Equivalence of control systems with linear systems on Lie groups and homogeneous spaces

DOI:10.1051/cocv/2009027 www.esaim-cocv.org

EQUIVALENCE OF CONTROL SYSTEMS WITH LINEAR SYSTEMS ON LIE GROUPS AND HOMOGENEOUS SPACES

Philippe Jouan

Abstract. The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only if the vector fields of the system are complete and generate a finite dimensional Lie algebra.

A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms.

An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine.

Affine vector fields on homogeneous spaces can be characterized by their Lie brackets with the projec- tions of right invariant vector fields.

A linear system on a homogeneous space is a system whose drift part is affine and whose controlled part is invariant.

The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which has its own interest. The present proof makes use of geometric control theory arguments.

Mathematics Subject Classification.17B66, 57S15, 57S20, 93B17, 93B29.

Received December 5, 2008.

Published online July 31, 2009.

1. Introduction

The main purpose of this paper is to characterize the class of control systems

˙

p=f(p) + m j=1

u_jg_j(p) (1.1)

which are globally diffeomorphic to a linear system on a Lie group or a homogeneous space. They turn out to be the systems whose vector fields are complete and generate a finite dimensional Lie algebra.

We say that a vector field on a connected Lie group islinearif its flow is a one parameter group of automor- phisms. Linear vector fields on Lie groups are nothing else than the so-called infinitesimal automorphisms in the Lie group literature (see for instance [3]). They were first considered in a control theory context by Markus,

Keywords and phrases. Lie groups, homogeneous spaces, linear systems, complete vector field, finite dimensional Lie algebra.

1 LMRS, CNRS UMR 6085, Universit´e de Rouen, avenue de l’Universit´e, BP 12, 76801 Saint- ´Etienne-du-Rouvray, France.

Philippe.Jouan@univ-rouen.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2009

on matrix Lie groups (see [9]), and then in the general case by Ayala and Tirao (see [2]). They are the natural extension to Lie groups of the linear fields on vector spaces and, for this reason, are still called linear.

There is another way, of course equivalent, to define linear vector fields on a connected Lie groupG. Let us denote bygits Lie algebra, that is the set of right invariant vector fields. Then a vector fieldX is linear if and only if

∀Y ∈g [X, Y]∈g (1.2)

and moreover satisfies X(e) = 0, whereestands for the identity ofG. In case whereX satisfies condition (1.2) but not X(e) = 0 it will be said affine (it is in that case equal to the sum of a linear vector field and a left invariant one). This second definition was introduced in [2] for affine vector fields (on Lie groups). Its interest is twofold. On the one hand it does not require the knowledge of the flow of X. On the other one it can be extended to homogeneous spaces.

An affine vector field on a homogeneous space G/H is the projection, if it exists, of an affine vector field on G. Theorem 4.1 provides a characterization of affine vector fields on homogeneous spaces in terms of Lie brackets with the invariant ones, similar to (1.2).

A system defined on a homogeneous space,

˙

x=F(x) + m j=1

u_jY_j(x), (1.3)

is called linear if the fieldF is affine and theY_j’s invariant. Linear systems on Lie groups and invariant systems appear as particular cases of this general setting.

The extension of affine vector fields and linear systems to homogeneous spaces is motivated by the fact that the class of systems diffeomorphic to a linear one on a Lie group is rather restrictive, while the class of systems diffeomorphic to linear systems on homogeneous spaces is much wider. More accurately we have (Thm. 6.1, Sect.6):

We assume the family {f, g1, . . . , g_m}to be transitive. Then system(1.1)is diffeomorphic to a linear system on a Lie group or a homogeneous space if and only if the vector fields f, g1, . . . , g_m are complete and generate a finite dimensional Lie algebra.

In this statement, the transitivity assumption means that there is only one orbit, equal to the state space, under the action of the system. This is not really a limitation since by the Orbit theorem (see Sect.2) we know that we can always consider the restriction of the system to the orbit through a given point.

The proof makes appeal to Theorem5.1(Sect.5):

Let Γ be a transitive family of vector fields on a connected manifold M. If all the vector fields belonging to Γ are complete, and if Γ generates a finite dimensional Lie algebra L(Γ), then M is diffeomorphic to a homogeneous space G/H, where G is a simply connected Lie group whose Lie algebra is isomorphic to L(Γ).

The tangent mapping of this diffeomorphism induces an isomorphism fromL(Γ)onto the Lie algebra of invariant vector fields on G/H, and all the vector fields belonging toL(Γ) are complete.

Theorem 5.1 is very closely related to a theorem of Palais ([10], Thm. III, p. 95), but the new proof given here uses control theory ideas like transitivity, rank condition, normal accessibility, Sussmann’s Orbit Theorem.

An important consequence of Theorem 5.1 is that the families of vector fields under consideration are Lie determined.

Thus linear systems on Lie groups and homogeneous spaces appear as models for a wide class of systems, and will certainly play an important role in studying topics such as controllability, stabilization, optimal control and observability. Some results about controllability and observability of linear systems on Lie groups are already known, see [1,2,4,6].

The paper is organized as follows.

In Section2 the definitions and facts from control theory used in the sequel are recalled.

In Section 3 the various definitions of linear and affine vector fields on Lie groups are stated, and their equivalence proved (see Thm. 3.1), as well as their properties, in particular their completeness which is one

of the main ingredients used herein. Some of the proofs can be found in the literature (for instance [2–4]) and are in that case quoted, but to the author’s knowledge, Theorem3.1is nowhere completely proved.

