Factoring and decomposing a class of linear functional systems

Linear Algebra and Its Applications

Volumn 428, Issue 1, 2008, Pages 324-381

  Cluzeau, Thomas ^a   Quadrat, Alban ^b  

^a ENSIL   (France)

^b INRIA   (France)

Author keywords

Constructive homological algebra; Equivalences of systems; Factorization and decomposition problems; Galois transformations; Linear functional systems; Module theory; Morphisms; Quadratic conservation laws; Quadratic first integrals of motion; Symbolic computation

Indexed keywords

COMPUTATION THEORY; CONTROL THEORY; FACTORIZATION; MATHEMATICAL MODELS; PROBLEM SOLVING; TIME DELAY;

LINEAR FUNCTIONAL SYSTEMS; MATHEMATICAL PHYSICS; SHIFT OPERATORS; TIME-DELAY OPERATORS;

LINEAR ALGEBRA;

EID: 36148954300     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2007.07.008     Document Type: Article

References (78)

1. Abramov S.A., and Barkatou M.A. Rational solutions of first order linear difference systems. Proceedings of ISSAC'98 (1998), ACM Press
2. Abramov S.A., and Bronstein M. On solutions of linear functional systems. Proceedings of ISSAC'01 (2001), ACM Press 1-6
3. M. Barakat, D. Robertz, homalg: an abstract package for homological algebra, submitted for publication. .
4. M.A. Barkatou, On the reduction of matrix pseudo-linear equations, Technical Report RR 1040, Rapport de Recherche de l'Institut IMAG, 2001.
5. Barkatou M.A. On rational solutions of systems of linear differential equations. J. Symbolic Comput. 28 (1999) 547-567
6. Barkatou M.A. Rational solutions of matrix difference equations: the problem of equivalence and factorization. Proceedings of ISSAC'99 (1999), ACM Press
7. Barkatou M.A., Cluzeau T., and Weil J.-A. Factoring partial differential systems in positive characteristic. In: Wang D. (Ed). Differential Equations with Symbolic Computation (DESC Book) (2005), Birkhaüser
8. Barkatou M.A., and Pflügel E. On the equivalence problem of linear differential systems and its application for factoring completely reducible systems. Proceedings of ISSAC'98 (1998), ACM Press 268-275
9. Beke E. Die irreduzibilität der homogenen linearen differentialgleichungen. Math. Ann. 45 (1894) 278-294
10. Bjork J.-E. Rings of Differential Operators (1979), North-Holland Publishing Company
11. R. Bomboy, Réductibilité et résolubilité des équations aux différences finies. Ph.D. thesis, University of Nice-Sophia Antipolis, France, 2001.
12. Briançon J., and Maisonobe P. Idéaux de germes d'opérateurs différentiels à une variable. Enseignement Math. 30 (1984) 7-38
13. Bronstein M. An improved algorithm for factoring linear ordinary differential operators. Proceedings of ISSAC'94 (1994), ACM Press 336-340
14. F. Castro, Calculs effectifs pour les idéaux d'opérateurs différentiels, Travaux en Cours, 24, Herman, 1987.
15. Chyzak F., and Salvy B. Non-commutative elimination in Ore algebras proves multivariate identities. J. Symbolic Comput. 26 (1998) 187-227
16. Chyzak F., Quadrat A., and Robertz D. Effective algorithms for parametrizing linear control systems over Ore algebras. Appl. Algebra Engrg. Comm. Comput. 16 (2005) 319-376
17. F. Chyzak, A. Quadrat, D. Robertz, OreModules: A symbolic package for the study of multidimensional linear systems. Applications of Time-Delay Systems, in: J. Chiasson, J.-J. Loiseau (Eds.), LNCIS, vol. 352, 2007, pp. 233-264. .
18. T. Cluzeau, Algorithmique modulaire des équations différentielles linéaires, Ph.D. thesis, University of Limoges, France, 2004.
19. Cluzeau T. Factorization of differential systems in characteristic p. Proceedings of ISSAC'03 (2003), ACM Press 58-65
