solution, for example: $$u = \pi(\theta,\dot{\theta}) = 2mgl\sin\theta.$$
      But these plots we've been making tell a different story.  How would you
      shape the natural dynamics - at each point pick a $u$ from the stack of
      phase plots - to stabilize the vertical fixed point <em>with minimal
      torque effort</em>?  This is exactly the way that I would like you to
      think about control system design.  And in the <a href="dp.html">dynamic
      programming chapter</a>, we'll start to devel