# Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators

## TL;DR

This paper investigates the convergence of discretized solutions to evolution inclusions with time-dependent maximal monotone operators, establishing conditions for well-posedness and solution uniqueness using variational analysis.

## Contribution

It introduces a novel convergence analysis for discretized solutions of time-dependent differential inclusions with maximal monotone operators, extending existing theory.

## Key findings

- Discretized solutions converge to the unique continuous solution.
- Provided conditions for well-posedness of related nonsmooth differential equations.
- Applied passivity and linear matrix inequalities to establish solution properties.

## Abstract

This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in the modeling of certain 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. Under certain mild assumptions on the regularity with respect to the time argument, we construct a sequence of functions parameterized by the sampling time that corresponds to the discretization of the continuous-time system. 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).