Finite difference approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions

Mathematical formulation of nonlinear optimal control problems for semilinear elliptic equations with discontinuous coefficients and discontinuous solutions are examined. Finite difference approximations of optimization problems are constructed, and the approximation error is estimated with respect to the state and the cost functional. Weak convergence of the approximations with respect to the control is proved. The approximations are regularized using Tikhonov regularization.