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ξ
+
(
x,t
)
|
x
=0
=g
0
(
ξ
−
(0
,t
))
ξ
−
(
x,t
)
(7)
|
x
=
L
=g
L
(
ξ
+
(
L,t
))
where
ξ
+
(
x
)=
2
(
v
0
(
x
)+
a
0
(
x
)
−
a
0
(
x
)
p
(
α
)
dα
),
ξ
0
−
(
x
)=
2
(
v
0
(
x
)
0
p
(
α
)
dα
),
g
0
(
u
)andg
L
(
u
) are continuously differentiable functions of
u
in a neighborhood
of 0
0
R
, and the initial functions
ξ
+
(
x
)and
ξ
0
∈
(
x
) also satisfy the corresponding
−
boundary compatibility conditions.
2.3 Stability Analysis
Theorem 1.
Consider the Saint-Venant equations (5) with initial conditions (6)
satisfying the boundary compatibility conditions and the boundary conditions
selected as
ξ
+
(
x,t
)
|
x
=0
=
m
0
ξ
−
(0
,t
)
,
ξ
−
(
x,t
)
|
x
=
L
=
m
1
ξ
+
(
L,t
)
.
(8)
Suppose that
|
m
0
m
1
|
<
1
(9)
(
a
0
(
x
),
v
0
(
x
))
Then if
||
1
is suciently small, there exists a unique continuously
differentiable solution (
a
(
x,t
),
v
(
x,t
))
T
on (
x,t
)
||
∈
[0
,L
]
×
[0
,
+
∞
) to the problem
that is defined for all positive
t
and satisfies the estimate
1
<Ce
−αt
(
a
0
(
x
)
,v
0
(
x
))
1
(
a
(
·
,t
)
,v
(
·
,t
))
where
C>
0and
α>
0.
Proof:
This Theorem is a generalized result of Theorem 2 in [12].
According to Theorem 1, we can develop the boundary control laws for the
trapezoidal canal as follows.
Conclusion 1.
If the boundary conditions are designed by the following bound-
ary conditions
1+
m
0
1
g
Ab
(
A
)
a
(0
,t
)
,
V
(0
,t
)=
V
−
−
m
0
g
Ab
(
A
)
a
(
L,t
)
,
V
(
L,t
)=
V
+
1+
m
1
1
(10)
−
m
1
|
m
0
m
1
|
<
1
,
the result of Theorem 1 still holds.
3 Modelling and Control of Multi-reach Open Channels
3.1 Modelling
The aim of this section is to generalize the idea of the boundary control of a
single canal to the open-channel networks made up of multireaches in cascade.
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