Applications of Lefschetz numbers in control theory
Peter Saveliev
We develop some applications of techniques of the Lefschetz coincidence theory in control theory. The topics are existence of equilibria and their robustness, controllability and its robustness.
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\begin{document}
\title[Lefschetz numbers in control theory]{Applications of Lefschetz numbers in control theory}
\author{Peter Saveliev}
\address{Department of Mathematics, Marshall University }
\email{saveliev@member.ams.org}
\subjclass{93B, 55M20, 55H25}
\keywords{control system, equilibrium, controllability, robustness, fixed point theory,
Lefschetz number, coincidence point, homology theory, algebraic topology}

\begin{abstract}
We develop some applications of techniques of the Lefschetz coincidence theory
in control theory. The topics are existence of equilibria and their
robustness, controllability and its robustness.

\end{abstract}
\maketitle


\section{Introduction.}

The goal of this paper is to provide examples of what Lefschetz coincidence
theory can contribute to control theory. We discuss existence of equilibria
and their robustness, controllability and its robustness.

We develop some topological techniques, already available in dynamics, in the
control theoretic setting. A (discrete) dynamical system on a manifold $M$ is
simply a map $f:M\rightarrow M.$ Then $x\in M$ and $f(x)$ are the current and
next states of the system respectively. An equilibrium of the system is a
fixed point of $f.$The problem of detecting equilibria can be treated via the
more general Coincidence Problem\textit{ }\cite[VI.14]{Bredon}, \cite[Ch.
7]{Vick}, \cite{Gorn-book}: \textquotedblleft Given two maps $f,g:N\rightarrow
M$ between two $n$-dimensional manifolds, what can be said about the
coincidence set $C$ of all $x$ such that $f(x)=g(x)$?\textquotedblright%
\ Indeed, the equilibrium set $C=\{x\in M:f(x)=x\}$ is the coincidence set of
$f$ and the identity map $g:M\rightarrow M.$ The famous Lefschetz coincidence
theorem states that if the Lefschetz number $\lambda_{fg}$ is not equal to
zero then there is at least one coincidence, i.e., $C\neq\varnothing.$ Using
this and other invariants one can find out whether a dynamical system has an
equilibrium or a periodic point.

In case of a \textit{controlled} dynamical system, the next state $f(x,u)$
depends not only on the current state, $x\in M,$ but also on the
\textit{input}, $u\in U.$ A discrete time control system is given by the space
of inputs $U$, the space of states $M$, the \textquotedblleft
state-input\textquotedblright\ space $N=M\times U,$ a map $f:N=M\times
U\rightarrow M,$ and the projection $g:N=M\times U\rightarrow M$ (in general
$N$ is a fiber bundle and $g:N\rightarrow M$ is the bundle projection). Then,
just as above, the equilibrium set $C=\{x\in M:f(x,u)=x$ for some $u\in U\}$
of the system is the coincidence set of the pair $(f,g).$ However, since the
dimensions of $N$ and $M$ are not equal anymore, the Lefschetz \textit{number}
is replaced with the Lefschetz \textit{homomorphism} \cite{Sav1} which does a
better job at detecting coincidences.

Another application of the coincidence theory approach is controllability. A
system is called controllable if any state can be reached from any other
state, i.e., for each pair of states $x,y\in M$ there are inputs
$u_{0},...,u_{r}\in U$ such that $x_{1}=f(u_{0},x),x_{2}=f(u_{1}%
,x_{1}),...,y=x_{r+1}=f(u_{r},x_{r}).$ Therefore controllability is equivalent
to surjectivity of the composition of several iterations of $f.$ On the other
hand a map is surjective if it has a coincidence with any constant map.

The state space $M$ is often a manifold, as opposite to a Euclidean space,
when it appears in robotics. For example, $M=\mathbf{T}^{n}=(\mathbf{S}%
^{1})^{n},$ the $n$-dimensional torus, is the space of all possible states of
a robotic arm with $n$ revolving joints \cite[p. 1]{NvdS}; or $M=\mathbf{R}%
^{3}\times SO(3)$ is the space of positions of a rigid body \cite[Chapter
2]{Latombe}. Typically, we have $N=M\times U.$ However nontrivial bundles are
also common. For example, consider a spherical pendulum with a gas jet control
which is always directed in the tangent space. Then its state space is
$M=\mathbf{S}^{2}$, the $2$-sphere, while the state-input space $N$ is the
tangent bundle $T\mathbf{S}^{2}$ of $\mathbf{S}^{2},$ which is an
$\mathbf{R}^{2}$-bundle over $M$ not isomorphic to $M\times\mathbf{R}^{2}$
\cite[p. 17]{NvdS}. In spite of the abundance of such examples \cite{BL},
\cite{Latombe}, \cite{NvdS} topological techniques have not thus far found
broad applications in control theory. The only recent examples known to the
author are \cite{Jon} - \cite{Kappos4}.

The topological approach provides the following advantages. Consider a control
system as a triple $(M,N,f)$ of topological spaces $M,N$ and a continuous map
$f$ as described above. Since our knowledge of the model is inevitably
imprecise, we have to deal with perturbations of the system. As perturbations
may be understood as variations of unknown parameters of the system their
effect on the behavior of the system is also unknown. However, if the system
depends continuously on these parameters, the change of $M,N,$ and $f$ is also
continuous. This means that we are to consider spaces homeomorphic to $M,N$
and maps homotopic to $f.$ An appropriate tool to deal with this degree of
generality is homology theory. Indeed, the homology groups $H_{\ast
}(M),H_{\ast}(N)$ of $M,N$ and the homology homomorphism $f_{\ast}:H_{\ast
}(N)\rightarrow H_{\ast}(M)$ of $f$ remain constant under homeomorphisms of
$M,N$ and homotopies of $f.$ They can also be rigorously and effectively
computed \cite{Misch}, \cite{KMM}.

Further, the perturbations of $f$ are normally assumed \textquotedblleft
small\textquotedblright\ (in particular, this is the basis of the notion of
structural stability). However unless actual estimates are available, we don't
know how \textquotedblleft small\textquotedblright\ are the perturbations of
the real system. Therefore in order to take into account the \textquotedblleft
worst possible scenario\textquotedblright\ we consider large, but still
continuous, perturbations of the system$.$ As an example, a constant external
force, such as gravitation, in any of the above robotic systems may be treated
as such a perturbation. Thus the use of homology theory provides answers with
a new, for control theory, degree of robustness. Providing results of this
nature is the first objective of this paper. We apply Lefschetz coincidence
theory to prove existence of equilibria (Theorem \ref{SuffEqDis}) and
controllability (Theorem \ref{SuffControl}) for systems determined by maps
homotopic to $f$.

The second objective of this paper is to study robustness of these properties
under arbitrarily small\ perturbations because sometimes they produce a
dramatic change in the properties of the system. This change may be the loss
of an equilibrium (Theorem \ref{NecEqDis}) or the loss of controllability
(Theorem \ref{NecContr}).

The paper is organized as follows. Some preliminaries from algebraic topology
are outlined in the Section \ref{Prel}. In Section
\ref{classical Lefschetz theory} we review the classical theory of Lefschetz
numbers and show its inadequacy for control theory. In Section
\ref{Lefschetz homomorphism} we consider the necessary generalization, the
Lefschetz homomorphism, of the Lefschetz number and state several relevant
results about existence of coincidences. In Section
\ref{Removability of coincidences} we state some results about removability of
coincidences. In Section \ref{Equilibria of discrete systems} we provide
sufficient conditions of existence of equilibria of a discrete system and
their robustness. In Section \ref{Controllability} we provide sufficient
conditions of controllability of a discrete system and its robustness. In
Section \ref{Controllability of continuous systems} we discuss how our
coincidence results can be applied to existence of equilibria and
controllability of continuous time control systems. Notions of control theory
are defined as needed, for details see \cite{NvdS}, \cite{Sastry},
\cite{Sontag}.

The author thanks the referees for their comments that have helped
significantly improve this paper.

\section{Preliminaries from algebraic topology.\label{Prel}}

The terminology we use is standard \cite{Bredon}. Suppose $N$ is a topological
space and $A\subset N$ is a subspace. The singular \textit{homology group}
$H_{k}(N,A)$ of $N$ relative to $A$ over $\mathbf{Q}$ or any other field is
defined as follows. If $\Delta_{k}$ is the standard $k$-simplex, $k=0,1,2...$,
any map $\sigma:\Delta_{k}\rightarrow N$ is called a singular $k$-simplex in
$N$. We let $C_{k}(N,A)$ be the vector space over $\mathbf{Q}$ generated by
all singular $k$-simplices of $N$ whose images are not completely in $A.$ Then
the boundary operator $\partial_{k}:C_{k}(N,A)\rightarrow C_{k-1}(N,A)$ is
defined in the natural way and we let $H_{k}(N,A)=\ker\partial_{k}%
/\operatorname{Im}\partial_{k}.$ Further, let $C^{k}(N,A)$ be the dual of
$C_{k}(N,A),$ i.e., the vector space of all linear functions from $C_{k}(N,A)$
to $\mathbf{Q.}$ Then $\partial_{k}$ generates the coboundary operator
$\partial^{k}:C^{k}(N,A)\rightarrow C^{k+1}(N,A)$ and we let $H^{k}%
(N,A)=\ker\partial^{k}/\operatorname{Im}\partial^{k}$ be the
\textit{cohomology group} of $N$ relative to $A.$ Also $H_{k}(N)=H_{k}%
(N,\varnothing),$ $H^{k}(N)=H^{k}(N,\varnothing).$ If $(N,A)$ is a simplicial
complex, its simplicial homology and cohomology is defined in the same way
starting with $C_{k}(N,A)$ generated by all simplices of $(N,A).$ The homology
and cohomology groups $H_{k}(N,A;G),$ $H^{k}(N,A;G)$ over any group $G$ can be
defined in a similar fashion.

