• 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¥ Ïdom(Ric), but Ric can be extended to obtain X¥ and the controller u=-B2*X¥x is stabilizing and makes ||Tzw||¥ =g1.The stability property will also not hold for any g£g1 , and no controller whatsoever exists which makes ||Tzw||¥ <g1.