Approximation and regularization of optimal control problems for a non-selfadjoint elliptic equation with variable coefficients

The problem of minimizing a quadratic functional, defined on solutions of the Dirichlet problem for a non-selfadjoint equation of elliptic type with variable coefficients in an arbitrary convex domain, is considered. The controls are the coefficients of the equation and its free term. The question of whether such problems are well-posed in the weak topology is investigated. Error bounds are derived for difference approximations with respect to the state variables, as well as bounds on the rate of convergence of the approximations with respect to the functions. Weak convergence with respect to the control is proved.