MATH




Lefschetz and Nielsen numbers in Control Theory.

Peter Saveliev

Department of Mathematics, Marshall University

saveliev@marshall.edu

Suppose $M$ is a manifold and $f:M\rightarrow M$ is a map. Then $x\in M$ is the current state of a discrete dynamical system on $M$ and $f(x)$ is the next state.
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An equilibrium $x\in M$ is a fixed point of $f,$ $f(x)=x.$
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The Lefschetz number of $f$ is defined by:
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Lefschetz coincidence theorem: if $\lambda _{f}\neq 0$ then any $f^{\prime }$ homotopic to $f$ has at least one fixed point.

If $\lambda _{f}\neq 0$ then any perturbation of the system has at least one equilibrium.




Suppose $M$ is a manifold and MATH is a map.

$x\in M$ is the current state of a discrete control system on $M$ and $f(x,u)$ is the next state, where $u\in U$ is the input.
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An equilibrium pair $(x,u)\in M\times U$ is a coincidence point of $f$ and the projection MATH $f(x,u)=p(x,u)=x.$
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(1) $M=\QTR{bf}{T}^{n}$ is the configuration space of a robotic arm with $n$ revolving joints;

(2) MATH is the configuration space of a rigid body in space;

(3) $N=T\QTR{bf}{S}^{2}$ is the state-input space of a spherical pendulum with a gas jet control which is always directed in the tangent space.







How to define the Lefschetz coincidence number so that if MATH then there is at least one coincidence?
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(1) Suppose $M,N$ are compact connected orientable $n$-manifolds and $f,g:N\rightarrow M$ are maps. Then the Lefschetz number of $f,g$ is defined by:
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where $D_{N}$ and $D_{M}$ are the Poincaré duality isomorphisms for manifolds $M$ and $N.$

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(2) Suppose MATH open, $g:N\rightarrow M$ is Vietoris: (i) $g$ is proper (i.e. $g^{-1}(B)$ is compact for any compact $B\subset Y),$ (ii) $g$ has acyclic fibers. Then
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(existence of $g_{\ast }^{-1}$ guaranteed by the Vietoris Mapping Theorem).

Suppose $N$ is an arbitrary topological space, $A\subset N$, $M$ is an orientable compact connected manifold, $\dim M=n$, and MATH $g:N\rightarrow M$ are maps .
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$E=H_{\ast }(M)$ is equipped with the cap product MATH, one can define the "Lefschetz class" $L(h)\in E$ of an endomorphism $h$ of any degree.
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Theorem

If MATH is a homomorphism of degree $m$ then
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where MATH is a basis for $H_{k}(M)$ and MATH the corresponding dual basis for $H^{k}(M).$


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Proposition

If the degree of $h$ is zero, MATH.

For a given $z\in H_{k}(N,A),$ suppose $h_{fg}^{z}$ is defined as the composition
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i.e., MATH
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The Lefschetz homomorphism MATH $k=0,1,...,$ is a homomorphism of degree $(-n)$ defined as
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MATH

Proposition

Since MATH where $O_{N}$ is the fundamental class of $N,$we recover the Lefschetz number, MATH










Theorem

(Existence of coincidences) If $\Lambda_{fg}\neq 0$ then any pair of maps MATH homotopic to $f,g$ has a coincidence.





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Corollary

(Existence of equilibria) Given a control system MATH suppose the homomorphism MATH of degree $k$ is defined by MATH for each $v\in H_{k}(U).$

If for some $v\in H_{\ast }(U)$
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where MATH is a basis for $H_{k}(M)$ and MATH the corresponding dual basis for $H^{k}(M),$ then every perturbation of the system has an equilibrium.








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Example: Suppose $M=\QTR{bf}{S}^{n},$ then
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where $\overline{O}_{M}$ is the dual of $O_{M}.$ The second term vanishes unless $|v|=0.$ Therefore the system has an equilibrium if MATH for some MATH
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Example: If MATH is the multiplication of a compact Lie group and $p$ is the projection, then for $u\in H_{m}(M)$
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Therefore the system has a equilibrium.







