Page 74 - Textos de Matemática Vol. 36

64 Chapter 2. Global aspects
In Minimal Cost Variance Control problems there appears a coupled sys- tem of Riccati type differential equations quite similar to the one from memo- ryless feedback Nash games. We consider the following system (see [SWSp95]):
M˙ =−ATM−MA−Q+MBR−1BTM−γ2VBR−1BTV, V˙ =−ATV −VA+2γVBR−1BTV +MBR−1BTV
+ V B R − 1 B T M − 4 M E W˜ E T M , S1 := BR−1BT , S12 := γ2BR−1BT , S2 := 2γBR−1BT , S21 := 4EW˜ ET
withM(tf)=Qf, V(tf)=0.Introducing
and
M 0 Qf 0 K := 0 V , Kf := 0 0 ,
we represent this system as a single generalized Riccati differential equation K˙ = − A˜ T K − K A˜ − Qˆ + K S˜ K + J K S˜ J K
+ KJS˜KJ − JKJS0JKJ − KSˆJKJ − JKJSˆK,
with terminal values K (tf ) = Kf , KfT = Kf . Here we used the same notation
as above for A˜, S˜, S0 and, additionally,
ˆ Q 0 ˆ S2 0
Q:= 0 0 , S:= 0 0 .
In this situation we can apply Theorems 3.6.10, 3.6.11 with
g1(K) = JKJS0JKJ−KSˆJKJ−JKJSˆK, g2(K) = JS˜KJ, g3(K) = JKS˜J,
where g1T(K) = JKTJS0JKTJ − JKTJSˆKT − KTSˆJKTJ = g1(KT), and g3T (K) = JS˜KT J = g2(KT ).
Finally we show the applicability of Theorems 2.5.10, 2.5.11 to coupled generalized Riccati differential equations occurring in Markovian Jump Linear Quadratic Control problems. In [AFJ94] we investigated the generalized Riccati differential equation
N−1
K˙ =−ATK−KA−Q+KSK−CiTKCi, K(tf)=0,
i=1
where Q, S are symmetric matrices. In order to apply Theorems 2.5.10, 2.5.11
we set here
andg2 =g3 =0.
N−1
g1(K) = CiT KCi, g1T (K) = g1(KT ),
i=1