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In this article a family of second order ODEs associated to inertial gradientdescend is studied. These ODEs are widely used to build trajectories convergingto a minimizer $x^*$ of a function $F$, possibly convex. This family includesthe continuous version of the Nesterov inertial scheme and the continuous heavyball method. Several damping parameters, not necessarily vanishing, and aperturbation term $g$ are thus considered. The damping parameter is linked tothe inertia of the associated inertial scheme and the perturbation term $g$ islinked to the error that can be done on the gradient of the function $F$. Thisarticle presents new asymptotic bounds on $F(x(t))-F(x^*)$ where $x$ is asolution of the ODE, when $F$ is convex and satisfies local geometricalproperties such as {\L}ojasiewicz properties and under integrability conditionson $g$. Even if geometrical properties and perturbations were already studiedfor 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 andperturbed algorithms if $F$ satisfies some {\L}ojasiewicz properties especiallyin the setting of stochastic algorithms.

Nesterov's acceleration and Polyak's heavy ball method in continuous time: convergence rate analysis under geometric conditions and perturbations