Synchronizing automata with random inputs

We study the problem of synchronization of automata with random inputs. We present a series of automata such that the expected number of steps until synchronization is exponential in the number of states. At the same time, we show that the expected number of letters to synchronize any pair of the famous Cerny automata is at most cubic in the number of states

On observability in networked control systems with packet losses

This paper deals with observability properties of networked control systems subject to packet losses. We employ a switching systems perspective in which available information on the packet loss signal, e.g., there can be at most a pre-specified number of consecutive losses, is modelled through an automaton. Based on this perspective we address several natural extensions envisioned in [7]. Our specific contributions are as follows. Firstly, we show that the method introduced in [7] in the context of controllability of linear systems subject to packet losses extends to the question of observability. The proposed characterization is necessary and sufficient as well as algorithmically verifiable. For the observability problem, our proof is valid also for non-invertible matrices, thereby improving upon the previous results in [7]. Secondly, we show that the model employed for our analysis encompasses the model of wireless control networks with switching delays introduced in [6] (though at a cost of exponential encoding). We raise several open questions related to the algebraic nature of the problem under consideration

Stability of linear problems: joint spectral radius of sets of matrices

It is wellknown that the stability analysis of step-by-step numerical methods for differential equations often reduces to the analysis of linear difference equations with variable coefficients. This class of difference equations leads to a family F of matrices depending on some parameters and the behaviour of the solutions depends on the convergence properties of the products of the matrices of F. To date, the techniques mainly used in the literature are confined to the search for a suitable norm and for conditions on the parameters such that the matrices of F are contractive in that norm. In general, the resulting conditions are more restrictive than necessary. An alternative and more effective approach is based on the concept of joint spectral radius of the family F, r(F). It is known that all the products of matrices of F asymptotically vanish if and only if r (F) < 1. The aim of this chapter is that to discuss the main theoretical and computational aspects involved in the analysis of the joint spectral radius and in applying this tool to the stability analysis of the discretizations of differential equations as well as to other stability problems. In particular, in the last section, we present some recent heuristic techniques for the search of optimal products in finite families, which constitute a fundamental step in the algorithms which we discuss. The material we present in the final section is part of an original research which is in progress and is still unpublished