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

Nonlinear Iwasawa decomposition of control flows

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.
Citation: Fritz Colonius, Paulo Régis C. Ruffino. Nonlinear Iwasawa decomposition of control flows. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 339-354. doi: 10.3934/dcds.2007.18.339
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