Carathéodory existence of admissible states for bounded measurable controls

Let $t_0<t_1$, let the dynamics $F:\mathbb R^n\times\mathbb R^m\to\mathbb R^n$ satisfy the uniform joint Lipschitz bound

for some $M\ge 0$, and let the running cost $\ell:\mathbb R^n\times\mathbb R^m\to\mathbb R$ be jointly continuous. Let $u$ be a measurable control on $[t_0,t_1]$ that is essentially bounded and takes values in the constraint set $\Omega$ at almost every time. Then there is a state trajectory $x$ such that $(u,x)$ is an admissible pair: $x(t_0)=x_{\mathrm{init}}$, $x$ is absolutely continuous on $[t_0,t_1]$,

and $t\mapsto \ell(x(t),u(t))$ is integrable on $[t_0,t_1]$.

This is the Carathéodory (generalized Picard) existence theorem for the control system with a fixed measurable control. The right-hand side $f(t,x)=F(x,u(t))$ is measurable in $t$, globally Lipschitz in $x$ with the constant weight $M$, and integrably bounded on $[t_0,t_1]$ because $u$ is essentially bounded, so a unique absolutely continuous solution exists on the whole interval. The statement supplies the perturbed trajectories needed for needle variations of an admissible pair and, more generally, the state map $u\mapsto x(u)$ on bounded measurable controls.

Formalization Note. Controls and states are functions on all real times and are constrained only on $[t_0,t_1]$; measurability, the essential bound, and the constraint $u(t)\in\Omega$ are imposed with respect to Lebesgue measure restricted to $[t_0,t_1]$. Continuity of $F$ is not assumed separately, since it follows from the Lipschitz bound. The cases $n=0$ and $m=0$ are allowed.