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March  2007, 18(2&3): 339-354. doi: 10.3934/dcds.2007.18.339

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, Brazil

Received  December 2005 Revised  June 2006 Published  March 2007

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.
Keywords: Control flows, isometries, group of affine transformations, non-linear Iwasawa decomposition..
Mathematics Subject Classification: Primary: 37J4.
Citation: Fritz Colonius, Paulo Régis C. Ruffino. Nonlinear Iwasawa decomposition of control flows. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 339-354. doi: 10.3934/dcds.2007.18.339
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