Moduli and canonical forms for linear dynamical systems II: The topological case

Abstract

In this paper we study real linear dynamical systems$\dot x = Fx + Gu,y = Hx,x \in R^n$ = state space,u ∈ R ^m = input space,y ∈ R ^p = output space, under the equivalence relation induced by base change in state space; or in other words we study triples of matrices with real coefficients (F, G, H) of sizesn × n, n × m, p × n respectively, under the action(F, G, H.) →(TFT ⁻¹,TG, HT ⁻¹) ofGL _n (R), the group of invertible realn × n matrices. One of the central questions studied is: “do there exist continuous canonical forms for this equivalence relation?”. After various trivial obstructions to the existence of such forms have been removed the answer is very roughly: no ifm ≥ 2, p ≥ 2, yes ifm = 1, orp = 1. For a precise statement cf. theorem 1.7.

Existence or nonexistence of continuous canonical forms is related to the existence of a universal family of real linear dynamical systems. More precisely continuous canonical forms exist if such a universal family exists and if the underlying vector bundle of this family is the trivial vector bundle. In the case studied we show that a universal family in the appropriate sense does exist.

The methods used are purely (differential) topological and in particular do not involve any algebraic geometry. There is a corresponding algebraic theory over any fieldk instead ofR which is the subject of part III of this series of papers.