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\title{Is it possible to recognize Local Controllability
in a finite number of differentiations?}

\author{Andrei A. Agrachev\\
Steklov Mathematical Institute\\
ul. Gubkina 8, Moscow 117966\\
Russia}

\date{October 15, 1998}

\begin{document}
\maketitle

\section{Description of the problem}

Let $f_1,\ldots ,f_k,\ k\ge2$, be real--analytic vector
fields defined on a neighborhood of the origin in $R^n$, and
$t>0$. The point $x\in R^n$ is called attainable from
the origin for a time less than $t$ and with no more
than $N$ switchings, if there exists a subdivision
$0=t_0<t_1<\ldots<t_{N+1}<t$ of the segment $[0,t]$ and solutions
$\xi_j(t)$, $t\in [t_j,t_{j+1}]$ of the differential equations
$\dot x=f_{i_j}(x)$, for some $i_j\in\{1,\ldots,k\}$, such that
$\xi_0(0)=0$, $\xi_{j-1}(t_j)=\xi_j(t_j)$ for $j=1,\ldots,N$,
$\xi_N(t_{N+1})=x$. Let ${\cal A}_t(N)$ be the set of all such points $x$;
the set ${\cal A}_t=\bigcup\limits_{N>0}{\cal A}_t(N)$ is called
the attainable set for a time no greater than $t$.

\bigskip The family of vector fields $\{f_1,\ldots,f_k\}$ is said to be
{\em Small Time Locally Controllable} (STLC) at $0\in R^n$, if
${\cal A}_t$ contains an open neighborhood of $0$ for any positive
$t$, i.~e. $0\in int\,{\cal A}_t\quad \forall t>0$.

\bigskip \sl Let $\{f_1,\ldots,f_k\}$ be an STLC family of vector fields.
Is it true that:
\begin{description}
\item[(a)] $\exists\,N_1$ such that any family of vector fields
with the same Taylor polynomials of order $N_1$ at $0$ as the Taylor
polynomials of $f_1,\ldots,f_k$ is STLC ?
\item[(b)] $\exists\,N_2$ such that ${\cal A}_t$ contains a ball
of radius $t^{N_2}$ centered at $0$, for any small enough $t>0$ ?
\item[(c)] $\exists\,N_3$ such that $0\in{\cal A}_t(N_3)$, for
any $t>0$ ?
\end{description}

\rm The property (c) implies both (a) and (b). The answer
to question (c) is positive, if $n\le 2$, and is negative for
some STLC couples $\{f_1,f_2\}$ of polynomial vector fields, if 
$n\ge 4$. Question (c) is open for $n=3$. Questions (a) and (b)
are open for all $n\ge 3$ for both analytical and polynomial
families of vector fields.

\section{Motivations and references}
The local controllability problem is classical in Geometric Control
Theory. If one has a controlled system, the first question is: in
what direction one can move? Small Time Local Controllability is
just a possibility of moving in any direction, slower or faster.
Problems (a), (b), (c) have the following meaning: 

\begin{itemize}
\item Is it always possible to recognize an STLC family
in a finite number of differentiations?
\item Is it possible to move not too slow in any direction?
\item How complicated is the control strategy one really needs?
\end{itemize}
If $\{f_1,\ldots,f_k\}$ is an STLC family, then
$\{-f_1,\ldots,-f_k\}$ is also STLC (this is true though not evident); 
hence $\{f_1,\ldots,f_k\}$ is small time stabilizable 
at $0$ by open-loop controls. See \cite{co} for the connection with the
stabilizability by a feedback control.

There are many partial results. If the cone
$\{\sum_{i=1}^k\alpha_if_i \mid \alpha_i\ge 0\}$ is a linear space,
then a simple characterization of the STLC families given in \cite{na} 
provides the positive answers to all the questions.
The symmetrized family $\{f_1,\ldots,f_k,-f_1,\ldots,-f_k\}$
automatically satisfies the previous condition. If this symmetrized 
family is STLC, then the
attainable sets for the original family $\{f_1,\ldots,f_k\}$
have a nonempty interior, 
although the family may fail to be STLC (\cite{ju}). If $\{f_1,\ldots,f_k\}$
is STLC, then for any $t>0$ there exists $N_t$ such that
$0\in int\,{\cal A}_t(N_t)$ (do not confuse with property (c), where
$N$ doesn't depend on $t$~!).
The positive answer to all the questions in the 2-dimensional case
follow from the results of \cite{pe}; 
a counterexample to property (c) in the
4-dimensional case was obtained in \cite{ka}. 

This is a long story, many efforts were made 
and rather strong sufficient as well as some
necessary conditions for Small Time Local Controllability 
in a space of arbitrary dimension were obtained, 
see [4---8], [10---12].
Unfortunately (or maybe fortunataly for you, the reader of this chapter),
the gap between the necessary and sufficient conditions is still big 
enough to keep open the above formulated fundamental questions. 



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\bibitem{co} J.-M. Coron, ``Stabilizaton of controllable systems", in 
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\end{thebibliography}

\end{document}







