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\def\T{{\mathbf T}}
\def\cost{{\cal J}}
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\def\fpd#1#2{\frac{\partial #1}{\partial #2}}
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\def\ovl#1{\overline{#1}}
\def\til#1{\widetilde{#1}}
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\def\etal{et al.}

\title{Autonomous optimal control problems}
\author{B. Langerock \\ Department of Mathematical Physics and Astronomy,\\ Ghent
University, Krijgslaan 281 S9,B-9000 Ghent Belgium \\
{\ttfamily \small bavo.langerock@rug.ac.be}\\
{\ttfamily \small http://maphyast.rug.ac.be}}

\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}

\begin{document}
\maketitle

\begin{abstract}
A geometric version of the maximum principle for autonomous
optimal control problems is derived and applied to the
length-minimising problem in sub-Riemannian geometry and to
Lagrangian mechanics on Lie-algebroids.
\end{abstract}

\noindent {\bf Keywords:}  autonomous geometric control theory

\noindent {\bf 2000 MSC:} 49Kxx, 53Cxx

\section{Introduction} \label{intro}
The development of a differential geometric setting for optimal
control theory has been carried out, among other, by H.J. Sussmann
in \cite{sussmann4}, where a coordinate-free formulation of the
maximum principle is given.

In \cite{control} we have given a proof of a coordinate-free
version of the maximum principle for (time-dependent) optimal
control systems with fixed endpoint conditions, relying on an
approach due to L.S. Pontryagin \etal in \cite{Pont}. As a side
result of our approach, we were able to give some necessary and
sufficient conditions for the existence of so-called {\it
(strictly) abnormal extremals} (for an example of a stricly
abnormal extremal, we refer to \cite{Mont}). In this paper, it is
our goal to prove the maximum principle for autonomous optimal
control problems and apply it to sub-Riemannian geometry
(\cite{stri,stri2}) and Lagrangian systems on Lie-algebroids
(\cite{marti,martiActa,wein}).

The outline of the paper is as follows. In the remainder of this
section we first recall the notion of a geometric control
structure as described in \cite{control} in a time-dependent
setting, as well as the notions of an optimal control problem with
fixed and variable endpoints, respectively. We then consider an
adapted version of these notions for the autonomous case. In
Section \ref{S2} we briefly review the approach to the maximum
principle presented in \cite{procwil} and \cite{control} and use
it as a starting point to derive a version of the maximum
principle for autonomous optimal control problems. Section
\ref{S3} contains some specific results for linear autonomous
optimal control problems and in Section \ref{S4} we discuss some
applications.

We start by recalling the definition of {\em a geometric control
structure} (see \cite{control} for more details). A geometric
control structure is a triple $(\tau,\nu,\rho)$ where $(i)$ $\tau:
M \to \R$ and $(ii)$ $\nu: U \to M$ are fibre bundles, with
typical fibres denoted by, respectively, $Q$ and $C$, where $Q$ is
called the {\em configuration space} and $C$  the {\em control
domain}. The control bundle $U$ is related to the first jet bundle
of $\tau$ by means of a bundle map $(iii)$ $\rho: U \to J^1\tau$,
with $\tau_{1,0}\circ \rho = \nu$ (where $\tau_{1,0}: J^1\tau \to
M$ denotes the standard projection, see for instance
\cite{saunders}). This is represented schematically in the
following commuting diagram:

\begin{center}
  \setlength{\unitlength}{.8cm}
\begin{picture}(4.73,5)
\put(0.996,0.297){\parbox[t]{0.487\unitlength}{$\R$}}
\put(1.17,2.14){\vector(0,-1){1.40}}
\put(0.975,2.44){\parbox[t]{0.551\unitlength}{$M$}}
\put(1.19,4.22){\vector(0,-1){1.34}}
\put(1.04,4.51){\parbox[t]{0.445\unitlength}{$U$}}
\put(1.65,4.68){\vector(1,0){1.72}}
\put(3.69,4.66){\parbox[t]{0.911\unitlength}{$J$$^{1}$$\tau{}$}}
\put(3.57,4.25){\vector(-4,-3){1.91}}
\put(2.42,5.02){\parbox[t]{0.381\unitlength}{$\rho{}$}}
\put(0.699,1.48){\parbox[t]{0.318\unitlength}{$\tau{}$}}
\put(0.699,3.67){\parbox[t]{0.403\unitlength}{$\nu{}$}}
\put(3.12,3.43){{$\tau{}$$_{1,0}$}}
\end{picture}
\end{center}

