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

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, $Φ: \mathcal H \rightarrow \mathbb R$ and $Ψ: \mathcal H \rightarrow \mathbb R$ two convex differentiable functions, $γ$ a positive damping parameter, and $ε(t)$ a function of $t$ which tends to zero as $t$ goes to $+\infty$, we consider the second-order differential equation $\ddot{x}(t) + γ\dot{x}(t) + \nabla Φ(x(t)) + ε(t) \nabla Ψ(x(t)) = 0.$ This system models the emergence of various collective behaviors in game theory, as well as the asymptotic control of coupled nonlinear oscillators. Assuming that $ε(t)$ 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 $Ψ$ over the set $C$ of minimizers of $Φ$. As key assumptions, we suppose that $\int_{0}^{+\infty}ε(t) dt = + \infty$ and that, for every $p$ belonging to a convex cone $\mathcal C$ depending on the data $Φ$ and $Ψ$ $\int_{0}^{+\infty} \left[Φ^* \left(ε(t)p\right) -σ_C \left(ε(t)p\right)\right]dt < + \infty$ where $Φ^*$ is the Fenchel conjugate of $Φ$, and $σ_C$ is the support function of $C$. An application is given to coupled oscillators.

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