further analysis, it will be convenient to express our system and its variables in the vector
form. Let:
2 3
pz =6 7; (5.15)
4 5
q
and
z_ = f(z) + ug(z) (5.16)
where 2 3 
pln p
f(z) =6 q 7 (5.17)  4 5
2bp  + d p
3 q
and 2 3
0g(z) =6 7: (5.18)
4 5
Gq
Notice that 2 3   
 ln pDf(z(t)) =6 q + 1 bq p23 3 pd 7; (5.19)
4 52
p 3)q dp(
and 2 3
0 0Dg(z(t)) =6 7: (5.20)
4 5
0 G
In this thesis, h
;
i denotes the standard inner product on R2. In order to simplify the
calculations, we will make use of the following proposition.
Proposition 13 [5] Let h be a continuously di¤erentiable vector …eld and de…ne
(t) = h(t);h(z(t))i: (5.21)
Then the derivative of along a solution to the system equation (5.16) for control u and a
28
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