What kind of sentence is "Do your work, John"

I am having an arguement with my children about whether or not this is imperative. It seemed to be, but then we found rules that imperative sentences must have "you" as the subject. It doesn't appear to be declarative, and is decidedly not either of interrogative or an exclamation. I have found mention of third party imperative, but I cannot tell if that is the correct description for it.

Solution 1:

Normal Imperatives and Vocative Address

Yes, that’s an imperative sentence whose implied subject is you.

[You] Do your work, John.

The John at the end is a vocative use, a noun of direct address, just like:

Your parents aren’t home, John.

In neither case is John the subject, but in the first case the implicit you and John co-refer to the same person.

Third-person imperatives

The third-party imperative you mention is today most often done through a periphrastic construction using let or may:

• Let your sister find him.
• May he never find her.

Formerly it was also possible to create third-person imperatives through a present subjunctive (now in form a bare, uninflected infinitive) as many of our neighboring languages still permit:

• God save the queen.
• Devil take the hindmost.
• God rest you merry gentlemen.

But even that familiar line from the carol goes on to a normal periphrastic imperative:

• Let nothing you dismay.

Meaning “Let nothing dismay you.”

These frozen phrases are no longer productive in present day English, and their grammar is no longer apparent to most modern speakers, any more than is that of things like these:

• Till death us do part.
• Come Saturday morning, I’m going away with a friend.