Convergence and stability analysis for iterative dynamics with application in balanced resource allocation: A trajectory distance based Lyapunov approach

This paper addresses the convergence and stability analysis for iterative processes such as numerical iterative algorithms by using a novel trajectory distance based approach. Iterative dynamics are widespread in distributed algorithms and numerical analysis. However, efficient analysis of convergence and sensitivity of iterative dynamics is quite challenging due to the lack of systematic tools. For instance, the trajectories of iterative dynamics are usually not continuous with respect to the initial condition. Hence, the classical dynamical systems theory cannot be applied directly. In this paper, a trajectory distance based Lyapunov approach is proposed as a means to tackling convergence and sensitivity to the initial condition of iterative processes. Technically the problem of convergence and sensitivity is converted into finiteness of trajectory distance and semistability analysis of discrete-time systems. A semidefinite Lyapunov function based trajectory distance approach is proposed to characterize convergence and semistability of iterative dynamics. Two examples are provided to elucidate the proposed method. Finally, the proposed framework is used to solve the convergence and stability of iterative algorithms developed for balanced resource allocation and damage mitigation problems under adversarial attacks.