Homology and cohomology groups $\{H_{k}(N,A):k=0,1,2...\},$ $\{H^{k}%
(N,A):k=0,1,2...\}$ over fields are (graded) vector spaces with the following
properties. The Betti numbers, $b_{k}=\dim H_{k}(N),$ for $k=0,1,2,$ are the
numbers of path components, \textquotedblleft tunnels\textquotedblright, and
\textquotedblleft voids\textquotedblright\ of $N,$ respectively. In case of a
path connected $N,$ the identities of $H_{0}(N)=H^{0}(N)=\mathbf{Q}$ are
denoted by $1.$ If $N$ is contractible, it is \textit{acyclic}, i.e.,
$H_{k}(N)=H^{k}(N)=0$ for $k>0.$ If $N$ is an $n$-dimensional simplicial
complex, $H_{k}(N)=0$ for all $k>n.$ If $M$ is a compact connected orientable
$n$-dimensional manifold with boundary $\partial M$ then $H_{n}(M,\partial
M)=H^{n}(M,\partial M)=\mathbf{Q.}$ The identities of these two groups are the
\textit{fundamental classes} $O_{M}$ and $\overline{O}_{M}$ of $M$
respectively. Further, there is the \textit{Poincar\'{e} duality} isomorphism
$D_{M}:H^{k}(M,\partial M)\rightarrow H_{n-k}(M)$ given by the cap product
with the fundamental class $O_{M}$. The \textit{cap product} is the
homomorphism $\frown:H^{k}(N,A)\otimes H_{m}(N,A)\rightarrow H_{m-k}(N)$ given
by $x\frown a=(1\times x)\Delta a$, where $\Delta$ is a diagonal
approximation. Then $a\in H_{k}(N,A)$ and $x\in H^{k}(N,A)$ are called
\textit{dual} if $x\frown a=\left\langle x,a\right\rangle =x(a)=1.$ In
particular, $O_{M}$ and $\overline{O}_{M}$ are dual. By the K\"{u}nneth
Theorem, $H_{k}(M\times U)=\sum_{i+j=k}H_{i}(M)\otimes H_{j}(U),$
$k=0,1,2,...$.

Suppose $B$ is a subspace of the topological space $M$ and $f:N\rightarrow M$
is a map, then $f:(N,A)\rightarrow(M,B)$ is a \textit{map of pairs} if
$f(A)\subset B.$ In this case $f$ generates the natural homomorphism from
$C_{k}(N,A)$ to $C_{k}(M,B)$. This homomorphism generates $f_{\ast}%
:H_{k}(N,A)\rightarrow H_{k}(M,B),$ the \textit{homology homomorphism }of
$f$\textit{, }and\textit{ }$f^{\ast}:H^{k}(M,B)\rightarrow H^{k}(N,A),$ the
\textit{cohomology homomorphism }of $f$. Two maps $f,g:(N,A)\rightarrow(M,B)$
are called \textit{homotopic} if $f$ can be continuously \textquotedblleft
deformed\textquotedblright\ into $g$, i.e., there is a map $F:[0,1]\times
(N,A)\rightarrow(M,B)$ such that $F(0,\cdot)=f$ and $F(1,\cdot)=g.$ If $f$ and
$g$ are homotopic then $f_{\ast}=g_{\ast}.$ In particular, if $f$ is homotopic
to a constant map then $f_{\ast}$ is trivial, i.e., $f_{\ast}:H_{k}%
(N,A)\rightarrow H_{k}(M,B)$ is zero for $k=1,2,...,$ or simply $f_{\ast}=0$.
In the case of $n$-manifolds, the homomorphism $f_{\ast}:H_{n}(N,\partial
N)\rightarrow H_{n}(M,\partial M)$ is the multiplication by $\deg f$, the
degree of $f.$ The $k$th homotopy group $\pi_{k}(N)$ of $N$ is the group of
homotopy classes of maps of $k$-spheres to $N$.

\section{Review of Lefschetz theory. \label{classical Lefschetz theory}}

In this section $M$ and $N$ are orientable compact connected manifolds with
boundaries $\partial M,\partial N,$ and $\dim M=\dim N=n$.

Consider the\textbf{\ }Fixed Point Problem: \textquotedblleft If
$f:M\rightarrow M$ is a map, what can be said about the set of points $x\in M$
such that $f(x)=x$?\textquotedblright\ Applications of fixed point theorems
(Kakutani, Banach, etc.) to control problems are abundant, \cite{Balach},
\cite{Carmichael}, \cite{CNZ}, \cite{Id}, \cite{Klamka}, \cite{Nistri}.
However the methods we suggest in this paper go far beyond those.

One may associate to $f$ an integer $\lambda_{f}$ called the Lefschetz number
\cite{Brown}:%
\[
\lambda_{f}=\sum_{k}(-1)^{k}Trace(f_{\ast k}),
\]
where $f_{\ast k}:H_{k}(M)\rightarrow H_{k}(M)$ is induced by $f.$ The
\textit{Lefschetz fixed point theorem} states that if $\lambda_{f}\neq0,$ then
$f$ has a fixed point.

The Coincidence Problem is concerned with a similar question about two maps
$f,g:N\rightarrow M$ and their coincidences\textit{ }$x\in N,f(x)=g(x).$ One
of the main tools is the Lefschetz coincidence number $\lambda_{fg}$ defined
similarly to $\lambda_{f}$ as the alternating sum of traces of a certain
endomorphism on the homology group of $M.$ Algebraically, if $h:E_{\ast
}\rightarrow E_{\ast}$ is a (degree $0)$ endomorphism of a finitely generated
graded vector space $E_{\ast}=\{E_{k}\},$ given by $h_{k}:E_{k}\rightarrow
E_{k},$ then its Lefschetz number\textit{ }is $L(h)=\sum_{k}(-1)^{k}%
Trace(h_{k}).$ To apply this formula in the topological setting we let
$E_{\ast}=H_{\ast}(M)$, then the Lefschetz number is defined as $\lambda
_{fg}=L(g_{\ast}D_{N}f^{\ast}D_{M}^{-1}),$ where $D_{M}:H^{k}(M,\partial
M)\rightarrow H_{n-k}(M),$ $D_{N}:H^{k}(N,\partial N)\rightarrow H_{n-k}(N)$
are the Poincar\'{e} duality isomorphisms. Observe that for $f^{\ast}%
:H^{k}(M,\partial M)\rightarrow H^{k}(N,\partial N)$ to be well defined the
map $f$ has to be boundary preserving, $f:(N,\partial N)\rightarrow(M,\partial
M).$

A \textit{Lefschetz type coincidence theorem} states that if $\lambda_{fg}%
\neq0$ then the pair $(f,g)$ (and any pair homotopic to them) has a
coincidence$.$ The converse is false in general. When $\lambda_{fg}=0,$ the
maps $f,g$ may have coincidences but under certain circumstances they can be
removed by homotopies of $f,g$ \cite{BS}.

Until recently such theorems have been mostly considered in the following two
settings. First \cite[VI.14]{Bredon}, \cite[Ch. 7]{Vick}, $f:N\rightarrow M$
is a map between two $n$-manifolds as above. This way the Lefschetz theorem
can be applied to detect equilibria of a dynamical system but it does not
apply to an even simplest control system because the dimensions of $N=M\times
U$ and $M$ have to be equal. Second \cite{Gorn-book}, $f:X\rightarrow M$\ is a
map from an arbitrary topological space $X$ to an open subset of
$\mathbf{R}^{n}$\ and all fibers $f^{-1}(y)$\ are acyclic. Here the dimensions
are also equal in the sense that $H_{\ast}(X)=H_{\ast}(M)$ (Vietoris Theorem).
Thus neither case is broad enough to cover control systems the input spaces
$U$ of which have nonzero dimension.

As an example from dynamics, consider the problem of existence of closed
orbits of a flow. The flow is given by a map $f:M\times\lbrack0,\infty
)\rightarrow M$ so that the initial position is $f(0,x)=x$ and $f(t,x)$ is the
position at time $t$. Closed orbits correspond to coincidences of $f$ and the
projection $p:M\times\lbrack0,\infty)\rightarrow M$\textit{. }More generally
one considers $f:M\times X\rightarrow M,$ where $X$ is a topological space.
This situation was studied in \cite{Knill}, \cite{GN2}, \cite{GNO},
\cite{Geoghegan} under the name \textquotedblleft parametrized fixed point
theory\textquotedblright. These results can be applied to detect equilibria
(Section \ref{Equilibria of discrete systems}), but the setting is not general
enough to study controllability (Section \ref{Controllability}). The author
\cite{Sav}, \cite{Sav1} extended some of the results of \cite{GNO} to the
general case of two arbitrary maps $f,g:N\rightarrow M$ from an arbitrary
topological space to a manifold. The content of these papers is briefly
outlined in the next section.

\section{Detecting coincidences. \label{Lefschetz homomorphism}}

In this section $N$ is an arbitrary topological space, $A\subset N$, $M$ is an
orientable compact connected manifold with boundary $\partial M$, $\dim M=n$,
and $f:(N,A)\rightarrow(M,\partial M),$ $g:N\rightarrow M$ are maps.

