Second-order differential equations with asymptotically small dissipation and piecewise flat potentials

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Stochastic Heavy ball (2016, result): “Consistent with [CEG09b], (1) is equivalent to the following integro-differential equation: 9 xt “ ́ 1 kptq ż t 0 hpsq∇fpxsqds, (2) where h and k are two memory functions related to γ.”
Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity (2018, background): “Fast minimization of function Φ: Equation (1) is a particular case of the inertial gradient-like system (3) ẍ(t) + a(t)ẋ(t) +∇Φ(x(t)) = 0, with asymptotic vanishing damping, studied by Cabot, Engler and Gadat in [17, 18].”