Section: New Results

Algebraic Analysis Approach to Linear Functional Systems

Participants : Alban Quadrat [Disco] , Rosane Ushirobira [Non-A] .

Artstein's transformation of linear time-delay systems

Artstein's classical results show that a linear first-order differential time-delay system with delays in the input is equivalent to a linear first-order differential system without delays thanks to an invertible transform which includes integral and delay operators. Within a constructive algebraic approach, we show how Artstein's reduction can be found again and generalized as a particular isomorphism problem between the finitely presented modules defined by the two above linear systems over the ring of integro-differential time-delay operators. Moreover, we show that Artstein's reduction can be obtained in an automatic way by means of symbolic computation, and thus can be implemented in computer algebra systems.

Algebraic analysis for the Ore extension ring of differential time-varying delay operators

No algebraic (polynomial) approach seems to exist for the study of linear differential time-delay systems in the case of a (sufficiently regular) time-varying delay. Based on the concept of skew polynomial rings developed by Ore in the 30s, we construct the ring of differential time-delay operators as an Ore extension and to analyze its properties. A characterization of classical algebraic properties of this ring, such as noetherianity, its homological and Krull dimensions and the existence of Gröbner bases, are given in terms of the time-varying delay function. The algebraic analysis approach to linear systems theory allows us to study linear differential time-varying delay systems (e.g. existence of autonomous elements, controllability, parametrizability, flatness, behavioral approach) through methods coming from module theory, homological algebra and constructive algebra.