Consider we have an infinite dimensional optimization problem. We have state variable $x\in R^N$ and input $u \in R^M$ and we have have following optimization problem.
J = $\int_{0}^{tf } L(x,u)dt$
subject to
$\dot{x} = f(x,u), ~~~g(x,u) \le 0$
I know for the standard optimization problem, we know that problem is convex if the objective function is convex and so are constraints. In the infinite dimensional problem, how we will proceed with the proof that problem is convex. Should I try to discretize the time and then proceed with the convexity definition.