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Article Dans Une Revue Journal of Differential Equations Année : 2015

On damped second-order gradient systems

Résumé

Using small deformations of the total energy, as introduced in [31], we establish that damped second order gradient systems $$ u^{\prime\prime}(t)+\gamma u^\prime(t)+\nabla G(u(t))=0, $$ may be viewed as quasi-gradient systems. In order to study the asymptotic behavior of these systems, we prove that any (nontrivial) desingularizing function appearing in KL inequality satisfies $\varphi(s)\ge c\sqrt s$ 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 solutions 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.
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Dates et versions

hal-01088713 , version 1 (28-11-2014)
hal-01088713 , version 2 (02-04-2015)
hal-01088713 , version 3 (01-07-2015)
hal-01088713 , version 4 (18-01-2017)
hal-01088713 , version 5 (21-11-2018)

Identifiants

Citer

Pascal Bégout, Jérôme Bolte, Mohamed Ali Jendoubi. On damped second-order gradient systems. Journal of Differential Equations, 2015, 259 (7 & 8), pp.3115-3143 & 4412. ⟨10.1016/j.jde.2015.04.016⟩. ⟨hal-01088713v5⟩
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