Suppose the state of a system is given by $X_t$ such that $X_0 = x \in (0,1)$ and $\dot X_t = [1 - \alpha_t C] X_t (1-X_t)$ where $C > 1$ and $\alpha_t \in [0,1]$.
An agent chooses $(\alpha_t)_{t \ge 0}$. Let $A$ denote the space of all measurable maps $\mathbb R_+ \to [0,1]$. Define, $\tau:= \inf\{t \ge 0: X_t \notin (a,b)\}$ where $0 < a <b <1$ and $x \in (a,b)$.
An agent solves the following: \begin{align*} J(x, \alpha):=& \int_0^\tau e^{-t} c \alpha_t dt + e^{-\tau}\mathbb{1}_{X_\tau = b}\\ V(x):=& \inf_{\alpha \in A} J(x,\alpha) \end{align*}
I want to say that it is without loss of generality to restrict attention to feedback controls, i.e., we can take $A$ to be a space of all controls $\alpha$ such that $\alpha_t = \phi(X_t)$ for some function $\phi: x \mapsto [0,1]$. To be clear, of course $\phi$ would depend on $\alpha$.
From what I understand, there are theorems like the measurable selection theorems that often let one do this stuff. But, I don't know enough about the underlying theory.