In a Hilbert framework, we introduce a new class of second-order dynamical systems that combine viscous and geometric damping but also a time rescaling process for nonsmooth convex minimization. A main feature of these systems is to produce trajectories that lie in the graph of the Fenchel subdifferential of the objective. Moreover, they do not incorporate any regularization or smoothing processes. This new class originates from some combination of a continuous Nesterov-like dynamic and the Minty representation of subdifferentials. These models are investigated through first-order reformulations that amount to dynamics involving three variables: two solution trajectories (including an auxiliary one) and another one associated with subgradients. We prove the weak convergence towards equilibria for the solution trajectories, as well as properties of fast convergence to zero for their velocities. Remarkable convergence rates (possibly of exponential-type) are also established for the function values. We additionally state notable properties of fast convergence to zero for the subgradients trajectory and for its velocity. Some numerical experiments are performed so as to illustrate the efficiency of our approach. The proposed models offer a new and well-adapted framework for discrete counterparts, especially for structured minimization problems. Inertial algorithms with a correction term are then suggested relative to this latter context.

在希尔伯特框架中，我们引入了一类新型二阶动力系统，该系统结合了粘性阻尼和几何阻尼，还结合了非光滑凸最小化的时间重新缩放过程。这些系统的一个主要特征是产生位于目标芬切尔次微分图中的轨迹。此外，它们不包含任何正则化或平滑过程。这个新类源自连续的 Nesterov 式动态和次微分的 Minty 表示的某种组合。这些模型通过一阶重构进行研究，相当于涉及三个变量的动力学：两个解轨迹（包括辅助轨迹）和另一个与次梯度相关的轨迹。我们证明了解轨迹向平衡的弱收敛性，以及速度快速收敛到零的特性。还为函数值建立了显着的收敛速度（可能是指数型的）。我们还指出了次梯度轨迹及其速度快速收敛到零的显着特性。进行了一些数值实验以说明我们方法的效率。所提出的模型为离散对应模型提供了一个新的且适应性强的框架，特别是对于结构化最小化问题。然后相对于后一种情况建议具有校正项的惯性算法。