Let $u$ denote a smooth section of $\tau \circ \nu$, i.e. $u:
I=[a,b] \to U$ such that $\tau(\nu(u(t)))=t$ (we assume that $I$
is a compact interval in $\R$ and that $u$ admits a smooth
extension to an open interval containing $I$). Then $u$ is a {\em
control} if $\rho \circ u = j^1c$ with $c = \nu \circ u$ the {\em
base section of $u$}. Given any section $c$ of $\tau$, then $c$ is
called a {\em controlled} section if $c$ is the base section of a
control. We say that the control $u$ {\em takes the point $c(a)$
to the point $c(b)$}, and is represented schematically by
$c(a)\stackrel{u}{\to}c(b)$. In order to fix the ideas we first
investigate locally the notion of smooth controls. Let $(t,x^i)$
denote bundle adapted coordinates on $M$, and, similarly, let
$(t,x^i,u^a)$ denote bundle adapted coordinates on $U$. Then, a
control $u$ locally satisfies:
\[\rho^i(t,c^i(t),u^a(t)) = \dot c^i(t),\] for all $t$. It is
easily seen that these equations correspond to the definition of a
control in \cite{Pont}.

However, it turns out that the class of smooth controls should be
further extended to sections admitting (a finite number of)
discontinuities in the form of certain `jumps' in the fibres of
$\nu$, such that the corresponding base section is piecewise
smooth. For instance, assume that $u_1:[a,b] \to U$ and $u_2:[b,c]
\to U$ are two smooth controls with respective bases $c_1$ and
$c_2$, such that $c_1(b)=c_2(b)$. The {\em composite control} $u_2
\cdot u_1:[a,c] \to U$ of $u_1$ and $u_2$ is defined by:
\[
u_2\cdot u_1 (t) = \left\{ \begin{array}{ll}  u_1(t)& \quad t \in [a,b],\\
 u_2(t) & \quad t \in \ ]b,c].\end{array}\right.
\]
It is readily seen that, although in general $u_2 \cdot u_1$ is,
discontinuous at $t=b$, the base section $\nu \circ (u_2\cdot
u_1)$ is continuous. This definition can easily be extended to any
finite number of smooth controls, yielding what we shall call in
general a {\em control} (a detailed definition can be found in
\cite{control}).

In the remainder of this section we focus our attention on {\em
optimal control problems}. Assume that a {\em cost} function $L$
on the control space $U$ is given, i.e. $L \in \cinfty{U}$. With
any control $u:[a,b]\to M$ we are now able to define its {\em
cost} $\cost(u)$:
\[
\cost(u) = \int^b_a L(u(t))dt.
\]
A control $u$ taking $x=\nu(u(a))$ to $y=\nu(u(b))$ is said to be
{\em optimal} if, for any other control $u'$ taking $x$ to $y$ we
have
\[
\cost(u)\le \cost(u').
\]
The problem of finding the optimal controls taking a given point
to another given point is called an {\em optimal control problem
with fixed endpoint conditions}.

On the other hand, assume that two immersed submanifolds $i: S_i
\to M$ and $j: S_f \to M$ are given. A control $u$ taking a point
$x \in i(S_i)$ to a point $y \in j(S_f)$ is said to be {\em
$(S_i,S_f)$-optimal} if, given any other control $u'$ taking $x'
\in i(S_i)$ to $y' \in j(S_f)$, then $\cost(u)\le \cost(u')$. The
problem of finding the $(S_i,S_f)$-optimal controls taking a point
in $S_i$ to a point in $S_f$ is called an {\em optimal control
problem with variable endpoint conditions} (see \cite{procwil}).

An {\em autonomous geometric control structure} consists of a pair
$(\tilde \nu,\tilde \rho)$ where $\tilde \nu: C \to Q$ is a bundle
and $\tilde \rho: C \to TQ$ a bundle map fibred over the identity
on $Q$. We can then consider the following geometric control
structure $(\tau,\nu,\rho)$, in the sense defined above,
associated with $(\tilde \nu,\tilde \rho)$:
\begin{enumerate}
\item $\tau: M = \R \times Q \to \R : (t,q)\mapsto \tau(t,q)=t$,
\item $\nu: U = \R \times C \to M:(t,p)\mapsto \nu(t,p) =
(t,\tilde \nu(p))$, and \item $\rho: U\to J^1\tau: (t,p) \mapsto
(t,\tilde \rho(p))$, since $J^1\tau \cong \R \times TQ$.
\end{enumerate}

Let $\tilde u:[a,b]\to C$ denote a curve in the control domain
$C$. Then $\tilde u$ is called {\em $\tilde \rho$-admissible} if
$\tilde \rho(\tilde u(t)) = \dot{\tilde c}(t)$, where $\tilde c(t)
= \tilde \nu(\tilde u(t))$ is called the base of $\tilde u$.

