Continuity of states for bounded admissible inputs

Let \(A\) generate a strongly continuous semigroup on an arbitrary Banach space \(X\), and let \(B\) be a bounded operator from an arbitrary Banach input space into the extrapolation space \(X_{-1}\). We prove that \(L^\infty\)-admissibility for all strongly measurable inputs implies that the finite-time control operator norms tend to zero with the time horizon. Consequently every state driven by a bounded strongly measurable input is continuous in the norm of \(X\), including at time zero. The argument applies the scalar Grothendieck property to a separate multiplier operator for each fixed input. A resolvent construction gives this operator a separable range in \(X\); the Orlicz--Pettis theorem and a disjoint-interval construction then give the uniform small-time estimate.