Page 72 - Textos de Matemática Vol. 36

62 Chapter 2. Global aspects In the next theorem we describe another type of invariant domains for the
equation (2.67).
Theorem2.5.11. LetP :R+ →R+ beanyfunction.Ifforsomeε>0,α,β∈R, α<βandforallt≤tf andy∈Cn with|y|=1thefollowinginequalitieshold
y∗(−2αA(t) + α2S(t) − Q(t))y ≤ −P (|α|) − ε, y∗(−2βA(t) + β2S(t) − Q(t))y ≥ P (|β|) + ε,
where for g1, g2, g3 as in (2.69) there holds
||g1(K) − Kg2(K) − g3(K)K|| ≤ P (||K||)
(2.72)
(2.73)
for all Hermitian matrices K ∈ Cn×n and all t ∈ (−∞, tf ], then the set I(α,β):={K =K∗ ∈Cn×n| αIn ≤K ≤βIn}
is negative invariant.
Proof. As in the proof of Theorem 2.5.10 let us denote by λ(t) either the minimal or maximal eigenvalue of the Hermitian solution K(t) to (2.67), respectively, and let y(t) denote a corresponding eigenvector with |y(t)| = 1. As in the previous proof we derive from (2.73)
dλ dt
≤ y∗(−2λA + λ2S − Q)y + P (||K||). (2.74) This together with (2.72) yields
dλ < 0 (2.75) dt
as λ (here considered to be the minimal eigenvalue) approaches the value α from above. Analogously, from the inequality
= y∗K˙ y = −2y∗KAy − y∗Qy + y∗KSKy + y∗g(K)y
= −2λy∗Ay + λ2y∗Sy − y∗Qy − y∗(g1(K) − Kg2(K) − g3(K)K)y
dλ ≥ y∗(−2λA + λ2S − Q)y − P (||K||), dt
we obtain together with (2.72)
dλ > 0 dt
as λ (here considered to be the maximal eigenvalue) approaches the value β from below. From inequalities (2.75), (2.76) we conclude similarly as in the proof to Theorem 2.5.10 that I(α,β) is a negative invariant set for equation (2.67).
(2.76)