Barabanov Norms Research Articles

The Barabanov Norm is Generically Unique, Simple, and Easily Computed

Every irreducible discrete-time linear switching system possesses an invariant convex Lyapunov function (Barabanov norm), which provides a very refined analysis of trajectories. Until recently that notion remained rather theoretical apart from special cases. In 2015 N.Guglielmi and M.Zennaro showed that many systems possess at least one simple Barabanov norm, which moreover, can be efficiently computed. In this paper we classify all possible Barabanov norms for discrete-time systems. We prove that, under mild assumptions, such norms are unique and are either piecewise-linear or piecewise quadratic. Those assumptions can be verified algorithmically and the numerical experiments show that a vast majority of systems satisfy them. For some narrow classes of systems, there are more complicated Barabanov norms but they can still be classified and constructed. Using those results we find all trajectories of the fastest growth. They turn out to be eventually periodic with special periods. Examples and numerical results are presented.

Barabanov norms, Lipschitz continuity and monotonicity for the max algebraic joint spectral radius

We present several results describing the interplay between the max algebraic joint spectral radius (JSR) for compact sets of matrices and suitably defined matrix norms. In particular, we extend a classical result for the conventional algebra, showing that the max algebraic JSR can be described in terms of induced norms of the matrices in the set. We also show that for a set generating an irreducible semigroup (in a cone-theoretic sense), a monotone Barabanov norm always exists. This fact is then used to show that the max algebraic JSR is locally Lipschitz continuous on the space of compact irreducible sets of matrices with respect to the Hausdorff distance. We then prove that the max algebraic JSR is locally Hoelder continuous on the space of compact sets of nonnegative matrices. Finally, we prove a strict monotonicity property for the max algebraic JSR that echoes a fact for the classical JSR. The single matrix characterisation of the max algebraic JSR plays a vital role in our proofs.

Geometric and asymptotic properties associated with linear switched systems

Consider a continuous-time linear switched system on Rn associated with a compact convex set of matrices. When it is irreducible and its largest Lyapunov exponent is zero there always exists a Barabanov norm associated with the system. This paper deals with two types of issues: (a) properties of Barabanov norms such as uniqueness up to homogeneity and strict convexity; (b) asymptotic behavior of the extremal solutions of the linear switched system. Regarding Issue (a), we provide partial answers and propose four related open problems. As for Issue (b), we establish, when n=3, a Poincaré–Bendixson theorem under a regularity assumption on the set of matrices. We then revisit a noteworthy result of N.E. Barabanov describing the asymptotic behavior of linear switched system on R3 associated with a pair of Hurwitz matrices {A,A+bcT}. After pointing out a gap in Barabanov's proof we partially recover his result by alternative arguments.

Canonical Construction of Polytope Barabanov Norms and Antinorms for Sets of Matrices

Barabanov norms have been introduced in Barabanov (Autom. Remote Control, 49 (1988), pp. 152--157) and constitute an important instrument in analyzing the joint spectral radius of a family of matrices and related issues. However, although they have been studied extensively, even in very simple cases it is very difficult to construct them explicitly (see, e.g., Kozyakin (Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), pp. 143--158)). In this paper we give a canonical procedure to construct them exactly, which associates a polytope extremal norm---constructed by using the methodologies described in Guglielmi, Wirth, and Zennaro (SIAM J. Matrix Anal. Appl., 27 (2005), pp. 721--743) and Guglielmi and Protasov (Found. Comput. Math., 13 (2013), pp. 37--97)---to a polytope Barabanov norm. Hence, the existence of a polytope Barabanov norm has the same genericity of an extremal polytope norm. Moreover, we extend the result to polytope antinorms, which have been recently introduced to compute the lower spectral radi...

A New Sufficient Condition for the Uniqueness of Barabanov Norms

The joint spectral radius of a bounded set of $d \times d$ real or complex matrices is defined to be the maximum exponential rate of growth of products of matrices drawn from that set. Under quite mild conditions such a set of matrices admits an associated vector norm, called a Barabanov norm, which can be used to characterize those sequences of matrices which achieve this maximum rate of exponential growth. In this note we continue an earlier investigation into the problem of determining when the Barabanov norm associated to such a set of matrices is unique. We give a new sufficient condition for this uniqueness and provide some examples in which our condition applies. We also give a theoretical application which shows that the property of having a unique Barabanov norm can in some cases be highly sensitive to small perturbations of the set of matrices.

A relaxation scheme for computation of the joint spectral radius of matrix sets

The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In this paper, an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.

Criteria for the stability of the finiteness property and for the uniqueness of Barabanov norms

A set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product. The extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research. In this article, we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set, but also for all sufficiently nearby sets of equal cardinality. We also prove a theorem giving conditions under which the Barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar, and show that in certain cases these conditions are also persistent under small perturbations.

Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets

The problem of construction of Barabanov norms for analysis ofproperties of the joint (generalized) spectral radius ofmatrix sets has been discussed in a number of publications. In[18, 21] the method of Barabanovnorms was the key instrument in disproving the Lagarias-WangFiniteness Conjecture. The related constructions wereessentially based on the study of the geometrical properties ofthe unit balls of some specific Barabanov norms. In thiscontext the situation when one fails to find among currentpublications any detailed analysis of the geometricalproperties of the unit balls of Barabanov norms looks a bitparadoxical. Partially this is explained by the fact thatBarabanov norms are defined nonconstructively, by an implicitprocedure. So, even in simplest cases it is very difficult tovisualize the shape of their unit balls. The present work maybe treated as the first step to make up this deficiency. In thepaper an iteration procedure is considered that allows to buildnumerically Barabanov norms for the irreducible matrix sets andsimultaneously to compute the joint spectral radius of thesesets.