Consider Optimization problem in the form $$\min ~ f(x) \quad \quad \text{subject to}\quad \quad F(x) \in C \subseteq Y $$
Where $F: X \to Y$ is continuously differentiable between banach spaces $X$ and $Y$ and $f$ is lipschitz function, and $C$ is closed convex set in $Y$. I'm wondering is there any class of optimal control problems fitted in above framework ?
I'm asking this, because KKT condition is available for above problem in literatures, and yet expert people in optimal control take different approach to ge necessary optimality condition! To my understanding an optimal control problem is just an Infinite dimensional optimization problem, why we can't employ one single approach to solve both.
My Guess:
I think the assumption $F$ be differentiable (in Frechet sense) is strong, and might make trouble to cover large class of optimal control problems.