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%From: stefani@facec.cce.unifi.it (Gianna  Stefani)
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%Cc: sussmann@hamilton.rutgers.edu   (Hector J. Sussmann)

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\markboth{GIANNA STEFANI}{HIGHER ORDER VARIATIONS: HOW CAN THEY BE
DEFINED}
\title{HIGHER ORDER VARIATIONS: HOW CAN THEY BE DEFINED IN ORDER
TO HAVE GOOD PROPERTIES?}
\author{GIANNA STEFANI\thanks{University of Naples. email: {\tt 
stefani@facec.cce.unifi.it}}}

\maketitle
\setcounter{page}{1}

\begin{abstract}
Good properties of variations in connection with their use are
discussed. A definition for variations at a point of a trajectory is
proposed with properties sufficiently good to state a higher order
maximum principle for the minimum time problem and for
an optimal control problem with constraints
on the end-point. The definition allows to define variations
on the base of the relations at a point in the Lie Algebra associated to
the system.
\end{abstract}

%\begin{keywords} Tangent vectors, higher order variations,
%maximum principles. \end{keywords}

\section {Introduction}
Consider a control system on an n-dimensional manifold $M$
$$ \dot x(t)=f(x(t),u(t)) \ , \ x(0)=\xi _0, \eqno(\Sigma)$$
where $f: M\times R^m \rightarrow TM$ is a
smooth function and the set ${\cal U}$ of admissible controls
is the class of those integrable maps $u$ from the compact interval
$J=[0,T]$ into a subset $\Omega $ of $R^m$ for which
the time dependent vector field
$(t,\xi) \mapsto f(\xi,u(t))$ and its first derivative
with respect to the state is locally $L^1$-bounded.
Let $t \rightarrow  x(t,\xi_0,u)$ denote the
solution of $(\Sigma)$ relative to the control $u$ and let
$R(\xi_0,t)$ be the reachable set at time $t$.

Consider a reference control $\hat u$ and suppose that the corresponding
reference trajectory $t \rightarrow \hat x(t) \equiv x(t,\xi_0, \hat u)$
is defined on $J$.
There are classical problems in optimal control theory linked to the study
of the reachable sets ``near" a point $\hat x(t)$ of the
reference trajectory.

A first problem, related to minimum time, is to decide whether
the point $\hat x(t)$ belongs to the interior of the reachable set
at time $t$. In fact if it does,
then the trajectory cannot be of minimum time after $t$.

A second problem is to decide whether $\hat x$ minimizes a cost
of the final point, say $\beta _{ 0 }(x(T))$, possibly in presence of
constraints $\beta _1(x(T))=\cdots=\beta_r(x(T))=0$.
This problem can be transformed in the following way.
If we define $\xi_1=\hat x(T)$ and 
$\beta\equiv (\beta_0, \ldots, \beta_r):
M \rightarrow R^{r+1}$, then the reference trajectory 
$\hat x$, solves
the original problem if and only if it satisfies the constraints and
$\beta(R(\xi_0,T))$ does not intersect the half-line
$\left \{\beta(\xi_1)-\lambda(1,0,\ldots,0) \ : \
\lambda >0\right \}$. In particular $\beta(\xi_1)$ cannot be
interior to $\beta (R(\xi_0,T))$.

If $\xi_0$ is an equilibrium point and $\hat x(t)\equiv\xi_0$,
then the first problem reduces to decide whether
$\xi_0 \in int\,R(\xi_0,T)$
and, if we allow $T$ to vary, then we deal with the so
called small time local
controllability (STLC) property, i.e. $\xi_0$
belongs to the interior of the reachable set at each time $t>0$.

A way to attempt the above problems is to study the {\em tangent vectors\/}
to a subset $A$ of a manifold $N$ at a point $y$.
The simplest way to define tangent vectors is
to define them as tangent directions to curves starting at $y$ and
lying in $A$.

\begin{definition}
We say that $v \in T_yN$ is tangent to $A$ at $y$
and we write $v \in T_yA$, if and only if there exists $\gamma:
[0,\bar \epsilon] \rightarrow A$, such that $\gamma(\epsilon)=
y+\epsilon v+o(\epsilon)$.
\end{definition}

With this definition $T_yA$ is
a cone, possibly nonconvex.
In general $T_yA$ has not the following ``good property"
\begin{equation}\label{int}
T_yA=T_yN \Rightarrow y \in int\, A,
\end{equation}
so that we have to select particular subsets of $T_yA$
to obtain it. The choice of such sets may
depend on the problem.

