Section: New Results
Foundations of the behavioural approach
Participant : Alban Quadrat.
Within the algebraic analysis approach to behaviours  ,  , in  , we propose to consider a system not only as a behaviour  , where is the finitely presented left -module defined by the matrix defining the system and the signal space, but as the set of all the 's, where , where is the global dimension of . In this new framework, using Yoneda product, the left -homomorphims of  and the internal symmetries of the behaviour  are generalized to the full system In particular, a system-theoretic interpretation of the Yoneda product is given.
In  , we study the construction of a double complex leading to a Grothendieck spectral sequence converging to the obstructions 's for the existence of a chain of successive parametrizations starting with the behaviour , where is the Auslander transpose of . These obstructions can be studied by means of a long process starting with the -obstructions 's for the solvability of certain inhomogeneous linear systems defined by the algebraic obstructions 's measuring how far is for being a projective left -module. Hence, the algebraic properties of the left -module , defining the behaviour , 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 . 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 -module ). Finally, setting , the above results can be used to find again the characterization of the grade/purity filtration of by means of a Grothendieck spectral sequence. See Section 6.1 and  ,  ,  .
Within the algebraic analysis approach to behaviours  ,  , in  , we explain how the concept of inverse image of a finitely presented left -module , defining the behaviour  , can be used to study the problem of characterizing the restriction of the behaviour to a non characteristic submanifold of . In particular, we detail the explict construction of inverse images of left -modules for standard maps.