Controllability on the Group of Diffeomorphisms

www.elsevier.com/locate/anihpc

Controllability on the group of diffeomorphisms

A.A. Agrachev

, M. Caponigro

aSteklov Mathematical Institute, Gubkina str. 8, 119991 Moscow, Russia bSISSA, Via Beirut 2-4, 34151 Trieste, Italy

Received 5 December 2008; received in revised form 10 July 2009; accepted 14 July 2009 Available online 5 August 2009

Abstract

Given a compact manifoldM, we prove that any bracket generating family of vector fields onM, which is invariant under multiplication by smooth functions, generates the connected component of the identity of the group of diffeomorphisms ofM.

©2009 Elsevier Masson SAS. All rights reserved.

Résumé

SoitMune variété compacte, nous montrons que toute famille de champs de vecteurs satisfaisant la condition du rang et étant invariante par multiplication par fonctions lisses engendre la composante connexe de l’identité du groupe DiffM.

©2009 Elsevier Masson SAS. All rights reserved.

MSC:93B05; 58D05

Keywords:Controllability; Diffeomorphisms; Geometric control

1. Introduction

In this paper we give a simple sufficient condition for a family of flows on a smooth compact manifoldM to generate the group Diff0(M)of all diffeomorphisms ofMthat are isotopic to the identity.

If all flows are available then the result follows from the simplicity of the group Diff0(M)(see [9]). Indeed, flows are just one-parametric subgroups of Diff0(M)and all one-parametric subgroups generate a normal subgroup. In other words, any isotopic to the identity diffeomorphism ofMcan be presented as composition of exponentials of smooth vector fields.

In this paper we prove that a stronger result holds for a proper subset of the space of smooth vector fields onM.

Our main result is as follows.

Theorem 1.1.LetF⊂VecMbe a family of smooth vector fields and letGrF= {e^t1f1◦ · · · ◦e^tkfk: t_i ∈R, f_i ∈F, k∈N}.

* Corresponding author. Fax: +39 (0)40 3787 528.

E-mail addresses:agrachev@sissa.it (A.A. Agrachev), caponigro@sissa.it (M. Caponigro).

0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2009.07.003

IfGrFacts transitively onMthen there exist a neighborhoodOof the identity inDiff0(M)and a positive integerm such that everyP ∈Ocan be presented in the form

P =e^a1f1◦ · · · ◦e^amfm,

for somef₁, . . . , f_m∈Fanda₁, . . . , a_m∈C^∞(M).

In particular, if F is a bracket generating family of vector fields then any diffeomorphism in Diff0(M) can be presented as composition of exponentials of vector fields in F rescaled by smooth functions. In fact, a stronger result is valid. The theorem states that every diffeomorphism sufficiently close to the identity can be presented as the composition ofmexponentials, where the numbermdepends only onF.

The structure of the paper is the following. In Section 2 we fix the notation used throughout the paper and we make some remarks about tools used in the sequel. In Section 3 we state some simple corollaries of the main result and explain its meaning for geometric control theory. Then we start the proof of Theorem 1.1 showing, in Section 4, an auxiliary result concerning local diffeomorphisms in Rⁿ. Namely, given n vector fields over Rⁿ,X1, . . . , Xn, linearly independent at the origin, we find a closed neighborhoodV of the origin inRⁿsuch that the image of the map

φ:(a₁, . . . , a_n)→e^a1X1◦ · · · ◦e^anXn|V

fromC₀^∞(Rⁿ)ⁿtoC₀^∞(V )ⁿhas nonempty interior. In Section 5 we show how to reduce the proof of Theorem 1.1 to the mentioned auxiliary fact using a geometric idea that goes back to the Orbit Theorem of Sussmann [8].

2. Preliminaries

LetMbe a smoothn-dimensional compact connected manifold. Throughout the paper smooth meansC^∞. We denote by VecMthe Lie algebra of smooth vector fields onMand by Diff0Mthe connected component of the identity of the group of diffeomorphisms ofM.

IfV is a neighborhood of the origin inRⁿwe setC₀^∞(V )= {a∈C^∞(V ): a(0)=0}. Similarly, ifU is an open subset ofMthenC_q^∞(U, M)is the Fréchet manifold of smooth mapsF :U→Msuch thatF (q)=q. All the spaces above are endowed with the standardC^∞topology.

