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\begin{document}
\noindent  A.A.~Agrachev and J.-P.~Gauthier, {\bf On the subanalyticity 
of Carnot-Carath\'eodory \mbox{distances}}.
{\it Annales de l'Institut Henri Poincar\'e; Analyse non-lin\'eaire} 
{\bf 18},  No. 3, 2001, pp.~359 -382.  (Item No. 1 831 660)

\medbreak\noindent FEATURED REVIEW

\medbreak\noindent 
This paper proves a number of spectacularly strong results on 
\mbox{subanalyticity} properties of Carnot-Carath\'eodory distances
arising from generic real analytic 
\mbox{subriemannian} metrics. The main
theorems say that: (T1)~{\em generically, the 
\mbox{germ} at a point $q_0$ of
the function $q\mapsto\rho(q)\,\equaldef\, {\rm dist}(q,q_0)$ is
subanalytic if the \mbox{dimension} $n$ of the manifold and the dimension $k$
of the distribution satisfy $n\le(k-1)k+1$}, (T2)~{\em generically
(and, in fact, on the complement of a set of distributions of 
\mbox{infinite}
\mbox{codimension}), small balls $\{q:\rho(q)\le r\}$ are subanalytic if
$k\ge 3$}, and (T3)~{\em \mbox{generically}, the germ of $\rho$ at $q_0$ is
not subanalytic if 
\mbox{$n\ge(k-1)\Big(\frac{k^2}{3}+\frac{5k}{6}+1\Big)$}}.
Furthermore, several of the paper's
results are much more concrete than the word 
\mbox{``generically''}
might suggest, since they are proved by exhibiting a very explicit set
of conditions on the distribution---which happen to be generic---under
which the conclusion holds.  For example, (T1) is a consequence of the
much stronger result asserting that (T4)~{\em for every real analytic
subriemannian structure whose distribution is ``medium fat,''
$\rho$ is subanalytic at $q_0$}, together with the 
observation (proved in \cite{agrasar})
that medium fatness is a generic property if
$n\le(k-1)k+1$.  Statement T4 is a far-reaching generalization 
of earlier work on the subject, where subanalyticity was proved
for ``strongly bracket-generating'' (also known as ``fat'') 
distributions, a very restrictive class introduced by 
R.~Strichartz, cf. \cite{strich-a,strich-b}. 

These results continue the work by several authors (e.g.,
\mbox{Sussmann} \cite{sussioffe}, Ge \cite{ge}, Agrachev-Sarychev
\cite{agrasar}, Jacquet \cite{jacquet}) 
who had investigated the subanalyticity of
Carnot-Carath\'eodory distances and proved similar conclusions for
much more \mbox{restrictive} classes of metrics.  In a broader sense,
the paper belongs to a program of research espoused by some
differential-geometric control theorists, who proposed to study
properties of \mbox{optimal} controls going beyond the first-order conditions
given by the Pontryagin \mbox{Maximum} \mbox{Principle}, with the objective of
establishing compactness results that might lead to \mbox{conclusions}
about the structure of the value function.  What makes this paper a
\mbox{truly} remarkable tour de force is how far-reaching the new
results are, and how the proofs combine  a rich variety of techniques
in \mbox{ingenious} and often unexpected ways.

I will now state the main definitions and results more precisely,
and then outline the main ideas of some the proofs.

If $\nu=+\infty$ or $\nu=\omega$, a {\em distribution of class $C^\nu$}
on a manifold $M$ of class $C^\nu$ is a linear subbundle 
$\Delta$ of the tangent bundle $TM$ of $M$. The 
{\em dimension}---or {\em rank}---
of $\Delta$ is its fiber dimension. A distribution $\Delta$ is
{\em bracket-generating} if the Lie algebra $L(\Delta)$ 
of vector fields generated by the set $\Gamma^\infty(\Delta)$ 
of all sections of $\Delta$ of class $C^\infty$ satisfies the
``H\"ormander condition'' that $\{V(q):V\in L(\Delta)\}=T_qM$ 
at every $q\in M$, where $T_qM$ is the tangent space of $M$ at $q$.
A {\em subriemannian manifold of class $C^\nu$} is a triple 
$\mathcal{M}=(M,\Delta,g)$, where $M$ is a manifold of class $C^\nu$, 
$\Delta$ is a bracket-generating distribution of class $C^\nu$ on $M$,
and $g$ is a Riemannian metric on $\Delta$ (i.e., a map 
$q\mapsto g_q$ such that $g_q$ is a symmetric positive definite
bilinear form on $\Delta(q)$ for each $q$) of class $C^\nu$.
An {\em admissible path} is a Lipschitz integral arc of 
$\Delta$. Clearly, an admissible path $\gamma:[a,b]\mapsto M$
has a well-defined {\em length} $\|\gamma\|$, given by
$\|\gamma\|\equaldef\int_a^b\sqrt{g_{\gamma(t)}(\dot\gamma(t),
\dot\gamma(t))}\,dt$.
If $M$ is connected, then any two points $q$, $q'$ can be 
joined by an admissble path $\gamma$, so we can define
the {\em distance} $d_{\mathcal{M}}(q,q')$ to be the infimum
of the lengths of all admissible paths from $q$ to $q'$
Given $q$, if $q'$ is sufficiently close to $q$, then the 
infimum defining $d_{\mathcal{M}}(q,q')$ is in fact a minimum.

