Asymptotics for solutions of problem on optimally distributed control in convex domain with small parameter at one of higher derivatives

Abstract

We consider a problem on optimally distributed control in a planar strictly convex domain with a smooth boundary and a small parameter at one of the higher derivatives in the elliptic operator. On the boundary of the domain the homogeneous Dirichlet condition is imposed, while the control is additively involved in an inhomogeneity. As a set of admissible controls we use a unit ball in the corresponding space of square integrable functions. The solutions of the studied boundary value problem are treated in the generalized sense as elements of some Hilbert space. As the optimality criterion, we employ the sum of squared norm of the deviation of a state from a prescribed one and the squared norm of the control with some coefficient. Such structure of the optimality criterion allows, if this is needed, to strengthen the role of the first or the second term in this criterion. In the first case it is more important to achieve a prescribed state, while in this second case it is more important to minimize the resource expenses. We study in details the asymptotics of the problem generated by the differential operator with a small coefficient at one of the higher derivatives, to which a zero order differential operator is added.