• The Inria's Research Teams produce an annual Activity Report presenting their activities and their results of the year. These reports include the team members, the scientific program, the software developed by the team and the new results of the year. The report also describes the grants, contracts and the activities of dissemination and teaching. Finally, the report gives the list of publications of the year.

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## 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 ${\mathrm{ext}}_{D}^{0}\left(M,ℱ\right)$ [107] , where $M$ is the finitely presented left $D$-module defined by the matrix defining the system and $ℱ$ the signal space, but as the set of all the ${\mathrm{ext}}_{D}^{i}\left(M,ℱ\right)$'s, where $0\le i\le n$, 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 ${\mathrm{ext}}_{D}^{0}\left(M,ℱ\right)$ [94] are generalized to the full system ${\left\{{\mathrm{ext}}_{D}^{i}\left(M,ℱ\right)\right\}}_{i=0,...,n}$ 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 ${\mathrm{tor}}_{D}^{i}\left(N,ℱ\right)$'s for the existence of a chain of successive parametrizations starting with the behaviour ${\mathrm{ext}}_{D}^{0}\left(M,ℱ\right)$, where $N$ is the Auslander transpose of $M$. These obstructions ${\mathrm{tor}}_{D}^{i}\left(N,ℱ\right)$ can be studied by means of a long process starting with the $ℱ$-obstructions ${\mathrm{ext}}_{D}^{j}\left({\mathrm{ext}}_{D}^{k}\left(N,D\right),ℱ\right)$'s for the solvability of certain inhomogeneous linear systems defined by the algebraic obstructions ${\mathrm{ext}}_{D}^{k}\left(N,D\right)$'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 ${\mathrm{ext}}_{D}^{0}\left(M,ℱ\right)$, and the functional properties of the signal space $ℱ$ can be simultaneously used to study the obstructions for the existence of a chain of successive parametrizations starting with the behaviour ${\mathrm{ext}}_{D}^{0}\left(M,ℱ\right)$. 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 $ℱ$). Finally, setting $ℱ=D$, 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 ${\mathrm{ext}}_{D}^{0}\left(M,ℱ\right)$ [107] , can be used to study the problem of characterizing the restriction of the behaviour ${\mathrm{ext}}_{D}^{0}\left(M,ℱ\right)$ to a non characteristic submanifold of ${ℝ}^{n}$. In particular, we detail the explict construction of inverse images of left $D$-modules for standard maps.