An introduction to constructive algebraic analysis and its applications
Journées Nationales de Calcul Formel. 3 – 7 Mai 2010, Les cours du CIRM, no. 2 (2010), pp. 281-471.
DOI : 10.5802/ccirm.11
Quadrat, Alban 1

1 INRIA Sophia Antipolis, 2004 Route des Lucioles BP 93, 06902 Sophia Antipolis cedex, France.
@article{CCIRM_2010__1_2_281_0,
     author = {Quadrat, Alban},
     title = {An introduction to constructive algebraic analysis and its applications},
     booktitle = {Journ\'ees Nationales de Calcul Formel. 3 {\textendash} 7 Mai 2010},
     series = {Les cours du CIRM},
     pages = {281--471},
     publisher = {CIRM},
     number = {2},
     year = {2010},
     doi = {10.5802/ccirm.11},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ccirm.11/}
}
TY  - JOUR
AU  - Quadrat, Alban
TI  - An introduction to constructive algebraic analysis and its applications
BT  - Journées Nationales de Calcul Formel. 3 – 7 Mai 2010
AU  - Collectif
T3  - Les cours du CIRM
PY  - 2010
SP  - 281
EP  - 471
IS  - 2
PB  - CIRM
UR  - http://www.numdam.org/articles/10.5802/ccirm.11/
DO  - 10.5802/ccirm.11
LA  - en
ID  - CCIRM_2010__1_2_281_0
ER  - 
%0 Journal Article
%A Quadrat, Alban
%T An introduction to constructive algebraic analysis and its applications
%B Journées Nationales de Calcul Formel. 3 – 7 Mai 2010
%A Collectif
%S Les cours du CIRM
%D 2010
%P 281-471
%N 2
%I CIRM
%U http://www.numdam.org/articles/10.5802/ccirm.11/
%R 10.5802/ccirm.11
%G en
%F CCIRM_2010__1_2_281_0
Quadrat, Alban. An introduction to constructive algebraic analysis and its applications, dans Journées Nationales de Calcul Formel. 3 – 7 Mai 2010, Les cours du CIRM, no. 2 (2010), pp. 281-471. doi : 10.5802/ccirm.11. http://www.numdam.org/articles/10.5802/ccirm.11/

[1] S. A. Abramov, M. Bronstein, “On solutions of linear functional systems”, in Proceedings of ISSAC’01, 1-6, ACM Press, 2001.

[2] M. Auslander, M. Bridger, Stable Module Theory, Memoirs of the American Mathematical Society, 94 American Mathematical Society, 1969.

[3] R. Baer, “Erweiterung von Gruppen und ihren Isomorphismen”, Mathematische Zeitschrift, 38 (1934), 375-416.

[4] M. Barakat, D. Robertz, “homalg: An abstract package for homological algebra”, J. Algebra Appl., 7 (2008), 299-317, homalg project: http://wwwb.math.rwth-aachen.de/homalg/.

[5] M. Barakat, “Spectral filtrations via generalized morphisms”, , 30/04/09, submitted for publication. | arXiv

[6] M. A. Barkatou, “On rational solutions of systems of linear differential equations”, J. Symbolic computation, 28 (1999), 547-567.

[7] M. A. Barkatou, “Factoring systems of linear functional systems using eigenrings”, in Computer Algebra 2006, Latest Advances in Symbolic Algorithms, Proceedings of the Waterloo Workshop, Ontario, Canada (10-12/04/06), I. Kotsireas and E. Zima (Eds.), World Scientific, 22-42.

[8] T. Becker, V. Weispfenning, Gröbner Bases. A Computational Approach to Commutative Algebra, Springer, NewYork, 1993.

[9] C. M. Bender, G. V. Dunne, L. R. Mead, Underdetermined systems of partial differential equations, J. Mathematical Physics, 41 (2000), 6388-6398.

[10] J. E. Bjork, Rings of Differential Operators, North Holland, 1979.

[11] J. E. Bjork, Analytic 𝒟-modules and Applications, Kluwer, 1993.

[12] Y. A. Blinkov, C. F. Cid, V. P. Gerdt, W. Plesken, D. Robertz, “The MAPLE Package “Janet”: I. Polynomial Systems”, In the proceedings of Computer Algebra in Scientific Computing CASC 2003, V. G. Ganzha, E. W. Mayr, E. V. Vorozhtsov (Eds.), 31-40, http://wwwb.math.rwth-aachen.de/Janet.

[13] A. Borel et al., Algebraic D-modules, Perspectives in Mathematics, vol. 2, Academic Press, 1987.

