Intersection Theory on Configuration Spaces and its Applications-I

Structures of Null Sets of Nonlinear State Equations

Abstract

We study global control problems for time-independent nonlinear state equations from a view point of an intersection theory on “configuration spaces of control systems”. Singularities of a feedback controlled system appear as the intersection of the null set of a state equation and an input manifold that is a geometrization of a feedback control law. We decompose the null set of the state equation into the null manifold, the degenerated null manifold and the set of critical points in relation to a feature of singularities of feedback controlled systems. We also discuss generic properties of control laws in the case that the controlled system has only simple or hyperbolic singularities.