence of the definition of H and Proposition 4.1 since h is the Lie algebra of H.  To show our first results we need to introduce the stable, unstable and central subgroup to linear system on H. Since h is D-invariant, it follows that D is a derivation on h. From eigenvalues of derivation D on h we can written M M M hα , hα , and h− = hα , h0 = h+ = α;Re(α)>0 α;Re(α)=0 8 α;Re(α)<0 where α are eigenvalues of the derivation D, such that h = h+ ⊕ h0 ⊕ h− and [hα , hβ ] = hα+β , with α + β = 0 if the sum is not an eigenvalue. Let us denote by H + , H 0 and H − the connected ϕt -invariant Lie subgroups H + , H 0 and H − with Lie algebras h+ , h0 and h− , respectively. To make the next proof clear, we remember the notation of attainable set from identity e: AH := {x ∈ H : φt (u, e) = x, for some t ≥ 0}. If we change X by −X we obtain the inverse linear flow ϕ∗t , which satisfies the following property ϕ∗t = ϕ−t . Thus, we have the inverse linear control system defined by m X dg uj Yj (g). = −X (g) + dt j=1 Its attainable set from identity is denoted by A∗H := {x ∈ H : φ∗t (u, e) = x, for some t ≥ 0}. Theorem 4.5 Let G be a solvable Lie group and H the Lie subgroup of G given above. If a derivation D of linear control field X has only eigenvalues with zero real part when it restricts to H, then ΣL is controllable on H . Proof: We first observe that H is solvable. Furthermore, it is true that ΣL satisfies ad-rank on H. Thus, Proposition 2.13 in [6] assures that AH and A∗H are open. Also, from Theorem 4.1 of [6] we conclude that ΣL is controllable in H.  Theorem 4.6 Let G be a nilpotent Lie group and H the Lie subgroup of G given above. Assume that Σ is bounded. Then ΣL is controllable on H if and only if H = H0 . Proof: It is a direct application of Theorem 4.5 in [6].  Other way to study controllability is by means of rank condition and Lie saturate, as presented at Theorem 12 in [8, ch.3]. We show that D-invariance is sufficient condition to controllability of ΣL in H. In fact, from Proposition 4 in [7] we have that a is included on Lie Saturate. Then a possible condition to controllability is a = h. Our next result show a condition to occur this equality. Proposition 4.7 a = h if and only if a is D-invariant. Proof: We first suppose that a is D-invariant. Thus, it is direct that Da ⊂ a. Consequently, h ⊂ a. We thus get h = a. Conversely, suppose that a = h. Then it is true that D(Y ) ∈ a for all Y ∈ a. It means that a is D-invariant.  Theorem 4.8 If a is D-invariant, then Σ is controllable on H. 9 Proof: If a is D-invariant, then a = h. From Proposition 4 in [7] it follows that h is the Lie saturate. On the other hand, the control flow φt satisfies the rankcondition. Therefore, by Theorem 12 in [8], the linear control system is controllable in H.  5 Inner Derivation Case In this section we apply formula (4) when D is inner, namely, there is X ∈ g such that D = ad(X) (see for instance [10]). In particular, the linear vector field X can be decomposed as X = X + dIX, where X is a right invariant vector field and dIX is the vector field induced by I(g) = g −1 . These facts allow us to describe the solution in a simpler way and to relate it to the solution of an associated invariant system. We begin by remembering, in this case, that the flow ϕt can be written as ϕt (g) = exp(tX)g exp(−tX) (see e.g. [3] or [7]). Note that the results stated along this section are, in particular, true for semisimple Lie groups since all derivations defined on their Lie algebras are inner (see for instance [10]). Given a linear control system ΣL , we yield the following right-invariant control system m X dg ΣI : uj Yj (g), = X(g) + dt j=1 where X is such that X = X + dIX. Replacing ϕt (g) = exp(tX)g exp(−tX) in the formula (4) of Theorem 3.4, we obtain the following description for the solutions of ΣL : Theorem 5.1 Let u be an admissible control function and suppose that it is constant on an interval [0, T ]. Then the solution of the linear system with control function u is the curve   m X tuj Yj  exp(−tX), 0 ≤ t ≤ T. (6) φt (u, e) = exp tX + j=1 Proof: We first simplify the notation by denoting Y = m X uj Yj . Replacing j=1         t it t it ϕit/n exp Y = exp X exp Y exp − X n n n n in formula (4) we see that φt (u, e) = = = n−1 Y   t ϕit/n exp Y lim n→∞ n i=0       n−1 Y it it t exp X exp Y exp − X lim n→∞ n n n i=0  n−1     t t (1 − n)t t lim exp Y exp X Y exp X . exp n→∞ n n n n 10         t t t t Y exp X exp − X exp − Y in the above expresn n n n sion we can assert that  n t t φ( u, e) = lim exp Y exp X exp (−tX) . n→∞ n n Inserting exp Using the Lie product Formula we conclude that   m X tuj Yj  exp(−tX), φt (u, e) = exp tX + j=1 and proof is complete. 