Singular Perturbation Analysis of Accelerated Inertial Systems for Non-Potential Operators in the High-Damping Regime

This article analyzes the asymptotic behavior of three classes of second-order inertial systems with vanishing damping when applied to non-potential operators. We consider the dynamics (V-AVD)_α (V-AVD) α , (V-DIN-AVD)α ,β (V-DIN-AVD) α , β , and (V-ISIHD)α ,β (V-ISIHD) α , β , where V: H → H V : H → H is a λ λ -cocoercive operator. By using a time-scale change t = √2(α +1)s t = 2 ( α + 1 ) s , we show that as the damping parameter α → +∞ α → + ∞ , the rescaled trajectories converge uniformly on compact time intervals to solutions of first-order differential equations. Specifically, we establish convergence to the monotone flow ẏ(s) + V(y(s)) = 0 y ˙ ( s ) + V ( y ( s ) ) = 0 for (V-AVD), to a Levenberg-Marquardt type equation ẏ(s) + V(y(s)) + β ₀ d/dsV(y(s)) = 0 y ˙ ( s ) + V ( y ( s ) ) + β 0 d ds V ( y ( s ) ) = 0 for (V-DIN-AVD) in finite dimensions under the condition λ > 2β ₀