Page 66 - Textos de Matemática Vol. 36

56 Chapter 2. Global aspects
Theorem 2.5.6. Let M11, M12, M21, M22 be piecewise continuous and locally in- tegrable on (−∞,T]. If for some matrices C ∈ Cn×n, with C∗ = C, D ∈ Cn×m
with
CM +DM CM +M∗ D+DM
Θ= 11 21 12 11 22 (2.53)
0 M∗ D 12
Θ(t) + Θ∗(t) ≤ 0
C + DW0 + W0∗D∗ > 0
Proof. Here we make use of the reduction of Riccati differential equations to a linear system as described in Theorem 2.2.1. From the remarks made there it is clear that a blow up of the solution W = W(·,W0) of the Riccati differential equation occurs at the moments where detQ(t) vanishes. To formulate non- blow-up conditions we introduce the quadratic Lyapunov-type function
V (t) := x∗(Q∗(t)CQ(t) + Q∗(t)DP (t) + P ∗(t)D∗Q(t))x. (2.56) Here 0 ̸= x ∈ Cn and C ∈ Cn×n,C∗ = C,D ∈ Cn×m are some parameters.
Evidently
Calculating the derivative of V (t) in (2.56), along a solution of L, we obtain
it holds that
for all t ≤ t0(≤ T), and if
(2.54)
(2.55) for some W0 ∈ Cm×n, then the solution W (t, W0) of RDE with W (t0, W0) = W0
exists for all t ≤ t0.
(suppressing t)
and therefore V˙ =
V0 := V (t0) = x∗(C + DW0 + W0∗D∗)x. (2.57) V˙ = 2x∗[Q∗C(M11Q+M12P)
+ (Q∗M∗ +P∗M∗ )DP +Q∗D(M Q+M P)]x 1112 2122
2x∗Q∗(CM11 + DM21)Qx
+ 2x∗Q∗(CM + M∗ D + DM ))Px + +2x∗P∗M∗ DPx.
121122 12
This can be represented as
1V˙ =(x∗Q∗,x∗P∗) 2
CM +DM CM +M∗ D+DM Qx × 11 21 12 11 22
0 M∗D Px 12
∗ ∗ ∗ ∗ Qx =(xQ,xP)Θ Px ,