Let $A, B$ be Banach spaces, and for any $a\in A$, $B_a\in B$ is a measurable subset. Consider the following optimization problem: $$L(a)=\inf_{b\in B_a}\ell(b),$$ where $\ell(b)$ is a infinite-times differentiable functional of $b$.

Question: prove that $L(a)$ is Frechet differentiable, moreover, $L(a)$ satisfies Taylor expansion $$L(a)-L(a_0)=L'(a_0)(a-a_0)+O(\|a-a_0\|_A^2),$$ where $\|\|_A$ is the norm of $A$.

We can assume that, when $a$ is moving, the set $B_a$ is smoothly moving. I am not sure how to describe such smoothly moving. Intuitively, this can make $L(a)$ smooth in $a$. But I am not sure how to prove this. Ideas or suggestions will be highly appreciated!