20. T. Cluzeau, A. Quadrat, Using morphism computations for factoring and decomposing general linear functional systems, in: Proceedings of MTNS 2006, Kyoto, Japan (24-28/07/06), INRIA report RR-5942, 2006. .
21. T. Cluzeau, A. Quadrat, Morphisms: a homological algebra package for factoring and decomposing linear functional systems, in preparation.
22. Coutinho S.C. A Primer of Algebraic D-Modules. London Mathematical Society Student Texts vol. 33 (1995), Cambridge University Press
23. F. Dubois, N. Petit, P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid, in: Proceedings of the 5th European Control Conference, Karlsruhe, Germany, 1999.
24. A. Fabiańska, A. Quadrat, Applications of the Quillen-Suslin theorem to multidimensional systems theory, in: H. Park, G. Regensburger (Eds.), Gröbner Bases in Control Theory and Signal Processing, Radon Series on Computation and Applied Mathematics, vol. 3, de Gruyter, in press.
25. Fuhrmann P.A. Algebraic system theory: an analyst's point of view. J. Franklin Inst. 301 (1976) 521-540
26. Fuhrmann P.A. Linear Operators and Systems in Hilbert Space (1981), McGraw-Hill
27. Fuhrmann P.A. A Polynomial Approach to Linear Algebra (1996), Springer-Verlag
28. Fuhrmann P.A. On behavior homomorphisms and system equivalences. Systems Control Lett. 44 (2001) 127-143
29. n(k). J. Symbolic Comput. 36 6 (2003) 845-853
30. Grigoriev D.Y. Complexity of irreducibility testing for a system of linear ordinary differential equations. Proceedings of ISSAC'90 (1990), ACM Press 225-230
31. Grigoriev D., and Schwarz F. Generalized Loewy-decomposition of D-modules. Proceedings of ISSAC'05 (2005), ACM Press
32. van Hoeij M. Factorization of differential operators with rational functions coefficients. J. Symbolic Comput. 24 (1997) 537-561
33. Jacobson N. Pseudo-linear transformations. Ann. Math. 38 (1937) 484-507
34. M. Kashiwara, Algebraic study of systems of partial differential equations. Master Thesis, Tokyo Univ. 1970, Mémoires de la Société Mathématiques de France 63, 1995 (English translation).
35. Kandri-Rody A., and Weispfenning V. Noncommutative Gröbner bases in algebras of solvable type. J. Symbolic Comput. 9 1 (1990) 1-26
36. Kredel H. Solvable Polynomial Rings (1993), Shaker
37. Lam T.Y. Lectures on Modules and Rings. Graduate Texts in Mathematics vol. 189 (1999), Springer
38. L. Landau, L. Lifschitz, Physique théorique. Tome 1: Mécanique, fourth ed., MIR, 1982.
39. L. Landau, L. Lifschitz, Physique théorique. Tome 6: Mécanique des fluides, second ed., MIR, 1989.
40. Levandovskyy V., and Schönemann H. Plural - a computer algebra system for non-commutative polynomial algebras. Proceedings of ISSAC'03,176-183 (2003), ACM Press.
41. V. Levandovskyy, Non-commutative computer algebra for polynomial algebras: Gröbner bases, applications and implementation, Ph.D. thesis, University of Kaiserslautern, 2005.
42. Li Z., Schwarz F., and Tsarëv S. Factoring systems of PDE's with finite-dimensional solution space. J. Symbolic Computat. vol. 36 (2003) 443-471
43. Logar A., and Sturmfels B. Algorithms for the Quillen-Suslin theorem. Journal of Algebra 145 (1992) 231-239
44. Macaulay 2 project, .
45. B. Malgrange, Systèmes à coefficients constants, Séminaire Bourbaki 1962/1963, 1-11.
46. McConnell J.C., and Robson J.C. Noncommutative Noetherian rings (2000), American Mathematical Society
47. H. Mounier, J. Rudolph, M. Petitot, M. Fliess, A flexible rod as a linear delay system, in: Proceedings of the 3rd European Control Conference, Roma, Italy, 1995.
48. Oaku T., and Takayama N. Algorithms for D-modules-restriction, tensor product, localization, and local cohomology groups. J. Pure Appl. Algebra 156 (2001) 267-308
49. Oaku T., Takayama N., and Tsai H. Polynomial and rational solutions of holonomic systems. J. Pure Appl. Algebra 164 (2001) 199-220
