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

Abstract

Abstract. We investigate the asymptotic properties as t → ∞ of the following differential equation in the Hilbert space H:where the map a : R + → R + is nonincreasing and the potential G : H → R is of class 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 ∞ 0 a(t) dt = ∞ guarantees that the energy function converges toward its minimum. The more stringent condition ∞ 0 e − t 0 a(s) ds dt < ∞ is necessary to obtain the convergence of the trajectories of (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.

Asymptotics for a gradient system with memory term

“…The dynamical system (6) has been intensively studied in [7,8]. Let us recall that the function ℰ defined by…”

“…More precisely, we study the following integro-differential equation: ℎ( ) as → +∞, this equation can be interpreted as an averaged gradient system. In the recent papers [7,8], special attention is devoted to the particular case corresponding to ℎ( ) = 1, ( ) = for ≥ 0, thus modelling a situation of uniform memory. When Φ is convex and has a set of non-isolated minima, it is proved in [7] that the nonstationary solutions cannot converge.…”

“…In the recent papers [7,8], special attention is devoted to the particular case corresponding to ℎ( ) = 1, ( ) = for ≥ 0, thus modelling a situation of uniform memory. When Φ is convex and has a set of non-isolated minima, it is proved in [7] that the nonstationary solutions cannot converge. This result is striking since the corresponding trajectories of the basic gradient system are convergent (at least weakly) under the same assumptions.…”

“…The quantity ∈ ℝ + ∪ {+∞} and the map : [0, ) → ℝ depend respectively on ℎ, and their derivatives. Equation (3) has been recently studied in [7,8], where the mechanical interpretation is emphasized via the use of a suitable energy function. When the map is constant, the underlying dynamical system is known under the terminology of "Heavy Ball with Friction" system.…”

“…In this paper, we exploit fully the recent results of [7,8] to derive new results with respect to the generalized averaged gradient system ( ). Special attention is devoted to the particular case corresponding to ℎ( ) = , ( ) = , for ≥ 1.…”