Optimal control of parabolic variational inequalities

Optimal control of parabolic variational inequalities is studied in the case where the spatial domain is not necessarily bounded. First, strong and weak solutions concepts for the variational inequality are proposed and existence results are obtained by a monotone and a finite difference technique. An optimal control problem with the control appearing in the coefficient of the leading term is investigated and a first order optimality system in a Lagrangian framework is derived.