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\topmatter
%
%
%
\title
{Feedback--Invariant Optimal Control Theory and Differential
Geometry ---  I.\break
Regular Extremals}
\endtitle
\author
A.~A.~Agrachev, \  R.~V.~Gamkrelidze
\endauthor
\NoRunningHeads
\flushpar
\abstract
Feedback--invariant approach to smooth optimal control
problems is considered. A Hamiltonian method of investigating regular
extremals is developed, analogous to the
differential--geometric method of investigation Riemannian
geodesics in terms of the Levi--Civita connection and the
curvature tensor.
\endabstract
\endtopmatter
\document


\


 \heading{\S0.    Introduction}\endheading


\

{\bf 1. Outline of the content.}\
This
\footnote"{}"{
{\it 1991 Mathematics Subject Classification.}
49K15, 53B05(B15), 53C22. \hfil\break
\indent{\it Key words.}  Hamiltonian, extremal, geodesic,
connection, curvature.}
is the first in a
series of forthcoming papers, devoted to the unification of the
Theory of Smooth Optimal Control Problems and that part of
Differential Geometry which is dealing with geodesics of
different kinds. The obtained results, we believe,   not merely
suggest    a dictionary for translating   the known results from
one language into another, but they really extend the scope of
applicability of both theories.
The key notions brought into interplay are   "Hamiltonian
system" in optimal control and the "curvature tensor"
in differential geometry.

Since the discovery of the Pontryagin maximum principle, cf.~
[11], finding   extremals in  problems of optimal control is
reduced to solving Hamiltonian systems of differential
equations.
Even in the classical case of Riemannian geometry, the maximum
principle approach to finding   geodesics leads to the final
result much simpler and shorter than the traditional method
of  using the Levi--Civita connection.  If we consider
more general geometric variational problems, dealing not only
with regular extremals (geodesics), but with singular extremals
as well, then we should admit that the maximum principle approach
has no serious alternative.

Turning now from geodesics to the curvature tensor,  which
describes quite deeply not only local but also global
behavior of geodesics  without even solving any differential
equations, we see that it is   obtained in a standard way
from the Levi--Civita connection, whereas a
Hamiltonian approach to the curvature tensor or its analogue
was never considered. The main content of this paper is
devoted to such an approach.

We shall give now a brief overview of the content by sections.
In \S1 the  notion of the
$\Cal L$--derivative is introduced. The intuitive meaning of
this notion, which plays in the sequel an important role, could
be described as follows.

Let
$f:U\longrightarrow M$
be a smooth mapping between two finite-dimensional manifolds,
with the differential
$f'_u:T_uU\longrightarrow T_{f(u)}M$
at
$u\in U$.
The point
$u$
is critical for
$f$
if and only if the image of
$f'_u$
is annihilated by some nonzero covector
$\lambda$,
the Lagrange multiplier,
$$
\lambda\,f'_u=0,\ \lambda\in T^*_{f(u)}M,\,\lambda\ne0.
$$
In this equation, the argument
$\lambda$
belongs to the symplectic manifold
$T^*M$,
the argument
$u$
--- to the manifold
$U$.
Linearization of the equation at the point
$(u,\lambda)$
gives us a linear system of   equations in variables
$\delta u\in T_uU,\,\delta\lambda\in  T_\lambda(T^*M)$.
Let
$\Cal L_{(u,\lambda)}(f)$
be the set of all
$\delta \lambda$
which satisfy the linear system (with some
$\delta u$).
The linear subspace
$\Cal L_{(u,\lambda)}(f)\subset T_\lambda(T^*M)$
is called the
$\Cal L$--derivative of
$f$
at
$(u,\lambda)$.
It turns out that
$\Cal L_{(u,\lambda)}(f)$
is always a Lagrangian subspace of the symplectic space
$T_\lambda (T^*M)$,
in particular,
$dim\,\Cal L_{(u,\lambda)}(f)=dim\,M$.
Thus the dimension of the
$\Cal L$--derivative is independent on the character of
the critical point; for a constant mapping we have
$\Cal L_{(u,\lambda)}(f)=T_\lambda(T^*_{f(u)}M)$.
The optimal control situation is more general, with an
infinite--dimensional
$U$.
In \S1 the
$\Cal L$--derivative for the infinite--dimensional case is
considered, and the important chain rule for the
$\Cal L$--derivative is formulated.

In \S2 we give
a feedback--invariant definition of a smooth control system
which includes as special cases many basic
differential--geometric structures.   The space of admissible
trajectories  is introduced  and the boundary--value mapping is
defined, which sends the trajectory into its boundary
points.
Critical points of the boundary--value
mapping are the  extremal trajectories, geodesics in
the geometric terminology. At the end of the section a
Hamiltonian characterization of  extremal trajectories is given
in form of a feedback--invariant analogue of the
maximum--principle.


In \S3 the
$\Cal L$--derivative of the boundary--value mapping and of its
particular case, of the endpoint mapping, is computed. We
also introduce   regular extremals  which are   trajectories
of    a fixed Hamiltonian system, defined in a region of
$T^*M$.
For regular extremals, the
$\Cal L$--derivative of the boundary--value mapping is computed
particularly simple. Let
$\tau\mapsto\lambda_\tau\in T^*M,\ 0\le\tau\le t$,
be a regular extremal,
$\lambda_\tau=P_\tau(\lambda_0)$,
where
$\tau\mapsto P_\tau$
is the Hamiltonian flow in
$T^*M$
such that all of its trajectories are regular extremals.
Then the
$\Cal L$--derivative is the graph of the linear symplectic
mapping
$P_{t*}:T_{\lambda_0}(T^*M)\longrightarrow T_{\lambda_t}(T^*M)$.

In \S4 Jacobi curves are introduced and investigated.
Jacobi curves are   curves in the Lagrangian Grassmannian
corresponding to   given extremals of a very general nature,
certainly including all regular extremals, and are constructed,
roughly, in the following way.
An arbitrary segment of an extremal is again an extremal. Hence,
varying the initial point of the
extremal with the fixed endpoint (or vice versa), we
obtain
$\Cal L$--derivatives
of the endpoint mapping, which are Lagrangian subspaces
in a fixed symplectic space depending on a time--variable,
thus obtaining the Jacobi curve.

We also develop here the differential geometry of regular curves
in a Lagrangian Grassmannian. Nonregular Jacobi curves, occurring
in problems with nonholonomic constraints, will be considered
in future publications.  The most important infinitesimal
invariants of a regular curve in the Lagrangian Grassmannian
are  the "derivative curve" and the "curvature tensor".
The curves of constant scalar curvature are characterized.
Formulas are derived  which relate the number of conjugate
points, Maslov index and curvature.

The obtained results are applied in \S5 to develop the
differential geometry of Hamiltonian systems on
$T^*M$,
and of differential equations of second order on
$M$.
To a trajectory of the Hamiltonian system, passing through a
point
$\lambda\in T^*M$,
 germ of a curve in the Lagrangian Grassmannian of the
symplectic space
$T_\lambda(T^*M)$
is assigned.  Infinitesimal invariants of these germs define a
(in general nonlinear) canonical connection   on
$T^*M$
associated to the Hamiltonian. The main result of this
section consists in deriving identities, connecting
the curvature of the canonical connection with the curvatures
of the germs of curves in the Lagrangian Grassmannian. A
similar theory  is developed for differential equations of the
second order for which we have to substitute
$T^*M$
by
$TM$,
and
the Lagrangian Grassmannian  by ordinary Grassmannian.
It turns out that the canonical connection of the equation of
the geodesic flow of a (pseudo)--Riemannian structure coincides
with the Levi--Civita connection  of this structure.

In \S6, for two--dimensional systems, the curvature of the
extremals of smooth control systems is expressed through
standard "state--invariants", the iterated Lie brackets of
vector fields.

\

\noindent{\bf 2. Preliminaries.}\
Here we introduce some formulas of "Chronological calculus" and certain
notions and relations related to linear symplectic spaces used in the
article, cf. [3,8], [1,7,9].

Assume
$M$
is a smooth, i.e. of class
$C^\infty$,
manifold, and
$C^\infty(M)$
is the algebra of smooth functions on
$M$.
We identify an arbitrary diffeomorphism
$P:M\longrightarrow M$
with the corresponding automorphism of the algebra
$C^\infty(M)$,
$$
a(\cdot)\mapsto a\circ P(\cdot)=a(P(\cdot)),\  a\in C^\infty(M).
$$
Under this identification, the action of
$P$
on
$a$,
i.e. the substitution  of
$P$
into
$a$,
is denoted by
$Pa$,
and the value of
$P$
at
$x\in M$
is denoted by
$xP$,
$xPa\overset\text{\rm def}\to=a(P(x))$.
As usual, smooth vector fields
$X$
on
$M$
are identified with the derivations of the algebra
$C^\infty(M)$,
hence they are
$\Bbb R$--linear
mappings
of
$C^\infty(M)$
satisfying the Leibniz rule,
$X(a_1a_2)=a_1Xa_2+a_1Xa_2$.
The Lie bracket,
$[X_1,X_2]=X_1\circ X_2-X_2\circ X_1$,
turn the
$\Bbb R$--linear
space of   vector fields into the Lie algebra,
$Vect\,M$.
For a given
$X$,
the inner derivation of the Lie algebra
$Vect\,M$
is defined,
$$
ad\,X:Vect\,M\longrightarrow Vect\,M,\ (adX)Y=[X,Y].
$$
Every diffeomorphism
$P$
defines an inner automorphism
$Ad\,P:Vect\,M\longrightarrow Vect\,M$, \linebreak
$(Ad\,P)X=P\circ X\circ P^{-1}$.
It is easily seen that the differential of the inverse diffeomorphism
$P^{-1}$,
denoted by
$P^{-1}_*$,
acts on vector fields according to the formula
($Ad\,P)X=P^{-1}_*X$.

We also consider nonstationary vector fields, i.e. measurable essentially
bounded mappings,
$t\mapsto X_t,\ t\in\Bbb R,\ X_t\in Vect\,M$,
and nonstationary flows, i.e. Lipschitz mappings
$t\mapsto P_t,\ t\in \Bbb R,\ P_t\in Diff\,M$.
Every nonstationary vector field defines the corresponding differential equation
$\dot x=X_t(x)$
on
$M$
with  an arbitrary initial condition,
$x(t_0)=x_0$.
If a solution of the differential equation exists for arbitrary
$x_0\in M,\ t\in \Bbb R$,
i.e. if the field
$X_t$
is complete, then it uniquely defines for
$t\in\Bbb R$
an absolutely continuous solution of the
operator differential equation
$$
{dP\over dt}=P_t\circ X_t,\ P_{t_0}=id, \eqno(0.1)
$$
which we call the {\sl flow on}
$M$,
defined by the nonstationary complete vector field
$X_t$,
and denote
$$
P_t=\overset\longrightarrow\to{exp}\int\limits_{t_0}^tX_\tau d\tau.
$$
We also call this flow the {\sl right chronological exponential
of}
$X_\tau$.
For stationary vector fields
$X_\tau\equiv X$,
the corresponding flows are denoted by
$P_t=e^{tX}$.
In the sequel, all  vector fields are  assumed to be complete. This will not
restrict the generality of our considerations.

The chronological exponential admits an asymptotical expansion as a Volterra
series,
$$
\overset\longrightarrow\to{exp}\int\limits_{t_0}^tX_\tau d\tau\approx
id+\int\limits_{t_0}^tX_\tau d\tau+
\ldots+\int\limits_{t_0}^td\tau_1\int\limits_{t_0}^{\tau_1}d\tau_2\ldots\int\lim
its_{t_0}^{\tau_{i-1}}
(X_{\tau_i}\circ
\ldots\circ X_{\tau_1})d\tau_i+\ldots\ \ \ .
$$
For a stationary field, we obtain
$$
e^{(t-t_0)X}\approx id+(t-t_0)X+\ldots+{(t-t_0)^i\over i!}X^i+\ldots\ \ \ .
$$
In the sequel, we shall need the following important {\sl variation
formula}, which represents the chronological exponential of the sum of two
vector
fields as a product of  two chronological exponentials,
$$
\aligned
\overset\longrightarrow\to{exp}\int\limits_{t_0}^t(X_\tau+&Y_\tau)d\tau=
\overset\longrightarrow\to{exp}\int\limits_{t_0}^tX_\tau d\tau\circ
\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
Ad\left(\overset\longrightarrow\to{exp}\int\limits_{t}^\tau X_\theta
d\theta\right) Y_\tau d\tau=\\
&\overset\longrightarrow\to{exp}\int\limits_{t_0}^tAd
\left(\overset\longrightarrow\to{exp}\int\limits_{t_0}^\tau X_\theta
d\theta\right)Y_\tau d\tau
\circ \overset\longrightarrow\to{exp}\int\limits_{t_0}^t X_\tau d\tau.
\endaligned
\eqno(0.2)
$$
Differentiating the expression
$$
Ad\left(\overset\longrightarrow\to{exp}\int\limits_{t_0}^\tau X_\theta
d\theta\right) Y=
\overset\longrightarrow\to{exp}\int\limits_{t_0}^\tau X_\theta d\theta\circ
Y\circ
\left(\overset\longrightarrow\to{exp}\int\limits_{t_0}^\tau X_\theta
d\theta\right)^{-1}
$$
with respect to
$\tau$,
we obtain the equality
$$
{d\ \over d\tau}Ad\left(\overset\longrightarrow\to{exp}\int\limits_{t_0}^\tau
X_\theta d\theta\right) Y=
Ad\left(\overset\longrightarrow\to{exp}\int\limits_{t_0}^\tau X_\theta
d\theta\right)
\circ ad\,X_\tau Y \ \ \forall Y\in Vect\,M.
$$
In other words, the expression
$Ad\left(\overset\longrightarrow\to{exp}\int\limits_{t_0}^\tau X_\theta
d\theta\right)$
satisfies an equation, similar to (0.1), where the field
$X_\tau$
is substituted by the operator
$ad\,X_\tau$.
This remark should justify the notation
$$
Ad\left(\overset\longrightarrow\to{exp}\int\limits_{t_0}^\tau X_\theta
d\theta\right)=
\overset\longrightarrow\to{exp}\int\limits_{t_0}^\tau ad\,X_\theta d\theta,\ \
Ad\left(e^{\tau X}\right)=e^{\tau ad\,X},
$$
which is also supported by the validity of the asymptotic expansions
$$
\aligned
\overset\longrightarrow\to{exp}\int\limits_{t_0}^t ad\,X_\tau &d\tau\approx
id+\int\limits_{t_0}^tad\,X_\tau
d\tau+\ldots+\int\limits_{t_0}^td\tau_1\int\limits_{t_0}^{t_1}d\tau_2
\ldots\int\limits_{t_0}^{\tau_{i-1}}(ad\,X_{\tau_i}\circ\ldots\circ
ad\,X_{\tau_1})d\tau_i+\ldots\\
&e^{\tau\,ad\,X}\approx id+\tau\,ad\,X+\ldots+{\tau^i\over i!}(ad\,X)^i+\ldots\
\ \ .
\endaligned
$$
In this notation, the variation formula (0.2) takes the form
$$
\aligned
\overset\longrightarrow\to{exp}\int\limits_{t_0}^t(X_\tau+&Y_\tau)d\tau=
\overset\longrightarrow\to{exp}\int\limits_{t_0}^tX_\tau d\tau\circ
\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
\left(\overset\longrightarrow\to{exp}\int\limits_{t}^\tau ad\,X_\theta
d\theta\right) Y_\tau d\tau=\\
&\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
\left(\overset\longrightarrow\to{exp}\int\limits_{t_0}^\tau ad\,X_\theta
d\theta\right)Y_\tau d\tau
\circ \overset\longrightarrow\to{exp}\int\limits_{t_0}^t X_\tau d\tau.
\endaligned
\eqno(0.3)
$$

We shall also need the Hamiltonian version of the variation
formula. Let
$TM,\ T^*M$
be tangent and cotangent bundles of
$M$,
with canonical projections denoted by one letter,
$\pi$.
Let
$\theta$
be the canonical 1--form on
$T^*M$, $<\theta_\lambda,v>=<\lambda,\pi_*v>\ \forall
\lambda\in T^*M,\,v\in T_\lambda(T^*M)$.
The 2--form
$\sigma=d\theta$
is the canonical symplectic structure on
$T^*M$.
Every smooth function
$h$,
defined on an open region of
$T^*M$,
defines a Hamiltonian vector field
$\overset\rightarrow\to h$
on the region by the formula
$$\overset\rightarrow\to h\,\rfloor\sigma=-dh.$$
The corresponding differential equation
$\dot\lambda=\overset\rightarrow\to h(\lambda)$
is the Hamiltonian system associated to the Hamiltonian
$H$.
The Poisson bracket of two Hmiltonians is given by the
relation
$$
\{h_1,h_2\}\overset\text{\rm def}\to=\sigma(
\overset\rightarrow\to h_1,\overset\rightarrow\to
h_2)=\overset\rightarrow\to h_1h_2.
$$
If the functions
$h_i$
are linear on fibres of
$T^*M$,
$h_i(\lambda)=<\lambda,X_i>,\ X_i\in Vect\,M,\ i=1,2$,
then
$$
\{h_1,h_2\}(\lambda)=<\lambda,[X_1,X_2]>.
$$
A measurable essentially bounded family of Hamiltonians
$h_t,\,t\in\Bbb R$,
will be called a {\sl nonstationary Hamiltonian.}
The corresponding {\sl Hamiltonian flow}
$\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
\overset\rightarrow\to h_\tau d\tau$
preserves the symplectic structure and satisfies the relation
$$
{\partial\ \over\partial
t}\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
\overset\rightarrow\to h_\tau d\tau g=
\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
\overset\rightarrow\to h_\tau d\tau\{h_t,g\}\ \ \forall g.
$$
Finally, the variation formula for Hamiltonian flows could be
reduced to the following relations,
$$
\aligned
\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
(\overset\rightarrow\to h_\tau +\overset\rightarrow\to g)d\tau&=
\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
\overset\rightarrow\to h_\tau d\tau\circ
\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
(\overrightarrow{\overset\longrightarrow\to
{exp}\int\limits_{t}^\tau
\overset\rightarrow\to h_\theta
d\theta)}g_\tau d\tau=\\
& \overset\longrightarrow\to{exp}\int\limits_{t_0}^t
(\overrightarrow{\overset\longrightarrow\to
{exp}\int\limits_{t_0}^\tau
\overset\rightarrow\to h_\theta
d\theta)}g_\tau d\tau\circ \overset\longrightarrow\to{exp}
\int\limits_{t_0}^t
\overset\rightarrow\to h_\tau d\tau
\endaligned
$$
We introduce  now some notions and formulate certain facts of
linear symplectic geometry used in the sequel. For details cf.
[1,7,9].

Let
$\Sigma$
be a
$2n$--dimensional symplectic space with the skew-symmetric form
$\sigma(\cdot,\cdot)$,
for example, the cotangent space
$T_\lambda(T^*M)$.
For every subspace
$S\subset\Sigma$
put
$S^\angle=\{\zeta\in\Sigma\Bigr|\ \sigma(S,\zeta)=0\}$,
hence
$dim\,S+dim\,S^\angle=2n$.
The subspace
$S$
is called isotropic if
$S^\angle\subset S$.
An
$n$--dimensional subspace
$\Lambda\subset\Sigma$
is called Lagrangian if
$\Lambda^\angle=\Lambda$.
The set of all Lagrangian subspaces is organized into a
smooth
$n(n+1)\over2$--dimensional manifold, the Lagrangian
Grassmannian,
$L(\Sigma)=\{\Lambda\subset\Sigma\Bigr|\
\Lambda^\angle=\Lambda\}$.

