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I'm trying to solve for the optimal policy function $h^*(x)$ over the domain of all policy functions $u_t = h(x_t)$ for the following problem. \begin{equation} \min_{h(x)} \, \mathbb{E}[\sum_{t=0}^{\infty} \, f(x_t,u_t)] \end{equation} where $f(x_t,u_t) = x_t^2u_t^2$ and subject to the following constraints \begin{equation} x_{k+1} = g(x_k,u_k,\epsilon_k) \forall k \end{equation} where $\epsilon_k$ is the stochastic noise that can attain values from a finite discrete set \begin{equation} \sum_{r=0}^{N}u_r \geq M \end{equation} Can someone suggest methods to solve for the optimal policy function $u_t = h^*(x_t)$ with the given constraints as described above?

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    This is a fairly general question, as you didnt mention dependencies of the $\epsilon_k$ or assumptions on $g$. Also the infinite time horizon makes it harder. However a solution to this Problem can probably be found in this [Book](http://dspace.mit.edu/handle/1721.1/4852#files-area). – 2017-02-07
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    @Vincent.W. The function $g$ is linear in $(x_k, u_k, \epsilon_k)$ and for starters assume $\epsilon_k$ can take only two different values with equal probability. Also, for simplicity you may regard this process in finite time steps i.e. $t$ varies from $0$ to $N$. – 2017-02-07
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    @Vincent.W. Could you point out the relevant sections in the text pertaining to this problem as I'm already finding the text to be a bit abstract to read? – 2017-02-07

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