s underactuated)
    if we impose a torque-limit constraint, $|u|\le u_{max}$.  Looking at the
    phase portraits again, you can now visualize the control problem.  Via
    feedback, you are allowed to change the direction of the vector field at
    each point, but only by a fixed amount.  Clearly, if the maximum torque is
    small (smaller than $mgl$), then there are some states which cannot be
    driven directly to the goal, but must pump up energy to reach the goal.
    Furthermore, if the torque-limi