Nonlinear Iwasawa decomposition of control flows

Fritz Colonius; Paulo Régis C. Ruffino

doi:10.3934/dcds.2007.18.339

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2007, Volume 18, Issue 2&3: 339-354. Doi: 10.3934/dcds.2007.18.339

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Nonlinear Iwasawa decomposition of control flows

Fritz Colonius^1, and
Paulo Régis C. Ruffino^2,

1.
Institut für Mathematik, Universität Augsburg, 86135 Augsburg
2.
Departamento de Matemática, Universidade Estadual de Campinas, 13.081-970 - Campinas - SP

Received: November 30, 2005

Revised: May 31, 2006

Published: May 2007

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Abstract

Let $\varphi(t,\cdot,u)$ be the flow of a control system on a Riemannian manifold $M$ of constant curvature. For a given initial orthonormal frame $k$ in the tangent space $T_{x_{0}}M$ for some $x_{0}\in M$, there exists a unique decomposition $\varphi_{t}=\Theta_{t}\circ\rho_{t}$ where $\Theta_{t}$ is a control flow in the group of isometries of $M$ and the remainder component $\rho_{t}$ fixes $x_{0}$ with derivative $D\rho_{t}(k)=k\cdot s_{t}$ where $s_{t}$ are upper triangular matrices. Moreover, if $M$ is flat, an affine component can be extracted from the remainder.

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Mathematics Subject Classification: Primary: 37J40.

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