Nonlinear Iwasawa decomposition of control flows
Fritz Colonius Paulo Régis C. Ruffino
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.
Topological conjugacy for affine-linear flows and control systems
Fritz Colonius Alexandre J. Santana
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
Fundamental semigroups for dynamical systems
Fritz Colonius Marco Spadini
Algebraic semigroups describing the dynamic behavior are associated to compact, locally maximal chain transitive subsets. The construction is based on perturbations and associated local control sets. The dependence on the perturbation structure is analyzed.
keywords: perturbation structure algebraic semigroups. Chain transitive subsets local control sets
Invariance entropy, quasi-stationary measures and control sets
Fritz Colonius

For control systems in discrete time, this paper discusses measure-theoretic invariance entropy for a subset Q of the state space with respect to a quasi-stationary measure obtained by endowing the control range with a probability measure. The main results show that this entropy is invariant under measurable transformations and that it is already determined by certain subsets of Q which are characterized by controllability properties.

keywords: Invariance entropy quasi-stationary measures control sets coder-controllers transitivity set

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