In our cases we have to study the sets $R(\xi_0,T)$ and
$\beta(R(\xi_0,T))$. It is clear that if $v \in T_{\xi_1}R(\xi_0,T)$
then $T_{\xi_1}\beta (v)\in T_{\beta(\xi_1)}\beta (R(\xi_0,T))$,
therefore we are mainly interested in studying subsets of tangent
vectors to $R(\xi_0,T)$.

A natural way to obtain tangent vectors
in the reachable sets is to consider control variations
$$\epsilon \rightarrow u_{ \epsilon } \ , \ u_0=\hat u$$
of the reference control $\hat u$ and the curve
$$\epsilon \mapsto\gamma(\epsilon)= x(T,\xi _{ 0 },u_\epsilon)
\in R(\xi_0,T).$$
If $ \gamma(\epsilon)\equiv
\xi_1+\epsilon v+o(\epsilon),$
then $v\in T_{\xi_1}R(\xi_0,T)$. Such vectors $v$ with 
``good properties"
are sometimes called {\em variations}.

In particular consider the so-called ``Pontryagin variations"
obtained in the following way.
Take the controls $u_{ \epsilon t \omega  } : [0,T] \rightarrow
\Omega $ obtained replacing the reference control $\hat u$
by the constant value $\omega$
in the interval $[t ,t +\epsilon ]$.
The final point of the relative trajectory gives rise to a curve
$$ \gamma :\epsilon \mapsto x(T,\xi _{ 0 },u_{\epsilon t \omega})$$
in the reachable set $R(\xi _{ 0 },T)$. If $t$ is
a Lebesgue point of $t\mapsto f(\hat x(t),\hat u(t))$, see \cite{lm67}, 
then the tangent vector to this
curve can be thought as the transport along the reference trajectory of a
 ``trajectory's variation'' $v_t$ produced at time $t ,$ given by
$$v_t= f(\hat x(t ),\omega )
-f(\hat x(t ),\hat u(t )).$$
Namely, let $\xi\mapsto \hat\Phi_{t ,T}$ be the flow from time $t$
to time $T$ of the reference time-dependent vector field
$(t,\xi)\mapsto f(\xi,\hat u(t)).$
If $t$ is a Lebesgue point of $\hat u$, then
$$T_{\hat x(t)}\hat\Phi_{t,T}
\Bigl( f(\hat x(t ),\omega )-
f(\hat x(t ),\hat u(t )) \Bigr) $$
belongs to $T_{\xi_1}R(\xi_0,T)$ for all $\omega \in \Omega$.

The Pontryagin variations are a subclass of the so-called
{\em needle variations}, for which
the support (i.e. the set in which the control
differs from the reference one) goes to zero with the
variational parameter $\epsilon$.

\begin{definition}
We call $v_t \in T_{\hat x(t)}$ a needle trajectory
variation of order $k$ at time $t$ if there exists
a control variation $\epsilon \mapsto u_{ \epsilon t}$ 
such that the relative trajectory gives rise
to a continuous curve
 $$ \gamma :\epsilon \mapsto
\hat\Phi_{t+\epsilon^{1/k},t}
(x(\epsilon^{1 / k},\hat x(t),u_{\epsilon t}))
\equiv \hat x(t)+
\epsilon v_t + o(\epsilon).$$
\end{definition}

We easily obtain as before that
$T_{\hat x(t)}\hat\Phi_{t,T}( v_t)$
is a tangent vector to
the reachable set $R(\xi _{ 0 },T)$.

Let us review some results which apply in the above considered cases.

\subsection {STLC Problem}
In \cite{frankcon} a sufficient condition of STLC for a multivalued
inclusion is given using a suitable definition of variations. If we apply
that definition to system $(\Sigma$) we obtain the following:

\begin{definition}
We say that $v \in T_{\xi_0}M$ is a STLC variation of order $k$ if
and only if for all $\epsilon$ sufficiently small
$$\xi_0+\epsilon v+o(\epsilon) \in R(\xi_0,\epsilon^ {1/ k}).$$
\end{definition}

The result in \cite{frankcon}, applied to our case is the following:

\begin{theorem}
Let $\Omega$ be bounded. If the convex cone generated by
the STLC variations of any order defined above
is the whole tangent space, then
$$\xi_0 \in int\, R(\xi_0,[0,t]) \ , \ \forall t>0.$$
\end{theorem}

Notice that no continuity far from zero is required in the
definition of STLC variations. Moreover the above variations are tangent
vectors to the sets $R(\xi_0,[0,t])$, for all $t>0$, but only
particular tangent vectors are variations.

