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58 Chapter 2. Global aspects
If 0 ̸= x ∈/ Ker(C + DW0 + W0∗D∗), then from (2.61) it follows that V(t,x)>0 forall t∈(−∞,t0]. (2.62)
If 0 ̸= x ∈ Ker(C + DW0 + W0∗D∗), then for 0 < δ < t0 − t we get from (2.61)
t0 t
Q(s) Q∗(s), P∗(s)(Θ(s)+Θ∗(s)) P(s) ds x
V(t,x)=−x∗ t0
≥−x∗
t0 −δ
Since Q(t0) = I, by continuity Q−1(t) exists in (t0 − δ, t0] for sufficiently small δ, hence, together with Theorem 2.2.1, i.e., with W(t) = P(t)Q−1(t), we infer from (2.63)
∗∗∗∗I
V(t,x)≥−δx Q1(s0) I W (s0) (Θ(s0)+Θ (s0)) W(s0) Q1(s0)x (2.64)
for some intermediate s0 ∈ (t0 − δ, t0 ]. With W (t0 ) = W0 and the rank condition (2.60) we infer
∗ ∗ ∗ I
−δx I W0 (Θ(t0)+Θ (t0)) W x>0.
0
Again by continuity and for sufficiently small δ we infer from (2.64) V(t,x)>0 forall t∈(−∞,t0].
This, together with (2.62) completes the proof since we now conclude, as in the proof of the previous theorem, that det Q(t) ̸= 0 for all t ∈ (−∞, t0].
In [FJS97], [FJS00] global existence has been concluded under the weaker assumption C + DW0 + W0∗D∗ ≥ 0 using an argument from [Feuc94], but it seems that there is some gap in that argumentation. However, from the proof to Corollary 2.5.7 it also becomes clear that the sufficient conditions for global existence in (2.59) and (2.60) are not the weakest possible.
On the other hand, by choosing C = 0 and D = 0 in (2.53) and (2.59), respectively, conditions (2.54) and (2.59) are fulfilled for any Riccati differential equation. The rank condition (2.60) is not fulfilled and, indeed, for instance the real solutions of the scalar Riccati differential equation x˙ = 1 + x2 all have a finite escape time, since they are of the type x(t) = tan(t + c) for some real constant c depending on the initial value. This shows that (2.54) and (2.59) cannot be sufficient without any further assumptions.
Q(s) Q∗(s), P∗(s)(Θ(s)+Θ∗(s)) P(s) ds x.
(2.63)