Lyapunov's first theorem

Lyapunov's first theorem provides a way to determine whether a system will behave in a stable manner over time. It states that if there is a carefully chosen "energy-like" function (called a Lyapunov function) that always decreases as the system evolves, then the system will tend to settle into a steady state rather than diverge or behave unpredictably. This concept helps engineers and scientists verify the stability of complex systems, such as control systems or mechanical devices, by showing that they will naturally move toward a stable configuration without needing to simulate every possible scenario.