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DISCO - 2013



Section: New Results

Foundations of the behavioural approach

Participant : Alban Quadrat.

Within the algebraic analysis approach to behaviours [91] , [113] , in [34] , we propose to consider a system not only as a behaviour ext D0(M,) [107] , where M is the finitely presented left D-module defined by the matrix defining the system and the signal space, but as the set of all the ext Di(M,)'s, where 0in, where n is the global dimension of D. In this new framework, using Yoneda product, the left D-homomorphims of M [94] and the internal symmetries of the behaviour ext D0(M,) [94] are generalized to the full system { ext Di(M,)}i=0,...,n In particular, a system-theoretic interpretation of the Yoneda product is given.

In [117] , we study the construction of a double complex leading to a Grothendieck spectral sequence converging to the obstructions tor Di(N,)'s for the existence of a chain of successive parametrizations starting with the behaviour ext D0(M,), where N is the Auslander transpose of M. These obstructions tor Di(N,) can be studied by means of a long process starting with the -obstructions ext Dj( ext Dk(N,D),)'s for the solvability of certain inhomogeneous linear systems defined by the algebraic obstructions ext Dk(N,D)'s measuring how far M is for being a projective left D-module. Hence, the algebraic properties of the left D-module M, defining the behaviour ext D0(M,), and the functional properties of the signal space can be simultaneously used to study the obstructions for the existence of a chain of successive parametrizations starting with the behaviour ext D0(M,). These results can be used to find again the different situations studied in the literature (e.g., cases of an injective or a flat left D-module ). Finally, setting =D, the above results can be used to find again the characterization of the grade/purity filtration of M by means of a Grothendieck spectral sequence. See Section  6.1 and [86] , [87] , [30] .

Within the algebraic analysis approach to behaviours [91] , [113] , in [116] , we explain how the concept of inverse image of a finitely presented left D-module M, defining the behaviour ext D0(M,) [107] , can be used to study the problem of characterizing the restriction of the behaviour ext D0(M,) to a non characteristic submanifold of n. In particular, we detail the explict construction of inverse images of left D-modules for standard maps.