A suitable generalization to unbounded control of the above definition
and result has been obtained recently by R.M. Bianchini \cite{rosa93}.

\subsection {Minimum cost without constraints}
It is clear that if $(-1)$ is tangent to $\beta_0(R(\xi_0,T))$
at $\beta_0(\xi_1)$, then $\hat x$ does not minimize
$\beta _{ 0 }(x(T))$ if there are no constraints on the final point.
In this case all the tangent vectors to $\beta_0(R(\xi_0,T))$
have to be considered and we obtain:

\begin{theorem}
If $\hat x$ minimizes $\beta _{ 0 }(x(T))$ over all the solutions of
the system $(\Sigma$), then
$$-T_{\xi_1}\beta_0 (v) \leq 0 \ , \ \forall v
\in T_{\xi_1}R(\xi_0,T).$$
\end{theorem}

In particular if we apply the above result to the
Pontryagin variations,
then we obtain
\begin{equation}\label{*}
-T_{\xi_1}\beta_0 \circ T_{\hat x(t)}
\hat\Phi_{t,T}\Bigl(
f(\hat x(t ),\omega )-
f(\hat x(t ),\hat u(t ))\Bigr) \leq 0 \ , \
\forall \omega \in \Omega \ , \ a.e. \ t \in [0,T].
\end{equation}
The covector $\hat p(t) \equiv -T_{\xi_1}\beta_0
\circ T_{\hat x(t)}\hat\Phi_{t,T}$
is the solution of the adjoint equation, given in a chart by
\begin{equation}\label{adj}
\dot p(t)=-p(t)\,{{\partial f}\over{\partial x}}
(\hat x(t),\hat u(t)) \ ,
\end{equation}
with boundary condition
\begin{equation}\label{boundary}
p(T)=-T_{\xi_1}\beta_0.
\end{equation}
Therefore the inequality (\ref{*}) leads to
the Pontryagin Maximum Principle, i. e.\\
{\em If $\hat x$ minimizes $\beta _{ 0 }(x(T))$ over
all the solutions of
the system $(\Sigma$), then the solution
 $\hat p: [0,T] \rightarrow (R^{ n })^*$ of the adjoint equation
(\ref{adj}) with boundary condition (\ref{boundary})
 is such that , for all $\omega \in \Omega $ and almost all $t\in [0,T]$,
$$\hat p(t)f(\hat x(t),\hat u(t)) \geq \hat p(t)f(\hat x(t),\omega).$$ }
Every further tangent vector to $R(\xi_0,T)$ at $\xi_1$
gives a strengthening of the Pontryagin Maximum Principle.
In particular needle variations give conditions along the trajectory,
see also \cite{frank89}.

In this case where no constraints are imposed on the end
point one can obtain tangent vectors to
$\beta_0(R(\xi_0,T))$ besides the ones coming from
$R(\xi_0,T)$ (see for example \cite{kno81}, \cite{ls90}).

\subsection{Minimum time and
minimum cost with constraints}
The case of minimum time and the one of
minimum cost with
constraints are very similar because in both cases we
want to select
 a subset $\cal K$ of tangent vectors to $A$ at $y$
with the property (\ref{int}). We shall deal with a property
little stronger than (\ref{int}), but which is basically
equivalent to Maximum Principle when
applied to needle variations. Consider
the following property which says that $y\in int\,A$
if the convex cone generate by ${\cal K}$ is the whole
tangent space $T_yN$
\begin{equation}\label{maxpri}
y \in \partial A \ \Rightarrow \ \exists p^* \in T^*_yN
\ s.t. \ p^*\neq 0 \ and\ p^*v\leq 0 \ ,\ \forall v \in {\cal K}.
\end{equation}

For sake of simplicity, in what follows we consider the case when
$N=M \ , \ y=\xi_1 \ , \ A=R(\xi_0,T)$.

