In this research, we are pursuing the robustness of a self-excited vibrational system with negative damping. In practice, this is manifested as conductor galloping of overhead power lines, which is an oscillation of the lines caused by strong winds. Improved transmission tower designs are needed which are capable of combating excessive stresses exerted on the tower by the galloping power lines. Our model of this self-excited system shows that the oscillations can be controlled by adding a boundary velocity feedback controller at the transmission tower. Using a decomposition method, we show there is a closed form analytical solution which predicts the system can be stabilized under certain conditions. Through this research, power transmission systems that are more reliable and resistant to galloping can be engineered.