Page 52 - Textos de Matemática Vol. 36

42 Chapter 2. Global aspects hence, if step nk + 1 would be the first step such that Gnk+1(ak) = ∅. This
results in the desired decomposition of Γk.

Remark 2.4.5. (i) The question about G0ζ(ak) ̸= ∅, i.e., concerning existence of solutions of ARE, has been settled only in special cases. Mainly in the case of HARE under control-theoretical conditions. In the non-symmetric case, however, only necessary controllability conditions for existence have been obtained in particular cases (see [AFJ93a], [AFJ93b]). A sufficient condition was obtained in [AFIJ03],Theorem 2.3.2.
(ii) The general case of multiple eigenvalues not all having full geometric mul- tiplicity was discussed in a similar way in [FrJa95].
Instead of the description of the structure of the set of solutions Γ of ARE as was provided in Theorem 2.4.4, there is also the possibility of using the more geometric approach of Graßmann manifolds. This was used in [Shay86] in the case of symmetric Riccati differential equations and exploited in the non- symmetric case in [FrJa91]. Because of time limitations in this lecture, for the interested reader, we refer to [AFIJ03].
Now we study the dynamic behavior of solutions of Riccati differential equations.
Let C ∈ Vn(C(m+n)×n) be such that Q0 in the representation formula (Theorem
2.3.1) is regular, ε > 0 and φ ∈ {φ1,...,φ2h}. Then W(·;WC) can have an
infinite number of poles in the sector {t ∈ C|φ−ε < argt < φ+ε} and, in
contrast to Lemma 2.4.1, lim wlα(reiφ;WC) may fail to exist. This behavior r→∞
was already demonstrated in Example 2.3.6. According to Lemma 2.4.1 the phase portrait of equation RDE is (as t → ∞) very simple if we restrict t to the interior of a sector Sk, 1 ≤ k ≤ 2h. On the other hand the behavior of W(·;WC) along one of the rays Bφ = {t ∈ C|argt = φ} may be very complicated if φ ∈ {φ1,...,φ2h}.
Obviously it is sufficient to describe W(·;WC) along one of the rays Bφ. Without loss of generality, and since it is the practically interesting case, we restrict ourselves to the description of the phase portrait of equation RDER, which means that we now restrict t so that t ∈ R. On the Graßmann manifold Gn(Cm+n) and for G ∈ Gn(Cm+n) we define the sets
Ms(S0) = {G ∈ Gn(Cm+n)|S(t,G) → S0 and
Mu(S0) = {G ∈ Gn(Cm+n)|S(t,G) → S0
for t → ∞} for t → −∞}
(2.36)
which are called stable and unstable sets, respectively, for G of the flow S(t, G). The next four theorems describe the most important features of the dy- namic behavior of solutions of RDER. The situation would become more difficult
if we did not restrict ourselves to the semi–simple case (see [FrJa95]).
ζ