
()
()
()
1
1
11 1
1
Co , , ,
∂+
⎛⎞
⎜⎟
∂
⎜⎟
⎛⎞
⎛⎞⎛ ⎞
⎜⎟
∂+
⎜⎟
⎜⎟⎜ ⎟
⎜⎟
⎜⎟
⎜⎟⎜ ⎟
∈∀
∂
⎜⎟
⎜⎟
⎜⎟⎜ ⎟
⎜⎟
⎜⎟
⎜⎟⎜ ⎟
⎜⎟
⎜⎟
⎝⎠⎝ ⎠
⎝⎠
∂+
⎜⎟
⎜⎟
∂
⎝⎠
d
M
d
M
mmM
md
fx x
x
FF
gxx
GG
x
x
GG
gxx
x
(3.10)
where Co denotes the convex hull of polytope with M vertices defi ned in
(3.10), then the state trajectories x
~
(t) of the shifted gene network in (3.1) will
belong to the convex combination of the state trajectories of the following
M linearized synthetic gene networks derived from the vertices of the
polytope in (3.10) (Boyd et al. 1994, Chen and Wu 2009)
()
1
, 1, ,
=
=++ =
∑
m
jiiji
i
dx NF x v dt M G xdW E j M
(3.11)
By the global linearization theory (Boyd et al. 1994), if (3.10) holds,
then every trajectory ...