•If g2³g1>g0 then X¥(g1) ³X¥(g2) and Y¥(g1) ³Y¥(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 g1be the largest gfor 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.