50. Oberst U. Multidimensional constant linear systems. Acta Appl. Math. 20 (1990) 1-175
51. Palamodov V.P. Linear Differential Operators with Constant Coefficients (1970), Springer
52. Pillai H.K., and Shankar S. A behavioural approach to control of distributed systems. SIAM J. Control Optim. 37 (1999) 388-408
53. Pillai H.K., Wood J., and Rogers E. On homomorphisms of n-D behaviours. IEEE Trans. Circ. Syst. 46 (2002) 732-742
54. Polderman J.W., and Willems J.C. Introduction to Mathematical Systems Theory. A Behavioral Approach. Texts in Applied Mathematics vol. 26 (1998), Springer
55. Pommaret J.-F. Partial Differential Control Theory (2001), Kluwer
56. Pommaret J.-F., and Quadrat A. Generalized Bezout identity. Appl. Algebra Engrg. Comm. Comput. 9 (1998) 91-116
57. Pommaret J.-F., and Quadrat A. Localization and parametrization of linear multidimensional control systems. Systems Control Lett. 37 (1999) 247-260
58. Pommaret J.-F., and Quadrat A. Algebraic analysis of linear multidimensional control systems. IMA J. Control Optim. 16 (1999) 275-297
59. J.-F. Pommaret, A. Quadrat, Equivalences of linear control systems, in: Proceedings of the 14th Mathematical Theory of Networks and Systems, Perpignan, France, 2000.
60. Pommaret J.-F., and Quadrat A. A functorial approach to the behaviour of multidimensional control systems. Int. J. Appl. Math. Comput. Sci. 13 (2003) 7-13
61. van der Put M., and Singer M.F. Galois Theory of Linear Differential Equations. Grundlehren der Mathematischen Wissenschaften vol. 328 (2003), Springer
62. A. Quadrat, D. Robertz, Parametrizing all solutions of uncontrollable multidimensional linear systems, in: Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 2005.
63. A. Quadrat, D. Robertz, On the Monge problem and multidimensional optimal control, in: Proceedings of MTNS 2006, Kyoto, Japan (24-28/07/06).
64. A. Quadrat, D. Robertz, Computation of bases of free modules over the Weyl algebras, J. Symbolic Comput., in press.
65. D. Robertz, Formal computational methods for control theory, Ph.D. thesis, RWTH Aachen, Germany, 2006. .
66. Rotman J.J. An Introduction to Homological Algebra (1979), Academic Press
67. Schwarz F. A factorization algorithm for linear ordinary differential operators. Proceedings of ISSAC'89 (1989), ACM Press 17-25
68. Singer M.F. Testing reducibility of linear differential operators: a group theoretic perspective. Appl. Algebra Engrg. Comm. Comput. 7 (1996) 77-104
69. Stafford J.T. Module structure of Weyl algebras. J. London Math. Soc. 18 (1978) 429-442
70. N. Takayama, Kan: A system for computation in algebraic analysis. .
71. Tsai H., and Walther U. Computing homomorphisms between holonomic D-modules. J. Symbolic Comput. 32 (2001) 597-617
72. Tsarëv S.P. Some problems that arise in the factorization of linear ordinary differential operators. Programmirovanie 1 (1994) 45-48
73. Translation in Programming and Comput. Software 20 (1994) 27-29
74. Wood J. Modules and behaviours in nD systems theory. Multidimens. Systems Signal Process 11 (2000) 11-48
75. Wu M. On the factorization of differential modules. In: Wang D. (Ed). Differential Equations with Symbolic Computation (DESC Book) (2005), Birkhaüser
76. M. Wu, On solutions of linear functional systems and factorization of modules over Laurent-Ore algebras, Ph.D. thesis, University of Nice-Sophia Antipolis, France, 2005.
77. Zerz E. Topics in Multidimensional Linear Systems Theory. Lecture Notes in Control and Information Sciences vol. 256 (2000), Springer
78. Zerz E. An algebraic analysis approach to linear time-varying systems. IMA J. Math. Control Inform. 23 (2006) 113-126