Computation of Koszul homology and application to involutivity of partial differential systems
| Autor: | Chenavier, Cyrille, Cluzeau, Thomas, Quadrat, Alban |
|---|---|
| Přispěvatelé: | XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs (OURAGAN), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité) |
| Rok vydání: | 2022 |
| Předmět: |
Koszul homology
[INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC] Formal integrability ACM: I.: Computing Methodologies/I.1: SYMBOLIC AND ALGEBRAIC MANIPULATION/I.1.2: Algorithms/I.1.2.0: Algebraic algorithms [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA] [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA] Multidimensional systems Behaviours Systems of partial differential equations Spencer cohomology [SPI.AUTO]Engineering Sciences [physics]/Automatic ACM: G.: Mathematics of Computing/G.1: NUMERICAL ANALYSIS/G.1.8: Partial Differential Equations Control and Systems Engineering Cartan's involutivity [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ACM: I.: Computing Methodologies/I.1: SYMBOLIC AND ALGEBRAIC MANIPULATION/I.1.1: Expressions and Their Representation |
| Zdroj: | SSSC 2022-8th IFAC Symposium on System Structure and Control SSSC 2022-8th IFAC Symposium on System Structure and Control, Sep 2022, Montréal, Canada |
| ISSN: | 2405-8963 |
| DOI: | 10.1016/j.ifacol.2022.11.302 |
| Popis: | International audience; The formal integrability of systems of partial differential equations plays a fundamental role in different analysis and synthesis problems for both linear and nonlinear differential control systems. Following Spencer's theory, to test the formal integrability of a system of partial differential equations, we must study when the symbol of the system, namely, the top-order part of the linearization of the system, is 2-acyclic or involutive, i.e., when certain Spencer cohomology groups vanish. Combining the fact that Spencer cohomology is dual to Koszul homology and symbolic computation methods, we show how to effectively compute the homology modules defined by the Koszul complex of a finitely presented module over a commutative polynomial ring. These results are implemented using the OreMorphisms package. We then use these results to effectively characterize 2-acyclicity and involutivity of the symbol of a linear system of partial differential equations. Finally, we show explicit computations on two standard examples. |
| Databáze: | OpenAIRE |
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