# optimization over moving domains

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!

$$\newcommand\R{\mathbb R}$$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 $$-18\sqrt3$$.