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\begin{document}
\begin{center}
{\bf SYMMETRIES OF WEAK-CONTROLLABLE SYSTEMS}
\end{center}
\begin{center}
{V. S. KALNITSKY}
\end{center}
\section{Introduction}
Hereafter all objects belong to the $C^\infty$-category. The
completeness of the vector field means that every trajectory is
defined on whole $\Bbb R$. In other words, the flow forms
one-parameter group of transformations.
After the basic facts about extendability of trajectories in
O.D.E. theory, it turns out that in the others theories dealing
with vector fields there are no difficult theorems about
completeness. The notion of completeness stands in the row of
properties which either have place obviously or one can say
nothing of important about it. The Palais example shows that the
set of complete fields is not even the linear space. Because of
this the theory of infinite dimensional Lie algebras and groups
are different ones.
From the other side in the affine and semi-Riemannian geometry the
incompleteness is unavoidable phenomenon. An example of
incomplete spray of affine connection on closed manifold
(nonmetrizable) one can find even on one-dimensional manifold\,---
on the circle. Moreover, many physically justified models with
incomplete geodesic flows appeared.
Such situation might be explained by the absence of "an uniform
reason" for completeness. There are several direct arguments of
that sort: explicit integration; boundedness in infinity of the
vector length (in particular, the compactness of support); the
existence of transitive group of transformation. We used in the
article the more effective indirect methods which ascertain the
completeness of field in the presence of another complete field.
Namely, when one considers the symmetries. The typical examples of
such sort theorems are the Palais and Kobayashi ones. We will show
the tight relation between the completeness of symmetries and
different notions of controllability of Nonlinear Control Systems.
The main results are the theorems~\ref{theorem22}
and~\ref{theorem33}
\section{Controllable Systems}
Let $\varphi^X_t$ be the flow of the field $X$. Consider the set
of vector fields $\{Y_\alpha\}$.
\begin{definition}\label{def1}
The point $z'$ is {\it accessible} from the point $z$, $z'\sim z$,
if there exist the set of numbers $\{t_1,\dots,t_N\}$ and set of
indices $\{\alpha_1,\dots,\alpha_N\}$ such, that
\begin{equation}\label{eq1}
z'=\varphi^{Y_{\alpha_1}}_{t_1}\circ\dots\circ\varphi^{Y_{\alpha_N}}_{t_N}(z).
\end{equation}
\end{definition}
Geometrically the accessibility means that one can reach the point
$z$ by the arcs of trajectories of fields $\{Y_\alpha\}$.
Accessibility is the equivalence on the manifold and each point
$z$ define the class of equivalence $\frak O(z)$. Due to the
Sussmann-Nagano theorem~\cite{suss} the orbit $\frak O(z)$ is
connected immersion submanifold of $M$. The set $\{Y_\alpha\}$ is
called {\it full controllable} if the class of equivalence equals
$M$. Consider the system $\{Y_\alpha\}$ of complete fields. If
these fields generate the finite dimensional Lie algebra $\frak
g$, then according the Palais theorem this algebra consists of
complete fields and their flows possess the structure of connected
Lie group $G$ acting on manifold. For the system of complete
fields $\{Y_\alpha\}$ generating the finite dimensional Lie
algebra the notions of full-controllability and transitive action
of group $G$ are equivalent. In general situation the system of
complete fields does not generate the finite dimensional Lie
algebra and the linear combination of complete fields can be
incomplete, however the conclusion of Sussmann theorem is true.
It is useful to introduce the notion of controllability {\it on
the set}. The system is full-controllable on the set $A$ if there
exists the point $x_0\in M$ such that $A\subset\frak O(x_0)$.
\begin{proposition}\label{cor1}
If the system of complete fields $\{Y_\alpha\}$ generates the
finite dimensional Lie algebra, then it is full-controllable on
each orbit of action of corresponding Lie group.
\end{proposition}
In the paper~\cite{Lew} the sufficient conditions for full
controllability are found. These conditions are applicable for
linear systems and systems satisfying the Palais theorem
conditions. The further extension of the notion of controllability
leads to criteria of topological nature.
\begin{definition}\label{def2}
The system $\{Y_\alpha\}$ is {\it weak-controllable} from the
point $x_0$ on the set $A$, if
\begin{equation}
\forall U_{x_0}, \forall z\in A=>\exists z'\in U_{x_0}: z'\sim z,
\end{equation}
where $U_{x_0}$ is the neighborhood of $x_0$.