The tangent space
$T_\Lambda L(\Sigma),\ \Lambda\in L(\Sigma)$,
is identified in a natural way with the space of
quadratic forms  defined on the
$n$--dimensional space
$\Lambda$.
Indeed, let
$t\mapsto\Lambda_t$
be a germ of a smooth curve in
$L(\Sigma)$.
We correspond to the tangent vector
${d\  \over dt}\Lambda$
the quadratic form
$\dot\Lambda_0:\lambda_0\mapsto\sigma(\lambda_0,{d\ \over
dt}\lambda_0)$,
where
$t\mapsto\lambda_t$
is a germ of a smooth curve in
$\Sigma,\ \lambda_t\in\Lambda_t$.
It is easy to show that
$\dot\Lambda_0(\lambda_0)$
is correctly defined, i.e. it does   depend on
${d\ \over dt}\Lambda_0,\ \lambda_0$,
but is independent on the choice of the germs
$\Lambda_\cdot,\ \lambda_\cdot$\ \ .

The symplectic group
$Sp(\Sigma)$
is the group of linear transformations of
$\Sigma$
preserving
$\sigma$,
hence transforming Lagrangian subspaces into Lagrangian
subspaces.
$Sp(\Sigma)$
acts on
$L(\Sigma)$
transitively, thus
$L(\Sigma)$
is a homogeneous space for the group
$Sp(\Sigma)$.

Let
$h$
be a quadratic form (quadratic Hamiltonian) on
$\Sigma$,
then
$e^{t\overset\rightarrow\to h}\in Sp(\Sigma)$.
Put
$\Lambda_t=e^{t\overset\rightarrow\to h}(\Lambda_0)$,
then
$\dot\Lambda_0=2h\Bigr|_{\lambda_0}$.
A smooth curve in
$L(\Sigma)$,
$t\mapsto\Lambda_t$,
is called {\sl   monotonically nondecreasing (nonincreasing) }
if
$\dot\Lambda_t\ge0\ (\dot\Lambda_t\le 0)$.
The subset
$$
\Cal M_{\Lambda_0}=\left\{\Lambda\in L(\Sigma)\Bigr|\
\Lambda\bigcap\Lambda_0\ne0\right\}\subset\Sigma
$$
is called the {\sl train} of the Lagrangian subspace
$\Lambda_0$.
$\Cal M_{\Lambda_0}$
is an algebraic hypersurface in
$L(\Sigma)$,
smooth beyond some set of codimension 3 in
$\Sigma$,
hence
$\Cal M_{\Lambda_0}$
is a pseudomanifold. Moreover, the hypersurface
$\Cal M_{\Lambda_0}$
carries a natural coorientation, defined in such a way that
the monotonically  increasing curves intersect
$\Cal M_{\Lambda_0}$
in the positive direction, and monotonically decreasing curves
--- in the negative direction.

Thus for every continuous curve
$t\mapsto \Lambda(t),\ t\in[t_0,t_1]$,
such that
$\Lambda(t_i)\notin\Cal M_{\Lambda_0},\ i=0,1$,
the  intersection number,
$\Lambda(\cdot)\,\cdot\,\Cal M_{\Lambda_0}$,
is defined, and
$\Lambda(\cdot)\,\cdot\,\Cal M_{\Lambda_0}\ge0\,(\le0)$,
if the curve
$\Lambda(\cdot)$
is nondecreasing (nonincreasing).

The curve is called {\sl simple} if there exists
$\Delta\in L(\Sigma)$
such that
$\Lambda(t)\bigcap\Delta=0\ \forall t\in[t_0,t_1]$.
If the curve
$\Lambda(\cdot)$
is simple, then
$\Lambda(\cdot)\,\cdot\,\Cal M_{\Lambda_0}\le n$.
Finally, if
$\Lambda(\cdot)$
is closed,
$\Lambda(t_0)=\Lambda(t_1)$,
then the intersection number
$\Lambda(\cdot)\,\cdot\,\Cal M_{\Lambda_0}$
does not depend on
$\Lambda_0$
and is denoted
$Ind\,\Lambda(\cdot)$.
This  is  the Maslov index of the closed curve.

\

\heading \S1. $\Cal L$--derivatives of smooth mappings \endheading

\

\noindent
We start with some   definitions and constructions relevant to critical
points of smooth mappings. The exposition is carried out for infinite
dimensional case, sufficiently general for handling variational and
control problems  discussed further.

The differential   of a scalar--valued function on a Banach
space (evaluated at an arbitrary point) is an element of
the dual  space. In the finite--dimensional situation we
can make no difference between the initial space and its dual, but in
the infinite--dimensional case the dual might be less comprehensible
than the initial space.  A standard example --- the space
$L_\infty[0,1]$
of admissible controls in optimal control problems, which is very natural
and simplest possible to be considered in most situations, but its dual
is pretty involved.  Due to the restrictive nature of the functionals involved
in smooth control problems, the final results, if appropriately  formulated,
do not use the dual space at all, though some cumbersome analytical efforts are
needed for eliminating  the dual space in
final formulations.  Meanwhile, a natural modification of
some basic definitions makes it possible to avoid all artificial
complications connected with this phenomenon.  The simple trick consists in
considering the initial space as a dual to some ''acceptable'' Banach
space, in our example, considering from the beginning the space of controls
$L_\infty[0,1]$
as the   dual to
$L_1[0,1]$,
and appropriately defining (stiffening) the notion of the differential of a
smooth  mapping on such a space.  Formally, we proceed in the following way.

\

\noindent
Let
$B$
be a Banach space,
$B^*$
--- its dual.  We shall always suppose the natural (isometric)
inclusion
$B\subset B^{**}$.

A differentiable scalar--valued function
$a$
on
$B^*$
(a "nonlinear functional")
is said {\sl to be of
class}
$*$--$C^1$, or {\sl
$*$%
--differentiable of class}
$C^1$,
if it is of class
$C^1$
in the usual sense and its differential
$d_xa$
at an arbitrary point
$x\in B$,
which is an element of the second dual,
$d_xa\in B^{**}$,
belongs in fact to
$B$,
$$
d_xa\in B(\subset B^{**})\ \forall x\in B^*.
$$
Equivalently,  we can say that
$a$
is of class
$*$--$C^1$
if
$da$
is a continuous mapping from
$B^*$
to $B$,
$$
da:B^*\longrightarrow   B,\ x\mapsto d_xa\in B.
$$
A scalar--valued function
$a$
{\sl of class
$*$--$C^1$
is said to be  of class}
$*$--$C^k$,
if it is of class
$C^k$
(in the usual sense), and for
$\forall \xi \in B^*$
the scalar valued function on
$B^*$,
$$
<\xi,da>:B^*\longrightarrow \Bbb R,\ x\mapsto <\xi,d_xa>,
$$
where
$<\cdot,\cdot>$
is   the   pairing between
$B^*$ and
$B$,
is of   class
$*$--$C^{k-1}$
(definition by induction).

Finally, we say that  a {\sl differentiable mapping of class
$C^k$
(in the usual sense),
$\Phi:B_1^*\longrightarrow B_2^*$,
is of class
$*$--$C^k$,}
if for every  scalar--valued function
$a$
of class
$*$--$C^k$
on
$B_2^*$,
the composition
$$
a\circ\Phi:B_1^*\overset\Phi\to\longrightarrow B_2^*\overset a
\to\longrightarrow\Bbb R
$$
is of class
$*$--$C^k$
on
$B^*_1$.
We shall also say that
$\Phi$
{\sl is
$*$%
--differentiable of class
$C^k$.}
Evidently, the composition of two
$*$%
--differentiable mappings of  class
$C^k$
is again a
$*$%
--differentiable mapping of class
$C^k$.
The chain rule for differentiation implies that the mapping
$\Phi$ is of class
$*$--$C^k$,
if  every composition
$$
\eta\circ\Phi:B^*_1\overset\Phi\to \longrightarrow B^*_2
\overset \eta\to \longrightarrow\Bbb R,
\ x\mapsto <\eta,\Phi(x)>,\ \forall\eta\in B_2,
$$
is a scalar--valued function of class
$*$--$C^k$
on
$B^*_1$.

A Banach manifold, modeled on a space
$B^*$,
is said to be
$*$%
--{\sl smooth of class}
$C^k$,
or {\sl  of class}
$*$--$C^k$,
if the mappings
$B^*\longrightarrow B^*$,
induced by the overlapping neighborhoods of corresponding atlases,
are of class
$*$--$C^k$.
Between two such manifolds, mappings of class
$*$--$C^k$
are well defined in an obvious way.  For a fixed
$k$,
all
$*$--$C^k$
manifolds and all
$*$--$C^k$
mappings between them  form a category.  Cotangent spaces in this  category
consist,  {\sl   by   definition,} of  differentials of
$*$--smooth
scalar valued functions at corresponding points, hence they are isomorphic to
$B$,
{\sl not to}
$B^{**}$.
Tangent spaces are isomorphic to
$B^*$.

Let
$U$
be a
$*$%
--smooth manifold modeled on a Banach space
$B^*$,
$f:U\longrightarrow M$
be a
$*$%
--smooth mapping into an $n$--dimensional manifold
$M$
of appropriate class.
A pair
$$
(u,\lambda),\ \ u\in U,\  \lambda\in \left(T^*_{f(u)}M\right)\setminus\{0\},
$$
will be called a {\sl Lagrangian point of}
$f$
if
$
\lambda f'_u=0,
$
where
$\lambda  f'_u$
is the composition of the differential
$f'_u:T_uU\longrightarrow T_{f(u)}M$
with the linear functional
$\lambda:T_{f(u)}U\longrightarrow\Bbb R$,
i.e., is the image of
$\lambda$
under the adjoint
$f^{'*}_u$,
$$\lambda f'_u\overset\text{\rm def}\to=\lambda\circ f'_u=
f^{'*}_u\lambda
:T_uU \longrightarrow\Bbb R.
$$
Thus for an arbitrary pair
$(u,\lambda),\ u\in U,\ \lambda\in T^*_{f(u)}M$,
we have
$$
\lambda f'_u\in T^*_uU\approx B,
$$
and, if
$(u,\lambda)$
is a Lagrangian point,
$\lambda f'_u$
is the zero element of the fibre
$T^*_uU$.
The first component
$u$
of every Lagrangian point
$(u,\lambda)$
of  $f$
is a critical point of
$f$,
the second component
$\lambda$
is a {\sl Lagrange multiplier}
associated with the critical point
$u$.

Consider the induced bundle
$f^*(T^*M)$
over
$U$
defined by the mapping
$f:U\longrightarrow M $
and the cotangent bundle
$T^*M$,
$$
f^*(T^*M)=\left\{(u,\lambda)\Bigr|\ u\in U,\, \lambda\in
T^*_{f(u)}M\right\} =\mathop\bigcup\limits_{u\in U}T^*_{f(u)}M.
$$
Besides the canonical projection
$(u,\lambda)\mapsto u$,
it defines canonically two additional mappings
$$
f^{'*}:(u,\lambda)\mapsto\lambda f'_u,\ \ \varphi:(u,\lambda)\mapsto\lambda,
$$
represented in  the diagram
$$
\matrix
{}&f^*(T^*M)&{}\\
{}&\overset f^{'*}\to\swarrow\overset \varphi\to \searrow&{}\\
{}&T^*U\ \ \ \ \ T^*M&{}
\endmatrix.
$$
Identifying
$U$
with the trivial section of
$T^*U,\ U\subset T^*U$,   we can assert that the set of the
Lagrangian points of
$f$
is identical with the preimage of
$U$
under the mapping
$f^{'*}$
less the trivial section of
$f^*(T^*M)$.
Furthermore,
every tangent space to
$T^*U$
at an arbitrary point of the trivial section is canonically
represented as a direct  sum of its {\sl horizontal and vertical subspaces,}
$$
T_uT^*U=T_uU\oplus T^*_uU,\ \forall u\in U\subset T^*U.
$$
Denote  by
$$
Ver_u:T_uT^*U\longrightarrow T^*_uU,\  u\in U\subset T^*U,
$$
the projector onto the vertical  subspace.  Let
$(u,\lambda)$
be a Lagrangian point of
$f$.
The linear mapping
$$
f''_{(u,\lambda)}\overset\text{\rm def}\to=Ver_u\circ(f^{'*})'_
{(u,\lambda)}:T_{(u,\lambda)}f^*(T^*M)\overset(f^{'*})'_{(u,\lambda)}\to
\longrightarrow T_uT^*U\overset Ver_u\to\longrightarrow T^*_uU
$$
contains a complete information about the second order approximation
of
$f$
at
$u$.

Note that
$$
ker\,\varphi'_{(u,\lambda)}\approx ker\,f'_u\subset T_uU.
$$
Thus
$
f''_{(u,\lambda)}\Bigr|_{ker\,\varphi'_{(u,\lambda)}}
$
is a well--defined mapping from
$ker\,f'_u$
to
$T^*_uU$.
We call this mapping the {\sl second derivative of
$f$
at the Lagrangian point}
$(u,\lambda)$
and denote in  the sequel by
$$
D^2_{(u,\lambda)}f:ker\,f'_u\longrightarrow T^*_uU.
$$
This definition needs some clarification.  At the first site,
the natural choice for the second derivative is the mapping
$(f^{'*})'_{(u,\lambda)}$
defined at all points
$(u,\lambda)\in f^*(T^*M)$.
But such a definition would be completely useless, since, by
virtue  of the implicit function theorem,
in some neighborhood of every regular point, local coordinates
could be introduced in which
$f$
is linear, hence it is senseless to consider in such points
second derivatives. Concerning the horizontal component of the
mapping
$(f^{'*})'_{(u,\lambda)}$
at a Lagrangian point, it is easy to see that it coincides with
the differential of the canonical projection
$(u,\lambda)\mapsto u$
and has no connections with the differential properties
of the mapping
$f$.

Define, finally,
$$
\Cal L_{(u,\lambda)}(f)=\varphi'_{(u,\lambda)}(ker\,f''_{(u,\lambda)})
\subset T_\lambda(T^*M).
$$

The choice of   arbitrary local coordinates in
$U$
leads to the representation
$$
T_{(u,\lambda)}f^*(T^*M)=T_\lambda(T^*_{f(u)}M)\oplus T_uU=T_\lambda(
T^*_{f(u)}M)\oplus B^*.
$$
If   local coordinates are introduced in
$M$
as well, then we obtain the representation
$$
\lambda=(p,f(u)),\ p\in\Bbb R^{n*},\ T^*_{f(u)}M=\Bbb R^{n*},\
T_{(\lambda,u)}f^*(T^*M)=\Bbb R^{n*}\oplus B^*.
$$
The mappings
$\varphi'_u$
and
$f''_{(u,\lambda)}$
take the form
$$
\varphi'_u:(\xi,v)\mapsto(\xi,f'_uv),\ \xi\in \Bbb R^{n*},\,v\in B^*;\
f''_{(u,\lambda)}:(\xi,v)\mapsto\xi f'_u+pf''_uv,\eqno(1.1)
$$
where
$pf''_u$
is the second derivative at
$u$
of the
$*$%
--smooth function
$pf:B^*\longrightarrow \Bbb R$.
Thus the linear mapping
$pf''_u:B^*\longrightarrow B$
is symmetric (selfadjoint).  Denote
$$
F=\left\{\xi f'_u\Bigr|\,\xi \in \Bbb R^{n*}\right\},\
E=\left\{pf''_uv\Bigr|\, v\in B^*\right\}.
$$
Note that
$F$
is the image of the tangent space to the fibre of the bundle
$f^*(T^*M)$
under the mapping
$f''_{(u,\lambda)}$.
At the same time the subspace
$E$
depends on the choice of the local coordinates in
$U$.
Denote
$$
\delta_{(u,\lambda)}=dim(F\cap\overline E)-dim
(F\cap E).
$$
The following Proposition shows that the number
$ \delta_{(u,\lambda)}$
is independent on the coordinate choice.

\

\noindent
{\bf Proposition 1.1.}\ {\sl
$\Cal L_{(u,\lambda)}(f)$
is an isotropic subspace of dimension
$(n-\delta_{(u,\lambda)}f)$
in the
($2n$%
--dimensional)  tangent space
$T_\lambda(T^*M)$
to the cotangent bundle
$T^*M$
with the natural symplectic structure.}

\


\noindent
\underbar{Proof}.\ First we proof the isotropy.  The choice of   local
coordinates in
$M$
identifies
$T_\lambda(T^*M)$
with
$\Bbb R^{n*}\oplus\Bbb R^n$,
and the canonical symplectic form
$\sigma$
takes the form
$$
\sigma\left((\xi_1,\eta_1),(\xi_2,\eta_2)\right)=\xi_1\eta_2-\xi_2\eta_1,
\ \xi_i\in \Bbb R^{n*},\,\eta_i\in \Bbb R^n.
$$
We must prove the implication
$$
(\xi_i,\eta_i)\in \Cal L_\lambda(f)\Longrightarrow \xi_1\eta_2=\xi_2\eta_1.
$$
We have
$\eta_i=f'_uv_i$,
where
$v_i\in B^*,\ \xi_if'_u+pf''_uv_i=0$.
Hence
$$
\xi_1\eta_2=\xi_1f'_uv_2=-<v_2,pf''_uv_1>,
$$
and the identity to be proved follows from the symmetry of the operator
$pf''_u$.

Now we turn to the dimension of
$\Cal L_{(u,\lambda)}(f)$.
Formula (1.1) and the symmetry of
$pf''_u$
imply that
$$
ker\,f''_{(u,\lambda)}\approx coker\, \,f'_u\oplus E\cap F\oplus E^\bot,\
E^\bot=ker\,pf''_u=ker\,f''_{(u,\lambda)}\cap B^*.
$$
Furthermore,
$ker\,f'_u=F^\bot$.
Hence
$$
\Cal L_{(u,\lambda)}(f)\approx coker\,f'_u\oplus E\cap F\oplus E^\bot /E^\bot
\cap F^\bot.
$$
Thus
$$
Dim\,\Cal L_{(u,\lambda)}(f)=corank\,f'_u+dim\,F-
\delta_{(u,\lambda)}f=
corank\,f'_u+rank\,f'_u-
\delta_{(u,\lambda)}f.
$$

If
$dim\,\Cal L_{(u,\lambda)}(f)=n$,
then
$\Cal L_{(u,\lambda)}(f)$
is a Lagrangian subspace in
$T_\lambda(T^*M)$.
In this case we call
$\Cal L_{(u,\lambda)}$
the {\sl Lagrangian derivative } of
$f$,
or {\sl $\Cal L$--derivative},
at the Lagrangian point
$(u,\lambda)$.


The $\Cal L$--derivative, i.e., the image
$im\,\varphi'_{(u,\lambda)}\Bigr|_{ker\,f''_{(u,\lambda)}}$,
could be considered as a dual object to the second derivative.
Note that
$$
ker\,D^2_{(u,\lambda)}f=ker\,\varphi'_{(u,\lambda)}\cap
ker\,f''_{(u,\lambda)}.
$$
In coordinates we obtain
$$
D^2_{(u,\lambda)}f:v\mapsto pf''_uv,\ v\in ker\,f'_u.
$$
The following assertion is a result of direct calculations.

\

\noindent
{\bf Lemma.}\ {\sl The relations (1), (2) below are equivalent:}
$$
ker\, D^2_{(u,\lambda)}f=0\tag1
$$
$$
\overline{imf''_{(u,\lambda)}}=T^*_uU.\tag2
$$
Suppose that
$D^2_{(u,\lambda)}f$
is injective and
$U$
is finite--dimensional. Then the implicit function theorem implies
that the germ at
$(u,\lambda)$
of the set of Lagrangian  points is a germ of a smooth
$n$%
--dimensional manifold.  The restriction of
$\varphi$
on this germ is a Lagrange immersion  into
$T^*M$,
and
$\Cal L_{(u,\lambda)}(f)$
is the tangent space to the obtained germ of a Lagrange
submanifold.