We now translate concepts defined in the autonomous geometric
control structure $(\tilde \nu,\tilde \rho)$ to known concepts in
the associated geometric control structure $(\tau,\nu,\rho)$. For
instance, it is an easy exercise to see that every $\tilde
\rho$-admissible curve $\tilde u : [a,b] \to C$ determines a
smooth control $u:[a,b] \to U$ with $u(t)=(t,\tilde u(t))$. On the
other hand, if we assume that $u:[a,b] \to U$ is a smooth control,
then $u$ can be written as $u(t)=(t,\tilde u(t))$ since $u$ is a
section of $\tau\circ \nu$. The curve $\tilde u:[a,b] \to C$ is
$\tilde \rho$-admissible since $\rho(u(t)) = (t,\tilde \rho(\tilde
u(t))$ and $j^1_t c = (t,\dot{\tilde c}(t))$ (where
$c(t)=(t,\tilde c(t))$ denotes the base of $u$). This
correspondence between $\tilde \rho$-admissible curves and smooth
controls is easily extended to the more general notion of
composite controls. Similar to the control setting, we say that
the $\tilde\rho$-admissible curve $\tilde u$ takes $\tilde c(a)$
to $\tilde c(b)$.

Consider a function $\tilde L$ on $C$. Then we can define a cost
function on $U$ by considering $L= p_C^*\tilde L$, where $p_C:U\to
C$ denotes the natural projection. We define a functional on the
set of $\tilde\rho$-admissible curves by:
\[
\cost(\tilde u) = \int^b_a \tilde L(\tilde u(t))dt,
\]
where we have used the same notation as in the previous section
since $\cost(\tilde u) = \cost(u)$, with $u$ the control
associated with $\tilde u$.

A $\tilde\rho$-admissible curve $\tilde u:[a,b]\to C$, taking $p$
to $q$, is called {\em optimal} if, given any
$\tilde\rho$-admissible curve $\tilde u':[a,b]\to C$ taking $p$ to
$q$, then $\cost(\tilde u) \le \cost(\tilde u')$. Note that
$\tilde u$ is optimal with respect to $\tilde L$ iff the
associated control $u(t)=(t,\tilde u(t))$ is optimal with respect
to $L$.

We say that $\tilde u$ is {\em strong optimal} if, given any other
control $\tilde u': [a',b']\to C$ taking $p$ to $q$, then
$\cost(\tilde u) \le \cost(\tilde u')$. Consider $S_p=\R$ and
$i_p: S_p \to \R \times Q : s \mapsto (s,p)$. Similarly we put
$S_q =\R$ with $j_q : S_q \to \R \times Q: s \mapsto (s,q)$. Using
the notations from above, we can write $S_i=S_p$ and $S_f=S_q$. It
is now easily seen that a $\tilde \rho$-admissible curve $\tilde
u$, taking $p$ to $q$ is strong optimal iff the associated control
$u$ is $(S_p,S_q)$-optimal with respect to $p^*_C\tilde L$. The
problem of finding the optimal $\tilde \rho$-admissible curves
taking $p$ to $q$, is called the {\em autonomous optimal control
problem}.

\section{The Maximum principle} \label{S2}

\subsection{Non-autonomous optimal control problems}
We now proceed towards the formulation of the maximum principle
for non-auton\-omous optimal control problems, proven in
\cite{control}, providing necessary conditions for optimal
controls. We first define the notion of a multiplier of a control
$u$. For that purpose, we construct a $1$-parameter family of
closed two-forms on $U\times_M V^*\tau$ (where $V\tau =\ker T\tau$
is the {\em vertical} subbundle of $TM$ and $V^*\tau$ its dual).
Let $\tilde \omega$ be the closed two-form on the fibred product
$U\times_M T^*M$, obtained by pulling back the canonical
symplectic form on $T^*M$ by the projection $U\times_M T^*M\to
T^*M$. Next, for any real number $\lambda$ we can define a section
$\sigma_\lambda$ of the fibration $U\times_MT^*M \to U \times_M
V^*\tau$ in the following way. Take $u_m \in U_m, \eta_m \in
V^*_m\tau$ and put $\sigma_{\lambda}(u_m,\eta_m) =(u_m,
\alpha_m)$, where $\alpha_m \in T^*_mM$ is uniquely determined by
the conditions $(i)$ $\langle \alpha_m, \T(\rho(u_m))\rangle +
\lambda L(u_m) = 0$ and $(ii)$ $\alpha_m$ projects onto $\eta_m$.
As usual, $\T:J^1\tau\to TM$ represents the total time derivative
defined by $\T(j^1_tc)=T_tc(\partial_t)$, for $j^1_tc \in J^1\tau$
arbitrary and with $\partial_t$ the standard vector field on $\R$.
The mapping $\sigma_{\lambda}$ is smooth, as can be easily seen
from the following coordinate expression: putting
$u_m=(t,x^i,u^a)$ and $\eta_m = p_i\,dx^i_{|m}$, a straightforward
computation gives

