Nonlinear Iwasawa Decomposition of Control Flows

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Introduction
Dynamical systems in a …nite dimensional di¤erentiable manifold M (including deterministic, random, stochastic and control systems) are globally described by the corresponding trajectories in the group Dif(M ) of global di¤eomorphisms of the manifold M .In most interesting examples and applications, the manifold M has a Riemannian metric endowed with the corresponding geometric structure: orthonormal frame bundle O(M ) over M , Levi-Civita horizontal lift, covariant derivative of tensors, geodesics, among other structures whose constructions depend intrinsically on this metric.
Once a di¤erentiable manifold is endowed with a Riemannian metric, one can distinguish the elements in the group of di¤eomorphisms Dif(M ) which preserve this metric, the group I(M ) of isometries of M .In general the group Dif(M ) is an in…nite dimensional Lie group, while the group of isometries I(M ) is …nite dimensional.This group carries geometric and topological properties of M .Roughly speaking, what we describe in this paper is a factorization of a ‡ow ' t (a one-parameter family of di¤eomorphisms) into a component t which lies in this …nite dimensional subgroup of isometries I(M ) and another component (the remainder) t which …xes a given point on M and-via its derivative-contains the long time stability behavior (Lyapunov exponents) of the system.The title of the paper is motivated by the classical Iwasawa decomposition for linear maps which is the unique factorization, via Gram-Schmidt orthonormalization, of a matrix as a product of an orthogonal and an upper triangular matrix; hence one has a decomposition into an isometry and a matrix containing the expansion/contraction terms.
A similar decomposition has appeared in Liao [13] for stochastic ‡ows, with hypotheses on the vector …elds of the systems.A geometrical condition on the manifold M (constant curvature), instead of on the vector …elds was established in Ru¢ no [16], with some examples also in [17].This paper intends to apply the same technique to show that this decomposition also holds in the context of control ‡ows.
We remark that a main interest in this kind of decomposition for (random) dynamical systems is the fact that characteristic asymptotic parameters of the systems (Lyapunov exponents and rotation numbers) appear separately in each of the components of the decomposition.For details on the de…nitions of these asymptotic parameters we refer to the articles by Liao [12], Ru¢ no [17], Arnold and Imkeller [2] and the references therein.For control ‡ows these questions need further study.
The paper deals with control ‡ows which include a shift on time-varying vector …elds.In more detail, the main result of this paper, the non-linear Iwasawa decomposition for control ‡ows, can be described as follows: Assume certain geometrical conditions on the vector …elds or that the manifold M has constant curvature (cf.Theorem 5.1), and …x an initial condition x 0 2 M and an initial orthonormal frame k of the tangent space T x 0 M .Then there exists a unique factorization where t corresponds to a control ‡ow in the group of isometries, t …xes the starting point x 0 for all t 0, i.e., t (x 0 ; X) x 0 , and the derivative in the space parameter satis…es D (k) = k s t where s t are upper triangular matrices.Adding some other restrictions on the vector …elds (or assuming that M is ‡at, cf.Corollary 5.2) one can go further in the decomposition and factorize the remainder t of equation ( 1) to get a (dynamically) weaker remainder (using the same notation t ): where t are isometries, t are in the group of a¢ ne transformations of M (hence so does t t ), and the new remainders t are di¤eomorphisms which again …x x 0 for all t 0, but the derivative with respect to the space parameter x is given by the identity D t Id Tx 0 M .In decomposition (2) we have extracted the a¢ ne component from the previous remainder in (1).Hence, in this second factorization, the dynamics of t is reduced locally to the identity, up to …rst order.
Section 2 provides an overview of control ‡ows and Section 3 recalls geometric preliminaries for non-expert readers (these sections can be skipped by those who are familiar with these topics).Section 4 derives the nonlinear Iwasawa decomposition and proves that the isometric part is, by itself, a control ‡ow, with appropriate vector …elds.Section 5 characterizes the manifolds for which the required assumptions are always satis…ed.Finally, Section 6 adapts some examples in [16] and [17] to the context of control ‡ows in all simply connected manifolds of constant curvature: Euclidean spaces R n , spheres, and a hyperbolic space.

