Not a lot of people seem to remember all of the games had health bars for bosses. Health bars for bosses in the first game were always on top of the player's health bar but are at the bottom center since DMC2.
Royalguard's attack Release is at it's strongest when the bar on the style gauge is lit up completely.
DTE or Devil Trigger Explosion is only executable if you have more than three DT orbs. While holding down the DT button, the extra orbs will glow red. When all maximum orbs are red, release to unleash a devastating explosion of demonic energy as you you transform. To get extra DT orbs, buy the Devil Orbs at the Time Goddess Statues