(please explain here the theory)
Essentially is an optimization method that take in consideration strong mathematical definition of the restrictions on the expected behaviour of the closed loop and the strict stability of it.
It is very appreciated due of its strong mathematical foundation, the mathematical way to express engineering restrictions, its optimization/optimatility aspect and the ability to include both classic and robust control concepts in a single design framework.
The "H" stand for Hardy space. "Infinity" stand of the fact that it design to accomplish minimax restrictions in the frequency domain. The infinity norm represent the maximum value of a vector.