A distance function $d_{\mathcal{M}}$ arising from a  
subriemannian manifold $\mathcal{M}=(M,\Delta,g)$ of class
$C^\nu$ with $M$ connected is a {\em Carnot-Carath\'eodory}
distance of class $C^\nu$.  

Finally, if $M$ is a real-analytic manifold, then 
a subriemannian structure on $M$ with a $k$-dimensional 
distribution is a section of class $C^\infty$
of the bundle $SBR^k(M)$ over $M$ whose fiber 
$SBR_q^k(M)$ at each point $q$ of $M$ is the  set 
of pairs $(S,g)$ consisting of a $k$-dimensional subspace 
of the tangent space $T_q$ and a positive definite
quadratic form $g$ on $S$. The set ${\rm Sec}^\infty(SBR^k(M))$
of all such sections can be endowed with the $C^\infty$ 
Whitney topology. A property of real analytic subriemannian
structures holds {\em generically} if there is an open dense
subset $\mathcal{G}$ of ${\rm Sec}^\infty(SBR^k(M))$
such that the property holds for all real analytic 
members of $\mathcal{G}$.

I now move on to the proofs. To convey the flavor, and show the rich and
varied ideas and techniques that are brought together by the authors, I
shall limit myself to one of the results, namely, (T2).

Since the results are local, we may take $M$ to be $\rr^n$ and work
on a \mbox{neighborhood} $\mathcal{N}$ of $q_0$ on which there is a 
$g$-orthonormal basis ${\bf X}=(X_1,\ldots,X_k)$ of $C^\omega$ sections of
the distribution $\Delta$. The admissible curves are then exactly the
Lipschitz curves $\xi:I_\xi\mapsto\mathcal{N}$ defined 
on some compact interval $I_\xi=[a_\xi,b_\xi]$ that satisfy 
$\dot\xi(t)=\sum_{i=1}^ku^i(t)X_i(\xi(t))$ for a.e.~$t\in I_\xi$
for some ``control'' 
\mbox{${\bf u}_\xi=(u_\xi^1,\ldots,u_\xi^k)\in L^\infty(I_\xi,\rr^k)$},
which is then \mbox{uniquely} determined by $\xi$.
Clearly, if $\xi:[a,b]\mapsto\mathcal{N}$ is admissible, then 
the \mbox{subriemannian} 
length $\Lambda_{\mathcal{M}}(\xi)$ of $\xi$ is the $L^1$ norm of 
${\bf u}_\xi$. 
Furthermore,  $\xi$ can be reparametrized
so as to yield an admissible curve $\gamma:[0,1]\mapsto\mathcal{N}$ 
which is \mbox{PCAL}
(parametrized by \mbox{constant} times arc-length), i.e., having the
\mbox{property} that 
\mbox{$\|{\bf u}_\gamma(t)\|=\Lambda_{\mathcal{M}}(\xi)=\Lambda_{\mathcal{M}}
(\gamma)$} for a.e. $t\in[0,1]$. Then the $L^2$ norm of ${\bf u}_\gamma$ is 
also equal to $\Lambda_{\mathcal{M}}(\gamma)$. If $\gamma
:[0,1]\mapsto\mathcal{N}$ is a PCAL length-minimizer, and 
\mbox{$\zeta:[0,1]\mapsto\mathcal{N}$} is an arbitrary admissible
curve with the same \mbox{endpoints}, then 
\mbox{$\|{\bf u}_\gamma\|_{_{L^2}}=\|{\bf u}_\gamma\|_{_{L^1}}=
\Lambda_{\mathcal{M}}(\gamma)
\le\Lambda_{\mathcal{M}}(\zeta)
=\|{\bf u}_\zeta\|_{_{L^1}}\le\|{\bf u}_\zeta\|_{_{L^2}}$}, so 
$\gamma$ is also an energy-minimizer, i.e., $\gamma$ minimizes  
the $L^2$ norm of the control among all
\mbox{admissible} curves $\zeta:[0,1]\mapsto\mathcal{N}$ 
with the same endpoints as $\gamma$.  Conversely,
if $\gamma$ is an energy-minimizer, then $\gamma$ must be PCAL
(for otherwise its PCAL \mbox{reparametrization} $\eta$
would satisfy \mbox{$\|u_\eta\|_{_{L^2}}=\|u_\eta\|_{_{L^1}}=
\|u_\gamma\|_{_{L^1}}<\|u_\gamma\|_{_{L^2}}$}), 
and $\gamma$ is a lenght-minimizer.
(\mbox{Indeed}, let \mbox{$\zeta\!:\![0,1]\!\mapsto\!\mathcal{N}$} be 
admissible with the same endpoints as $\gamma$. Let $\eta$ be its \mbox{PCAL}
\mbox{reparametrization}. Then 
\mbox{$\,\Lambda_{\mathcal{M}}(\gamma)\!=\!\|{\bf u}_\gamma\|_{_{L^1}}\!=
\!\|{\bf u}_\gamma\|_{_{L^2}}\!\le\!\|{\bf u}_\eta\|_{_{L^2}}\!=\!
\Lambda_{\mathcal{M}}(\eta)\!=\!\Lambda_{\mathcal{M}}(\zeta)$}.$\,$)
\ Let $\mathcal{Y}\equaldef L^2([0,1],\rr^k)$, and let
$\Omega$ be the set of all ${\bf u}\in\mathcal{Y}$ such that
the Cauchy \mbox{problem}  \mbox{$\dot q(t)=\sum_{i=1}^ku^i(t)X_i(q(t))$},
$q(0)=q_0$, has a solution $\xi_{{\bf u}}:[0,1]\mapsto
\mathcal{N}$ (which is \mbox{obviously} unique).
Then $\Omega$ is open 
in $\mathcal{Y}$, and the map ${\bf u}\mapsto\xi_{{\bf u}}$
is continuous from $(\Omega,{\rm weak})$ to
$C^0([0,1],\rr^n)$. Define the ``endpoint map''
$f:\Omega\mapsto\mathcal{N}$ by 
$f({\bf u})=\xi_{{\bf u}}(1)$ for ${\bf u}\in\Omega$.
For $R>0$, let $B_R=\{{\bf u}\in\mathcal{Y}:
\|{\bf u}\|_{_{L^2}}\le R\}$, $U_R=\{{\bf u}\in\mathcal{Y}:
\|{\bf u}\|_{_{L^2}}=R\}$, 
$B_R^{\mathcal{M}}\equaldef\{q\in M:\rho(q)\le R\}$,
$S_R^{\mathcal{M}}\equaldef\{q\in M:\rho(q)=R\}$. 
Fix $R$ such that $B_R\subseteq\Omega$. Then $B_R^{\mathcal{M}}
\subseteq\mathcal{N}$, and  $B_R^{\mathcal{M}}$ is compact, because
$B_R^{\mathcal{M}}=f(B_R)$, $B_R$ is weakly compact,
and $f$ is continuous from $(\Omega,{\rm weak})$ to $\rr^n$.

