Asymptotics of the solution of a bisingular optimal distributed control problem in a convex domain with a small parameter multiplying a highest derivative

We consider an optimal distributed control problem in a strictly convex planar domain with a smooth boundary and a small parameter multiplying a highest derivative of an elliptic operator. A zero Dirichlet condition is set on the boundary of the domain, and control is additively involved in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. The solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with some coefficient. Due to this structure of the optimality criterion, the role of the first or second term of the criterion can be strengthen, if necessary. It is more important to achieve a given state in the first case and to minimize the resource cost in the second case. The asymptotics of the problem generated by the sum of a second-order differential operator with a small coefficient at a highest derivative and a zero-order differential operator is studied in detail.