arXiv subjects

Raphael M. Jungers

Publications and source records attributed to Raphael M. Jungers.

27 records · Page 2Linked to original sources

arXiv2012

Lifted polytope methods for stability analysis of switching systems

We describe new methods for deciding the stability of switching systems. The methods build on two ideas 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 appealing theoretical properties of our methods like a finiteness computational result which extends a known result for unlifted sets of matrices, and provide numerical examples of their good behaviour.

math.OC↗

arXiv2012

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.

math.OC↗

arXiv2012

Convex Optimization methods for computing the Lyapunov Exponent of matrices

We introduce a new approach to evaluate the largest Lyapunov exponent of a family of nonnegative matrices. The method is based on using special positive homogeneous functionals on $R^{d}_+,$ which gives iterative lower and upper bounds for the Lyapunov exponent. They improve previously known bounds and converge to the real value. The rate of convergence is estimated and the efficiency of the algorithm is demonstrated on several problems from applications (in functional analysis, combinatorics, and lan- guage theory) and on numerical examples with randomly generated matrices. The method computes the Lyapunov exponent with a prescribed accuracy in relatively high dimensions (up to 60). We generalize this approach to all matrices, not necessar- ily nonnegative, derive a new universal upper bound for the Lyapunov exponent, and show that such a lower bound, in general, does not exist.

math.OC↗

arXiv2012

When is a set of LMIs a sufficient condition for stability?

We study stability criteria for discrete time switching systems. We investigate the structure of sets of LMIs that are a sufficient condition for stability (i.e., such that any switching system which satisfies these LMIs is stable). We provide an exact characterization of these sets. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies the stability of a switching system.

math.OC↗

arXiv2007

Distance distribution in random graphs and application to networks exploration

We consider the problem of determining the proportion of edges that are discovered in an Erdos-Renyi graph when one constructs all shortest paths from a given source node to all other nodes. This problem is equivalent to the one of determining the proportion of edges connecting nodes that are at identical distance from the source node. The evolution of this quantity with the probability of existence of the edges exhibits intriguing oscillatory behavior. In order to perform our analysis, we introduce a new way of computing the distribution of distances between nodes. Our method outperforms previous similar analyses and leads to estimates that coincide remarkably well with numerical simulations. It allows us to characterize the phase transitions appearing when the connectivity probability varies.

cond-mat.stat-mech↗

arXiv2007

Overlap-free words and spectra of matrices

Overlap-free words are words over the binary alphabet $A=\{a, b\}$ that do not contain factors of the form $xvxvx$, where $x \in A$ and $v \in A^*$. We analyze the asymptotic growth of the number $u_n$ of overlap-free words of length $n$ as $ n \to \infty$. We obtain explicit formulas for the minimal and maximal rates of growth of $u_n$ in terms of spectral characteristics (the lower spectral radius and the joint spectral radius) of certain sets of matrices of dimension $20 \times 20$. Using these descriptions we provide new estimates of the rates of growth that are within 0.4% and $0.03 %$ of their exact values. The best previously known bounds were within 11% and 3% respectively. We then prove that the value of $u_n$ actually has the same rate of growth for ``almost all'' natural numbers $n$. This ``average'' growth is distinct from the maximal and minimal rates and can also be expressed in terms of a spectral quantity (the Lyapunov exponent). We use this expression to estimate it. In order to obtain our estimates, we introduce new algorithms to compute spectral characteristics of sets of matrices. These algorithms can be used in other contexts and are of independent interest.

cs.DM↗

arXiv2007

Linear time algorithms for Clobber

We prove that the single-player game clobber is solvable in linear time when played on a line or on a cycle. For this purpose, we show that this game is equivalent to an optimization problem on a set of words defined by seven classes of forbidden patterns. We also prove that, playing on the cycle, it is always possible to remove at least 2n/3 pawns, and we give a conformation for which it is not possible to do better, answering questions recently asked by Faria et al.

cs.GT↗

arXiv2007

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.

cs.MA↗

arXiv2007

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 $2 \times 2$ 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.

math.DS↗