\[\sigma_{\lambda}(t,x^i,u^a,p_i) = \left(t,x^i,u^a,-\rho^i(t,x^i,u^a)p_i
-\lambda L(t,x^i,u^a), p_i \right).
\]
We can now use $\sigma_{\lambda}$ to pull-back the closed two-form
$\tilde\omega$ to a closed two-form on $U\times_M V^*\tau$, which
will be denoted by $\omega_{\lambda} = \sigma^*_\lambda \tilde
\omega$. Herewith, we can introduce the following definition of a
multiplier.

\begin{definition}{Definition}
Given a control $u: [a,b] \to U$, a pair $(\eta,\lambda)$
consisting of a piecewise smooth section $\eta$ of $V^*\tau$ along
$c = \nu \circ u$ and a real number $\lambda$, is called a {\rm
multiplier} of $u$ if the following conditions are satisfied:
\begin{enumerate}
\item \label{max1}$i_{(\dot u(t),\dot \eta(t))} \omega_{\lambda}
=0$ on every smooth part of the curve $(u(t),\eta(t))$, \item
\label{max2}given any $t_0 \in [a,b]$, and if we put
$\sigma_{\lambda}(u(t_0),\eta(t_0))= (u(t_0),\alpha_0)$ then the
function $u'\mapsto\langle \alpha_0, \T(\rho(u')) \rangle +
\lambda L(u')$, defined on $\nu^{-1}(c(t_0))$, attains a global
maximum for $u'=u(t_0)$, \item \label{max3} $(\eta(t),\lambda)
\neq (0,0)$ for all $t \in [a,b]$.
\end{enumerate}
\end{definition}

We then have the following result (cf. \cite{control}).

\begin{theorem}{Theorem}\label{MP}
 Assume that $x\stackrel{u}{\to} y$ and that
$u$ is optimal. Then there exists a multiplier $(\eta, \lambda)$
with $\lambda \le 0$.
\end{theorem}

In \cite{procwil} we have considered optimal control problems with
variable endpoints. Assume that $S_i$ and $S_f$ denote two
immersed submanifolds of $M$, where either $S_i$ or $S_f$ reduce
to a point (note that if we assume $u$, with $x
\stackrel{u}{\to}y$, to be $(S_i,S_f)$-optimal, then $u$ is also
$(S_i',S_f')$-optimal where $S_i'=S_i$ and $S_f'=\{y\}$ or
$S_f'=S_f$ and $S_i'=\{x\}$). Let us denote the annihilator of a
linear subspace $W$ in a vector space $V$ by $W^0$.
\begin{theorem}{Theorem} \label{MPVE}
Let $x \stackrel{u}{\to} y$ with $x\in S_i$ and $y\in S_f$. If $u$
is $(S_i,S_f)$-optimal then there exists a multiplier
$(\eta,\lambda)$ such that
\begin{enumerate}
\item $\lambda \le 0$, \item $\sigma_\lambda(u(a),\eta(a)) \in
(Ti(T_xS_i))^0$, if $S_f=\{y\}$ or \\
$\sigma_\lambda(u(b),\eta(b)) \in (Tj(T_yS_f))^0$, if $S_i=\{x\}$.
\end{enumerate}
\end{theorem}
\subsection{Autonomous optimal control problems}
Consider an autonomous optimal control problem as defined in
Section \ref{intro} and how it was related to a non-autonomous
optimal control problem. Let $\omega_Q$ denote the canonical
symplectic form on $T^*Q$, and consider the closed two-form
$\tilde \omega_Q$ on $C\times_Q T^*Q$, which is the pull-back of
$\omega_Q$ under the projection $p_{T^*Q}:C\times_Q T^*Q \to
T^*Q$. We can now prove the following version of the maximum
principle for autonomous optimal control problems.

\begin{theorem}{Theorem} \label{MPAUT}
    If a $\tilde \rho$-admissible curve $\tilde u:[a,b]\to C$ is
    optimal with respect to $\tilde L$ then there exists a
    piecewise smooth one-form
    $\tilde \eta(t)$ along $\tilde c(t)=\tilde \nu(\tilde u(t))$
    and a real number $\lambda \le 0$ such that:
    \begin{enumerate}
\item\label{AUT1} $i_{(\dot{\tilde u}(t),\dot{\tilde \eta}(t))}
\tilde \omega_Q = - h_\lambda(\tilde u(t),\tilde\eta(t))$ on every
smooth part of the curve $(\tilde u,\tilde\eta)(t)$, with
$h_\lambda \in \cinfty{C\times_Q T^*Q}$ defined by
$h_\lambda(\tilde u'_p,\zeta_p) = \langle \zeta_p,\tilde
\rho(\tilde u'_p)\rangle + \lambda L(\tilde u'_p)$ for arbitrary
$(\tilde u'_p ,\zeta_p) \in C\times_Q T^*Q$,

