The Heavy-Ball ODE with Time-Varying Damping: Persistence of Excitation and Uniform Asymptotic Stability
Jorge I. Poveda, Andrew R. Teel · 2020
We study the uniform asymptotic stability properties of the heavy-ball optimization dynamics with general time-varying damping. Unlike existing results in the literature, which have focused mainly on standard convergence results, we study a stronger limiting notion called uniform asymptotic stability, which is instrumental for the design of feedback-based algorithms. Given that recent results in the literature have shown that a class of heavy-ball optimization dynamics with vanishing damping fails to satisfy this limiting notion, we study sufficient and necessary conditions on the time-varying coefficients such that uniform asymptotic stability for the set of minimizers of the cost function is achieved. Our main results show that such conditions are related to the notion of persistence of excitation, which is commonly used in adaptive control and system identification. Moreover, we show that the persistence of excitation condition is not necessary for a class of high-resolution accelerated optimization dynamics with Hessian-driven damping. Our results are established by using a nested Matrosov theorem that has not been used before in the analysis of accelerated optimization algorithms.