×

A new insight into Serre’s reduction problem. (English) Zbl 1318.93029

Summary: The purpose of this paper is to study the connections existing between Serre’s reduction of linear functional systems – which aims at finding an equivalent system defined by fewer equations and fewer unknowns – and the decomposition problem – which aims at finding an equivalent system having a diagonal block structure – in which one of the diagonal blocks is assumed to be the identity matrix. In order to do that, we further develop results on Serre’s reduction problem and on the decomposition problem obtained in M. .S. Boudellioua, A. Quadrat [”Serre’s reduction of linear functional systems” Math. Comput. Sci. 4, No. 2-3, 289-312 (2010; Zbl 1275.16003)], t. Cluzeau, A. Quadrat [”Factoring and decomposing a class of linear functional systems”, Linear Algebra Appl., 428, 1, 324-381, (2008)]. Finally, these techniques are used to analyze the decomposability of linear systems of partial differential equations studied in hydrodynamics such as the Oseen equations.

MSC:

93B25 Algebraic methods
93B17 Transformations
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
13C05 Structure, classification theorems for modules and ideals in commutative rings
34K06 Linear functional-differential equations
13N10 Commutative rings of differential operators and their modules
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 1275.16003
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barkatou, M. A., Factoring systems of linear functional equations using eigenrings, (Kotsireas, I.; Zima, E., Latest Advances in Symbolic Algorithms. Latest Advances in Symbolic Algorithms, Proc. of the Waterloo Workshop, Ontario, Canada (10-12/04/06) (2007), World Scientific), 22-42
[2] Boudellioua, M. S.; Quadrat, A., Serre’s reduction of linear functional systems, Math. Comput. Sci., 4, 2-3, 289-312 (2010) · Zbl 1275.16003
[3] Cluzeau, T.; Quadrat, A., Factoring and decomposing a class of linear functional systems, Linear Algebra Appl., 428, 1, 324-381 (2008) · Zbl 1131.15011
[4] Cluzeau, T.; Quadrat, A., OreMorphisms: a homological algebraic package for factoring, reducing and decomposing linear functional systems, (Topics in Time-Delay Systems. Topics in Time-Delay Systems, Lecture Notes in Control and Inform. Sci., vol. 388 (2009), Springer: Springer Berlin), 179-194, OreMorphisms project
[5] Cluzeau, T.; Quadrat, A., Isomorphisms and Serre’s reduction of linear systems, (Proceedings of the 8th International Workshop on Multidimensional \((n\) D) Systems \((nDs)\). Proceedings of the 8th International Workshop on Multidimensional \((n\) D) Systems \((nDs)\), Erlangen, Germany (2013))
[6] Cluzeau, T.; Quadrat, A., A new insight into Serre’s reduction problem (2014), INRIA research report 8629 · Zbl 1318.93029
[7] Chyzak, F.; Quadrat, A.; Robertz, D., Effective algorithms for parametrizing linear control systems over Ore algebras, Appl. Algebra Engrg. Comm. Comput., 16, 319-376 (2005) · Zbl 1109.93018
[8] Chyzak, F.; Quadrat, A.; Robertz, D., OreModules: a symbolic package for the study of multidimensional linear systems, (Chiasson, J.; Loiseau, J.-J., Applications of Time-Delay Systems. Applications of Time-Delay Systems, Lecture Notes in Control and Information Sciences, vol. 352 (2007), Springer), 233-264, OreModules project · Zbl 1248.93006
[9] Dolean, V.; Nataf, F.; Rapin, G., New constructions of domain decomposition methods for systems of PDEs, C.R. Acad. Sci. Paris, Ser. I, 340, 693-696 (2005) · Zbl 1071.65166
[10] Eisenbud, D., Commutative Algebra: With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150 (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0819.13001
[11] Fabiańska, A.; Quadrat, A., Applications of the Quillen-Suslin theorem to multidimensional systems theory, (Park, H.; Regensburger, G., Gröbner Bases in Control Theory and Signal Processing. Gröbner Bases in Control Theory and Signal Processing, Radon Series on Computation and Applied Mathematics, vol. 3 (2007), de Gruyter), 23-106, The QuillenSuslin project · Zbl 1197.13011
[12] Kashiwara, M., Algebraic Study of Systems of Partial Differential Equations, Mémoires de la Société Mathématique de France, vol. 63 (1995), English translation (Kyoto 1970)
[13] Lam, T. Y., Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189 (1999), Springer · Zbl 0911.16001
[14] Lin, Z.; Boudellioua, M. S.; Xu, Li, On the equivalence and factorization of multivariate polynomial matrices, (Proceedings of the 2006 IEEE International Symposium on Circuits and Systems. Proceedings of the 2006 IEEE International Symposium on Circuits and Systems, (ISCAS 2006), Kos, Greece (2006))
[15] Landau, L.; Lifschitz, L., Physique théorique, Tome 6: Mécanique des fluides (1989), MIR
[16] Malgrange, B., Systèmes différentiels à coefficients constants, Sém. Bourbaki 1962/63, 1-11 (1962)
[17] Manitius, A., Feedback controllers for a wind tunnel model involving a delay: analytical design and numerical simulations, IEEE Trans. Automat. Control, 29, 1058-1068 (1984)
[18] McConnell, J. C.; Robson, J. C., Noncommutative Noetherian Rings (2000), American Mathematical Society · Zbl 0644.16008
[19] Quadrat, A.; Robertz, D., Computation of bases of free modules over the Weyl algebras, J. Symbolic Comput., 42, 1113-1141 (2007), Stafford project · Zbl 1137.16002
[20] Quadrat, A.; Robertz, D., On the Baer extension problem for multidimensional linear systems (2007), INRIA research report n. 630
[21] Quadrat, A.; Robertz, D., A constructive study of the module structure of rings of partial differential operators, Acta Appl. Math., 133, 187-234 (2014) · Zbl 1311.16019
[22] Quadrat, A., An introduction to constructive algebraic analysis and its applications, (CIRM, Les cours du CIRM. Les cours du CIRM, Journées Nationales de Calcul Formel, vol. 1 (2010)), 281-471 (2010), INRIA report 7354
[23] Rotman, J. J., An Introduction to Homological Algebra (2009), Springer · Zbl 1157.18001
[24] van der Put, M.; Singer, M. F., Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften., vol. 328 (2003), Springer-Verlag: Springer-Verlag Berlin · Zbl 1036.12008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.