A nice example in \cite{bre85} shows that tangent vectors
generated by general control variations do not have the property
(\ref{maxpri}). Following \cite{bre85}, we introduce the
summability property which is easily seen to imply (\ref{maxpri}).

\begin{definition}
A subset ${\cal K}$ of $T_{\xi_1}R(\xi _{ 0 },T)$ has the summability
property if for all finite subset
$\left \{v_{ 1 },\cdots ,v_{ p }\right \}$ of ${\cal K}$ there exists a
constant $\bar\epsilon>0$ and a continuous map
$$\varphi : [0,\bar\epsilon] \times
\left \{(c_{ 1 },\ldots ,c_{ p }) : c_i\geq 0 \ , \ \sum c_i=1 \right \}
\rightarrow R(\xi _{ 0 },T)$$
such that
$$\varphi (\epsilon,c_{ 1 },\ldots ,c_{ p })=
\xi _1 +\epsilon ( c_{ 1 }v_{ 1 }+\cdots +c_{ p }v_{ p})
+ o(\epsilon).$$
\end{definition}

In \cite{bre85} systems linear with respect to the control are considered
and an order for tangent vectors is defined in
connection with the $L^1$-norm of the control variation.
It is shown that the subsets
consisting of tangent vectors of order one and at most
one tangent vector of higher order have the summability
property.
Notice that the control variations considered in \cite{bre85}
have no restriction on the support.

In general we can obtain a Maximum Principle each time
that we can select a set ${\cal V}_t \ , \ t\in [0,T)$, of needle
variations in such a way that the set
$${\cal K}=\left \{T_{\hat x(t)}
\hat\Phi_{t,T}({\cal V}_t) : t \in [0,T)\right \}$$
has the summability property. Namely, if this is the
case we have that if
$\xi_1 \in \partial R(\xi _{ 0 },T)$
then there exists a nonzero solution
$\hat p: [0,T] \rightarrow (R^{ n })^*$
of the adjoint equation (\ref{adj})
such that
$$\hat p(t)v_t \leq 0 \ ,\ \forall v_t \in
{\cal V}_t \ , \ t\in [0,T).$$
It is possible to prove that the needle variations of order one
(like the Pontryagin variations) have the sommability property,
by concatenating the controls.
They give rise to the Pontryagin Maximum Principle for this case.

For nonlinear systems it is possible that Pontryagin Maximum
Principle is not sufficient
to single out a unique candidate and additional conditions may be
required.
To obtain such additional conditions one can add some new variations.

The main difficulties one meets in order to prove
 the ``summability property'' arise when two variations are obtained
at the same time. In fact if $v_t$ and $v_{t_1}$ are variations
at different times, then it is possible to sum them simply
concatenating the two control variations. If $v$ and $v_1$
are higher order variations at the same time, concatenating
their control variations, a new principal part may appear.

To our knowledge there is no example which proves that there are
needle variations which do not have the summability
property (the example in \cite{bre85} concerns variations
with finite support). Nevertheless this 
property has not been proved for general higher order
needle variations.

The higher order maximum principles developed
 in \cite{kre77} and \cite{kno81} are based on needle variations
that are required to be obtained continuously on a small
but finite arc of trajectory, so that each variation can be thought
as produced at different times. Basically if $v\in T_{\hat x(t)}M$
is a variation at $t$ as defined either in \cite{kre77}
or in \cite{kno81}, then there is a positive constant $h$
and a continuous map $w:[t-h,t+h] \rightarrow TM$
such that $w(t)=v$ and $w(\tau)$ is a variation at $\tau$,
for all $\tau\in [t-h,t+h].$ In particular if the convex cone
generated by the variations at a time is the whole tangent
space, then the same holds for neighboring times.

In the next section a definition of variations
developed in \cite {bssiam93} is presented (see also \cite{stmp}).
Such variations verify the summability property and
may occur at an isolated time. This aim is obtained by requiring
that a variation at time
 $t $ can be obtained with control variations of time length
$\epsilon $ and
 starting times $t +\gamma \epsilon $, depending on a new
parameter $\gamma.$

The original motivation to this definition was to extend the known
results
 of controllability at a point to the case of nonstationary trajectories.

 In particular our aim was to obtain higher order variations at time
$t $ using
 the relations at $\hat x(t )$ in the Lie Algebra of the vector
fields associated
to the system, see \cite{susscont} and \cite{bssiam90}.
This kind of ``thin conditions''
which depend
only on a finite number of derivatives at $\hat x(t ),$ cannot give
variations on an arc of trajectory (see Example 3.3 below).
Of course if the relations in the Lie Algebra are verified
on an arc of trajectory,
then the variations are produced in the same arc and some additional
properties can be developed.

