  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.

Analysis of PDEs