  In this article a family of second order ODEs associated to inertial gradient
descend is studied. These ODEs are widely used to build trajectories converging
to a minimizer $x^*$ of a function $F$, possibly convex. This family includes
the continuous version of the Nesterov inertial scheme and the continuous heavy
ball method. Several damping parameters, not necessarily vanishing, and a
perturbation term $g$ are thus considered. The damping parameter is linked to
the inertia of the associated inertial scheme and the perturbation term $g$ is
linked to the error that can be done on the gradient of the function $F$. This
article presents new asymptotic bounds on $F(x(t))-F(x^*)$ where $x$ is a
solution of the ODE, when $F$ is convex and satisfies local geometrical
properties such as {\L}ojasiewicz properties and under integrability conditions
on $g$. Even if geometrical properties and perturbations were already studied
for most ODEs of these families, it is the first time they are jointly studied.
All these results give an insight on the behavior of these inertial and
perturbed algorithms if $F$ satisfies some {\L}ojasiewicz properties especially
in the setting of stochastic algorithms.
