# zbMATH — the first resource for mathematics

Localization and parametrization of linear multidimensional control systems. (English) Zbl 0948.93016
Summary: We study the relation existing between the parametrization of differential operators by potential-like arbitrary functions and the localization of differential modules, while applying these results to the parametrization of linear multidimensional control systems. We show that the localization of differential modules is a natural way to generalize some well-known results on transfer matrix, classically obtained by using Laplace transform, to time-varying ordinary differential control systems and to partial differential control systems with variable coefficients. In particular, we show that the parametrizations obtained by localization are simpler than those obtained by formal duality but are worse in the sense of Palamodov–Kashiwara’s classification of differential modules.

##### MSC:
 93B25 Algebraic methods 93B15 Realizations from input-output data 16D80 Other classes of modules and ideals in associative algebras
Full Text:
##### References:
 [1] H. Blomberg, R. Ylinen, Algebraic Theory for Multivariable Linear Sytems, Academic Press, London, 1983. · Zbl 0556.93016 [2] E. Cartan, Les Systèmes Différentiels Extérieurs et Leurs Applications Géométriques, Hermann, Paris, 1945. · Zbl 0063.00734 [3] H. Goldschimdt, Prolongation of linear partial differential equations: I. A conjecture of Elie Cartan, Ann. Sci. Ecole. Norm. Sup. 4 série 1 (1968) 417-444. [4] Fliess, M., Some basic structural properties of generalized linear systems, Systems control lett., 15, 391-396, (1990) · Zbl 0727.93024 [5] M. Fliess, Controllability revisited, in: A.C. Antoulas (Ed.), The Influence of R.E. Kalman, Springer, Berlin, 1991, pp. 463-474. · Zbl 0777.93005 [6] Fliess, M., Une interprétation algébrique de la transformation de Laplace et des matrices de transfert, Linear algebra appl., 203-204, 429-442, (1994) · Zbl 0802.93010 [7] M. Fliess, H. Mounier, Controllability and observability of linear delay systems: an algebraic approach, ESAIM COCV 3 (1998) 301-314. · Zbl 0908.93013 [8] Fröhler, S.; Oberst, U., Continuous time-varying linear systems, Systems control lett., 35, 97-110, (1998) · Zbl 0909.93041 [9] M. Janet, Leçons sur les Systèmes d’Équations aux Dérivées Partielles, Cahiers Scientifiques IV, Gauthier-Villars, Paris, 1929. · JFM 55.0276.01 [10] T. Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980. [11] R.E. Kalman, On the general theory of control systems, in: Proc. 1st Int. Congress on Automatic Control, Moscow, 1960, Butterworths, London, 1961. · Zbl 0112.06303 [12] M. Kashiwara, Algebraic study of systems of partial differential equations, Mémoires de la Société Mathématiques de France No. 63 (1995). · Zbl 0877.35003 [13] S. Kleon, U. Oberst, Transfer operators and state spaces for discrete multidimensional linear systems, Seminary Rapport, University of Hagen 63 (1998) 409-426. · Zbl 0979.93061 [14] E.R. Kolchin, Differential algebra and algebraic groups, Pure Appl. Math. No. 54, Academic Press, New York, 1973. · Zbl 0264.12102 [15] P. Maisonobe, C. Sabbah, $$D$$-modules Cohérents et Holonomes, Travaux en cours No. 45, Hermann, Paris, 1993. [16] B. Malgrange, Cohomologie de Spencer (d’après Quillen), Sém. Math. Orsay, 1966. [17] B. Malgrange, Séminaire “Opérateurs différentiels et pseudodifférentiels”, Université Scientifique et Médicale de Grenoble, 1975-1976. [18] H. Mounier, Propriétés structurelles des systèmes linéaires à retards: aspects théoriques et pratiques, Thèse de doctorat, Université de Paris XI, Orsay, 1995. [19] Oberst, U., Multidimensional constant linear systems, Acta appl. math., 20, 1-175, (1990) · Zbl 0715.93014 [20] V.P. Palamodov, Linear Differential Operators with Constant Coefficients, Grundlehren der Mathematischen Wissenschaften, vol. 168, Springer, New York, 1970. · Zbl 0191.43401 [21] J.F. Pommaret, Partial Differential Equations and Group Theory: New Perspectives for Applications, Kluwer, Dordrecht, 1994. · Zbl 0808.35002 [22] J.F. Pommaret, Duality for control systems revisited, in: M. Guglielmi (Ed.), Proc. IFAC Conf. “System Structure and Control”, Nantes, 5-7 July 1995, Ecole Centrale de Nantes, France, pp. 438-443. [23] Pommaret, J.F., Dualité différentielle et applications, C.R. acad. sci. Paris, Série I, 320, 1225-1230, (1995) · Zbl 0863.93016 [24] J.F. Pommaret, Partial Differential Control Theory, Kluwer, Dordrecht, to appear. · Zbl 0755.93048 [25] J.F. Pommaret, A. Quadrat, Formal obstructions to the controllability of partial differential control systems, in: A. Sydow (Ed.), IMACS World Congress, vol. 5, Berlin, August 1997, pp. 209-214. [26] J.F. Pommaret, A. Quadrat, Generalized Bezout identity, AAECC 9 (2) (1998) 91-116. · Zbl 1073.93516 [27] J.F. Pommaret, A. Quadrat, Algebraic analysis of linear multidimensional control systems, IMA J. Math. Control and Inform., to appear. · Zbl 1158.93319 [28] D. Quillen, Formal properties of overdetermined systems of linear partial differential equations, Thesis, Harvard, 1964. · Zbl 1295.35005 [29] Spencer, D.C., Overdetermined systems of partial differential equations, Bull. amer. math. soc., 75, 1-114, (1965) [30] Ch. Riquier, La Méthode des Fonctions Majorantes et les Systèmes d’Équations aux Dérivées Partielles, Mémorial Sci. Math. XXXII, Gauthier-Villars, Paris, 1910. [31] J.F. Ritt, Differential Algebra, AMS Colloq. Publ., vol. 33, American Mathematical Society, New York, 1950. · Zbl 0037.18402 [32] J.J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979. · Zbl 0441.18018 [33] Wood, J.; Rogers, E.; Owens, H., Formal theory of matrix primeness, Math. control signals systems, 11, 40-78, (1998) · Zbl 0898.93008 [34] D.C. Youla, G. Gnavi, Notes on n-dimensional system theory, IEEE Trans. Circuits Systems CAS-26 (2) (1979) 105-111. · Zbl 0394.93004 [35] D.C. Youla, P.F. Pickel, The Quillen-Suslin theorem and the structure of n-dimensional elementary polynomial matrices, IEEE Trans. Circuits Systems CAS-31 (6) (1984) 513-518. · Zbl 0553.13003 [36] Zerz, E., Primeness of multivariate polynomial matrices, Systems control lett., 29, 3, 139-145, (1996) · Zbl 0866.93053
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.