SIAM Journal of Control and Optimization, 44 (2005) 5, 1677-1690, also an invited talks at the Joint Mathematics Meetings in January 2004, Phoenix, and at International Conference on Nielsen Theory and Related Topics, Memorial University of Newfoundland, St John's, Canada, June 28 - July 2, 2004.
The goal of this paper is to develop some applications the Lefschetz theory techniques, already available in dynamics, in control theory. A dynamical system on a manifold M is determined by a map f:M→M. The current state, xÎ M, of the system determines the next state, f(x). The equilibria are fixed points of f, f(x)=x. More generally, one deals with coincidences of a pair of maps f,g:N→M, f(x)=g(x), between manifolds of the same dimension. The main tool is the Lefschetz number λfg defined in terms of the homology of M,N,f: if λfg≠0 then there is at least one coincidence. In the control situation, the next state f(x,u) in determined by the current one, xÎ M, as well as by the input, uÎU. It is described by a map f:N=U×M→M and its equilibria are coincidences of f and the projection g:N=U×M→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.