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Think of a strategy game where you can spend your energy on either (1) accumulating resources or (2) earning points. The objective is to maximize the overall points earned by time $T$ when the game terminates. Points once earned, are never spent. Energy available at any time is constant.

At any point of time, we need to decide what % of our energy should be spent on acquiring resources v/s earning points. Resource accumulation is important because (1) rate of points earned is proportional to current level of resources and (2) rate of acquiring resources is also proportional to current level of resources (eg: if you have more money, you can make better machines/factories).

How should the allocation $a(t)$ change over time so that we end up with maximum points earned by time T?

Let's call $a(t)$ = proportion of energy that is spent for acquiring resources at time $t$ ($0 <= a(t) <= 1$)

$R(t)$ = Resources accumulated by time t

$R'(t) = k*R(t)*a(t)$

$P(t)$ = Points earned by time $t$

$P'(t) = m*R(t)*(1-a(t))$

$T$ = Time of death (known in advance with full certainty)

Here, we can see that $R(t) = C*exp(k*\int_0^t a(u)*du)$

I am unable to proceed from here (how to integrate $P'(t)$?). The final goal is to find the function $a(t)$ which maximizes $P(T)$.

Intuitively I feel that you'd want to acquire resources in the early stage of the game and later, you would use those resources to earn points. So $a(t)$ should decrease over time, reaching $0$ at $t = T$. But I'm looking for a proof.