This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in optimization algorithms and the modeling of physical systems. The differential inclusion is described by a time-dependent set-valued mapping having the property that, for a given time instant, the set-valued mapping describes a maximal monotone operator. By successive application of a proximal operator, we construct a sequence of functions parameterized by the sampling time that corresponds to the discretization of the continuous-time system. Under certain mild assumptions on the regularity with respect to the time argument, and using appropriate tools from functional and variational analysis, this sequence is then shown to converge to the unique solution of the original differential inclusion. The result is applied to develop conditions for well-posedness of differential equations interconnected with nonsmooth time-dependent complementarity relations, using passivity of underlying dynamics (equivalently expressed in terms of linear matrix inequalities).

本文研究了随时间变化的微分包含的解，这些包含在优化算法和物理系统建模中的效用。微分包含由时间相关的集值映射描述，该映射具有以下特性：对于给定的时刻，集值映射描述最大单调算子。通过连续应用近端算子，我们构造了一系列由采样时间参数化的函数，该采样时间对应于连续时间系统的离散化。在关于时间参数的规律性的某些温和假设下，并使用来自函数和变分分析的适当工具，然后显示该序列收敛到原始微分包含的唯一解。