  We consider final state observability estimates for bi-continuous semigroups
on Banach spaces, i.e. for every initial value, estimating the state at a final
time $T>0$ by taking into account the orbit of the initial value under the
semigroup for $t\in [0,T]$, measured in a suitable norm. We state a sufficient
criterion based on an uncertainty relation and a dissipation estimate and
provide two examples of bi-continuous semigroups which share a final state
observability estimate, namely the Gauss-Weierstrass semigroup and the
Ornstein-Uhlenbeck semigroup on the space of bounded continuous functions.
Moreover, we generalise the duality between cost-uniform approximate
null-controllability and final state observability estimates to the setting of
locally convex spaces for the case of bounded and continuous control functions,
which seems to be new even for the Banach spaces case.