Control Flows
In this section we describe some basic facts on control ‡ows.We consider a control system in a complete connected d-dimensional Riemannian manifold M given by a family F of smooth vector …elds F X (M ).We assume that the linear span of F is a …nite dimensional subspace E X (M ), i.e., F is contained in a …nite dimensional a¢ ne subspace of X (M ).The timedependent vector …elds taking values in F , i.e. measurable curves in F , are Throughout we will assume that all corresponding (nonautonomous) di¤erential equations have unique (absolutely continuous) global solutions ' t (x 0 ; X); t 2 R; with ' 0 (x 0 ; X) = x 0 .Then system (4) de…nes a ‡ow on here t is the shift on F given by ( t X) (s) = X t+s ; s 2 R. We call this the associated (non-parametric) control ‡ow (cp.also [5]).It is closely related to control ‡ows as considered in [4] with the shift on the space U of control functions, i.e. the space of measurable curves where U is the control range.Here the time dependent vector …elds are parametrized by the control functions and it has to be assumed that the system is control-a¢ ne and the control rangeU is compact and convex.In fact, the time-dependent vector …elds in F (and hence the control ‡ow ( 5)) can be parametrized as follows.
Proposition 2.1 (i) Let F X (M ) be a compact and convex subset of the m-dimensional subspace E X (M ) spanned by F .Then there exist a convex and compact subset U R m , 0 2 U and m + 1 vector …elds X 0 ; : : : ; X m 2 X (M ) such that (ii) Conversely, consider a control-a¢ ne system onM of the form where m 2 N; X 0 ; :::; X m 2 X (M ), u 2 U with control range U R m convex and compact.Then is a convex and compact subset of a …nite dimensional space E X (M ) of vector …elds and Proof: Starting from (ii): Clearly, for a compact and convex set U R m , the set F in ( 7) is a convex and compact subset of a …nite dimensional vector space in X (M ).The vector space E spanned by the vector …elds X 0 ; X 1 ; : : : ; X m has dimension bounded by m + 1. Conversely, let F be a convex and compact set generating an m dimensional space E X (M ).Fixing X 0 2 F and a base X 1 ; :::; X m of E one …nds that every element X 2 F can uniquely be written as with coe¢ cientsu i 2 R. We may assume that X 1 ; : : : ; X m 2 F , since E is generated by F .Clearly, the corresponding set U of coe¢ cients forms a convex and compact subset of R m (with 0 2 U ).It remains to show that for every X 2 F one can …nd a measurable selection u with This follows from Filippov' s Theorem, see e.g.Aubin and Frankowska [3], Theorem 8.2.10.
Remark: If F is contained in an (m 1)-dimensional a¢ ne subspace ofE, then, in the second part of the proof, one can restrict the linear combination to m vector …elds, instead of (m+1): just take e.g.X 0 = X 1 in the arguments above.
This proposition shows that the nonparametric control ‡ows are just a concise way of writing the control ‡ows corresponding to control-a¢ ne systems as considered, e.g., in [4]; here one uses the shift on the space U of admissible control functions instead of the shift on the space of time dependent vector …elds.Nonparametric control ‡ows inherit all properties of control ‡ows; in fact they can also be considered as the special case (here the right hand side of the di¤erential equation denotes the vector …eld u(t) evaluated at x:) For a …xed control function u( ), these equations reduce to ordinary di¤erential equations, hence one can apply all the techniques of existence and uniqueness of solution and di¤erential dependence on parameters.The family t is a continuous skew-product ‡ow on F M when F L 1 (R; E) is endowed with the weak topology.Note that the Mcomponent of satis…es the cocycle property ' t+s (x; X) = ' t (' s (x; X); s X): When the time-dependent control vector …eld X is implicit in the context, for sake of simplicity in the notation, we shall write simply ' t instead of ' t ( ; X).

Geometric Preliminaries
In this section, we recall some geometric constructions; a general reference is Kobayashi and Nomizu [9].
We shall denote the linear frame bundle over a d-dimensional smooth manifold M by GL(M ).It is a principal bundle over M with structural group Gl(d; R).A Riemannian structure on M is determined by a choice of a subbundle of orthonormal frames O(M ) with structural subgroup O(d; R).We shall denote by : GL(M ) ! M and by o : O(M ) ! M the projections of these frame bundles onto M .The canonical Iwasawa decomposition given by the Gram-Schmidt orthonormalization in the elements of a frame k = (k 1 ; : : : ; k d ) de…nes a projection ?: k 7 !k ?: GL(M ) ! O(M ) such that GL(M ) is again a principal bundle over O(M ) with structural group S Gl(d; R), the subgroup of upper triangular matrices with positive elements in the diagonal.The principal bundles described above factorize as = o ?.
