  Within this chapter, we discuss control in the coefficients of an obstacle
problem. Utilizing tools from H-convergence, we show existence of optimal
solutions. First order necessary optimality conditions are obtained after
deriving directional differentiability of the coefficient to solution mapping
for the obstacle problem. Further, considering a regularized obstacle problem
as a constraint yields a limiting optimality system after proving, strong,
convergence of the regularized control and state variables. Numerical examples
underline convergence with respect to the regularization. Finally, some
numerical experiments highlight the possible extension of the results to
coefficient control in phase-field fracture.
