A rigorous asymptotic expansion for a control problem in a thin domain

We study the asymptotic behavior of a control problem for a quadratic cost functional and a linear elliptic equation in a thin domain of wide ε. For each n∈N, we obtain an asymptotic expansion of the control, the direct state, and the adjoint state, giving an approximation of order εⁿ⁺¹. Since this is a singular perturbation problem, one of the main difficulties is the existence of boundary layers, and then the appearance of boundary terms in the asymptotic expansions. We show how this type of problems can be solved by an iterative scheme, which reduces the question to obtain an approximation of order ε. To carry out this procedure, the problem needs to be reformulated in such a way that its structure does not change with the iterates.