We recall that for a frame k in GL(M ) a connection determines a direct sum decomposition of the tangent space at k into horizontal and vertical subspaces which will be denoted by Throughout this paper we restrict attention to the Levi-Civita connection.
The covariant derivative of a vector …eld X at x is a linear map denoted by rX(x) : In terms of …bre bundles, the covariant derivative is de…ned as a derivative along horizontal lift of trajectories, hence it has a purely vertical component.Considering the right action of the structural group in the frame bundle GL(M ), via adjoint, we can associate to rX an element in the structural group Gl(d; R) of the principal bundle GL(M ) given by the matrix which acts on the right such that rX(k) = k X(k).Note that, di¤erent from rX, the right action of the matrix X(k) does depend on k.
The natural lift of X to GL(M ) is the unique vector …eld X in GL(M ) such that L X(k) = 0, where is the canonical R d -valued 1-form on GL(M ) de…ned by (Hk( )) = for all 2 R d .This natural lift is given by: where M is the derivative of the local 1-parameter group of di¤eomorphisms t associated to the vector …eld X.Note that it describes the in…nitesimal behavior of the linearized ‡ow of X in a basis k of the space T x 0 M .Naturally, X is equivariant by the right action of Gl(d; R) in the …bres.
The next lemma guarantees that the left action of the linearized ‡ow is also well de…ned in the subbundle O(M ).In fact, this is a well expected result since the left action of Gl(d; R) is well de…ned even in smaller quotient space, e.g. in the associated ‡ag manifolds, see e.g.[18].In any case, for the reader' s convenience we shall present a proof of this simpler version which is all that we need here.
Lemma 3.1 The projection ?: GL(M ) ! O(M ) is invariant for the linearized ‡ow in the sense that, for all k 2 GL(M ), Proof: This is a consequence of the commutativity of the right action of Gl(d; R) (in particular, in this case, the action of the subgroup S of upper triangular matrices) on GL(M ) with any other linear left actions (in particular, in this case, the linearized ‡ow).In fact, consider the Iwasawa decomposition k = k ?s (k) for some s (k) 2 S. Hence, Equality (10) follows by the uniqueness of the Iwasawa decomposition.
The vertical component V X(k) at k 2 1 (x 0 ) is given by the covariant derivative rX(k) (see e.g.Elworthy [6, Chap.II, §2], or Kobayashi and Nomizu [9, Chap.III, §1]).In terms of Lie algebra, consider the canonical Iwasawa decomposition of the Lie algebra of matrices gl(d; R) = G = K S into a skew-symmetric and upper triangular component, respectively.By projecting in each of these two components, we write (recall (8)) With this notation, we have the decomposition: where Again, we have the decomposition of ( X) ?(k) into horizontal and vertical components: In terms of the left action of (rX) we shall denote V ( X) ?(k) = (rX(k)) ?k, where (rX(k)) ? is a skew-symmetric map: T x M !T x M .The characterization of (rX(k)) ? in terms of its left action on O(M ) is the content of the following lemma.Although the formula looks quite intricate, it helps to understand the corresponding right action of X(k).
The image of the j-th component k j under the matrix (rX(k)) ? is given by Proof: For a di¤erentiable function t 7 !V t : R 2 R d with V t 6 = 0 for all t 2 ( ; ) and derivative _ V t one has For the sake of simplicity, …x a basis in T x M and denote by A the matrix which represents the linear transformation rX(x).Formula ( 12) with t = 0 will be used in each coordinate of ; where each component comes from the orthogonalization process: One easily checks, by induction in j and using that kV j 0 k = 1 for all j, that the derivatives satisfy: which gives, by formula ( 12) One sees the skew-symmetry of (rX(k)) ?by checking that We shall consider the connected Lie group of di¤eomorphisms Dif(M ) which is generated by the exponential of the Lie algebra of smooth, bounded vector …elds X (M ).The exponential of vector …elds here means the associated ‡ow.We shall denote by A(M ) the Lie subgroup of a¢ ne transformations of M whose elements are given by maps 2 Di (M ) such that their derivatives D preserve horizontal trajectories in T M .This is equivalent to saying that a¢ ne maps are those which preserve geodesics.Its Lie algebra a(M ) is the set of in…nitesimal a¢ ne transformations characterized by vector …elds X such that the Lie derivative of the connection form ! on GL(M ) satis…es L X != 0. Thus X is an in…nitesimal a¢ ne transformation if for all vectors …elds Y : where the tensor A X = L X r X and R is the curvature (see e.g.Kobayashi and Nomizu [9, Chap.VI, Prop.2.6]).For a …xed k 2 GL(M ), the linear map is injective, see e.g.[9, Theorem VI.2.3].We denote by a(k) its image in T k GL(M ).