Finally, we call {\sl the Hessian of the mapping}
$f$
at
$(u,\lambda)$
the quadratic form
$$
Hess_{(u,\lambda)}f:ker\,f'_u\longrightarrow \Bbb R,\
Hess_{(u,\lambda)}f(v)=<v,D^2_{(u,\lambda)}fv>.
$$
Negative and positive indices of the quadratic form
$Hess_{(u,\lambda)}f(v)$\footnote"*"{
The negative (positive) index of a quadratic form
$Q$
on a linear space
$B,\ ind_-Q,\,(ind_+Q)$,
is the maximal dimension of subspaces in
$B$
where
$Q$
is negative (positive).},
(which are nonnegative integers or
$+\infty$) are important characteristics of the Lagrangian point
$(u,\lambda)$.
In particular, for the optimization problems they give essential
information about the configuration of the image of a small
neighborhood of the point  under
$f$.
We formulate here only the simplest assertion of this kind. For deeper
results in this respect and particularities cf. [2,5,6].

\

\noindent{\bf Proposition 1.2.}\   {\sl Let
$\widetilde\gamma$
be a nonconstant germ of a smooth curve on
$M$
with the initial point
$f(u)$.
Assume that
$im\,\gamma\cap f(O_u)=f(u)$
for a certain representative
$\gamma$
of the germ
$\widetilde\gamma$
and certain neighborhood
$O_u$
of
$u$
in
$U$.
Then there exists a Lagrangian point
$(u,\lambda)$
such that}
$$
<\lambda,\dot\gamma(0)>\le0 \ \text{\rm \sl and}\
ind_-\,Hess_{(u,\lambda)}f<corank\,f'_u.
$$

\

There is an intimate tie between the indices of the Hessian and the
Lagrangian derivative. Certainly,  the Lagrangian
derivative  at a fixed point could not give any estimates for the
indices of the Hessian in that point, but it is possible to express
the   increments of the indices along a one--parametric family of
Lagrangian points through the Maslov index of the corresponding family
of Lagrangian derivatives. Cf. [7] and \S4 of this paper.

We emphasize that the isotropic subspace
$\Cal L_{(u,\lambda)}(f)$
is called
$\Cal L$%
--derivative only in case when its dimension is $n$, i.e. when it is a
Lagrangian subspace.
This is always the case if
$U$
is finite--dimensional, but by far not always in infinite--dimensional case.
It turns out that if one
of the indices of
$Hess_{(u,\lambda)}f$ is finite then
$\Cal L_{(u,\lambda)}(f)$
could always be extended in a natural way to an
$n$%
--dimensional Lagrangian subspace, which should be called the
$\Cal L$%
--derivative. Below we give a precise formulation of this result.  In this
paper the result is not used, therefore the proof will be given in subsequent
publications.


Let
$(u,\lambda)$
be a Lagrangian point of
$f$
and
$N$
be a germ of a submanifold  in
$U$
at
$u$,
hence
$(u,\lambda)$
is a Lagrangian point of
$f\Bigr|_N$.
If
$N$
is finite--dimensional then
$\Cal L_{(u,\lambda)}(f\Bigr|_N)$
is a Lagrangian subspace. Denote by
$\Cal N$
the set of all such germs partially ordered by inclusion. Then
$\left\{\Cal L_{(u,\lambda)}(f\Bigr|_N)\right\}_{N\in\Cal N}$
is a generalized sequence of points of the Lagrangian Grassmannian
$L(T_\lambda(T^*M))$.

\

\noindent
{\bf Theorem.} {\sl The limit\ \
$\Cal N$-$lim\,\Cal L_{(u,\lambda)}(f\Bigr|_N)$
exists iff one of the indices of
$Hess_{(u,\lambda)}f$
is finite.}

\

This limit is the precise definition of the
$\Cal L$%
--derivative at the Lagrangian point
$(u,\lambda)$.
It contains the isotropic subspace
$\Cal L_{(u,\lambda)}(f)$.
But it is not enough to prove the existence of the limit, we must compute it.
Introducing local coordinates, we can assume  that
$U$
is a Banach space and
$u$
its origin.
Let
$U_0\subset U$
be an arbitrary linear subspace which is dense in
$U$,
and
$\Cal N_0$
be the set of all finite--dimensional subspaces
$\Cal N_0\subset U_0$,
partially ordered by inclusion.  Thus,
$\Cal N_0\subset\Cal N$.
The following assertion is an essential addition to the Theorem, making possible
to
explicitly compute the    limit indicated in the Theorem.

\

\noindent
{\bf Proposition 1.3.}\ {\sl Under the hypothesis of the Theorem  the limit
$$
\Cal N_0\text{-}lim\,\Cal L_{(u,\lambda)}(f\Bigr|_{N_0})=\Cal N\text{-}lim\,\Cal
L_{(u,\lambda)}(f\Bigr|_N). $$
}

\

\noindent
\underbar{Remark}.\  According to the definition of the
$\Cal L$%
--derivative we have
$$
T_\lambda(imf'_u)^\bot\subset\Cal L_{(u,\lambda)}(f).
$$
In particular,
$$
T_\lambda(\Bbb R \lambda)\subset\Cal L_{(u,\lambda)}(f).
$$
Thus the subspace
$\Cal L_{(u,\lambda)}(f)$
contains a one--dimensional subspace which depends only on
$\lambda$
and not on
$f$.
This makes possible to make all our constructions in the
$2(n-1)$%
--dimensional symplectic space
$T_\lambda(\Bbb R \lambda)^\angle/T_\lambda(\Bbb R\lambda)$
and not in the
$2n$%
--dimensional space
$T_\lambda(T^*M)$,
considering
$\Cal L_{(u,\lambda)}(f)$
as a (
$(n-1)$%
--dimensional ) Lagrangian subspace in
$T_\lambda(\Bbb R\lambda)^\angle/T_\lambda(\Bbb R \lambda)$.
Certainly, the same reduction could be described in the language of contact
geometry,
considering not
$T^*M$
but its projectivization
$\Bbb P\,T^*M$,
which possesses the natural structure of a
$(2n-1)$%
--dimensional contact manifold. In a certain sense the contact formulation is
more
natural, though we shall not use it here for the following reasons.  Even
considering
$\Bbb P\,T^*M$
instead of
$T^*M$,
we would be led to consider homogeneous coordinates on projective spaces
$\Bbb P\,T_q^*M$,
thus constantly returning to the same space
$T^*M$.

The following assertion formulates the "chain rule" for
$\Cal L$%
--derivatives. It easily follows from the  definitions and has
many useful consequences.

\

{\bf Proposition 1.4.}\ {\sl  Suppose
$f_i:U\longrightarrow M_i$
are  $*$--smooth mappings,
$u\in U,\ \lambda_i\in T^*_{f_i(u)}M_i$, $i=1,2,3$.
Suppose further that
$(u,(-\lambda_0,\lambda_1))$
is a Lagrangian point for the mapping
$(f_0,f_1):U\longrightarrow M_0\times M_1$,
$(u,(-\lambda_1,\lambda_2))$
is a Lagrangian point for
$(f_1,f_2):U\longrightarrow M_1\times M_2$,
and
$$\aligned
&(-\eta_1,\eta_2)\in\Cal L_{(u,(-\lambda_0,\lambda_1))}(f_0,f_1)\subset
T_{-\lambda_0}(T^*M_0)
\times T_{\lambda_1}(T^*M_1),\\
&(-\eta_0,\eta_1)\in\Cal L_{(u,(-\lambda_1,\lambda_2))}(f_1,f_2)\subset
T_{-\lambda_1}(T^*M_1)
\times T_{\lambda_2}(T^*M_2),
\endaligned
$$
then
$(u,(-\lambda_2,\lambda_0))$
is a Lagrangian point of the mapping
$(f_2,f_0):U\longrightarrow M_2\times M_0$, and
$(-\eta_2,\eta_0)\in\Cal L_{(u,(-\lambda_2,\lambda_0))}(f_2,f_0)$.
}

\

Suppose now that
$M_i=M,\ i=1,2,3,$
and that the projections
$$\aligned
&\pi_i^{(0,1)}:(-\eta_0,\eta_1)\mapsto\eta_i,\ (-\eta_0,\eta_1)\in \Cal
L_{(u,(-\lambda_0,\lambda_1))}(f_0,f_1),\ i=0,1\\
&\pi_j^{(1,2)}:(-\eta_1,\eta_2)\mapsto\eta_j,\ (-\eta_1,\eta_2)\in \Cal
L_{(u,(-\lambda_1,\lambda_2))}(f_1,f_2),\ j=1,2,
\endaligned
$$
are invertible mappings of
$\Cal L_{(u,(-\lambda_0,\lambda_1))}$
on
$T_{\lambda_i}(T^*M),\ i=0,1$,
and of
$\Cal L_{(u,(-\lambda_1,\lambda_2))}$
on
$T_{\lambda_j}(T^*M),\ j=1,2$,
respectively.  Then, Proposition 1.3 implies that the mappings
$$
\pi_k^{(2,0)}:(-\eta_2,\eta_0)\mapsto\eta_k,\ (-\eta_2,\eta_0)\in\Cal
L_{(u,(-\lambda_2,\lambda_0))}
(f_2,f_0),\ k=2,0,
$$
 are also invertible.

 Set
 $$
 \Phi_{ij}=\pi_i^{(i,j)}\circ(\pi_j^{(i,j)})^{-1},\ \Phi_{ji}=\Phi_{ij}^{-1}.
 $$
 The property for the  subspaces
 $\Cal L_{(u,(-\lambda_i,\lambda_j))}(f_i,f_j)$
 to be Lagrangian  is equivalent to the fact that the mappings
 $\Phi_{ij}:T_{\lambda_j}(T^*M)\longrightarrow T_{\lambda_i}(T^*M)$
 are symplectic.  Proposition 1.3 also implies the important composition rule
 $$
\Phi_{20}=\Phi_{21}\circ\Phi_{10}.
$$
 We shall meet  below concrete applications of this rule to   control systems
 considering the boundary--value mappings on admissible curves, cf. \S3.

 The projections
 $\pi^{(0,1)}_i,\ i=0,1$,
 are invertible only if
 $f_0$
 and
 $f_1$
 are submersions at
 $u$.
 The symplectic mapping
 $\Phi_{10}$
 which represents the
 $2n$%
 --dimensional
 $\Cal L$%
 --derivative of the mapping
 $(f_0,f_1)$
 represents also the
 $n$%
 --dimensional
 $\Cal L$%
 --derivative of the mapping
 $f_1\Bigr|_{f_0=const}$,
the restriction of
$f_1$
to the level of
$f_0$
through
$u$.
Direct calculations imply
$\Cal L_{(u,\lambda_1)}(f_1\Bigr|_{f_0=const})=
\Phi_{10}(T_{\lambda_0}(T^*_{f_0(u)}M))$.




\heading{\S2.  Smooth control systems and basic
structures of Differential Geometry}\endheading

{\bf 1. Definition of smooth control systems.}
Suppose a smooth (locally trivial) fibre bundle  over a
smooth  $n$--dimensional manifold
$M$
is given,
$$
pr:\ W\longrightarrow M,\eqno(2.1)
$$
with the typical fibre
$U$,
a smooth $r$--dimensional manifold.
Furthermore, suppose an arbitrary smooth fibrewise
mapping of
$W$
into the tangent bundle
$TM$
is defined over the identity mapping of
$M$,
$$
f:W\longrightarrow TM,\ f(W_x)\subset T_xM\ \forall x\in M;\
W_x=pr^{-1}\{x\}.\eqno(2.2)
$$
We call the data (1.1)--(1.2) a {\sl smooth control system},
(2.1) (or
$W$)
is called the {\sl control space} of the system, the typical
fibre
$U$
is called the {\sl space of control parameters};
$M$
is the {\sl state space}, the cotangent bundle
$T^*M$
is the {\sl phase space} of the system.


{\sl Morphisms} between two control systems
$f_i:W_i\longrightarrow TM_i,\ i=1,2$,
are, by definition, arbitrary commutative diagrams
$$
\matrix
W_1&\overset f_1\to\longrightarrow&TM_1\\
\Phi\downarrow&{}&F_*\downarrow \\
W_2&\overset f_2\to\longrightarrow&TM_2,\\
\endmatrix\eqno(2.3)
$$
where
$\Phi$
is a smooth fibrewise mapping of control
spaces,
$F:M_1\longrightarrow M_2$
is a  diffeomorphism.  We denote the morphism (2.3) by
$(\Phi,F_*)$.
If
$(\Phi',F'_*)$
is a second morphism,
$$
\matrix
W_2&\overset f_2\to\longrightarrow&TM_2\\
\Phi'\downarrow&{}&F'_*\downarrow \\
W_3&\overset f_3\to\longrightarrow&TM_3,\\
\endmatrix
$$
then their composition is defined,
$(\Phi'\circ\Phi,F'_*\circ F_*)$,
which again is a  morphism between
$f_1:W_1\longrightarrow TM_1$,
and
$f_3:W_3\longrightarrow TM_3$.
The identity morphisms are defined in an obvious way.
Thus a {\sl category of smooth control systems} is
introduced.

If
$\Phi$
in (2.3) is a diffeomorphism and
$F_*=id,\ (M_1=M_2)$,
then the   morphism
$(\Phi,id)$
is called a {\sl feedback
transformation}, and the corresponding control systems
are said to be {\sl feedback equivalent.}   Feedback
transformations are smooth fibre
transformations  of the control space over the identity map.   If
$W_1=W_2$,
then the feedback transformations are also called {\sl
guage transformations}.
Two feedback equivalent control systems are equivalent in our
category, hence the given definition of a control system is
''feedback--invariant''.

According to the usual  "state--invariant"   definition, a smooth control
system,
$$
\dot x=f(x,u)\in T_xM,\  (x,u)\in M\times U,\eqno(2.4)
$$
is a family of smooth vector fields on the state manifold
$M$,
indexed by a control parameter
$u\in U$.
The control space is the direct product
$W=M\times U$
with the canonical trivialization,
$$
pr:\ M\times U\longrightarrow M,\ (x,u)\mapsto x,\eqno(2.5)
$$
and the mapping
$W\longrightarrow TM$
is given by
$f$.
Evidently, this definition, though invariant under coordinate
transformations in
$M$,
is not feedback--invariant.

A measurable essentially bounded curve in the control space,\footnote"*"
{A measurable mapping
$w:[0,t]\longrightarrow W$
is said to be essentially bounded if the closure of the image under
$w$
of a subset of
$[0,t]$
of full measure is compact. For simplicity, we shall assume in the
sequel that the closure of the image of
$w$
is compact.}
$$
\xi:[0,t]\longrightarrow W,\eqno(2.6)
$$
is called an {\sl admissible control space trajectory} of the system
(2.1)--(2.2)
if its projection  on
$M$,
$$
x=pr\,\xi:\ [0,t]\longrightarrow M,\eqno(2.7)
$$
is a Lipschitz curve in M satisfying for almost all
$\tau$
the differential equation
$$
{dx\over d\tau}(\tau)=f\circ\xi(\tau),\ 0\le\tau\le t.\eqno(2.8)
$$
The curve
$x(\tau)$
is called an {\sl admissible state space trajectory.}
Admissible control space trajectories will be also called
{\sl admissible controls.}



The set of all measurable essentially
bounded mappings $w$
will be considered as a space with the following natural topology.
Consider a metric
$\rho$,
compatible with the topology of
$W$,
and define the
$\varepsilon$--neighborhood
of a given
$\widetilde w$
by the relation
$$
\Cal O(\varepsilon,\widetilde w)=\left\{w\Bigr|\mathop {ess\,sup}_{0\le\tau\le
t}
\rho(w(\tau),\widetilde w(\tau))=
{\parallel \rho(w,\widetilde w)\parallel}_\infty<\varepsilon\right\}.
$$
Since the closure of the image of
$\tilde w$
is compact, the introduced topology is independent on the choice of the metric.

Denote by
$\Omega^t$
the space of all admissible controls (2.6), (considered as a   subspace of the
space of all measurable essentially bounded curves
$[0,t]\longrightarrow W$).

{\bf Proposition 2.1} {\sl The space
$\Omega^t$
of all admissible controls can be given, in a natural way, the
structure of a
$*$--smooth Banach manifold modeled on the direct product}
$$
\Bbb R^n\times(L^r_1[0,t])^*=\Bbb R^n\times L^r_\infty[0,t],\
n=dim\,M,\, r=dim\,U.
$$


\noindent{\sl Proof.} First we proof that the space of essentially
bounded mappings
$u:[0,t]\longrightarrow U$
is, in a natural way, a
$*$--smooth Banach manifold modeled on
$L^r_\infty[0,t]$.

For this we define coordinate mappings of
$\varepsilon$--neighborhoods of an arbitrary element
$\widetilde u:[0,t]\longrightarrow U$
into
$L^r_\infty[0,t]$, assuming that a Riemannian
metric
$\rho$
is fixed on
$U$.


Denote by
$Q$
the (compact) closure of the image of
$\widetilde u$
and consider a finite cover of
$Q$
by
$\varepsilon$--balls
$B_\varepsilon(p_i),\ i=1,\ldots,s$,
centered at
$p_i\in
Q\subset \mathop\bigcup\limits_{\alpha=1}^sB_\varepsilon(
p_\alpha)=B
$.
Set
$\varepsilon$
small enough to secure the following two conditions.
\roster
\item
The exponential mapping,
$$
exp: TU\longrightarrow U\times U,
$$
is invertible on
$
\bigcup\limits_{p\in Q}(p,B_\varepsilon(p))\subset Q\times B
$,
$$
exp^{-1}:\bigcup\limits_{p\in Q}(p,B_\varepsilon(p))\longrightarrow
\bigcup\limits_{p\in Q}T_pU.
$$
\item
The tangent subbundles
$TB_\varepsilon(p_i)\subset TU,\ i=1,\ldots,s,$
are trivial, with trivializations
$$
\zeta_i:TB_\varepsilon(p_i)\longrightarrow\Bbb R^r.
$$
\endroster
Introduce the (measurable) mapping
$$
\zeta:TB\longrightarrow\Bbb R^r,\ \zeta(z)=\zeta_i(z),\ i=\mathop{min}
\limits_\alpha\left\{\alpha\Bigr|\,z\in TB,\,pr(z)\in
B_\varepsilon(p_\alpha)\right\}.
$$
For every measurable curve
$u(\tau),0\le\tau\le t$,
in the
$\varepsilon$--neighborhood of
$\widetilde u$,
$
u(\tau)\in B_\varepsilon(\widetilde u(\tau))$
$0\le\tau\le t$,
we can define a measurable curve
$v(\tau),\,0\le\tau\le t$,
in
$\Bbb R^r$
according to the correspondence
$$
u(\tau)\mapsto\zeta\circ exp^{-1}(\widetilde u(\tau),u(\tau))=v(\tau)
\in\Bbb R^r,
$$
which is an injection and satisfies the relation
$$
\mathop{ess\,sup}_{0\le\tau\le
t}|v(\tau)|={\parallel v\parallel}_\infty \le\delta(\varepsilon)\rightarrow
0\,(\varepsilon\rightarrow 0).
$$
Conversely, every measurable curve
$v(\tau),\,0\le\tau\le t$,
in
$\Bbb R^r$,
satisfying the relation
$
{\parallel v\parallel}_\infty\le\delta,
$
with
$\delta$
sufficiently small,
could be obtained in this way. Indeed, denote the restriction of
$\zeta$
to
$T_pB,\,p\in Q$,
by
$\zeta_p$.
Then
$$
u(\tau)=exp\circ\zeta^{-1}_{\widetilde
u(\tau)} v(\tau)=(\zeta\circ exp^{-1})^{-1}v(\tau)
\in B_\varepsilon(\widetilde u(\tau)).
$$
From here the assertion is easily deduced.  We now turn to the proof of
the Proposition.

We shall show that there is a natural one--to--one correspondence
between admissible controls, sufficiently close to a given admissible
control
$\widetilde \xi$,
and arbitrary  pairs
$(x(0),u)$,
where
$x(0)$
is sufficiently close to the initial condition
$\widetilde x(0)=pr\,\widetilde\xi(0)$
and
$u:[0,t]\longrightarrow U$
is sufficiently close (in  the $ess\,sup$ topology) to the mapping
$\widetilde u:[0,t]\longrightarrow U$ corresponding to
$\widetilde \xi$.