Given an autonomous vector fieldf ∈VecMwe denote byt→e^tf, witht∈R, the flow onMgenerated byf, which is a one-parametric subgroup of Diff0(M).

If f_τ is a nonautonomous vector field, using “chronological” notation (see [1]), we denote by ^−−→exp_t

0f_τdτ the

“nonautonomous flow” at timetof the time-varying vector fieldf_τ.

Given a family of vector fieldsF⊂VecMwe associate toFthe subgroup of Diff0(M) GrF=

e^t1f1◦ · · · ◦e^tkfk: t_i∈R, f_i∈F, k∈N .

LieFis the Lie subalgebra of VecMgenerated byFand the algebra of vector fields in LieF evaluated atq∈Mis LieqF= {V (q): V ∈LieF}.

Definition 1.A familyF∈VecMis calledbracket generating, orcompletely nonholonomic, if LieqF=T_qM, for everyq∈M.

A classical result in Control Theory is the Rashevsky–Chow Theorem (see [7,2]) that gives a sufficient condition for controllability.

Theorem 2.1(Rashevsky–Chow). LetMbe a compact connected manifold. IfFis bracket generating thenGrFacts transitively onM.

Another classical result, due to Lobry [5], claims that Gr{f₁, f₂}acts transitively onMfor a generic pair of smooth vector fields(f₁, f₂). Namely, the set of pairs of vector fields(f₁, f₂)such that Gr{f₁, f₂}acts transitively onMis an open dense (in theC^∞topology) subset of the product space VecM×VecM.

Moreover, ifMhas the structure of a semisimple Lie group then the set of pairs(f₁, f₂)of left-invariant vector fields such that Gr{f₁, f₂}acts transitively onMis an open dense subset in the Cartesian square of the Lie algebra (see [4]).

3. Corollaries of the main result

A direct consequence of Theorem 1.1 is the following.

Corollary 3.1.LetF⊂VecM, ifGrFacts transitively onM, then Gr

af: a∈C^∞(M), f∈F

=Diff0(M).

Last corollary shows the relation between the Rashevsky–Chow Theorem and Theorem 1.1. Indeed, if F is a bracket generating family of vector fields, then, by Rashevsky–Chow Theorem, for every pair of pointsq0, q1∈M there existt1, . . . , tk∈Randf1, . . . , fk∈Fsuch that

q₀=q₁◦e^t1f1◦ · · · ◦e^tkfk,

and, by Corollary 3.1, for every diffeomorphismP ∈Diff0(M)there exista₁, . . . , a∈C^∞(M)andg₁, . . . , g∈F such that

P =e^a1g1◦ · · · ◦e^ag.

In other words, we have that controllability of a system of vector fields on the manifold implies a certain “control- lability” on the group of diffeomorphisms. Namely, if it is possible to join every two points of the manifoldMby exponentials of vector field inF then we can realize every diffeomorphism as composition of exponentials of vector fields inFrescaled by suitable smooth functions.

Let us reformulate Corollary 3.1 in terms of control systems. Consider the control system onM

˙ q=

k

i=1

u_i(t, q)f_i(q), q∈M, (1)

where{f1, . . . , f_k}is a bracket generating family of vector fields andu1, . . . , u_k aretime varying feedbackcontrols, that is

u_i: [0,1] ×M→R,

such thatu_i(t, q)is piecewise constant intfor everyq and smooth with respect toq for everyt. Then Corollary 3.1 states that for everyP ∈Diff0(M)there exist time-varying feedback controlsu₁, . . . , u_k, such thatq(1)=P (q(0)) for any solutionq(·)of system (1); in other words,

P =^−−→exp 1 0

k

i=1

ui(t,·)fidt.

The next corollary is stated from a geometric viewpoint, in terms of completely nonholonomic vector distributions.

Corollary 3.2.Let⊂T Mbe a completely nonholonomic vector distribution. Then every diffeomorphism ofMthat is isotopic to the identity can be written ase^f1◦ · · · ◦e^fk, wheref₁, . . . , f_kare sections of.

4. An auxiliary result

Proposition 4.1.LetX₁, . . . , X_n∈VecRⁿbe such that span

X₁(0), . . . , Xn(0)

=Rⁿ.