By I(M ) A(M ) we denote the Lie group of isometries of M .Its Lie algebra i (M ) is the space of Killing vector …elds or in…nitesimal isometries, characterized by the skew-symmetry of the covariant derivative, i.e., a vector …eld X is Killing if and only if hrX(Z); W i = hZ; rX(W )i; for all vectors Z; W in a tangent space T x M .Then, by Lemma 3.2, for any orthonormal frame k we have that (rX(k)) ?= rX and ( X) ?(k) = X(k).
For a …xed k 2 O(M ), the linear map is a restriction of the map i 1 de…ned above, hence it is also injective.We denote by i (k) its image in T k O(M ).
Since the dynamics can be described as trajectories in Lie groups (of di¤eomorphisms, isometries, a¢ ne transformations, etc.), whenever convenient, we shall change from the usual dynamical terminology into the Lie group terminology.For example, vector …elds are identi…ed with Lie algebra elements which will generate right invariant vector …elds in the Lie group Dif(M ); furthermore, if belongs to Dif(M ), one identi…es the derivative D : T M !T M (which sends vector …elds into vector …elds in M ) with the derivative of the left action L : T Dif(M ) ! T Dif(M ).In fact, if = e X , then, given another vector …eld

Decompositions of Control Flows
This section describes conditions on the vector …elds of the control system for the existence of the decomposition into isometric or a¢ ne transformations.
We start with a theorem which, under certain conditions on the vector …elds X 2 F , factorizes the control ‡ow ' t of equation ( 4) in the form ' t = t t such that t is a control ‡ow in the a¢ ne transformations group, and the remainder t …xes the initial point and has trivial derivatives (identity).
Let k be an element in GL(M ) which is a basis for T x 0 M , i.e. (k) = x 0 .We shall assume the following hypothesis on the vector …elds X 2 F determining the control system (4) (recall formula (9)): (H1) For every X 2 F , the lifted vector …eld (X) is tangent to the orbit of the frame k under the group of a¢ ne maps (acting on the bundle GL(M )).
Since the operation of lifting vector …elds commutes with the translation by a di¤eomorphism, an a¢ ne transformation maps tangent spaces to the orbit onto tangent spaces.Hence hypothesis (H1) is equivalent to Observe that in the …nite dimensional case (classical a¢ ne control system), this condition holds if it holds for the vector …elds X 0 ; :::; X m in the representation (6).Intuitively, a vector …eld X satis…es hypothesis (H1' ) (hence (H1)) if the associated ‡ow carries x 0 and its ' in…nitesimal neighborhood' (i.e., a basis in T x 0 M ) along trajectories which ' instantaneously' coincide with the trajectories of an in…nitesimally a¢ ne transformation.Theorem 4.1 Suppose that every vector …eld X 2 F of the control system (4) satis…es the hypothesis (H1) (or equivalently (H1' )) for a …xed frame k 2 GL(M ), and let x 0 = (k).Then the associated control ‡ow ' t factorizes uniquely as where t is a control ‡ow in the group of a¢ ne transformationsA(M ), and the remainder t satis…es t (x 0 ) x 0 and D t = Id (Tx 0 M ) for all t 0.
Proof: Since the linear map i 1 of equation ( 13) is injective, by hypothesis (H1), for each X 2 F we can uniquely de…ne the in…nitesimal a¢ ne transformation X a which satis…es X a (k) = X(k).Hence, by the comments after Lemma 3.1, one easily sees that X a (x 0 ) = X(x 0 ) and rX a (x 0 ) = rX(x 0 ): Let t be the solution of the following equation in the Lie group A(M ), with where the elements [ ] a in the Lie algebra a(M ) act on the right in A(M ).
