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 ${\mathrm{ext}}_D^0(M,ℱ)$ [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 ${\mathrm{ext}}_D^i(M,ℱ)$'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 ${\mathrm{ext}}_D^0(M,ℱ)$ [94] are generalized to the full system ${\left\{{\mathrm{ext}}_D^i(M,ℱ)\right\}}_{i=0,...,n}$ 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 ${\mathrm{tor}}_D^i(N,ℱ)$'s for the existence of a chain of successive parametrizations starting with the behaviour ${\mathrm{ext}}_D^0(M,ℱ)$, where $N$ is the Auslander transpose of $M$. These obstructions ${\mathrm{tor}}_D^i(N,ℱ)$ can be studied by means of a long process starting with the $ℱ$-obstructions ${\mathrm{ext}}_D^j({\mathrm{ext}}_D^k(N,D),ℱ)$'s for the solvability of certain inhomogeneous linear systems defined by the algebraic obstructions ${\mathrm{ext}}_D^k(N,D)$'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 ${\mathrm{ext}}_D^0(M,ℱ)$, 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 ${\mathrm{ext}}_D^0(M,ℱ)$. 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 ${\mathrm{ext}}_D^0(M,ℱ)$ [107] , can be used to study the problem of characterizing the restriction of the behaviour ${\mathrm{ext}}_D^0(M,ℱ)$ 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.