For $0<r<R$, let
$\Omega_r^{min}$ be the set of energy-minimizing controls of norm $r$, that is,
the set of all ${\bf u}\in\Omega$ such that 
$\|{\bf u}\|_{_{L^2}}=r$ and
$\rho(f({\bf u}))=r$. Then
$\Omega_r^{min}$ is strongly compact and 
$S_r^{\mathcal{M}}=f(\Omega_r^{min})$. (Indeed, if 
$\hat{\bf u}=\{{\bf u}_j\}_{j=1}^\infty$ is a sequence 
in $B_R$, then $\hat{\bf u}$ has a subsequence 
$\tilde{\bf u}=\{{\bf u}_{j(\ell)}\}_{\ell=1}^\infty$ 
that weakly converges to a control ${\bf u}\in B_R$, and then
$f({\bf u}_{j(\ell)})\to f({\bf u})$ as $\ell\to\infty$. 
If $0<r<R$ and $q\in S_r^{\mathcal{M}}$, then 
we can pick the ${\bf u}_j$ such that 
$\|{\bf u}_j\|_{_{L^2}}\le r+2^{-j}$ and 
$f({\bf u}_j)=q$. It follows that $\|{\bf u}\|_{_{L^2}}\le r$ and
$f({\bf u})=q$. But then $\|{\bf u}\|_{_{L^2}}=r$, because $\rho(q)=r$, 
so ${\bf u}\in\Omega_r^{min}$, and then $q\in f(\Omega_r^{min})$, showing 
that  $S_r^{\mathcal{M}}=f(\Omega_r^{min})$.  If 
$\hat{\bf u}$ is a sequence in $\Omega_r^{min}$ then the weak limit
${\bf u}$ of the weakly convergent  subsequence $\tilde{\bf u}$ 
will satisfy $\|{\bf u}\|_{_{L^2}}=r$, because
$\rho(f({\bf u}))=\lim_{\ell\to\infty}\rho(f({\bf u}_{j(\ell)}))=r$.
So $\tilde{\bf u}$ converges strongly to ${\bf u}$, proving the 
compactness of $\Omega_r^{min}$.)

Since $S_r^{\mathcal{M}}=f(\Omega_r^{min})$, the subanalyticity
of $S_r^{\mathcal{M}}$ would follow if we could \mbox{somehow}
replace $\Omega_r^{min}$ by a compact subanalytic subset of some
finite-dimensional space $\rr^s$.  To do this, a more detailed
analysis of the minimizing controls is needed. Let 
$f_r$ denote the restriction of $f$ to the sphere 
$U_r$. Then $U_r$ is a $C^\infty$ Hilbert \mbox{submanifold}
of $\mathcal{Y}$ of codimension one, and $f_r$ is a map of class $C^\infty$,
\mbox{depending} \mbox{smoothly} on $r$ for $0<r<R$ (i.e., the map
$]0,R[\times U_1\ni(r,{\bf u})\mapsto f(r{\bf u})$ is smooth).
Let ${\rm Crit}(f_r)$ denote the set of critical points of 
$f_r$. Then \mbox{$\Omega_r^{min}\subseteq{\rm Crit}(f_r)$}. (\mbox{Reason}: 
if ${\bf u}\in\Omega_r^{min}$ but
${\bf u}\notin{\rm Crit}(f_r)$, then we can let
${\bf u}_s(t)=s{\bf u}(st)$ for $0<s\le 1$, $0\le t\le 1$; 
then $\|{\bf u}_s\|_{_{L^2}}=sr$, 
$f({\bf u}_s)=\xi_{{\bf u}}(s)$, and ${\bf u}_s\to{\bf u}$ as
$s\uparrow 1$; therefore ${\bf u}_s\notin{\rm Crit}(f_{sr})$ 
if $1-s$ is small enough; if we pick such an $s$, then 
$f({\bf u}_{s+h})\in f(U_{sr})$ if $0<h<1-s$ and $h$ is small
enough, so  $\rho(f({\bf u}_{s+h}))\le sr$ for such $h$,
contradicting the fact that 
${\bf u}_{s+h}\in\Omega_{(s+h)r}^{min}$, since 
${\bf u}\in\Omega_{r}^{min}$.)

