Digital regulator for DCM Boost - How does it relate to duty cycle?
And so my question is, how does my controller, GC, relate to duty
  cycle?


You need to implement a PWM block into your system as shown below: -



I've shown it in red after your gain stage but I'm not really sure what that gain stage is modeling and, depending on what that gain stage is in terms of modeled physical components, the PWM boost stage may need to precede it.

So, it comes down to these parameters: -


Power uplift needed to raise the load to a voltage higher than \$V_{IN}\$
This is the \$I_{OUT}\cdot (V_{OUT} - V_{IN})\$ terms in your D equation that I corrected for you in comments a few hours ago
Your output inductance
Your output load resistance = \$\dfrac{V_{OUT}}{I_{OUT}}\$
The inverse of your PWM switching frequency, \$T_S\$