In Section4affine vector fields on homogeneous spaces are introduced, and characterized by their Lie brackets with the projections of right invariant vector fields (Thm.4.1).

Section 5 is devoted to the previously mentioned Theorem5.1, and to a corollary (see Cor.5.1) where the transitivity assumption is relaxed.

The main result, Theorem 6.1, is stated and proved in Section 6, which is ended by Corollary 6.1 where systems diffeomorphic to a linear system on a simply connected Lie group are characterized.

Section 7begins by some examples of linear vector fields. Then we consider a well known system and show that it is equivalent to a linear one on a homogeneous space of the group Heisenberg. The equivalence is computed.

Throughout the paper (in fact from Sect.5) the vector fields under consideration are only assumed to beC^k, k≥1, because the class of differentiability does not matter. The important properties are the completeness of the vector fields and the fact that they generate a finite dimensional Lie algebra.

On the other hand the simply connected spaces will be assumed to be connected. However this definition of simply connectedness, that can be found for instance in [5], is not universal and will be recalled in some statements.

2. Preliminaries

In this section some standard definitions and facts from control theory are reviewed.

Let Γ ={g_i; i∈I}be a family ofC^k vector fields on a connectedC^k+1manifoldM, withk≥1 (theg_i’s are not required to be complete).

Let us denote by (γ_tⁱ) the flow ofg_i. Theorbit of Γ through a pointp∈M is the set of pointsq for which there exist vector fieldsg_i1, . . . , g_ir ∈Γ, and real numberst1, . . . , t_r such that

γⁱ_tr^r◦. . .◦γ_tⁱ₁¹(p) is defined and equal toq. Let us recall the Sussmann’s orbit theorem:

Orbit Theorem. The orbit of Γthrough each pointpof M is a connected immersed submanifold ofM. In the original proof of Sussmann the vector fields are assumed to beC^∞(see [11]), but a proof forC^k vector fields, with k≥1, can be found in [7].

This family of vector fields is said to betransitiveif the orbit through each pointpofM is equal toM, that is ifM is the only orbit of Γ.

LetV^k(M) stand for the space ofC^k vector fields onM. It is not a Lie algebra wheneverk <+∞. But it may happen that all the Lie brackets of elements of Γ of all finite lengths exist and are alsoC^k. In that case the subspace ofV^k(M) spanned by these Lie brackets is a Lie algebra and we will say thatthe family Γ generates a Lie algebra. This last will be denoted by L(Γ).

Let us assume that Γ generates a Lie algebra, and let us consider therank of Γ at each pointp∈M, that is the dimension of the subspace of T_pM, the tangent space to M atp, spanned by the vectorsγ(p), γ ∈ L(Γ).

The so-calledrank condition asserts that the family Γ is transitive as soon as its rank is maximum, hence equal to dimM, at each point.

To finish let us recall the definition of Lie-determined systems (see for instance [7]): the family Γ is said to be Lie-determined if at each pointp∈M, the rank of Γ atpis equal to the dimension of the orbit of Γ throughp.

3. Linear and affine vector fields on Lie groups

LetGbe aconnected Lie group, andgits Lie algebra, that is the set of right invariant vector fields. The set of analytic vector fields onGis denoted byV^ω(G), and the normalizer ofginV^ω(G) is by definition

N = norm_Vω(G)g={F∈V^ω(G); ∀Y ∈g [F, Y]∈g}·

Definition 3.1. A vector fieldF onGis said to be affineif it belongs toN.

Such a vector fieldF is said to be linear if it moreover verifiesF(e) = 0, whereestands for the identity ofG.

In other words the restriction ofad(F) to g, also denoted byad(F), is a derivation ofg. From the Jacobi identity, it is clear thatN is a Lie subalgebra ofV^ω(G), and that the mapping F −→ad(F) is a Lie algebra morphism fromN intoD(g), the set of derivations of g.

Proposition 3.1 (see[2]). The kernel of the mappingF −→ad(F)is the set of left invariant vector fields. An affine vector field F can be uniquely decomposed into a sum

F=X+Z whereX is linear and Z left invariant.

This proposition is no longer true whenever the groupGis not connected.

Proof. LetZ be an affine vector field whose bracket with any element ofgvanishes. Its flowz_tcommutes with the one of anyY ∈g. This writes

∀x∈G z_t(exp (sY)x) = exp (sY)z_t(x)

for allt, s∈Rfor which this makes sense. Fixx∈G. There existY1. . . Y_k∈gsuch that x= exp (Y1). . .exp (Y_k),

thanks to the connectedness ofG. Therefore

z_t(x) =z_t(exp (Y1). . .exp (Y_k)) = exp (Y1). . .exp (Y_k)z_t(e) =xz_t(e) fortsufficiently small, and

Z_x = _d^d_t|t=0z_t(x)

= _d^d_t|t=0xz_t(e)

=T_eL_x.Z_e

whereT_eL_xstands for the tangent mapping at the identityeof the left translationL_x. This proves thatZ is left invariant and, the converse being obvious, since the bracket of a left-invariant vector field with a right-invariant one vanishes, the first part of the proposition.

For the second one let Z be the left invariant vector field defined by Z_e = F(e). Then the vector field

X =F−Z is clearly linear.

Theorem 3.1. Let X be a vector field on a connected Lie group G. The following conditions are equivalent:

(1) X is linear;

(2) the flow ofX is a one parameter group of automorphisms ofG;

(3) X verifies

∀x, x∈G Xxx =T L_x.Xx+T R_x.Xx. (3.1) The second item implies that a linear vector field on a connected Lie group is complete.

