Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic behavior for a class of damped second-order gradient systems via Lyapunov method

Renan J. S. Isneri, Eric B. Santiago, Severino H. da Silva

TL;DR

The paper analyzes the long-time behavior of damped second-order gradient systems $\ddot{u}+a\dot{u}+\nabla W(u)=0$ under coercivity and near-minimum convexity assumptions on $W$, introducing a strict Lyapunov functional $V_a$ that yields uniform asymptotic stability of any global minimizer $u_*$ for $a\in(0,a_0]$, with exponential decay when $W$ satisfies a quadratic near-minimum control. It further examines the conservative limit $a=0$, showing energy conservation leads to bounded but nonconvergent trajectories and establishes a singular limit as $a\to0^+$, where decay rates vanish. Complementary nonlinear semigroup analysis proves the existence of a global attractor $\mathcal{A}$, with $\mathcal{A}=\omega(K)$ for an absorbing set $K$ independent of $a$, and reduces to the equilibrium when $W$ has a unique minimizer. Numerical simulations across quadratic, double-well, and exponential potentials illustrate the theory, highlighting underdamped, critically damped, overdamped, and conservative regimes and providing tangible insight into the phase-space geometry. The results offer a robust framework for uniform stabilization, attractor structure, and the impact of damping on gradient-like dynamics with broad relevance to optimization and physics.

Abstract

In this work we study the asymptotic behavior of a class of damped second-order gradient systems $$ \ddot{u}(t) + a\dot{u}(t) + \nabla W(u(t)) = 0, $$ under assumptions ensuring local convexity of the potential near equilibrium and coercivity at infinity. By introducing a Lyapunov functional adapted to the geometry of the system, we establish uniform asymptotic stability of the equilibrium for all $a \in (0,a_0]$, together with exponential decay when the potential satisfies a quadratic control near its minimum. Furthermore, complementary arguments based on semigroup theory reveal the existence of a global attractor. We also present numerical simulations for some $W$ potentials that illustrate the behavior of trajectories near equilibrium, in both dissipative and conservative regimes.

Asymptotic behavior for a class of damped second-order gradient systems via Lyapunov method

TL;DR

The paper analyzes the long-time behavior of damped second-order gradient systems under coercivity and near-minimum convexity assumptions on , introducing a strict Lyapunov functional that yields uniform asymptotic stability of any global minimizer for , with exponential decay when satisfies a quadratic near-minimum control. It further examines the conservative limit , showing energy conservation leads to bounded but nonconvergent trajectories and establishes a singular limit as , where decay rates vanish. Complementary nonlinear semigroup analysis proves the existence of a global attractor , with for an absorbing set independent of , and reduces to the equilibrium when has a unique minimizer. Numerical simulations across quadratic, double-well, and exponential potentials illustrate the theory, highlighting underdamped, critically damped, overdamped, and conservative regimes and providing tangible insight into the phase-space geometry. The results offer a robust framework for uniform stabilization, attractor structure, and the impact of damping on gradient-like dynamics with broad relevance to optimization and physics.

Abstract

In this work we study the asymptotic behavior of a class of damped second-order gradient systems under assumptions ensuring local convexity of the potential near equilibrium and coercivity at infinity. By introducing a Lyapunov functional adapted to the geometry of the system, we establish uniform asymptotic stability of the equilibrium for all , together with exponential decay when the potential satisfies a quadratic control near its minimum. Furthermore, complementary arguments based on semigroup theory reveal the existence of a global attractor. We also present numerical simulations for some potentials that illustrate the behavior of trajectories near equilibrium, in both dissipative and conservative regimes.
Paper Structure (10 sections, 10 theorems, 147 equations)

This paper contains 10 sections, 10 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Damped System Stability
  3. The Conservative Case $a=0$
  4. A singular limit as $a\to 0^{+}$
  5. Existence of Attractors
  6. Numerical Simulations
  7. Quadratic Potential
  8. Double-Well Potential
  9. Exponential Potential
  10. Implementation

Key Result

Theorem 1.1

Assume that the potential $W$ satisfies W1-W4. Then, for every $\varepsilon > 0$ and $a_0 > 0$, there exists $\delta > 0$ such that for all $a \in (0, a_0]$, and any solution $u_a$ of Equation with initial data satisfying the following hold: If, additionally, W2 and W3 hold with $\delta = \lambda = +\infty$, then there exists $R > 0$ such that, for all $a \in (0, a_0]$, and any solution $u_a$ of

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • ...and 14 more