To define the correspondence consider the projection
$$
\widetilde x(\tau)=pr\,\widetilde\xi(\tau),\ \tau\in[0,t],
$$
and take a "tubular"
$\varepsilon$--neighborhood of
$\widetilde x$,
$$
T:[0,t]\times B_\varepsilon\longrightarrow M,\ T(\tau,0)=\widetilde
x(\tau),\ \tau\in[0,t],\eqno(2.9)
$$
where
$B_\varepsilon\subset \Bbb R^n$
is an $\varepsilon$--ball centered at the origin,
$T$
is a diffeomorphism for every fixed
$\tau$,
Lipschitz in
$\tau$,
$$
T(\tau,\cdot)=T_\tau:B_\varepsilon\longrightarrow M; \ x=T_\tau (q),\
q=T^{-1}_\tau (x),\ q\in B_\varepsilon,x\in T_\tau(B_\varepsilon).
$$
Since
$[0,t]\times B_\varepsilon$
is contractible, the induced bundle
$T^*(W)$
is trivial,
$$
T^*(W)=\mathop
\bigcup_{(\tau,q)}\left((\tau,q),W_{T(\tau,q)}\right)\approx\left(
[0,t]\times
B_\varepsilon\right)\times U,\ \ (\tau,q)\in[0,t]\times B_\varepsilon.
$$
Every trivialization
$$
\vartheta:\left([0,t]\times B_\varepsilon\right)\times U\longrightarrow
T^*(W) $$
generates a continuous family of diffeomorphisms
$$
\vartheta(\tau,q):U\longrightarrow W_{T(\tau,q)},\
\tau\in[0,t],q\in B_\varepsilon,u\in U,
$$
smooth in
$q$
and Lipschitz in
$\tau$.
Introduce the mapping
$$
(\tau,x,u)\mapsto f_\tau(x,u)\overset\text{\rm
def}\to=f\circ\vartheta(\tau,T^{-1}_\tau (x))(u),\ \ \tau\in[0,t],
\,x\in T_\tau B,\,u\in U,
$$
which is smooth in
$x,u$
and Lipschitz in
$\tau$.

Every admissible state space trajectory
$x(\tau)=pr\,\xi(\tau),\ 0\le\tau\le t$,
in the tubular neighborhood (2.9) of
$\widetilde x$,
$x(\tau)\in T(\tau,B_\varepsilon)$,
is a solution of the equation
$$
{dx\over d\tau}=f_\tau(x,u(\tau)),\ 0\le \tau\le t,\eqno(2.10)
$$
where
$u(\tau)$
is uniquely defined on
$[0,t]$,
(up to  a set of measure zero), by the relation
$$
u(\tau)=\vartheta^{-1}(\tau,T^{-1}_\tau (x(\tau)))(\xi(\tau)),\ 0\le\tau \le
t.\eqno(2.11)
$$
For any preassigned
$\delta>0$,
all sufficiently small
$\varepsilon>0$,
and all admissible controls
$\xi$, sufficiently close (in the $ess\,sup$ topology) to
$\widetilde\xi$,
the inequality
${\parallel\rho(u,\widetilde u)\parallel}_\infty<\delta$
holds, where
$\widetilde u$
corresponds to
$\widetilde \xi$
according to (2.11).
Conversely, for every
$u(\tau)$,
satisfying the last inequality for a sufficiently small
$\delta$,
there exists a solution
$x(\tau),\,0\le\tau\le t$,
of the equation (2.10) with the initial condition
$| x(0)-\widetilde x(0)|<\delta$.
This solution is an admissible state space trajectory corresponding to
the admissible control space trajectory
$\xi$,
obtained by inverting the  relation (2.11),
$$
\xi(\tau)=\vartheta(\tau,T^{-1}_\tau (x (\tau)))(u(\tau)),\
0\le\tau\le t.
$$
$\xi$
is arbitrarily close to
$\widetilde\xi$
for sufficiently small
$\delta>0$.
Thus, every pair
$$
(x(0),u),\ u:[0,t]\longrightarrow U,\ {\parallel\rho(u,\widetilde
u)\parallel}_\infty<\delta,\ | x(0)-\widetilde x(0)|<\delta, $$
uniquely defines, for sufficiently small
$\delta$,
an admissible control in a certain neighborhood of
$\widetilde \xi$,
and all such controls could be obtained in this way. This proves the
Proposition.

Definition of smooth control systems introduced here is general
enough to include as special cases basic
differen\-tial--geometric structures.  Below we give several
examples.

An extensive and important class of control systems is defined
by (locally trivial) smooth subbundles of the tangent bundle
$TM$,
considered as control spaces
$W$,
and the corresponding inclusion maps
$$
f:W\subset TM.
$$
Many standard geometric structures are reduced to such
systems, the structure type depending on the choice of the
typical fibre
$U$
of the control space which, in this case, is a submanifold of
$\Bbb R^n$,
$$
U\subset \Bbb R^n,\ n=dim\,M.
$$
\roster
\item
 $U$
is an ellipsoid with   center at the origin.  We obtain the
Riemannian geometry. Admissible trajectories in the state space
are arbitrary Lipschitz curves
$x(t)$
of length
$t$,
parametrized by the arc length.
\item
U is a strongly convex body in
$\Bbb R^n$,
symmetric with respect to the origin.  This is the case of the
Finsler geometry.
\item
$U$
is a hyperboloid,  symmetric with respect to the origin --- the
case of pseudo--Riemannian geometry.
\item
$U$
is a linear subspace in
$\Bbb R^n$
of an arbitrary dimension. We come to the theory of
distributions (in the differential--geometric sense).
Admissible curves are the integral curves of the
distribution.
\item
$U$
is the intersection of an ellipsoid   centered at the origin
with a linear subspace. We obtain the sub--Riemannian
geometry.
\item
The ''affine'' versions (i.e. translates with respect to the
origin) of the structures (1)--(5). Though not very popular in
geometry, they are of utmost importance in applications to
Mechanics and Mathematical Physics.
\endroster


Examples (1)--(6)  could be generalized  in the following way.
Suppose the control space (2.1) is a (locally trivial) subbundle
of an arbitrary vector bundle over
$M$,
$E\longrightarrow M$,
with the typical fibre coinciding with one of the
above mentioned types (1)--(6), and
$f$
in (2.2) is the restriction on
$W$
of a certain fibrewise mapping
$E\longrightarrow TM$,
linear on fibres; in the examples considered
$f$
was an embedding.  These broader classes  of systems include
''singular'' versions of geometric structures (1)--(6) with
degenerations at certain points.

\

{\bf 2. The boundary--value mapping.} The mapping
$$
F_\tau: \Omega^t\longrightarrow M,\  F_\tau(\xi)=pr\circ\xi(\tau),
$$
which evaluates the admissible state--space trajectory
$x=pr\circ \xi$
at the moment
$\tau$,
is a $*$--smooth submersion for
$\forall\tau\in[0,t]$.
At the same time, the {\sl   boundary--value mapping}
$$
(F_0,F_t):\Omega^t\longrightarrow M\times M,\ (F_0,F_t)(\xi)=
(F_0(\xi),F_t(\xi))
$$
is, in general,  not a submersion. Critical points of boundary--value
mappings are called  {\sl extremal controls} of the control system.

Denote by
$\Omega^t_{x_0}$
the set of admissible controls
$\xi\in\Omega^t$
subject to the condition
$x(0)=pr\circ \xi(0)=x_0$.
In other words,
$\Omega^t_{x_0}$
is the level set
over the point
$x_0\in M$
of the submersion
$F_0$.
Evidently,
$\Omega^t_{x_0}$
is a $*$--smooth Banach manifold modeled on
$L^r_\infty[0,t]$.

We introduce the {\sl endpoint  mapping}
$$
F_{0t}=F_t\Bigr|_{\Omega^t_{x_0}}:\Omega^t_{x_0}\longrightarrow M.
$$
Critical points of
$F_{0t}$
are exactly the extremal controls in
$\Omega^t_{x_0}$.


Let
$\sigma$
be   the natural symplectic form on
$T^*M$
and put
$$
H(\lambda,z)=\lambda f(z)\overset\text{\rm def}\to=\lambda\circ f(z),\ z\in
W,\,\lambda\in T^*_{pr(z)}M.
$$
$H$
is a smooth function on the direct product of the fibred manifolds
$W,\,T^*M$
over
$M$.
We call it the {\sl Hamiltonian of the control system  (2.1)--(2.2).}

\

\noindent{\bf Proposition 2.2.}  {\sl The triple
$$
(\xi,(-\lambda_0,\lambda_t)),\ \xi\in\Omega^t,\
\lambda_0\in T^*_{pr\,\xi(0)}M,\ \lambda_t\in T^*_{pr\,\xi(t)}M
$$
is a Lagrangian point of the boundary value mapping
$(F_0,F_t)$
iff there exists a Lipschitz curve
$$
\tau\mapsto\lambda_\tau\in T^*_{pr\, \xi(\tau)}M,\ \tau\in[0,t],
$$
satisfying the condition
$$
j^*_2\left({d\lambda_\tau\over d\tau}\rfloor\sigma\right)=-dH,
$$
where
$j_1,j_2$
are the natural projections,
}
$$
\matrix
W& \times&T^*M\\
\  &  M & \\
\overset j_1\to\swarrow&\ &\overset j_2\to\searrow\\
W& &T^*M.
\endmatrix
$$

\

\noindent {\sl Proof.} If the curve
$\tau\mapsto\lambda_\tau$
exists then it is unique (for a given Lagrangian point).  Indeed,
according to Proposition 2.1, (see (2.10)), for an appropriate trivialization of
$W$
along the trajectory
$pr\,\xi$
the admissible control
$\xi$
is represented as
$
\tau\mapsto(\widetilde u(\tau),\widetilde x(\tau))\in U\times M,
$
and the controls close to
$\xi$
are exactly the solutions of the equation
$$
{dx\over d\tau}=f_\tau(x,u(\tau)),\ 0\le\tau\le t,\ u\in U,\ x\in M,
$$
close to
$(\widetilde u(\tau),\widetilde x(\tau))$.


The Lipschitz curve
$\tau\mapsto\lambda_\tau\in T^*_{\widetilde x(\tau)}M$
satisfies the assumptions of the Proposition 2.2 if and only if it
satisfies the equations
$$
{d\lambda_\tau \over d\tau}=\overset\rightarrow\to
H{_\tau}(\lambda_\tau,\widetilde u(\tau)),\eqno(2.12)
$$
$$
{\partial
H_\tau\over\partial u}(\lambda_\tau,\widetilde u(\tau))=0,\eqno(2.13)
$$
where
$\tau\mapsto\overset\rightarrow\to H_\tau(\lambda,u)$
is the Hamiltonian vector field on
$T^*M$
corresponding to the Hamiltonian
$$
\lambda\mapsto\lambda f_\tau(x,u)=H_\tau(\lambda,u),\ \lambda\in T^*_xM,\ x\in
M.
$$
To prove the     Proposition, we shall compute explicitly the
differential of the mapping
$F_t$,
using the variation formula, cf. Introduction.  We have
$$
\aligned
F_t(x_0,u(\cdot))=&x_0 \,\overset\longrightarrow\to{exp}\int
_0^tf_\tau(\cdot,u(\tau))d\tau=\\
&x_0\,\overset\longrightarrow\to{exp}
\int_0^t\widetilde f_\tau d\tau\circ
\overset\longrightarrow\to{exp}\int_0^t
\overset\longrightarrow\to{exp}\int_t^\tau ad\,\widetilde f_\theta d\theta
(f_\tau(\cdot,u(\tau))-\widetilde {f}_\tau)d\tau,
\endaligned
$$
where
$\widetilde f_\tau(x)=f_\tau(x,\widetilde u(\tau))$.
Thus,
$$
\aligned
&\lambda F'_t\Bigr|_{\widetilde u}(\delta u,\delta x_0)=\int_0^t\lambda
\overset\longrightarrow\to{exp}\int_t^\tau ad\,\widetilde f_\theta
d\theta{\partial f_\tau\over\partial u}\Bigr|_{\widetilde u(\tau)}\delta
u(\tau)d\tau+\lambda(\overset\longrightarrow\to{exp}\int_0^t\widetilde
f_\tau d\tau)_*\delta x_0=\\
&\int_0^t\overset\longrightarrow\to{exp}\int_t^\tau\overset\rightarrow
\to{\widetilde H}_\theta d\theta{\partial H_\tau\over\partial
u}(\lambda,\widetilde u(\tau))\delta u(\tau)d\tau+\lambda(\overset
\longrightarrow\to{exp}\int_0^t\widetilde f_\tau d\tau)_*\delta x_0,
\endaligned
$$
where
$\widetilde H_\theta(\lambda)=H_\theta(\lambda,\widetilde u(\theta))$.
Hence,
$$
\lambda_tF'_t\Bigr|_{\widetilde u}(\delta u,\delta x_0)=\int_0^t{\partial H_\tau
\over\partial u}(\lambda_\tau,\widetilde u(\tau))\delta u(\tau)d\tau+
\lambda_0\delta x_0,
$$
where
$\tau\mapsto \lambda_\tau$
satisfies (2.12).  The equality
$\lambda_tF'_t\Bigr|_{\tilde u}-\lambda_0=0$
implies (2.13).


The curves
$\tau\mapsto(\xi(\tau),\lambda_\tau)$
in Proposition 2.2 will be called {\sl extremals}.










\heading{\S3.  $\Cal L$--Derivatives of Endpoint Mappings.
Regular Extremals}\endheading

We preserve the notations of the previous section. Put
$$
g_\tau(\lambda,u)=\lambda(\overset\longrightarrow\to {exp}\int\limits_t^\tau
ad\,\widetilde f_\theta d\theta(f_\tau(\cdot,u)-\widetilde f_\tau)).
$$
All Lagrangian points
$(\xi',\lambda'_t)$
of the mapping
$F_{0t}$,
sufficiently close to
$(\xi,\lambda_t)$,
are characterized by the condition: there exists a Lipschitz curve
$\tau\mapsto\lambda'_\tau \in T^*_{x'(\tau)}M$
such that
$$
\dot\lambda'_\tau=\overset\rightarrow\to  g_\tau(\lambda'_\tau,u'(\tau)),\
{\partial\
\over\partial u}g_\tau({(\lambda'_\tau,u'(\tau))}=0, \eqno(3.1)
$$
where
$\xi'(\tau)=(x'(\tau),u'(\tau)),\ x'(0)=\widetilde x(t)$.
Note also, that
$g_\tau\Bigr|_{u=\widetilde u(\tau)}\equiv0$.\hfil\break
Linearising (3.1), we obtain

\

\noindent{\bf Proposition 3.1}.\  {\sl The relations
$$
\eta_t\in\Cal L_{(\xi,\lambda_t)}(F_{0t})\subset T_{\lambda_t}(T^*M)
$$
are equivalent to the relations:
$$
\aligned
&\text{\sl there exist curves}\ \  \tau\mapsto\eta_\tau\in
T_{\lambda_t}(T^*M),\
\tau\mapsto v(\tau)
\in T_{\widetilde u(\tau)}U\ \ \text{\rm \sl such that}\\
&\dot\eta_\tau={\partial \ \over\partial  u}\overset\rightarrow\to g_\tau
v(\tau),\  {\partial^2g_\tau\over\partial u^2}(v(\tau),\cdot)=
-<d_{\lambda_\tau}({\partial g_\tau\over\partial u}\Bigr|_{\widetilde u(\tau)}
\cdot),\eta_\tau>, \endaligned
$$
$$
\eta_0\in T_{\lambda_t}(T^*_{\widetilde
x(t)}M),\ \ 0\le\tau\le t.
$$
}
The last two equations can be written in a more symmetric form as
$$
\dot\eta_\tau={\partial\overset\rightarrow\to g_\tau\over\partial u}v(\tau),\
{\partial^2g_\tau\over\partial u^2}(v(\tau),\cdot)
=\sigma({\partial\overset\rightarrow\to g_\tau\over\partial u}\cdot,
\eta_\tau)\eqno(3.2)
$$
In \S1, from general considerations, we derived that
$\Cal L_{(\xi,\lambda_\tau)}(F_{0t})$
is an isotropic space. This follows easily also from (3.2).  Indeed,
$$
\aligned
\sigma(\eta^1_t,\eta^2_t)=&\int\limits_0^t{d\
\over d\tau}\sigma(\eta^1_\tau,\eta^2_\tau)d\tau;
\ \ {d\ \over d\tau}\sigma(\eta^1_\tau,
\eta^2_\tau)=\sigma({\partial\overset\rightarrow\to g_\tau\over\partial
u}v^1(\tau),\eta^2_\tau)+\sigma(\eta^1_\tau,{\partial\overset\rightarrow\to
g_\tau\over\partial u}v^2(\tau))=\\
&{\partial^2g_\tau\over\partial u^2}(v^2(\tau),v^1(\tau))
 -
{\partial^2g_\tau\over\partial u^2}(v^1(\tau),v^2(\tau))=0.
\endaligned
$$
We use further the following notation:
$\dsize{{\partial^2g_\tau\over\partial u^2}(\cdot,\cdot)}$
denotes a symmetric bilinear form on
$\dsize{T_{\widetilde u(\tau)}}U$;
the corresponding quadratic form is denoted by
$\dsize{{\partial^2g_\tau\over\partial u^2}(\cdot)}$.
Furthermore, \linebreak
$\dsize{{\partial^2g_\tau\over\partial u^2}:T_{\widetilde
u(\tau)}}U\longrightarrow  T^*_{\tilde u(\tau)}U$
is a selfadjoint linear mapping.
If it is nondegenerate, then the mapping
$\dsize{\left({\partial^2g_\tau\over\partial u^2}\right)^{-1}:T^*_{\widetilde
u(\tau)}U\longrightarrow T_{\widetilde
u(\tau)}}U$
is defined and is also selfadjoint. The corresponding quadratic form on
$T^*_{\widetilde u(\tau)}U$
is denoted by
$\dsize{\left({\partial^2g_\tau\over\partial u^2}\right)^{-1}(\cdot)}$.
Observe also that the expression
$$
{\partial^k g_\tau\over\partial u^k}\Bigr|_{\lambda_\tau}={\partial^k\over
\partial u^k}\lambda_\tau f_\tau,\ k=1,2,\ldots\eqno(3.3)
$$
is the $k$--th derivative of the function
$\lambda f$,
restricted on the fibre
$W_{\widetilde x(\tau)}$
of the bundle
$W$.
The first derivative has an invariant meaning, independent on the local
trivialization of
$W$
and on the choice of coordinates in the typical fibre. The first derivative
vanishes along the extremals, hence at these  points the second derivative has
an  invariant meaning, which is the Hessian of the function
$u\mapsto\lambda_\tau f_\tau(\widetilde x(\tau),u)$
at the point
$\widetilde u(\tau)$.

An extremal
$\tau\mapsto(\xi(\tau),\lambda_\tau)$
is called {\sl regular} if the quadratic form
$\dsize{{\partial^2\ \over\partial u^2}\lambda_\tau f_\tau\Bigr|_{\widetilde
u(\tau)}}$
is nondegenerate at every
$\tau\in[0,t]$.
The following proposition is an evident consequence of the relations (3.2).

\

\noindent{\bf Proposition 3.2.}\   {\sl Assume
$\tau\mapsto(\xi(\tau),\lambda_\tau)$
is a regular extremal.
Then the relation
$\eta_t\in\Cal L_{(\xi,\lambda_t)}(F_t)$
is equivalent to the requirement that the solution
$\eta_\tau,\ 0 \le\tau\le t$,
of the linear Hamiltonian system on
$T_{\lambda_t}(T^*M)$
with the nonstationary quadratic Hamiltonian
$$
q_\tau(\eta)=-{1\over2}\left({\partial^2g_\tau\over\partial
u^2}\right)^{-1}\left(\sigma({\partial\overset\rightarrow\to g_\tau\over\partial
u}\cdot,\eta)\right), \ \eta\in T_{\lambda_t}(T^*M),
$$
and the "end--condition",
$\eta_t$
satisfies the "initial condition"
$\eta_0\in T_{\lambda_t}(T^*_{\widetilde x(t)}M)$.
}

\

\noindent{\bf Corollary.}\  {\sl If the Lagrangian point
$(\xi,\lambda_t)$
defines a regular extremal, then
$\Cal L_{(\xi,\lambda_t)}(F_{0t})$
is a Lagrangian subspace, hence an
$\Cal L$--derivative of the mapping
$F_{0t}$
at
$(\xi,\lambda_t)$.
}

\

In the sequel only regular extremals will be considered.

