Strong Stationarity for Optimal Control of a Nonsmooth Coupled System: Application to a Viscous Evolutionary Variational Inequality Coupled with an Elliptic PDE
Livia M. Betz · SIAM Journal on Optimization · 2019
This paper is mainly concerned with an optimal control problem governed by a nonsmooth coupled system of equations. The nonsmooth nonlinearity is Lipschitz continuous and directionally differentiable, but not Gâteaux differentiable. We derive a strong stationary optimality system, i.e., an optimality system which is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. The abstract result is then applied to prove strong stationarity for optimal control of a coupled system consisting of a viscous evolutionary variational inequality (EVI) and an elliptic PDE. To this end, we show that EVIs with viscosity can be formulated as nonsmooth ODEs in Hilbert space in a general setting. The nonsmooth nonlinearity appearing in the ODE turns out to be the solution operator of an elliptic variational inequality, for which we can give an explicit formula.