Effective algebraic analysis approach to linear systems over Ore algebras. Physical systems are typically modeled by systems of functional equations; these can be differential equations, differential time-delay systems, discrete-time systems, etc. The algebraic analysis approach studies such systems from a purely algebraic viewpoint, using D-module theory and homological algebra techniques. The different types of systems can be uniformly described by using the general concept of Ore algebras. Recently, we have developed the Mathematica package OreAlgebraicAnalysis that implements (1) Groebner basis techniques for finitely presented left modules over Ore algebras, (2) algorithms for deciding module-theoretic properties, such as torsion-freeness, reflexivity, projectiveness, freeness, and (3) algorithms from homological algebra, such as computation of free resolutions and projective dimension. The OreAlgebraicAnalysis package makes use of the very generic implementation of Ore algebras in the package HolonomicFunctions, that also provides the computation of Groebner bases in such rings. We present and demonstrate these packages by means of examples from control theory. Joint work with Alban Quadrat, Maris Tonso.
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- Bachelier, Olivier; Cluzeau, Thomas; David, Ronan; Yeganefar, Nima: Structural stabilization of linear 2D discrete systems using equivalence transformations (2017)