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Types of Damping

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If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...

Second Order systems II

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Related Experiment Video

Updated: Jul 1, 2025

Characterizing Dissipative Elastic Metamaterials Produced by Additive Manufacturing

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Published on: June 28, 2024

Second Order Splitting Dynamics with Vanishing Damping for Additively Structured Monotone Inclusions.

Radu Ioan Boţ¹, David Alexander Hulett¹

¹Faculty of Mathematics, University of Vienna, Vienna, Austria.

Journal of Dynamics and Differential Equations

|March 4, 2024

Summary

This study introduces a novel splitting system for finding operator zeros in Hilbert spaces. The method ensures convergence to solutions and fast velocity reduction, with applications in convex optimization.

Keywords:

Asymptotic stabilization Damped inertial dynamics Lyapunov analysis Monotone inclusions Splitting system Vanishing viscosity

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Area of Science:

Optimization Theory
Functional Analysis
Numerical Analysis

Background:

Addresses the challenge of finding zeros for the sum of maximally monotone and cocoercive operators.
Builds upon existing methods by introducing a novel second-order dynamical system with vanishing damping.

Purpose of the Study:

To analyze the asymptotic behavior of trajectories generated by a time-dependent forward-backward splitting system.
To establish weak convergence of trajectories to the solution set of the operator equation.
To demonstrate fast convergence of velocities to zero.

Main Methods:

Utilizes a second-order dynamical system with vanishing damping.
Employs a time-dependent forward-backward splitting operator.
Analyzes the system's behavior in a real Hilbert space framework.

Main Results:

Proves weak convergence of generated trajectories to the set of zeros of A + B.
Demonstrates that the velocities of the trajectories converge rapidly to zero.
Derives fast convergence rates for a specific convex optimization problem as a special case.

Conclusions:

The proposed splitting system is effective for finding zeros of sums of monotone and cocoercive operators.
The system offers theoretical guarantees for convergence and fast velocity reduction.
Numerical experiments validate the theoretical findings and demonstrate practical applicability.