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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: A bisingular problem of optimal boundary control for solutions of an elliptic equation in a bounded domain with a smooth boundary is considered. The coefficient of the Laplacian is assumed to be small, and integral constraints are imposed on the control. A complete asymptotic expansion in powers of the small parameter is obtained for the solution of the problem. [ABSTRACT FROM AUTHOR]

The concordance method of asymptotic expansions applied for constructing uniform asymptotic expansions of singularly-perturbed partial differential equations and systems is presented. [ABSTRACT FROM AUTHOR]