        \item\label{AUT2} given any $t \in I$, the function $\tilde u'\mapsto
        h_\lambda(\tilde u',\tilde \eta(t))$ on $C_{c(t)}=\tilde \nu^{-1}(c(t))$ attains a global
        maximum for $\tilde u'=\tilde u(t)$,
        \item \label{AUT3}$h_\lambda(\tilde u(t),\tilde\eta(t)) = \mbox{const.}$ for all $t$ and,
        \item \label{AUT4}$(\tilde\eta(t),\lambda) \neq 0$ for all $t \in I$.
    \end{enumerate}
    If $\tilde u$ is strong optimal then condition
    (\ref{AUT3}) is to be replaced by $h_\lambda(\tilde u(t),\tilde\eta(t)) =0$.
\end{theorem}
\noindent {\bf Proof. }
    Since we already know from Section \ref{intro} that if
    $\tilde u$ is optimal, then the associated control $u$ is also optimal
    in the non-autonomous setting. This correspondence admits us
    to apply Theorem \ref{MP}. The remainder of this proof
    consists of translating the necessary conditions from the
    time-dependent setting to the autonomous setting.

    First, note that $V^*\tau \cong \R\times T^*Q$ and that the section $\sigma_\lambda$
    can be written as $\sigma_\lambda(u',\zeta) = - h_\lambda(p_C(u'),\zeta) dt +
    \zeta$ where $(u',\zeta) \in U \times_M V^*\tau$. Recalling the definition of a
    multiplier then we know that there exists a piecewise smooth
    one-form $\eta(t)=(t,\tilde \eta(t))$ and a $\lambda\le 0$ such that,
    of $u$ denotes the control associated with $\tilde u$,
    \begin{enumerate}
        \item \label{maxvgl1} $i_{(\dot u(t),\dot \eta(t))} \omega_\lambda =0$ on
        every smooth part of the curve $(u,\eta)(t)$,
        \item \label{maxvgl2} given any
        $t \in I$, then the function $u'\mapsto\langle \sigma_\lambda(u(t),\eta(t)),
        \T(\rho(u')) \rangle + \lambda L(u')$ defined on $\nu^{-1}(c(t))$
        attains a global maximum for $u'=u(t)$,
        \item \label{maxvgl3}
        $(\eta(t),\lambda) \neq 0$ for all $t \in I$.
    \end{enumerate}
The closed two form $\omega_\lambda$ equals (with a slight abuse
of notations)    $\tilde \omega_Q -dh_\lambda \wedge dt$. The
function
    \[
    u'\mapsto \langle \sigma_\lambda(u(t),\eta(t)), \T(\rho(u')) \rangle + \lambda
    L(u'),
    \]
    equals $-h_\lambda(\tilde u(t),\tilde \eta(t)) + h_\lambda(p_C(u'),\tilde \eta(t))$. We
    conclude that $h_\lambda(p_C(u'),\tilde\eta(t))$ also attains a global
    maximum for $u' =u(t)$ or for $\tilde u'=p_C(u')=\tilde u(t)$.

    Given any tangent vector $X = \tilde X + X^t \fpd{}{t} \in T(U \times_M V^*\tau)$,
    with $\tilde X \in T(C\times_Q T^*Q)$, then $i_X \omega_\lambda = i_{\tilde
    X} \tilde \omega_Q - dh_\lambda(\tilde X) dt + X^t dh_\lambda$. If we assume that $X = (\dot
    u(t),\dot{\eta}(t))$, then $X^t =1$ and $\tilde X = (\dot{\tilde
    u}(t),\dot{\tilde\eta}(t))$, and equation (\ref{maxvgl1}) is equivalently
    rewritten as: $i_{(\dot{\tilde u}(t),\dot{\tilde \eta}(t))} \tilde \omega_Q =
    - dh_\lambda$.

