A Lyapunov function is a mathematical tool used to analyze the stability of dynamical systems. It is a scalar function that helps determine whether a system will return to equilibrium after a disturbance. If the function decreases over time, it indicates that the system is stable. In control theory, Lyapunov's stability theorem provides criteria for stability using these functions. By constructing a suitable Lyapunov function, engineers can assess the behavior of systems, ensuring they remain stable under various conditions. This concept is widely applied in fields like robotics, aerospace, and economics.