Proof. (1)⇒(3). The vector field X is linear, hence for all Y ∈ g, the Lie bracket [X, Y] is right invariant.

Therefore we have

∀x∈G [X, Y] =R_x∗[X, Y] = [R_x∗X, R_x∗Y] = [R_x∗X, Y].

This proves firstly that [R_x∗X, Y] is right invariant for allY ∈g, hence that the vector fieldR_x∗X is affine, and secondly that ad(R_x∗X) = ad(X) hence following Proposition3.1that

R_x∗X =X+Z (3.2)

whereZ is left invariant. This last is characterized by

Z_e= (R_x∗X)_e=T R_x.X_x⁻¹ hence for allx∈G

Z_x =T L_x T R_x.X_x⁻¹ =T R_xT L_x.X_x⁻¹.

Considering (R_x∗X)_x =T R_x.X_xx⁻¹, equality (3.2) evaluated at the pointx becomes T R_x.X_xx⁻¹ =Xx+T R_xT L_x.X_x⁻¹.

It remains to applyT R_x−1 to obtain

X_xx⁻¹ =T R_x−1.X_x+T L_x.X_x⁻¹ and to replacex⁻¹ byxto obtain equality (3.1).

(3)⇒(2) (see also [3]). Let us denote byϕ_tthe flow ofX, defined on a domain ofR×G. The curve t−→ϕ_t(x)ϕ_t(x)

is defined on an open interval containing 0, and takes the valuexx att= 0. Moreover

d

dtϕ_t(x)ϕ_t(x) =T L_ϕt(x).X_ϕt(x)+T R_ϕt(x).X_ϕt(x)

=X_ϕt(x)ϕt(x). This proves that

ϕ_t(x)ϕ_t(x) =ϕ_t(xx)

as soon as the left-hand side exists. It remains to show that X is complete. To begin with, notice that equality (3.1), evaluated atx=x=e, impliesX_e= 0.

Letx∈Gandt∈R. SinceX_e vanishes,ϕ_t is defined on an open neighborhoodV_tofe. The groupGbeing connected, it is generated by this neighborhood. Therefore there exist x1, . . . , x_n∈V_t such thatx=x1. . . x_n, and

ϕ_t(x) =ϕ_t(x1. . . x_n) =ϕ_t(x1). . . ϕ_t(x_n)

is well defined. This proves thatX is complete and thatϕ_tis an automorphism ofGfor allt∈R.

(2)⇒(1). Let (ϕ_t, t∈R) be a one parameter group of automorphisms ofG, andX its infinitesimal generator.

For all right invariant vector fieldsY, we have [X, Y]_e= d

dt|t=0T_ϕt(e)ϕ_−t.Y_ϕt(e)= d

dt|t=0T_eϕ_−t.Y_e (3.3)

sinceϕ_t(e) =efor allt∈R. Consideringϕ_−t◦R_ϕt(x)=R_x◦ϕ_−twe have at any pointx:

[X, Y]_x =_d^d_t|t=0T_ϕt(x)ϕ_−t.Y_ϕt(x)

=_d^d_t|t=0T_ϕt(x)ϕ_−tT_eR_ϕt(x).Y_e

=_d^d_t|t=0T_eR_xT_eϕ_−t.Y_e

=T_eR_x.[X, Y]_e.

The vector fieldX is therefore affine, and consequently linear since ϕ_t(e) =efor allt∈R.

Notations. Here and in the following the flow of a linear vector fieldX will be denoted by (ϕ_t)_t∈R. To a given linear vector fieldX, one can associate the derivationD ofgdefined by:

∀Y ∈g DY =−[X, Y],

that is D =−ad(X). The minus sign in this definition comes from the formula [Ax, b] = −Abin Rⁿ. It also enables to avoid a minus sign in the equality

∀Y ∈g ∀t∈R ϕ_t(expY) = exp(e^tDY), stated in the forthcoming Proposition3.2.

Example: the inner derivations

LetX ∈gbe a right invariant vector field. We denote byI the diffeomorphism of Gdefined by x−→x⁻¹. The vector fieldI_∗X is left invariant and equal to−X_e ate. Therefore the vector field

X =X+I∗X

is linear. Indeed X belongs to N, because for all Y ∈g, we have [X, Y] = [X+I_∗X, Y] = [X, Y] ∈ g, and satisfies moreoverX(e) = 0. The derivation associated toX is inner since it is equal to−ad(X) =−ad(X) (see Sect.7.1.1for the example of linear vector fields on matrix Lie groups whose derivation is inner).

This shows also that given an inner derivation D =−ad(X), there always exists a linear vector field onG whose associated derivation is D. This is no longer true in the general case, but holds whenever Gis simply connected. This fact is crucial in the sequel.

Theorem 3.2. The groupG is assumed to be (connected and) simply connected. LetD be a derivation of its Lie algebra g. Then there exists one and only one linear vector field on Gwhose associated derivation isD.

Proof. The proof is essentially contained in [3], Lemme 4, p. 250.

Important remark. Under the assumption that the group G is connected, it was stated in Proposition3.1 that an affine vector field F can be decomposed into F = X +Z, with X linear and Z left invariant. This decomposition is natural because ad(X) = ad(F), but it may be useful to decomposeF into a linear part and a right invariant one. SinceI∗Z is right invariant andZ+I∗Z linear, we can write

F=X − I ∗Z withX=X+Z+I∗Z.

Of course ad(X) = ad(X) + ad(I∗Z)= ad(X) = ad(F) (except ifZ belongs to the center ofg).

This remark will be used in Section4.

