Page 78 - Textos de Matemática Vol. 36

68 Chapter 3. Numeric solutions K1
a Nash equilibrium is the existence of a solution K to the algebraic Nash 2
Riccati equation:
AT 0 X1 X1 Q1 X1
X1
S1 S2 X , (3.3)
0=− 0 AT X − X A− Q + X 22222
where Si := Bi R−1 BiT for i = 1, 2, which is moreover stabilizing, i.e. ii
A − S1K1 − S2K2 is a stable matrix.
Remark 3.2.2. The given results are adapted to our setting of positive systems, since usually the definition of the Nash strategy is given by minimizing the cost functional, where negative (semi-) definiteness has to be substituted by positive (semi-) definiteness. This means that concavity has to be replaced then by convexity.
As we see later the concavity approach, especially the assumption on the matrices Qi, is not appropriate in the case of positive systems. Therefore we use an approach presented in [Krem03] (see also [AFIJ03] Chapter 9 and [KrSt02]), which relaxes the conditions on Qi. For our use we cite a weaker version to avoid the concept of playability:
Theorem 3.2.3. The open loop Nash game, as defined above, with stable matrix A has a unique Nash equilbirium if the following conditions are satisfied:
1. Rii are negative definite for i = 1, 2.
2. The standard algebraic Riccati equations
Qi +ATXi +XiA−XiSiXi =0, i=1,2, (3.4) have for i = 1, 2 a stabilizing solution Ei, i.e. A−SiEi, i = 1, 2, are stable.
Here again Si := BiR−1BiT . ii
3. The algebraic open loop Nash Riccati equation (3.3) has a (unique) left-
K1
K
right stabilizing solution
feedback form as
u∗=−R−1BTKx∗, i=1,2, (3.5) i iiii
with x∗ being the solution of the closed loop equation
x˙ =(A−S1K1 −S2K2)x, x(0)=x0.
A−S1K1 −S2K2 are both stable.
−K1S2 AT −K2S2
2
, i.e. it fulfills
AT −K1S1
and −S1K2
In this case the Nash equilibrium strategy (u∗1,u∗2) can explicitly be given in