On damped second-order gradient systems

Pascal Bégout (1) , Jérôme Bolte (2) , Mohamed Ali Jendoubi (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 GREMAQ - Groupe de recherche en économie mathématique et quantitative
3 Faculté des Sciences de Bizerte [Université de Carthage]

Using small deformations of the total energy, as introduced in [28], we establish that damped second order gradient systems u ′′ (t) + γu ′ (t) + ∇G(u(t)) = 0, may be viewed as quasi-gradient systems. In order to study the asymptotic behavior of these systems, we prove that any (non trivial) desingularizing function appearing in KL inequality satisfies ϕ(s) O(√ s)) whenever the original function is definable and C 2 . Variants to this result are given. These facts are used in turn to prove that a desingularizing function of the potential G also desingularizes the total energy and its deformed versions. Our approach brings forward several results interesting for their own sake: we provide an asymptotic alternative for quasi-gradient systems, either a trajectory converges, or its norm tends to infinity. The convergence rates are also analyzed by an original method based on a one dimensional worst-case gradient system. We conclude by establishing the convergence of damped second order systems in various cases including the definable case. The real-analytic case is recovered and some results concerning convex functions are also derived.

Mots clés: dissipative dynamical systems; gradient systems; inertial systems; Kurdyka-{\L}ojasiewicz inequality; global convergenc

Domaines: Equations aux dérivées partielles [math.AP]

Pascal Bégout, Jérôme Bolte, Mohamed Ali Jendoubi. On damped second-order gradient systems. 2014. ⟨hal-01088713v1⟩