Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups

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