Can Linear Supply-Demand Equilibria Be Understood as a Feedback-Control Process? I'm not an economist, but rather an applied mathematician working in control theory. I've recently been watching Berkeley's intro to econ course in a personal project to better understand economics and finance. The first few lectures are on linear supply-demand curves and how they create price equilibria. 

To wit, suppose $\sigma(p)$ and $\delta(p)$ and the quantity supplied and demanded at price $p$ resp. Then the assumption of linearity implies

$$ \sigma(p) = S_\sigma p+\sigma_0,\ \ \delta(p) = -S_\delta p+\delta_0$$

(this is my notation). The numbers $S_{\sigma,\delta}$, $\delta_0,\sigma_0$ are presumed positive and correspond to the slope (what I believe is called "elasticity" in econ jargon) and the intercepts, which I've interpreted as the amount supplied and demanded at $p = 0$ (e.g. how much people are willing to supply or would demand if the good were free). Clearly we assume $p \geq 0$.

The equilibrium is met when supply and demand are equal--$\delta = \sigma$--and give the equilibrium price
$$
p_e = \frac{\delta_0-\sigma_0}{S_\delta+S_\sigma}.
$$   

This much is trivial to a real economist. 

Now to the interesting part:

Through the lecturer explaining how an out of equilibrium price eventually comes to equilibirium, I recognized the basic idea of feedback at work: the change in price is forced by the signed difference between supply and demand. I propose this is modeled by the equation

$$
\dot{p} = k(\delta-\sigma),
$$

or exactly a feedback control law ($P$-controller) with feedback gain $k$. Furthermore, since we assume the supply and demand curves are linear,   the entire equation is now a linear differential equation

$$
\dot{p} = k(\delta_0-\sigma_0)-k(S_\delta+S_\sigma)p.
$$

The equilibrium occurs when $\dot{p} = 0$ and is identical to that obtained using the graphical method. 

But wait--there's more! Since we now have a differential equation, we can get a time-domain solution:
$$
p(t) = e^{-t/\tau}p_i+p_e(1-e^{-t/\tau})
$$
Here $p_i$ is the initial (out-of-equilibrium) price, $\tau = \frac{1}{k(S_\delta+S_\sigma)}$, and $p_e$ is the equilibrium price. 

This implies that not only can we predict what equilibrium prices will be,  but we can also, subject to understanding $k$, predict the amount of time it will take to settle into the equilibrium. In fact, the form of this equation means that equilibrium is asmyptotic ($p_e = p(t= \infty)$) and percentages--63%, 86%, 95%, etc. for $t/\tau = 1,2,3,$ etc.--of equilibrium will be obtained. $t/\tau = 6$ is a good approximation of actual equilibrium (99.8%), so we gain the ability to state how long it will take for a price initially at $p_i$ to settle to $p_e$ (call this $t_e$, the "equilibirum time")

$$
t_e \approx \frac{6}{k(S_\delta+S_\sigma)}.
$$

Thoughts on $k$

So what is $k$ really? The short answer is I don't know. Mathematically it is a feedback gain, but that's not really so helpful. 

It has the units of price/(quantity$\cdot$time), so I have pulled it into two factors $k = f_t/S_X$. I suppose $f_t$ to be the frequency at which transactions are made of the good. It makes sense that a less often traded good will reach equilibrium more slowly than a more often traded good. $S_X$ has the units of $S_{\delta,\sigma}$ but I have no good interpretation.  I've considered that we might have $S_X \approx S_\delta + S_\sigma$ to produce $\tau \approx 1/f_t$ but haven't got much farther than this.

The Bottom Line

I think I've found an interesting interpretation of supply-demand dynamics in terms of control theoretic principles, namely proportional feedback. Unfortunately I am so unknowledgeable in econ that I'm unable to know whether or not this is a fruitful line of inquiry which I should pursue, or whether I have just reconstructed a longstanding theory. 

Can anyone help provide some context?
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.