Approximations to Problems of Optimal Control of Leading Coefficients of Elliptic Equations in Nondivergence Form with an Unbounded Nonlinearity in the Coefficients

We consider problems of optimal control of processes described by the Dirichlet boundary value problem for elliptic equations with mixed derivatives and unbounded nonlinearity. The controls are contained in the coefficients multiplying the highest derivatives of the state equation. Finite-difference approximations to nonlinear optimization models are constructed and investigated, and to find an approximate solution of a nonlinear boundary value problem for the state, an iteration process implementing the problem is constructed. The convergence of the iteration process used to prove the existence and uniqueness of the solution of a nonlinear difference scheme approximating the original boundary value problem for the state is studied rigorously. We establish the mesh $$W_{2,0}^2(\omega )$$ -norm estimates consistent with the smoothness of the desired solution for the rate of convergence of difference schemes that approximate the nonlinear state equation. The convergence of the approximations to the optimal control problems with respect to state, functional, and control is investigated; the regularization of the approximations is carried out.