# reference request – Solution to dynamic program-type recursion

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?