Lyapunov theory

Lyapunov theory provides a method to analyze whether a system will stay stable over time. Imagine a ball in a bowl: if you gently push it, it rolls back to the bottom, indicating stability. Lyapunov’s approach involves creating a special mathematical function, like a "energy" measure, that decreases as the system evolves. If this function always drops or stays constant, the system tends to settle into a steady state and remains stable. This method helps engineers determine if systems—like robots, aircraft, or economic models—will behave predictably and safely under small disturbances.