Page 58 - Textos de Matemática Vol. 36

48
Chapter 2. Global aspects
ii)
iii)
If rank A = n − 1, then it follows with Lemma 2.4.2, i) that at least one element of W(·;WC) tends to infinity as t → ∞, supposing that
According to Theorem 2.4.6, ii) any solution W (·; WC ) with C =  .  cm+n
and
cm+n
converges at an exponential rate towards Wd, as t → ∞. Analogously, if
cn
Hence, the basin of attraction of Wd with respect to convergence in Cˆm×n (Cˆ the extended complex plane C ∪ {∞} with the chordal metric) is open and dense in Cm×n, moreover, from (2.44) follows that its complement in Cm×n is of measure zero.
This basin of attraction with respect to the chordal metric we denote by GBA(Wd):={W0 ∈Cm×n|(chordal−lim)t→∞W(t,W0)=Wd},
GBA(Wrd) := {W0 ∈ Cm×n|(chordal − lim)t→−∞W (t, W0) = Wrd}
and call it the generalized basin of attraction of the dichotomic or reverse dichotomic solution, respectively. From our previous considerations we immediately obtain
and
c1 .
cm+1 .
det . ̸=0. cm+n
cm+1 .
det .  ̸= 0 (2.44)
det  .  ̸= 0, towards the reverse dichotomic solution, as t → −∞.
GBA(Wd) = {W0 = WC|det .  ̸= 0} cm+n
GBA(Wrd) := {W0 = WC|det .  ̸= 0}. cn
In the case of the Hermitian algebraic Riccati equation HARE and the associated differential equation, because of the symmetry of the spectrum of M the dichotomy (and reverse dichotomy) condition says that there is no eigenvalue on the imaginary axis.
cm+1 .
c1 .
 c1  .