Artemis BJJ (MYGYM Bristol), Can Sönmez, Bristol, UK - 06/06/2016
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.
To take the back, fire your underhooking arm up into their armpit. You're trying to knock them forwards, while simultaneously scooting your body down towards their legs. At that point, pull your 'paw' arm back, so that you can base on that elbow, swiftly pushing up onto the hand. That should give you the balance to reach around to their lat with what was your underhooking arm. For further control, swing your leg over their back too. Establish a hook by digging your heel inside their knee. Finally, get a seatbelt grip (one arm under their armpit, the other over their shoulder, locking your hands together) and roll towards your non-hooking foot for standard back control.
Teaching & Sparring Notes: This is definitely the one I should always teach first. I had a good question from Chris about that scoot down the body, when you bump with your arm. I'd say that's a combination of a strong knock into the armpit, scooting down your hips and that grip on the lat. Something I should codify more, though.
In sparring, I had a play with some top techniques I saw on an old Carlos Machado DVD, where he gets a cross collar grip. That's a different approach (I normally always go for a cross face, from everywhere), so it made for an interesting change in sparring. Main thing to watch out for there is that you keep the hip pressure down, so they can't easily get the closed guard back.