  This paper analyzes two eXtended finite element methods (XFEMs) for linear
quadratic optimal control problems governed by Poisson equation in non-convex
domains. We follow the variational discretization concept to discretize the
continuous problems, and apply an XFEM with a cut-off function and a classic
XFEM with a fixed enrichment area to discretize the state and co-state
equations. Optimal error estimates are derived for the state, co-state and
control. Numerical results confirm our theoretical results.