[14] M. S. Boudellioua, A. Quadrat, “Serre’s reduction of linear functional systems”, INRIA report 7214, to appear in Mathematics in Computer Science, 2010.

[15] H. Cartan, S. Eilenberg, Homological Algebra, Princeton University Press, 1956.

[16] F. Chyzak, A. Quadrat, D. Robertz, “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”, in the book Applications of Time-Delay Systems, J. Chiasson and J. -J. Loiseau (Eds.), Lecture Notes in Control and Information Sciences (LNCIS) 352, Springer, 2007, 233-264, OreModules project: http://wwwb.math.rwth-aachen.de/OreModules.

[18] F. Chyzak, B. Salvy, “Non-commutative elimination in Ore algebras proves multivariate identities”, J. Symbolic Comput., 26 (1998), 187-227.

[19] T. Cluzeau, A. Quadrat, “Factoring and decomposing a class of linear functional systems”, Linear Algebra Appl., 428 (2008), 324-381.

[20] T. Cluzeau, A. Quadrat, “OreMorphisms: A homological algebraic package for factoring and decomposing linear functional systems”, in the book Topics in Time-Delay Systems: Analysis, Algorithms and Control, J.-J. Loiseau, W. Michiels, S.-I. Niculescu and R. Sipahi (Eds.), Lecture Notes in Control and Information Sciences (LNCIS) 388, Springer, 2009, 179-196, OreMorphisms project: http://www-sop.inria.fr/members/Alban.Quadrat/OreMorphisms/index.html.

[21] T. Cluzeau, A. Quadrat, “Serre’s reduction of linear partial differential systems based on holonomy”, to appear in the proceedings of MTNS 2010, Budapest (Hungary) (05-07/07/10).

[22] T. Cluzeau, A. Quadrat, “Duality, Auslander transpose, adjoints and stable ranks: A constructive approach”, in preparation, 2010.

[23] R. Courant, D. Hilbert, Methods of Mathematical Physics, Wiley Classics Library, Wiley, 1989.

[24] S. C. Coutinho, M. P. Holland, “Module structure of rings of differential operators”, Proc. London Math. Soc., 57 (1988), 417-432.

[25] G. Culianez, Formes de Hermite et de Jacobson: Implémentations et applications, internship (INSA de Toulouse) under the supervision of A. Quadrat, INRIA Sophia Antipolis (06-07/05), Jacobson project: http://www-sop.inria.fr/members/Alban.Quadrat/Stages.html.

[26] F. Dubois, N. Petit, P. Rouchon, “Motion planning and nonlinear simulations for a tank containing a fluid”, in the proceedings of the 5th European Control Conference, Karlsruhe (Germany), 1999.

[27] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, 1994.

[28] K. Eriksson, D. Estep, P. Hansbo, C. Johnson, Computational Differential Equations, Cambridge University Press, 1996.

[29] A. Fabiańska, A. Quadrat, “Applications of the Quillen-Suslin theorem in multidimensional systems theory”, in the book Gröbner Bases in Control Theory and Signal Processing, H. Park and G. Regensburger (Eds.), Radon Series on Computation and Applied Mathematics 3, de Gruyter publisher, 2007, 23-106, QuillenSuslin project: http://wwwb.math.rwth-aachen.de/QuillenSuslin/.

[30] N. Fitchas, A. Galligo, “Nullstellensatz effectif et conjecture de Serre (Théorème de Quillen-Suslin) pour le calcul formel”, Math. Nachr., 149 (1990), 231-253.

[31] M. Fliess, “Some basic structural properties of generalized linear systems”, Systems & Control Letters, 15 (1990), 391-396.

[32] M. Fliess, J. Lévine, P. Martin, P. Rouchon, “Flatness and defect of nonlinear systems: introductory theory and examples”, Int. J. Control, 61 (1995), 1327-1361.

[33] S. Fröhler, U. Oberst, “Continuous time-varying linear systems”, Systems and Control Letters, 35 (1998), 97-110.

[34] A. Galligo, “Some algorithmic equations on ideals of differential operators”, in EUROCAL’85, vol. 2 (Linz 1985), Lecture Notes in Computer Science 204, Springer, 413-412.

[35] E. Goursat, “Sur une généralisation du problème de Monge”, Ann. Fac. Sciences de Toulouse, 22 (1930), 249-295.

[36] E. Goursat, “Sur quelques équations de Monge intégrables explicitement”, Annali della Scuola Normale di Pisa, Classe di Scienze 2e série, 1 (1932), 35-59.