The inclusion $\Omega_r^{min}\subseteq{\rm Crit}(f_r)$, which is a special
case of the ``Pontryagin Maximum Principle'' of optimal 
control theory, is the basic necessary condition for a 
control ${\bf u}$ to be an energy-minimizer.  Controls that
belong to $\cup_{r}{\rm Crit}(f_r)$ are called ``extremal.'' 
For ${\bf u}\in\Omega\backslash\{0\}$, let
$E_{{\bf u}}=\{{\bf v}\in\mathcal{Y}:\langle{\bf v},{\bf u}\rangle=0\}$,
and write $A_{{\bf u}}=Df({\bf u})(E_{{\bf u}})$,
$B_{{\bf u}}=Df({\bf u})(\mathcal{Y})$. Then a control
${\bf u}\in U_r$ is extremal iff $A_{{\bf u}}\neq\rr^k$.   This
can happen in three mutually exclusive ways: (1)~$A_{{\bf u}}\neq\rr^k$
but $B_{{\bf u}}=\rr^k$, (2)~$A_{{\bf u}}=B_{{\bf u}}\neq\rr^k$,
(3)~$A_{{\bf u}}\neq B_{{\bf u}}\neq\rr^k$. If ${\bf u}$ satisfies
(1) or (3) then ${\bf u}$ (or its corresponding trajectory
$\xi_{{\bf u}}$) is a {\em normal extremal}. 
If ${\bf u}$ satisfies (2) or (3) then ${\bf u}$ 
is an {\em abnormal extremal}. 
(Thus if  ${\bf u}$ satisfies (3) then it is both normal and
abnormal. If ${\bf u}$ satisfies (1) then it is {\em strictly
normal}, and if it satisfies (2) then it is {\em strictly abnormal}.)
Clearly, a control ${\bf u}$ is extremal iff there exists
a vector $\lambda\in\rr^k\backslash\{0\}$ such that
$\langle\lambda,Df({\bf u})v\rangle=0$ for all 
$v\in E_{{\bf u}}$. In that case, there exists a unique $\nu\in\rr$ such that 
$\langle\lambda,Df({\bf u})v\rangle=\nu\langle{\bf u},v\rangle$
for all $v\in\mathcal{Y}$. Then ${\bf u}$ is normal if 
$\lambda$ can be chosen so that $\nu\neq0$, and abnormal 
if $\lambda$ can be chosen so that $\nu=0$.   

The functional $\mathcal{Y}\ni v\mapsto\langle\lambda,Df({\bf
u})v\rangle$ can be represented as follows. 
Let $\rr_k$ be the space of $k$-dimensional real row vectors.  Let
$\mu_\lambda:[0,1]\mapsto\rr_k$ be the unique solution $\mu$ of
the ``adjoint \mbox{equation}''
\mbox{$\dot\mu(t)=-\sum_{i=1}^ku_\xi^i(t)\mu(t)\cdot\frac{\partial
X_i}{\partial q} (\xi(t))$} such that $\mu(1)=\lambda$. Then 
$\langle\lambda,Df({\bf u})v\rangle=\int_0^1(\sum_{i=1}^k
v_i(t)\,\langle\mu_\lambda,X_i(\xi_{{\bf u}}(t))\rangle\,dt$ for
all $v\in\mathcal{Y}$. So
{\em ${\bf u}$ is extremal iff there \mbox{exists} an
``extremal adjoint vector'' (EAV) for ${\bf u}$, i.e., 
a nontrivial solution $\mu$ of the adjoint equation for which
there is a real constant $\nu$ (the ``abnormal multiplier''
associated to $\mu$) such that 
$\langle\mu(t),X_i(\xi_{{\bf u}}(t))\rangle= \nu
u^i(t)$ for $i=1,\ldots,k$ for a.e. $t$.} 
Furthermore, {\em ${\bf u}$
is normal (resp. abnormal) 
if $\mu$ can be chosen so that $\nu\neq0$ (resp. $\nu=0$).}

