# A functorial approach to the behaviour of multidimensional control systems

Jean-François Pommaret; Alban Quadrat

International Journal of Applied Mathematics and Computer Science (2003)

- Volume: 13, Issue: 1, page 7-13
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topPommaret, Jean-François, and Quadrat, Alban. "A functorial approach to the behaviour of multidimensional control systems." International Journal of Applied Mathematics and Computer Science 13.1 (2003): 7-13. <http://eudml.org/doc/207626>.

@article{Pommaret2003,

abstract = {We show how to use the extension and torsion functors in order to compute the torsion submodule of a differential module associated with a multidimensional control system. In particular, we show that the concept of the weak primeness of matrices corresponds to the torsion-freeness of a certain module.},

author = {Pommaret, Jean-François, Quadrat, Alban},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {weak primeness; algebraic analysis; multidimensional systems; rings of differential operators; torsion and extension functors; controllability; behavioural framework; linear multidimensional system; implicit form; torsion submodule; weak primeness property},

language = {eng},

number = {1},

pages = {7-13},

title = {A functorial approach to the behaviour of multidimensional control systems},

url = {http://eudml.org/doc/207626},

volume = {13},

year = {2003},

}

TY - JOUR

AU - Pommaret, Jean-François

AU - Quadrat, Alban

TI - A functorial approach to the behaviour of multidimensional control systems

JO - International Journal of Applied Mathematics and Computer Science

PY - 2003

VL - 13

IS - 1

SP - 7

EP - 13

AB - We show how to use the extension and torsion functors in order to compute the torsion submodule of a differential module associated with a multidimensional control system. In particular, we show that the concept of the weak primeness of matrices corresponds to the torsion-freeness of a certain module.

LA - eng

KW - weak primeness; algebraic analysis; multidimensional systems; rings of differential operators; torsion and extension functors; controllability; behavioural framework; linear multidimensional system; implicit form; torsion submodule; weak primeness property

UR - http://eudml.org/doc/207626

ER -

## References

top- Fornasini E. and Valcher M.E. (1997): nD polynomial matrices with applications to multidimensional signal analysis. - Multidim. Syst. Signal Process., Vol. 8, No. 4, pp. 387-408. Zbl0882.93038
- Kashiwara M. (1970): Algebraic Study of Systems of Partial Differential Equations. - Memoires de la Societe Mathematiques de France, No. 63 (1995).
- Malgrange B. (1966): Ideals of Differential Functions. - Oxford: Oxford University Press. Zbl0177.17902
- Oberst U. (1990): Multidimensional constant linear systems. - Acta Appl. Math., Vol. 20, pp. 1-175. Zbl0715.93014
- Pillai H.K. and Shankar S. (1999): A behavioural approach to control of distributed systems. - SIAM J. Contr. Optim., Vol. 37, No. 2, pp. 388-408. Zbl0919.93020
- Pommaret J.F. (2001): Partial Differential Control Theory. -Dordrecht: Kluwer.
- Pommaret J.F. and Quadrat A. (1999a): Localization and parametrization of linear multidimensional control systems. -Syst. Contr. Lett., Vol. 37, pp. 247-260. Zbl0948.93016
- Pommaret J.F. and Quadrat A. (1999b): Algebraic analysis of linear multidimensional control systems. - IMA J. Contr.Optim., Vol. 16, pp. 275-297. Zbl1158.93319
- Pommaret J.F. and Quadrat A. (2000): Equivalences of linear control systems. - Proc. Int. Symp. Mathematical Theory of Networks and Systems, MTNS 2000, Perpignan, France, (on CD-ROM). Zbl0971.93017
- Quadrat A. (1999): Analyse algebrique des systèmes decontrôle lineaires multidimensionnels. - Ph. D. Thesis, Ecole des Ponts et Chaussees, Marne-La-Vallee, France.
- Quadrat A. (2003): The fractional representation approach to synthesis problems: An algebraic analysis viewpoint, I. (weakly) doubly coprime factorizations, II. internal stabilization. - SIAM J. Contr. Optim., (to appear). Zbl1035.93017
- Rotman J.J. (1979). An Introduction to Homological Algebra. - New York: Academic Press. Zbl0441.18018
- Shankar S. (2001): The lattice structure of behaviours. -SIAM J. Contr. Optim., Vol. 39, No. 6, pp. 1817-1832. Zbl0983.93025
- Smith M.C. (1989): On stabilization and the existence of coprime factorizations. - IEEE Trans. Automat. Contr., Vol. 34, No. 9, pp. 1005-1007. Zbl0693.93057
- Willems J.C. (1991): Paradigms and puzzles in the theory of dynamical systems. - IEEE Trans. Automat. Contr., Vol. 36, No. 3, pp. 259-294. Zbl0737.93004
- Wood J., Rogers E. and Owens D.H. (1998): A formal theory of matrix primeness. - Math. Contr. Signals Syst., Vol. 11, No. 1, pp. 40-78. Zbl0898.93008
- Wood J. (2000): Modules and behaviours in nD systems theory. - Multidim. Syst. Signal Process., Vol. 11, pp. 11-48. Zbl0963.93015

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.