The generalization of the Lefschetz number is based on the fact that since the
finitely generated graded vector space $E=H_{\ast}(M)$ is equipped with the
cap product $\frown:E^{\ast}\otimes E_{\ast}\rightarrow E_{\ast}$, one can
define the Lefschetz class\ $L(h)\in E_{\ast}$ of a graded endomorphism $h$
given by $h_{k}:E_{k}\rightarrow E_{k+m}$ of any degree $m$ not just of degree
$0$ as in the classical case.

\begin{definition}
\label{Lef-formula}\cite[Proposition 2.2]{Sav1} If $h:H_{k}(M)\rightarrow
H_{k+m}(M),$ $k=0,1,2...,$ is a graded homomorphism of degree $m$ then the
\textit{Lefschetz class}\ $L(h)\in H_{m}(M)$ is defined as
\[
L(h)=\sum_{k}(-1)^{k(k+m)}\sum_{j}x_{j}^{k}\frown h(a_{j}^{k}),
\]
where $\{a_{1}^{k},...,a_{m_{k}}^{k}\}$ is a basis for $H_{k}(M)$ and
$\{x_{1}^{k},...,x_{m_{k}}^{k}\}$ the corresponding dual basis for $H^{k}(M).$
\end{definition}

It is easy to see that if the degree $m$ of $h$ is zero, $L(h)=\sum
_{k}(-1)^{k}Trace(h_{k})$.

For a given $z\in H_{s}(N,A),$ suppose the homomorphism $h_{fg}^{z}$ is
defined as the composition%
\[
H_{i}(M)^{\underrightarrow{\quad D_{M}^{-1}\quad}}H^{n-i}(M,\partial
M)^{\underrightarrow{\quad f^{\ast}\quad}}H^{n-i}(N,A)^{\underrightarrow
{\quad\frown z\quad}}H_{s-n+i}(N)^{\underrightarrow{\quad g_{\ast}\quad}%
}H_{s-n+i}(M),
\]
i.e.,%
\[
h_{fg}^{z}(x)=g_{\ast}((f^{\ast}D_{M}^{-1}(x))\frown z)).
\]
Its degree is $m=s-n.$ 

\begin{definition}
The \textit{Lefschetz homomorphism} $\Lambda_{fg}:H_{s}(N,A)\rightarrow
H_{s-n}(M),$ $k=0,1,...,$ of the pair $(f,g)$ is defined by
\[
\Lambda_{fg}(z)=L(h_{fg}^{z}).
\]

\end{definition}

The degree of the homomorphism $h_{fg}^{z}$ is zero if $z\in H_{n}(N,A).$ If,
moreover, $N$ is a orientable compact connected manifold of dimension $n,$ we
have $H_{n}(N,\partial N)=\mathbf{Q.}$ Its identity is the fundamental class
$O_{N}\in H_{n}(N,\partial N)$ of $N.$ Since $D_{N}(x)=x\frown O_{N},$ we
recover the classical \textit{Lefschetz number}, $\lambda_{fg}=\Lambda
_{fg}(O_{N}).$

\begin{theorem}
\label{Lef-type}\cite[Theorem 6.1]{Sav1} \textbf{(Existence of coincidences)}
If\textit{ }$\Lambda_{fg}\neq0$\textit{\ then any pair of maps }$f^{\prime
},g^{\prime}$\textit{\ homotopic to }$f,g$ has a coincidence.
\end{theorem}

Especially important for the control theory applications are the following two
corollaries. They are applied to existence of equilibria (Section
\ref{Equilibria of discrete systems}) and controllability (Section
\ref{Controllability}) respectively. Observe that the second corollary is
about a map of pairs and the first is not.

\begin{corollary}
\label{Proj}\textbf{(Existence of fixed points)} (cf. \cite{GNO}) Let
$g:M\times U\rightarrow M$ be a map. Given $v\in H_{s}(U),$ suppose the
homomorphism $g_{v}:H_{i}(M)\rightarrow H_{i+s}(M),i=0,1,...,$ of degree $s$
is defined by
\[
g_{v}(x)=(-1)^{(n-i)s}g_{\ast}(x\otimes v),
\]
$x\in H_{i}(M).$ Then, if
\[
L(g_{v})\neq0\text{ for some}\ v\in H_{s}(U)
\]
then any map $g^{\prime}:M\times U\rightarrow M$ homotopic to $g$ has a fixed
point $x$, $g^{\prime}(x,u)=x$ for some $u.$
\end{corollary}

\begin{proof}
Let $(N,A)=(M,\partial M)\times U$ and apply the above theorem to the pair
$p,g,$ where $p:(M,\partial M)\times U\rightarrow(M,\partial M)$ is the
projection. Also according to Corollary 5.7 in \cite{Sav1}, $\Lambda
_{pg}(O_{M}\otimes v)=L(g_{v})$.
\end{proof}

\begin{corollary}
\label{Surj}\textbf{(Sufficient condition of surjectivity)} If
\[
f_{\ast}:H_{n}(N,A)\rightarrow H_{n}(M,\partial M)=\mathbf{Q}\mathit{\ }%
\text{\textit{is nonzero}}%
\]
\textit{ then any }map $f^{\prime}:(N,A)\rightarrow(M,\partial M)$
\textit{homotopic to }$f$ is onto.
\end{corollary}

\begin{proof}
Apply the theorem to the pair $f,c,$ where $c$ is any constant map (as in
Section 5 in \cite{Sav} and Proposition 6.8 in \cite{Sav1}).
\end{proof}

In case of manifolds of equal dimensions the condition of this corollary is
equivalent to the nonvanishing of the degree $\deg f$ \cite[p. 186]{Bredon} of
$f$.

\section{Removing coincidences.\label{Removability of coincidences}}

In this section $M$ is a compact orientable connected manifold with boundary
$\partial M$, $\dim M=n,$ $N$ is a manifold, $f,g:N\rightarrow M$ are maps.

When $\dim N=\dim M=n>2,$ the vanishing of the Lefschetz number $\lambda_{fg}$
implies that the coincidence set can be removed by homotopies of $f,g$
\cite{BS}. If $\dim N=n+m,m>0,$ this is no longer true even if $\lambda_{fg}$
is replaced with $\Lambda_{fg}$. Some progress has been made for $m=$ $1.$ In
this case the secondary obstruction to the removability of a coincidence set
was considered in \cite{Fuller}, \cite{DG}, \cite{Jez}. These results can be
used to study removability of equilibria when the dimension of the input space
is $1$. However the conditions on $f$ and $g$ are hard to verify. Necessary
conditions of the global removability for arbitrary $m$ were considered in
\cite[Section 5]{GJW} with $N$ a torus and $M$ a nilmanifold. For some $m>1,$
a partial converse of Theorem \ref{Lef-type} is provided by the author
\cite{Sav2}. A version of this theorem is given below.

Suppose $F$ is an isolated subset of the coincidence set of $f,g$ and
$f(F)=g(F)=\{x\},$ $x\in M\backslash\partial M.$ Let $D$ be a open
neighborhood of $x$ such that $D\cap\partial M=\varnothing.$ Choose a
neighborhood $W$ of $F$ in $N$ with no coincidences such that $f(W)\subset D$
and $g(W)\subset D.$ Suppose $V\subset\overline{V}\subset W$ is another
neighborhood of $F,$ then there is an open neighborhood $B\subset\overline
{B}\subset D$ of $x$ such that $f(W\backslash V)\subset D\backslash B.$
Therefore $f:(W,W\backslash V)\rightarrow(D,D\backslash B)$ is a map of pairs.

\begin{theorem}
\label{Remov}\textbf{(Local removability of coincidences) }Suppose the
following property is satisfied
\[
\text{(*) \ }H^{k+1}(W,W\backslash V;\pi_{k}(\mathbf{S}^{n-1}))=0\text{ for
}k\geq n+1.
\]
Suppose also that
\[
f_{\ast}:H_{n}(W,W\backslash V)\rightarrow H_{n}(D,D\backslash B)=\mathbf{Q}%
\mathit{\ }\text{\textit{is zero.}}%
\]
\textit{Then there is a homotopy of }$f$ \textit{constant on the compliment of
}$V$ to a map $f^{\prime}$\textit{ such that the new pair has no coincidences
in }$V$. 
\end{theorem}

Since $D$ is arbitrary we can say that the homotopy can be chosen
\textit{arbitrarily small.}

\begin{proof}
According to the proof of Theorem 2 in \cite{Sav2} the coincidence subset $F$
can be removed by a homotopy of $f$ constant on $N\backslash V$ provided the
local cohomology index $I_{fg}^{W}(\tau)$ vanishes. This index is defined as
follows. Since $F\subset V$ is the set of all coincidences in $W,$ the map
$(f,g):(W,W\backslash V)\rightarrow D^{\times}=(D\times D,D\times D\backslash
d(D))$, where $d(D)$ is the diagonal of $D\times D,$ is well defined.
Therefore the homomorphisms $(f,g)_{\ast}:H_{k}(W,W\backslash V)\rightarrow
H_{k}(D^{\times})$ and $(f,g)^{\ast}:H^{k}(D^{\times})\rightarrow
H^{k}(W,W\backslash V)$ are also well defined. Now let $I_{fg}$ be the
homology coincidence homomorphism defined by $I_{fg}=(f,g)_{\ast}%
:H_{k}(W,W\backslash V)\rightarrow H_{k}(D^{\times}).$ Let $I_{fg}^{W}%
(\tau)=(f,g)^{\ast}(\tau)\in H^{n}(W,W\backslash V)$ be the cohomology
coincidence index \cite[Section 2]{Sav2}, where $\tau$ is the identity of
$H^{n}(D^{\times})=\mathbf{Q}$. By Theorem 6.1 in \cite{Sav1}, $\Lambda
_{fg}(z)=\pi_{\ast}(\tau\frown I_{fg}(z)),$ where $\pi:D\times D\rightarrow D$
is the projection on the first factor. Then, for any $z\in H_{n}(W,W\backslash
V)$%
\begin{align*}
\Lambda_{fg}(z) &  =\pi_{\ast}(\tau\frown(f,g)_{\ast}(z))=\pi_{\ast
}(f,g)_{\ast}((f,g)^{\ast}(\tau)\frown z)\\
&  =\left\langle (f,g)^{\ast}(\tau),z\right\rangle =\left\langle I_{fg}%
^{W}(\tau),z\right\rangle .
\end{align*}
Therefore $I_{fg}^{W}(\tau)=0$ if and only if $\Lambda_{fg}(z)=0$ for all
$z\in H_{n}(W,W\backslash V).$ Finally, observe that $g|_{W}$ is homotopic to
a constant map. Therefore $f_{\ast}=0$ if and only if $\Lambda_{fg}(z)=0$ for
all $z\in H_{n}(N,A)$ (Section 5 in \cite{Sav}).
\end{proof}

Condition (*) ensures that only the primary obstruction to removability, i.e.,
the Lefschetz number, can be nonzero. Further investigation of necessary
conditions of removability of coincidences will require computing higher order
obstructions. The case when $f(F)$ is not a single point is best addressed in
the context of Nielsen theory via Wecken type theorems \cite{Sav4}. In
general, the homotopy of $f$ cannot be always chosen arbitrarily small.

