I have the following dynamic programming principle-type problem.

Suppose that we are given a sequence $beta_1,dots,beta_nin (0,infty)$, some target $yin (0,infty)$ with $y>sum_{t=1}^N beta_t$, an $rgeq 1$, and a usc convex function $f:(0,infty)rightarrow (0,infty)$ which is continuous at $0$.

Define the set of controls $A$ to consist of all $n$-tuples $alpha_1,dots,alpha_nin (0,infty)$ satisfying the recursive bound:

$$

begin{aligned}

y_T &leq y\

y_t &:= t^{-r} (alpha_t + sum_{s=max{t-T-1,1}}^{t-1} beta_s + f(y_s)\

y_0 & :=0

end{aligned}

$$

I want to maximize:

$$

max_{(alpha_1,dots,alpha_n)in A},sum_{n=1}^N alpha_n qquad boldsymbol{(1)}.

$$

*(If it helps, am most interested in $f(y)=Ly^{s}$ for some $sin (0,1)$*)

- Does there exist a maximizer of $(1)$ in A? Or an $epsilon$-maximizer?
- If this type of problem is studied in the literature, what are some good references?