\end{definition}
Analogously we denote the set of accessibility $\frak O(U)=\{z\in
M|\exists z'\in U: z\sim z'\}$. Now we can rewrite the
definition~\ref{def2} in the form
\begin{equation}
A\subset\bigcap_{U_{x_0}}\mathfrak O(U_{x_0})
\end{equation}
It is clear that weak-controllability follows from the full
controllability. The inverse is not true as the next example
shows. Let us consider the system consisting of one field
$x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$ on the
plane $(x,y)$. This system is weak-controllable from the origin on
the whole plane, but it is not full-controllable.
The $\Omega$-limit of the vector field $Y$ flow on the set $A$ is
the set
\begin{equation}
\Omega^Y(A)=\bigcap_{z\in A}Cl\,\gamma(z),
\end{equation}
where $\gamma(z)$ is the trajectory passing through the point $z$.
Consider the system $\{Y\}$ consisting of one vector field.
\begin{theorem}\label{theorem2}
The system $\{Y\}$ is weak-controllable if and only if
$\Omega^Y(A)\ne\varnothing$
\end{theorem}
\begin{proof}
Weak-controllability of the system from the point $x_0$ means
that this point is the limit point of each trajectory intersecting
the set $A$, i.e. $x_0\in\Omega^Y(A)$.
\end{proof}
\begin{corollary}
The system $\{Y\}$ is weak-controllable from any point on any
subset of everywhere dense trajectories.
\end{corollary}
In particular, if all trajectories are dense on manifold, the
system is weak-controllable from any point of $M$. Let us consider
now the system $\{Y_\alpha\}$ of complete fields generating the
finite dimensional Lie algebra $\frak g$ and corresponding Lie
group $G$. In view of proposition~\ref{cor1} the set $\frak O(z)$
coincides with the $G$-orbit $O(z)$ for any point $z$. In this
case the weak-controllability on $A$ means that there exists the
point $x_0$ which belongs to the closure of each orbit
intersecting the set $A$, $x_0\in\Omega(A)=\bigcap_{z\in
A}Cl\,O(z)$. It is clear that nonemptiness of the set $\Omega(A)$
is the criterion of weak-controllability.
\section{Controllability and completeness of symmetries}
In this section we bind the notions introduced in section 2 with
completeness of symmetries. The first example of such connection
could be the following sufficient condition of completeness. We
will distinguish the positive and negative $\Omega$-limits of
flow on the set $A$
\begin{equation}
\Omega^{\pm}_X(A)=\bigcap_{z\in A}Cl\,\gamma^{\pm}(z).
\end{equation}
If for every trajectory $\gamma$ on manifold holds
$\Omega^{+}_X(\gamma)\ne\varnothing$ and
$\Omega^{-}_X(\gamma)\ne\varnothing$, the field $X$ is complete.
Indeed, let us consider the point $x_0\in\Omega^{+}_X(\gamma)$.
This point has a neighborhood $U^\varepsilon$, in every point of
which the trajectory is extendable on interval $[0,\varepsilon)$.
The point $x_0$ is the limit point for the positive
semi-trajectory of any point $z\in\gamma$. Therefore, there
exists such point $T>0$, that $\varphi^X_T(z)\in U^\varepsilon$.
Hence, the point $\varphi^X_{T+\varepsilon/2}(z)$ is defined. It
means that for every point of this trajectory it is extendable on
$\varepsilon/2$ in positive direction. Analogously one can prove
the extendability in negative direction. For instance, any flow on
closed manifold satisfies above condition. Let us formulate now
the main result of this section.
\begin{theorem}\label{theorem22}
The system of complete fields $\{Y_\alpha\}$ is
weak-controllable. If the field $X$ commutes with all fields of
the system, it is complete.
\end{theorem}
\begin{proof}
Let's say the point $z$ has $\varepsilon$-property with respect
to field $X$, if the trajectory $\gamma(z)$ is extendable on
interval $(-\varepsilon,\varepsilon)$. Let $Y$ be complete and
the field $X$ commute with it, $[Y,X]=0$. If the point $z$ has
$\varepsilon$-property with respect to the field $X$, then all
points $\varphi^Y_t(z)$ have $\varepsilon$-property. Indeed, let
us fix the number $T$ and consider the curve
$\varphi^X_s\left(\varphi^Y_T(z)\right)=\varphi^Y_T\left(\varphi^X_s(z)\right).$
The equality holds by virtue of commutativity and the right part
of equality is defined for all $s\in(-\varepsilon,\varepsilon)$.