Suppose $M$ and $N$ are smooth manifolds and MATH are smooth maps, $A\ $is a closed subset of $N$.
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If $\dim N=n+m,m>0,$ the vanishing of the Lefschetz number $\Lambda _{fg}$ does not guarantee that the coincidence set can be removed by homotopies of $f,g$.
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A closed submanifold $F$ of $N$ satisfies condition (A) if one of the following three conditions holds:

(1) $m=4$ and $n\geq6;$

(2) $m=5$ and $n\geq7;$

(3) $m=12$ and MATH.




Theorem

(Removability of coincidences) Suppose condition (A) is satisfied for $F=C_{fg},$ the coincidence set of $f,g$, MATH MATH and MATH

If for all MATH
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then there is a homotopy of $f$ to a smooth map $f^{\prime }$ such that the new pair has no coincidences.

The homotopy can be chosen arbitrary small and constant on a compliment of a neighborhood of $F.$




Suppose the control system is given by a fiber bundle MATH and a map $f:N\rightarrow M.$ Here $U$ is the space of inputs, $M$ is the space of states of the system.




Corollary

(Robustness of equilibria) Suppose $f$ is smooth, and there is only one equilibrium state, MATH Suppose condition (A) is satisfied for the equilibrium manifold MATH and MATH.

If
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where MATH then there is an arbitrary small perturbation of the system which has no equilibria.




Example: MATH $n$ odd, has degree $1$.




Corollary

(Sufficient condition of surjectivity) If
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then any map MATH homotopic to $f$ is onto.




The system given by MATH is called controllable if any state can be reached from any other state.

Given a submanifold $L$ of $M,$ let MATH be the restriction of $f.$ Then the system is called robustly controllable from $L$ if there is $\varepsilon >0$ such that for any map $f_{0}$ $\varepsilon $-homotopic to $f^{\prime },$ maps $f_{1},...,f_{s}$ $\varepsilon $-homotopic to $f,$ and for each $y\in M$ there are $x\in L$ and inputs MATH such that
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The system is called strongly robustly controllable from $L$ if this condition is satisfied for all perturbations of $f.$

Theorem

(Sufficient condition of robust controllability) Suppose MATH Suppose that there are MATH MATH such that
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where $a_{s}=O_{M},$ the fundamental class of $M.$

Then the system is strongly robustly controllable from $L$.




The restrictions of $f,$ MATH MATH MATH are onto, where $M_{0},M_{1},...$ are submanifolds of $M$ with MATH







Example: MATH MATH and MATH A robotic arm with $n$ joints where only the first joint can be controlled directly and the next state of a joint is "read" from the current state of the previous joint. Then this system is controllable. Indeed after $n$ iterations with inputs $u_{1},...,u_{n}$ the system's state is $(u_{n},...,u_{1}).$
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Theorem

Suppose MATH where $K_{i}$ are manifolds of dimensions $n_{i}.$ Suppose MATH, where MATH Suppose for $i=1,...,s,$ maps MATH where $K_{0}=U,$ are given by
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If all MATH are nonzero, $i=1,...,s,$ are onto then the system is controllable.









Suppose the control system has the output function $h:M\rightarrow V.$

For each state $a\in M,$ let $z=H_{u}(a)$ be the output of the state achieved from $a$ after the inputs $u_{1}...,u_{s}$ are applied, i.e., map MATH is defined for each $u=(u_{1}...,u_{s})$ by
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Then the set of states indistinguishable from $a$ is
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The system is observable if $I_{a}=\{a\}$ for all $a.$




Given $a\in M$ and $u\in U^{s},$ let $b_{u}=H_{u}(a).$ Then MATH

Similarly to the classical Nielsen theory we minimize the number of components of $I_{a}$ under homotopies of $f$ and $h.$

Two points $x,y\in I_{a}$ are Nielsen equivalent if there is a path $\alpha $ between $x$ and $y$ so that for all $s$ and all $u\in U^{s},$ $H_{u}\alpha $ is homotopic relative to the end-points to the constant path $\beta _{u}=b_{u}.$

An essential class is one that cannot be removed by homotopies of $f$ and $h.$

The parametric Nielsen number $N^{U}(H_{u})$ is the number of essential classes.
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Theorem

If every perturbation of the system is observable then $N^{U}(H_{u})=1$.




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