Consider the following ODE $$ \dot x(t) = f(x(t),u(t)), t \in [0,T]. $$ where $f$ is smooth, $u(t) \in \mathbb R$ is measurable, and $x(t)=[x_1(t),..,x_n(t)]^T \in \mathbb R^n$, with an initial condition $x(0)$
I am interested in the functional: $$\varphi: u \mapsto \int_0^T x_n(t) .$$
Are there some results that can be used to prove that $\varphi$ is convex?