  Let $A$ be an abelian variety defined over a number field $F$. We prove a
control theorem for the fine Selmer group of the abelian variety $A$ which
essentially says that the kernel and cokernel of the natural restriction maps
in a given $\mathbb{Z}_p$-extension $F_\infty/F$ are finite and bounded. We
emphasise that our result does not have any constraints on the reduction of $A$
and the ramification of $F_\infty/F$. As a first consequence of the control
theorem, we show that the fine Tate-Shafarevich group over an arbitrary
$\mathbb{Z}_p$-extension has trivial $\Lambda$-corank. We then derive an
asymptotic growth formula for the $p$-torsion subgroup of the dual fine Selmer
group in a $\mathbb{Z}_p$-extension. However, as the fine Mordell-Weil group
needs not be $p$-divisible in general, the fine Tate-Shafarevich group needs
not agree with the $p$-torsion of the dual fine Selmer group, and so the
asymptotic growth formula for the dual fine Selmer groups do not carry over to
the fine Tate-Shafarevich groups. Nevertheless, we do provide certain
sufficient conditions, where one can obtain a precise asymptotic formula.
