Infinite matrix products : from the joint spectral radius to combinatorics

This thesis is devoted to the analysis of problems that arise when long products of matrices taken in a given set are constructed. A typical application is the stability of switched linear systems. The stability of a discrete-time linear system is a classical engineering problem that has been well understood for long: the dynamics can be expressed in terms of the eigenvalues of the matrix ruling the system. A more complicated problem arises when the dynamical system can switch, that is, if the matrix changes over time. If this matrix is taken from a given set but can be chosen arbitrarily in this set at every time, the stability problem turns to the computation of a quantity, the joint spectral radius of the set of matrices, introduced in the early sixties. While this quantity appears to be hard to compute, it has acquired more and more importance during the last decades, and new applications of the joint spectral radius in engineering or mathematics are frequently discovered. It has for instance been proved useful for the analysis of regularity of fractals, for the continuity of wavelets, or for autonomous agents detection in sensor networks. In the first part of this thesis, we present a theoretical survey of the joint spectral radius, including old and new results. The joint spectral subradius, which is its stabilizability counterpart, is also considered. In a second part, we study some applications related to long products of matrices. We first analyse in detail a problem in coding theory, that has been recently shown to involve a joint spectral radius computation. We then propose a new application of the joint spectral radius (and related quantities) to a classical problem in number theory, namely the counting of overlap-free words. We then turn to problems related with autonomous agents detection: we analyse the trackability of sensor networks, and introduce and analyse a new notion, namely the observability of sensor networks.(FSA 3) -- UCL, 200