Constructive computation of flat outputs of a class of multidimensional linear systems with variable coefficients

The purpose of this paper is to give a constructive algorithm for the computation of bases of finitely presented free modules over the Weyl algebras of differential operators with polynomial or rational coefficients. In particular, we show how to use these results in order to recognize when a multidimensional linear system defined by partial differential equations with polynomial or rational coefficients is flat and, if so, to compute flat outputs and the injective image representations of the system. These new results are based on recent constructive proofs of a famous result in non-commutative algebra due to J. T. Stafford [27]. The different algorithms have been implemented in the package STAFFORD [25] based on OREMODULES [2]. These results allow us to achieve the general solution of the socalled Monge problem for multidimensional linear systems defined by partial differential equations with polynomial or rational coefficients. Finally, we constructively answer an open question posed by Datta [5] on the possibility to generalize the results of [13] to multi-input multi-output polynomial time-varying controllable linear systems. We show that every controllable ordinary differential linear system with at least two inputs and polynomial coefficients is flat. Keywords— Flat multidimensional linear systems, injective image representation, constructive computation of bases of free modules, Stafford’s results, non-commutative algebra. I. A PEDESTRIAN INTRODUCTION TO THE MONGE PROBLEM