## Main Article Content

## Abstract

Functional systems arise in the treatment of systems of partial differential equations, delay-differential equations, multidimensional equations, etc. The problem of reducing a linear functional system to a system containing fewer equations and unknowns was first studied by Serre. Finding an equivalent presentation of a linear functional system containing fewer equations and fewer unknowns can generally simplify both the study of the structural properties of the linear functional system and of different numerical analysis issues, and it can sometimes help in solving the linear functional system. In this paper, Fuhrmann's equivalence is used to present a constructive result on the reduction of under-determined linear functional systems to a single equation involving a single unknown. This equivalence transformation has been studied by a number of authors and has been shown to play an important role in the theory of linear functional systems.

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## Keywords

## Article Details

* * References

- Rosenbrock, H.H. State space and multivariable theory, 1970, Nelson-Wiley, London, New York.
- Kailath, T. Linear Systems, 1980, Prentice-Hall.
- Youla, D.C. and Gnavi, G. Notes on -dimensional system theory. IEEE Trans. Circuits and Systems, 1979, 26(2), 105–111.
- Youla, D.C. and Pickel, P.F. The Quillen-Suslin theorem and the structure of n-dimensional elementary polynomial matrices. IEEE Trans. Circuits and Systems, 1984, 31, 513–517.
- Zerz, E. Topics in Multidimensional Linear Systems Theory, 2000, Springer, London.
- Fuhrmann, P.A. On strict system equivalence and similarity. Int. J. Control, 1977, 25(1), 5-10.
- Pugh, A.C., McInerney, S.J., Hou, M. and Hayton, G.E. 1996. A transformation for 2-D systems and its invariants. In Proceedings of the 35th IEEE conference on decision and control, pages 2157–2158, Kobe (Japan).
- Pugh, A.C., McInerney, S.J. and El-Nabrawy, E.M.O. Zero structures of -D systems. Int. J. Control, 2005,78(4), 277–285.
- Serre, J.P. Sur les modules projectifs. Séminaire Dubreil-Pisot, 1960/61, 2, 23-34, In Jean-Pierre Serre, Oeuvres, Collected Papers, 1960-1971, 2, 23-34.
- Quillen, D. Projective modules over polynomial rings. Invent. Math., 1976, 36, 167-171.
- Suslin, A.A. Projective modules over polynomial rings are free. Soviet Math Dokl., 1976, 17(4), 1160–1164.
- Frost, M.G. and Boudellioua, M.S. Some further results concerning matrices with elements in a polynomial ring. Int. J. Control, 1986, 43(5), 1543-1555.
- Boudellioua, M.S. and Quadrat, A. Serre's reduction of linear functional systems. Mathematics in Computer Science, 2010, 4(2), 289–312.
- Boudellioua, M.S., Further results on the equivalence to Smith form of multivariate polynomial matrices. Control and Cybernetics, 2013, 42(2), 543–551.
- Boudellioua, M.S., On the simplification of systems of linear multidimensional equations. In The Sage Days 24 Workshop on Symbolic Computation in Differential Algebra and Special Functions, 2010, Hagenberg (Austria).
- Lin, Z. and Bose, N.K., A generalization of Serre's conjecture and related issues. Linear Algebra and its Applications, 2001, 338, 125–138.
- Fabianska, A. and Quadrat, A. Applications of the Quillen-Suslin theorem in multidimensional systems theory. In H. Park and G. Regensburger, editors, Gröbner Bases in Control Theory and Signal Processing, Radon Series on Computation and Applied Mathematics 3, 2007, pages 23–106. de Gruyter publisher.
- Pommaret, J.F. and Quadrat, A. Formal elimination for multidimensional systems and applications to control theory. Mathematics of Control, Signal and Systems, 2000, 13, 193–215.
- Levandovskyy, V. and Zerz, E. Obstructions to genericity in the study of parametric problems in control theory. In H. Park and G. Regensburger, editors, Gröbner Bases in Control Theory and Signal Processing, Radon Series on Computation and Applied Mathematics 3, 2007, pages 127–149. de Gruyter publisher.
- Chyzak, F., Quadrat, A. and Robertz, D. OreModules: A symbolic package for the study of multidimensional linear systems. In J. Chiasson and J.J. Loiseau, editors, Applications of Time-Delay Systems, LNCIS 352, 2007, pages 233–264. Springer, http://wwwb.math.rwth-aachen.de/OreModules/.

