Let $c,\,\mathcal{P}_0,\,\mathcal{P}_1,\,\mathcal{P}_2,\ldots$ be a sequence of positive real numbers.

Let $N\in\{1,\,2,\,3,\ldots\}$ and let $t\in\{0,\,1,\,2,\ldots\}$, with $N$ and $t$ fixed.

For all $j\in\{1,\ldots,\,N\}$ and $t\in\{0,\,1,\,2,\ldots,\}$, let $q_j^{(t)}=p_j^{(t)}\cdot(1-p_j^{(t-1)})\cdot(1-p_j^{(t-2)})\cdot\ldots\cdot(1-p_j^{(0)})$, with $p_j^{(k)}\in(0,\,1)$ and $p_j^{(k)}$ is a function of $\mathcal{P}_k$, for all $k\in\{0,\,1,\,2,\ldots,\,t\}$.

For all $t\in\{0,\,1,\,2,\ldots\}$ and with $\delta\in(0,\,1]$ known, let $$f(\mathcal{P}_0,\,\mathcal{P}_1,\ldots,\,\mathcal{P}_t)=\left\{\prod_{j=1}^{N}(1-q_j^{(t)})\right\}\cdot\left\{\delta^0\cdot(\mathcal{P}_0-c)\cdot\sum_{j=1}^{N}q_j^{(0)}+\cdots+\delta^t\cdot(\mathcal{P}_t-c)\cdot\sum_{j=1}^{N}q_j^{(t)}\right\}$$

**Problem:** Determine the values of $\mathcal{P}_0,\,\mathcal{P}_1,\,\mathcal{P}_2,\ldots$ that maximize
$$F(\mathcal{P}_0,\,\mathcal{P}_1,\,\mathcal{P}_2,\ldots)=\sum_{t=0}^{+\infty}f(\mathcal{P}_0,\,\mathcal{P}_1,\ldots,\,\mathcal{P}_t),$$
subject to $\mathcal{P}_t>c$, for all $t\in\{0,\,1,\,2,\ldots\}$.

To solve this optimization problem, I am trying to use dynamic programming. However, I do not know whether I can apply dynamic programming in this optimization problem. I was wondering if you could tell me whether I can use this technique and how I can do that, please. After knowing that, I can write the respective program in ${\tt R}$-software.

If you have any question, please let me know. Thank you very much for your help and suggestions.