[37] E. Goursat, “Sur une équation de Monge à deux variables indépendantes”, Annali della Scuola Normale di Pisa, Classe di Scienze 2e série, 4 (1935), 15-33.

[38] J. Hadamard, “Sur l’équilibre des plaques élastiques circulaires libres ou appuyées et celui de la sphère isotrope”, Annales Scientifiques de l’Ecole Normale Supérieure, 18 (1901), 313-324.

[39] A. Hillebrand, W. Schmale, “Towards an effective version of a theorem of Stafford”, J. Symbolic Comput., 32 (2001), 699-716.

[40] M. Janet, Leçons sur les systèmes d’équations aux dérivées partielles, Cahiers scientifiques IV, Gauthier-Villars, 1929.

[41] J. Johnson, “Systems of n partial differential equations in n unknowns functions: the conjecture of M. Janet”, Trans. Amer. Math. Soc. 242 (1978), 329-334.

[42] T. Kailath, Linear Systems, Prentice-Hall, 1980.

[43] R. E. Kalman, P. L. Falb, M. A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, 1969.

[44] 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).

[45] M. Kashiwara, T. Kawai, T. Kimura, Foundations of algebraic analysis, Princeton Mathematical Series, vol. 37, Princeton University Press, Princeton, 1986.

[46] E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, 1973.

[47] V. Kolmanovskii, V. Nosov, Stability of Functional Differential Equations. Academic Press, 1986.

[48] H. Komatsu, “Relative cohomology of sheaves of solutions of differential equations”, in Hyperfunctions and Pseudo-Differential Equations, Lectures Notes in Mathematics 287, Springer, 1971, 192-263.

[49] V. Kucera, Discrete Linear Control: The Polynomial Equation Approach, Wiley, Chichester 1979.

[50] L. Landau, L. Lifschitz, Physique théorique, Tome 1: Mécanique, 4th edition, MIR.

[51] L. Landau, L. Lifschitz, Physique théorique, Tome 2: Théorie des champs, 4th edition, MIR, 1989.

[52] L. Landau, L. Lifschitz, Physique théorique, Tome 6: Mécanique des fluides, 2nd edition, MIR, 1989.

[53] L. Landau, L. Lifschitz, Physique théorique, Tome 7: Elasticité, 2nd edition, MIR, 1990, second edition.

[54] T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer, 1999.

[55] T. Y. Lam, Serre’s Problem on Projective Modules, Springer Monograph in Mathematics, Springer Verlag, 2006.

[56] P. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Comm. Pure Appl. Math., 21 (1968), 467-490.

[57] A. Leykin, “Algorithmic proofs of two theorems of Stafford”, J. Symbolic Comput., 38 (2004), 1535-1550.

[58] V. Levandovskyy, Non-commutative Computer Algebra for Polynomial Algebras: Gröbner Bases, Applications and Implementation, PhD Thesis, University of Kaiserslautern, 2005.

[59] V. Levandovskyy, E. Zerz, “Obstructions to genericity in the study of parametric problems in control theory”, in Gröbner Bases in Control Theory and Signal Processing, H. Park and G. Regensburger (Eds.), Radon Series on Computation and Applied Mathematics, 3, de Gruyter publisher, 2007, 127-149, Singular project: http://www.singular.uni-kl.de/.

[60] Z. Lin, N. K. Bose, “A generalization of Serre’s conjecture and related issues”, Linear Algebra and Its Applications, 338 (2001), 125-138.

[61] A. Logar, B. Sturmfels, “Algorithms for the Quillen-Suslin theorem”, Journal of Algebra, 145 (1992), 231-239.

[62] H. Lombardi, I. Yengui, “Suslin’s algorithms for reduction of unimodular rows”, Journal of Symbolic Computation, 39 (2005), 707-717.

[63] H. Lombardi, “Dimenion de Krull explicite. Applications aux théorèmes de Kronecker, Bass, Serre et Forster”, Notes de cours, 14/07/05. http://hlombardi.free.fr/publis/publis.html.

[64] H. Lombardi, C. Quitté, Algèbre Commutative, Méthodes constructives (Modules projectifs de type fini), book, to appear http://hlombardi.free.fr/publis/publis.html.

[65] S. MacLane, Homology, Springer Verlag, 1995.

[66] P. Maisonobe, C. Sabbah, 𝒟-modules cohérents and holonomes, Hermann, 1993.

[67] B. Malgrange, “Systèmes à coefficients constants”, Séminaire Bourbaki 1962/63, 1-11.

[68] B. Malgrange, “Ouverts concaves et théorèmes de dualité pour les systèmes différentiels à coefficients constants”, Comm. Math. Helv., 46 (1971), 487-499.

