Admissible Input Elements for Systems in Hilbert Space and a Carleson Measure Criterion

Abstract

We study the control system \[\dot x = Ax + bu,\quad x \in X,\quad u{\text{ scalar}},\] where A generates a semigroup on the Hilbert space X, but, in general, the control input element $b \notin X$. Many boundary value control systems, point control force situations, etc., can be studied in this context. We define and analyze “admissible” input elements b and develop sufficient conditions for b to be admissible in terms of the Carleson measure theorem of $H^p $-theory.