Let $h_i:T^*\mathcal{N}\mapsto\rr$ be
the momentum functions of the vector fields $X_i$,
given by $h_i(q,p)\equaldef\langle p,X_i(q)\rangle$ for
$q\in\mathcal{N}$, $p\in T_q^*\mathcal{N}$. Define
$h=\frac{1}{2}\sum_{i=1}^kh_i^2$. If $\mu$ is an EAV
for ${\bf u}$, with abnormal multiplier $\nu$,
then the derivative of $h$ along the curve 
\mbox{$t\mapsto\Xi(t)=(\xi_{{\bf u}}(t),\mu(t))$} in 
$T^*\mathcal{N}$ is the sum 
$\sigma(t)=\sum_{i=1}^k\sum_{j=1}^kh_i(\Xi(t))u_j(t)
\langle\mu(t),[X_j,X_i](\xi(t))\rangle$. But $\sigma(t)\equiv 0$,
because 
$\sigma(t)=\nu\sum_{i=1}^k\sum_{j=1}^ku_i(t)u_j(t)
\langle\mu(t),[X_j,X_i](\xi(t))$. So
$h$ is constant along $\Xi$. If $\nu\neq 0$, then we can normalize
$\mu$ so that $\nu=1$, and then the constant value $c_{h,\Xi}$ of
$h$ along $\Xi$ is just $\|{\bf u}(t)\|$.  So $\xi_{{\bf u}}$
is PCAL, and it is easily verified that the curve
$\Xi$ is an integral curve of the 
Hamilton vector field $\vec{h}$ arising from $h$.
It follows that {\em an admissible curve
$[0,1]\ni t\mapsto\xi(t)\in\mathcal{N}$ is a normal extremal iff
it is the projection on $\mathcal{N}$ of a curve
$[0,1]\ni t\mapsto\Xi(t)\in T^*\mathcal{N}$ in the cotangent
bundle of $\mathcal{N}$ which is an integral curve
of $\vec{h}$ such that the constant value $c_{h,\Xi}$
of $h$ along $\Xi$ is nonzero, and in that case
the control ${\bf u}_{\xi}=(u_\xi^1,\ldots,u_\xi^k)$ is given 
by $u_\xi^i(t)=h_i(\Xi(t))$, so $\|{\bf u}\|_{_{L^2}}=\sqrt{2c_{h,\Xi}}$.}

Now, if $0<r<R$ we let 
$C(r)=\{p\in T_{q_0}^*\mathcal{N}:\sum_{i=1}^kh_i(q_0,p)^2=r^2\}$. 
For each $p\in C(r)$ we let $\tilde\Gamma_p$ be the 
maximal integral curve of $\vec{h}$ such that $\Gamma_p(0)=(q_0,p)$.
Then the domain of $\tilde\Gamma_p$ contains the interval
$\Big[0,\frac{R}{r}\Big]$. Let $\Gamma_p$ be the restriction
of $\tilde\Gamma_p$ to $[0,1]$. Write $\tilde\Gamma_p=(\tilde\gamma_p,
\tilde\mu_p)$, $\Gamma_p=(\gamma_p,\mu_p)$, so 
$\gamma_p$ is a normal \mbox{extremal} \mbox{for} to
a control ${\bf v}_p$, and $\mu_p$ is a field of covectors along $\gamma_p$. 
Let $\varphi_r$ be the map $C(r)\ni p\mapsto f({\bf v}_p)$. 
Let \mbox{$C_{min}(r)=\{p\in C(r):\rho(\gamma_p(1))=r\}$}.
Then  \mbox{$S_r^{\mathcal{M}}=\varphi_r(C_{min}(r))$}.
If it was
true that (I)~{\em \mbox{every} energy-minimizer is a normal \mbox{extremal}},
and (II)~{\em  if \mbox{$0<r<R$}, then $C_{min}(r)$ contains a bounded
subset $C_b(r)$ such that \mbox{$S_r^{\mathcal{M}}=\varphi_r(C_b(r))$}},
then the \mbox{subanalyticity}
of $S_r^{\mathcal{M}}$ would \mbox{follow}.  (Indeed,
fix $\delta$ such that $\delta>0$ and 
\mbox{$B\equaldef\{q:\|q-q_0\|\le\delta\}\subseteq
B_{\frac{r}{2}}^{\mathcal{M}}$}.  
Let $r'$ be such that $0<r'<r$ and $B_{r'}^{\mathcal{M}}\subseteq
B$. Pick \mbox{compact} subanalytic \mbox{subsets} $K$, $K'$ of
$C(r)$, $C(r')$, respectively,  such that 
\mbox{$C_b(r)\subseteq K$} and $C_b(r')\subseteq K'$. Let
\mbox{$J_1=\{p\in K:\gamma_p(1)\in B\}$}, 
\mbox{$J_2=\{p\in K:(\exists p')(\exists t)
(p'\in K'\wedge 0< t<\frac{r}{r'}\wedge\gamma_{p}(1)=
\tilde\gamma_{p'}(t))\}$}. Then 
$K\backslash(J_1\cup J_2)$ is a subanalytic subset of $K$, so
$\varphi_r(K\backslash(J_1\cup J_2))$ is subanalytic, and  
\mbox{$\varphi_r(K\backslash(J_1\cup J_2))=S_r^{\mathcal{M}}$}, so
$S_r^{\mathcal{M}}$ is subanalytic as well.) 

Conditions (I) and (II) are obviously true if $k=n$, i.e., in the Riemannian
case, but they are false in general. Furthermore, 
to prove subanalyticity of small subriemannian spheres 
it suffices to prove (I) and (II). 
The way  Agrachev and Gauthier do it is
a remarkable technical achievement, which elegantly brings
combines a variety of methods from optimal
control theory that had been developed for totally
different purposes. 

