40 research outputs found
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
Mather sets for sequences of matrices and applications to the study of joint spectral radii
The joint spectral radius of a compact set of d-times-d matrices is defined
?to be the maximum possible exponential growth rate of products of matrices
drawn from that set. In this article we investigate the ergodic-theoretic
structure of those sequences of matrices drawn from a given set whose products
grow at the maximum possible rate. This leads to a notion of Mather set for
matrix sequences which is analogous to the Mather set in Lagrangian dynamics.
We prove a structure theorem establishing the general properties of these
Mather sets and describing the extent to which they characterise matrix
sequences of maximum growth. We give applications of this theorem to the study
of joint spectral radii and to the stability theory of discrete linear
inclusions.
These results rest on some general theorems on the structure of orbits of
maximum growth for subadditive observations of dynamical systems, including an
extension of the semi-uniform subadditive ergodic theorem of Schreiber, Sturman
and Stark, and an extension of a noted lemma of Y. Peres. These theorems are
presented in the appendix
On the Finiteness Property for Rational Matrices
We analyze the periodicity of optimal long products of matrices. A set of
matrices is said to have the finiteness property if the maximal rate of growth
of long products of matrices taken from the set can be obtained by a periodic
product. It was conjectured a decade ago that all finite sets of real matrices
have the finiteness property. This conjecture, known as the ``finiteness
conjecture", is now known to be false but no explicit counterexample to the
conjecture is available and in particular it is unclear if a counterexample is
possible whose matrices have rational or binary entries. In this paper, we
prove that finite sets of nonnegative rational matrices have the finiteness
property if and only if \emph{pairs} of \emph{binary} matrices do. We also show
that all {pairs} of binary matrices have the finiteness property.
These results have direct implications for the stability problem for sets of
matrices. Stability is algorithmically decidable for sets of matrices that have
the finiteness property and so it follows from our results that if all pairs of
binary matrices have the finiteness property then stability is decidable for
sets of nonnegative rational matrices. This would be in sharp contrast with the
fact that the related problem of boundedness is known to be undecidable for
sets of nonnegative rational matrices.Comment: 12 pages, 1 figur
A Gel'fand-type spectral radius formula and stability of linear constrained switching systems
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.Comment: 16 pages; to appear in Linear Algebra and its Application
Computationally efficient approximations of the joint spectral radius
The joint spectral radius of a set of matrices is a measure of the maximal
asymptotic growth rate that can be obtained by forming long products of
matrices taken from the set. This quantity appears in a number of application
contexts but is notoriously difficult to compute and to approximate. We
introduce in this paper a procedure for approximating the joint spectral radius
of a finite set of matrices with arbitrary high accuracy. Our approximation
procedure is polynomial in the size of the matrices once the number of matrices
and the desired accuracy are fixed
An explicit counterexample to the Lagarias-Wang finiteness conjecture
The joint spectral radius of a finite set of real matrices is
defined to be the maximum possible exponential rate of growth of long products
of matrices drawn from that set. A set of matrices is said to have the
\emph{finiteness property} if there exists a periodic product which achieves
this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that
every finite set of real matrices satisfies the finiteness
property. However, T. Bousch and J. Mairesse proved in 2002 that
counterexamples to the finiteness conjecture exist, showing in particular that
there exists a family of pairs of matrices which contains a
counterexample. Similar results were subsequently given by V.D. Blondel, J.
Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample
to the finiteness conjecture has so far been given. The purpose of this paper
is to resolve this issue by giving the first completely explicit description of
a counterexample to the Lagarias-Wang finiteness conjecture. Namely, for the
set \mathsf{A}_{\alpha_*}:= \{({cc}1&1\\0&1), \alpha_*({cc}1&0\\1&1)\} we
give an explicit value of \alpha_* \simeq
0.749326546330367557943961948091344672091327370236064317358024...] such that
does not satisfy the finiteness property.Comment: 27 pages, 2 figure
Extremal sequences of polynomial complexity
The joint spectral radius of a bounded set of real matrices is
defined to be the maximum possible exponential growth rate of products of
matrices drawn from that set. For a fixed set of matrices, a sequence of
matrices drawn from that set is called \emph{extremal} if the associated
sequence of partial products achieves this maximal rate of growth. An
influential conjecture of J. Lagarias and Y. Wang asked whether every finite
set of matrices admits an extremal sequence which is periodic. This is
equivalent to the assertion that every finite set of matrices admits an
extremal sequence with bounded subword complexity. Counterexamples were
subsequently constructed which have the property that every extremal sequence
has at least linear subword complexity. In this paper we extend this result to
show that for each integer , there exists a pair of square matrices
of dimension for which every extremal sequence has subword
complexity at least .Comment: 15 page