Lyapunov stability theory

Lyapunov stability theory is a way to determine if a system’s behavior remains close to a desired state over time. Imagine balancing a pencil upright; if small nudges don't cause it to fall away significantly, it’s stable. Mathematically, Lyapunov functions are like energy measures that decrease or stay constant when the system is near equilibrium, confirming stability. If such a function exists, the system is stable because small disturbances won’t lead to big deviations, ensuring predictable and controlled behavior around the equilibrium point.