Proposition 3.2. For all t∈R

T_eϕ_t= e^tD and consequently

∀Y ∈g ∀t∈R ϕ_t(expY) = exp(e^tDY).

Proof.

(1) Let us first prove the equality

d

dtT_eϕ_t.Y_e=DT_eϕ_t.Y_e.

This equality has been previously stated at t = 0 (see equality (3.3) in the proof of Thm. 3.1). In general

d

dtT_eϕ_t.Y_e = _d^d_s|s=0T_eϕ_t+s.Y_e

= _d^d_s|s=0T_eϕ_s.T_eϕ_t.Y_e

=DT_eϕ_t.Y_e.

(2) From the equality proved above, the first formula of the proposition is immediate. For the second one, remark thatϕ_t is a Lie group morphism. Therefore

ϕ_t(expY) = exp(T_eϕ_t.Y)

= exp(e^tDY).

To finish this section notice the following proposition:

Proposition 3.3. An affine vector field on a connected Lie group is complete.

Proof. This proposition is a consequence of the forthcoming Theorem5.1, but it can be proved in an elementary way. Indeed letF be an affine vector field and X+Z its decomposition into a linear vector field X and a left invariant one Z. Let us denote by t −→e(t) the maximal trajectory ofF through the identity e, defined on an interval ]a, b[. One can verify, using the third characterization of linear vector fields (see Thm. 3.1), that t −→ϕ_t(x)e(t) is the trajectory ofF through the point x∈G, also defined on ]a, b[. Let us assumeb <+∞

and let us choosex=e(b/2). Then

t−→ϕ_t−b 2

e

b 2

e

t−b

2

is the trajectory of F throughx=e(b/2) at t=b/2. Therefore the trajectory t−→e(t) can be extended up

to 3b/2, a contradiction.

4. Affine vector fields on homogeneous spaces

LetH be a closed subgroup ofG. The homogeneous spaceG/H is the manifold of left cosets ofH, and we denote by Π the projection of G ontoG/H. For each right invariant vector field Y ∈ g, the projection Π_∗Y of Y ontoG/H is always well defined, and will be referred to as an invariant vector field onG/H. It is well known that the set of such vector fields, Π_∗g={Π_∗Y; Y ∈g}, is a Lie algebra and that Π_∗ is a Lie algebra morphism fromgonto Π_∗g.

LetX be a linear vector field onG. We investigate the existence onG/H of a vector field Π-related to X. Such a vector field exists if and only if

∀x∈G, ∀y∈H, ∀t∈R Π(ϕ_t(xy)) = Π(ϕ_t(x)).

But Π(ϕ_t(xy)) =ϕ_t(x)ϕ_t(y)H, and the preceding condition is equivalent to

∀y∈H, ∀t∈R ϕ_t(y)∈H.

ThusX is Π-related to a vector field onG/H if and only ifH is invariant under the flow ofX, therefore if and only if X is tangent toH.

In the particular case where H is a discrete subgroup ofG, this amounts to the condition thatX vanishes everywhere onH, or thatH is included in the set of zeroes ofX.

Assume now H to be connected, and denote by hits Lie algebra. Since the elements of H are products of exponentials the invariance of H underX writes

∀Y ∈h, ∀t∈R ϕ_t(expY) = exp(e^tDY)∈H.

This is equivalent to ∀Y ∈h, ∀t∈R, e^tDY ∈h, and finally to the invariance of hunder D.

Proposition 4.1. Let H be a closed subgroup ofG,G/H the homogeneous space of left cosets ofH, andΠthe projection of GontoG/H.

A linear vector field X on GisΠ-related to a vector field on G/H if and only ifH is invariant under X. If H is discrete, this condition holds if and only ifH is included in the set of zeroes of X.

If H is connected it is equivalent to the invariance of its Lie algebra h under the derivation D associated toX.

Under these conditions, the projection ofX ontoG/Hwill be denoted by Π_∗X. Let us now consider an affine vector fieldF onG. It is equal to

F =X+Y

where X is linear andY right invariant. This decomposition (see “important remark” in Sect.3) is chosen in order to ensure that the projection Π_∗Y of Y ontoG/H is well defined. Then F is Π-related to a vector field onG/Hif and only if Π_∗X exists. In that case Π_∗F = Π_∗X+ Π_∗Y will stand for the projection ofF ontoG/H.

Proposition 4.2. Let H be a closed subgroup ofG,G/H the homogeneous space of left cosets ofH, andΠthe projection of GontoG/H.

LetF be an affine vector field onGandF =X+Y its decomposition into a linear vector fieldX and a right invariant oneY.

Then F is Π-related to a vector field on G/H if and only if this holds for X, hence if and only if H is invariant underX.

The next task is to define and characterize affine vector fields on connected homogeneous spaces. In general a homogeneous space is defined as a manifold on which a Lie group acts smoothly and transitively. For our purpose it is more convenient to use the following equivalent definition: a (connected) homogeneous space M is a manifold diffeomorphic to a quotient G/H, whereGis a (connected) Lie group andH a closed subgroup ofG.

There are two remarks to make about the choice of the Lie groupG.

(1) We can assumeGto be simply connected. If not letG be the universal covering ofG,σthe projection ofG ontoGandH =σ⁻¹(H). ThenG/H is diffeomorphic toG/ H.

(2) We can assume that dim Π_∗g= dimg. If not letkbe the kernel of Π_∗ and letK be the connected Lie subgroup of Gwhose Lie algebra is k. The subgroupK is normal and included in H because kis an ideal ofgincluded in the Lie algebrahofH (it is easy to see that Y ∈kif and only if∀x∈G,∀t∈R, x⁻¹exp(tY)x∈H). NowG/K is a simply connected (becauseK is connected) Lie group and

G/H∼(G/K)/(H/K).

