Page 83 - Textos de Matemática Vol. 36

3.5. Example 73 3.5 Example
With the nomenclature of this paper we consider the following system, defining a positive system according to Theorem 3.1.3:
 −3.0 0 0 3.5   .5   1 0 0 0  A:= 0. −7 0 0 , B1:=0, B2:=0 1 0 0.
 0 . . 3 − 7 1 . 3   0 .   0 0 1 0 
5 0 1−16 0
The coefficients of the cost-functional for the first player are:
0002
and for the second player: 50000
Q2:= 0 4 0 0  0 0 5 0 
and
−40 0 0 0 R22:= 0 −1 0 0.
 0 0 − 1 0  0 0 0 −40

S1 :=

0−100 00−10 
5000 Q1 :=  0 0 0 0 
 0 0 0 0  0003
and R11 := −1,
0 0 0 0
This gives the matrices S1 and S2 as:

−1 0 0 0  40 
10
−.25 −0. −0. −0. −0. 0 −0. 0
 −0. −0. −0. −0.  −0. 0 −0. 0


 and S2:=  

Obviously, this game does not fulfill
Theorem 3.2.3. On the other hand
to condition (C), since the positivity of Qi, i.e. Qi ≥ 0, contradicts the demand of negative semidefiniteness of Qi there. Therefore positive system resemble an interesting class of systems from the gametheoretic viewpoint.
P1
Checking further the assumptions of Theorem 3.4.1, we see for P = P
. 0 0 0 −1 
condition (C) and hence we have to consider Theorem 3.4.1 does not work, if one is tied
2
with