In this study, we introduce numerical methods for minimum energy among three dynamic systems governed by a class of integro-differential equation with Abel type weakly singular kernels. These types of equations are developed from a class of integro-differential equations originated from an aeroelasticity problem. By weighting energy criteria for three systems, we intend to find the most stability status among systems for various initial conditions and tracking targets. By interchanging the differentiation and integration of the integro-differential equation, part of the numerical scheme is constructed. Promising numerical results are provided.