Asymptotics of a solution to an optimal boundary control problem with performance index defined on the boundary

In this paper, we study the optimal control problem of the value of the solution to an elliptic equation in a bounded domain with a smooth boundary by means of a flow through the domain boundary. We consider the operator of the equation, which is the sum of the Laplace operator with a small coefficient and a zero-order operator. The control is constrained by an integral relation. As a performance index, we employ the sum of the squared norm of the deviation of a state from a prescribed state on the domain boundary and the squared norm of the control. We obtain a complete asymptotic expansion of the solution to the problem in powers of the small parameter.