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March  2017, 7(1): 41-52. doi: 10.3934/mcrf.2017003

A discrete hierarchy of double bracket equations and a class of negative power series

Nancy López Reyes 1,, and  Luis E. Benítez Babilonia 2,

1. 

Instituto de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia, Calle 67 No. 53 -108, Medellin, Colombia
2. 

Departamento de Ciencias Básicas, Universidad del Sinú, Cra 1w No. 38-153, Barrio Juan XXⅢ, Montería, Colombia

* Corresponding author:nancy.lopez@udea.edu.co

Received  October 2015 Revised  September 2016 Published  December 2016

Fund Project: The first author is supported by UdeA under SUI Project (Acta No.701,2015-03-11).

The space of negative power series of $z$ on $\{z\in \mathbb{C}:|z|>1\}$ can also be parametrized by means of a system of double bracket differential equations. To show this parametrization we introduce a group factorization for equation system. This work, for the case of a double bracket system, is a continuation of an earlier study discussed in The discrete KP hierarchy and the negative power series on the complex plane. Comp. and App. Math. 32 (2013), 483-493 for the case of one bracket system.

Keywords: Double bracket equations, realization theory, linear system, infinite-dimensional system.
Mathematics Subject Classification: Primary: 15A30, 37K10, 93B05, 93B15.
Citation: Nancy López Reyes, Luis E. Benítez Babilonia. A discrete hierarchy of double bracket equations and a class of negative power series. Mathematical Control & Related Fields, 2017, 7 (1) : 41-52. doi: 10.3934/mcrf.2017003
References:
[1]

L. Benitez-Babilonia, R. Felipe and N. López Reyes, Algebraic analysis of a discrete hierarchy of double bracket equations, Diff. Equ. and Dyn. Syst., 17 (2009), 77-90.  doi: 10.1007/s12591-009-0006-x.  Google Scholar

[2]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems: Foundations and Applications, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[3]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Systems Theory, Texts in Applied Mathematics 21, Springer-Verlag, New York, 1995. doi: 978-1-4612-8702-5.  Google Scholar

[4]

R. Felipe, Algebraic aspects of Brockett type equations, Physica D, 132 (1999), 287-297.  doi: 10.1016/S0167-2789(99)00025-1.  Google Scholar

[5]

R. Felipe and F. Ongay, Algebraic aspects of the discrete KP hierarchy, Linear Alg. and its Appl., 338 (2001), 1-17.  doi: 10.1016/S0024-3795(01)00365-2.  Google Scholar

[6]

R. Felipe and N. López Reyes, The finite discrete KP hierarchy and the rational functions Disc. Dyna. in Natu. and Soci., 2008 (2008), Article ID 792632, 10pp. doi: 10.1155/2008/792632.  Google Scholar

[7]

R. Felipe and N. López Reyes, Integrability of a double bracket system, Rev. Integración, 31 (2013), 15-23.   Google Scholar

[8]

B. Jacob and H. J. Zwart, Properties of the realization of inner functions, Math. Cont. Sign. Syst., 15 (2002), 356-379.  doi: 10.1016/S0167-6911(01)00113-X.  Google Scholar

[9]

N. López Reyes, R. Felipe and T. Castro Polo, The discrete KP hierarchy and the negative power series on the complex plane, Comp. and Appl. Math., 32 (2013), 483-493.  doi: 10.1007/s40314-013-0031-9.  Google Scholar

[10]

Y. Nakamura, Geometry of rational functions and nonlinear integrable systems, SIAM J. Math. Anal., 22 (1991), 1744-1754.  doi: 10.1137/0522108.  Google Scholar

[11]

T.-Y. Tam, Gradiente flows and double bracket equations, Diff. Geom. Appl., 20 (2004), 209-224.  doi: 10.1016/j.difgeo.2003.10.008.  Google Scholar

[12]

H. J. Zwart, Transfer functions for infinite-dimensional systems, Syst. Cont. Lett., 52 (2004), 247-255.  doi: 10.1016/j.sysconle.2004.02.002.  Google Scholar

show all references

References:
[1]

L. Benitez-Babilonia, R. Felipe and N. López Reyes, Algebraic analysis of a discrete hierarchy of double bracket equations, Diff. Equ. and Dyn. Syst., 17 (2009), 77-90.  doi: 10.1007/s12591-009-0006-x.  Google Scholar

[2]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems: Foundations and Applications, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[3]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Systems Theory, Texts in Applied Mathematics 21, Springer-Verlag, New York, 1995. doi: 978-1-4612-8702-5.  Google Scholar

[4]

R. Felipe, Algebraic aspects of Brockett type equations, Physica D, 132 (1999), 287-297.  doi: 10.1016/S0167-2789(99)00025-1.  Google Scholar

[5]

R. Felipe and F. Ongay, Algebraic aspects of the discrete KP hierarchy, Linear Alg. and its Appl., 338 (2001), 1-17.  doi: 10.1016/S0024-3795(01)00365-2.  Google Scholar

[6]

R. Felipe and N. López Reyes, The finite discrete KP hierarchy and the rational functions Disc. Dyna. in Natu. and Soci., 2008 (2008), Article ID 792632, 10pp. doi: 10.1155/2008/792632.  Google Scholar

[7]

R. Felipe and N. López Reyes, Integrability of a double bracket system, Rev. Integración, 31 (2013), 15-23.   Google Scholar

[8]

B. Jacob and H. J. Zwart, Properties of the realization of inner functions, Math. Cont. Sign. Syst., 15 (2002), 356-379.  doi: 10.1016/S0167-6911(01)00113-X.  Google Scholar

[9]

N. López Reyes, R. Felipe and T. Castro Polo, The discrete KP hierarchy and the negative power series on the complex plane, Comp. and Appl. Math., 32 (2013), 483-493.  doi: 10.1007/s40314-013-0031-9.  Google Scholar

[10]

Y. Nakamura, Geometry of rational functions and nonlinear integrable systems, SIAM J. Math. Anal., 22 (1991), 1744-1754.  doi: 10.1137/0522108.  Google Scholar

[11]

T.-Y. Tam, Gradiente flows and double bracket equations, Diff. Geom. Appl., 20 (2004), 209-224.  doi: 10.1016/j.difgeo.2003.10.008.  Google Scholar

[12]

H. J. Zwart, Transfer functions for infinite-dimensional systems, Syst. Cont. Lett., 52 (2004), 247-255.  doi: 10.1016/j.sysconle.2004.02.002.  Google Scholar

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