Continuous time-varying linear systems

Abstract

We discuss implicit systems of ordinary linear differential equations with (time-) variable coefficients, their solutions in the signal space of hyperfunctions according to Sato and their solution spaces, called time-varying linear systems or behaviours, from the system theoretic point of view. The basic result, inspired by an analogous one for multidimensional constant linear systems, is a duality theorem which establishes a categorical one–one correspondence between time-varying linear systems or behaviours and finitely generated modules over a suitable skew-polynomial ring of differential operators. This theorem is false for the signal spaces of infinitely often differentiable functions or of meromorphic (hyper-)functions or of distributions on R . It is used to obtain various results on key notions of linear system theory. Several new algorithms for modules over rings of differential operators and, in particular, new Gröbner basis algorithms due to Insa and Pauer make the system theoretic results effective.