    Since $(\tilde u,\tilde \eta)$ solves the implicit
    Hamiltonian system with Hamiltonian $h_\lambda$, the function
    $h_\lambda$ is constant on every smooth part of the curve
    $(\tilde u(t),\tilde \eta(t))$. Thus, it remains
    to prove that $h_\lambda(\tilde u(t),\tilde \eta(t))$ is
    continuous.
    Consider therefore a discontinuous point (at $t=t_0$)
    of $\tilde u(t)$, and assume that we have
    fixed an adapted coordinate chart containing it. Then,
    \[
    h_\lambda(\tilde c^i(t),\tilde u^a(t),\tilde \eta_i(t))
    \ge h_\lambda(\tilde c^i(t),w^a,\tilde \eta_i(t)),
    \]
    for all $w^a$. If we consider this inequality and
    take successively the limit from the left and from the right for  $t \to t_0$,
    we obtain the continuity of $h_\lambda(\tilde c^i(t),\tilde u^a(t),\tilde
    \eta_i(t))$ as a function of $t$. It now remains to prove, in the case
    of strong optimality, that $h_\lambda$ is zero on $(\tilde
    u(t),\tilde{\eta}(t))$. We make use of Theorem \ref{MPVE}. From the fact that
    \[
    \sigma_\lambda(u(a),\eta(a))= -h_\lambda(\tilde u(a),\tilde \eta(a)) dt + \tilde \eta_i(a) dx^i \in
    (TS_p)^0=V^*\tau,
    \]
    we obtain $ h_\lambda(\tilde u(a),\tilde \eta(a))= 0$. \rule{5pt}{5pt}

Before proceeding to the following section, we introduce some
additional definitions. Assume that a $\tilde \rho$-admissible
curve $\tilde u$ is given. A couple $(\tilde \eta,\lambda)$, where
$\tilde \eta$ denotes a one form along the base of $\tilde u$ and
a real number $\lambda$, is called a {\em local multiplier} if the
conditions (\ref{AUT1}), (\ref{AUT3}) and (\ref{AUT4}) from
Theorem \ref{MPAUT} are satisfied. If, in addition, condition
(\ref{AUT2}) is satisfied then $(\tilde \eta,\lambda)$ is called a
{\em global multiplier}. Note that the implicit Hamiltonian system
in condition (\ref{AUT1}), implies that $h_\lambda$ attains a
local extremum which justifies the above definitions. It is well
known from literature, that the $\tilde \rho$-admissible curve
$\tilde u$ is called a {\em global (local) extremal} if it admits
a global (local) multiplier $(\tilde \eta,\lambda)$ with $\lambda
\le 0$. Furthermore if $\lambda=0$, then $\tilde u$ is called an
{\em abnormal} extremal, and if $\lambda < 0$, then $\tilde u$ is
called a {\em normal} extremal. Using these definitions, Theorem
\ref{MPAUT} says that any optimal $\tilde \rho$-admissible curve
is a global extremal. Note that, given any global multiplier
$(\tilde \eta, \lambda)$, then for any $\alpha
> 0$, the pair $(\alpha \tilde \eta, \alpha \lambda)$ is also a multiplier.
Therefore, we shall henceforth always assume that the multiplier
$(\tilde \eta,\lambda)$ is `normalised', in the sense that
$\lambda$ equals $0$, $1$ or $-1$.

\section{Linear autonomous control problems} \label{S3}
In the following we shall concentrate on {\em linear autonomous
geometric optimal control structures}, i.e. autonomous geometric
optimal control structures satisfying the additional conditions
that $\tilde \nu:C\to Q$ is a linear bundle and $\tilde \rho$ is a
linear bundle map.

We first consider the maximality condition derived in Theorem
\ref{MPAUT}. Fix any $\zeta^0 \in T^*Q$ and let
$x_0=\pi_Q(\zeta^0)$. The function $\tilde u \mapsto
h_\lambda(\tilde u,\zeta^0)$ for any $\tilde u \in C_{x_0}$,
attains a local extremum at $\tilde u=\tilde u_0$ iff
\[
\left.\fpd{}{u^a}\right|_{\tilde u^a_0} \left(\tilde \rho^i_a u^a
\zeta^0_a + \lambda L(x^i_0,u^a,\zeta^0_a) \right) = 0,
\]
or equivalently $\tilde \rho^*(\zeta) = -\lambda \F L(u)$, where
$\F L: C \to C^*$ denotes the fibre derivative of $L$ and
$\tilde\rho^*:T^*Q \to C^*$ is the dual of the linear bundle map
$\tilde \rho$.

If $\tilde u$ is an abnormal local extremal (i.e. there exists a
local multiplier $(\tilde \eta,\lambda)$ with $\lambda=0$), then
$\tilde\rho^*(\tilde\eta(t))=0$ for any $t$ or, equivalently,
$\tilde \eta(t) \in
 (\tilde\rho(C_{q(t)}))^0$ (the annihilator space of the image of
$C_{q(t)}$ under $\tilde \rho$). Moreover, in this specific case,
the function $\tilde u'\mapsto h_0(\tilde u',\tilde\eta(t))$
equals $0$ for all $\tilde u' \in C_{q(t)}$, which implies that
$\tilde u$ is a global abnormal extremal. We conclude that for
linear autonomous control problems, the abnormal local extremals
are global abnormal extremals.

