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May  2011, 10(3): 847-857. doi: 10.3934/cpaa.2011.10.847

Topological conjugacy for affine-linear flows and control systems

Fritz Colonius 1, and  Alexandre J. Santana 2,

1. 

Institut für Mathematik, Universität Augsburg, 86135 Augsburg
2. 

Departamento de Matemática, Universidade Estadual de Maringá, Maringá-PR, 87020-900, Brazil

Received  March 2009 Revised  September 2009 Published  December 2010

Hyperbolic affine-linear flows on vector bundles possess unique bounded solutions on the real line. Hence they are topologically skew conjugate to their linear parts. This is used to show a classification of inhomogeneous bilinear control systems.
Keywords: affine-linear flows, bilinear control systems., Topological conjugacy.
Mathematics Subject Classification: Primary: 37B55, 37N35; Secondary: 93B1.
Citation: Fritz Colonius, Alexandre J. Santana. Topological conjugacy for affine-linear flows and control systems. Communications on Pure & Applied Analysis, 2011, 10 (3) : 847-857. doi: 10.3934/cpaa.2011.10.847
References:
[1]

A. A. Agrachev and Y. L. Sachkov, "Control Theory from a Geometric Viewpoint,", Springer-Verlag, (2004).   Google Scholar

[2]

B. Aulbach and T. Wanner, Integral manifolds for Carathéodory type differential equations in Banach spaces,, in, (1996), 45.   Google Scholar

[3]

V. Ayala, F. Colonius and W. Kliemann, On topological equivalence of linear flows with applications to bilinear control systems,, J. Dynamical and Control Systems, 13 (2007), 337.  doi: doi:10.1007/s10883-007-9021-9.  Google Scholar

[4]

L. Baratchart, M. Chyba and J.-P. Pomet, A Grobman-Hartman theorem for control systems,, J. Dynamics and Differential Equations, 19 (2007), 95.   Google Scholar

[5]

F. Colonius and W. Kliemann, "The Dynamics of Control,", Birkh\, (2000).   Google Scholar

[6]

Nguyen Dinh Cong, "Topological Dynamics of Random Dynamical Systems,", Oxford Math. Monogr., (1997).   Google Scholar

[7]

D. L. Elliott, "Bilinear Control Systems. Matrices in Action,", Applied Mathematical Sciences, 169 (2009).   Google Scholar

[8]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,", CRC Press, (1999).   Google Scholar

show all references

References:
[1]

A. A. Agrachev and Y. L. Sachkov, "Control Theory from a Geometric Viewpoint,", Springer-Verlag, (2004).   Google Scholar

[2]

B. Aulbach and T. Wanner, Integral manifolds for Carathéodory type differential equations in Banach spaces,, in, (1996), 45.   Google Scholar

[3]

V. Ayala, F. Colonius and W. Kliemann, On topological equivalence of linear flows with applications to bilinear control systems,, J. Dynamical and Control Systems, 13 (2007), 337.  doi: doi:10.1007/s10883-007-9021-9.  Google Scholar

[4]

L. Baratchart, M. Chyba and J.-P. Pomet, A Grobman-Hartman theorem for control systems,, J. Dynamics and Differential Equations, 19 (2007), 95.   Google Scholar

[5]

F. Colonius and W. Kliemann, "The Dynamics of Control,", Birkh\, (2000).   Google Scholar

[6]

Nguyen Dinh Cong, "Topological Dynamics of Random Dynamical Systems,", Oxford Math. Monogr., (1997).   Google Scholar

[7]

D. L. Elliott, "Bilinear Control Systems. Matrices in Action,", Applied Mathematical Sciences, 169 (2009).   Google Scholar

[8]

C. Robinson, "Dynamical Systems. Stability, Symbolic Dynamics, and Chaos,", CRC Press, (1999).   Google Scholar

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