The variations defined in the next section are more general than
the ones defined in \cite{kre77}. The comparison with the ones
defined in \cite{kno81} is more difficult, nevertheless the variation
which appear in Example 3.3 is not a variation as defined in
\cite{kno81}.

\section {The variational cone}
In this section we give a definition of variation which is a slight
modification of the one given in \cite{bssiam93},
where more general time-dependent control
 systems and admissible control functions are considered.

\begin{definition}
A vector $v\in T_{\hat x(t)}M$ is
a {\em right variation\/} of order $k$ of
$(\hat u,\hat x)$ at time $t\in [0,T)$ if there are
positive numbers $\bar \gamma , \bar c, \bar \epsilon $
and a three-parameters control variation
$$\eta : [0,\bar \gamma ]\times [0,\bar c]\times [0,\bar \epsilon ]
\rightarrow {\cal U}$$
such that the maps
$$(\xi,\gamma,c,\epsilon)\mapsto \varphi_{\gamma,c,\epsilon}(\xi)
\equiv x(\epsilon^{1 / k},\xi,\eta(\gamma,c,\epsilon))$$
$$(\xi,\gamma,c,\epsilon)\mapsto T_\xi \varphi_{\gamma,c,\epsilon}$$
are continuous for $\xi$ belonging to a neighborhood of $\hat x(t)$ and
$$\hat\Phi_{t+\epsilon^{1/k}(1+\gamma),t}
(\varphi_{\gamma, c ,\epsilon}
(\hat x(t+\gamma \epsilon^{1/ k})))=
\hat x( t)+
\epsilon \,c\, v+ o(\epsilon )$$
where
$$\lim_{\epsilon\rightarrow 0}{{o(\epsilon)}\over \epsilon}=0$$
uniformly with respect to $(\gamma,c)$.
\end{definition}

\begin{remark}
Here only right variations are considered,
 i.e. relative to control variations which take place after $t$.
In an analogous way left variations could be considered, see \cite{bssiam93}.
\end{remark}

\begin{remark}
Notice that with this definition a topology on ${\cal U}$
is not needed. In \cite{bssiam93}
the continuity of the map $\varphi$ is obtained
through the continuity of $\eta$ with respect to a suitable
topology introduced in the set of admissible control ${\cal U}$.
This topology may depend on the map $f$. For example, if
$f$ is not linear with respect to the control, then in general
the $L^1$-topology cannot be chosen.
\end{remark}

If we denote by
${\cal V}_t$ the set of variations at time $t$,
it is easy to see that ${\cal V}_t$ is a
positive cone which may be not convex. For the properties
of ${\cal V}_t$ we refer to \cite {bssiam93} where
the proofs of the results described in this section are also
given. It is easily seen that the slight different definition
of variation does not change the proofs.

To explain the meaning of the parameters $\gamma$ and
$c$ let us remark that the role of $\gamma$ is to allow
``place" to concatenate the control variations at the same
time. Basically we require the same trajectory
variation in an interval of time which goes to zero as
the time-length of the control variation. The role of $c$
is more technical and it is the same as the one of the $c_i$'s
in the summability property.

The following one is the main result on the variations
proved in \cite{bssiam93} in a little different form.

\begin{theorem}
The subset ${\cal K}$ of $T_{\xi_1}R(\xi_0,T)$ defined by
\begin{equation}\label{cono}
{\cal K}= convex \ hull \left \{ T_{\hat x(t)}\hat\Phi_{t,T}
\left({\cal V}_t \right) : t\in [0,T) \right \},
\end{equation}
has the summability property.
Therefore if
$${\cal K}=T_{\xi_1}M$$
then
$$\xi_1 \in int R(\xi_0,T).$$
\end{theorem}

\begin{remark}
Pontryagin variations are of this type, namely
$$f(\hat x(t), \omega ) - f(\hat x(t), \hat u(t))$$
is a right (left) variation of order one at each time $t$ that is
a right (left) Lebesgue point for $\hat u$.
Therefore this kind of variations gives a strengthening of
the Pontryagin Maximum Principle.
\end{remark}

\section{Higher order variations induced by the relations
in the Lie Algebra}
In what follows we give a result proved in \cite{bssiam93} which states
that suitable relations
in the Lie Algebra associated to $(\Sigma )$ at a point
$\hat x(t)$ generate variations at $t$.
The result concerns affine systems and a trajectory
relative to a control taking
values in the interior of $\Omega $.

