Lyapunov's stability theorem is a fundamental concept in control theory and dynamical systems. It provides criteria to determine the stability of an equilibrium point in a system. Specifically, it states that if there exists a continuous function, called a Lyapunov function, that decreases over time and is positive definite, then the equilibrium point is stable. This theorem is essential for analyzing systems without solving differential equations directly. It helps engineers and scientists ensure that systems, such as robotic systems or aerospace vehicles, behave predictably and remain close to desired states despite small disturbances.