optimization over moving domains

Question

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!

Answer 1

$\newcommand\R{\mathbb R}$The answer is no.

E.g., let $A=B=\R$, $B_a=[1-a^2,2]$ for $a\in A$, and $\ell(b)=b^3-3b$ for $b\in B$. Then $\ell(b)$ and $B_a$ are perfectly smooth, but $L$ is not differentiable.

Indeed, here $L(a)=-2+a^4\min(0,3-a^2)$ for $a\in A=\R$. The left derivative of $L$ at $\sqrt3$ is $0$ and the right derivative of $L$ at $\sqrt3$ is $-18\sqrt3$.