Performance Analysis of Positive Systems and Optimization Algorithms with Time-delays

Abstract: Time-delay dynamical systems are used to model many real-world engineering systems, where the future evolution of a system depends not only on current states but also on the history of states. For this reason, the study of stability and control of time-delay systems is of theoretical and practical importance. In this thesis, we develop several stability analysis frameworks for dynamical systems in the presence of communication and computation time-delays, and apply our results to different challenging engineering problems.The thesis first considers delay-independent stability of positive monotone systems. We show that the asymptotic stability of positive monotone systems whose vector fields are homogeneous is independent of the magnitude and variation of time-varying delays. We present explicit expressions that allow us to give explicit estimates of the decay rate for various classes of time-varying delays. For positive linear systems, we demonstrate that the best decay rate that our results guarantee can be found via convex optimization. We also derive a set of necessary and sufficient conditions for asymptotic stability of general positive monotone (not necessarily homogeneous) systems with time-delays. As an application of our theoretical results, we discuss delay-independent stability of continuous-time power control algorithms in wireless networks.The thesis continues by studying the convergence of asynchronous fixed-point iterations involving maximum norm pseudo-contractions. We present a powerful approach for characterizing the rate of convergence of totally asynchronous iterations, where both the update intervals and communication delays may grow unbounded. When specialized to partially asynchronous iterations (where the update intervals and communication delays have a fixed upper bound), or to particular classes of unbounded delays and update intervals, our approach allows to quantify how the degree of asynchronism affects the convergence rate. In addition, we use our results to analyze the impact of asynchrony on the convergence rate of discrete-time power control algorithms in wireless networks.The thesis finally proposes an asynchronous parallel algorithm that exploits multiple processors to solve regularized stochastic optimization problems with smooth loss functions. The algorithm allows the processors to work at different rates, perform computations independently of each other, and update global decision variables using out-of-date gradients. We characterize the iteration complexity and the convergence rate of the proposed algorithm, and show that these compare favourably with the state of the art. Furthermore, we demonstrate that the impact of asynchrony on the convergence rate of the algorithm is asymptotically negligible, and a near-linear speedup in the number of processors can be expected.