Many examples of control systems are found in modern-day industrial applications, ranging from the Segway to cruise control in an automobile. The self-stabilizing inverted pendulum serves as an archetypal problem by which the robustness of such control systems, which use feedback to control themselves, is measured. In this project, I analyze several simple control systems involving the inverted pendulum, and attempt to find their stability solution. I have also built a computer simulation in Mathematica, a computational software package, which demonstrates fundamental concepts in control theory, as well as the difficulty of manual control and the usefulness of feedback in self-stabilizing systems. This simulation provides a useful learning tool to users who otherwise have little experience with the theory of control. Finally, I investigate theoretical problems related to the so-called radius of stability, which is a number that tells us how far we may deviate from equilibrium before the control system fails. I hypothesize that as further variables are introduced into a control model, the radius of stability will decrease accordingly. Since control systems are widely used in modern day society, it is important that we understand the stability of such systems, thereby maximizing the safety and effectiveness of their applications.