Especially important for the control theory applications are the following
corollaries. They are applied to disappearance under perturbations of
equilibria (Section \ref{Equilibria of discrete systems}) and controllability
(Section \ref{Controllability}) respectively.

\begin{corollary}
\label{NecParam}\textbf{(Removability of fixed points)} Suppose the conditions
of the theorem are satisfied for $N=M\times U$, where $U$ is a manifold, $x\in
M\backslash\partial M$ an isolated fixed point of $f:M\times U\rightarrow M$
(i.e., $f(x,u)=x$ for some $u\in U),$  $F=\{x\}\times\{u\in U:g(x,u)=x\}$.
Then \textit{there is a homotopy of }$f$ to a map $f^{\prime}$\textit{ such
that} $f^{\prime}$ has no fixed points in \textit{a neighborhood of }$F.$ The
homotopy can be chosen \textit{arbitrarily small and constant on the
compliment of a neighborhood of }$F.$
\end{corollary}

\begin{proof}
If $g:M\times U\rightarrow M$ is the projection then $F$ is the coincidence
set of $f,g.$
\end{proof}

\begin{corollary}
\label{NecSurj}\textbf{(Necessary condition of surjectivity)} Suppose the
conditions of the theorem are satisfied for $F=f^{-1}(x)$ of $f:N\rightarrow
M$. \textit{Then there is a homotopy of }$f$ to a map $f^{\prime}$ which is
not onto; specifically, $x\notin f^{\prime}(N)$. The homotopy can be chosen
\textit{arbitrarily small and constant on the compliment of  a neighborhood of
}$F.$
\end{corollary}

\begin{proof}
If $g$ is the constant map then $F$ is the coincidence set of $f,g.$
\end{proof}

These two corollaries are partial converses of Corollaries \ref{Proj} and
\ref{Surj} respectively.

A submanifold $F$ of $N$ satisfies condition (*) if one of the
following\ three conditions holds \cite[Section 4]{Sav2}:

\begin{itemize}
\item \textit{(a1) }$M$\textit{ is a surface, i.e., }$n=2;$\textit{ or}

\item \textit{(a2) }$F$\textit{ is acyclic, i.e., }$H_{k}(F)=0$%
\textit{\ for\ }$k=1,2,...$\textit{; \ or}

\item \textit{(a3) every component of }$F$\textit{ is a homology }%
$m$\textit{-sphere, i.e., }$H_{k}(F)=0$\textit{\ for\ }$k\neq0,m,$\textit{ for
the following values of }$m$\textit{ and }$n$\textit{:}
\end{itemize}

\textit{\qquad(1) }$m=4$\textit{ and }$n\geq6;$

\textit{\qquad(2) }$m=5$\textit{ and }$n\geq7;$

\textit{\qquad(3) }$m=12$\textit{ and }$n=7,8,9,$ or $n\geq14$\textit{.}

\section{Existence of equilibria.\label{Equilibria of discrete systems}}

In this section $M$\textit{ }is a compact orientable connected manifold with
boundary $\partial M$, $\dim M=n$, $U$ is a topological space.

A \textit{discrete time control system} $D_{g}$ is given by a map $g:M\times
U\rightarrow M$, with $U$ the \textit{space of inputs}, $M$ the \textit{space
of states} of the system.

We say that $D_{g^{\prime}}$ is a \textit{perturbation} of $D_{g}$ if
$g^{\prime}$ homotopic to $g.$ To justify this definition recall that a system
$D_{g^{\prime}}$ is normally called a perturbation of $D_{g}$ if $g^{\prime}$
is \textquotedblleft close enough\textquotedblright\ to $g$ in terms of the
distance between $g(z)$ and $g^{\prime}(z)$. However, if $g^{\prime}$ is a
simplicial approximation of $g$ \cite[p. 251]{Bredon} then $g$ and $g^{\prime
}$ are homotopic. Thus we permit large but continuous perturbations of the
system. Properties preserved under such perturbations may be called
\textit{strongly robust}.

As before suppose $\{a_{1}^{k},...,a_{m_{k}}^{k}\}$ is a basis for $H_{k}(M)$
and $\{x_{1}^{k},...,x_{m_{k}}^{k}\}$ the corresponding dual basis for
$H^{k}(M).$

\begin{theorem}
\textbf{\label{SuffEqDis}(Existence of equilibria) }If
\[
L(g_{v})=(-1)^{ns}\sum_{k}(-1)^{k}\sum_{j}x_{j}^{k}\frown g_{\ast}(a_{j}%
^{k}\otimes v)\neq0\text{ for some }v\in H_{s}(U)
\]
then every perturbation of the discrete time system $D_{g}$ has an equilibrium.
\end{theorem}

\begin{proof}
In light of Corollary \ref{Proj} we only need to show that the above formula
for the Lefschetz class $L(g_{v})$ of $g_{v}(x)=(-1)^{(n-i)s}g_{\ast}(x\otimes
v),$ $x\in H_{i}(M),$ is true. Since the degree of $g_{v}$ is $s$ and
$a_{j}^{k}\in H_{k}(M),$ we substitute $m=s$ and $i=k$ in Definition
\ref{Lef-formula}:%
\begin{align*}
L(g_{v}) &  =\sum_{k}(-1)^{k(k+s)}\sum_{j}x_{j}^{k}\frown(-1)^{(n-k)s}g_{\ast
}(a_{j}^{k}\otimes v)\\
&  =\sum_{k}(-1)^{k^{2}+ns}\sum_{j}x_{j}^{k}\frown g_{\ast}(a_{j}^{k}\otimes
v),
\end{align*}
and the formula follows.
\end{proof}

The following a generalization of a well known theorem about dynamical systems.

\begin{corollary}
Suppose $D_{g}$ is a perturbation of the constant system $D_{p},$ i.e.,
$p(x,u)=x$ for all $u.$ If the Euler characteristic of $M$ is nonzero,
$\chi(M)\neq0,$ then $D_{g}$ has an equilibrium.
\end{corollary}

\begin{proof}
Since $p_{\ast}(a_{j}^{k}\otimes v)=a_{j}^{k}$ if $v=1$ and $0$ otherwise, we
have%
\begin{align*}
L(g_{v}) &  =\sum_{k}(-1)^{k}\sum_{j}x_{j}^{k}\frown p_{\ast}(a_{j}^{k}\otimes
v)\\
&  =\sum_{k}(-1)^{k}\sum_{j}1\\
&  =\sum_{k}(-1)^{k}m_{k}\\
&  =\chi(M).
\end{align*}

\end{proof}

\begin{corollary}
\textbf{\label{SuffEqDisCor}}Suppose $M=\mathbf{S}^{n},$ and suppose one of
the following conditions is satisfied:

(1) $g_{\ast}(d\otimes1)\neq(-1)^{n+1}d,$ where $d$ is the identity of
$H_{n}(\mathbf{S}^{n});$ or

(2) $g_{\ast}(1\otimes v)\neq0$ for some $v\in H_{n}(U).$

Then every perturbation of the discrete time system $D_{g},$ $g:$
$\mathbf{S}^{n}\times U\rightarrow\mathbf{S}^{n},$ has an equilibrium.
\end{corollary}

\begin{proof}
Let's compute $L(g_{v})$ for an arbitrary $v\in H_{s}(U).$ As $a_{j}^{k}\in
H_{k}(M),$ we have $a_{j}^{k}\otimes v\in H_{k+s}(M\times U)$ and $g_{\ast
}(a_{j}^{k}\otimes v)\in H_{k+s}(M).$ Since $H_{i}(M)=H_{i}(\mathbf{S}^{n})=0$
for all $i\neq0,n,$ we have $g_{\ast}(a_{j}^{k}\otimes v)=0$ except for the
following two cases. (1) Choose $v=1\in H_{0}(U),s=0,$ then either
$k=0,a_{j}^{0}=1,x_{j}^{0}=1,$ or $k=n,a_{j}^{n}=d,x_{j}^{n}=\overline{d}.$
(2) Choose $v\in H_{n}(U),s=n,$ then $k=0,a_{j}^{0}=1,x_{j}^{0}=1.$ Here
$\overline{d}$ is the dual of $d,$ $\overline{d}\frown d=1.$ Thus we have%
\begin{align*}
\text{(1) }L(g_{1}) &  =(-1)^{n0}(1\frown g_{\ast}(1\otimes1)+(-1)^{n}%
\overline{d}\frown g_{\ast}(d\otimes1))\\
&  =1+(-1)^{n}\overline{d}\frown g_{\ast}(d\otimes1);\\
\text{(2) }L(g_{v}) &  =(-1)^{nn}(1\frown g_{\ast}(1\otimes v))\\
&  =(-1)^{n}g_{\ast}(1\otimes v).
\end{align*}
Now, if either $L(g_{1})$ or $L(g_{v})$ is nonzero, then $D_{g}$ has an
equilibrium by the theorem.
\end{proof}

Condition (1) means that the degree of $\overline{g}(\cdot)=g(\cdot
,u_{0}):\mathbf{S}^{n}\rightarrow\mathbf{S}^{n}$ is not equal to $(-1)^{n+1}.$

If $U=M$ is a compact Lie group and $g:M\times M\rightarrow M$ is the
multiplication, then $D_{g}$ has a equilibrium \cite[Example 2.3]{GNO}. For
more examples, see \cite{GNO}, \cite{Sav}, \cite{Sav1}.

