Discrete regularization of optimal control problems on ill-posed monotone variational inequalities

The paper considers the problem of minimizing an abstract functional which depends on the control and on the solution of an ill-posed variational inequality (v.i.) with a bounded monotone exact operator with the μ-property (these properties are not needed for approximate operators, nor is the μ-property in the pseudomonotone case). Solvability of the problem is shown. For its regularization the original v.i. is regularized by means of v.i. with a small step. A number of generalizations are indicated.