Then there exist a compact neighborhoodV of the origin inRⁿ and an open subsetV ofC₀^∞(V )ⁿ such that every F ∈Vcan be written as

F =e^a1X1◦ · · · ◦e^anXn|V, for somea₁, . . . , a_n∈C₀^∞(Rⁿ).

In order to prove this result we need the following lemma.

Lemma 4.2.LetX1, . . . , Xn∈VecRⁿbe such that span

X1(0), . . . , Xn(0)

=Rⁿ,

and letU0be a neighborhood of the identity inC₀^∞(Rⁿ)ⁿ. Then there exist a neighborhoodV of the origin inRⁿand a neighborhoodU of the identity inC₀^∞(V )ⁿsuch that for everyF ∈U there existϕ₁, . . . , ϕ_n∈U0such that

F =ϕ₁◦ · · · ◦ϕ_n|V,

whereϕ_kpreserves the1-foliation generated by the trajectories of the equationq˙=X_k(q)fork=1, . . . , n.

Proof. SinceX₁, . . . , X_nare linearly independent at 0 then there exists a neighborhood of the originV ⊂Rⁿsuch that

span

X₁(q), . . . , X_n(q)

=Rⁿ, for everyq∈ ¯V .

Now, there exists a ballB⊂Rⁿcontaining 0∈Rⁿsuch that, for everyq∈V, the map

(t₁, . . . , t_n)→q◦e^t1X1◦ · · · ◦e^tnXn (2)

is a local diffeomorphism fromBto a neighborhood ofq. Let Uε=

F ∈C₀^∞(V )ⁿ: F−I _C1< ε ,

whereI denotes the identical map andεis to be chosen later. Ifεis sufficiently small, then, for everyF ∈Uε and q ∈V,F (q)belongs to the image of map (2). Therefore, given everyF ∈Uε, it is possible to associate with every q∈V ann-uple of real numbers(t1(q), . . . , tn(q))∈Bsuch that

F (q)=q◦e^t1(q)X1◦ · · · ◦e^tn(q)Xn.

We claim that there existsη(ε)such thatη(ε)→0 asε→0 and t_{i C}1< η(ε)for everyi=1, . . . , nand forF ∈Uε. Indeed F −I _C0< ε implies that t_{i C}0 < cε, fori=1, . . . , nand for some constantc. Moreover, ifq∈V, for everyξ∈Rⁿwe have

D_qF ξ=

e^t1(q)X1◦ · · · ◦e^tn(q)Xn

∗ξ+ n

i=1

e^t1(q)X1◦ · · · ◦dti

dq ·ξ X_i◦ · · · ◦e^tn(q)Xn. Therefore D_qF ξ−ξ _C0< εimplies t_{i C}1< η(ε), whereη→0 asε→0.

Now consider, for everyk=1, . . . , n, the map Φ_k(q)=q◦e^t1(q)X1◦ · · · ◦e^tk(q)Xk.

Note that Φ₀=I and Φ_n=F. For every k,Φ_k is a smooth diffeomorphism being smooth and invertible by the Implicit Function Theorem. Indeed, for everyq∈V the differential ofΦ_katqis

D_qΦ_kξ =

e^t1(q)X1◦ · · · ◦e^tk(q)Xk

∗ξ+ k

i=1

e^t1(q)X1◦ · · · ◦ dt_i

dqξ X_i◦ · · · ◦e^tk(q)Xk.

DenoteT (ξ )=DqΦkξ−ξ. Ifεis sufficiently small we have T 0<1. ThereforeDqΦkis of the formI+T, with T contraction, and thus invertible.

Finally callU=Uεand define for everyk=1, . . . , n, the smooth maps ϕ_k(q)=q◦e^{tk(Φk−1−1(q))Xk},

then the statement follows. 2

Thanks to the last lemma our problem is to find an appropriate exponential representation of every of the func- tionsϕ_k.

The main idea of the proof of Proposition 4.1 lies in the fact that a linear diffeomorphism is the exponential of a linear vector field. So, our goal is to find a change of coordinates that linearizesϕ_kalong trajectories of the equation

˙

q=X_k(q).