Put
$$
\Cal D=\left\{(w,\lambda)\in W\negthinspace \times_M (T^*M)
\Bigr|
\ \ {\partial\ \over\partial u}(\lambda f)\Bigr|_w=0,\
{\partial^2\ \over\partial u^2}(\lambda f)\Bigr|_w\ \text{\rm
is invertible.}\right\}
$$

\

\noindent{\bf Proposition 3.3} \roster
\item
{\sl All regular extremals are contained  in
$\Cal D$.
\item
$\Cal D$ is a smooth submanifold of dimension $2n$.
\item
Through every point of
$\Cal D$
passes a unique \underbar{continuous} regular extremal.}
\endroster

\

\noindent{\sl Proof.} \roster
\item
is evident.
\item
follows from implicit function theorem.
\item
follows from implicit function theorem and the relations (2.12),  (2.13).
\endroster

The mapping
$\varphi:(w,\lambda)\mapsto\lambda,\ (w,\lambda)\in\Cal D,$
is locally one--to--one, and its   image,
$\Cal D_r$,
(possibly empty), is open in
$T^*M$.
For many important problems, which include all examples of \S2,
this mapping is globally
one--to--one, hence is a diffeomorphism.  In this case the smooth
function is defined,
$$
h(\lambda)=\lambda f(\varphi^{-1}(\lambda)),\ \lambda\in\Cal D_r,
$$
which will be called the {\sl master--Hamiltonian} of the
corresponding control system.
Evidently, the restrictions of
$h$
to   fibres
$\Cal D\bigcap T^*_xM$
are positively homogeneous functions of  degree one.
If the master--Hamiltonian exists,
then the regular extremals are exactly the curves
$\tau\mapsto(\varphi^{-1}(\lambda_\tau),\lambda_\tau)$,
where
$\lambda_\tau$
is an arbitrary trajectory  of the Hamiltonian system
$$
\dot\lambda=\overset\rightarrow\to h(\lambda),\ \lambda\in\Cal D_r.
$$
All regular extremals are smooth.

We now describe the domains
$\Cal D_r$ and the  master--Hamiltonians for control
systems (1) --- (6)
enumerated in \S2. The corresponding computations are
straightforward.
For Riemannian   geometry we have
$\Cal D_r=T^*M\setminus M$, and the restriction of the
master--Hamiltonian,
$h\Bigr|_{T_x^*M}$
is the square root of a positive quadratic form on
$T^*_xM$.
For the Finsler structure,
$\Cal D_r$
again coincides with
$T^*M\setminus M$, and
$h\Bigr|_{T^*_xM}$
is the support function to the unit Finsler ball in
$T_xM$,
hence, is convex.
For pseudo--Riemannian geometry of a given signature,
$h\Bigr|_{T^*_xM}$
is the square root of a quadratic form of the same signature,
$\Cal D_r\bigcap T^*_xM$
is the positive cone of the quadratic form.
For a distribution,
$\Cal D_r=\emptyset$.
Finally, in case of sub--Riemannian geometry we have
$\Cal D_r\bigcap T^*_xM=T^*_xM\setminus(span\,W_x)^\perp$,
the master--Hamiltonian
$h\Bigr|_{T^*_xM}$
is the square root of a nonnegative quadratic form, with
kernel equal to
$(span\,W_x)^\perp$.
For the "affine versions" of the considered structures
the domains
$\Cal D_r$
remain unchanged, and to the Hamiltonians scalar functions
are added,   which are linear on  fibres.


Returning to the
$\Cal L$--derivative, we note that, for regular extremals, the
projection of the Lagrangian subspace
$$
\Cal L_{(\xi,(-\lambda_0,\lambda_t))}(F_0,F_t)\subset T_{\lambda_0}
(T^*M)\times T_{\lambda_t}(T^*M)
$$
onto the factors in the right--hand side is one--to--one.  Thus we are
in a situation discussed at the end of \S1.  Hence the symplectic
mappings are defined,
$\Phi_{t_1t_0}:T_{\lambda_0}(T^*M)\longrightarrow T_{\lambda_t}(T^*M)$,
satisfying the conditions
$$
\Phi_{t_2t_0}=\Phi_{t_2t_1}\circ\Phi_{t_1t_0},\
\Cal L_{(\xi,\lambda_t)}(F_{0t})=\Phi_{t0}(T_{\lambda_0}(T^*_{q_0}M)).
$$
It is easily seen that
$\Phi_{t_1t_0}=\left(e^{(t_1-t_0)\overset\rightarrow\to h}\right)_*$,
where
$h$
is the master--Hamiltonian.




\heading
{\S4. Jacobi  Curves}
\endheading
For an arbitrary
$\lambda\in T^*M$
consider the hyperplane
$(\Bbb R\lambda)^\angle$
in the symplectic space
$T_\lambda(T^*M)$\footnote"\dag"{
We identify
$\lambda$
with the corresponding tangent vector to the linear space
$T^*_{\pi(\lambda)}M$.
}
and consider the factor space
$
\Sigma_\lambda=T_\lambda(T^*M)/(\Bbb R\lambda)^\angle,
$
which is a symplectic space of dimension
$2(n-1)$.
Let
$L(\Sigma_\lambda)$
be the corresponding Lagrangian Grassmannian, the manifold of the
Lagrangian
subspaces  in
$\Sigma_\lambda$.
At the same time
$L(\Sigma_\lambda)$
is the manifold of Lagrangian subspaces in
$T_\lambda(T^*M)$
containing
$\lambda$.

Let
$\tau\mapsto (\widetilde\xi(\tau),\lambda_\tau)$
be a regular extremal,
$\widetilde x(\tau)=pr\,\widetilde\xi(\tau), 0\le\tau\le t$.
Denote by
$F_{\tau,t}:\Omega^{t-\tau}_{\widetilde x(\tau)}\longrightarrow M$
the end--point mapping, defined on admissible state space trajectories,
starting at
$\widetilde x(\tau)$.
For every
$\tau\in[0,t]$
the
$\Cal L$--derivative
$\Cal L_{(\xi,\lambda_t)}(F_{\tau,t})\in L(\Sigma_{\lambda_t})$
is defined. Consider the curve
$$
J_{(\xi,\lambda)}:\tau\mapsto\Cal L_{(\xi,\lambda_t)}(F_{\tau,t})
$$
in the Lagrangian Grassmannian
$L(\Sigma_{\lambda_t})$,
which will be called the {\sl Jacobi curve} associated with the regular
extremal
$\tau\mapsto(\xi(\tau),\lambda_\tau)$.

Note that the line
$\Bbb R\lambda_t$
belongs to the kernel of the quadratic forms
$q_\tau$
from Proposition 3.2.   Hence the Hamiltonians
$q_\tau$
are correctly defined  on
$\Sigma_{\lambda_t}$.
Every linear Hamiltonian field on
$\Sigma_{\lambda_t}$
defines a vector field on
$L(\Sigma_{\lambda_t})$
which we  also call Hamiltonian.  From the Proposition 3.2 and the
variation
formula for Hamiltonian systems, cf. Introduction, it follows that
$J_{(\xi,\lambda)}$
is a trajectory of the Hamiltonian system on
$L(\Sigma_{\lambda_t})$, defined by the Hamiltonian
$-(\overset\longrightarrow\to{exp}\int\limits_\tau^t\overset\rightarrow
\to q_0d\theta)q_\tau$.
Furthermore, the relations at the end of \S3 imply
$$
J_{(\xi,\lambda)}(\tau)=\left(e^{(t-t_1)\overset\rightarrow \to h}
\right)_*
J_{(\xi,\lambda_\cdot) |_{[t_0,t_1]}}(\tau),\  0\le
t_0\le\tau\le t_1\le t.\eqno(4.1)
$$
The Jacobi  curve belongs to the Lagrangian Grassmannian, which is a
homogeneous space for the symplectic group. We shall consider two
curves in
a Lagrangian Grassmannian to be {\sl equivalent} if one is transformed
into the
other by a symplectic transformation.  From (4.1) the following basic
assertion follows:  the germ of the Jacobi curve
$J_{(\xi,\lambda)}$
at
$\tau$
is defined, (up to the equivalence),  by the germ at
$\tau$
of the extremal
$(\xi,\lambda_\cdot)$.
To fully appreciate this fact, we should emphasize that the Jacobi curve is a
curve in a special remarkable homogeneous space of the symplectic group,
whereas the extremal belongs to   a smooth manifold  with a completely
incomprehensible group of transformations.

We remind that the tangent vectors to the Lagrangian Grassmannian
at the point
$\Lambda\in L(\Sigma)$
are quadratic forms on
$\Lambda\subset\Sigma$.
Descending from germs of curves to 1--jets, we obtain

\

\noindent{\bf Proposition 4.1.}\ {\sl
$\dsize{{d\ \over d\tau}J_{(\xi,\lambda)}(\tau)}$
is a quadratic form of rank
$dim\,U$
and signature \linebreak
$\dsize{sgn{\partial^2\ \over\partial u^2}\lambda_\tau f_\tau
\Bigr|_{\widetilde
\xi(\tau)}}$.
}

\

Before investigating the germs of Jacobi curves, we remind  an important
result, related to the curve  as a whole, namely, to the indices
 of the Hessian of the end--point mapping.  First of all, the
negative (positive)  index  of
$
Hess_{(\xi,\lambda_t)}F_t
$
is finite if and only if
$\dsize{{\partial^2\ \over\partial u^2}\lambda_\tau
f_\tau\Bigr|_{\widetilde \xi(\tau)}>0\, (<0)}$,
i.e. when the Jacobi curve
$J_{(\xi,\lambda)}$
is monotonically nondecreasing (nonincreasing).  Suppose the finiteness
condition is satisfied, and let
$\overline J_{(\xi,\lambda)}$
be the closed curve in
$L(\Sigma_\lambda)$,
obtained by adding to
$J_{(\xi,\lambda)}$
of an arbitrary nondecreasing (nonincreasing)  simple curve connecting
$J_{(\xi,\lambda)}(t)$
with
$J_{(\xi,\lambda)}(0)$.
\footnote"\dag"
{A curve in a Lagrangian Grassmannian is a family of Lagrangian subspaces
of a symplectic space. The curve is said to be simple if there exists
a Lagrangian subspace, transversal to all subspaces of the family.}
Then,
$$
\pm ind_{\mp}\,Hess_{(\xi,\lambda_t)}F\pm rank\,
F'_{0t}\bigr|_\xi=Ind\,\overline J_{(\xi,\lambda)},
\eqno(4.2)
$$
where
$dim\,U\le rank\, F'_{0t}\Bigr|_\xi\le n-1$.
Here,
$Ind$
is the Maslov index of a closed curve on a Lagrangian Grassmannian, cf.
Introduction.
Details and proofs could be found in [1,4].

Now we turn to the geometry of germs of a curve on the Lagrangian
Grassmannian
$L(\Sigma)$
of a given
$2(n-1)$--dimensional symplectic space
$\Sigma$.

\

\noindent{\bf Lemma.}\ {\sl Take an arbitrary
$\Lambda_0\in L(\Sigma)$.
The set
$$
\Lambda_0^\pitchfork=\left\{\Lambda\in L(\Sigma)\Bigr|\
\Lambda_0\bigcap\Lambda=0\right\}
$$
can be given invariantly the structure of an affine space over the vector space
of linear selfadjoint mappings of
$\Lambda^*_0$
into
$\Lambda_0$.}

\noindent{\sl Proof.}  First, we remark that the set of all
$(n-1)$--dimensional subspaces in
$\Sigma$, transversal to
$\Lambda_0$,
has the structure of an affine space over the space of all linear mappings
from
$\Sigma/\Lambda_0$
into
$\Lambda_0$,
and this affine structure does not depend on the symplectic structure in
$\Sigma$.
Indeed, if
$\Sigma=\Lambda_0\oplus\Delta$,
then the subspace
$\Delta$
intersects every coset
$(z+\Lambda_0)\in\Sigma/\Lambda_0$
exactly at one point.  Define the mapping
$(\Delta_1-\Delta_0):\Sigma/\Lambda_0\longrightarrow\Lambda_0$
by the formula
$(\Delta_1-\Delta_0)(z+\Lambda_0)=\Delta_1\bigcap(z+\Lambda_0)-
\Delta_0\bigcap(z+\Lambda_0)$.
It is easy to see that the introduced operation of difference of two
subspaces defines the desired  affine structure.  Furthermore, the
symplectic structure on
$\Sigma$
defines a nondegenerate pairing between
$\Lambda_0$
and
$\Sigma/\Lambda_0$,
hence we can identify
$\Sigma/\Lambda_0$
with
$\Lambda^*_0$.
Since the subspaces
$\Delta_0,\Delta_1\in \Lambda_0^\pitchfork$
are Lagrangian, their difference,
$(\Delta_1-\Delta_0):\Lambda_0^*\longrightarrow\Lambda_0$
is selfadjoint. By counting dimensions it is easily seen that every
selfadjoint mapping from
$\Lambda_0^*$
into
$\Lambda_0$
is realized as such a difference.

Let
$\tau\mapsto\Lambda(\tau),\ \tau\in[0,t]$,
be a smooth curve in
$L(\Sigma)$.
We call  the curve
$\Lambda(\cdot)$
{\sl regular} if
$\dot\Lambda(\tau)$
is a nondegenerate quadratic form on
$\Lambda(\tau)$
for every
$\tau$.
In this article we restrict to considering only regular curves, postponing
more general cases, (highly important and informative), to further
publications.  The Jacobi curve of a regular extremal is a regular curve
if and only if
$dim\,U=n-1$,
cf. Proposition 4.1.

Let
$\Lambda(\cdot)$
be a regular curve, and consider its germ at an arbitrary point
$t $. We have
$\Lambda(\tau)\in\Lambda(t )^\pitchfork$
for all
$\tau\ne t $, sufficiently close to
$t $.
More precisely,
$\tau\mapsto\Lambda(\tau)$
defines the germ of the curve in the affine space
$\Lambda(t )^\pitchfork$
with a simple pole at
$t $.
We shall give a coordinate representation of this fact.

Let
$\Sigma=\left\{(p,q)\Bigr|\ p,q\in\Bbb R^{(n-1)}\right\}$,
and the symplectic form has the canonical expression
$\sigma((p_1,q_1),(p_2,q_2))$
$=<p_1,q_2>-<p_2,q_1>$.
Without restricting generality, we can assume that
$\Lambda(t )=\left\{(p,0)\Bigr|\ p\in\Bbb R^{(n-1)}\right\}$.
Then for every
$\tau$
close to
$t $,
$\Lambda(\tau)$
is represented as
$\Lambda(\tau)=\left\{(p,S_\tau p)\Bigr|\ p\in\Bbb R^{(n-1)}\right\}$,
where
$S_\tau$
is a smooth family of symmetric
$(n-1)\times(n-1)$--matrices,
$S_{t }=0$.
The regularity of the curve
$\Lambda(\cdot)$
means that
$det\,\dot S_\tau\ne0$.
Every
$\Delta\in\Lambda_0^\pitchfork$
has the form
$\Delta=\left\{(A_\Delta q,q)\Bigr|q\in\Bbb R^{(n-1)}\right\}$,
where
$A_\Delta$
is a
$(n-1)\times(n-1)$ symmetric matrix, and the mapping
$\Delta\mapsto A_\Delta$
from
$\Lambda_t^\pitchfork$
onto the space of symmetric matrices defines coordinates on
$\Lambda_t^\pitchfork$,
compatible with the invariant affine structure.  In these
coordinates the  curve
$\tau\mapsto\Lambda(\tau)\in\Lambda_t^\pitchfork$
has the expression
$$
\tau\mapsto S^{-1}_\tau={1\over \tau-t}\dot S^{-1}_t-{1\over2}\dot S^{-1}_t
\ddot S_t\dot S^{-1}_t-
{(\tau-t)\over3}\left(((2\dot S_t)^{-1}
\ddot S_t)\spdot-((2\dot S_t)^{-1}\ddot S_t)^2\right)\dot S^{-1}_t+
O((\tau-t)^2).\eqno(4.3)
$$
To write down the Laurent series of the curve in the affine space we have
to use some coordinates, but the coefficients of the series have a clear
invariant meaning.  Indeed,  translation of the affine space by a
vector of the corresponding linear space leaves all coefficients of the
series unchanged, with the exception of the free term, to which the
translating vector is added. Thus, all coefficients of the Laurent
series, except the free term, are   elements of the linear space,
and the free term is an element of the affine space. For a regular curve
$\Lambda(\cdot)$
in
$L(\Sigma)$
we obtain
$$
\Lambda(\tau)\equiv{1\over\tau-t}\Lambda_{-1}(t)+\Lambda_0(t)+
\sum\limits_{i=1}^\infty(\tau-t)^i\Lambda_i(t),
$$
where
$\Lambda_0(t)\in\Lambda(t)^\pitchfork$,
and
$\Lambda_{-1}(t),\Lambda_1(t),\Lambda_2(t),\ldots$
are selfadjoint linear mappings from
$\Lambda^*(t)$
into
$\Lambda(t)$.
Note that
$\Lambda_{-1}(t)=\left(\dot\Lambda(t)\right)^{-1}$.
Put
$$
R(t)=-3\Lambda_1(t)\circ\left(\Lambda_{-1}(t)\right)^{-1}=-3\Lambda_1(t)
\circ\dot\Lambda(t).
$$
Then,
$R(t):\Lambda(t)\longrightarrow\Lambda(t)$
is a linear operator, symmetric with respect to the (pseudo)--Euclidean
structure on
$\Lambda(t)$,
defined by the quadratic form
$\dot\Lambda(t)$.
According to (4.3), the operator
$R(t)$
is expressed in coordinates as the Schwarz derivative
$$
R(t)=\left((2\dot S_t)^{-1}\ddot S_t\right)\spdot-\left((2\dot
S_t)^{-1}\ddot S_t\right)^2\eqno(4.4)
$$
The curve
in
$L(\Sigma)$,
$t\mapsto\Lambda_0(t)$,
is called the {\sl derivative curve} of
$\Lambda(\cdot)$.
The operator
$R(t):\Lambda(t)\longrightarrow\Lambda(t)$
is called the  {\sl curvature operator}
of the curve
$\Lambda(\cdot)$
at the point
$t$.
Straightforward calculations in coordinates show that the
derivative curve of a regular curve is smooth, though not necessarily
regular.  We have
$\Sigma=\Lambda(t)\oplus\Lambda_0(t)$.
Hence
$$
\Lambda_0(t)\cong\Sigma/\Lambda(t)={\Lambda(t)}^*,\ \
\Lambda(t)\cong \Sigma/\Lambda_0(t)={\Lambda_0(t)}^*.
$$
Hence,
$\dot\Lambda(t)$
also is a linear mapping from
$\Lambda(t)$
into
$\Lambda_0(t)$,
and
$\dot\Lambda_0(t)$
is a linear mapping from
$\Lambda_0(t)$
into
$\Lambda(t)$.
Their calculation in coordinates leads us to the following important
identity
$$
R(t)=-\dot\Lambda_0(t)\circ\dot\Lambda(t),\eqno(4.5)
$$
which could be considered as another equivalent definition of the
curvature operator.  Since
$\dot\Lambda(t)$
is nondegenerate, we have
$\dot\Lambda_0(t)=0\iff R(t)=0.$
We shall call a regular curve {\sl flat} if it satisfies   one of the
equivalent conditions,
$\dot\Lambda_0(t)\equiv0\iff R(t)\equiv0$.
If
$R(t)\equiv\varkappa\,id,\ \varkappa\in\Bbb R$,
then the curve
$\Lambda(\cdot)$
is said to have a constant curvature
$\varkappa$.

