PYTHON · NUMPY

The same energy is the proof

See that the energy you descended is, unchanged, a Lyapunov certificate: it provably decreases along the closed loop.

01Challenge

Show that V̇ ≤ 0 everywhere — the controller's own energy certifies that the pole cannot diverge.

02Model

The field commits an action, then (maybe) scores it. Energy First needs no second object: the very energy you descended is the certificate. If the controller works, V must fall along the loop — and here it falls by construction.

03Guided practice

V̇ = dV/dt along the closed loop = ω·ω̇ + Kp·θ·θ̇, with θ̇=ω and ω̇ = −Kp·θ − Kd·ω. The cross terms cancel and V̇ = −Kd·ω² ≤ 0 — a guarantee, not a sample.

04Feedback

PASS WHEN PASS when max V̇ ≤ 0 over the sampled states — the energy is a valid Lyapunov certificate.

If your run fails, check:

V̇ works out to −Kd·ω² — always ≤ 0. The same energy is both the objective and the proof.
There is no second network: the certificate is the object you already descended.

05Retrieve & space

Why is a score over a window NOT a certificate?
What single term makes V̇ negative?

06Mastery & project

One energy, both roles: descending it is control, and its decrease is the stability guarantee — no bolted-on verifier.

Add a disturbance torque and find the largest one for which V̇ stays ≤ 0.

← To act is to descend an energy Structure, not proof →

PYTHON · NUMPY

The same energy is the proof

Sample thousands of states and measure V̇ under the closed loop. It PASSes when V never increases.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you