Strong Stationarity for the Control of Viscous History-Dependent Evolutionary Vis Arising in Applications

Abstract

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