convex optimization with nonlinear continuous dynamic constraint (finite horizon)

Question

Recently, I am reading a paper: http://www.sciencedirect.com/science/article/pii/S0167691112000874
and I am confusing about the following.

If I say the following is a convex optimization problem:

\begin{equation}\begin{aligned} & {\underset{u,t_f}{\min}} & & \int_0^{t_f} l(g(u(t)))dt\\ & \text{s.t.} & & \dot{x}(t)=f(t,x(t),g(u(t))) \text{ a.e. on }[0,t_f] \\ \end{aligned} \end{equation}

where

1. $g(u(t))\in \mathbb{R}$ is a measure of the control effort, where $u(t)\in \mathbb{R}^3$.
2. a.e. means almost everywhere.

So if I say that optimization problem is convex, does it mean $l(\cdot)$ is a convex function, $f(\cdot)$ is a convex function and $g(\cdot)$ is also a convex function?

Also, how to deal with the finite horizon continuous time dynamic constraint? discretize it?

Thanks for suggestions.