[69] B. Malgrange, “Lettre à P. Rouchon”, 26/01/00, private communication with A. Quadrat, 14/02/00.

[70] F. Malrait, P. Martin, P. Rouchon, “Dynamic feedback transformations of controllable linear time-varying systems”, in Lecture Notes in Control and Inform. Sci. 259, Springer, 2001, 55-62.

[71] J. C. McConnell, J. C. Robson, Noncommutative Noetherian Rings, American Mathematical Society, 2000.

[72] G. Monge, “Où l’on fait voir que les équations aux différences ordinaires, pour lesquelles les conditions d’intégrabilité ne sont pas satisfaites, sont suscesptibles d’une véritable intégration, et que c’est de cette intégration que dépend celle des équations aux différences partielles élevées”, Mémoire de l’Académie Royale des Sciences, Paris, 1784, 502-576.

[73] H. Mounier, Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques, PhD thesis, University of Paris XI, 1995.

[74] H. Mounier, J. Rudolph, M. Petitot, M. Fliess, “A flexible rod as a linear delay system”, in Proceedings of 3rd European Control Conference, Rome (Italy), 1995, 3676-3681.

[75] H. Mounier, P. Rouchon, J. Rudolph, “Some examples of linear systems with delays”, European Journal of Automation, 31 (1997), 911-925.

[76] H. Mounier, J. Rudolph, M. Fliess, P. Rouchon, “Tracking control of a vibrating string with an interior mass viewed as delay system”, ESAIM COCV, 3 (1998), 315-321.

[77] T. Oaku, N. Takayama, “Algorithms for D-modules-restriction, tensor product, localization, and local cohomology groups”, J. Pure Appl. Algebra, 156 (2001), 267-308.

[78] U. Oberst, “Multidimensional constant linear systems”, Acta Appl. Math., 20 (1990), 1-175.

[79] N. Petit, P. Rouchon, “Dynamics and solutions to some control problems for water-tank systems”, IEEE Trans. Automat. Control, 47 (2002), 595-609.

[80] H. K. Pillai, S. Shankar, “A behavioural approach to control of distributed systems”, SIAM Journal on Control and Optimization, 37 (1999), 388-408.

[81] J. W. Polderman, J. C. Willems, Introduction to Mathematical Systems Theory. A Behavioral Approach, TAM 26, Springer, 1998.

[82] J.-F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, 1978.

[83] J.-F. Pommaret, Differential Galois Theory, Gordon and Breach, 1983.

[84] J.-F. Pommaret, Partial Differential Equations and Group Theory: New Perspectives for Applications, Kluwer, 1994.

[85] J.-F. Pommaret, Partial Differential Control Theory, Kluwer Academic Publishers, Mathematics and Its Applications, 2001.

[86] J.-F. Pommaret, “Algebraic analysis of control systems defined by partial differential equations”, in the book Advanced Topics in Control Systems Theory, F. Lamnabhi-Lagarrigue, A. Loria, E. Panteley (Eds), Lecture Notes in Control and Information Sciences (LNCIS) 311, Springer, 2005, 155-223.

[87] J.-F. Pommaret, A. Quadrat, “Generalized Bézout identity”, Appl. Algebra Engrg. Comm. Comput., 9 (1998), 91-116.

[88] J.-F. Pommaret, A. Quadrat, “Localization and parametrization of linear multidimensional control systems”, Systems & Control Letters, 37 (1999), 247-260.

[89] J.-F. Pommaret, A. Quadrat, “Algebraic analysis of linear multidimensional control systems”, IMA J. Math. Control Inform., 16 (1999), 275-297.

[90] J.-F. Pommaret, A. Quadrat, “Formal elimination for multidimensional systems and applications to control theory”, Math. Control, Signal and Systems, 13 (2000), 193-215.

[91] J.-F. Pommaret, A. Quadrat, “Equivalences of linear control systems”, Proceedings of MTNS 2000, Perpignan (France).

[92] J.-F. Pommaret, A. Quadrat, “A functorial approach to the behaviour of multidimensional control systems”, Int. J. Appl. Math. Comput. Sci., 13 (2003), 7-13.

[93] J.-F. Pommaret, A. Quadrat, “A differential operator approach to multidimensional optimal control”, Int. J. Control, 77 (2004), 821-836.

[94] M. van der Put, M. F. Singer, Galois Theory of Linear Differential Equations, Grundlehren der mathematischen Wissenschaften, 328, Springer, 2003.

[95] A. Quadrat, “Extended Bézout identities”, Proceedings of European Control Conference, Porto (Portugal), (04-07/09/01).

