  There have been many recent efforts to study accelerated optimization
algorithms from the perspective of dynamical systems. In this paper, we focus
on the robustness properties of the time-varying continuous-time version of
these dynamics. These properties are critical for the implementation of
accelerated algorithms in feedback-based control and optimization
architectures. We show that a family of dynamics related to the continuous-time
limit of Nesterov's accelerated gradient method can be rendered unstable under
arbitrarily small bounded disturbances. Indeed, while solutions of these
dynamics may converge to the set of optimizers, in general, this set may not be
uniformly asymptotically stable. To induce uniformity, and robustness as a
byproduct, we propose a framework where we regularize the dynamics by using
resetting mechanisms that are modeled by well-posed hybrid dynamical systems.
For these hybrid dynamics, we establish uniform asymptotic stability and
robustness properties, as well as convergence rates that are similar to those
of the non-hybrid dynamics. We finish by characterizing a family of
discretization mechanisms that retain the main stability and robustness
properties of the hybrid algorithms.
