I have the following dynamic programming principle-type problem.
Suppose that we are given a sequence $\beta_1,\dots,\beta_n\in (0,\infty)$, some target $y\in (0,\infty)$ with $y>\sum_{t=1}^N \beta_t$, an $r\geq 1$, 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 $\alpha_1,\dots,\alpha_n\in [0,\infty)$ satisfying the recursive bound: $$ \begin{aligned} y_T &\leq y\\ y_t &:= t^{-r} (\alpha_t + \sum_{s=\max\{t-T-1,1\}}^{t-1} \beta_s + My_s\\ y_0 & :=0 \end{aligned} $$ I want to maximize: $$ \max_{(\alpha_1,\dots,\alpha_n)\in A}\,\sum_{n=1}^N \alpha_n \qquad \boldsymbol{(1)}, $$ for some $M>0$.
- Is there a known closed-form for the maximizer of $(1)$ in A?
- If this type of problem is studied in the literature, what are some good references?
