# Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions

## TL;DR

This paper introduces a new class of state-dependent maximal monotone differential inclusions, establishes solution existence and uniqueness, and characterizes nonsmooth Lyapunov pairs, with applications to sweeping processes and Lur'e systems.

## Contribution

It is the first to define state-dependent maximal monotone differential inclusions and analyze their solutions and stability properties.

## Key findings

- Existence and uniqueness of solutions established.
- Characterization of nonsmooth Lyapunov pairs provided.
- Applicable to state-dependent sweeping processes and Lur'e systems.

## Abstract

In this paper, we introduce for the first time a class of state-dependent maximal monotone differential inclusions. Then the existence and uniqueness of solutions are obtained by using an implicit discretization scheme and a kind of hypo-monotonicity assumption respectively. In addition, a characterization for nonsmooth Lyapunov pairs associated with such systems is provided. Our result can be applied to study state-dependent sweeping processes and Lur'e dynamical systems. It is new even the involved maximal monotone operators depend only on the time.