Let $(\Sigma )$ be of the form
$$\dot x=f_{ 0 }(x)+\sum ^{ m }_{ i =1} u_{ i }f_{ i }(x),$$
and let $\Omega $ be the hypercube
$$H_\rho =\Bigl\{ (\omega _{ 1 },..,\omega _{ m })\in
R^{ m } : | \omega _{ i }| \leq \rho , i=1,..,m \Bigr \} ,$$
possibly $\rho =+\infty ,$ i.e. $H_\infty =R^{ m }.$
We choose $\hat u \equiv 0$ so that $\hat x(t)=exp\, t f_{ 0 }
( \xi_{ 0 }).$

The same arguments can be used to analyze
trajectories relative
to $C^\infty $ controls,
by modifying the system in a standard way.

In order to state the result we need the notations introduced in
\cite{bssiam90}, \cite{bssiam93}.
Let $Lie X$ be the free Lie Algebra on $R$ generated by
the noncommutative indeterminate $X=
\Bigl \{ X_{ 0 },\ldots ,X_{ m } \Bigr \}  .$
${\cal S}$ will denote the ideal of $Lie X$ generated
by $\left\{X_1,\ldots ,X_{ m }\right\}.$
Replacing $X_{ i }$ by $f_{ i }$ in an element $\chi \in
Lie X,$
we obtain a vector field which will be denoted by $\chi _f$.
For any subset A of $Lie X$,
$A_{f}= \Bigl \{\chi _f : \chi \in  A \Bigr \}$
will denote a subset of the Lie Algebra
$Lie f$ generated by
$\Bigl \{ f_{ 0 },\ldots ,f_{ m }\Bigr \}.$

By means of a set  $l= \{l_{ 0 },\ldots ,l_{m }\}$ of nonnegative
integers we define
a weight on $Lie X$ which will induce a weight on $Lie f$.
Let $\Lambda $ be a bracket in ${\cal S}.$ We denote the ''length''
of $\Lambda $ with respect to $X_{ i }$
( i.e. the number of times that $X_{ i }$ appears in
$\Lambda $ ) by $| \Lambda | _{ i }.$
The weight of $\Lambda $ is defined by
$$\| \Lambda \| _{ l }=\sum _{i=0}^{m}l_{ i }
| \Lambda | _{ i },$$
 and the subspace of ${\cal S}_f$ consisting of the elements of weight
not greater than $i$ is given by
$$V_{ i }^{ l }= span \Bigl \{ \Lambda _f
 : \Lambda \in {\cal S} \ ,\  \| \Lambda \| _{ l }\leq i \Bigr \}.$$
 An element $\chi \in {\cal S}$ is called
$l - homogeneous$ if it is a linear combination
of brackets with the same weight, which will be called the weight
of $\chi .$

Following \cite{susscont} we say that an $l -homogeneous$
element $\chi \in {\cal S}$
is $l -neutralized$ for $(\Sigma )$ at $\xi $
if $\chi _f$ is a linear combination at $\xi $
of brackets with less weight. In other words

\begin{definition}
 $\chi $ is
$l -neutralized$ at $\xi$ if
there is an $i <\| \chi \|_l$  such
that $\chi _f(\xi )\in V_i ^l(\xi ).$
\end{definition}

For more details and examples see \cite{bssiam90}, \cite{bssiam93}.

To characterize the elements of $Lie f$ which give rise
to variations,
we introduce the set of obstructions relative to a weight $l,$
 see \cite{bssiam90}, \cite{bssiam93}.
 Let
$${\cal B}= span \Bigl \{\Lambda \in {\cal S}
: | \Lambda | _0 \  is \ odd \ ,\  | \Lambda | _{ i }
\ is \ even \ ,\  i=1,\ldots ,m \Bigr \}$$
$${\cal B} ^l _S=\Bigl \{\chi \in
{\cal B} : \chi \ is \ symmetric \ w.r.t. \ the
\ X_i's \ with \ the
\ same \ weight \Bigr \} .$$
The set of obstructions relative to the weight $l$
is the set:
$${\cal B}^* _l= Lie\{X_{ 0 } ,
{\cal B}^{ l }_S) \}\cap {\cal S}.$$
The following result states that an element of ${\cal S}_f$
determines a variation
at time $t$ if every obstruction with less or equal weight is neutralized
at $\hat x(t)$.
 
