CONTINUOUS TIME-VARYING LINEAR-SYSTEMS

We discuss implicit systems of ordinary linear differential equations with (time-) variable coefficients, their solutions in the signal spac e of hyperfunctions according to Sate and their solution spaces, calle d time-varying linear systems or behaviours, from the system theoretic point of view. The basic result, inspired by an analogous one for mul tidimensional constant linear systems, is a duality theorem which esta blishes a categorical one-one correspondence between time-varying line ar systems or behaviours and finitely generated modules over a suitabl e skew-polynomial ring of differential operators. This theorem is fals e for the signal spaces of infinitely often differentiable functions o r of meromorphic (hyper-)functions or of distributions on R. It is use d to obtain various results on key notions of linear system theory. Se veral new algorithms for modules over rings of differential operators and, in particular, new Grobner basis algorithms due to Insa and Pauer make the system theoretic results effective. (C) 1998 Elsevier Scienc e B.V. All rights reserved.