Asymptotic expansion for the solution of an optimal boundary control problem in a doubly connected domain with different control intensity on boundary segments

An optimal boundary control problem for solutions of an elliptic equation in a bounded domain with a smooth boundary is considered. The coefficient multiplying the Laplacian is assumed to be small, and integral constraints are imposed on the control. Its own intensity of control is specified on each of the boundary components. A complete asymptotic expansion in powers of the small parameter is obtained for the solution of the problem.