An integro-differential operator approach to linear differential systems

Une approche par opérateurs intégro-diférentiels des systèmes différentiels linéaires

Alban Quadrat (1, 2)

1 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

In this paper, we initiate a new algebraic analysis approach to linear differential systems based on rings of integro-differential operators. Within this algebraic analysis approach, we first interpret the method of variations of constants as an operator identity. Using this result, we show that the module associated with a state-space representation of a linear system is the same as the one associated with its standard convolution representation. This finitely presented module over the ring of integro-differential operators is proved to be stably free. Finally, we show how the reachability property can be expressed within this algebraic analysis approach.

Mots clés: Linear systems; Polynomial methods; Algebraic analysis; Rings of integro-differential operators; Parametrization; Reachability; Behaviour theory

Domaines: Automatique / Robotique; Calcul formel [cs.SC]; Anneaux et algèbres [math.RA]; Algèbres d'opérateurs [math.OA]

Alban Quadrat. An integro-differential operator approach to linear differential systems. MTNS 2022 - 25th International Symposium on Mathematical Theory of Networks and Systems, Sep 2022, Bayreuth, Germany. ⟨10.1016/j.ifacol.2022.11.054⟩. ⟨hal-03908541⟩