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In this paper we deal with a general second order continuous dynamical systemassociated to a convex minimization problem with a Fr\`echet differentiableobjective function. We show that inertial algorithms, such as Nesterov'salgorithm, can be obtained via the natural explicit discretization from ourdynamical system. Our dynamical system can be viewed as a perturbed version ofthe heavy ball method with vanishing damping, however the perturbation is madein the argument of the gradient of the objective function. This perturbationseems to have a smoothing effect for the energy error and eliminates theoscillations obtained for this error in the case of the heavy ball method withvanishing damping, as some numerical experiments show. We prove that the valueof the objective function in a generated trajectory converges in order O(1/t^2)to the global minimum of the objective function. Moreover, we obtain that atrajectory generated by the dynamical system converges to a minimum point ofthe objective function.

An extension of the second order dynamical system that models Nesterov's convex gradient method