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arXiv · 2512.20751

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
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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.

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Renan J. S. Isneri, Eric B. Santiago, Severino H. da Silva. 2025-12-23. Asymptotic behavior for a class of damped second-order gradient systems via Lyapunov method. https://arxiv.org/abs/2512.20751

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