Hence, the left part is defined too. Let now all points of the set
$U^\varepsilon_X$ have $\varepsilon$-property with respect to $X$.
Under this condition all points of the accessibility set $\frak
O(U^\varepsilon_X)$ of the system $\{Y_\alpha\}$ of complete
fields possess this property. The system is weak-controllable on
manifold, therefore there exists the point $x_0$ form every
neighborhood of which all points of manifold are accessible. The
point $x_0$ in turn has the neighborhood
$U^\varepsilon_X$ possessing
$\varepsilon$-property and, hence, $\frak O(U^\varepsilon_X)=M$.
Finally, two mutually inverse map $\varphi^X_{\pm\varepsilon/2}$
are defined globally on manifold. It means that the field $X$ is
complete.
\end{proof}
\section{The Kobayashi Theorem Extension}
Consider the manifold $M$ provided with symmetrical affine
connection. It means that the spray $S$ is given on the tangent
bundle $TM$. The field $Y$ on $TM$ is called the polynomial
symmetry of spray if $[S,Y]=0$ and $[Y,L]=kL$, $k\geqslant0$,
where $L$ is the Liouville field. The number $k$ is the rank of
the symmetry. All polynomial spray symmetries are complete
lifts~\cite{Maart} on $TM$ of symmetrical tensor fields of type
$(1,k)$ on $M$. The symmetry is covariant constant if it is the
complete lift of covariant constant tensor field. The two known
results about spray symmetries are the following theorems.
\begin{theorem}
{\rm (Kobayashi \cite{Kob})} If the spray is complete then all
symmetries of rank 0 are complete.
\end{theorem}
\begin{theorem}
{\rm (\cite{Kaln1})} If the spray is complete then all covariant
constant symmetries of rank 1 are complete.
\end{theorem}
The space of all polynomial spray symmetries is, in general,
infinite dimensional. It is however known~\cite{Kaln1} that for
each rank the space is finite dimensional. The space of all
covariant constant symmetries is commutative Lie algebra. By
virtue of the above two theorems we can consider the system
$\{Y_\alpha\}$ of covariant constant symmetries of ranks~0 and 1
which are complete if the spray is.
\begin{theorem}\label{theorem33}
If the system $\{Y_\alpha\}$ is weak-controllable on
$TM\backslash M$ and the spray is complete, then all covariant
constant spray symmetries are complete.
\end{theorem}
\begin{proof}
In view of theorem~\ref{theorem22} all symmetries are complete on
$TM\backslash M$ and the null section $M$ of tangent bundle $TM$
consists of null points of complete lifts.
\end{proof}
\begin{note}
In the article of the author~\cite{Kaln2} the wide class of
covariant constant symmetries of all ranks is described which are
complete if the spray is. Hence, one can weaken the condition of
the theorem~\ref{theorem33}.
\end{note}
The described approach allows to inverse the statement about the
spray completeness.
\begin{theorem}\label{theorem1}
If the set of complete spray symmetries (even not polynomial) is
weak-controllable on $TM\backslash M$, then the spray is complete.
\end{theorem}
\subsection*{Acknowledgements}
This work is supported by VNP of Ministry of Education of RF
3.1N4733
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\footnotesize
\bibitem{suss} Sussmann, H.J. Orbits of families of vector fields
and integrability of distributions //~{\it Trans. Amer. Math. Soc.
80(1973), 171--188}
\bibitem{Kob} Kobayashi, Sh., Transformation groups in differential geometry, 1986.
\bibitem{Lew} Lewis, A. D. and Murray, R. M. Controllability of simple
mechanical control systems, {\it SIAM Journal on Control and
Optimization}, 35(3), 766--790.
\bibitem{Maart} R. Maartens, D. Taylor //arXiv:physics/9712019v1,
1997.
\bibitem{Kaln1} Kalnitsky V.S., Algebra of generalized Jacobi fields
//~{\it J. Math. Sci.} (New-York) 91 (1998), no. 6, pp. 3476--3491.
\bibitem{Kaln2} Kalnitsky V.S., Completeness of nilpotent fields //~{\it Diff. Geom. Fig. Manif.} V.~36 (2005)
\end{thebibliography}
\end{document}