\begin{theorem} \label{brack}
\cite{bssiam93}. Let $l= \Bigl \{
l_{ 0 },\ldots ,l_{ m } \Bigr \}$ be a set of nonnegative
 weights. If $\rho <+\infty ,$ then we also suppose $l_{ 0 }\leq
l_{ i } \ ,\  i=1,\ldots ,m.$
If $\Phi $ is a bracket in ${\cal S}$ such that each
$\Lambda \in {\cal B}^*_{ l}$ with $\Vert
\Lambda \Vert _l \leq \Vert \Phi
\Vert _l$ is  $l -neutralized$
at $\hat x(t)$ , then $\Phi _f(\hat x(t))$ is a variation at $t$.
\end{theorem}

Applying the above theorem it is possible to prove that
particular brackets induce variations at any time, see
\cite{bssiam90}, \cite{bssiam93}. As an example we have
that for all $t\in [0,T)$
$${\cal C}_t\equiv \left\{ad^k_{f_0}f_i (\hat x(t))
:k\geq 0 \ ,\ i=1\ldots m \right\}\subset{\cal V}_t$$
and
$$\left[ f_{ i },f_j\right] (\hat x(t))
\subset{\cal V}_t\ ,\ i,j=1\ldots m .$$
Such vectors are variations also in the sense of \cite{kre77}
and \cite{kno81}.

We are going
now to provide an example of a vector field which defines a
variation $v$ at $t=0$, but the set of variations at positive times is
separated from $v$. In particular, otherwise than what happens
in \cite{kre77} and \cite{kno81}, the convex cone generated
by the variations is the whole tangent space at an isolated point.

\begin{example} \label{exa}
Let us consider the following system on  $R^3$
\begin{eqnarray*}
\dot x & = & 1 + u\\
\dot y & = & ux\\
\dot z & = & u(x^3 y+y^2)
\end{eqnarray*}
with $\xi_0=(0,0,0)$ and $|u| \leq 1$ .
Let the reference control be $\hat u\equiv 0$ so that the reference
trajectory is given by $\hat x(t)=(t,0,0)$.
We have that the convex cone generated by ${\cal C}_0$
is given by
$$span \Bigl \{ {\partial \over {\partial x}},
{\partial \over {\partial y}} \Bigr \}.$$
Consider the bracket
$$\chi_f=[ad_{f_0}f_1,ad_{f_1}^2 f_0]
=(2-6x){\partial \over {\partial z}}.$$
$\pm \chi_f(\xi_0)$ define variations at $0$. 
In fact they satisfy the
hypotheses of Theorem \ref{brack} with $l=\{2,3\}$,
for details see \cite{bssiam90},\cite{bssiam93}.
Therefore
$$convex \, cone \, {\cal V}_0=R^3.$$
To show that the convex cone generated by ${\cal V}_t, \ t$
 ``near $0$", cannot be $R^3$, consider $\bar \xi=\hat x(\bar t)
 \ , \ 0<\bar t < {1/ 3}$ and define
$$\hat y :t \mapsto exp \, t  f_0 (\bar \xi).$$
The convex cone generated by the variations at
$\bar t$ for $(\hat u,\hat x)$ is the same as the one
generated by the variations
at $0$ for $(\hat u,\hat y)$. If this cone would be $R^3$,
then $\hat y(t)$
would be interior to $R(\bar\xi,t)$ for all $t>0$. But
$$ad_{f_1}^2 f_0=
-(2x^3+6xy-2y){\partial \over {\partial z}},$$
moreover also for the couple $(\hat u,\hat y)$
the convex cone generated by ${\cal C}_0$
is equal to $span \,\left\{ {\partial \over {\partial x}},
{\partial \over {\partial y}} \right\}$ for all $t>0$.
Therefore
$$ad_{f_1}^2 f_0(\bar\xi)=(6-2 \bar t^3){\partial \over
{\partial z}} \not\in span \,{\cal C}_0.$$
The results in \cite{stmtns} imply that for each $t>0$
sufficiently small $\hat y(t)=exp \, t \, f_0 (\bar \xi)$
belongs to the boundary of $R(\bar \xi,t)$.  A contradiction.
\end{example}

\begin{thebibliography}{99}

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\bibitem{frank89}
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\bibitem{frankcon}
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\bibitem{ls90}
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\bibitem{stmtns}
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\bibitem{stmp}
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\end{thebibliography}
%\endcontribution
\end{document}



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