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

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.