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In a Hilbert space $\mathcal H$, we study the asymptotic behaviour, as timevariable $t$ goes to $+\infty$, of nonautonomous gradient-like dynamicalsystems involving inertia and multiscale features.  Given $\mathcal H$ a general Hilbert space, $\Phi: \mathcal H \rightarrow\mathbb R$ and $\Psi: \mathcal H \rightarrow \mathbb R$ two convexdifferentiable functions, $\gamma$ a positive damping parameter, and $\epsilon(t)$ a function of $t$ which tends to zero as $t$ goes to $+\infty$, weconsider the second-order differential equation $$\ddot{x}(t) + \gamma\dot{x}(t) + \nabla \Phi (x(t)) + \epsilon (t) \nabla \Psi (x(t)) = 0. $$ Thissystem models the emergence of various collective behaviors in game theory, aswell as the asymptotic control of coupled nonlinear oscillators. Assuming that$\epsilon(t)$ tends to zero moderately slowly as $t$ goes to infinity, we showthat the trajectories converge weakly in $\mathcal H$. The limiting equilibriaare solutions of the hierarchical minimization problem which consists inminimizing $\Psi$ over the set $C$ of minimizers of $\Phi$. As key assumptions,we suppose that $ \int_{0}^{+\infty}\epsilon (t) dt = + \infty $ and that, forevery $p$ belonging to a convex cone $\mathcal C$ depending on the data $\Phi$and $\Psi$ $$ \int_{0}^{+\infty} \left[\Phi^* \left(\epsilon (t)p\right)-\sigma_C \left(\epsilon (t)p\right)\right]dt < + \infty $$ where $\Phi^*$ isthe Fenchel conjugate of $\Phi$, and $\sigma_C $ is the support function of$C$. An application is given to coupled oscillators.

Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects