The H2 control problem is to find a proper, real rational controller K that stabilizes G internally and minimizes the H2 norm of the transfer matrix Tzw from w to z, see the figure below
It is well-known that a system with LQR controller has at least 60o phase margin and 6 dB gain margin. However, it is not clear whether these stability margins will be preserved if the states are not available and the output feedback H2 (or LQG)controller has to be used.
for more information on stability margins of the H2 control, see the book, Essentials of Robust Control / chapter 13 / page 265 / section 13.6.
on the other hand, Optimal H-infinity Control: Find all admissible controllers K(s) such that infinity norm of Tzw is minimized. additionally,
Suboptimal H-infinity Control: Given gamma > 0, find all admissible controllers
K(s), if there are any, such that infinity norm of Tzw < gamma.