On damped second-order gradient systems

Abstract

Using small deformations of the total energy, as introduced in \cite{MR1616968}, we establish that damped second order gradient systems \begin{gather*} u^\pp(t)+\gamma u^\p(t)+\nabla G(u(t))=0, \end{gather*} 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 \vphi(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