[96] A. Quadrat, “The fractional representation approach to synthesis problems: an algebraic analysis viewpoint. Part I: (Weakly) doubly coprime factorizations”, SIAM J. Control Optim., 42 (2003), 266-299.

[97] A. Quadrat, “Purity filtration of general linear systems of partial differential equations”, submitted for publication, INRIA report, 2010.

[98] A. Quadrat, “Purity filtration of n-dimensional linear systems”, to appear in the proceedings of MTNS 2010, Budapest (Hungary) (05-07/07/10).

[99] A. Quadrat, “Extendability of multidimensional linear systems”, to appear in the Proceedings of MTNS 2010, Budapest (Hungary) (05-07/07/10).

[100] A. Quadrat, “Systems and Structures: An algebraic analysis approach to mathematical systems theory”, book in preparation, 2010.

[101] A. Quadrat, D. Robertz, “Parametrizing all solutions of uncontrollable multidimensional linear systems”, Proceedings of 16th IFAC World Congress, Prague (Czech Republic), (04-08/07/05).

[102] A. Quadrat, D. Robertz, “On the Monge problem and multidimensional optimal control”, Proceedings of MTNS 2006, Kyoto (Japan), (20-24/07/06).

[103] A. Quadrat, D. Robertz, “Computation of bases of free modules over the Weyl algebras”, J. Symbolic Comput., 42 (2007), 1113-1141, Stafford project (http://wwwb.math.rwth-aachen.de/OreModules).

[104] A. Quadrat, D. Robertz, “On the Baer extension problem for multidimensional linear systems”, INRIA report 6307, 2007, http://hal.inria.fr/inria-00175272, submitted for publication.

[105] A. Quadrat, D. Robertz, “Baer’s extension problem for multidimensional linear systems”, Proceedings of MTNS 2008, Virginia (USA), (28/07-01/08/08).

[106] A. Quadrat, D. Robertz, “Controllability and differential flatness of linear analytic ordinary differential systems”, to appear in the Proceedings of MTNS 2010, Budapest (Hungary) (05-07/07/10).

[107] D. Quillen, “Projective modules over polynomial rings”, Invent. Math., 36 (1976), 167-171.

[108] J. F. Ritt, Differential Algebra, Dover, 1996.

[109] D. Robertz, “Janet bases and applications”, in the book Groebner Bases in Symbolic Analysis, M. Rosenkranz, D. Wang, D. (eds.), Radon Series on Computation and Applied Mathematics 2, de Gruyter publisher, 2007, 139-168, JanetOre project: http://wwwb.math.rwth-aachen.de/~daniel/index.html.

[110] J. J. Rotman, An Introduction to Homological Algebra, Springer, 2nd edition, 2009.

[111] J. T. Stafford, “Module structure of Weyl algebras”, J. London Math. Soc., 18 (1978), 429-442.

[112] J.-P. Serre, “Faisceaux algébriques cohérents”, Annals of Mathematics, 61 (1955), 197-278.

[113] J.-P. Serre, “Sur les modules projectifs”, Séminaire Dubreil-Pisot, vol. 2, 1960/1961, in Oeuvres, Collected Papers, Vol. II 1960-1971, Springer, 1986, 23-34.

[114] M. F. Singer, “Testing reducibility of linear differential operators: a group theoretic perspective”, Appl. Algebra Engrg. Comm. Comput., 7 (1996), 77-104.

[115] A. Suslin, “The projective modules are free over polynomial rings”, Dklady-Soviet Math., 229 (1976), 1063-1066.

[116] H. Tsai, U. Walther, “Computing homomorphisms between holonomic D-modules”, J. Symbolic Comput. 32 (2001), 597-617.

[117] K. Washizu, Variational Methods in Elasticity & Plasticity, Pergammon Press, 3rd, 1982.

[118] J. Wood, “Modules and behaviours in nD systems theory”, Multidimensional Systems and Signal Processing, 11 (2000), 11-48.

[119] N. Yoneda, “On Ext and exact sequences”, J. Fac. Sci. Univ. Tokyo Sect. I, 8 (1960), 507-576.

[120] P. Zervos, Le problème de Monge, Mémorial des sciences mathématiques, fascicule LIII, Gauthier-Villars, 1932.

[121] E. Zerz, Topics in Multidimensional Linear Systems Theory, Lecture Notes in Control and Information Sciences 256, Springer, 2000.

[122] E. Zerz, “An algebraic analysis approach to linear time-varying systems”, IMA J. Math. Control Inform., 23 (2006), 113-126.

Cité par Sources :