##### Tools

This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Applications of Lefschetz numbers in control theory by Saveliev

The goal of this paper is to develop some applications the Lefschetz fixed point theory techniques, already available in dynamics, in control theory. A dynamical system on a manifold $M$ is determined by a map $f:M\rightarrow M$. The current state, $x\in M$, of the system determines the next state, $f(x)$. The equilibria are the fixed points of $f$: $$f(x)=x.$$ More generally, one studies coincidences of a pair of maps $f,g:N\rightarrow M$: $$f(x)=g(x),$$ between two manifolds of the same dimension. The main tool is the Lefschetz number $\lambda _{fg}$ defined in terms of the homology of $M,N,f$: if $\lambda_{fg}≠0$ then there is at least one coincidence. In the control situation, the next state $f(x,u)$ is determined by the current state, $x\in M$, as well as by the input, $u\in U$. It is described by a map $f:N=U×M \rightarrow M$ and its equilibria are the coincidences of $f$ and the projection $g:N=U×M \rightarrow M$. The Lefschetz coincidence theory has to be generalized because in this case the dimensions of $N$ and $M$ are not equal. In particular, the Lefschetz number has to be replaced with the Lefschetz homomorphism, see Lefschetz coincidence theory for maps between spaces of different dimensions. The latter detects equilibria when the former fails. In this paper the Lefschetz number and homomorphism are applied to detection of equilibria, controllability, and their robustness.