# GT Pequan

# Computer algebra methods for the study of the stability and the stabilization of multidimensional systems

11/24/2016Speaker(s) : Yacine Bouzidi (Inria Lille-Nord Europe)

A multidimensional system (also called n-D systems) is a system in which information propagates in more than one independent direction (usually the time axis for standard 1-D systems). Multidimensional systems naturally arise in the study of partial difference equations, differential time-delay systems, partial differential equations, images, filters... At the core of the study of such systems is the analyse of the stability as well as the stabilization which can be done, Within the fractional representation approach, by means of computations on matrices with entries in the integral domain of structurally stable rational fractions, namely the ring of rational functions which have no poles in the closed unit polydisc U^n = {z = (z1 , . . . , zn ) ∈ C^n | |z1 | ≤ 1, . . . , |zn | ≤ 1} . In this talk, we present some techniques for computing efficiently and exactly in this ring. More precisely, we propose algorithms for solving the two following problems which are useful in the study of stability and stabilization of n-D systems. 1. Given a polynomial D(z1 , . . . , zn ) ∈ Q[z1 , . . . , zn ], check that VC (D(z1, . . . , zn)) ∩ Un = ∅ 2. Given a zero-dimensional ideal I := (p1 , . . . , pr) ⊂ Q[z1, . . . , zn ] check that VC (I) = {z ∈ C^n | p1(z) = · · · = pr (z) = 0} ∩ U^n = ∅, and if so, compute a polynomial s ∈ I such that V (s) ∩ U^n = ∅.

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