We recall that, in the Lie algebra terminology, X t here means X t ( t ), the right invariant vector …eld evaluated at t .Equation ( 16) is obviously a control system in A(M ) and the solution t generates a control ‡ow on A(M ): Indeed, it is generated by the convex and compact set of vector …elds on A(M ) which is contained in the …nite dimensional vector space obtained by considering all X 2 E. Using that t 1 t = Id M one easily …nds that the control system for the inverse We de…ne t = 1 t ' t .In the Lie group of di¤eomorphisms of M we have the following equation for t : In the last line we use the right invariance of the X and the fact that ), which (by commutativity of right and left action) yields D 1 t (X t ( t )) ( t ).That is, it is a direct application of the formula L g (X)(h) = L g (X(g 1 h)) for right invariant vector …elds in a Lie group (with L g = D 1 ; h = t ; g = 1 ).By de…nition of X a (equation ( 15)) and equation (17) we have that not only _ t (x 0 ) = 0 but also that D 1 t X t [D 1 t (X t )] a ( t ) = 0, hence the derivative of the linearization d dt D t (u) = 0.This establishes the properties of each component of the factorization of ' t = t t stated in the theorem.
For uniqueness, suppose that 0 t 0 t = t t where 0 t and 0 t also satisfy the properties stated.This implies that 1 t 0 t (x 0 ) = x 0 for all t 0. Besides, the derivative D x 0 ( 1 Remark.We emphasize that the a¢ ne transformation system t does depend on the choice of the initial frame k. Remark.Observe that, in general, t is not a control system in Di (M ) since the vector …elds involved in the equation do not depend exclusively on X t and on the point t .On the other hand, the control ‡ow t may be considered as a skew product ‡ow in F A(M ).This follows at once from its de…nition.Then ( t ; t ) is a skew product ‡ow in the …ber bundle F A(M ) M !A(M ) M with base ‡ow t .In the linear case, this is well known and was used, e.g., by Johnson, Palmer and Sell [7] in their proof of the Oseledets theorem for linear ‡ows on vector bundles.For the next theorem, …x an element k 2 O(M ).We shall assume the following hypothesis on the vector …elds X 2 F of the system (recall that i(k) denotes the image of the map i 2 de…ned in ( 14)): (H2) For every X 2 F , the lifted vector …eld (X) is tangent to the orbit of the frame k under the group of isometries (acting on the bundle O(M )).
Again, the operation of lifting vector …eld (in Gl(M )) and its orthonormalization ?commute with the translation by an isometry ; hence maps tangent spaces to the orbit onto tangent spaces and hypothesis (H2) is equivalent to Intuitively, a vector …eld X satis…es hypothesis (H2) if the associated ‡ow carries x 0 and its ' in…nitesimal neighborhood'(i.e., an orthonormal basis in T x 0 M ) along trajectories which ' instantaneously' coincide with trajectories of a Killing vector …eld (in…nitesimal isometry).That is, a vector …eld X violates (H2), if there is no isometry which rotates and translates the ' in…nitesimal neighborhood'of x 0 into the same directions as the ‡ow induced by X does.
The nonlinear Iwasawa decomposition is described in the following theorem.
Theorem 4.2 Suppose that for a certain frame k 2 O(M ) with x 0 = o (k), all vector …elds X 2 F of the control system (4) satisfy hypothesis (H2) (hence (H2' )).Then, the associated control ‡ow ' t has a unique decomposition where t is a control ‡ow in the group of isometries I(M ), t (x 0 ) = x 0 and D x 0 t (k) = k s t for all t 0, where s t lies in the group of upper triangular matrices.
