• If g2³ g1 >g0 then *X*¥(g1) ³ *X*¥(g2) and *Y*¥(g1) ³ *Y*¥(g2) . Thus *X*¥ and *Y*¥ are decreasing fun. of g, as is r(*X**¥**Y**¥*).

• At g=g0 , any one of the 3 conditions can
fail. It is most likely that condition (iii) will fail first.

• To understand this, consider (i) and let g1 be the largest g for which *H*¥ fails to be in *dom(Ric),* because it fails to have either the stability property or
the complementary property. The same remarks will apply to (ii) by duality.

• If the stability property fails at g=g1 , then *H*¥ Ï*dom(Ric),* but *Ric* can be extended to obtain *X*¥ and the controller *u=-**B**2*****X*¥*x* is stabilizing and makes *||T**zw**||**¥** *=g1.The stability property will also not hold for any g£g1 , and no controller whatsoever exists which makes *||T**zw**||**¥** *<g1.