In such circumstances "Idenitfied [position]" normally suffices and, by definition, it wll be a radar control service.
Not all identified aircraft get a
control service. VFR aircraft in D/E/F/G may get varying radar services, many not 'controlled', IFR aircraft in F/G will not get a controlled service but may get a radar service where they are under continuous surveillance, or just a 'snapshot' service, but even then they'll be identified for the 'snapshot' period.