We can therefore restrict ourselves to homogeneous spacesG/HwhereGis simply connected andH is a closed subgroup ofGsuch that

dim Π_∗g= dimg.

Notice that whenever the subgroup H is not normal then the projections of the right invariant vector fields belonging to the Lie algebra h of H do not vanish in general. More accurately they vanish at the point H,

but not necessarily on G/H (see the second item in the proof of Thm. 4.1). Therefore the dimension ofG/H may be strictly less than the dimension of its Lie algebra of invariant vector fields, as shown for instance by Examples7.2and7.3.

Definition 4.1. LetGbe a simply connected Lie group, andH a closed subgroup ofGsuch that dim Π_∗g= dimg, where Π stands for the projection ofGontoG/H. A vector field f onG/H is said to be affine if it is Π-related to an affine vector field ofG.

It is clear that Π_∗gis invariant for the Lie bracket with any affine vector field Π_∗F. Let us now state and prove the converse statement, and thus characterize the affine vector fields on homogeneous spaces.

Theorem 4.1. Let G be a (connected and) simply connected Lie group, and H a closed subgroup of G such that dim Π_∗g= dimg, whereΠstands for the projection of GontoG/H.

A vector field f onG/H is affine if and only if

∀Y ∈g [f,Π_∗Y]∈Π_∗g, that is if and only ifΠ_∗gis ad(f)-invariant.

Proof. The necessary part being clear, let us prove the converse and, for this purpose, let us begin by some preliminary remarks.

(1) We can assume without loss of generalityf(H) = 0 since we can add tof an invariant vector field Π_∗Y that verifies Π_∗Y(H) =−f(H).

(2) Let us notice that a right invariant vector field Y belongs to the Lie algebra h of H if and only if Π_∗Y(H) = 0. Indeed

Y ∈h ⇐⇒ ∀t∈R exp(tY)∈H

⇐⇒ ∀t∈R exp(tY)H =H

⇐⇒Π_∗Y(H) = 0.

By assumptionf induces a derivation D on Π_∗g, defined byD =−ad(f). Sincegand Π_∗gare isomorphic, we can define the derivationDongby the equality Π_∗◦D=D◦Π_∗. MoreoverGbeing simply connected there exists a (unique) linear vector fieldX onGassociated to D.

LetY ∈h. We have

Π_∗[Y,X] = Π_∗(DY) =D(Π _∗Y) = [Π_∗Y, f].

But Π_∗Y(H) = 0 becauseY ∈h, and by assumptionf(H) = 0. Therefore [Π_∗Y, f](H) = 0, and [Y,X] belongs toh.

This proves that h is invariant under D. Let H0 denote the connected component of e in H. Following Proposition4.1, the linear vector fieldX is related to a vector fieldfonG/H0by the projection ofGontoG/H0.

IfH0=H, thenf=f. Indeedfandf verify

(i) ∀g∈Π_∗g [f , g] = [f, g]

(ii)f(H) =f(H) = 0,

and according to the forthcoming Lemma4.1this implies the equality offandf.

IfH is not connected thenG/H0 is a covering space ofG/H, andf can be lift to a vector fieldf onG/H0. By the previous method we obtainf=f. Moreoverf is related tof by the projection ofG/H0 ontoG/H, and therefore to X. More accurately the equality between fandf implies thatfvanishes onH/H0, because so doesf. Therefore the connected components ofH are invariant underX, and according to Proposition4.1,

X is Π-related to a vector field onG/H which is nothing else thanf.

Lemma 4.1. Let (g_i)_i∈I be a transitive family of vector fields (see Sect. 2) on a connected manifold M, and letf be a vector field on M that satisfies

(i) ∀i∈I [f, g_i] = 0 (ii)∃x0∈M f(x0) = 0.

Then f = 0.

Proof. Let ϕⁱ_t denote the flow of g_i. Let x∈M. By assumption there existi1, . . . , i_r ∈I and t1, . . . , t_r ∈R such that

x=ϕⁱ_t^r_r◦. . .◦ϕⁱ_t¹₁(x0).

The flowψ_toff commutes withϕⁱ_t,∀i∈I. Hence for alltsufficiently small ψ_t(x) =ϕⁱ_t^r_r◦. . .◦ϕⁱ_t¹₁(ψ_t(x0))

=ϕⁱ_t^r_r◦. . .◦ϕⁱ_t¹₁(x0)

=x.

Thereforef(x) = 0.

Example. The affine vector fields on the sphereSⁿare described in Section7.1.2and this example is generalized in Section7.1.3.

The next proposition is obvious but useful in the sequel.

Proposition 4.3. An affine vector field on a homogeneous space is complete.

We can now state the definition of general linear systems. They are the systems

˙

x=F(x) + m j=1

u_jY_j(x) (4.1)

on homogeneous spacesG/H, where the fieldF is affine and theY_j’s invariant. Linear systems on Lie groups, obtained when the subgroupH is normal, and invariant systems, obtained when the vector fieldF is invariant, are two particular cases of this general setting.

5. Finite dimensional algebras of vector fields

Let Γ ={g_i; i∈I}be a family of vector fields on a connected manifoldM. All the vector fieldsg_ibelonging to Γ are assumed to beC^k, for a common k≥1, andM is therefore at leastC^k+1.

Recall that the family Γ is said to generate a Lie algebra if all the Lie brackets of elements of Γ of all finite lengths exist and are alsoC^k, and that we define in that case the Lie algebraL(Γ) as the subspace of V^k(M) spanned by these Lie brackets.