In the control setting Corollary \ref{NecParam} reads as follows. 

\begin{theorem}
\textbf{\label{NecEqDis}(Removability of equilibria) }Suppose $U$ is a
manifold and suppose $x\in M\backslash\partial M$ is an isolated equilibrium
of $D_{g}.$ Suppose condition (*) is satisfied for $F=\{x\}\times\{u\in
U:g(x,u)=x\}$ and
\[
f_{\ast}:H_{n}(W,W\backslash V)\rightarrow H_{n}(D,D\backslash B)=\mathbf{Q}%
\mathit{\ }\text{\textit{is zero,}}%
\]
where $V\subset\overline{V}\subset W$ and $B\subset\overline{B}\subset
D\subset M\backslash\partial M$ are neighborhoods of $F$ and $x$ respectively.
Then this equilibrium can be removed by an \textit{arbitrarily} small
perturbation restricted to a neighborhood of $F$.
\end{theorem}

\section{Controllability.\label{Controllability}}

In this section $M$ is a compact orientable connected manifold with boundary
$\partial M,$ $\dim M=n$, $U$ is a topological space.

Suppose a discrete system $D_{f}$ is given by $f:M\times U\rightarrow M.$ The
system $D_{f}$ is called \textit{controllable} \cite{Sontag} if any state can
be reached from any other state by means of $f$, i.e., for each pair of states
$x,y\in M$ there are inputs $u_{0},...,u_{r}\in U$ such that $x_{1}%
=f(u_{0},x),x_{2}=f(u_{1},x_{1}),...,y=x_{r+1}=f(u_{r},x_{r}),$ notation
$x\rightsquigarrow_{f}y.$

Below this notion is generalized in three, nontypical but topologically
appropriate, ways. First, we consider the possibility of an arbitrary state
reached not from any given state but from a state in a particular subset $L$
of $M.$ Second, as before we permit arbitrary, not necessarily small,
perturbations of $f.$ Third, instead of looking into controllability of a new,
perturbed, system $D_{g},$ where $g$ is homotopic to $f$, we allow for
consecutive applications of possibly different maps each homotopic to $f$.

\begin{definition}
Given $L\subset M,$ let $f^{\prime}:L\times U\rightarrow M$ be the restriction
of $f.$ Then the system is called \textit{strongly robustly controllable from
}$L$ if for any map $f_{0}$ homotopic to $f^{\prime},$ any maps $f_{1}%
,...,f_{r}$ homotopic to $f,$ and for each $y\in M$ there is $x\in L$ and
inputs $u_{0},...,u_{r}\in U$ such that
\[
x_{1}=f_{0}(x,u_{0}),x_{2}=f_{1}(x_{1},u_{1}),...,y=x_{r+1}=f_{r}(x_{r}%
,u_{r}).
\]

\end{definition}

Then the system is controllable if it controllable from any point.

It is clear that controllability is equivalent to surjectivity of several
iterations of $f.$ To deal with surjectivity we apply Corollary \ref{Surj}
which requires $f$ to be a map of pairs. For this purpose in this section we
make the following assumption about $D_{f}$. If the initial state lies at the
boundary $\partial M$ of $M$ then the next state, regardless of the input,
lies within a certain neighborhood of $\partial M$. For simplicity we make a
topologically equivalent assumption,
\[
f(\partial M\times U)\subset\partial M.
\]


Next, let $U^{\prime}$ be the set of controls that take any given state to the
boundary of $M,$ i.e.,
\[
U^{\prime}=\{u\in U:f(x,u)\in\partial M\text{ for all }x\in M\}.
\]
Then $f(M\times U^{\prime})\subset\partial M.$ Combining this with the above
assumption we conclude that $f$ is a map of pairs, $f:(M,\partial
M)\times(U,U^{\prime})\rightarrow(M,\partial M).$ Let $L^{\prime}%
=L\cap\partial M,$ then $f^{\prime}:(L,L^{\prime})\times(U,U^{\prime
})\rightarrow(M,\partial M)$ is also a map of pairs$.$

The following theorem translates the above \textquotedblleft
reachability\textquotedblright\ condition into the language of homology: any
element of $H_{n}(M,\partial M)=\mathbf{Q}$ can be reached from some $a_{0}\in
H_{\ast}(L,L^{\prime})$ by means of $f_{\ast}$.

\begin{theorem}
\label{SuffControl}\textbf{(Sufficient condition of robust controllability)
}Suppose that there are $a_{0}\in H_{p}(L,L^{\prime}),$ $v_{0}\in H_{s_{0}%
}(U,U^{\prime}),...,v_{r}\in H_{s_{r}}(U,U^{\prime})$ such that
\[
a_{1}=f_{\ast}^{\prime}(a_{0}\otimes v_{0}),a_{2}=f_{\ast}(a_{1}\otimes
v_{1}),...,a_{r+1}=f_{\ast}(a_{r}\otimes v_{r})\in H_{n}(M,\partial
M)\backslash\{0\}.
\]
Then the discrete time system $D_{f}$ is strongly \textit{robustly
}controllable from $L$.
\end{theorem}

Here, if $a_{i}\in H_{n_{i}}(M,\partial M),i=0,1,2,...,r,$ then $n_{0}%
=p,n_{1}=p+s_{0},n_{2}=n_{1}+s_{1},...,n_{r+1}=n_{r}+s_{r}=n.$ Thus we have a
sequence of homology classes $a_{0},..,a_{r}$ of $(M,\partial M)$
\textquotedblleft climbing\textquotedblright\ dimensions from $p$ to $n.$

\begin{proof}
The result of consecutive applications of $f$ is defined as a map
$F:(L,L^{\prime})\times(U,U^{\prime})^{r+1}\rightarrow(M,\partial M)$ given
by
\[
F(x,u_{0},...,u_{r})=f(...f(f^{\prime}(x,u_{0}),u_{1}),...,u_{r}),
\]
i.e., it is given by the composition
\begin{align*}
F &  :(L,L^{\prime})\times(U,U^{\prime})\times...\times(U,U^{\prime
})^{\underrightarrow{~\ \ \ f^{\prime}\times Id\ \ \ ~~}}\\
&  (M,\partial M)\times(U,U^{\prime})\times...\times(U,U^{\prime
})^{\underrightarrow{~\ \ \ f\times Id\ \ \ ~~}}...\text{.}%
\end{align*}
Then $x\rightsquigarrow_{f}F(x,u_{0},...,u_{r}).$ Suppose a map $f_{0}$ is
homotopic to $f^{\prime}$ and maps $f_{1},...,f_{r}$ are homotopic to $f.$ The
result of consecutive applications of $f_{0},...,f_{r}$ is defined as a map
$G:(L,L^{\prime})\times(U,U^{\prime})^{r+1}\rightarrow(M,\partial M)$ given
by
\[
G(x,u_{0},...,u_{r})=f_{r}(...f_{1}(f_{0}(x,u_{0}),u_{1}),...,u_{r}).
\]
Therefore strong robust controllability from $L$ means that $G:L\times
U^{r+1}\rightarrow M$ is onto. By Corollary \ref{Surj} if
\[
F_{\ast}:H_{n}((L,L^{\prime})\times(U,U^{\prime})\times...\times(U,U^{\prime
}))\rightarrow H_{n}(M,\partial M)=\mathbf{Q}%
\]
is nonzero then every map homotopic to $F$ is onto. Since $G$ is clearly
homotopic to $F,$ all we need to prove is that $F_{\ast}$ is nonzero. By the
K\"{u}nneth theorem $F_{\ast}$ is given by the composition
\begin{align*}
F_{\ast} &  :H_{\ast}(L,L^{\prime})\otimes H_{\ast}(U,U^{\prime}%
)\otimes...\otimes H_{\ast}(U,U^{\prime})^{\underrightarrow{~\ \ \ f_{\ast
}^{\prime}\otimes Id\ \ \ ~~}}\\
&  H_{\ast}(M,\partial M)\otimes H_{\ast}(U,U^{\prime})\otimes...\otimes
H_{\ast}(U,U^{\prime})^{\underrightarrow{~\ \ \ f_{\ast}\otimes Id\ \ \ ~~}%
}...\text{.}%
\end{align*}
Now the condition of the theorem implies that $f_{\ast}(...f_{\ast}(f_{\ast
}^{\prime}(a_{0}\otimes v_{0})\otimes v_{2})\otimes...\otimes v_{r})\neq0$ for
some $a_{0}\in H_{p}(L,L^{\prime})$ and some $v_{0}\in H_{s_{1}}(U,U^{\prime
}),...,v_{r}\in H_{s_{r}}(U,U^{\prime})$ such that $p+s_{1}+...+s_{r}=n.$
Therefore $F_{\ast}(a_{0}\otimes v_{0}\otimes v_{2}\otimes...\otimes
v_{r})\neq0$.
\end{proof}

Moreover, it is clear that what we have is the \textquotedblleft finite time
reachability\textquotedblright, i.e., every state can be reached in a finite
number, $r+1,$ of steps and that number is common for all states.

