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and notice that
z
D
0
is an equilibrium. Select the Lyapunov function
.
z
/
D
z
T
J
K
.
0
/
1
z
:
V
(B.8)
Since
J
K
.
0
/
1
is also positive definite, conditions (a)and(b) of Definition A.4 are
satisfied. Let
z
.t/ be a trajectory of system (B.7), then
d
dt
V
.
z
.t//
D
z
.t/
T
J
K
.
0
/
1
z
.t/
C
z
.t/
T
J
K
.
0
/
1
z
.t/
D
2
z
.t/
T
J
K
.
0
/
1
z
.t/
D
2
z
.t/
T
J
K
.
0
/
1
J
K
.
0
/
J
g
.
x
/
z
.t/
D
2
z
.t/
T
J
g
.
x
/
z
.t/
D
z
.t/
T
J
g
.
x
/
C J
g
.
x
/
T
z
.t/ < 0;
unless
z
.t/
D
0
: Hence the zero equilibrium of system (B.7) is asymptotically stable
implying that all eigenvalues of its coefficient matrix have negative real parts.
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