Proof of Proposition 4.1. LetV,U, andU0be as in Lemma 4.2. We denote byXk the set of allϕ∈U0such that ϕpreserves 1-foliation generated by the equationq˙=X_k(q). EveryF ∈U can be written asF=ϕ₁◦ · · · ◦ϕ_n|V. Now consider the open subset ofC₀^∞(V )ⁿ

V⊆

F ∈U: F =ϕ₁◦ · · · ◦ϕ_n|V, ϕ_k∈Xk, (D_qϕ_k)X_k(q)=X_k(q), q∈ϕ_k⁻¹(0), k=1, . . . , n .

Since every F ∈U is close to the identity, then so is ϕ_k for every k. Moreover, ϕ_k(0)=0 and X_k transversal to the hypersurface ϕ_k⁻¹(0) at any point. Therefore we may rectify the field X_k in a neighborhood of the ori- gin in such a way that, in new coordinates, ϕ_k(x₁, . . . , x_k−1,0, xk+1, . . . , x_n)=0 andX_k = _∂x^∂_k. Set x:=x_k and y:=(x₁, . . . , x_k−1, x_k+1, . . . , x_n).

Since the following argument does not depend onk=1, . . . , nthe subscriptkis omitted.

Letα(y)=log(_∂x^∂ ϕ(0, y)). Note that by the definition ofV we haveα(y)=0 for everyy. In what follows we treaty as an(n−1)-dimensional parameter and, for the sake of readability, we omit it. We will show, step by step, that the argument holds for every value of the parameteryand all maps and vector fields under consideration depend smoothly ony. Consider the homotopy fromϕto the identity

ϕ_t(x)=e^α(t−1)ϕ(t x)/t, t∈ [0,1].

There exists a nonautonomous vector fielda(t, x)_∂x^∂ such that

ϕ_t=^−−→exp t

0

a(τ,·)∂

∂xdτ.

It is easy to see that ^∂a_∂x(t,0)=α. Leta(t, x)=αx+b(t, x)x withb(t,0)=0. We want to find a time-dependent change of coordinatesψ (t, x)that linearizes the flow generated bya(t, x). Namely ifx(t )is a solution ofx˙=a(t, x) andz(t )=ψ (t, x(t ))then we wantz(t )˙ =αz(t ). We can supposeψ (t,0)=0 and writeψ (t, x)=xu(t, x), where u(0, x)=1. On one hand we have

d dtz= d

dt

xu(t, x)

= ˙xu(t, x)+xx˙∂ u

∂x(t, x)+x∂ u

∂t(t, x)=a(t, x)u(t, x)+xa(t, x)∂ u

∂x(t, x)+x∂ u

∂t(t, x), and, on the other hand,

d

dtz=αz=αxu(t, x).

Therefore, we can finduby solving x a(t, x)∂ u

∂x(t, x)+∂ u

∂t(t, x)+b(t, x)u(t, x)

=0.

The first-order linear PDE a(t, x)∂ u

∂x(t, x)+∂ u

∂t (t, x)+b(t, x)u(t, x)=0 (3)

can be solved by the method of characteristics. The characteristic lines of (3) are of the formξ_t =(t, ϕ_t(x₀))with initial data(0, x0). Note that these characteristic lines depend smoothly onyand are well defined for everyy. Alongξ_t, Eq. (3) becomes the linear (parametric with parametery) ODE

˙

u= − ˜b(t )u,

whereb(t )˜ =b(ξ_t). Now we can defineu(ξ_t)=e^{−0tb(τ ) dτ˜} . This formula, being applied to all characteristics, defines a smooth solution to Eq. (3). In particularu(t,0)=1 sinceb(t,0)=0.

We have constructed a time-dependent change of coordinatesψ (t, x)such that

ψ (t,·)◦^−−→exp t

0

a(τ,·)∂

∂xdτ◦ψ (t,·)⁻¹=e^{t αx∂x∂} , for everyt∈ [0,1].

Recall that^−−→exp1

0a(τ,·)_∂x^∂ dτ=ϕ. Therefore ϕ=ψ (1,·)⁻¹◦e^αx∂x∂ ◦ψ (1,·)=e^{ψ(1,·)∗αx∂x∂} .

