A Gel'fand-type spectral radius formula and stability of linear constrained switching systems

2011/07/01 by Dai, Xiongping
#15B52 #37N35 #93D20 #Dynamical Systems (math.DS) #FOS: Electrical engineering #FOS: Mathematics #Optimization and Control (math.OC) #Rings and Algebras (math.RA) #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · doi:10.48550/arxiv.1107.0124

Abstract

Using ergodic theory, in this paper we present a Gel'fand-type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup \bS⁺ restricted to a subset that need not carry the algebraic structure of \bS⁺. This generalizes the Berger-Wang formula. Using it as a tool, we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one-sided Markov shift associated to \bS.

Related

• Stabilizability and lower spectral radius for linear switched systems with singular matrices
• Gripenberg-like algorithm for the lower spectral radius
• A convergent hierarchy of non-linear eigenproblems to compute the joint\n spectral radius of nonnegative matrices
• A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory
• Certifying unstability of Switched Systems using Sum of Squares Programming