This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in which, at each point in time, the control can be chosen from a Banach space $\mathcal U$ but the state space lies in $\mathbb R$. More formally, let $g,f: \mathbb R \times \mathcal U \rightarrow \mathbb R$ be functionals, and consider the problem \begin{align} \max_x & \int_0^s g(x(t),u(t)) \, dt \\ \text{s.t.} & \frac{d x(t)} {d t} = f(x(t), u(t)) \\ & x(0) = x_0 \,, u(t) \in \mathcal U \,. \end{align} We may assume further than $g$ and $f$ are continuously differentiable in $x$ and continuous in $u$. It would even suffice is $g$ and $f$ are independent on the state! Further regularity conditions are welcomed too.

- Does the Maximum Principle hold in this setting?
- Do you know of any reference that proves the result?

I am aware of some more general results in which the state space in Banach, but I am looking for a simpler result that takes advantage of the fact that the state space is $\mathbb R$.

Thanks in advance!