Decay rates for stabilization of linear continuous-time systems with random switching
Fritz Colonius Guilherme Mazanti
Mathematical Control & Related Fields 2018, 8(0): 1-20 doi: 10.3934/mcrf.2019002

For a class of linear switched systems in continuous time a controllability condition implies that state feedbacks allow to achieve almost sure stabilization with arbitrary exponential decay rates. This is based on the Multiplicative Ergodic Theorem applied to an associated system in discrete time. This result is related to the stabilizability problem for linear persistently excited systems.

keywords: Random switching almost sure stabilization arbitrary rate of convergence Lyapunov exponents persistent excitation Multiplicative Ergodic Theorem
Nonlinear Iwasawa decomposition of control flows
Fritz Colonius Paulo Régis C. Ruffino
Discrete & Continuous Dynamical Systems - A 2007, 18(2&3): 339-354 doi: 10.3934/dcds.2007.18.339
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.
Invariance entropy, quasi-stationary measures and control sets
Fritz Colonius
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 2093-2123 doi: 10.3934/dcds.2018086

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
Topological conjugacy for affine-linear flows and control systems
Fritz Colonius Alexandre J. Santana
Communications on Pure & Applied Analysis 2011, 10(3): 847-857 doi: 10.3934/cpaa.2011.10.847
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
Discrete & Continuous Dynamical Systems - A 2006, 14(3): 447-463 doi: 10.3934/dcds.2006.14.447
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

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