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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.

  1. Does the Maximum Principle hold in this setting?
  2. 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!

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