We have a continuous variable $s$ which changes over discrete time $t$ according to the difference equation:
$$ s_t = s_{t-1} + \lambda - \frac{Kn_t}{N} $$
where $0 \leq s \leq 1$, $n = 1, 2, ..., N$, $t = 1, 2, ..., M$, $s_1 = 1$, and $n_1 = 0$. Here, $n_t$ represents the number of agents within a group of size $N$ which is performing a task at time $t$. $s_t$ gives the need for the task at that time, and $0$ represents a situation where the task does not need to be performed while $1$ represents a case of maximal need. $\lambda$ is a constant which represents the growth rate of the task, an $K$ is a constant which represents how much work an agent can accomplish in a single timestep. It must be the case that $K \geq \lambda$ so that the task need does not increase at a rate faster than it an be removed.
My goal is to find a function of $n_t$ which drives $s_t$ to $0$ while simultaneously minimizing $\sum_{t=1}^M n_t$. That is, I want to know the function where the task can be accomplished with the least amount of work. I believe that the limiting function of the more general function is $n_t = \frac{N \lambda}{K}$ as $t \rightarrow M$, but I'm not sure how to derive the more general formula. Is this possible, and if so, how do we derive it?