Consider the control system $$ \dot x(t) = f(x(t),u(t)) $$ where $x(.)$ takes its values in $\mathbb{R}^n$ and $u$ takes its values in $\mathbb{R}^m$ and $f$ is a smooth vector field. For $T>0$, denote $x(.;x_0)$ the trajectory defined on $[0,T]$ starting from $x_0$ at $t=0$.
For an admissible trajectory $x(.),u(.)$, the cost is defined by $$ g:\mathbb{R}^n\rightarrow \mathbb{R} $$ $$ z\mapsto \int_0^T L(x(s;z),u(s)) ds $$ where $L:\mathbb{R}^n\times \mathbb{R}^m$ is smooth.
Assume $g'(z)$ exists, how to compute it ? Should I considered a topology on a set of curves ?