Theorem 5.1. Let Γ be a family ofC^k vector fields on a connected manifoldM. If (i) all the vector fields belonging to Γare complete;

(ii) Γgenerates a finite dimensional Lie algebraL(Γ);

(iii) the familyΓ is transitive;

then M is C^k+1 diffeomorphic to a homogeneous spaceG/H, where G is a (connected and) simply connected Lie Group whose Lie algebra is isomorphic to L(Γ), andH is a closed subgroup of G.

By this diffeomorphism L(Γ)is related to the Lie algebra of invariant vector fields onG/H, andΓto a subset of this Lie algebra.

Moreover all the vector fields belonging toL(Γ) are complete.

Proof.

(1) Let G be a (connected and) simply connected Lie group whose Lie algebra g is isomorphic to L(Γ), Lin short, and let us denote by

L: L −→g this Lie algebra isomorphism.

(2) In the productM×Gconsider the distribution spanned by the family of vector fields{(g, L(g)); g∈ L}. This distribution is involutive, and its rank is constant, equal to dim(G) = dim(L). Henceforth it is completely integrable (the proof of the Frobenius theorem forC^k vector fields on aC^k+1 manifold with k≥1 can be found for instance in [8]).

Let us fix an arbitrary pointp0 inM, and letS be the leaf of the foliation through the point (p0, e) (whereestands for the identity element ofG). We denote by Π1 (resp. Π2) the projection ofS ontoM (resp. ontoG). We are going to prove that Π2 is a diffeomorphism.

(3) Notations. Forg_i∈Γ ={g_i; i∈I}we denote by Y_i =L(g_i) the corresponding vector field onG. The flows ofg_i andY_i are respectively denoted by

(t, p)−→γ_tⁱ(p) and (t, x)−→exp(tY_i)x.

(4) First of all let us show that Π2 is onto. SinceL is a Lie algebra isomorphism and Γ generatesL, the Lie algebrag ofG is generated by the family L(Γ) = {Y_i; i ∈I}. This family is therefore transitive onGwhich is connected. Letx∈G. There exists an integer r, indicesi1, . . . , i_r∈Iand real numbers t1, . . . , t_rsuch that

x= exp(t_rY_ir). . .exp(t1Y_i1).

Thanks to the completeness assumption of the elements of Γ the point p=γ_tⁱ_r^r◦. . .◦γ_tⁱ¹₁(p0)

is well defined. Moreover the vector fields (g_i, Y_i) are tangent to S and the point (p, x) belongs to S.

Its projection ontoGisx, and this proves the surjectivity of Π2.

(5) Let us now prove that Π2 is a covering map. Since the family L(Γ) = {Yi; i ∈I} generates the Lie algebragofG, the identityeis normally accessible frome(for the family{±Yi; i∈I}) (see [7] for the notion of normal accessibility, in particular Cor. 1, p. 154). Letnbe the dimension of G. We can find indicesi1, . . . , i_n∈I and real numberst1, . . . , t_n>0 such that the mapping

(s1, . . . , s_n)−→exp((t_n+s_n)Y_in). . .exp((t1+s1)Y_i1) is a local diffeomorphism at (0, . . . ,0)∈Rⁿ.

Letx0= exp(−t1Y_i1). . .exp(−t_nY_in). Then the mapping Ψ defined by (s1, . . . , s_n)−→exp((t_n+s_n)Y_in). . .exp((t1+s1)Y_i1)x0

is also a local diffeomorphism, and satisfies Ψ(0, . . . ,0) =e.

We can choose a neighbourhood of 0 inRⁿ sent diffeomorphically by Ψ onto an open and connected neighbourhoodV ofein G. Let us denote by ˜Ψ the similar mapping fromRⁿ×M intoM, that is

(s1, . . . , s_n;p)−→γ_tⁱⁿ_n+sn◦. . .◦γ_tⁱ¹_1+s1◦γ_−tⁱ¹₁◦. . .◦γ_−tⁱⁿ_n(p).

Letxbe a given point inG, and for everyp∈M such that (p, x)∈S letσ_p be the mapping σ_p: V x −→ S

y −→ ( ˜Ψ(τ)(p), y)

where τ = Ψ⁻¹(yx⁻¹). The neighbourhood V x of x is evenly covered by {σ_p(V x); p ∈ M and (p, x) ∈ S}. Indeed it is clear that Π2◦σ_p is the identity of V x and that the sets σ_p(V x) cover Π⁻₂¹(V x). Let us show that they are mutually disjoint. If not we can find τ, τ, p, p, such that σ_p(Ψ(τ).x) =σ_p(Ψ(τ).x). But

σ_p(Ψ(τ).x) =σ_p(Ψ(τ).x) ⇐⇒( ˜Ψ(τ)(p),Ψ(τ).x) = ( ˜Ψ(τ)(p),Ψ(τ).x)

=⇒Ψ(τ).x= Ψ(τ).x

=⇒τ=τ

=⇒Ψ(τ)(p) = ˜˜ Ψ(τ)(p)

=⇒p=p.

This proves that Π2is a covering ofGbyS (notice thatS is connected and locally connected). ButG is simply connected and Π2 is therefore a diffeomorphism. In particular, given a pointx∈G, there is one and only one pointp∈M for which (p, x)∈S.

