Page 73 - Textos de Matemática Vol. 36

2.5. Global existence results 63
Now we shall describe briefly some possible applications of Theorems 2.5.10 and 2.5.11. First we apply our results to the Riccati matrix differential equation HRDE and some of its generalizations as described in (2.65)
K˙ =−A∗K−KA−Q+KSK, K(tf)=Kf.
Here we clearly have g1 = g2 = g3 = 0. From Theorem 2.5.10 we infer that if Kf > 0 and if x∗Q(t)x ≥ α|x|2, then the solution K(t) stays positive for t ≤ tf , hence a blow up of the solution only can occur if one of the eigenvalues of K(t) tends to +∞ in finite time. If now also the second condition in (2.72) is fulfilled for some β with P = 0, then we obtain boundedness, i.e., K(t) ≤ βI if K(tf ) ≤ βI.
As our next example we discuss coupled Riccati type differential equations as they appear in the theory of Nash games with memoryless perfect state (or closed loop) information structure (see [StHo69]) in the two player case:
K˙1 =−ATK1 −K1A−Q1 +K1S1K1 +K1S2K2 +K2S2K1 −K2S12K2, K˙2 =−ATK2 −K2A−Q2 +K2S2K2 +K2S1K1 +K1S1K2 −K1S21K1,
where K1 (tf ) = K1f , K2 (tf ) = K2f . With the following notation
˜ A 0 ˜ Q1 0 ˜ S1 0 A=0A,Q=0Q,S=0S,
22
S12 0 K1 0 K1f 0 0In S0= 0 S ,K= 0 K ,Kf= 0 K ,J= I 0 ,
21 2 2f n
where A ∈ Rn×n and Q, S, S0 ∈ Rn×n are symmetric, the coupled system then
can be written as a single equation
K˙ =−A˜TK−KA˜−Q˜+KS˜K+JKS˜JK+KJS˜KJ−JKJS0JKJ,
with terminal values K(tf ) = Kf , KfT = Kf . Here we can apply Theorems 2.5.10, 2.5.11 with
g1(K) = JKJS0JKJ, where g1T (K) = JKT JS0JKT J = g1(KT ), and
g 2 ( K ) = J S˜ K J , g 3 ( K ) = J K S˜ J ,
where g3T (K) = JS˜KT J = g2(KT ). Notice that this equation also has been studied in [AFJ94], [JaKu98], [SiCr73a].