In a Hilbert framework ℌ, we study the convergence properties of a Newton-like inertial dynamical system governed by a general maximally monotone operator A ℌ: → 2^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 Attouch-Peypouquet (Math. Prog. 2019), 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.

在希尔伯特框架ℌ，我们研究了由一般最大单调算子支配的牛顿式惯性动力系统的收敛性一ℌ：→2 ^ℌ。当A等于凸下半连续固有函数的次微分，动态对应于Nesterov加速梯度方法的连续版本中引入的Hessian驱动阻尼。结果，振荡被显着衰减。根据Attouch-Peypouquet（Math.Prog.2019）引入的技术，最大单调算子由其Yosida近似替换，并适当调整了正则化参数。牛顿式校正项（对应于凸极小化情况下的Hessian驱动项）的动力学介绍提供了一个良好的演化系统，对于该系统，我们将获得所生成轨迹朝向A的零点的弱收敛性。。我们还获得了速度趋于零的快速收敛。结果可以容忍错误，扰动的存在。然后，我们将结果专门化为运算符A是凸下半连续函数的次微分的情况，并获得快速优化结果。