To begin with, one needs some results about the second variation
of an extremal control, especially the theory of the ``index.''
If $F$ is a map of class $C^2$ from a Banach manifold $\mathcal{U}$ 
to a finite-dimensional manifold $M$, and $\bar u$ is a critical point
of $F$, then $F$ has a well defined {\em Hessian} (or ``intrinsic second
derivative'') ${\rm Hes}_{\bar u}F$ at $\bar u$. By definition,
${\rm Hes}_{\bar u}F$ is a quadratic map from ${\rm ker}\,DF(\bar u)$
to ${\rm coker}\,DF(\bar u)$, given by 
$$
{\rm Hes}_{\bar u}F(v)=2\lim_{t\downarrow 0}
t^{-1}\Theta\Big(F(\gamma(\sqrt{t}))-F(\bar u)\Big)
\quad{\rm for}\quad v\in{\rm ker}\,DF(\bar u)\,,
$$
if $\Theta:T_{F(\bar u)}M\mapsto{\rm coker}\,DF(\bar u)$ 
is the canonical projection, and 
$\gamma:[-\eps,\eps]\mapsto\mathcal{U}$
is any $C^2$ curve such that $\gamma(0)=\bar u$ and
$\dot\gamma(0)=v$.  If
$\lambda:T_{F(\bar u)}M\mapsto\rr$ is a nontrivial linear
functional that annihilates ${\rm im}\,DF(\bar u)$, then
we can define the {\em index} ${\rm ind}_F(\bar u,\lambda)$ 
to be the supremum of the numbers ${\rm dim}\,V-{\rm dim}\,{\rm coker}
\,DF(\bar u)$, taken over all linear subspaces $V$ of
${\rm ker}\,DF(\bar u)$ such that the quadratic form
$\lambda\cdot{\rm Hes}_{\bar u}F$ is positive definite
on $V$.  Then $-{\rm dim}\,M\le {\rm ind}_F(\bar u,\lambda)\le+\infty$.
(If $\xi$ is a PCAL extremal of length $r$, and
$\lambda$, $\nu$ are a covector in
$T_{\xi(1)}^*M\backslash\{0\}$ and a real number
such that $\langle\lambda,Df_r({\bf u}_\xi)v\rangle=
\nu\langle{\bf u}_\xi,v\rangle$ for
all $v\in\mathcal{Y}$, then
$(\lambda\cdot{\rm Hes}_{{\bf u}_\xi}f_r)(v)=
\langle\lambda,D^2f({\bf u}_\xi)(v)\rangle-\nu\|v\|^2$ for all
$v\in\mathcal{Y}$.)

The index ${\rm ind}(\xi,\lambda,\nu)
\equaldef{\rm ind}_{f_r}({\bf u}_\xi,\lambda)$
of the extremal $\xi$, relative to the multipliers
$\lambda$, $\nu$, is then well defined. The key facts about the
index are then (F1)~{\em the function $({\bf u},\lambda,\nu)
\mapsto{\rm ind}(\xi_{{\bf u}},\lambda,\nu)$ is lower semicontinuous},
and (F2)~{\em if ${\bf u}$ is a minimizer then
${\rm ind}(\xi_{{\bf u}},\lambda,\nu)<0$ for some 
$(\lambda,\nu)$}.  Furthermore, there is a class
of abnormal extremals, called ``Goh  extremals,'' 
with important special properties. Precisely, a {\em Goh multiplier}
for an extremal $\xi$ is a multiplier $(\lambda,0)$ such that
the adjoint vector $\mu_\lambda(t)$ annihilates
the vectors $[X_i,X_j](\xi(t))$ for all $i,j,t$.  It can  then
be proved that 
(F3)~{\em if ${\rm ind}(\xi_{{\bf u}},\lambda,0)<0$ then 
$(\lambda,0)$ is a Goh multiplier for $\xi$}. A {\em Goh extremal}
is an extremal that admits a Goh multiplier. Then
{\em if there do not exist Goh extremals it follows that the spheres
$S_r^{\mathcal{M}}$ are subanalytic for small $r$.}
(The argument is as follows. First, every minimizer
must be a normal extremal, because if a minimizer
$\xi$ was strictly abnormal then (F2) tells us that 
$\xi$ admits a multiplier $(\lambda,\nu)$ such that
${\rm ind}(\xi,\lambda,\nu)<0$, and then the strict 
abnormality of $\xi$ implies that $\nu=0$, so 
$(\lambda,0)$ must be a Goh multiplier for $\xi$, and we have reached
a contradiction. Therefore (I) holds. To prove (II), we 
define $C_b(r)$ to be the set of all $p\in C_{min}(r)$ such 
that ${\rm ind}(\gamma_p,\mu_p(1),\nu)<0$ 
for some $\nu$. Then $\varphi_r(C_b(r))=S_r^{\mathcal{M}}$ 
thanks to (F2). Let us prove that $C_b(r)$ is compact. 
Pick a sequence $\{p_j\}_{j=1}^\infty$ in $C_b(r)$. Then
the controls ${\bf v}_{p_j}$ belong to $\Omega_r^{min}$,
which is compact. Therefore, by passing to a subsequence,
we may assume that ${\bf v}_{p_j}$ converge to a limit
${\bf v}$, which gives rise to a minimizing trajectory 
$\xi$. If  the sequence $\{p_j\}_{j=1}^\infty$ is unbounded, then
we may assume, after passing to a subsequence,
that $\|p_j\|\to\infty$. If we let $\tilde p_j=\frac{p_j}{\|p_j\|}$,
then we may assume, after passing to a subsequence, that
the $\tilde p_j$ converge to a limit $\tilde p$.
Since each $\mu_{\tilde p_j}$ is an adjoint vector for $\xi_j$ with 
multiplier $\frac{1}{\|p_j\|}$, the function $\mu_{\tilde p}$
is an adjoint vector for $\xi$ with multiplier $0$, i.e.,
an abnormal multiplier. The lower semicontinuity of the index 
implies that ${\rm ind}(\xi,\mu_p(1),0)<0$. But then 
$\xi$ is a Goh extremal, and we have reached a contradiction.
So $\{p_j\}_{j=1}^\infty$ is bounded, and then
it has a subsequence that converges to a $p\in C(r)$. The corresponding
control ${\bf v}_p$ is then ${\bf v}$, which is a minimizer,
so $p\in C_{min}(r)$. Finally, the lower semicontinuity of the index
implies that $p\in C_b(r)$.)

