Let $T>0$, $(U,d)$ be a metric space and $\mathcal{V}:=\{u:[0,T]\to U\,|\, u\text{ is measurable}\}$. Consider the control system $$\begin{cases}\dot{x}(t)=f(t,x(t),u(t)),\quad \text{a.e. }t\in[0,T],\\x(0)=x_0,\end{cases}$$ where $f:[0,T]\times\mathbb{R}^n\times U\to \mathbb{R}^n$ is measurable.
According to [1, P. 102], for any $u\in\mathcal{V}$, the above control system has a unique solution $x:[0,T]\to\mathbb{R}^n$ if the following conditions hold:
$U$ is a separable metric space.
There exist a constant $L>0$ and a modulus of continuity $\omega: [0,\infty) \to [0,\infty)$ such that
2.1) $|f(t,x,u) - f(t,\hat{x},\hat{u})|\leq L|x-\hat{x}|+\omega(d(u,\hat{u})),\quad \forall t\in[0,T],x,\hat{x}\in \mathbb{R}^n,u,\hat{u}\in U,$
2.2) $|f(t,0,u)|\leq L,\quad \forall (t,u)\in[0,T]\times U.$
My question is about the proof of this statement ([1] does not provide the proof).
Can one extend the standard Picard–Lindelöf theorem to prove it? specially, since condition 2.1 is basically a Lipschitz condition for $f$. I cannot also see what role separability of $U$ and the condition 2.2 play in the existence and uniqueness of the solution. Any help or hint with the proof is appreciated.
[1] Yong, Jiongmin; Zhou, Xun Yu, Stochastic controls. Hamiltonian systems and HJB equations, Applications of Mathematics. 43. New York, NY: Springer. xx, 438 p. (1999). ZBL0943.93002.