\

\noindent{\bf Proposition 4.2.} {\sl Germs of two regular curves of
constant curvature
$\varkappa$
are equivalent iff the signatures of their velocities are equal.

A regular curve is flat iff its Laurent series in the powers of
$(\tau-t)$
has no positive power terms    for every
$t$.}

\

\noindent{\sl Proof.} Introduce in
$\Sigma$
coordinates in which the symplectic structure has canonical form and
$$
\Sigma=\left\{(p,q)\Bigr|\  p,q\in\Bbb R^{n-1}\right\},\ \
\Lambda(t)=\left\{(p,0)\Bigr|\ p\in\Bbb R^{n-1}\right\},\ \
\Lambda_0(t)=\left\{(0,q)\Bigr|\ q\in\Bbb R^{n-1}\right\}.
$$
Then,
$\Lambda(\tau)=\left\{(p,S_\tau p)\Bigr|\ p\in\Bbb R^{n-1}\right\}$,
where
$S(t)=\ddot S(t)=0$.
If
$\Lambda(\cdot)$
is a curve of constant curvature
$\varkappa$,
then
$$
\left((2\dot S_\tau)^{-1}\ddot S_\tau\right)\spdot=
\left((2\dot S_\tau)^{-1}\ddot S_\tau\right)^2+\varkappa \,id.
$$
Solving the matrix differential equation with the initial condition at
$\tau=t$, we obtain
$$S_\tau=\left\{
\aligned
&|2\varkappa|^{-{1\over2}}tg\left(|2\varkappa|^{1\over2}(\tau-t)\right)
\dot S_t,\,\ \varkappa>0,\\
&(\tau-t)\dot S_t,\ \varkappa=0,\\
&|2\varkappa|^{-{1\over2}}th\left(|2\varkappa|^{1\over2}(\tau-t)\right)
\dot S_t,\ \varkappa<0.
\endaligned\right.\eqno(4.6)
$$
Furthermore, under symplectic transformations, which leave fixed
$\Lambda(t)$
and
$\Lambda_0(t)$,
the matrix
$\dot S_t$
is transformed as the matrix of a quadratic form, and since it is, by
assumption, nondegenerate, the signature is its only invariant.

We give now another equivalent  definitions  of the derivative curve
and the curvature operator, which are more geometric and justify the
choice of the term  "curvature". We shall use a  natural approach to
constructing differential geometry of curves on   arbitrary homogeneous
manifolds. The structure of a homogeneous space, i.e. a transitive action
of a given Lie group, singles out a class of "distinguished" curves
--- the orbits of one--parameter subgroups of the group.  Consider an
arbitrary germ of a smooth curve, and find a "distinguished" curve which
has the same jet of the maximal possible order, as the corresponding jet
of the given jet.  On the space of "distinguished" curves the group
is, in general, nontransitive.  The invariants of the approximating
"distinguished" curves are the most important differential invariants of
the initial germ.  This is how the curvature and torsion appear in
$\Bbb R^3$.  Certainly, every homogeneous space brings its own
specific  features into the general methodology.

One--parameter  subgroups of
$Sp(\Sigma)$
are the flows of linear {\sl stationary} Hamiltonian systems.
They define  the family of distinguished curves on
$L(\Sigma)$.
Elementary calculations imply to the  following

\

\noindent{\bf Proposition 4.3.}\ {\sl Let
$\Lambda(\cdot)$
be a regular curve in
$L(\Sigma)$,
and
$h$
be a quadratic form on
$\Sigma$,
such that the 2--jet of the curve
$\tau\mapsto e^{(\tau-t)\overset\rightarrow\to h}(\Lambda(t))$
coincides with the 2--jet of the curve
$\Lambda(\cdot)$
at
$t$,
and the subspace
$\Lambda(t)_h^\perp=\left\{y\in\Sigma\Bigr|\ h(y,\Lambda(t))=0\right\}$
is Lagrangian.  Then
$\Lambda(t)^\perp_h=\Lambda_0(t)$. If, in addition,  the 3--jet of the
curve
$\tau\mapsto e^{(\tau-t)\overset\rightarrow\to h}\Lambda(t)$
coincides with the 3--jet of the curve
$\Lambda(\cdot)$
at
$t$,
then the form
$h$
is uniquely defined, where
$\dot\Lambda(t)=2h\Bigr|_{\Lambda(t)},\
\dot\Lambda_0(t)=2h\Bigr|_{\Lambda_0(t)}$.
}

\

The curvature operator is defined by the germ of the curve, but it
also enables to make conclusions about global properties of the curve.

The points
$t_0,t_1$ are said to be {\sl conjugate} for the curve
$\Lambda(\cdot)$
if
$\Lambda(t_0)\bigcap \Lambda(t_1)\ne0$;
the number
$dim\left(\Lambda(t_0)\bigcap\right.$
$\left.\Lambda(t_1)\right)$
is  the {\sl multiplicity} of the conjugate pair.

The following Proposition is a direct consequence of elementary
facts of symplectic geometry, and we formulate it as a separate
assertion for the convenience of references.

\

\noindent{\bf Proposition 4.4.}\ {\sl  Let
$\Lambda:[0,T]\longrightarrow L(\Sigma)$
be a smooth curve,
$\dot\Lambda(t)>0\ \forall t,\  \Lambda(0)\bigcap\Lambda(T)=0$,
and
$\overline\Lambda(\cdot)$
is a closed curve obtained from
$\Lambda(\cdot)$
by adding to it of a regular simple nondecreasing curve. Then every
$t\in[0,T]$
is conjugate only to a finite number of points, and
$$
Ind\,\overline\Lambda(\cdot)=n-1+\sum\limits_{0<t<T}dim(\Lambda(0)
\bigcap\Lambda(t))=n-1+\sum\limits_{0<t<T}dim(\Lambda(t)\bigcap
\Lambda(T)).
$$
}
The last assertion of this section is the "comparison theorem", which
estimates the index through the curvature.

\

\noindent{\bf Theorem.}\   {\sl Assume
$\Lambda(\cdot)$
is a smooth curve in
$L(\Sigma)$
and
$\dot\Lambda(t)>0$
for
$\forall  t$.
If
$R(t)\le C\,id$
for some
$C\ge0$
and
$\forall t$,
then
$|t_1-t_0|\ge{\pi\over\sqrt C}$
for every pair of conjugate points
$t_0,t_1$.
In particular, if
$R(t)\le0$,
then there are no conjugate points.

Assume
$tr\,R(t)\ge(n-1)c$
for some
$c>0$
and for
$\forall t$,
then for arbitrary
$t_0\le t$
the interval
$[t,t+{\pi\over\sqrt c}]$
contains a point conjugate to
$t_0$.
Both estimates are exact.
}

\

\noindent{\sl Proof.}\ We start with the case
$R(t)\le0$.
The absence of   conjugate points under this assumption easily follows
from the standard facts about Lagrangian Grassmannian contained, for
example, in [7].  In this case, local coordinates exist from
the standard affine atlas in which
$\Lambda(t_0)$
is represented by the zero matrix, and
$\Lambda_0(t_0)$
--- by a symmetric positive matrix. Let
$S_t$
be the matrix corresponding to
$\Lambda(t)$,
$S^0_t$
be the matrix corresponding to
$\Lambda_0(t)$.
Then
$\dot S_t>0$
and
$det(S^0_t-S_t)\ne0$,
since
$\Lambda(\cdot)$
is monotonically increasing and
$\Lambda_0(t)\bigcap\Lambda(t)=0$.
The operator
$R(t)$
is represented by the matrix
$(S^0_t-S_t)^{-1}\dot S^0_t(S_t^0-S_t)^{-1}\dot S_t$.
Hence,
$\dot S^0_t\le0$.
The given relations hold until
$\Lambda(t)$
and
$\Lambda_0(t)$
remain in the coordinate neighborhood. However, the relations imply
that
$0\le S_t<S^0_t\le S^0_{t_0}$,
hence,
$\Lambda(t)$
and
$\Lambda_0(t)$
do not leave the coordinate neighborhood at all.

Now assume that
$R(t)\le C\,id$.
We shall make use of the following formula, a direct
consequence of (4.4).
Assume
$\varphi:\Bbb R\longrightarrow\Bbb R$
is a smooth function,
$\dot\varphi(\tau)>0$,
and
$R_\varphi(\tau)$
is the curvature operator of the curve
$\tau\mapsto\Lambda(\varphi(\tau))$.
Then
$$
R_\varphi(\tau)=\dot\varphi^2(\tau)R(\varphi(\tau))+
\left({({\ddot\varphi\over2\dot\varphi})\spdot}(t)-({\ddot\varphi
(t)\over2\dot\varphi(t)})^2\right)\,id\eqno(4.7)
$$
Put
$$
\varphi_{\overline t}(\tau)={1\over\sqrt C}\left(arc\,tg(\sqrt C\tau)
+{\pi\over2}\right)+\overline t,\ \
\varphi_{\overline t}(\Bbb R)=(\overline t,\overline t+{\pi\over
\sqrt C}).
$$
We obtain,
$$
R_{\varphi_{\overline t}}(\tau)={1\over(C\tau^2+1)^2}\left(R(\varphi_
{\overline t}(\tau))-C\,id\right)\le0.
$$
Hence the curve
$\tau\mapsto\Lambda(\varphi_{\overline t}(\tau))$
has no conjugate points on the interval
$(\overline t,\overline t+{\pi\over\sqrt C})$.

Assume now that
$tr\,R(t)\ge(n-1)c$.
We shall prove that, if
$\Delta\bigcap\Lambda(\tau)=0$
for  some
$\Delta\in L(\Sigma)$
and
$\forall\tau\in[t,\overline t]$,
then
$\overline t-t<{\pi\over\sqrt c}$.

Indeed, if such a
$\Delta$
exists, then
$\Lambda\Bigr|_{[t,\overline t]}$
is completely contained in a coordinate neighborhood, therefore the
curvature operator
$R(\tau)$
is defined by the formula (4.4).  Put
$Z(\tau)=(2\dot S_\tau)^{-1}\ddot S_\tau$, $z(\tau)=tr\,Z(\tau),\
\tau\in[\tau,t]$.
Then,
$$
\dot Z(\tau)=Z^2(\tau)+R(\tau),\ \ \dot z(\tau)=tr\,Z^2(\tau)+tr\,R(\tau).
$$
Since for an arbitrary symmetric
$(n-1)\times(n-1)$--matrix
$A$
we have
$tr\,A^2\ge{1\over n-1}(tr\,A)^2$,
the inequality
$\dot z\ge\dsize{z^2\over n-1}+(n-1)c$
holds.  Hence,
$z(\tau)\ge x(\tau),\ t\le\tau\le\overline t$,
where
$x(\cdot)$
is a solution of the equation
$$
\dot x={x^2\over n-1}+(n-1)c,
$$
i.e.
$x(\tau)=(n-1)\sqrt c\,tg(\sqrt c(\tau-\tau_0))$.

The function
$z(\cdot)$,
together with
$x(\cdot)$,
are bounded on the interval
$[t,\overline t]$.
Hence,
$\overline t-t<{\pi\over\sqrt c}$.

To verify that the estimates are exact, it is enough to consider  curves
of constant curvature.

Applying the theorem to the Jacobi curve
$J_{(\xi,\lambda)}$,
one can obtain explicit estimates of the index of
$Hess_{(\xi,\lambda_t)}f$
through the curvature of the Jacobi curve in case of a finite index.
Indeed, formula (4.2) and Proposition 4.4 imply the following form   of
the classical Morse formula,
$$
ind\,Hess_{(\xi,\lambda_t)}f=\sum\limits_{0<\tau<t}dim(J_{(\xi,\lambda)}
(\tau)\bigcap J_{(\xi,\lambda)}(t)).
$$
In other words, the more conjugate points, the bigger the index. If there
are no conjugate points at all, then
$Hess_{(\xi,\lambda_t)}f$
is sign--definite.






\heading
{\S5. Canonical Connections of Hamiltonian Systems and of Differential
Equations of Second Order.}
\endheading

\noindent{\bf 1. Nonlinear connections on fibre bundles.}\
Assume a smooth (locally trivial) fibre bundle
$E=\bigcup\limits_{x\in M}E_x$
over
$M$
is given, with the canonical projection
$\pi:E\longrightarrow M$.
In the tangent bundle
$TE$
the {\sl "vertical"}  subbundle is defined,
$$
T^{ver}E=\bigcup\limits_{e\in E}T_eE_{\pi(e)}
\subset TE=\bigcup\limits_{e\in E}T_eE,\ \ ker\,\pi_*=T^{ver}E,
\ im\,\pi_*=TM.
$$
Any direct complement to
$T^{ver}E$
in
$TE$
will be called a {\sl (nonlinear) connection} on
$E$,
vector fields on
$TE$
with values in the
direct complement will be called {\sl horizontal},
vector fields with values in
$T^{ver}E$
will be called {\sl vertical.}


Assume a connection is fixed on
$E$.
Then, for every
$e\in E$,
the restriction
$\pi_*\Bigr|_{T_eE}$
defines a one--to--one mapping of the space of horizontal tangent vectors
at
$e$ onto
$T_{\pi(e)}M$.
Hence there exists a uniquely defined mapping,
$X\mapsto\nabla_X,\  X\in Vect\,M$,
of the space of vector fields on
$M$
into the space of horizontal fields on
$E$,
satisfying the relation
$\pi_*\nabla_X=X\  \forall X\in Vect\,M$.
Evidently, the correspondence
$X\mapsto\nabla_X$
is a
$C^\infty(M)$--linear (~or tensorial) mapping:
$$
\nabla_{X+Y}=\nabla_X+\nabla_Y,\ \nabla_{a X}=(a\circ\pi)\nabla_X
\ \ \forall a\in C^\infty(M).
$$
For every vertical field
$V$,
the commutator
$[\nabla_X,V]$
is a vertical field, and the mapping
$X\mapsto[\nabla_X,V]$
is
$C^\infty(M)$--linear (tensorial).
In particular,
the restriction
$[\nabla_X,V]\Bigr|_{E_x}$
is uniquely defined by
$X(x)$
and
$V$.
To emphasize this remark explicitly, as well as for some technical
reasons which will be clear below, we omit the brackets in the  commutator
$[\nabla_X,V]$,
and call the expression
$\nabla_XV\overset\text{\rm def}\to=[\nabla_X,V]$
the   {\sl covariant derivative} of the (vertical) field
$V$
along
$X$.
For every
$v\in T_xM$
the covariant derivative,
$\nabla_vV\in Vect\,E_x$,
is correctly defined.



Every horizontal vector field
is represented as
$e\mapsto\nabla_{\xi(e)},\ e\in E$,
where
$\xi(e)\in T_{\pi(e)}M$.
The restriction of the mapping
$e\mapsto\xi(e)$
to
$E_x$
is a vector--function with  values in the vector space
$T_xM$.
Since we can act on every smooth vector--function  by an arbitrary vector
field,  defined on the domain of definition of the function,  by
differentiating the vector--function along the corresponding directions,
the action of vertical fields on the mapping
$e\mapsto\xi(e)$
is correctly defined.
The following evident, though very useful, formula  gives the
decomposition of the  commutator of a horizontal and
a vertical field into
the horizontal and vertical components,
$$
[\nabla_\xi,V](e)=(\nabla_{\xi(e)}V-\nabla_{V\xi})(e).\eqno(5.1)
$$
The commutator of vertical fields is vertical, at the same time the
commutator of horizontal fields might not be horizontal. The description of
the vertical component of the commutator of two horizontal vector fields
leads us to the important notion of the curvature of a connection.
Put
$$
R^\nabla(X,Y)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]},\  \forall X,Y\in
Vect\,M.
$$
Evidently, the field
$R^\nabla(X,Y)$
is vertical, and the mapping
$X\wedge Y\mapsto R^\nabla(X,Y)$
is tensorial, (is a homomorphism of a
$C^\infty(M)$--module
$Vect\,M\wedge Vect\,M$ into the module of vertical vector fields).
In particular,
$R^\nabla(X,Y)\Bigr|_{E_x}$
is depending only on
$X(x),Y(x)$,
hence for
$\forall v_1,v_2\in T_xM$
the field
$R^\nabla(v_1,v_2)\in Vect\,E_x$
is correctly defined.  Now let
$e\mapsto\nabla_{\xi_i(e)},\ i=1,2$,
be arbitrary horizontal fields.  It is easy to show that the field
$e\mapsto R^\nabla(\xi_1(e),\xi_2(e))$
is the vertical component of the field
$[\nabla_{\xi_1},\nabla_{\xi_2}]$,
i.e.
$[\nabla_{\xi_1},\nabla_{\xi_2}]-R^\nabla(\xi_1,\xi_2)$
is a horizontal field.

For a nonstationary field
$X_\tau$ on
$M$
the flow
$t\mapsto
\overset\longrightarrow\to{exp}\int\limits_{t_0}^t\nabla_{X_\tau}d\tau
$
consists of fibrewise diffeomorphisms of the bundle
$E$.
Let
$
x(t)=x_0\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
\nabla_{X_\tau}d\tau,
$
hence
$t\mapsto x(t)$
is a trajectory of the flow on
$M$,
defined by the field
$X_\tau$.
Since the mapping
$X\mapsto\nabla_X$
is tensorial, the diffeomorphism
$$
\overset\longrightarrow\to{exp}\int\limits_{t_0}^{t_1}\nabla_{X_\tau}d\tau
\Bigr|_{E_x(t_0)}:E_x(t_0)\longrightarrow E_x(t_1)
$$
depends only on the connection and on  the  curve
$x(\tau),0\le\tau\le t$,
and is independent on the values of the field
$X_\tau$
off the curve
$x(\tau)$.
This diffeomorphism is called the {\sl parallel translation  along the
curve}
$x(\tau),0\le\tau\le t$.

Assume that
$E$
is a linear bundle. The connection is called {\sl linear} if the fields
$\nabla_X$
preserve the space of functions, linear on fibres.  If the connection
is linear, then the parallel translation
$\overset\longrightarrow\to{exp}\int\limits_{t_0}^t
\nabla_{X_\tau}d\tau\Bigr|_{E_{x(t_0)}}
$
is a linear mapping. We shall consider below nonlinear connections on
linear bundles.

So far we were concerned only with main definitions related to
connections, and all assertions were almost trivial. They could be
checked by introducing local coordinates, or algebraically, identifying
vector fields on
$M$
or
$E$
with corresponding derivations of algebras
$C^\infty(M)$
or
$C^\infty(E)$,
vertical fields on
$E$
--- with the annihilator  of the subalgebra  in
$C^\infty(E)$
of functions, constant on fibres.

\

\noindent{\bf 2. Connections associated with Hamiltonians.}\ Assume now
that the bundle
$E$
is not  arbitrary, but rather a region in
$T^*M$,
$E_x=E\cap T^*_xM,\ x\in M$.
Denote by
$h:E\longrightarrow\Bbb R$
a smooth Hamiltonian on
$E$,
by
$Dh$
--- the {\sl vertical differential} of
$h$,
$$
D_\lambda h\overset\text{\rm def}\to=d_\lambda(h\Bigr|_{E_x}),
\ \lambda\in E_x, x\in M.
$$
Thus
$D_\lambda h\in(T^*_xM)^*=T_xM$,
hence
$Dh:E\longrightarrow TM$
is a smooth fibrewise mapping.  Note that
$D_\lambda h=\pi_*\overset\rightarrow\to h(\lambda)$.
We assume that
$D_\lambda h\ne0,\ \lambda \in E$.
Since
$E_x$
is a region in a linear space, the {\sl second vertical derivative} is
also  well defined,
$D^2_\lambda h=d^2_\lambda(h\Bigr|_{E_x})$,
and is a quadratic form on
$T^*M$.

