We consider the optimal boundary control of aeroacoustic noise, governed by the two-dimensional unsteady compressible Euler equations. The control is the time- and space-varying wall normal velocity on a subset of an otherwise solid wall. The objective functional to be minimized is a measure of acoustic amplitude. Optimal transpiration boundary control of aeroacoustic noise introduces challenges beyond those encountered in direct aeroacoustic simulations or in many other optimization problems governed by compressible Euler equations. One nontrivial issue that arises in the optimal control problem is the formulation and implementation of transpiration boundary conditions. Because suction and blowing on the boundary are allowed, portions of the boundary may change from inflow to outflow, or vice versa, and different numbers of boundary conditions must be imposed at inflow vs outflow boundaries. Another important issue is the derivation of adjoint equations, which are required to compute the gradient of the objective function with respect to the control. Among other things, this is influenced by the choice of boundary conditions for the compressible Euler equations. The approaches to meet these challenges are described and results presented for three model problems. These problems are designed to validate the transpiration boundary conditions and their implementation, study the accuracy of gradient computations, and assess the performance of the computed controls.