The theorem involves multiple iterations of $f_{\ast}$ while it is preferable
to have a condition involving only $f_{\ast}$ itself. Let's consider a case
when this is possible.

Consider first a simple example, $U=\mathbf{S}^{1},$ $U^{\prime}=\varnothing,$
$M=\mathbf{T}^{n}=(\mathbf{S}^{1})^{n}$, and $f:\mathbf{S}^{1}\times
\mathbf{T}^{n}\rightarrow\mathbf{T}^{n}$ is given by $f(u,x_{1},...,x_{n}%
)=(u,x_{1},...,x_{n-1}).$ This may serve as a model for a robotic arm with $n$
joints where only the first joint can be controlled directly and the next
state of a joint is \textquotedblleft read\textquotedblright\ from the current
state of the previous joint. The system is obviously controllable. Indeed
after $n$ iterations with inputs $u_{1},...,u_{n}$ the system's state is
$(u_{n},...,u_{1}).$ Whether the system is robustly controllable is not as
obvious. The affirmative answer is provided by the theorem as follows. Let $L$
be a point, $p=0.$ Now, with $d$ the identity of $H_{1}(\mathbf{S}^{1})$ we
choose%
\begin{align*}
v_{0} &  =v_{1}=...=v_{n}=d\in H_{1}(\mathbf{S}^{1}),\text{ and}\\
a_{0} &  =1\in H_{0}(\mathbf{T}^{n}),\\
a_{1} &  =d\in H_{1}(\mathbf{T}^{n}),\\
a_{2} &  =d\otimes d\in H_{2}(\mathbf{T}^{n}),\\
&  ...\\
a_{n} &  =d\otimes...\otimes d\in H_{n}(\mathbf{T}^{n}).
\end{align*}


More generally, suppose the state space $M$ has the product structure,
$M=K_{1}\times...\times K_{s},$ where $K_{i}$ are manifolds of dimensions
$k_{i}.$ Suppose $f=(h_{1},...,h_{s})$, where $h_{i}:U\times M\rightarrow
K_{i}.$ For $i=1,...,s,$ define maps $h_{i}^{a}:K_{i-1}\rightarrow K_{i},$
where $K_{0}=U,$ by $h_{i}^{a}(x_{i-1})=h_{i}(a_{0},...,a_{i-2},x_{i-1}%
,a_{i},...,a_{s}).$ If all $h_{i}^{a}$ are onto then the system is
controllable. According to Corollary \ref{Surj} it suffices to require that
all $h_{i\ast}^{a}:H_{k_{i}}(K_{i-1})\rightarrow H_{k_{i}}(K_{i})$ are
nonzero, $i=1,...,s$.

The above theorem can be informally understood as follows. If there are some
submanifolds $M_{1},...,M_{r},$ $\dim M_{i}=n_{i},$ of $M$ such that
$M_{0}=L,M_{1}=f(M_{0}\times U),$ $M_{2}=f(M_{1}\times U),...,M=f(M_{r}\times
U)$ then the system is controllable. It means that the restrictions
$f_{0}:L\times U\rightarrow M_{1},f_{1}:M_{1}\times U\rightarrow
M_{2},...,f_{r}:M_{r}\times U\rightarrow M$ of $f$ are surjective$.$ This
holds provided $f_{i\ast}(O_{M_{i}}\otimes O_{U})=q_{i}O_{M_{i+1}},$ where
$O_{M_{i}}\in H_{n_{i}}(M_{i})$ is the fundamental class of $M_{i}$, for some
$q_{i}\in\mathbf{Q}.$ Since each $O_{M_{i}}$ corresponds to $a_{i}=J_{i\ast
}(O_{M_{i}})\in H_{n_{i}}(M),$ where $J_{i}:$ $M_{i}\rightarrow M$ is the
inclusion, we arrive at the requirement of the theorem. The robustness of each
of these surjectivity conditions can be tested by means of Corollary
\ref{NecSurj}. As a special case we have the following.

\begin{theorem}
\label{NecContr}\textbf{(Necessary condition of robust controllability)
}Suppose\textbf{ }$U$ is a manifold and there is a fiber $F=f^{-1}(x),x\in M,$
of $f$ satisfying condition (*) and
\[
f_{\ast}:H_{n}(W,W\backslash V)\rightarrow H_{n}(D,D\backslash B)=\mathbf{Q}%
\mathit{\ }\text{\textit{is zero,}}%
\]
where $V\subset\overline{V}\subset W$ and $B\subset\overline{B}\subset
D\subset M\backslash\partial M$ are neighborhoods of $F$ and $x$ respectively.
Then there is an \textit{arbitrarily} small perturbation restricted to a
neighborhood of $F$ of the system $D_{f}$ which is not controllable from $M;$
specifically, $x$ is unreachable from any point.
\end{theorem}

\begin{proof}
Corollary \ref{NecSurj} implies that there is $g$ homotopic to $f$ such that
$x\notin g(M\times U).$ 
\end{proof}

\section{Continuous systems.\label{Controllability of continuous systems}}

In this section we outline, in less details than above, the possibilities of
applying Lefschetz numbers to continuous systems.

In this section $M$\textit{ }is a compact orientable connected smooth manifold
with boundary $\partial M$, $\dim M=n.$ Let $TM$ be the tangent bundle of $M,$
then $\dim TM=2n$.

A \textit{continuous time control system} $C_{h}$ \cite[p. 16]{NvdS} is
defined as a commutative diagram
\[%
\begin{tabular}
[c]{ccc}%
$Q$ & $^{\underrightarrow{\quad h\quad}}$ & $TM,$\\
$\downarrow^{p}$ & $\swarrow^{\pi_{M}}$ & \\
$M$ &  &
\end{tabular}
\ \ \ \ \ \ \ \
\]
where $p:Q\rightarrow M$ is a fiber bundle over $M$ and $\pi_{M}$ is the
projection$.$ Thus $C_{h}$ is a parametrized vector field on $M.$ 

We say that $x\in M$ is an \textit{equilibrium }of this system\textit{ }if
there is $y\in Q$ such that $h(y)=(x,0)\in TM,x=p(y)\in M.$ Detecting an
equilibrium can be restated as a coincidence problem. Suppose $i:M\rightarrow
TM$ is the inclusion and $p_{1}:Q\times M\rightarrow Q,$ $p_{2}:Q\times
M\rightarrow M$ are the projections. Define the maps $f,g:Q\times M\rightarrow
TM$ by $f=hp_{1},g=ip_{2}.$ Then a coincidence of the pair $f,g$ is an
equilibrium of the system $C_{h}.$ Therefore equilibria can be detected by
means of the coincidence results in Section \ref{Lefschetz homomorphism} and
their robustness can be studied by means of the results of Section
\ref{Removability of coincidences}.

We have a simpler coincidence problem when $M\ $is parallelizable, i.e., $TM$
is isomorphic to $M\times\mathbf{R}^{n}$. For example, $\mathbf{S}^{1},$
$\mathbf{S}^{3},$ $\mathbf{S}^{7}$ are parallelizable. Let $q:TM\simeq
M\times\mathbf{R}^{n}\rightarrow M$ be the projection. Then a coincidence of
the pair $qh,p$ is an equilibrium of the system $C_{h}$ and we can use Theorem
\ref{Lef-type} to detect equilibria and Theorem \ref{Remov} to study their
robustness. In fact $D_{qh}$ is a discrete control system associated with the
continuous system $C_{h}.$ In particular, when $Q=M\times U,$ the results of
Sections \ref{Equilibria of discrete systems} and \ref{Controllability} can be
applied to study equilibria and controllability of $C_{h}.$

For a general $M$ a discrete system $D_{f}$ associated to the continuous
system $C_{h}$ may be constructed as follows.

Let $\mathcal{A}$ be the topological space of \textit{admissible controls
}associated with $C_{h}$\textit{, }i.e., a set of
functions\ $z:[0,d]\rightarrow Q,$ for all $d\in\mathbf{R}.$ A map
$c_{z}:[0,d]\rightarrow M$ is called a \textit{trajectory} of the control
system if there exists a control\ $z\in\mathcal{A}$ satisfying: $pz=c_{z}$ and
$\dfrac{d}{dt}c_{z}=hz.$

We assume that $Q=M\times U,$ where $U$ is the topological space of all
possible inputs, and $p:Q=M\times U\rightarrow M$ is the projection. Then
$\mathcal{A}$ is the set of pairs $(c,p),$ where $c:[0,d]\rightarrow M$ is a
trajectory and $p:[0,d]\rightarrow U$ is a function representing the input$.$
To simplify things even further we consider only constant inputs. First we
assume that the system $C_{h}$ satisfies the following existence and
uniqueness property: for every $x\in M$ and any \textit{constant} input
$p(t)=u\in U$ there is a unique trajectory $c$ such that $c(0)=x$ and
$(c,p)\in\mathcal{A}$. Then the following end point map $f_{d}:M\times
U\rightarrow M$ is well defined. We let $f_{d}(x,u)=c(d),$ where
$c:[0,d]\rightarrow M$ is the above trajectory. Assume also that the map
$f=f_{d}$ is continuous. Then for each $d\geq0$ we have a discrete time
control system $D_{f}$.

