p-Admissible control elements for diagonal semigroups on lr-spaces

For linear systems described by x˙(t)=Ax(t)+bu(t), where A is a diagonal operator on the state space lr for some 1⩽r<∞ and b∉lr, we develop necessary and sufficient conditions for b to be p-admissible. This extends results by Ho, Russell and Weiss to the case r≠2.