41 research outputs found
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
Observable Graphs
An edge-colored directed graph is \emph{observable} if an agent that moves
along its edges is able to determine his position in the graph after a
sufficiently long observation of the edge colors. When the agent is able to
determine his position only from time to time, the graph is said to be
\emph{partly observable}. Observability in graphs is desirable in situations
where autonomous agents are moving on a network and one wants to localize them
(or the agent wants to localize himself) with limited information. In this
paper, we completely characterize observable and partly observable graphs and
show how these concepts relate to observable discrete event systems and to
local automata. Based on these characterizations, we provide polynomial time
algorithms to decide observability, to decide partial observability, and to
compute the minimal number of observations necessary for finding the position
of an agent. In particular we prove that in the worst case this minimal number
of observations increases quadratically with the number of nodes in the graph.
From this it follows that it may be necessary for an agent to pass through
the same node several times before he is finally able to determine his position
in the graph. We then consider the more difficult question of assigning colors
to a graph so as to make it observable and we prove that two different versions
of this problem are NP-complete.Comment: 15 pages, 8 figure
Resonance and marginal instability of switching systems
We analyse the so-called Marginal Instability of linear switching systems,
both in continuous and discrete time. This is a phenomenon of unboundedness of
trajectories when the Lyapunov exponent is zero. We disprove two recent
conjectures of Chitour, Mason, and Sigalotti (2012) stating that for generic
systems, the resonance is sufficient for marginal instability and for
polynomial growth of the trajectories. We provide a characterization of
marginal instability under some mild assumptions on the sys- tem. These
assumptions can be verified algorithmically and are believed to be generic.
Finally, we analyze possible types of fastest asymptotic growth of
trajectories. An example of a pair of matrices with sublinear growth is given
On asymptotic properties of matrix semigroups with an invariant cone
Recently, several research efforts showed that the analysis of joint spectral
characteristics of sets of matrices is greatly eased when these matrices share
an invariant cone. In this short note we prove two new results in this
direction. We prove that the joint spectral subradius is continuous in the
neighborhood of sets of matrices that leave an embedded pair of cones
invariant. We show that the (averaged) maximal spectral radius, as well as the
maximal trace, of products of length t, converge towards the joint spectral
radius when the matrices share an invariant cone, and addi- tionally one of
them is primitive
On Primitivity of Sets of Matrices
A nonnegative matrix is called primitive if is positive for some
integer . A generalization of this concept to finite sets of matrices is
as follows: a set of matrices is
primitive if is positive for some indices
. The concept of primitive sets of matrices comes up in a
number of problems within the study of discrete-time switched systems. In this
paper, we analyze the computational complexity of deciding if a given set of
matrices is primitive and we derive bounds on the length of the shortest
positive product.
We show that while primitivity is algorithmically decidable, unless it
is not possible to decide primitivity of a matrix set in polynomial time.
Moreover, we show that the length of the shortest positive sequence can be
superpolynomial in the dimension of the matrices. On the other hand, defining
to be the set of matrices with no zero rows or columns, we give
a simple combinatorial proof of a previously-known characterization of
primitivity for matrices in which can be tested in polynomial
time. This latter observation is related to the well-known 1964 conjecture of
Cerny on synchronizing automata; in fact, any bound on the minimal length of a
synchronizing word for synchronizing automata immediately translates into a
bound on the length of the shortest positive product of a primitive set of
matrices in . In particular, any primitive set of
matrices in has a positive product of length
A Characterization of Lyapunov Inequalities for Stability of Switched Systems
We study stability criteria for discrete-time switched systems and provide a
meta-theorem that characterizes all Lyapunov theorems of a certain canonical
type. For this purpose, we investigate the structure of sets of LMIs that
provide a sufficient condition for stability. Various such conditions have been
proposed in the literature in the past fifteen years. We prove in this note
that a family of languagetheoretic conditions recently provided by the authors
encapsulates all the possible LMI conditions, thus putting a conclusion to this
research effort. As a corollary, we show that it is PSPACE-complete to
recognize whether a particular set of LMIs implies stability of a switched
system. Finally, we provide a geometric interpretation of these conditions, in
terms of existence of an invariant set.Comment: arXiv admin note: text overlap with arXiv:1201.322
Feedback stabilization of dynamical systems with switched delays
We analyze a classification of two main families of controllers that are of
interest when the feedback loop is subject to switching propagation delays due
to routing via a wireless multi-hop communication network. We show that we can
cast this problem as a subclass of classical switching systems, which is a
non-trivial generalization of classical LTI systems with timevarying delays. We
consider both cases where delay-dependent and delay independent controllers are
used, and show that both can be modeled as switching systems with unconstrained
switchings. We provide NP-hardness results for the stability verification
problem, and propose a general methodology for approximate stability analysis
with arbitrary precision. We finally give evidence that non-trivial design
problems arise for which new algorithmic methods are needed