Consider a control system $$ \dot x(t) =f(x(t),u(t)) \qquad (\star) $$ where $f$ is a smooth vector field and $x\in \mathbb{R}^n$.
The endpoint mapping is defined by $$ E:\mathbb{R}^n\times \mathbb{R}^+\times U \to \mathbb{R}^n\\ (x_0,T,v) \mapsto x(T;x_0,v) $$ where $x(T;x_0,v)$ denotes the solution of $(\star)$ at time $T$ starting from $x_0$ and with $u=v$.
- $U=L^\infty([0,T],\mathbb{R})$. Wyy does $E$ is a smooth map?
- Is there a $L^p$ topology endowing $U$ such that this map is not smooth?
Examples and book references are welcome