  We study an optimal control problem governed by elliptic PDEs with interface,
which the control acts on the interface. Due to the jump of the coefficient
across the interface and the control acting on the interface, the regularity of
solution of the control problem is limited on the whole domain, but smoother on
subdomains. The control function with pointwise inequality constraints is
served as the flux jump condition which we called Neumann interface control. We
use a simple uniform mesh that is independent of the interface. The standard
linear finite element method can not achieve optimal convergence when the
uniform mesh is used. Therefore the state and adjoint state equations are
discretized by piecewise linear immersed finite element method (IFEM). While
the accuracy of the piecewise constant approximation of the optimal control on
the interface is improved by a postprocessing step which possesses
superconvergence properties; as well as the variational discretization concept
for the optimal control is used to improve the error estimates. Optimal error
estimates for the control, suboptimal error estimates for state and adjoint
state are derived. Numerical examples with and without constraints are provided
to illustrate the effectiveness of the proposed scheme and correctness of the
theoretical analysis.
