Fabian Wirth
The generalized spectral radius and extremal norms.
Preprint series: Berichte aus der Technomathematik 00-16, Universität Bremen, 2000.
MSC:
34D08 Ordinary Differential Equations: Lyapunov exponents
93C55 Control systems, guided systems: Discrete-time
Abstract:
The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm always exists. This approach lends itself easily to the analysis of further properties of the generalized spectral radius. We prove that the generalized spectral radius is locally Lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius. Sufficient conditions for the existence of extremal norms are obtained.

Keywords: Discrete inclusions, generalized spectral radius, joint spectral radius, extremal norms, irreducibility

Download:  (gzipped postscript version, 101 KB)   (pdf version, not yet available)

Home Fabian Wirth's publication page Research Group "Regelungssysteme"
Zentrum für Technomathematik (German)
Institute for Dynamical Systems
The Maths Department
The University of Bremen

Notes:
appeared in Linear Algebra and its Apllications