#### References

Rosenbrock, H.H. State space and multivariable theory, 1970, Nelson-Wiley, London, New York.

Kailath, T. Linear Systems, 1980, Prentice-Hall.

Youla, D.C. and Gnavi, G. Notes on -dimensional system theory. IEEE Trans. Circuits and Systems, 1979, 26(2), 105–111.

Youla, D.C. and Pickel, P.F. The Quillen-Suslin theorem and the structure of n-dimensional elementary polynomial matrices. IEEE Trans. Circuits and Systems, 1984, 31, 513–517.

Zerz, E. Topics in Multidimensional Linear Systems Theory, 2000, Springer, London.

Fuhrmann, P.A. On strict system equivalence and similarity. Int. J. Control, 1977, 25(1), 5-10.

Pugh, A.C., McInerney, S.J., Hou, M. and Hayton, G.E. 1996. A transformation for 2-D systems and its invariants. In Proceedings of the 35th IEEE conference on decision and control, pages 2157–2158, Kobe (Japan).

Pugh, A.C., McInerney, S.J. and El-Nabrawy, E.M.O. Zero structures of -D systems. Int. J. Control, 2005,78(4), 277–285.

Serre, J.P. Sur les modules projectifs. Séminaire Dubreil-Pisot, 1960/61, 2, 23-34, In Jean-Pierre Serre, Oeuvres, Collected Papers, 1960-1971, 2, 23-34.

Quillen, D. Projective modules over polynomial rings. Invent. Math., 1976, 36, 167-171.

Suslin, A.A. Projective modules over polynomial rings are free. Soviet Math Dokl., 1976, 17(4), 1160–1164.

Frost, M.G. and Boudellioua, M.S. Some further results concerning matrices with elements in a polynomial ring. Int. J. Control, 1986, 43(5), 1543-1555.

Boudellioua, M.S. and Quadrat, A. Serre's reduction of linear functional systems. Mathematics in Computer Science, 2010, 4(2), 289–312.

Boudellioua, M.S., Further results on the equivalence to Smith form of multivariate polynomial matrices. Control and Cybernetics, 2013, 42(2), 543–551.

Boudellioua, M.S., On the simplification of systems of linear multidimensional equations. In The Sage Days 24 Workshop on Symbolic Computation in Differential Algebra and Special Functions, 2010, Hagenberg (Austria).

Lin, Z. and Bose, N.K., A generalization of Serre's conjecture and related issues. Linear Algebra and its Applications, 2001, 338, 125–138.

Fabianska, A. and Quadrat, A. Applications of the Quillen-Suslin theorem in multidimensional systems theory. In H. Park and G. Regensburger, editors, Gröbner Bases in Control Theory and Signal Processing, Radon Series on Computation and Applied Mathematics 3, 2007, pages 23–106. de Gruyter publisher.

Pommaret, J.F. and Quadrat, A. Formal elimination for multidimensional systems and applications to control theory. Mathematics of Control, Signal and Systems, 2000, 13, 193–215.

Levandovskyy, V. and Zerz, E. Obstructions to genericity in the study of parametric problems in control theory. In H. Park and G. Regensburger, editors, Gröbner Bases in Control Theory and Signal Processing, Radon Series on Computation and Applied Mathematics 3, 2007, pages 127–149. de Gruyter publisher.

Chyzak, F., Quadrat, A. and Robertz, D. OreModules: A symbolic package for the study of multidimensional linear systems. In J. Chiasson and J.J. Loiseau, editors, Applications of Time-Delay Systems, LNCIS 352, 2007, pages 233–264. Springer, http://wwwb.math.rwth-aachen.de/OreModules/.