Lyapunov's theorem

Lyapunov's theorem provides a method to determine if a system will stabilize over time without solving complex equations. It involves identifying a special function, called a Lyapunov function, that acts like an energy measure. If this function consistently decreases along the system’s paths, it shows the system will settle into a stable state, or equilibrium. Essentially, it’s a way to confirm stability by checking a single “energy-like” quantity rather than analyzing every possible behavior, making stability analysis more practical and manageable.