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ξ
(
x,t
) =
ξ
−
(
x,t
)
ξ
+
(
x,t
)
.
With the new coordinates in terms of Riemann invariants, (11) is rewritten as
∂ξ
∂t
+
Λ
(
ξ
)
∂ξ
∂x
=
0
(14)
where
Λ
(
ξ
)=
diag
(
λ
1
−
(
ξ
+
,ξ
1
)
,λ
2
−
(
ξ
+
,ξ
2
)
,λ
1
+
(
ξ
+
,ξ
1
)
,λ
2
+
(
ξ
+
,ξ
2
)).
−
−
−
−
The initial conditions (12) are equivalent to
ξ
(
x,
0) =
ξ
0
(
x
)
(15)
The boundary conditions (13) are expressed in terms of Riemann invariants by
ξ
−
(
L,t
)
ξ
+
(0
,t
)
=
g
ξ
−
(0
,t
)
ξ
+
(
L,t
)
(16)
where
g
=(g
2
,
g
4
,
g
1
,
g
3
)
T
is a suitable function. Therefore (14), (15) and (16)
constitute the characteristic form of (11), (12) and (13) for cascaded open chan-
nels.
3.2 Stability Analysis
Theorem 2
[6]. Consider the Saint-Venant equations (14) with initial condi-
tions (15) satisfying the boundary compatibility conditions and the boundary
conditions (16). If
g
(
0
)))
<
1 (17)
and
ξ
0
1
is sucient small, then there exists a unique continuously differen-
tiable solution on (
x,t
)
ρ
(abs(
∇
∈
[0
,L
]
×
[0
,
+
∞
) which satisfies the estimate
C
1
[0
,L
]
<Ce
−αt
ξ
0
C
1
[0
,L
]
,
ξ
(
·
,t
)
∀
t
≥
0
where
C>
0and
α>
0.
In Theorem 2, we observe that the condition (17) is only related to the Jaco-
bian of
g
at the origin. Therefore we develop one specific form of the boundary
conditions with stability guarantee as the following conclusion.
Conclusion 2.
If the parameters
m
i
,(
i
=0
,
1
,
2) are chosen properly to satisfy
the stability condition (17) and the boundary conditions (16) are replaced with
the following explicit expressions
⎧
⎨
V
1
(0
,t
)=
V
1
−
g
A
1
b
1
(
A
1
)
(
A
1
(0
,t
)
A
1
)
1+
m
0
1
−m
0
−
1
−m
i
A
i
)
,
(
i
=1
,
2)
,
(18)
V
i
(
L,t
)=
V
i
+
1+
m
i
⎩
g
A
i
b
i
(
A
i
)
(
A
i
(
L,t
)
−
the result of Theorem 2 still holds.
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