On the other hand, if $\tilde u$ is a normal local extremal i.e.
$\lambda=-1$, then $\tilde \rho^*(\zeta) = \F L(u)$. We say that
$L$ is a {\em regular cost} if the fibre derivative of $L$ is
invertible. We say that a curve $\tilde\eta(t)$ in $T^*Q$ {\em
generates} a curve $\tilde u(t)$ in $C$, if $\tilde u(t) = (\F
L)^{-1}(\tilde \rho^*(\tilde\eta(t)))$. In this case, if
$\tilde\eta$ is piecewise smooth, then the curve $\tilde u$
generated by $\tilde\eta$ is also continuous. Therefore, a normal
local extremal is a piecewise smooth curve in $C$. Moreover, from
the following proposition it follows that it is a smooth curve.

\begin{theorem}{Theorem}
Assume that $L$ denotes a regular cost. Every normal local
extremal $\tilde u$ is generated by an integral curve
$\tilde\eta(t)$ of the hamiltonian vector field $X_G$ on $T^*Q$
associated with the function $G(\zeta) = h_\lambda(\F
L^{-1}(\tilde \rho^*(\zeta), \zeta)$, for $\zeta \in T^*Q$ and
with $\lambda=-1$. The converse also holds, i.e. every integral
curve of $X_G$ generates a normal local extremal.
\end{theorem}

\noindent {\bf Proof. } We assume that $\tilde u$ is a local
extremal and fix a local multiplier $(\tilde\eta,\lambda)$ with
$\lambda=-1$. Consider the function ${\cal L} : T^*Q \to C
\times_Q T^*Q$, defined by:
\[
{\cal L}(\zeta) = (\F L^{-1}(\tilde \rho^*(\zeta)),\zeta).
\]
Note that ${\cal L}$ is a section of the bundle
$p_{T^*Q}:C\times_Q T^*Q \to T^*Q$, with $p_{T^*Q}$ the projection
onto the second factor. Then it is easily seen that ${\cal L}^*
h_\lambda = G$ and that ${\cal L}^* \tilde \omega_Q = \omega_Q$.
Recall the implicit Hamiltonian system: a multiplier has to
satisfy: $i_{(\dot{\tilde u}(t),\dot{\tilde\eta}(t))}\tilde
\omega_Q= -dh_\lambda(\tilde u(t),\tilde\eta(t))$, and consider
the tangent vectors $X = (\dot{\tilde u}(t),\dot{\tilde\eta}(t)) =
T{\cal L}(\dot{\tilde\eta}(t))$ and $Y = T{\cal L}(w)\in
T(C\times_Q T^*Q)$ for $w \in T(T^*Q)$ arbitrary, then $\tilde
\omega_Q (X,Y) = \omega_Q(\dot{\tilde\eta}(t), w)$. By
substituting ${\cal L}^* h_\lambda = G$ we obtain that
$i_{\dot{\tilde\eta}(t)} \omega_Q = -d G(\tilde\eta(t))$ for every
smooth part of $\tilde\eta$. By uniqueness of solutions to
differential equations, it follows that $\tilde\eta$ is smooth
(and therefore we have that $\tilde u$ is smooth).

On the other hand, assume that $i_{\dot{\tilde\eta}(t)} \omega_Q =
-d G(\tilde\eta(t))$ and consider the smooth curve $\tilde u(t) =
(\F L)^{-1} (\tilde \rho^*(\tilde\eta(t)))$ in $C$. Then, by
reversing the above arguments, we obtain that $ \tilde \omega_Q
(X,Y) = -dh_\lambda(Y)$ with $X = (\dot{\tilde
u}(t),\dot{\tilde\eta}(t)) = T{\cal L}(\dot{\tilde\eta}(t))$ and
$Y = T{\cal L}(w)\in T(C\times_Q T^*Q)$ for $w \in T(T^*Q)$
arbitrary and $\lambda=-1$. It remains to check that this is also
valid for arbitrary $Y \in T(C\times_Q T^*Q)$. Since ${\cal L}$ is
a section of $p_{T^*Q}$, any element $Y$ in $T(C\times_QT^*Q)$ can
be written as $Y = T{\cal L}(w) + Z$ where $w=Tp_{T^*Q}(Y) \in
T(T^*Q)$ and $Z \in \ker Tp_{T^*Q}$. Since $\tilde \omega_Q =
p^*_{T^*Q} \omega_Q$ it is easily seen that $i_Z \tilde \omega_Q =
0$. From this we conclude that $i_{X} \tilde \omega_Q =
-dh_\lambda(\tilde u(t),\tilde\eta(t))$ for arbitrary $t$.
\rule{5pt}{5pt}