(6) The next task is to prove that the left translations ofGinduce a group action onM. Let y ∈Gand (p, x)∈ S. There exists an unique point q ∈ M such that (q, yx) belongs to S. Let us show that q depends only on pandy but not on a particular choice ofx. Thanks again to the transitivity of the familyL(Γ), there exist an integerr, indicesi1, . . . , i_r∈I and real numberst1, . . . , t_r such that

y= exp(t_rY_r). . .exp(t1Y1).

Then the equalityq=γ_tⁱ_r^r◦. . .◦γ_tⁱ₁¹(p) holds.

Indeed (γ_tⁱ_r^r◦. . .◦γ_tⁱ₁¹(p),exp(t_rY_r). . .exp(t1Y1)x) belongs toS, andγ_tⁱ_r^r◦. . .◦γ_tⁱ¹₁(p) is therefore the only pointqsuch that (q,exp(t_rY_r). . .exp(t1Y1)x) = (q, yx) belongs toS.

We will denote byρ_y the diffeomorphismγⁱ_tr^r◦. . .◦γ_tⁱ₁¹ ofM. Notice that

(ρ_y(p), yx) = Π⁻₂¹◦L_y◦Π2(p, x). (5.1) Hence the mapping

(y, p)−→ρ_y(p) is of classC^k+1 from G×M ontoM.

To finish we haveρ_y◦ρ_y =ρ_yy for ally, y∈G, according to equality (5.1).

Therefore (y, p)−→ρ_y(p) is a transitive andC^k+1 action of the Lie groupGon the manifoldM. Let H be the isotropy group ofp0, that is the set of pointsxofGsuch that (p0, x) belongs toS, andG/H the manifold of left cosets ofH. Then

xH∈G/H−→ρ_x(p0)

is a diffeomorphism fromG/H ontoM, denoted by Φ in the end of the proof.

(7) We are left to prove that Φ induces an isomorphism between the Lie algebra of invariant vector fields onG/H andL. Let us denote by Π the projection ofGontoG/H. Then the equality

Φ◦Π = Π1◦Π⁻₂¹ holds. Recall that (p0, e)∈S. Then∀x∈G

Φ◦Π(x) = Φ(xH) =ρ_x(p0)

= Π1(ρ_x(p0), x)

= Π1◦Π⁻₂¹(x).

LetY ∈g. Then Φ_∗(Π_∗Y) = (Π1◦Π⁻₂¹)_∗Y =L⁻¹(Y). This equality has two consequences. The first one is that Γ is under Φ⁻¹ equivalent to a set of invariant vector fields onG/H. The second one is that all the vector fields of L(Γ) are complete, since Φ is a diffeomorphism, and they are related by Φ∗ to

complete vector field of G/H.

Thanks to the Orbit Theorem, recalled in Section 2, we can relax the transitivity assumption, and obtain the following corollary.

Corollary 5.1. Let Γ be a family ofC^k vector fields on a connected manifoldM. If (i) all the vector fields belonging to Γare complete;

(ii) Γgenerates a finite dimensional Lie algebraL(Γ),

then all the vector fields belonging to L(Γ)are complete, and the family Γ is Lie-determined.

Proof. Let pbe a point of M and let us denote by S the orbit of Γ through p. By the orbit theorem S is a submanifold of M. Moreover every vector field belonging to Γ, hence every vector field belonging to L(Γ), is tangent toS.

Let Γ_S stand for the family of restrictions to S of the vector fields of Γ. Clearly Γ_S generates a finite dimensional Lie algebra L(ΓS), which is nothing else than the set of restrictions to S of the vector fields ofL(Γ). By definition the family ΓS is transitive onSand satisfies the assumptions of Theorem5.1. Therefore S is diffeomorphic to a homogeneous space G/H, whereGis a simply connected Lie Group whose Lie algebra is isomorphic to L(Γ_S), and H is a closed subgroup of G. By this diffeomorphism Γ_S is related to a set of invariant vector fields, andL(Γ_S) to the Lie algebra of invariant vector fields onG/H. Therefore

∀q∈S rankL(Γ_S)(q) = dimG/H = dimS

and the family Γ_S is Lie-determined.

6. Application to control systems

Consider the control affine system

(Σ) x˙ =f(x) + m j=1

u_jg_j(x)

wherexbelongs to then-dimensional connected manifoldM and wheref, g1, . . . , g_mareC^k vector fields onM, withk≥1. The controlu= (u1, . . . , u_m) belongs toR^m.

The family Γ ={f, g1, . . . , g_m}is assumed to generate a Lie algebra denoted byL.

We also denote by L0 the ideal ofL generated byg1, . . . , g_m. It is well known that L0 is the smallest Lie subalgebra ofLcontainingg1, . . . , g_mand closed for the Lie bracket withf:

X ∈ L0 =⇒ [f, X]∈ L0.

Thedimension ofL0 is its dimension as a real Lie algebra, and itsrank at a point p∈M is the dimension of the subspace {X(p); X ∈ L0}of the tangent space T_pM ofM at p. In the particular case where the rank ofL0 is constant, it will be referred to as rank (L0).

It is also known thatL=L0+Rf. If the pointpis a zero off then rank (L)(p) = rank (L0)(p), and in case where the ranks of LandL0 are constant, the existence of a zero off implies their equality.

IfL(resp.L0) is finite dimensional, thenG(resp.G0) will stand for a (connected and) simply connected Lie group whose Lie algebra is isomorphic toL(resp.L0).

Theorem 6.1. We assume the family {f, g1, . . . , g_m} to be transitive. Then system (Σ) is diffeomorphic to a linear system on a Lie group or a homogeneous space if and only if the vector fieldsf, g1, . . . , g_m are complete and generate a finite dimensional Lie algebra.

