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LIFTED POLYTOPE METHODS FOR COMPUTING THE JOINT SPECTRAL RADIUS
, 2014
"... We present new methods for computing the joint spectral radius of finite sets of matrices. The methods build on two ideas that previously appeared in the literature: the polytope norm iterative construction, and the lifting procedure. Moreover, the combination of these two ideas allows us to introd ..."
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We present new methods for computing the joint spectral radius of finite sets of matrices. The methods build on two ideas that previously appeared in the literature: the polytope norm iterative construction, and the lifting procedure. Moreover, the combination of these two ideas allows us to introduce a pruning algorithm which can importantly reduce the computational burden. We prove several theoretical properties of our methods, such as finiteness computational result which extends a known result for unlifted sets of matrices, and provide numerical examples of their good behavior.
THEME Modeling, Optimization, and Control of Dynamic SystemsTable of contents
"... 4.3. Switched systems 5 5. Software................................................................................. 6 6. New Results.............................................................................. 6 6.1. New results: geometric control 6 6.2. New results: quantum control 8 6.3. New res ..."
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4.3. Switched systems 5 5. Software................................................................................. 6 6. New Results.............................................................................. 6 6.1. New results: geometric control 6 6.2. New results: quantum control 8 6.3. New results: neurophysiology 8 6.4. New results: switched systems 9
Geometric and asymptotic properties associated with linear switched systems ∗
, 2014
"... Consider continuoustime linear switched systems on Rn associated with compact convex sets of matrices. When the system is irreducible and the largest Lyapunov exponent is equal to zero, there always exists a Barabanov norm (i.e. a norm which is non increasing along trajectories of the linear switc ..."
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Consider continuoustime linear switched systems on Rn associated with compact convex sets of matrices. When the system is irreducible and the largest Lyapunov exponent is equal to zero, there always exists a Barabanov norm (i.e. a norm which is non increasing along trajectories of the linear switched system together with extremal trajectories starting at every point, that is trajectories of the linear switched system with constant norm). This paper deals with two sets of issues: (a) properties of Barabanov norms such as uniqueness up to homogeneity and strict convexity; (b) asymptotic behaviour of the extremal solutions of the linear switched system. Regarding Issue (a), we provide partial answers and propose four open problems motivated by appropriate examples. As for Issue (b), we establish, when n = 3, a PoincaréBendixson theorem under a regularity assumption on the set of matrices defining the system. Moreover, we revisit the noteworthy result of N.E. Barabanov [5] dealing with the linear switched system on R3 associated with a pair of Hurwitz matrices {A,A+ bcT}. We first point out a fatal gap in Barabanov’s argument in connection with geometric features associated with a Barabanov norm. We then provide partial answers relative to the asymptotic behavior of this linear switched system. 1
Nonlinear
"... eigenvalue problems arising from growth maximization of positive linear dynamical systems ..."
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eigenvalue problems arising from growth maximization of positive linear dynamical systems
Nonlinear
"... eigenvalue problems arising from growth maximization of positive linear dynamical systems ..."
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eigenvalue problems arising from growth maximization of positive linear dynamical systems