Can Linear Supply-Demand Equilibria Be Understood as a Feedback-Control Process?
I think there might be an issue with this as currently constructed: 

$\delta(p)= \infty$, $\forall P \in \mathbb{R_+}$ and $P_e = \infty =\dot{p},$ $\forall P \in \mathbb{R_+}$ This is because $\delta_0 = \infty $. Further, $\sigma(p) = S_\sigma p+\sigma_0= S_\sigma p$ since $ \sigma_0 = 0$. 

In words - there should be infinite demand for any normal good whenever P is 0 and there should be zero supply of any good whenever P is 0. 

That said, your intuition here is solid but I do think you're working on reinventing the tatonement wheel. Tatonement, in a nutshell, adjusts prices up when there is excess demand and prices down when there is excess supply. 

If you are able to do this after watching a few lectures online, then I say keep going. You could make meaningful contributions once you have a better understanding of the field. 

edit: If you'd like to message me privately, I would be more than happy to pass along my graduate school notes/materials to help guide your studies. At minimum, I recommend you followup your current study with some more advanced micro theory courses.