Proof: The …rst part of the proof proceeds similarly to the proof of Theorem 4.1, replacing the group A(M ) by I(M ): Since the linear map i 2 of equation ( 14) is injective, for every X 2 F , we can take X i , the unique in…nitesimal isometry which satis…es X i (u) = ( X) ?(u).Analogously to equation ( 15), we have that We de…ne the following system in the group I(M ), with initial condition Note that the equation above is a control system in I(M ) and the solution t generates a control ‡ow on I(M ): Indeed, it is generated by the convex and compact set of vector …elds on I(M ) The control system for the inverse We de…ne t = 1 t ' t .In the Lie group of di¤eomorphisms of M we have the following equation for t (by the same arguments as for equation ( 17)): By the …rst part of equation ( 18) and equation ( 20) we have that _ t (x 0 ) = 0.Moreover, by the decomposition of formula (11) and the second part of equation ( 18) we have that, for a given k 2 O(M ), where S on the right hand side are upper triangular matrices.
As mentioned before, the canonical lift of a vector …eld gives the in…nitesimal behavior of the linearized ‡ow acting on a basis, that is by the de…nition in (9), Since the Lie algebra element on the right hand side is upper triangular and D 0 (k) = k, one can write D t (k) = k s t where s t are upper triangular matrices which solve the following left invariant di¤erential equation in the Lie group of upper triangular matrices: This establishes the derivative property of the remainder t .Uniqueness of the decomposition follows easily from the fact that the map i 2 is injective, analogous to uniqueness in Theorem 4.1.
Note that in Theorem 4.2, again, the decomposition depends on the initial orthonormal frame k 2 O(M ) and the ‡ow t may be viewed as a skew product ‡ow on F I(M ).Now, juxtaposing the decompositions established by Theorems 4.1 and 4.2, we have the following factorization of ' t into three components.
Corollary 4.3 Suppose all vector …elds X 2 F in the control system (4) satisfy conditions (H1) and (H2) for a certain frame k 2 O(M ), with x 0 = o (k).Then, for the associated control ‡ow ' t , one has the unique decomposition where each of the components t , t , t have the properties stated in Theorems 4.1 and 4.2.Moreover t t corresponds to a control system in the group of a¢ ne transformations.
Proof: Let ' t = 0 t t be the unique decomposition according to Theorem 4.1, where 0 t is a control system in the group of a¢ ne transformationsA(M ), t (x 0 ) = x 0 and D t = Id Tx 0 M for all t 0.
Let ' t = t 0 t be the unique decomposition according to Theorem 4.2, where t is a control system in the group of isometries I(M ) with 0 t (x 0 ) = x 0 and D x 0 0 t (k) = k s 0 t for a certain family s 0 t in the group of upper triangular matrices.
Take the process t and t of the statement of this corollary as de…ned above and de…ne the process t = 1 t 0 t .These assignments de…ne the decomposition.
It only remains to prove that there exists a family on the group of upper triangular matrices such that D t (k) = k s t .By the properties above, D 0 t = D' t , hence Thus the upper triangular matrix family s t of the statement is given by s 0 t .This con…rms the expected fact that although, in general, t is di¤erent from 0 t they have the same derivative behavior (which carries the Lyapunov information of the system).

Conditions on the Manifold
This section characterizes Riemannian manifolds such that every vector …eld satis…es hypotheses (H1) and (H2), respectively, and hence the corresponding decompositions hold.These manifolds are precisely Riemannian manifolds with constant curvature (simply connected or quotients of them) for the isometric decomposition and ‡at space for the a¢ ne transformations decomposition.In particular, the three-factor decomposition of Corollary 4.3 exists for every control system if and only if M is a ‡at space.More precisely, we have the following result.
Theorem 5.1 If M is simply connected with constant curvature (or its quotient by discrete groups), then for every control system (4) and every orthonormal frame k 0 2 O(M ), the control ‡ow admits a unique non-linear Iwasawa decomposition ' t = t t as in Theorem 4.2.Conversely, if every control ‡ow on M admits this decomposition, then the space M has constant curvature.
Proof: For a simply connected manifold M of constant curvature the dimension of I(M ) equals d(d + 1)=2.Hence the linear map i 2 de…ned in equation ( 14) is bijective.Therefore, hypothesis (H2) is always satis…ed for any set of vector …elds.
Conversely, assume that for all vector …elds X and for every orthonormal frame k 2 O(M ), the corresponding ‡ow t has the non-linear Iwasawa decomposition t = t t .Then, the trajectory We recall that d dt (D t (k)) j t=0 = ( X) ?(k): For any …xed k 2 GL(M ), the linear map : X !T k GL(M ) given by X 7 !X(k) is surjective because it concerns only the local behavior of X on M .Hence, the projection of its image by ?: As a particular case of the theorem above, we have the following conditions on M which guarantee that every system on it will have a ‡ow which factorizes into the three components stated in Corollary 4.3.