5.1  Application This section presents applications of the above results in case of linear systems with inner derivation. Initially, we consider a system defined in GL(n; R)+ in the same way as in [9], and describe the solution curve for this case. According to [4], there are four three-dimensional semisimple Lie groups, which are Sl(2), SU (2), SO(3) and SO(2, 1)0 , so we construct the solutions of linear systems in these Lie groups. 5.2 Lie group in GL(n; R)+ Let G = Gl(n; R)+ be the set of all n × n real matrices with positive determinant and g = gl(n; R) its Lie algebra. For A ∈ g, the vector field XA (g) = Ag − gA is linear and its associated flow is ϕt (g) = etA ge−tA . Given B1 , . . . , Bm ∈ g, consider the right-invariant fields Bj (g) = Bj g and the linear control system m X dg uj Bj (g). = Ag − gA + dt j=1 Applying formula (4) for a constant control u, we can write the solution at the identity we can write the solution as φt (u, e) = et(A+ P uj Bj) −tA e . Now, considering a concatenation of two constant controls u1 and u2 on an interval [0, t + s], as in Remark 1, the solution becomes φt (u, g) = es(A+ P u2j Bj) t(A+ e P u1j Bj) −tA e , where u1 = (u11 , . . . , u1n ) and u2 = (u21 , . . . , u2n ). Corollary 5.2 If span{B1 , B2 , . . . , Bn } is invariant by A, then the control system is controllable on span{B1 , B2 , . . . , Bn }. 11 5.3 Semisimple Lie groups of dimension 3 We only study linear system on special linear group Sl(2, R) because in others are the same accounts. Its Lie algebra have the following description    x y sl(2, R) = : x, y, z ∈ R . z −x From above description, we have that the eigenvalues λ1 , λ2 of an element Z ∈ g satisfy the condition λ1 + λ2 = 0. Given a linear system ΣL over Sl(2, R), for a control function u, constant on an interval [0, T ], we denote ΣI = X + m X uj (t)Yj , 0 ≤ t ≤ T, j=1 the matrix of the right-invariant associated system. With these notations, by Theorem 5.1, the solution of ΣL is φt (u, e) = exp (tΣI ) exp(−tX), 0 ≤ t ≤ T. We denote by λ1 , λ2 the eigenvalues of ΣI and by δ1 , δ2 the eigenvalues of X. Applying methods of functional calculus we have exp(tΣI ) = p(ΣI ) and exp(tX) = q(X), where p(z) = az+b and q(z) = cz+d are polynomials satisfying the following conditions: p(λi ) = aλi + b = etλi , for i = 1, 2. q(δj ) = cδj + d = etδj , for j = 1, 2. Using the fact that λ1 + λ2 = δ1 + δ2 = 0 we obtain p(z) = senh(tδ) senh(tλ) z + cosh(tλ) and q(z) = z + cosh(tδ), λ δ where λ and δ may be any of the correspondent eigenvalues. The solution of the linear system then is written as    senh(tλ) senh(tδ) φ( u, e) = ΣI + cosh(tλ)Id X + cosh(tδ)Id , λ δ for 0 ≤ t ≤ T . As mentioned above, on the other 3-dimensional, semisimple Lie groups SU (2), SO(3) and SO(2, 1), the argument to construct solutions of linear control systems are the same. Summarizing: • In Sl(2, R),   senh(tδ) senh(tλ) ΣI + cosh(tλ)Id X + cosh(tδ)Id ; λ δ • In SU (2),    sen(tλ) sen(tδ) ΣI + cos(tλ)Id X + cos(tδ)Id ; λ δ φt (u, e) = φt (u, e) = 12 • In SO(3) or SO(2, 1)0 , φt (u, e) =  cosh(tλ) − 1 2 ΣI λ2 +  senh(tλ) ΣI + Id · λ   cosh(tδ) − 1 2 senh(tδ) X + X + Id . δ2 δ References [1] Agrachev,A.A.; Sachkov, Y. L.; Control Theory from the Geometric Viewpoint, Springer Verlag, 2004. [2] Ayala, V; Tirao, J.; Linear control systems on Lie groups and controllability. Proc. Symp. Pure Math. 64 (1999), 47-64. [3] Ayala, V; Da Silva, A. J.; Ferreira, M.; Affine and bilinear systems on Lie groups. February 4, 2017. [4] Biggs, R.; Remsing, C.C.; Control systems on three-dimensional Lie groups: equivalence and controllability. Journal of Dynamical and Control Systems. July 2014. DOI: 10.1007/10883-014-9212-0. [5] Cardetti, F; Mittenhuber, D; Local controllability for linear control systems on Lie groups. 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