To every
$\lambda$
there corresponds a curve
$J_\lambda$
  in the Lagrangian Grassmannian
$L(T_\lambda(T^*M))$,
according to the formula
$$
J_\lambda(t)=(e^{-t\overset\rightarrow\to
h})_*T_{\lambda_t}(T^*_{\pi(\lambda_t)}M),\ \lambda_t=\lambda e^{t
\overset\rightarrow\to h}.
$$
If
$h$
is a composition of the master--Hamiltonian of a control system with
a  function of a real variable, for example, some power of the
master--Hamiltonian,  then the curve
$$
t\mapsto J_\lambda(t)\bigcap ker\,d_\lambda h\in
L\left({\overset\rightarrow \to h(\lambda)}^\angle/\overset\rightarrow\to
h(\lambda)\right)
$$
is the Jacobi curve of the corresponding geodesic, cf. \S4.
Let
$\pi(\lambda)=x$,
then
$J_\lambda(0)=T_\lambda(E_x)=T^*_xM$.
It is easily seen that
$\dot J_\lambda(0)=D^2_\lambda h$.

As in \S4, we restrict ourself to  regular curves,
$J_\lambda$,
postponing more general cases, (quite important and interesting for
optimal problems), until further publications.  Thus, we suppose that
$D^2_\lambda h$
is a nondegenerate quadratic form on
$T^*_xM$.

\

\noindent\underbar{Remark.} At first sight, the nondegeneracy requirement
might seem too excessive. For example, the master--Hamiltonian, being
positive--definite of   first degree, does not satisfy it. But, if the
master--Hamiltonian is not zero at
$\lambda$
and generates a regular Jacobi curve,
then the square of the master--Hamiltonian satisfies the condition.

To every regular curve
$t\mapsto J_\lambda(t)$
corresponds the derivative curve
$t\mapsto J_{\lambda0}(t)$.
According to its definition, cf. \S4,
$$
J_{\lambda0}\in L(T_\lambda(T^*M)),\ \ T_\lambda(T^*M)=
T_\lambda(E_x)\oplus J_{\lambda0}(0).
$$
Evidently,
$J_{\lambda0}(0)$
is smooth in
$\lambda$.
We call the Lagrangian bundle
$J_{\lambda0}(0),\lambda\in E$,
the {\sl canonical connection on
$E$,
associated with the Hamiltonian}
$h$.

Let
$\nabla_X$
be a horizontal field  for the canonical connection,
$\pi_*\nabla_X=X,\ X\in Vect\,M$.

\

\noindent{\bf Lemma 5.1.}\  {\sl Assume that the restriction
$h\Bigr|_{E_x}$
of the Hamiltonian
$h$
to an arbitrary  fibre
$E_x$
is a positively homogeneous function of degree
$r+1,\ r\ne0,-1$.
Then
$\overset\rightarrow\to h=\nabla_{D_\lambda h},\ \lambda\in E$,
hence the field
$\overset\rightarrow\to h$
is horizontal.}

\

\noindent{\sl Proof.}\  We identify
$\lambda\in E_x\subset T^*_xM$
with the corresponding tangent vector from
$T_\lambda E_x=T^*_xM$.
The homogeneity of the Hamiltonian implies the identity
$$
(s\lambda)e^{t\overset\rightarrow\to h}=s\left(\lambda e^{s^rt
\overset\rightarrow\to h}\right)\ \ \forall s>0,
$$
from which the relations follow,
$
\left(\lambda-rt\overset\rightarrow\to h(\lambda)\right)\in J_\lambda(t)
\subset T_\lambda(T^*M)
$.
Moreover, since the Hamiltonian flow preserves
$h$,
and
${\overset\rightarrow\to h(\lambda)}^\angle=ker\,d_\lambda h$,
we obtain,
$$
J_\lambda(t)=\Bbb R\left(\lambda-rt\overset\rightarrow\to
h(\lambda)\right)
\oplus J_\lambda(t)\bigcap{\overset\rightarrow\to h(\lambda)}^\angle.
\eqno(5.2)
$$
From here we conclude that
${\overset\rightarrow\to h(\lambda)}\in J_{\lambda0}(t)$,
hence
$\overset\rightarrow\to h$
is a horizontal field.  To complete the prove, we remark that
$\pi_*\overset\rightarrow\to h=D\,h$.


Denote by
$R_{J_\lambda}$
the curvature operator of the curve
$\lambda\mapsto J_\lambda(t)$
for
$t=0$.
Since
$J_\lambda(0)=T_\lambda E_x=T_x^*M$,
where
$\lambda\in E_x$,
$R_{J_\lambda}:T^*_xM\longrightarrow T^*_xM$
is a linear operator. By
$R^\nabla$
we denoted the curvature of the canonical connection. Despite of
completely different ways of definition of these two curvatures, they
are intimately connected and the use of the same term "curvature" in both
cases is completely justified.

\

\noindent{\bf Theorem 5.1.}\ {\sl Under the conditions of Lemma 5.1
the following identity holds,}
$$
R_{J_\lambda}l=R^\nabla(D_\lambda h,l\rfloor D^2_\lambda h),\
\forall\lambda\in E_x,\ x\in M,\ l\in T_x^*M.
$$

\

\noindent{\sl Proof.}\  Let
$x_\lambda(t)$
be the projection on
$M$
of the point
$\lambda e^{t\overset\rightarrow \to h}$,
and
$\overline l$
--- a vertical field, which has a restriction to
$E_{x_\lambda(t)}$
coinciding with the parallel translation of the constant field
$l$
on
$E_x$
along the curve
$x_(\cdot)$
for
$\forall t$.
Then,
$$
\nabla_{Dh}\overline l\Bigr|_{\lambda e^{t\overset\rightarrow\to
h}}=0.   \eqno(5.3)
$$
Since the action of
$
\left(e^{-t\overset\rightarrow\to
h}\right)_*
$
on the vector fields coincides with the the action of
$
e^{t\,ad\overset\rightarrow\to
h}
$,
cf. Introduction, we obtain
$
J_\lambda(t)=\left\{\left(e^{t\,ad\overset\rightarrow\to h}
\overline l\right)(\lambda)\Bigr|\ l\in T^*_xM\right\}.
$
Let
$\nabla_{A_tl}$
be the horizontal component of the vector
$
\left(e^{t\,ad\overset\rightarrow\to h}\overline l\right)(\lambda),
$
and
$B_tl$
be its vertical component, so that
$A_t:T^*_xM\longrightarrow T_xM$, $B_t:T^*_xM\longrightarrow T^*_xM$,
are linear mappings, where
$$J_\lambda(t)=\left\{B_tl+\nabla_{A_tl}\Bigr|\ l\in T^*_xM\right\},\
A_0=0,\ B_0=id.
$$
Thus the germ at zero of the curve
$J_\lambda$
is represented by the matrix curve
$t\mapsto S_t=A_tB_t^{-1}$,
$J_\lambda(t)=\left\{l+\nabla_{S_tl}\Bigr|\ l\in T^*_xM\right\}$,
and the curvature operator
$R_{J_\lambda}$
has, according to (4.4), the form
$$
R_{J_\lambda}=\left((2\dot S_0)^{-1}\ddot S_0\right)\spdot-
\left((2\dot S_0)^{-1}\ddot S_0\right)^2.\eqno(5.4)
$$
Formulas (5.1) and (5.3) imply
$$
{d\ \over dt}e^{t\,ad\overset\rightarrow\to h}\overline l(\lambda)=
e^{t\,ad\overset\rightarrow\to h}[\overset\rightarrow\to h,\overline l]
(\lambda)=e^{t\,ad\overset\rightarrow\to h}[\nabla_{Dh},\overline l]
(\lambda)=-\left(e^{t\,ad\overset\rightarrow\to h}\nabla_{
\overline l\rfloor D^2h}\right)(\lambda).
$$
Hence
$\dot A_0l=-\nabla_{l\rfloor D^2_\lambda h},\ \dot B_0=0$.
Furthermore, the derivative curve of
$J_\lambda$
at
$t=0$
has the form, cf. (4.3),
$
J_{\lambda0}=\left\{-{1\over2}\dot S_0^{-1}\ddot S_0\dot S_0^{-1}v
+\nabla_v\Bigr|\ v\in T_xM\right\}
$.
At the same time, according to the definition of the canonical connection,
$J_{\lambda0}$
consists of horizontal vectors. Therefore
$\ddot S_0=0$,
and the formula (5.4) takes the form
$
R_{J_\lambda}={1\over2}\dot S_0^{-1}\dddot S_0.
$
Furthermore, since
$A_0=0$
and
$\ddot S_0=0$,
we have
$\ddot A_0=0$.
Hence the vector field
$$
{d^2\ \over dt^2}e^{t\,ad\overset\rightarrow\to h}\overline l\Bigr|_{t=0}
=-{d\ \over dt}e^{t\,ad\overset\rightarrow\to h}\nabla_{\overline
l\,\rfloor D^2h}=[\nabla_{\overline l\,\rfloor D^2h}, \overset
\rightarrow \to h]=[\nabla_{\overline l\,\rfloor D^2h},\nabla_{Dh}]
$$
is vertical at
$\lambda$.
Since the vertical component of the commutator of two horizontal fields is
the curvature, we have
$$
[\nabla_{\overline l\,\rfloor D^2h},\nabla_{Dh}](\lambda)=R^\nabla(l
\rfloor\,D^2_\lambda h,D_\lambda h).
$$
Furthermore, the point
$\lambda$
is indistinguishable from any other point of the form
$\lambda e^{t\overset\rightarrow\to h}$,
hence the last identity is satisfied for all such points, we have only to
substitute
$l$
by the value of the field
$\overline l$
at
$\lambda e^{t\overset\rightarrow\to h}$.
We obtain,
$$
{d^2\ \over dt^2}e^{t\,ad\overset\rightarrow\to h}\overline l(\lambda)=
e^{t\,ad\overset\rightarrow\to h}R^\nabla(\overline l\,\rfloor D^2h,
Dh)(\lambda).
$$
Thus
$\ddot B_0l=R^\nabla(l\,\rfloor D_\lambda^2h,D_\lambda h)$,
and
$\dddot A_0l$
is the horizontal component of the vector
$$[\overset\rightarrow\to h,R^\nabla(\overline l\,\rfloor D^2h,Dh)]
(\lambda)=[\nabla_{Dh},R^\nabla(\overline l\,\rfloor D^2h,Dh)](\lambda).
$$
According to formula (5.1), we have
$\dddot A_0l=R^\nabla(D_\lambda h,l\,\rfloor D^2_\lambda
h)\,\rfloor D^2_\lambda h$.
Collecting the obtained formulas together, we can write,
$$
\dddot S_0 l=\dddot A_0 l-3\dot A_0\ddot B_0l=2R^\nabla
(l\,\rfloor D^2_\lambda h,D_\lambda h)\rfloor D_\lambda^2 h,\ \
R_{J_\lambda} l={1\over2}\dot A^{-1}_0\dddot S_0 l=R^\nabla (D_\lambda
h,l\,\rfloor D_\lambda^2h).
$$


\

\noindent{\bf 3. Connections associated with second order differential
equations.}\ We have considered above canonical connections associated
with  Hamiltonian systems, a natural class of differential equations
on the cotangent bundle. Now we describe   connections with similar
properties for differential equations of the second order, a natural
class of differential equations on the tangent bundle.

Assume
$E$
is a region in
$TM,\ E_x=E\bigcap T_xM,\ x\in M$.
We shall say that a vector field
$Z$
on
$E$
is a {\sl differential equation of the second order} or that it defines
a differential equation of the second order, if
$\pi_*Z(v)=v\ \forall v\in E$.
For
$s\in\Bbb R$
denote by
$s_*:T(TM)\longrightarrow T(TM)$
the differential of the homothety
$v\mapsto sv,\ v\in TM$.
The field
$Z$,
defining the differential equation of the second order, is called
a {\sl spray} if
$Z(sv)=s_*Z(v)\ \forall v\in E,s\in \Bbb R$
such that
$sv\in E$.
In local coordinates, the differential equation of the second order,
defined by a spray, has the form
$\ddot x=\varphi(x,\dot x)$,
where
$\varphi$
is homogeneous in
$\dot x$
of degree 2. We should note that no  other degrees of homogeneity
are preserved  under the coordinate change on
$M$.


For every differential equation of the second order
$Z$
and every
$ v\in E$
we define a curve
$I_v$
in the Grassmannian
$G_n(T_v(TM))$
of all
$n$--dimensional subspaces in
$T_v(TM))$
by the formula
$$
I_v(t)=\left(e^{-tZ}\right)_*T_{v_t}E_{\pi(v_t)},\ v_t=ve^{tZ}.
$$
Before moving further we shall make few remarks about curves in the
Grassmannian
$G_n(\Bbb R^{2n})$.
So far, we considered curves only in Lagrangian Grassmannians.
Definitions and properties mentioned below are similar to those in the
Lagrangian Grassmannian, and are proved even easier since no additional
symplectic structure should be taken in consideration.

Put
$$
K_0\in G_n(\Bbb R^{2n}),\ K_0^\pitchfork=\left\{K\in G_n(\Bbb
R^{2n})\Bigr|\ K_0\bigcap K=0\right\}.
$$
Then
$K_0^\pitchfork$
has a natural structure, (independent on the choice of a basis in
$\Bbb R^{2n}$),
of an affine space over the vector space
$Hom(\Bbb R^{2n}/K_0,K_0)$.
We already described this affine structure in the proof of the Lemma
in \S4.  Furthermore, the tangent space
$T_{K_0}G_n(\Bbb R^{2n})$
is naturally identified with   the space
$Hom(K_0,\Bbb R^{2n}/K_0)$
in the following way. Assume
$t\mapsto K_t$
is a smooth curve in
$G_n(\Bbb R^{2n})$.
We correspond to the tangent vector
$\dot K_0={d\ \over dt}K_t\Bigr|_{t=0}$
the mapping
$k_0\mapsto {d\ \over dt}k_t\Bigr|_{t=0}+K_0$,
where
$k_t\in K_t$.
It is easy to show that this mapping depends only on
$\dot K_0$ and does not depend on the choice of the curves
$K_t$
and
$k_t$.


The curve
$\tau\mapsto K_\tau$
in
$G_n(\Bbb R^{2n})$
is  {\sl regular} if its velocities
$dK_\tau\over d\tau$
are regular linear mappings from
$K_\tau$
into
$\Bbb R^{2n}/K_\tau\ \forall\tau$.
It is easy to show that the curve
$I_v$
in
$G_n(T_v(TM))$
is nondegenerate.  In particular, for
$\tau=0$
we have
$I_v(0)=I_v(T_xM)=ker\,\pi_*\Bigr|_{T_v(TM)}$,
where
$x=\pi(v)$.
Identifying the spaces
$T_v(T_xM)$
and
$T_v(TM)/T_v(T_xM)\approx \pi_*T_v(TM)$
 with
 $T_xM$, we obtain
 ${d\ \over d\tau}I_v(0)=id$.

 The germ at
 $t$
 of a regular curve
 $\tau\mapsto K(\tau)$
 in
 $G_n(\Bbb R^{2n})$
 defines a curve in the affine space
 $K(t)^\pitchfork$
 with a simple pole at
 $\tau=t$.
 In other words,
 $$
 \aligned
 &K(\tau)\approx {1\over \tau-t}K_{-1}(t)+K_0(t)+\sum\limits_{i=1}^\infty
 (\tau-t)^iK_i(t),\ \
 K_0(t)\in K(t)^\pitchfork,\\
 &K_i(t)\in Hom(\Bbb R^{2n}/K(t),K(t)),\,i\ne0;\  \
 K_{-1}(t)=(\dot K(t))^{-1}.
 \endaligned
 $$
 Put
$R(t)=-3K_1(t)\dot K(t)$.
The curve
$t\mapsto K_0(t)$
in
$G_n(\Bbb R^{2n})$
is called the {\sl derivative curve} of
$K(\cdot)$.
The operator
$R(t):K(t)\longrightarrow K(t)$
is called the {\sl curvature operator} of the curve
$K(\cdot)$
at
$t$.
In local coordinates the curvature operator is represented, as in the
Lagrangian case, by the matrix Schwarz derivative (4.4), with matrices
not necessarily symmetric.

Let
$t\mapsto I_{v0}(t)$
be the derivative curve of
$I_v$.
We have
$T_v(TM)=I_vE_x+I_{v0}(0)$,
and
$I_{v0}(0)$
smoothly depends on
$v$.
The subbundle in
$TE$
with the fibres
$I_{v0}(0),v\in E$,
is called the {\sl canonical connection associated with the field}
$Z$.
Below, in this subsection,
we assume that the symbol
$\nabla_X$
denotes the horizontal field for the defined canonical connection,
such that
$\pi_*\nabla_X=X,\ X\in Vect\,M$.

\

\noindent{\bf Lemma 5.2.}  {\sl If
$Z$
is a spray, then
$Z$
is a horizontal field for the canonical connection,
$Z(v)=\nabla_v\ \forall v\in E$.}

\

\noindent{\sl Proof.}\  We identify the vector
$v\in E_x\subset T_xM$
with the corresponding vertical tangent vector in
$T_vE_x=T_xM$.
Since
$Z$
is a spray, we have
$(sv)e^{tZ}=s(ve^{stZ}),\ s\in\Bbb R$.
From here, we obtain
$$
\left(v-tZ(v)\right)\in I_v(t)\subset T_v(TM).\eqno(5.5)
$$
The subspace
$I_v(t)$
is represented as
$I_v(t)=\left\{l+\nabla_{S_tl}\Bigr|\ l\in T_vE_x=T_xM\right\}$,
where
$S_t:T_xM\longrightarrow T_xM$
is a linear operator smooth in
$t$,
$S_0=0$.
Moreover,
$\dot S_0=id$.

From  (4.3) it follows that the value at
$t=0$
of the derivative curve  of
$I_v$
has the form
$$
I_{v0}=\left\{-{1\over 2}\ddot S_0l+\nabla_l\Bigr|\ l\in T_xM\right\}.
$$
Since
$I_{v0}$
consists of horizontal vectors, we have
$\ddot S_0=0$.
Let
$Z(v)=l_0+\nabla_v$,
i.e.,
$l_0$
is the vertical component of the vector
$Z(v)$.
Then according to (5.5),
$tv=S_t(tl_0-v)$.
Differentiating 2 times in
$t$
yields
$\ddot S_0v=l_0$.
Hence
$l_0=0,\ Z(v)=\nabla_v$.

Denote by
$R_{I_v}$
the curvature operator of the curve
$t\mapsto I_v(t)$
at
$t=0$.
Since
$I_v(0)=T_xM$,
where
$x=\pi(v)$,
the mapping
$R_{I_v}:T_xM\longrightarrow T_xM$
is a linear operator.  The symbol
$R^\nabla$
denotes the curvature of the canonical connection associated with the
field
$Z$.

\

\noindent{\bf Theorem 5.2.} \ {\sl If
$Z$
is a spray, then}
$$
R_{I_v}l=R^\nabla(v,l)\ \ \forall v\in E_x,\,x\in M,\,l\in T_xM.
$$


\

The proof is a repeating of the   proof of  Theorem 5.1
with corresponding simplifications.


 \

 \noindent{\bf 4. Linear Connections.}\  Here we shall consider in more
detail linear connections on the bundles
$TM$
and
$T^*M$.
Assume a linear connection is given on
$TM$,
hence  for every
$X\in Vect\,M$
a horizontal vector field,
$\nabla_X$,
on
$TM$
is given, which preserves the space of functions, linear on the
fibres of
$TM$.
In this case, the diffeomorphism
$e^{-t\nabla_X}$
 is a {\sl linear} mapping of the fibre
 $T_{xe^{tX}}M$
 onto the fibre
 $T_xM$
 for
 $\forall x\in M,\,t\in \Bbb R$.
 Considering the adjoint linear mappings
 $$
 \left(e^{-t\nabla_X}\Bigr|_{T_{xe^tX}M}\right)^*:
 T^*_xM\longrightarrow T^*_{xe^{tX}}M,
 $$
we obtain the {\sl "adjoint"} flow on the bundle
$T^*M$.
The generating vector field for this flow will be denoted by
$\nabla^*_X$.
The mapping
$X\mapsto\nabla^*_X,\ X\in Vect\,M$,
defines a linear connection on
$T^*M$,
the {\sl adjoint} to the connection
$X\mapsto\nabla_X$.
Evidently, we could start with an arbitrary linear connection
on
$T^*M$,
and define the adjoint connection on
$TM$,
obtaining the involution
$\nabla^{**}_X=\nabla_X$.
For appropriately chosen notation the expressions for
$\nabla_X$
and
$\nabla^*_X$
are indistinguishable.

