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I have a problem that I am trying to model, but I am unable to capture one particularity of the system that seems to be crucial. I will try to give an abstract description first and then the toy model hoping that someone might have an idea or a reference suggestion.

Description: There is a system and I can observe one of its parameters $u$ - $u$ is nonnegative and bounded by some system constraints. There is an external parameter $v$ that I can control and I want it to be as close as possible in value to the observed parameter $u$. Unfortunately, there is a delay in observing the value of $u$ and updating the value of $v$: it is a time constant $D>0$.

Problem: I have historical data on observables $u$. Let us say that I have a vector of $u$ data points in some interval $[0, T]$. I would like to analyse this data and find the optimal vector of $v$-values that would minimize the L1 distance between $u$ and $v$ points. Unfortunately, I do not know how to write the condition that there is a delay $D$ involved: if $u_0$ and $v_0$ both start at 10, and $u_1$ becomes 50, then $v_1$ and up to $v_{D}$ are still 10 and only $v_{D+1}$ is set to 50.

A toy example: imagine you have a very expensive and limited memory system which is in every time instance given a new data to hold. The amount of data to hold is variable and the old data is simply overwritten and not needed anymore.

If in the next time instance you need to write more data than you have currently allocated, you write in what you can and put the remainder on the queue (you are penalised for using the queue) and issue an allocation request (and you get the requested memory after $D$ time instances have passed - for simplicity, the let the delay be the same in allocation and deallocation cases).

If in the next time instance you need to write less than you currently have, then you write everything in (you get penalised for having unused memory) and issue a deallocation request on the difference (and the memory is freed after $D$ time instances have passed).

I am trying to model this problem in order to try to find a $v$-vector (for a given $u$-vector) that minimizes the penalties.

Does anyone have any suggestion on references where I could look up these things further? Or and idea how to approach it? Thank you!

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