Next, the system $C_{h}$ is called \textit{controllable} if any state can be
reached from any other state, i.e., for each pair of states $x,y\in M$ there
is a trajectory $c:[0,d]\rightarrow M$ such that $x=c(0),y=c(d).$

We make the same assumption about $D_{f}$ as in Section \ref{Controllability}:
if the initial state lies at the boundary $\partial M$ of $M$ then the next
state, regardless of the input, lies within a certain neighborhood $W$ of
$\partial M,$ or, alternatively, $f(\partial M\times U)\subset\partial M$. In
particular, this condition is satisfied if $h(x,u)$ is tangent to $\partial M$
for all $x\in\partial M$. Let $U^{\prime}$ be the set of controls that take
any given state to the the boundary $\partial M,$ i.e.,
\[
U^{\prime}=\{u\in U:f(x,u)\in\partial M\text{ for all }x\in M\}.
\]
Then $f$ is a map of pairs, $f:(M,\partial M)\times(U,U^{\prime}%
)\rightarrow(M,\partial M).$ Given a subset $L$ of $M,$ let $L^{\prime}%
=L\cap\partial M$ and let $f^{\prime}:(L,L^{\prime})\times(U,U^{\prime
})\rightarrow(M,\partial M)$ be the restriction of $f.$

\begin{theorem}
\textbf{(Sufficient condition of controllability) }Suppose that there are
$a_{0}\in H_{p}(L,L^{\prime}),$ $v_{0}\in H_{s_{0}}(U,U^{\prime}),...,v_{r}\in
H_{s_{r}}(U,U^{\prime})$ such that
\[
a_{1}=f_{\ast}^{\prime}(a_{0}\otimes v_{0}),a_{2}=f_{\ast}(a_{1}\otimes
v_{1}),...,a_{r+1}=f_{\ast}(a_{r}\otimes v_{r})\neq0.
\]
Then the continuous time system $C_{h}$ is controllable from $L$ by means of
piece-wise constant controls.
\end{theorem}

\begin{proof}
The discrete system $D_{f}$ is controllable from $L$ by Theorem
\ref{SuffControl}.
\end{proof}

It follows also that if for a small enough $\varepsilon>0$ a map
$k:Q\rightarrow TM$ satisfies $d(k(z),h(z))<\varepsilon$ for all $z\in Q$,
where $d$ is the distance on $TM,$ and the system $C_{k}$ satisfies all of the
above assumptions, then $C_{k}$ is also controllable. We can say then that
$C_{h}$ is \textit{robustly controllable}$.$

Consider the applicability of this theorem to local controllability or
controllability in a Euclidean space. In either case $M$ is the $n$-ball. Then
$H_{i}(M,\partial M)$ is nontrivial only in dimension $n$. As a result the
above \textquotedblleft chain\textquotedblright\ of homology classes
$a_{1},a_{2},...,a_{r+1}$ has to have only one \textquotedblleft
link\textquotedblright, $a_{1}=f_{\ast}^{\prime}(a_{0}\otimes v_{0})\in
H_{n}(M,\partial M)\backslash\{0\}$. Thus the theorem reduces to the claim of
one-step controllability provided $f_{\ast n}^{\prime}$ is nonzero. As a
result the similarity between the homology reachability condition of the
theorem and the Lie bracket condition \cite[Section 3.1]{NvdS} does not
materialize. I believe however that a generalization of Theorem
\ref{SuffControl} will provide a necessary connection.

Observe also that if $\partial M=\varnothing$, then $f=f_{d}$ is homotopic to
the constant map $f_{0}$ under the homotopy $H(t,x,u)=f_{t}(x,u),$ hence
$f_{\ast}=0$. Therefore the condition of the theorem is never satisfied.

Here's another approach to controllability. Let $\mathcal{A}^{\prime}$ be the
set of controls whose trajectories have one of the end points at the boundary
of $M,$ i.e.,%
\[
\mathcal{A}^{\prime}=\{z:[0,d]\rightarrow Q,z\in\mathcal{A},c_{z}%
(0)\in\partial M\text{ or }c_{z}(d)\in\partial M\}.
\]
Define $G(u)=(c_{z}(0),c_{z}(d))$, the end points of the trajectory
$c_{z}=pz:[0,d]\rightarrow M$ corresponding to $z$. Then $G:(\mathcal{A}%
,\mathcal{A}^{\prime})\rightarrow(M\times M,\partial(M\times M))$ is a well
defined map of pairs.

\begin{theorem}
\textbf{\label{SuffControlCont}(Sufficient condition of controllability) }If
\[
G_{\ast}:H_{2n}(\mathcal{A},\mathcal{A}^{\prime})\rightarrow H_{2n}(M\times
M,\partial(M\times M))=\mathbf{Q}\text{ is non-zero}%
\]
then the continuous time system $C_{h}$ is controllable.
\end{theorem}

\begin{proof}
By Corollary \ref{Surj} $G$ is onto.
\end{proof}

A\ similar condition is found in \cite{Nistri}, where a boundary operator
$l:AC([0,1],\mathbf{R}^{n})\times L^{\infty}([0,1],\mathbf{R}^{n}%
)\rightarrow\mathbf{R}^{p}$ is considered instead of $G.$ One of the
conditions of controllability is $\deg l_{0}\neq0,$ where $l_{0}$ is the
restriction of $l$ to some $p$-dimensional subspace and $\deg l_{0}$ its
topological degree.

\begin{thebibliography}{99}                                                                                               %


\bibitem {Balach}K. Balachandran and J. P. Dauer, Controllability of nonlinear
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\bibitem {Bredon}G. E. Bredon, \textquotedblleft Topology and
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\bibitem {Brown}R. F. Brown, \textquotedblleft The Lefschetz Fixed Point
Theorem\textquotedblright. Scott-Foresman, Chicago 1971.

\bibitem {Brown2}R. F. Brown, Fixed point theory. In \textquotedblleft History
of topology\textquotedblright, pp. 271--299, North-Holland, Amsterdam, 1999.

\bibitem {BS}R. F. Brown and H. Schirmer, Nielsen coincidence theory and
coincidence-producing maps for manifolds with boundary, \textit{Topology
Appl.} \textbf{46} (1992), 65-79.

\bibitem {BL}F. Bullo, A. D. Lewis, \textquotedblleft Geometric Control of
Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical
Control Systems\textquotedblright. Springer-Verlag, 2004.

\bibitem {Carmichael}N. Carmichael and M. D. Quinn, Fixed-point methods in
nonlinear control. \textit{IMA J. Math. Control Inform}. \textbf{5} (1988) 1, 41--67.

\bibitem {CNZ}G. Conti, P. Nistri, and P. Zecca, Controllability problems via
set-valued maps. Recent advances in mathematical theory of systems, control,
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\bibitem {DG}D. Dimovski and R. Geoghegan, One-parameter fixed point theory.
\textit{Forum Math.} \textbf{2} (1990) 2, 125--154.

\bibitem {Fuller}F. B. Fuller, The homotopy theory of coincidences.
\textit{Ann. of Math.} (2) \textbf{59} (1954), 219--226.

\bibitem {Geoghegan}R. Geoghegan, Nielsen Fixed Point Theory. In
\textquotedblleft Handbook of Geometric Topology\textquotedblright, R.
Daverman and R. Sher, Eds, Elsevier Publishers, 2001.

\bibitem {GN2}R. Geoghegan and A. Nicas, Trace and torsion in the theory of
flows. \textit{Topology} \textbf{33} (1994), 683--719.

\bibitem {GNO}R. Geoghegan, A. Nicas, and J. Oprea, Higher Lefschetz traces
and spherical Euler characteristics. \textit{Trans. Amer. Math. Soc.}
\textbf{348} (1996), 2039--2062.

\bibitem {GJW}D. Gon\c{c}alves, J. Jezierski, and P. Wong, Obstruction theory
and coincidences in positive codimension. \textit{Preprint}.

\bibitem {Gorn-book}L. G\'{o}rniewicz, \textquotedblleft Topological fixed
point theory of multivalued mappings\textquotedblright. Mathematics and its
Applications. 495. Dordrecht: Kluwer Academic Publishers, 1999.

\bibitem {Id}D. Idczak, Applications of the fixed point theorem to problems of
controllability. \textit{Bull. Soc. Sci. Lett. L\'{o}dz} \textbf{39} (1989) 3,
7 pp.

\bibitem {Jez}J. Jezierski, One codimensional Wecken type theorems.
\textit{Forum Math. }\textbf{5} (1993) 5, 421--439.

\bibitem {Jon}E. A. Jonckheere, \textquotedblleft Algebraic and Differential
Topology of Robust Stability\textquotedblright, Oxford University Press, 1997.

\bibitem {KMM}T. Kaczynski, K. Mischaikow, and M. Mrozek, \textquotedblleft
Computational Homology\textquotedblright, Springer-Verlag, 2004.

\bibitem {Kappos-Conley}E. Kappos, The Conley index and global bifurcations.
I: Concepts and theory. \textit{Int. J. Bifurcation Chaos Appl. Sci. Eng.}
\textbf{5} (1995) 4, 937-953.

\bibitem {Kappos-Morse}E. Kappos, The role of Morse-Lyapunov functions in the
design of nonlinear global feedback dynamics. In Zinober, Alan S. I. (ed.),
\textquotedblleft Variable structure and Lyapunov control.\textquotedblright%
\ Berlin: Springer-Verlag. Lect. Notes Control Inf. Sci. 193, 249-267 (1994).

\bibitem {Kappos4}E. Kappos, Necessary conditions for the design of global
feedback dynamics, Proc. NOLTA '95, Dec.95.

\bibitem {Klamka}J. Klamka, Schauder's fixed-point theorem in nonlinear
controllability problems. \textit{Control Cybernet.} \textbf{29} (2000) 1, 153--165.

