Consider the time evolution of a probability distribution $P(\alpha_{t})$ as follows.
\begin{equation} P(\alpha_{t}) = F[P(\alpha_{t-1}), U_t(y_{t-1})], \end{equation} where $y_{t-1} = \mathcal{P}\alpha_{t-1}$, with $\mathcal{P}$ being some projection matrix, and $U$ being a control variable.
I wish to find a set of functions $\{U_t(y_{t-1})\}_{t=1}^{t=T}$, such that $\langle f(\alpha_T)\rangle_{P(\alpha_T)} \to \max$, for some known function $f$.
My questions:
- Is such a scenario already explored? If so, can you share with me the literature?
- If it is trivial, can you share with me the Bellman equation that needs to be solved?
Additional info.:
- The initial conditions of $\alpha$ (and hence $y$) are known.
- $\alpha$'s belong to a countable and bounded space.