The aim of this survey is to present the main important techniques and tools from variational analysis used for first and second order dynamical systems of implicit type for solving monotone inclusions and non-smooth optimization problems. The differential equations are expressed by means of the resolvent (in case of a maximally monotone set valued operator) or the proximal operator for non-smooth functions. The asymptotic analysis of the trajectories generated relies on Lyapunov theory, where the appropriate energy functional plays a decisive role. While the most part of the paper is related to monotone inclusions and convex optimization problems in the variational case, we present also results for dynamical systems for solving non-convex optimization problems, where the Kurdyka-Łojasiewicz property is used.

本次调查的目的是介绍用于隐式单阶和二阶动力学系统的变分分析中用于解决单调包含和非光滑优化问题的主要重要技术和工具。对于非光滑函数，微分方程通过分解器（在最大单调集值运算符的情况下）或近端运算符表示。生成的轨迹的渐近分析依赖于Lyapunov理论，其中适当的能量功能起着决定性的作用。虽然本文的大部分内容与变分情况下的单调包含和凸优化问题有关，但我们还给出了使用Kurdyka-Łojasiewicz属性解决非凸优化问题的动力学系统的结果。