On the long time behavior of second order differential equations with asymptotically small dissipation

Abstract:

We investigate the asymptotic properties as $t\to \infty$ of the following differential equation in the Hilbert space $H$: \begin{equation*}(\mathcal {S})\qquad \qquad \qquad \ddot {x}(t)+a(t)\dot {x}(t)+ \nabla G(x(t))=0, \quad t\geq 0,\qquad \qquad \qquad \qquad \quad \end{equation*} where the map $a:\mathbb {R}_+\to \mathbb {R}_+$ is nonincreasing and the potential $G:H\to \mathbb {R}$ is of class $\mathcal {C}^1$. If the coefficient $a(t)$ is constant and positive, we recover the so-called “Heavy Ball with Friction” system. On the other hand, when $a(t)=1/(t+1)$ we obtain the trajectories associated to some averaged gradient system. Our analysis is mainly based on the existence of some suitable energy function. When the function $G$ is convex, the condition $\int _0^\infty a(t) dt =\infty$ guarantees that the energy function converges toward its minimum. The more stringent condition $\int _0^{\infty } e^{-\int _0^t a(s) ds}dt<\infty$ is necessary to obtain the convergence of the trajectories of $(\mathcal {S})$ toward some minimum point of $G$. In the one-dimensional setting, a precise description of the convergence of solutions is given for a general nonconvex function $G$. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense.