ASYMPTOTICS OF THE SOLUTION TO A BISINGULAR PROBLEM OF OPTIMAL DISTRIBUTED CONTROL IN A CONVEX DOMAIN WITH A SMALL PARAMETER IN ONE OF THE HIGHER DERIVATIVES

Abstract

The paper considers the problem of optimal distributed control in a strictly convex planar domain with a smooth boundary and a small parameter in one of the higher derivatives of the elliptic operator. In this problem, a zero Dirichlet boundary condition is imposed, and the control enters additively into the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square-integrable functions. The solutions to the resulting boundary value problems are treated in the generalized sense as elements of a certain Hilbert space. The optimality criterion is the sum of the square of the norm of the state deviation from a given state and the square of the norm of the control, with a weighting coefficient. This structure of the optimality criterion allows either the first or the second term to be emphasized, depending on the need. In the first case, achieving the desired state is prioritized, while in the second case, minimizing resource costs becomes more important. The asymptotics of the problem are studied in detail, arising from a second-order differential operator with a small coefficient in one of the higher derivatives, to which a zero-order differential operator is added.