Summary: In a Hilbert framework \(\mathcal{H}\), we study the convergence properties of a Newton-like inertial dynamical system governed by a general maximally monotone operator \(A:\mathcal{H}\rightarrow 2^{\mathcal{H}}\). When \(A\) is equal to the subdifferential of a convex lower semicontinuous proper function, the dynamic corresponds to the introduction of the Hessian-driven damping in the continuous version of the accelerated gradient method of Nesterov. As a result, the oscillations are significantly attenuated. According to the technique introduced by H. Attouch and J. Peypouquet [Math. Program. 174, No. 1–2 (B), 391–432 (2019; Zbl 1412.37083)], the maximally monotone operator is replaced by its Yosida approximation with an appropriate adjustment of the regularization parameter. The introduction into the dynamic of the Newton-like correction term (corresponding to the Hessian driven term in the case of convex minimization) provides a well-posed evolution system for which we will obtain the weak convergence of the generated trajectories towards the zeroes of \(A\). We also obtain the fast convergence of the velocities towards zero. The results tolerate the presence of errors, perturbations. Then, we specialize our results to the case where the operator \(A\) is the subdifferential of a convex lower semicontinuous function, and obtain fast optimization results.