On the Euler equation appearing in the synthesis of radiating systems

A problem appearing when the Euler equation is used in the theory of antenna synthesis is considered. Behavior of a solution to this equation at the ends of the synthesis interval contradicts the Meixner conditions, according to which this solution must tend to zero as a square root of the distance to the ends. It is shown that this contradiction arises if we seek solutions in the L ₂ space and disappears if an L ₂ subspace with limited energy norm is selected as the space of solutions. In this case, the Euler equation arises from a variational minimization problem for a functional involving the norm of current determined in the energy space.