Invariant Manifolds, Zero Dynamics and Stability

Abstract

We consider ordinary differential equations which can be written as coupled pairs

$$\dot x = g(t,x,y),\dot y = h(t,x,y)$$

(1.1)

with x ∈ ℝⁿ and y ∈ ℝ^m. It will be tacitly assumed throughout this paper that g, h and all mappings — as s and w — which appear in the sequel are everywhere defined smooth CN-functions of their respective variables for some appropriate integer N ≥ 1. That solutions of an ordinary differential equation exist on a given (finite) time interval will also be taken for granted. Concerning (1.1) our basic assumption is

$$h(t,x,0) = 0$$

(1.2)

so that y = 0 represents a global invariant manifold for the system (1.1). The differential equation

$$\dot x = g(t,x,0)$$

(1.3)

Then describes the dynamics which prevail within this basic invariant man-ifold. For shortness we refer to (1.3) as to the differential equation of the “zero dynamics” for (1.1) with (1.2).

This work has been supported by Deutsche Forschungsgemeinschaft Kn 164/3-3.