Difference approximations of optimization problems for semilinear elliptic equations in a convex domain with controls in the coefficients multiplying the highest derivatives

Finite difference approximations are proposed for nonlinear optimal control problems for a non-self-adjoint elliptic equation with Dirichlet boundary conditions in a convex domain $\Omega\subset\mathbb{R}^2$ with controls involved in the leading coefficients. The convergence of the approximations with respect to the state, functional, and control is analyzed, and a regularization of the approximations is proposed.