Intersection Theory on Configuration Spaces and its Applications-II

Local Intersection Numbers and Projective Dimensions of Null Manifolds

Abstract

This paper is concerned with global control problems for time-independent nonlinear state equations from a view point of an intersection theory on “configuration spaces of control systems”. In this paper, we show a relationship between parity of Morse indices of hyperbolic points on a controlled system and local intersection numbers of an input manifold and a null manifold. We also discuss this result for cases that the closure of a null manifold has self-intersections or intersections to the closure of other unit components. Moreover, we define the projective dimension of the null manifold to disscuss the assignability of hyperbolic points on controlled systems.