Section: New Results

Algebraic Analysis Approach to Linear Functional Systems

Serre's reduction problem

The purpose of this work is to study the connections existing between Serre's reduction of linear functional systems - which aims at finding an equivalent system defined by fewer equations and fewer unknowns - and the decomposition problem - which aims at finding an equivalent system having a diagonal block structure - in which one of the diagonal blocks is assumed to be the identity matrix. In order to do that, in [62] , we further develop results on Serre's reduction problem and on the decomposition problem. Finally, we show how these techniques can be used to analyze the decomposability problem of standard linear systems of partial differential equations studied in hydrodynamics such as Stokes equations, Oseen equations and the movement of an incompressible fluid rotating with a small velocity around a vertical axis.

A spectral sequence central in the behaviour approach

Within the algebraic analysis approach to multidimensional systems, the behavioural approach developed by J. C. Willems can be understood as a dual approach to the module-theoretic approach. This duality is exact when the signal space is an injective cogenerator module over the ring of differential operators. In particular, the obstruction to the existence of a parametrization of a multidimensional system is characterized by the existence of autonomous elements of the multidimensional system. In [52] , we consider the case of a general signal space and investigate the connection between the algebraic properties of the differential module defining the multidimensional system and the obstruction to the existence of parametrizations of the multidimensional system. To do so, we investigate a certain Grothendieck spectral sequence connecting the obstructions to the existence of parametrizations to the obstructions to the differential module - defining the multidimensional system - to be torsion-free, reflexive ...projective.

Restrictions of $n$-D systems and inverse images of $D$-modules

The problem of characterizing the restriction of the solutions of a $n$-D system to a subvector space of ${ℝ}^n$ has recently been investigated in the literature of multidimensional systems theory. For instance, this problem plays an important role in the stability analysis and in stabilization problems of multidimensional systems. In this work, we characterize the restriction of a $n$-D behaviour to an algebraic or analytic submanifold of ${ℝ}^n$. In [51] , within the algebraic analysis approach to multidimensional systems, we show that the restriction of a $n$-D behaviour to an algebraic or analytic submanifold can be characterized in terms of the inverse image of the differential module defining the behaviour. Explicit characterization of inverse images of differential modules is investigated. Finally, we explain Kashiwara's extension of the Cauchy-Kowalevsky theorem for general $n$-D behaviours and non-characteristic algebraic or analytic submanifolds.

Artstein's transformation of linear time-delay systems

Artstein's classical results show that a linear first-order differential time-delay system with delays in the input is equivalent to a linear first-order differential system without delays thanks to an invertible transform which includes integral and delay operators. Within a constructive algebraic approach, we show how Artstein's reduction can be found again and generalized as a particular isomorphism problem between the finitely presented modules defined by the two above linear systems over the ring of integro-differential time-delay operators. Moreover, we show that Artstein's reduction can be obtained in an automatic way by means of symbolic computation, and thus can be implemented in computer algebra systems.