Corollary 5.2 If M is ‡at, simply connected (or its quotient by discrete groups) then for every control system (4) and every orthonormal frame k 2 O(M ), the associated ‡ow ' t has a unique decomposition ' = t t t as described in Corollary 4.3.Conversely, if every ‡ow ' t has this decomposition then M is ‡at.
Proof: If M is ‡at and simply connected, then a direct check shows that the dimensions of the groups i (M ) and A(M ) are d(d + 1)=2 and d(d + 1) respectively.This implies that the injective maps i 1 and i 2 are bijective, hence hypotheses (H1) and (H2) are satis…ed for any set of vector …elds on M .
Conversely, assume that for all vector …elds X and for every orthonormal frame k 2 O(M ) the corresponding ‡ow t has the decomposition t = t t t with the properties asserted.Then, the trajectory k t in GL(M ) induced by t satis…es k t = D 0 t (k); where 0 t = t t .We recall that As before, for a …xed k 2 GL(M ), the linear map X 7 !X(k) is surjective because it concerns only the local structure of X on M .Hence, equality (22) implies that the dimension of the group of a¢ ne transformationsA(M ) equals d(d + 1), which implies that M is ‡at (see, e.g.Klingenberg [8] or Kobayashi and Nomizu [9, Thm.VI.2.3]).

Examples
Liao in [13] illustrates the isometric decomposition by working out one example in the sphere S n .The results in the above section enlarge the class of examples to many well known manifolds including projective spaces, hyperbolic manifolds, ‡at torus and many non-compact manifolds.In this section we shall describe calculations on all the three possible simply-connected cases.
We shall concentrate mainly on the isometric part t since this is the component which carries more intuitive motivation.Note that (for stochastic ‡ows) this is the component which presents the angular behavior (matrix of rotation, see e.g.[17], [2]), while t presents the stability behavior (see [13] or [12]).
The control system t in the group of isometries presented in Theorem 4.2 becomes well de…ned by equation (19).In this section we shall give a description of the calculation of the vector …elds X i involved in this equation in each one of the three possibilities of simply connected manifolds with constant curvature.In the case of ‡at spaces, the coe¢ cients X a of equation ( 16) for the system 0 t = t t (Theorem 4.1) will also be described.

Flat spaces
We recall that the group A(R d ) of a¢ ne transformations inR d (or any of its quotient space by discrete subgroup) can be represented as a subgroup of Gl(d + 1; R): and v is a column vector : It acts on the left in R d through its natural embedding on R d+1 given by x 7 !(1; x).The group of isometries is the subgroup of A(M ) where g 2 O(n; R).Given a vector …eld X, assume that the initial condition x 0 is the origin and that k is an orthonormal frame in the tangent space at x 0 .One can easily compute the vector …elds X a 2 a(R d ) and X i 2 i (R d ) using the properties established in equations ( 15) and ( 18): We shall …x k to be the canonical basis fe 1 ; : : : ; e d g of R d .Then the matrix (D 0 X(k)) ? is simply the skew-symmetric component (D 0 X) K .
In terms of the Lie algebra action of a(R d ), the vector …elds X a and X i are given by the action of the elements Let ' t be the ‡ow associated with the vector …eld X.One checks by inspection and by uniqueness that the component 0 t = t t in the group of a¢ ne transformations (Theorem 4.1) and the component t (Theorem 4.2) which solve equations ( 16) and ( 19), respectively, are given by and where D 0 ' t = (D 0 ' t ) ? (D 0 ' t ) k is the classical Iwasawa decomposition of the derivative D 0 ' t .We are representing both the isometries and the a¢ ne transformations as subgroups of the Lie group of matrices Gl(n + 1; R).Recall that in the group of matrices the di¤erential of left or right action coincides with the product of matrices itself, i.e., DL g h = gh for g; h 2 Gl(n + 1; R).Hence one sees that equation ( 16) is given simply by: Note that, in general, though the X a corresponds to the …rst two elements of the Taylor series of a vector …eld X, the factor t presents a strong nonlinear behavior (in time) due to the fact that the coe¢ cients of equation ( 16) are non-autonomous.

