= Solution
Choose $R_0$ with $K\subset\{(y,w):|y|\leq R_0,\ |w|\leq R_0\}$ and let $A_0=\max_{|w|\leq R_0}|a(w)|$, which is finite by <continuity>. Every point transported from $K$ satisfies
$$
|X(t)|\leq R_0+TA_0,\qquad |V(t)|\leq R_0
\quad(0\leq t\leq T).
$$
Hence any
$$
\boxed{R_T>\sqrt{(R_0+TA_0)^2+R_0^2}}
$$
contains the <support of a function> in the required <phase space> ball throughout $[0,T]$. Equivalently, the image of the compact set $[0,T]\times K$ under the <characteristic flow map> is compact. This uniform support control puts the solution in $\mathcal C_T$ and justifies the following differentiations under the <integral>.
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