Intersection Theory of Boundary Tangency Manifolds for State Feedback Systems

Abstract

This paper presents a new approach to the classification problem of topological structures of feedback controlled systems for time-invariant general nonlinear state equations. The generalized Poincaré-Hopf index theorem by C.C. Pugh and C. McCord leads us to a notion of “Boundary tangency manifolds of the state equation”. We show a topological method to judge whether the controlled system is circumscribed or inscribed at a point of the boundary of a compact and connected submanifold in the state space. For this purpose, we discuss the intersection theory for boundary tangency manifolds and an input manifold which is the geometrical object for a differentiable state feedback control law.