Linear control systems
Consider the linear control system where A is a d d-matrix, B is a d m-matrix, x(t) 2 R d and the controls u take values u(t) 2 U R m .Let us …x the initial condition x 0 = 0 and the orthonormal frame bundle k 0 = (e 1 ; : : : ; e d ), the canonical basis.The a¢ ne transformation decomposition is obvious: the vector …eldsAx and the columns of B are in the a¢ ne transformation Lie algebra, hence the solution ‡ow ' t already lives in A(R d ).
For the Iwasawa decomposition, the projection of each vector …eld in the Lie algebra of isometries provides the equation for the isometric component of the ‡ow, see equation ( 19).Hence the isometric component is the ‡ow (rotations and translations) associated to the control system _ x(t) = A ? x(t) + Bu(t); where A ? is the skew-symmetric matrix such that A ? k = d(e At k) ?dt j t=0 .If A is skew-symmetric, the decomposition is trivial because the original system already lives in the group of isometries of R d .

Bilinear control systems
Consider the bilinear control system where the A i are d d-matrices, x(t) 2 R d and (u i (t)) 2 U R m .Again, the a¢ ne transformation decomposition is obvious: the vector …eldsA i x are in the a¢ ne transformation Lie algebra, hence the solution ‡ow ' t already lives in A(R d ).
For the Iwasawa decomposition, let us …x the initial condition x 0 = 0 and the orthonormal frame bundle k 0 = (e 1 ; : : : ; e d ), the canonical basis.Then the isometric component t (pure rotations) is the ‡ow associated to the system _ x(t) = A ? 0 x(t) + m X i=1 u i (t)A ?i x(t):

Spheres S d
Let X be a vector …eld in the sphere S d .Assume that the starting point is the north pole N = (0; 0; : : : ; 1) 2 S d and that the orthonormal frame is the canonical basis k = (e 1 ; : : : ; e d ).One way to calculate X i is …nding the element A in the Lie algebra of skew-symmetric matrices so(d + 1) whose vector …eld e A induced in S d satis…es equations (18), i.e.where X(N ) t is the transpose of the column vector X(N ).
To complement this description of the vector X i , we refer the reader to the calculations in Liao [13] in terms of the partial derivatives of the components of X.In that (rather analytical) description, however, one misses the geometrical insight which our description (in terms of the action of the skew-symmetric matrix A) tries to provide.
North-south ‡ow: Let S 2 fN g be parametrized by the stereographic projection from R 2 which intersects S 2 in the equator.The north-south ‡ow is given by the projection on S 2 of the linear exponential contraction on R 2 , precisely: ' t (p) = e t 1 (p).It is associated to the vector …eld X(x) = x ( e 3 ), where x is the orthogonal projection into the tangent space T x S d .For a point (x; y; z) 2 S 2 , one checks that the ‡ow is given by ' t (x; y; z) = 1 cosh(t) z sinh(t) (x; y; z cosh(t) sinh(t)) : Let x 0 = e 1 and k = (e 2 ; e 3 ).For these initial conditions we have the decomposition: ' t = t t where t = 0 @ sech(t) 0 tanh(t) 0 1 0 tanh(t) 0 sech(t) 1 A and, using the double-angle formulas sinh(2t) = 2 sinh(t) cosh(t) and cosh(2t) = 2 cosh 2 (t)

Hyperbolic spaces
In the stochastic context this example has already been worked out in [17], where we deal with the hyperboloid H n in R n+1 with the metric invariant under the Lorentz group O(1; n).In this case, a global parametrization centered at N = (1; 0; : : : ; 0) 2 H n is given by the graph of the map x 1 = q 1 + P n+1 j=2 (x j ) 2 .We just recall the following formula for a vector …eld X(x) = a 1 (x) @ 1 + : : : + a n+1 (x) @ n+1 with respect to the coordinates above, at the point N = (1; 0; : : : ; 0) 2 H n and for an orthonormal frame k in T N M X i (k) = 0 B B @ 0 a 2 (N ) ::: a n+1 (N ) a 2 (N ) : [@ j a i ](k) ? a n+1 (N ) Note that, if k is the canonical basis in T N M , then ([@ j a i ](k)) ? is simply [(@ j a i )] K .