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7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 1/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 2/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 3/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 4/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 5/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 6/32
Ω = ( ) = Ω ( )
=
+
(
)
+
(
)
= Ω
(
)
( ) = ( ) ( )
(
) = (
)
(
)
=
( ) = Ω
=
( ) = Ω
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 7/32
(Ω)
(Ω)
=
(
)
(
)
( )
= ( ) ( )
( )
=
(
)
(
)
(
)
= ( ) ( )
( )
( )
( ) =
(
)
+
+
+
+
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 8/32
( )
(
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 9/32
∆ =
∆ =
= ∆
= ∆
=
=
= (
)
=
(
)
=
=
= (
)
= (
) = =
=
(
)
=
(
)
=
= (
)
= (
) =
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 10/32
= ∆
∆
=
=
(
) + ∆
∆
=
=
(
)
+ ∆ ∆
=
=
(
) + ∆
=
(
)
+ ∆
=
(
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 11/32
=
(
)
+ (∆ )
+ (
+
+
+
) + (∆ )
∆
=
+ ∆
=
=
+ ∆
=
=
(∆
)
(∆
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 12/32
= +
=
=
[ ( )
+ (∆ )
+ (
+
+
) + (∆ )
]
+
=
+
=
[
+ ∆
]
+
=
(
∆
)
+
=
(
+ ∆
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 13/32
=
( )
+
+
+ (
+
+
+
+
)
+(∆ )
+
+
+ ∆ ∆
(
)
( )
+
+
+
+
+
=
+(∆
)
+
+
+ ∆
∆
(
)
( )
+
+
+
+
+
=
+(∆ )
+
+
+∆ ∆
(
)
+
+ ∆
∆
(
)
=
+
+ ∆ ∆
(
)
=
∆ ∆
(
) + ( )
+ (
+
+
)
=
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 14/32
=
∆
∆
(
) +
(
)
+
+
=
=
∆ ∆
(
) + ( )
+
+
= =
∆
∆
(
) +
=
=
∆ ∆
(
) +
= =
∆ ∆
(
)
=
=
∆ ∆
(
) +
= =
∆
∆
(
)
=
∆ ∆
(
) +
= =
∆ ∆
(
) = =
=
= = =
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 15/32
=
(∆ )
+
+ ∆ ∆
= =
=
=
=
(∆ )
+
+ ∆
∆
= =
=
=
=
∆
+ ∆
=
=
∆
∆
=
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 16/32
(
)
(
)
(
)
(
)
(
;
(
)
(
)
(
)
(
))
(
;
(
)
(
)
(
)
(
))
(
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(
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( )
=
(
)
( )
=
(
)
(
)
=
(
)
(
)
=
(
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 17/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 18/32
( ;
(
+
)
(
+
)
(
+
)
(
+
))
(
;
(
+
)
(
+
)
(
+
)
(
+
))
(
+
)
(
+
)
(
+
)
(
+
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 19/32
(
)
=
(
+
)
(
)
(
)
=
(
+
)
(
)
(
)
=
(
+
)
(
)
(
)
=
(
+
)
(
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 20/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 21/32
Ω = [
]
=
= [
]
[
]
=
= Ω ( )
(
) =
(
) =
(
)
=
( ) = Ω
= = Ω
(
)
=
(
)
=
(
)
=
(
)
=
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 22/32
(
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 23/32
(
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 24/32
(
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 25/32
(
)
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 26/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 27/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 28/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 29/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 30/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 31/32
7/31/2019 optimal Control : Wave equation
http://slidepdf.com/reader/full/optimal-control-wave-equation 32/32
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