Indeed, every
$Y\in Vect\,M$
could be considered as a
cross--section of the vector bundle
$TM$,
and as such could be identified with the vertical vector field  on
$TM$,
constant on fibres. Hence, the covariant derivative of the field
$Y$
along the field
$X$
is defined, denoted by
$\nabla_XY$,
which is vertical and constant on fibres,
$\nabla_XY \in Vect\,M$.
Furthermore,
$Y$
could be considered as a scalar--valued function on
$T^*M$,
linear on fibres. The image
$\nabla^*_XY$
of this function under the action of the
vector field
$\nabla^*_X\in Vect\,T^*M$
is again linear on fibres, in other words,
$\nabla^*_XY\in Vect\,M$.
It is easily proved that
$$
\nabla_XY=\nabla^*_XY.\eqno(5.6)
$$

 For a linear connection on
 $TM$
 we define in a usual way the {\sl torsion}
 $$
 T^\nabla(X,Y)=\nabla_XY-\nabla_YX-[X,Y],\ T^\nabla(X,Y)\in Vect\,M.
 $$

\

\noindent{\bf Lemma 5.3.}\ {\sl The following identity holds,
$$
T^\nabla(X,Y)=\sigma(\nabla^*_X,\nabla^*_Y),\  X,Y\in Vect\,M
$$
where
$\sigma$
is the canonical symplectic structure on
$T^*M$.}

\

\noindent{\sl Proof.}\  Let
$\theta$
be the canonical 1--form on
$T^*M,\ \sigma=d\,\theta$.
Since
$\pi_*\nabla^*_X=X$
we have
$X=<\theta,\nabla^*_X>$,
where
$X$
in the left--hand side of the last identity is considered as a
function on
$T^*M$,
linear on fibres.  Taking into account that
$\theta$
vanishes on vertical fields, we obtain
$$
\aligned
T^\nabla(X,Y)=\nabla_XY-&\nabla_YX-[X,Y]= \nabla^*_X<\theta,\nabla^*_Y>-
\nabla^*_Y<\theta,\nabla^*_X>-<\theta,\nabla^*_{[X,Y]}>=\\
&\sigma(\nabla^*_X,\nabla^*_Y)+<\theta,R(X,Y)>=\sigma(\nabla^*_X,\nabla
^*_Y).
\endaligned
$$

\

\noindent{\bf Corollary.}\  {\sl Connection
$\nabla$
has a zero torsion iff
$\nabla^*$
defines a Lagrangian subbundle in
$T(T^*M)$.}

Every linear connection on
$TM$
defines a spray
$Z$ according to the formula
$Z(v)=\nabla_v(v)\ \forall v\in TM$.
The trajectories of this spray are called {\sl geodesics of the
connection}
$\nabla$.
Different connections can define identical sprays, but among them there
exists a unique connection with vanishing torsion.
Not every spray can be obtained in this way, only the sprays which are
quadratic on fibres.\footnote"*" {In local coordinates, the differential
equation of the second order defined by a spray has the form
$\ddot x=\varphi(x,\dot x)$,
and the property of
$\varphi$
to be quadratic in
$\dot x$
is independent on the choice of the coordinates in
$M$.
This is a natural class of sprays, since for a spray, quadratic on a
given fibre
$T_{x_0}M$,
the corresponding differential equation of the second order has, in
appropriate coordinates, the form
$\ddot x=\varphi(x,\dot x)$,
where
$\varphi(x_0,\dot x)=0\ \forall \dot x$.}

\

\noindent{\bf Proposition 5.1.}\ {\sl The canonical connection
associated with a spray, quadratic on fibres, is linear and has
a vanishing torsion.}

\

\noindent{\sl Proof.}\ Fix an arbitrary point
$x_0\in M$
and local coordinates
$x=(x^1,\ldots,x^n)$
in the neighborhood of
$x_0$,
such that the corresponding differential equation of the second order
in these coordinates has the form
$\ddot x=\varphi(x,\dot x),\ \varphi (x_0,\dot x)=0\ \forall \dot x$.
We obtain,
$$
I_{(x_0,\dot x)0} =span\left\{{\partial\
\over\partial x^1},\ldots,{\partial\ \over\partial
x^n}\right\}\ \forall \dot x.
$$
In other words, for the canonical connection we have
$\dsize\nabla_{{\partial\
\over\partial x^i}}={\partial\
\over\partial x^i}$,
hence
$\nabla$
is linear  and has a vanishing torsion.

Thus the canonical connection associated with a spray,
quadratic on fibres, is linear with vanishing torsion, and its geodesics
are the trajectories of the spray.  In geometry, the geodesics of a
connection on
$TM$
are considered as the "straightest" lines, whereas the extremals of
a variational problem as the "shortest" lines.  We see that the
trajectories of every spray, not necessarily linear on fibres,
could be considered as a system of " straightest" lines, corresponding to
the canonical connection on
$TM$,
generally not linear.  At the same time, the shortest lines, being the
trajectories of the master--Hamiltonian, generate the canonical
connection on
$T^*M$.

According to the classical Riemannian geometry, the "shortest"
lines for the Riemannian variational problem are also the
"straightest"  lines for the Levi--Civita connection.  Therefore,
it is natural to expect   that the Levi--Civita connection on
$TM$
is conjugate to the canonical connection  on
$T^*M$,
associated with the corresponding Hamiltonian.  Indeed, we have the
following

\

\noindent{\bf Proposition 5.2.}\ {\sl Assume
$Q:TM\longrightarrow T^*M$
is a selfadjoint isomorphism, defined by a pseudo--Riemannian structure
on
$M$.
Then the Levi--Civita connection of this structure is the adjoint
connection to the canonical connection, associated with the Hamiltonian
$h:\lambda\mapsto\dsize{1\over2}<\lambda,Q^{-1}\lambda>, \ \lambda\in
T^*M$.
}

\

\noindent{\sl Proof.}\  The equation
$\dot\lambda=\overset\rightarrow\to h(\lambda)$
defines the pseudo--Riemannian geodesic flow in
$T^*M$,
where
$\pi(\lambda)\spdot=Q^{-1}\lambda$
along every trajectory of this flow.  Hence the isomorphism
$Q^{-1}:T^*M\longrightarrow TM$
transforms the geodesic flow in
$T^*M$
into the geodesic flow in
$TM$,
where the  last flow is defined by  a spray
$Z$,
quadratic on fibres.  Let
$X\mapsto \nabla_X$
be the canonical connection for
$h$.
From the definition, it follows that
$X\mapsto Q_*^{-1}\nabla_X$
is the canonical connection for
$Z$.
Exploiting the fact that the parallel translation generated by the
canonical connection preserves the Hamiltonian,( the canonical
connection is tangent to the levels of the Hamiltonian), we obtain
$Q_*^{-1}\nabla_X=\nabla^*_X$.
From Proposition 5.1 follows now that
$\nabla^*_X$
is the Levi--Civita connection, cf. also the remark after the
Proof of Proposition 5.1.

\


\noindent\underbar{Remark.}\  For   pseudo--Riemannian structure,
the master--Hamiltonian has the form\hfil\break
$\lambda\mapsto<\lambda,Q^{-1}\lambda>^{1\over2}$,
which is defined, in general, not for all
$\lambda\in T^*M$.





\heading
{\S6. Two--Dimensional Control Systems}
\endheading
Consider a standard "state invariant" control  system
$$
\dot x=f(x,u),\ \ x\in M,\ u\in U.
$$
It is interesting to find   explicit expressions  of such a fundamental
"state" and "feedback" invariant as the curvature tensor through
classical "state"--invariants --- linear relations between iterated
Lie brackets of vector fields
$f(\cdot,u),\ u\in U$.
We already have all necessary  means to derive such expressions,
though some efforts are still needed, and the obtained
expressions turn out to be pretty complicated even in the simplest
two--dimensional case. We restrict ourself to the two--dimensional case
only.


Assume that
$dim\,M=2,\ dim\,U=1$,
hence
$U$
is
$\Bbb R$
or
$S^1=\Bbb R/2\pi\Bbb Z$.
Put
$$\Cal O=\left\{(x,u)\in M\times U\Bigr|\ {\partial f\over\partial
u}\wedge{\partial^2f\over\partial u^2}\Bigr|_{(x,u)}\ne0\right\}.
$$
All extremal controls, contained in the region
$\Cal O$,
correspond to regular extremals, and through every point of the region
passes exactly one extremal control, corresponding to a uniquely
determined extremal, (up to a nonzero factor for
$\lambda$).
Assume
$R(x,u),\ (x,u)\in\Cal O$
is the curvature tensor of the uniquely determined germ of extremal,
with the corresponding control through
$(x,u)$.
The tensor is a linear operator, and since it acts in the
one--dimensional space,
$(x,u)\mapsto R(x,u)$
is a real--valued function on
$\Cal O$.

\

\noindent{\bf Proposition 6.1.}\ {\sl Assume
$\alpha_{i_1\ldots i_k},\ \beta_{i_1\ldots i_k}$
are smooth functions on
$\Cal O$,
defined by the formula
$$
\left[{\partial^{i_1}f\over\partial u^{i_1}},
\left[{\partial^{i_2}f\over\partial u^{i_2}},\left[\ldots,
{\partial^{i_k}f\over\partial u^{i_k}}\right]
\ldots\right]\right] =
\alpha_{i_1\ldots i_k}{\partial f\over\partial u}+
\beta_{i_1\ldots i_k}{\partial^2 f\over\partial u^2}.
$$
Then}
$$
\aligned
R=&\alpha_{001}+\beta_{101}-{1\over2}\beta_{002}+{1\over2}\beta_{001}
\beta_3 -{1\over2}\beta_{01}\beta_{12}+{3\over2}\alpha_{01}\beta_{02}-
\alpha_{02}\beta_{01}+\beta_{01}\beta_{03}-\\
&2\alpha_{01}^2+{1\over4}\beta^2_{02}-\alpha_{01}\beta_{01}\beta_3-
{3\over2}\beta_{01}\beta_{02}\beta_3+\alpha_3\beta^2_{01}-{1\over2}
\beta^2_{01}\beta_4+{3\over4}\beta^2_{01}\beta^2_3
\endaligned
$$

\noindent{\sl Proof.} We shall use Proposition 3.2, which expresses
the $\Cal L$--derivative of the endpoint--mapping, hence the Jacobi
curve, through solutions of the linear Hamiltonian system.

Assume
$\tau\mapsto(x_\tau,u_\tau)$
is an extremal control in
$\Cal O$.
Put
$$
g_\tau(\lambda,u)=\lambda\overset\longrightarrow\to{exp}\int
\limits_t^\tau ad\,f(\cdot,u_\theta)d\theta f(\cdot,u)\Bigr|_{x},
\ \ g^{(i)}_\tau(\lambda)={\partial^i\ \over\partial u^i}
g_\tau(\lambda,u_\tau),\
(x,u)\in\Cal O,\ \lambda\in T^*_xM.
$$
 Let
 $\lambda_t\in T^*_{x_t}M$
 be the Lagrange multiplier corresponding to the given extremal control.
 Then
 $g^{(1)}_\tau(\lambda_t)=0$
 identically in
 $\tau$.
 Differentiating the last identity with respect to
 $\tau$
 yields the following useful formula
 $$
 {d\ \over d\tau}u_\tau={\left\{g^{(1)}_\tau,g^{(0)}_\tau\right\}\over
g^{(2)}_\tau}\Bigr|_{\lambda_t},
 $$
where
$\{\cdot,\cdot\}$
are the Poisson brackets.  Note that the inclusion
$(x_\tau,u_\tau)\in\Cal O$
implies that
$g^{(1)}_\tau(\lambda_t),g^{(2)}_\tau(\lambda_t)$
can not vanish simultaneously.  Assume, for definiteness, that
$g^{(2)}_\tau(\lambda_t)<0$.
Certainly, substituting
$\lambda_t$
by
$-\lambda_t$
and, accordingly,
$g^{(2)}_\tau(\lambda_t)$
by
$g^{(2)}_\tau(-\lambda_t)=-g^{(2)}_\tau(\lambda_t)$,
we do not change the curvature.  The quadratic Hamiltonian
$q_\tau$
from Proposition 3.2 has in our case the form
$$
q_\tau(\eta)=-{\sigma\left(\overset\rightarrow\to
g^{(1)}_\tau(\lambda_t),\eta\right)^2\over2g^{(2)}_\tau(\lambda_t)}.
$$
The Hamiltonian
$q_\tau$
is defined on a 4--dimensional symplectic space
$T_{\lambda_t}(T^*M)$,
though the vertical line
$\Bbb R\lambda_\tau\subset T^*_{x_\tau}\approx T_{\lambda_t}(T^*_{x_t}M)$
consists entirely of fixed points of the Hamiltonian system.  The Jacobi
curve is constructed with the help of the solutions of the reduced
system,
defined on
$\Sigma_{\lambda_t}=(\Bbb R\lambda_t)^\angle/\Bbb R\lambda_t$,
cf. the beginning of \S4.  There exists a unique covector
$e\in T_{\lambda_t}(T^*_{x_t}M)$
satisfying the conditions
$\sigma(e,\overset\rightarrow\to g_t^{(1)})=1$,
$\sigma(e,\overset\rightarrow\to g_t^{(2)})=0$.
Consider the restriction of the Hamiltonian
$q_\tau$
onto the symplectic subspace
$span(e,\overset\rightarrow\to g_t^{(1)})\subset(\Bbb
R\lambda_t)^\angle$.
Let
$\eta=ye+z\overset\rightarrow\to g^{(1)}_t$,
then
$$
q_\tau(\eta)={1\over2}(a_\tau y+b_\tau z)^2,
\  a_\tau={\sigma(\overset\rightarrow\to g_\tau^{(1)},e)\over
\sqrt{|g^{(2)}_\tau|}}, \ b_\tau={\sigma(\overset\rightarrow\to
g^{(1)}_\tau, \overset\rightarrow\to
g^{(1)}_t)\over\sqrt{|g^{(2)}_\tau|}},\ b_t=0.
$$
Nonstationary Hamiltonian
$q_\tau$
defines a linear flow on
$\Bbb R^2$ and, accordingly,
a flow  on
$L(\Bbb R^2)=\Bbb RP^1$.
The Jacobi curve
$\tau\mapsto J(\tau)$
is a trajectory of the flow on
$\Bbb RP^1$, inverse to the flow defined by the Hamiltonian
$q_t$,
$J(t)=\Bbb Re$.
Let
$\Phi=\left(\matrix\varphi_{11}&\varphi_{12}\\\varphi_{21}&\varphi_{22}
\endmatrix\right),\ \Phi(t)=id$,
be the fundamental matrix of the linear flow on
$\Bbb R^2$, inverse to the flow defined by the Hamiltonian
$q_\tau$.
Then
$$
\dot\Phi=\Phi\left(\matrix ab&b^2\\-a^2&-ab\endmatrix\right),\
det\,\Phi=1.\eqno(6.2)
$$
As a local coordinate on
$\Bbb RP^1$
take
$z\over y$,
then
$J(\tau)$
is represented by the
$1\times1$--matrix
$S_\tau=\dsize{\varphi_{21}(\tau)\over\varphi_{11}(\tau)}$.
For the curvature we obtain the expression
$$
R(x_t,u_t)=\left({\ddot S_t\over2\dot S_t}\right)\spdot-
\left({\ddot S_t\over2\dot S_t}\right)^2={\ddot a_t\over a_t}-a_t\dot b_t-
2\left({\dot a_t\over a_t}\right)^2.
$$
Note that
$$
{\partial\over \partial\tau}g^{(i)}_\tau=g^{(i+1)}_\tau
{du\over d\tau}+\left\{g^{(0)}_\tau,g^{(i)}_\tau\right\}=
{g^{(i+1)}_\tau\over g^{(2)}_\tau}
\left\{g^{(1)}_\tau,g^{(0)}_\tau\right\}+
\left\{g^{(0)}_\tau,g^{(i)}_\tau\right\}.
$$
Furthermore, the quantities
$g^{(i)}_t(\lambda)=\lambda\dsize{\partial^i\ \over\partial u^i}f$,
hence, the Poisson brackets of
$g^{(i)}_t$,
are expressed through the Lie brackets of the fields
${\partial^i\ \over\partial u^i}f$.
Therefore, the consecutive derivatives of the functions
$a_\tau,b_\tau$,
with respect to
$\tau$ for
$\tau=t$,
are expressed explicitly, though quite cumbersome, through
$\alpha_{i_1\ldots i_k}(x_t,u_t),\beta_{i_1\ldots i_k}(x_t,u_t)$.
Direct calculations  give the expression (6.1). Though
pretty awkward, this formula is strongly simplified in some
important special cases.  Consider two--dimensional Riemannian and
Lorentzian geometries.  In the Riemannian case we have
$$f(x,u)=(cos\,u)v_1(x)+(sin\,u)v_2(x),$$
where
$v_1,v_2$
is an arbitrary orthonormal  frame of the considered Riemannian structure.
It is easy to eliminate in (6.1) all indices
$\ge2$.
Indeed, from every such index we can subtract 2, at the same time
changing the sign of the corresponding coefficient. Taking into account
also the symmetries of the coefficients as the "structure constants",
we obtain
$$
R=\alpha_{001}+\beta_{101}-2(\alpha^2_{01}+\beta^2_{01}).
$$
Note that
$R$
is the Gaussian curvature of the Riemannian surface, according
to Proposition 5.2 and Theorem 5.1.

Let
$[v_1,v_2]=c_1v_1+c_2v_2$,
for some smooth functions
$c_1,c_2$. Then
$$
R=v_1c_2-v_2c_1-c_1^2-c_2^2.
$$
In the Lorentzian case
$$
f(x,u)=(ch\,u)v_1(x)+(sh\,u)v_2(x).
$$
Again, 2 could be subtracted from the indices in (6.1), this time without
changing  the coefficients. For the Gaussian curvature we obtain,
$$
R=\alpha_{001}-2(\alpha^2_{01}-\beta^2_{01})=v_1c_2+v_2c_1+c_1^2
-c_2^2.
$$

\Refs


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A. A. Agrachev
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Quadratic Mappings in Geometric Control Theory
\jour
Itogi Nauki i Tekhniki; Problemy Geometrii, VINITI, Acad. Nauk SSSR
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%\#
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1988
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111-205
\lang
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Moscow
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\by
A. A. Agrachev, R. V. Gamkrelidze
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Exponential Representation of Flows and
Chronological Calculus
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Matem. Sbornik
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A. A. Agrachev, R. V. Gamkrelidze
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The Morse index
and the Maslov index for extremals of controlled
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\ref \no5
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A. A. Agrachev,  R. V. Gamkrelidze
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The extremality
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Dokl.
Acad. Nauk SSSR
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A. A. Agrachev, R. V. Gamkrelidze
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The quasi-extremality
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Dokl.
Itogi Nauki. VINITI. Sovremennye problemy matematiki. Novejshie
dostigeniya
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J. Soviet Math., Plenum Publ. Corp., pp. 1849-1864, 1991.

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\by
A. A. Agrachev, R. V. Gamkrelidze
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Symplectic Methods
for optimization and control
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to
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\ref \no8
\by
A. A. Agrachev, R. V. Gamkrelidze and A. V. Sarychev
\paper
Local Invariants of Smooth Control
Systems
\jour
Acta Applicandae Mathematicae
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%\#
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1989
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%\lang
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%
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V. I. Arnold
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Mathematical Methods of Classical Mechanics
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3-ed, Nauka, Moscow
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B. Bonnard
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Feedback equivalence for nonlinear systems and the
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\ref \no11
\by
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Mathematical theory of optimal processes
%\#
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%\lang
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%\publaddr
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\endRefs

\enddocument




