To conclude the proof of (T2), I shall now sketch
the \mbox{beautiful} \mbox{argument} used
by Agrachev and Gauthier to show that for $k\ge 3$, \mbox{generically},
there are no Goh extremals.  Suppose $\xi$ is a Goh extremal,
so that there exists an adjoint vector $t\mapsto\mu(t)$ 
satisfying the conditions
\mbox{$\langle\mu(t),X_i(\xi(t))\rangle=0$} and 
$\langle\mu(t),[X_i,X_j](\xi(t))\rangle=0$ for all $t,i,j$. 
Assume, in addition, that ${\bf u}_{\xi}\in C^\infty$. 
\mbox{Differentiation} of the
$\frac{k(k-1)}{2}$ 
identities 
\mbox{$\langle\mu(t),[X_i,X_j](\xi(t))\rangle=0$}, for $i<j$, yields
$\sum_{\ell=1}^ku^\ell(t)
\langle\mu(t),[X_\ell,[X_i,X_j]](\xi(t))\rangle=0$. This is
a system of $\frac{k(k-1)}{2}$ equations in 
the $k$ unknowns $u^1(t),\ldots,u^k(t)$. If $k\ge 3$ then
$\frac{k(k-1)}{2}\ge k$, so the system \mbox{contains} at least
$k$ equations, and then the existence of a nontrivial solution
\mbox{requires} at least one condition involving the coefficients.
Successive differentiations yield more \mbox{conditions} involving Lie 
brackets of higher and higher orders. The final result,
\mbox{after} one carefully keeps track of all these identitites (as the 
authors do
in the paper), is that if $k\ge 3$ then the conditions for
existence of smooth Goh extremals are only satisfied 
on a set of infinite codimension.   So, generically, no smooth
Goh extremals can exist if $k\ge 3$.

Finally, one has to infer the \mbox{nonexistence} of Goh 
extremals from the nonexistence of \mbox{{\em smooth}} Goh extremals.  
At this point, the following
remarkable \mbox{result} from control theory comes to the rescue:
{\em for a control system 
\mbox{$\dot q=f(q,u)$}, $u\in U$, with
$U$ compact subanalytic and $f$ real analytic in $q$ and $u$, if two
points $\bar q$, $\hat q$ are such that there \mbox{exists} a trajectory
$[a,b]\ni t\mapsto q(t)$ \mbox{for} a measurable control
$[a,b]\ni t\mapsto u(t)\in U$ such that $q(a)=\bar q$ and $q(b)=\hat q$, 
then there exists a trajectory $[a,b]\ni t\mapsto\tilde q(t)$ such
that $\tilde q(\tilde a)=\bar q$ and $\tilde q(\tilde b)=\hat q$,
\mbox{for} a control $[\tilde a,\tilde b]\ni t\mapsto\tilde
u(t)\in U$ which is \mbox{analytic} on an open dense subset of $[\tilde
a,\tilde b]$}.  This rather surprising fact was proved in 1986 in
\mbox{Sussmann} \cite{sussreg} (cf. \mbox{also}
\cite{sussreg2} for a more detailed
proof) as part of a study of \mbox{regularity} properties of 
\mbox{optimal}
controls, and also by Gauthier and Kupka in 1996 in \cite{gauku} for
different \mbox{reasons}, arising from their work on
\mbox{observability}.  In
our situation, it is used to prove that {\em if there exist no
\mbox{smooth} nontrivial Goh controls then there exist no nontrivial Goh
\mbox{controls}}. This is done by 
\mbox{considering} the %$2n+1$-dimensional 
system consisting of the \mbox{equations}
$\dot q=\sum_{i=1}^ku^iX_i(q)$, 
\mbox{$\,\dot\mu=-\sum_{i=1}^k
u^i\mu\cdot\frac{\partial X_i}{\partial q}(q)$}, 
\mbox{$\ \dot
z=\sum_{i=1}^k\langle\mu,X_i(q)\rangle^2
+\sum_{i=1}^k\sum_{j=1}^k\langle\mu,[X_i,X_j](q)\rangle^2$},
$\,$for $q$, $\mu$, and a new scalar variable $z$.
If $\xi:[a,b]\mapsto\mathcal{N}$ is a Goh extremal, 
then $\xi$ gives rise, in an \mbox{obvious} way, to a
\mbox{solution} $\Xi$ of the new system, going from a point
$(\xi(a),\bar\mu,0)$ to a point $(\xi(b),\hat\mu,0)$.
It follows that there is a trajectory $\tilde\Xi$
going from $(\xi(a),\bar\mu,0)$ to $(\xi(b),\hat\mu,0)$,
for a control $\tilde u$ which is analytic on an open
dense subset of its domain. Clearly, $z\equiv 0$ along $\tilde\Xi$.
If $I$ is a nontrivial interval on which $\tilde u$ is analytic, then
the restriction of $\tilde\Xi$ to $I$ is a nontrivial
smooth Goh extremal, and the proof is complete.

