# Strong stationarity for the control of viscous history-dependent evolutionary VIs arising in applications

## TL;DR

This paper develops a framework for analyzing viscous history-dependent evolutionary variational inequalities (EVIs) in control problems, establishing strong stationarity conditions despite non-smoothness.

## Contribution

It extends previous work by formulating viscous history-dependent EVIs as non-smooth ODEs in Hilbert space and investigates their directional differentiability.

## Key findings

- Formulation of viscous history-dependent EVIs as non-smooth ODEs in Hilbert space
- Establishment of strong stationary conditions for viscous damage models
- Analysis of Hadamard directional differentiability of the solution map

## Abstract

This paper addresses optimal control problems governed by history-dependent EVIs with viscosity. One of the prominent properties of the state system is its non-smooth nature, so that the application of standard adjoint calculus is excluded. We extend the results from [7] by showing that history-dependent EVIs with viscosity can be formulated as non-smooth ODEs in Hilbert space in a general setting. The Hadamard directional differentiability of the solution map is investigated. Based on previous results, this allows us to establish strong stationary conditions for two different viscous damage models with fatigue.