Using small deformations of the total energy, as introduced in , we establish that damped second order gradient systems u′′(t)+γu′(t)+∇G(u(t))=0, Turn MathJax off 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 φ(s)⩾cs√ whenever the original function is definable and C2. 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.
Dissipative dynamical systems; Gradient systems; Inertial systems; Kurdyka–Łojasiewicz inequality; Global convergence;
Journal of Differential Equations, vol. 259, n. 7-8, 2015, pp. 3115–3143