   how that energy changes with time: \begin{align*} \dot{E} =&
      ml^2\dot\theta \ddot\theta + \dot\theta mgl\sin\theta \\ =& \dot\theta
      \left[ u - mgl\sin\theta \right] + \dot\theta mgl\sin\theta \\ =&
      u\dot\theta. \end{align*} In words, adding energy to the system is simple
      - apply torque in the same direction as $\dot\theta$.  To remove
      energy, apply torque in the opposite direction (e.g., damping).</p>

      <p>To swing up the pendulum, even with torque limits, let us use th