This concludes the outline of the proof of (T2). The proofs
of the other \mbox{two} main \mbox{results} require other ideas,
in particular a careful analysis of \mbox{nilpotent}
\mbox{approximations}
of subriemannian structures. Lack of space prevents me from 
\mbox{going} into the details and doing full justice to the technical
virtuosity of the \mbox{authors}, but the proof that has just been sketched
for (T2) should suffice to 
establish that this is a major, impressive, masterly
piece of work, which 
\mbox{represents} a significant step forward 
in our understanding of subriemannian metrics, and constitutes
a magnificent success story for the field of differential-geometric
control theory.

\bigbreak
\begin{center}
Reviewed by {\it H.~J.~Sussmann\footnote{Department of Mathematics; Rutgers
University; Piscataway, NJ 08854-8019, USA. 
E-mail: {\it \mbox{sussmann@math.rutgers.edu}}. 
Web site: {\it http://www.math.rutgers.edu/$\widetilde{\,\,}$sussmann}.
Supported in part by NSF Grant DMS98-03411-00798.}}
\end{center}

\begin{thebibliography}{99}

\bibitem{agrasar}{Agrachev,~A.A., and A.V.~Sarychev,
``Sub-Riemannian metrics: minimality of abnormal geodesics versus
subanalyticity.'' {\it J. ESAIM: Control, Optimization and Calculus of 
Variations} {\bf 4}, 1999, pp.~377-403.}

\bibitem{gauku}{Gauthier, J.-P., and I. Kupka,
``Observability for systems with more outputs than inputs
and asymptotic observers.'' {\it Math. Zeitschr.} {\bf 223},
\mbox{1996, pp.~47-78.}}

\bibitem{ge}{Ge Zhong, ``Horizontal path spaces
and Carnot-Carath\'eodory metric.'' {\it Pacific J. Math} {\bf 161},
1993, pp.~255-286.}

\bibitem{jacquet}{Jacquet, S., ``Subanalyticity of the sub-Riemannian 
distance.'' {\it J. Dynam. Control Systems} {\bf 5},  no. 3, 1999,
pp.~303--328.}

\bibitem{strich-a} 
{Strichartz,~R., ``Sub-Riemannian Geometry.''
{\it J. Diff. Geometry} {\bf 24}, 1983, \mbox{pp.~221-263}.}

\bibitem{strich-b}  
{Strichartz,~R., ``Corrections $\,$to $\,$`Sub-Riemannian $\,$Geometry'.''
$\,${\it J. $\,$Diff. \mbox{Geometry}} {\bf 30}, 1989, pp.~595-596.}

\bibitem{sussioffe}{Sussmann,~H.J., 
``Optimal control and piecewise analyticity of the distance
function.'' In {\it Optimization and Nonlinear Analysis (Haifa, 1990)}, 
A.~Ioffe {\&} \mbox{S.~Reich} \mbox{Eds.}, \underline{Pitman Res. Notes Math.}
{\bf 244}, Longman Sci. Tech., Harlow, 1992, \mbox{pp.~298-310}.}

\bibitem{sussreg}{Sussmann,~H.J., 
``Trajectory $\,$regularity $\,$and $\,$real $\,$analyticity: $\,$some 
$\,$recent 
\mbox{results}.''
In {\it Proc. 25th IEEE Conference on Decision and Control (Athens, Greece,
Dec. 1986)}, IEEE publications, New York, 1986, pp.~592-595.}

\bibitem{sussreg2}{Sussmann,~H.J., 
``Some $\,$optimal $\,$control $\,$applications 
$\,$of $\,$real $\,$\mbox{analytic}
\mbox{stratifications}
$\,$$\,$and $\,$$\,$desingularization.''  
$\,\,\,$\ In {\it $\,$$\,$Singularities
$\,$$\,$Symposium---\L{}ojasiewiz 70}, B.~Jakubczyk, W.~Paw\-lucki and
J.~Stasica Eds., Banach \mbox{Center} Publications Vol. 44, Polish Academy of
Sciences, Warsaw, Poland, 1998, \mbox{pp.~211-232}.}

\end{thebibliography}



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