Hence, we provide the desired exponential representation for every of the functionsϕ1, . . . , ϕnfrom Lemma 4.2 and the proposition follows. 2

5. Proof of Theorem 1.1 Set

P=Gr

af: a∈C^∞(M), f∈F and

Pq=

P ∈P: P (q)=q

, q∈M.

Lemma 5.1.Anyq∈Mpossesses a neighborhoodU_q⊂Msuch that the set

{P|Uq:P ∈Pq} (4)

has nonempty interior inC_q^∞(U_q, M).

Proof. According to the Orbit Theorem of Sussmann [8] (see also the textbook [1]), the transitivity of the action of GrFonMimplies that

TqM=span

P_∗f (q): P ∈GrF, f ∈F .

Take X_i =P_i∗f_i for i=1, . . . , nwithP_i ∈GrF andf_i ∈F in such a way thatX₁(q), . . . , X_n(q)form a basis ofT_qM. Then, for all smooth functionsa₁, . . . , a_n, vanishing atq, the diffeomorphism

e^a1X1◦ · · · ◦e^anXn=P₁◦e^(a1◦P1)f1◦P₁⁻¹◦ · · · ◦P_n◦e^(an◦Pn)fn◦P_n⁻¹ belongs to the groupPq. The desired result now follows from Proposition 4.1. 2 Corollary 5.2.The interior of set(4)contains the identical map.

Proof. LetA be an open subset ofC_q^∞(U_q, M) that is contained in (4) and takeP₀|Uq ∈A. ThenP₀⁻¹◦A is a neighborhood of the identity contained in (4). 2

Definition 2.GivenP ∈Diff(M), we set suppP = {x∈M: P (x)=x}.

Lemma 5.3.LetObe a neighborhood of the identity inDiff(M). Then for anyq∈Mand any neighborhoodUq⊂M ofq, we have

q∈int

P (q): P∈O∩P,suppP ⊂Uq

.

Proof. Considernvector fieldsX₁, . . . , X_nas in the proof of Lemma 5.1 and letb∈C^∞(M)be a cut-off function such that suppb⊂U_qandq∈intb⁻¹(1). Then the diffeomorphism

Q(s₁, . . . , s_n)=e^s1bX1◦ · · · ◦e^snbXn

belongs toO∩Pfor anyn-uple of real numbers(s₁, . . . , s_n)sufficiently close to 0. Moreover suppQ(s1, . . . , s_n)⊂U_q. On the other hand, the map

(s₁, . . . , s_n)→Q(s₁, . . . , s_n)(q)

is a local diffeomorphism in a neighborhood of 0. 2

The next lemma is due to Palis and Smale (see [6, Lemma 3.1]).

Lemma 5.4.Let

jU_j=Mbe a covering ofMby open subsets and letObe a neighborhood of identity inDiff(M).

Then the groupDiff0(M)is generated by the subset{P∈O: ∃j such that suppP ⊂Uj}.

Proof. The group Diff0(M)is a path-connected topological group. Therefore it is generated by any neighborhood of the identityO.

SinceMis compact we can assume that the covering{U_j}is finite, namelyU₁∪ · · · ∪U_k=M. Now letP ∈O and consider the isotopyH:M× [0,1] →Msuch thatH (0,·)=I andH (1,·)=P. Consider a partition of unity

{λ_j:M→R|suppλ_j⊂U_j}

subordinated to the covering{Uj}^k_j=1. Let suppλj=Vj and letμj:M→M×[0,1]the mapμj=(I, λ1+· · ·+λj).

ConsiderQj =H◦μj, thenQk=P andQj=Qj−1onM\Vj. Finally, settingPj=Qj ◦Q⁻_j−¹₁, we haveP = Pk◦ · · · ◦P1and suppPj⊂Uj. The lemma is proved. 2

Proof of the theorem. According to Lemma 5.4, it is sufficient to prove that, for everyq∈M, there exist a neighbor- hoodU_q⊂Mand a neighborhood of the identityO⊂Diff(M)such that any diffeomorphismP ∈O, whose support is contained in U_q, belongs to P. Moreover, Lemma 5.3 allows to assume that P (q)=q. Finally, Corollary 5.2 completes the proof. 2

Acknowledgements

The authors are grateful to Boris Khesin who asked them the question answered by this paper (see also [3]) and Mikhael Gromov whose advice allowed to radically simplify the proof of Proposition 4.1.

References

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