ar X iv : m at h / 04 02 34 6 v 1 [ m at h . O C ] 2 1 Fe b 20 04 Applications of Lefschetz numbers in control theory

The goal of this paper is to develop some applications of the Lefschetz fixed point theory techniques, already available in dynamics, in control theory. A dynamical system on a manifold M is a map f : M → M. The current state, x ∈ M, of the system determines the next state, f(x), and its equilibria are the 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 6= 0 then there is at least one coincidence. Moreover, since all properties established through homology are ”robust by nature”, any pair f , g homotopic to f, g has a coincidence as well. In the control situation, the next state f(x, u) depends on the current one, x ∈ M, as well as the input, u ∈ U. It is described by a map f : N = U×M → M and its equilibria are the coincidences of f and the projection g : N = U × M → M . Since in this case the dimensions of N and M are not equal, the Lefschetz number has to be replaced with the so-called Lefschetz homomorphism. In this paper the Lefschetz homomorphism is applied to detection of equilibria and controllability. The secondary objective is to study robustness of these properties; for example, we find out when small perturbations of the systems can lead to the loss of equilibria.