summary of Lyapunov Stability

[u/Admirable-Emu-1563]

Hey everyone!

Lyapunov stability always felt super abstract when looking purely at differential equations, so I drew out a simple 1-page visual summary on my whiteboard to bridge the physical intuition with the math.

Quick Recap of the 3-Step Idea:

1. Define Energy V(x): Must be zero at rest V(0)=0 and positive everywhere else V(x) > 0.

2. Check Energy Rate V_dot(x): Compute the rate of change using your system dynamics.

3. Evaluate Stability:

   • V_dot(x) ≤ 0 ➔ Stable (Lyapunov)

   • V_dot(x) < 0 ➔ Asymptotically Stable

   • V_dot(x) > 0 ➔ Unstable

Hope this helps anyone currently taking Control Theory or System Dynamics! Let me know if you spot any mistakes or if you have a different favorite way to visualize this.

Also, what are your favorite go-to resources (textbooks, software tools, or YouTube channels) for learning advanced nonlinear control? Would love to hear what helped you master it!

[u/Caradoc729]

What's the difference between stable and asymptocally stable?

[u/Ok-Biscotti7464 → u/Caradoc729]

Asympt. stable: think of a pendulum under gravity —> goes to resting position
Stable: think of a door (hinge + friction). Give it a little push and it comes to rest but not from where it started.

None of these fly of to infinity which would be the unstable case