\section{Applications} \label{S4}
\begin{example}
\textnormal{Consider a sub-Riemannian structure $(Q,D,h)$, where
$Q$ is a manifold, $D$ a regular, i.e. constant rank and smooth,
distribution on $Q$ and $h$ is a Riemannian bundle metric on $D$.
Let $i: D \to TQ$ denote the natural injection of $D$ into $TQ$
(note that $\tilde \nu:D\to Q$ can be considered as a linear
bundle on $Q$). We would like to solve the length-minimising
problem in the sub-Riemannian structure $(Q,D,h)$, i.e. we have to
solve the autonomous optimal control problem with control
structure $(Q,\tilde \nu,i)$ and cost $E \in \cinfty{D}$, where
$E(v)= \frac{1}{2} h(v,v)$. It is easily seen that $E$ is a
regular cost, i.e. $\F E = \flat_h$, with $\flat_h$ defined by
$h(v,w)=\langle \flat_h(v),w\rangle$ for arbitrary $v,w \in D$.
Let $\sharp_h$ denote the inverse of $\flat_h$. The function $G$,
introduced in the above theorem, takes the form: $G(\zeta) =
\langle \zeta,i(\sharp_h(i^*(\zeta)))\rangle - \frac{1}{2}
h(\sharp_h(i^*(\zeta)),\sharp_h(i^*(\zeta)))$. If we consider the
tensor $g \in TQ \otimes TQ$ defined by $g(\zeta,\xi) =
h(\sharp_h(i^*(\zeta)),\sharp_h(i^*(\xi)))$ with $\xi,\zeta \in
T^*Q$, then $G (\zeta)= \frac{1}{2} g(\zeta,\zeta)$. In \cite{sub}
we have further investigated the equations of a local extremal
using connections over a bundle map and we gave necessary and
sufficient conditions for abnormal extremals.}
\end{example}
In the next example we consider the case where $C$ is a
Lie-algebroid with anchor map $\tilde \rho$. We refer to
\cite{marti,martiActa,wein} where the importance of this specific
case for a generalisation of Lagrangian mechanics is thoroughly
investigated. It should be noted that we only derive the
Lagrangian equations, using the theory developed above.
\begin{example} \textnormal{ Assume that $C$ is a Lie-algebroid with anchor map
$\tilde \rho$ and assume that $L$ is a regular cost. It is a well
known fact that the Lie-algebroid structure determines a Poisson
structure on $C^*$, where the bracket on $\cinfty{C^*}$ is denoted
by $\{\cdot,\cdot\}_{C^*}$, whereas the Poisson bracket on
$\cinfty{T^*Q}$ is denoted by $\{\cdot,\cdot \}$. It is also well
known that both Poisson structures are $\tilde \rho^*$ connected,
i.e. if $\chi = \tilde \rho^*$, then, given arbitrary $f,g \in
\cinfty{C^*}$ the following equality holds: $\{\chi^*f , \chi^*g\}
= \chi^*\{ f,g\}_{C^*}$. This implies that any Hamiltonian vector
field $X_f$ on $C^*$ is $\rho^*$-connected to the Hamiltonian
vector field $X_{\chi^* f}$ on $T^*Q$. Consider the function $G$
on $T^*Q$, introduced in the previous section. Define $g \in
\cinfty{C^*}$ by $g(\alpha) = \langle \alpha , \F L^{-1}(\alpha)
\rangle - L(\F L^{-1}(\alpha))$. Then, it is easily seen that
$\chi^* g = G$. This guarantees that, given any integral curve
$\eta(t)$ of $X_G$, then $\tilde \rho^*(\eta(t))$ is an integral
curve of $X_g$ and, conversely, any integral curve $\alpha(t)$ of
$X_g$, through a point in the image of $\tilde \rho^*$ is the
projection under $\tilde \rho^*$ of an integral curve of $X_G$.
>From this we conclude that, in the case where $C^*$ is a
Lie-algebroid, the integral curves of $X_g$ through a point in the
image of $\rho^*$ are projections of normal local extremals.}
\end{example}

\section*{Acknowledgements}
This research is supported by a grant from the ``Bijzonder
onderzoeksfonds'' of Ghent University. I am indebted to F.
Cantrijn for the many discussions and the careful reading of this
paper.

%bibliography

\providecommand{\bysame}{\leavevmode\hbox
to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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\end{thebibliography}

\end{document}
