Control systems are mechanisms or processes designed to regulate the behavior of other systems using control loops. These systems are pivotal in various fields such as engineering, robotics, and industrial automation. In this chapter, we take a look back at the most important findings from the research on the system’s control mechanism, specifically looking at how the contraction mapping theorem is applied to control systems. Thereafter, we aim to extend the result of [1] to metric fixed point theory context. Following that, our objective is to extend the finding of [1] to metric space while keeping the \(\digamma \) -contraction mapping in mind.