More accurately, under this condition the rank ofL0 is constant, equal todim(M)ordim(M)−1, and:

(i) if rank (L0) = dim(M), in particular if there exists one point p0 ∈M such thatf(p0) = 0, then(Σ) is diffeomorphic to a linear system on a homogeneous space G0/H ofG0;

(ii) if rank(L0) = dim(M)−1, thenΣis diffeomorphic to an invariant system on a homogeneous spaceG/H of G.

Proof. Let us prove the sufficiency.

The Lie algebrasL andL0 are finite dimensional and generated by complete vector fields. By Corollary5.1 they are Lie-determined. As L is transitive, its rank is everywhere full. Moreover the rank of L0 is constant overM, equal to dim(M) or dim(M)−1. IndeedL0 being Lie-determined, its rank is everywhere equal to the dimension of the zero-time orbit, which is constant, equal to dim(M) or dim(M)−1 (see [7]).

Let us assume rank (L0) = dim(M). Then we can apply Theorem 5.1 to the family L0. The manifoldM is diffeomorphic to a homogeneous spaceG0/H ofG0, and if we denote by Φ this diffeomorphism, the tangent mapping Φ_∗ induces a Lie algebra isomorphism between L0 and the Lie algebra of invariant vector fields onG0/H. The vector field Φ_∗f satisfies

∀g∈ L0 [Φ_∗f,Φ_∗g] = Φ_∗[f, g]∈Φ_∗(L0).

Since Φ_∗(L0) is equal to the Lie algebra of invariant vector fields onG0/H, and according to Theorem4.1, the vector field Φ_∗f is affine, that is Φ_∗f is the projection ontoG0/Hof an affine vector fieldF ofG0. This vector field can be chosen to be linear if and only if there is one point p0 in M such that f(p0) = 0: in the proof of Theorem5.1, we can choosep0to be the projection of the identityeofG0. Clearly system (Σ) is diffeomorphic to the linear system

˙

x=F(x) + m j=1

u_jY_j(x)

onG0/H, whereF stands for Φ_∗f, andY_j= Φ_∗g_j is an invariant vector field forj = 1, . . . , m.

We assume now rank(L0) = dim(M)−1. We apply Theorem5.1toL: the manifoldM is diffeomorphic to a homogeneous spaceG/H ofG, and under this diffeomorphismLis isomorphic to the Lie algebra of invariant vector fields onG/H. System (Σ) is obviously diffeomorphic to an invariant system onG/H.

From Theorem 6.1 we can deduce the following corollary, stated in the C^∞ case in order to ensure the existence ofLandL0:

Corollary 6.1. The manifoldM is assumed to beC^∞ and simply connected.

The family {f, g1, . . . , g_m} is assumed to beC^∞, complete and transitive, and the vector fieldf to vanish at a point p0∈M.

Then system (Σ) is equivalent to a linear system on a Lie group if and only if dim(L0) = dim(M).

Proof. The necessity part is obvious. Let us prove the sufficient one.

Since dim(L0)<∞, Theorem6.1applies, and sincef vanishes at one point, we have rank (L0) = dim(M) = dim(L0). Therefore Σ is diffeomorphic to a linear system on a homogeneous spaceG0/HofG0, with the previous notations. Now the two conditions dim(L0) = dim(M) andM simply connected imply G0/H =G0. The assumption that M is simply connected cannot be relaxed. If not, M remains diffeomorphic to a homogeneous spaceG/H, whereH is discrete, butG/H is a Lie group if and only ifH is normal.

7. Examples 7.1. Examples of linear and affine vector fields

See [9]. LetGbe a connected matrix Lie group, that is a connected Lie subgroup ofGl(n;R), for somen. For any matrix X belonging to the tangent spaceT_IGat the identityI, identified to g, the mapping M −→XM defines a right invariant vector field. We can also associate to the inner derivation D = −ad(X) the linear vector fieldX defined by

X(M) =XM−M X.

Hence in the inner derivation case, a linear system onGwrites M˙ =XM−M X+

m j=1

u_jY_jM

whereM ∈G, and X, Y1, . . . , Y_m∈g.

7.1.2. Affine vector fields on the sphere Sⁿ,n≥2

The sphere Sⁿ is diffeomorphic to the homogeneous spaceSO_n+1/SO_n, where SO_n is identified with the closed subgroup

H =

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

⎛

⎜⎜

⎜⎜

⎜⎝

1 0 . . . 0 0

... N 0

⎞

⎟⎟

⎟⎟

⎟⎠

; N ∈SO_n

⎫⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎭ ofSO_n+1.

On the one hand the Lie algebraso_n+1 is semi simple, sincen+ 1≥3, so all its derivations are inner.

On the other hand the subgroup H is connected, and, following Proposition 4.1, a linear vector field X onSO_n+1is related to a vector field onSO_n+1/Hif and only if its Lie algebrahis invariant underD=−ad(X).

ButX being of the form

X(M) =XM−M X, M ∈SO_n+1

for someX ∈so_n+1, this condition turns out to

∀Y ∈h [X, Y]∈h

and a straightforward computation shows that it holds if and only ifX∈h.

The flow ofX is given by

ϕ_t(M) = e^tXMe^−tX and its projection ontoSO_n+1/H is equal to

e^tXMe^−tXH = e^tXM H

since∀t∈R, e^−tX ∈H. This proves that the vector fieldsX andXhave the same projection onSⁿ∼SO_n+1/H.

In conclusion the only affine vector fields on the sphereSⁿ are the invariant ones. They are the vector fields defined by

f(x) =Ax, x∈Sⁿ whereA∈so_n+1, that isA=−A.