Lyapunov functions

A Lyapunov function is a mathematical tool used to determine if a system will stay stable over time. Think of it as an energy measure associated with the system state: if this "energy" decreases or remains constant, the system tends to settle into a stable point, like a ball resting at the bottom of a bowl. Conversely, if the "energy" increases, the system may become unstable or diverge. Lyapunov functions help engineers and scientists analyze complex systems—like robotic arms, ecosystems, or economic models—to ensure they behave predictably and safely.