\bibitem {Knill}R. J. Knill, On the homology of the fixed point
set.\ \textit{Bull. Amer. Math. Soc.} \textbf{77} (1971), 184--190.

\bibitem {Latombe}J.-C. Latombe, \textquotedblleft Robot Motion
Planning\textquotedblright, Kluwer Academic Publishers, 1991.

\bibitem {Misch}K. Mischaikow, Topological Techniques for Efficient Rigorous
Computations in Dynamics. \textit{Acta Numerica }\textbf{11} (2003), 435-477.

\bibitem {NvdS}H. Nijmeijer and A. van der Schaft, \textquotedblleft Nonlinear
dynamical control systems\textquotedblright. New York, Springer-Verlag, 1990.

\bibitem {Nistri}P. Nistri, On a general notion of controllability for
nonlinear systems. \textit{Boll. Un. Mat. Ital.} C (6) \textbf{5} (1986) 1,
383--403 (1987).

\bibitem {Sastry}S. Sastry, \textquotedblleft Nonlinear systems. Analysis,
stability, and control.\textquotedblright\ Interdisciplinary Applied
Mathematics. 10. New York, NY: Springer, \ 1999.

\bibitem {Sav}P. Saveliev, A Lefschetz-type coincidence theorem. \textit{Fund.
Math.} \textbf{162} (1999), 65--89.

\bibitem {Sav1}P. Saveliev, The Lefschetz coincidence theory for maps between
spaces of different dimensions. \textit{Topology Appl.} \textbf{116} (2001) 1, 137-152.

\bibitem {Sav2}P. Saveliev, Removing coincidences of maps between manifolds of
different dimensions. \textit{Top. Meth. Nonlin. Anal.} \textbf{22} (2003) 1,
105-114.\textit{\ }

\bibitem {Sav4}P. Saveliev, Higher order Nielsen numbers. To appear in
\textit{Fixed Point Theory Appl..} Preprint: http://front.math.ucdavis.edu/math.GT/0202113

\bibitem {Sontag}E. D. Sontag, \textquotedblleft Mathematical control theory.
Deterministic finite dimensional systems.\textquotedblright\ 2nd ed. Texts in
Applied Mathematics. 6. New York, NY: Springer, 1998.

\bibitem {Vick}J. W. Vick, \textquotedblleft Homology Theory, An Introduction
to Algebraic Topology\textquotedblright. Springer-Verlag, 1994.
\end{thebibliography}


\end{document}
math0402346.extracted_bib
\begin{thebibliography}{99}                                                                                               %


\bibitem {Balach}K. Balachandran and J. P. Dauer, Controllability of nonlinear
systems via fixed-point theorems. \textit{J. Optim. Theory Appl.} \textbf{53}
(1987) 3, 345--352.

\bibitem {Bredon}G. E. Bredon, \textquotedblleft Topology and
Geometry\textquotedblright. Springer-Verlag, 1993.

\bibitem {Brown}R. F. Brown, \textquotedblleft The Lefschetz Fixed Point
Theorem\textquotedblright. Scott-Foresman, Chicago 1971.

\bibitem {Brown2}R. F. Brown, Fixed point theory. In \textquotedblleft History
of topology\textquotedblright, pp. 271--299, North-Holland, Amsterdam, 1999.

\bibitem {BS}R. F. Brown and H. Schirmer, Nielsen coincidence theory and
coincidence-producing maps for manifolds with boundary, \textit{Topology
Appl.} \textbf{46} (1992), 65-79.

\bibitem {BL}F. Bullo, A. D. Lewis, \textquotedblleft Geometric Control of
Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical
Control Systems\textquotedblright. Springer-Verlag, 2004.

\bibitem {Carmichael}N. Carmichael and M. D. Quinn, Fixed-point methods in
nonlinear control. \textit{IMA J. Math. Control Inform}. \textbf{5} (1988) 1, 41--67.

\bibitem {CNZ}G. Conti, P. Nistri, and P. Zecca, Controllability problems via
set-valued maps. Recent advances in mathematical theory of systems, control,
networks and signal processing, II (Kobe, 1991), 253--258, Mita, Tokyo, 1992.

\bibitem {DG}D. Dimovski and R. Geoghegan, One-parameter fixed point theory.
\textit{Forum Math.} \textbf{2} (1990) 2, 125--154.

\bibitem {Fuller}F. B. Fuller, The homotopy theory of coincidences.
\textit{Ann. of Math.} (2) \textbf{59} (1954), 219--226.

\bibitem {Geoghegan}R. Geoghegan, Nielsen Fixed Point Theory. In
\textquotedblleft Handbook of Geometric Topology\textquotedblright, R.
Daverman and R. Sher, Eds, Elsevier Publishers, 2001.

\bibitem {GN2}R. Geoghegan and A. Nicas, Trace and torsion in the theory of
flows. \textit{Topology} \textbf{33} (1994), 683--719.

\bibitem {GNO}R. Geoghegan, A. Nicas, and J. Oprea, Higher Lefschetz traces
and spherical Euler characteristics. \textit{Trans. Amer. Math. Soc.}
\textbf{348} (1996), 2039--2062.

\bibitem {GJW}D. Gon\c{c}alves, J. Jezierski, and P. Wong, Obstruction theory
and coincidences in positive codimension. \textit{Preprint}.

\bibitem {Gorn-book}L. G\'{o}rniewicz, \textquotedblleft Topological fixed
point theory of multivalued mappings\textquotedblright. Mathematics and its
Applications. 495. Dordrecht: Kluwer Academic Publishers, 1999.

\bibitem {Id}D. Idczak, Applications of the fixed point theorem to problems of
controllability. \textit{Bull. Soc. Sci. Lett. L\'{o}dz} \textbf{39} (1989) 3,
7 pp.

\bibitem {Jez}J. Jezierski, One codimensional Wecken type theorems.
\textit{Forum Math. }\textbf{5} (1993) 5, 421--439.

\bibitem {Jon}E. A. Jonckheere, \textquotedblleft Algebraic and Differential
Topology of Robust Stability\textquotedblright, Oxford University Press, 1997.

\bibitem {KMM}T. Kaczynski, K. Mischaikow, and M. Mrozek, \textquotedblleft
Computational Homology\textquotedblright, Springer-Verlag, 2004.

\bibitem {Kappos-Conley}E. Kappos, The Conley index and global bifurcations.
I: Concepts and theory. \textit{Int. J. Bifurcation Chaos Appl. Sci. Eng.}
\textbf{5} (1995) 4, 937-953.

\bibitem {Kappos-Morse}E. Kappos, The role of Morse-Lyapunov functions in the
design of nonlinear global feedback dynamics. In Zinober, Alan S. I. (ed.),
\textquotedblleft Variable structure and Lyapunov control.\textquotedblright%
\ Berlin: Springer-Verlag. Lect. Notes Control Inf. Sci. 193, 249-267 (1994).

\bibitem {Kappos4}E. Kappos, Necessary conditions for the design of global
feedback dynamics, Proc. NOLTA '95, Dec.95.

\bibitem {Klamka}J. Klamka, Schauder's fixed-point theorem in nonlinear
controllability problems. \textit{Control Cybernet.} \textbf{29} (2000) 1, 153--165.

\bibitem {Knill}R. J. Knill, On the homology of the fixed point
set.\ \textit{Bull. Amer. Math. Soc.} \textbf{77} (1971), 184--190.

\bibitem {Latombe}J.-C. Latombe, \textquotedblleft Robot Motion
Planning\textquotedblright, Kluwer Academic Publishers, 1991.

\bibitem {Misch}K. Mischaikow, Topological Techniques for Efficient Rigorous
Computations in Dynamics. \textit{Acta Numerica }\textbf{11} (2003), 435-477.

\bibitem {NvdS}H. Nijmeijer and A. van der Schaft, \textquotedblleft Nonlinear
dynamical control systems\textquotedblright. New York, Springer-Verlag, 1990.

\bibitem {Nistri}P. Nistri, On a general notion of controllability for
nonlinear systems. \textit{Boll. Un. Mat. Ital.} C (6) \textbf{5} (1986) 1,
383--403 (1987).

\bibitem {Sastry}S. Sastry, \textquotedblleft Nonlinear systems. Analysis,
stability, and control.\textquotedblright\ Interdisciplinary Applied
Mathematics. 10. New York, NY: Springer, \ 1999.

\bibitem {Sav}P. Saveliev, A Lefschetz-type coincidence theorem. \textit{Fund.
Math.} \textbf{162} (1999), 65--89.

\bibitem {Sav1}P. Saveliev, The Lefschetz coincidence theory for maps between
spaces of different dimensions. \textit{Topology Appl.} \textbf{116} (2001) 1, 137-152.

\bibitem {Sav2}P. Saveliev, Removing coincidences of maps between manifolds of
different dimensions. \textit{Top. Meth. Nonlin. Anal.} \textbf{22} (2003) 1,
105-114.\textit{\ }

\bibitem {Sav4}P. Saveliev, Higher order Nielsen numbers. To appear in
\textit{Fixed Point Theory Appl..} Preprint: http://front.math.ucdavis.edu/math.GT/0202113

\bibitem {Sontag}E. D. Sontag, \textquotedblleft Mathematical control theory.
Deterministic finite dimensional systems.\textquotedblright\ 2nd ed. Texts in
Applied Mathematics. 6. New York, NY: Springer, 1998.

\bibitem {Vick}J. W. Vick, \textquotedblleft Homology Theory, An Introduction
to Algebraic Topology\textquotedblright. Springer-Verlag, 1994.
\end{thebibliography}