Stability and stabilization of linear switched systems in finite and infinite dimensions

Motivated by previous work on the stabilization of persistently excited systems, this thesis addresses stability and stabilization issues for linear switched systems in finite and infinite dimensions. After a general introduction presenting the main motivations and important results from the literature, we analyze four problems.The first system we study is a linear finite-dimensional random switched system. The time spend on each subsystem $i$ is chosen according to a probability law depending only on $i$, and the switches between subsystems are determined by a discrete Markov chain. We characterize the Lyapunov exponents by applying Oseledets' Multiplicative Ergodic Theorem to an associated discrete-time system, and provide an expression for the maximal Lyapunov exponent. These results are applied to a switched control system, showing that, under a controllability hypothesis, almost sure stabilization can be achieved with arbitrarily large decay rates, a situation in contrast to deterministic persistently excited systems.We next consider a system of $N$ transport equations with intermittent internal damping, linearly coupled by their boundary conditions through a matrix $M$, which can be seen as a system of PDEs on a star-shaped network. We prove that, if the activity of the intermittent damping terms is determined by persistently exciting signals, then, under suitable hypotheses on $M$ and on the rationality of the ratios between the lengths of the network edges, such system is exponentially stable, uniformly with respect to the persistently exciting signals. The proof of this result is based on an explicit representation formula for the solutions of the system, which allows one to efficiently track down the effects of the intermittent damping.The following topic we address is the asymptotic behavior of non-autonomous difference equations. We obtain an explicit representation formula for their solutions in terms of their initial conditions and some time-dependent matrix coefficients, which generalizes the one for the system of $N$ transport equations. The asymptotic behavior of solutions is characterized in terms of the matrix coefficients. In the case of difference equations with arbitrary switching, we obtain a stability result which generalizes Hale--Silkowski criterion for autonomous systems. Using classical transformations of hyperbolic PDEs into difference equations, we apply our results to transport and wave propagation on networks.Finally, we generalize the previous representation formula to a controlled difference equation, whose controllability is then analyzed. Relative controllability is characterized in terms of an algebraic property on the matrix coefficients from the explicit formula, generalizing Kalman criterion. We also compare the relative controllability for different delays in terms of their rational dependence structure, and provide a bound on the minimal controllability time. Exact and approximate controllability for systems with commensurable delays are characterized and proved to be equivalent. We also describe exact and approximate controllability for two-dimensional systems with two delays not necessarily commensurable.