Artemis BJJ (MYGYM Bristol), Can Sönmez, Bristol, UK - 10/06/2015
In half guard, your first concern is to stop them flattening you out and starting their pass. They are generally going to want to establish an underhook on their trapped leg side, using the other arm to control under your head. In many ways, it is a similar position to standard side control. That will enable them to crush you to the mat, then exert lots of shoulder pressure to kill your mobility. Many of the same attacks from side control can also be viable from here, like an americana.
Naturally, you don't want them to reach that dominant position. Your goal is to get up on your side, with your own underhook around their back, on your trapped leg side. That is one of the main fights you'll have in half guard, so it is essential that you get used to working for that underhook.
If you can get the underhook, that accomplishes two things. First, it prevents them crushing their chest into yours, which would help them flatten you out. Second, it means you can press into their armpit to help disrupt their base, as well as help you get up onto your side. You can use your knee knocking into their bum at the same time to help with this too, as that should bump them forward.
For your leg positioning, the standard half guard is to have the inside leg wrapped around with your foot on the outside. Your other leg triangles over your ankle. This provides you with what SBG refer to as a 'kickstand': that outside leg is useful for bridging and general leverage. It's harder for them to flatten you out if you can resist with that kickstand structure.
After you've controlled a leg, got the underhook and onto your side, you want to block their arms. Almost a decade ago, Indrek Reiland put together an awesome video (made even more awesome by being free) about the fundamentals of half guard. The main principle I use from Reiland is what he calls the 'paw'.
By that, he means hooking your hand around their bicep, just above the elbow. You aren't gripping with your thumb: this is just a block, to prevent them getting a cross-face. Reiland emphasises that preventing that cross-face is the main principle. Therefore, if you can feel they are about to remove your paw by swimming their arm around, bring your underhooking hand through to replace your first paw with a second: this is what Reiland calls the 'double-paw' (as he says in the video, it's an approach he learned from SBG black belt John Frankl).
Similarly, if they manage to underhook your underhook, bring that arm over for a double-paw (this is also applicable from the start, if you're framing against their neck), then work to recover your underhook. Keep in mind with the double-paw that you need to make sure you don't leave space under your elbow. Otherwise, as Reiland demonstrates, they can they go for a brabo choke. Get the elbow of your top double-pawing arm to their nearest armpit, as that makes it easier to circle your arm around to their back.
If their base is too solid to go for the back, you can recover full guard instead. You still want the underhook: if you need to make space, keep bumping until you can at least get your elbow by their armpit. You can then use that to pry up some space, circling your arm around for the underhook. Switch your leg positioning so that your 'kickstand' steps over their leg, hooking underneath their lower leg with your instep. Keep your legs tight, or they will pull their leg free.
Curl towards their same side knee on your paw-arm side, until you can push it out with your elbow. Get the knee of your inside leg up past that knee, which will enable you to shove their knee back and free your leg. From there, swing both legs around their back and lock your ankles for closed guard. I like to also shift from a paw to an underhook around their arm, trapping it to my chest, but that isn't essential.
Teaching: I just focused on the easy guard recovery bit, rather than all the maintaining stuff. I could do the maintaining stuff next time, though it was a nice relaxed class, plenty of sparring too. Everyone got the technique, I think because it's basically just the second part of the elbow escape from mount. It's worth noting that you can use your hand to push knee, with the proviso that you need to watch your neck.
There was a question about how to proceed when you're totally stuck flat on your back. For that, I generally suggest the lockdown, because your hips are already immobilised anyway. You can then disrupt their base and move into a more advantageous half guard position (not that the lockdown can't be useful, but for beginners, I'd generally say get back to the 'kickstand' leg position and go for the usual underhook and paw position).