  In a Hilbert space $\mathcal H$, we study the asymptotic behaviour, as time
variable $t$ goes to $+\infty$, of nonautonomous gradient-like dynamical
systems 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 convex
differentiable functions, $\gamma$ a positive damping parameter, and $\epsilon
(t)$ a function of $t$ which tends to zero as $t$ goes to $+\infty$, we
consider the second-order differential equation $$\ddot{x}(t) + \gamma
\dot{x}(t) + \nabla \Phi (x(t)) + \epsilon (t) \nabla \Psi (x(t)) = 0. $$ This
system models the emergence of various collective behaviors in game theory, as
well as the asymptotic control of coupled nonlinear oscillators. Assuming that
$\epsilon(t)$ tends to zero moderately slowly as $t$ goes to infinity, we show
that the trajectories converge weakly in $\mathcal H$. The limiting equilibria
are solutions of the hierarchical minimization problem which consists in
minimizing $\Psi$ over the set $C$ of minimizers of $\Phi$. As key assumptions,
we suppose that $ \int_{0}^{+\infty}\epsilon (t) dt = + \infty $ and that, for
every $p$ 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 $\Phi^*$ is
the Fenchel conjugate of $\Phi$, and $\sigma_C $ is the support function of
$C$. An application is given to coupled oscillators.
