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optimal Control : Wave equation

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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

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7/31/2019 optimal Control : Wave equation

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7/31/2019 optimal Control : Wave equation

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7/31/2019 optimal Control : Wave equation

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7/31/2019 optimal Control : Wave equation

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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

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( )

(

)

7/31/2019 optimal Control : Wave equation

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∆ =

∆ =

= ∆

= ∆

=

=

= (

)

=

(

)

=

=

= (

)

= (

) = =

=

(

)

=

(

)

=

= (

)

= (

) =

7/31/2019 optimal Control : Wave equation

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= ∆

=

=

(

) + ∆

=

=

(

)

+ ∆ ∆

=

=

(

) + ∆

=

(

)

+ ∆

=

(

)

7/31/2019 optimal Control : Wave equation

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=

(

)

+ (∆ )

+ (

+

+

+

) + (∆ )

=

+ ∆

=

=

+ ∆

=

=

(∆

)

(∆

)

7/31/2019 optimal Control : Wave equation

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= +

=

=

[ ( )

+ (∆ )

+ (

+

+

) + (∆ )

]

+

=

+

=

[

+ ∆

]

+

=

(

)

+

=

(

+ ∆

)

7/31/2019 optimal Control : Wave equation

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=

( )

+

+

+ (

+

+

+

+

)

+(∆ )

+

+

+ ∆ ∆

(

)

( )

+

+

+

+

+

=

+(∆

)

+

+

+ ∆

(

)

( )

+

+

+

+

+

=

+(∆ )

+

+

+∆ ∆

(

)

+

+ ∆

(

)

=

+

+ ∆ ∆

(

)

=

∆ ∆

(

) + ( )

+ (

+

+

)

=

7/31/2019 optimal Control : Wave equation

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=

(

) +

(

)

+

+

=

=

∆ ∆

(

) + ( )

+

+

= =

(

) +

=

=

∆ ∆

(

) +

= =

∆ ∆

(

)

=

=

∆ ∆

(

) +

= =

(

)

=

∆ ∆

(

) +

= =

∆ ∆

(

) = =

=

= = =

7/31/2019 optimal Control : Wave equation

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=

(∆ )

+

+ ∆ ∆

= =

=

=

=

(∆ )

+

+ ∆

= =

=

=

=

+ ∆

=

=

=

7/31/2019 optimal Control : Wave equation

http://slidepdf.com/reader/full/optimal-control-wave-equation 16/32

(

)

(

)

(

)

(

)

(

;

(

)

(

)

(

)

(

))

(

;

(

)

(

)

(

)

(

))

(

)

)

(

)

)

( )

=

(

)

( )

=

(

)

(

)

=

(

)

(

)

=

(

)

7/31/2019 optimal Control : Wave equation

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7/31/2019 optimal Control : Wave equation

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( ;

(

+

)

(

+

)

(

+

)

(

+

))

(

;

(

+

)

(

+

)

(

+

)

(

+

))

(

+

)

(

+

)

(

+

)

(

+

)

7/31/2019 optimal Control : Wave equation

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(

)

=

(

+

)

(

)

(

)

=

(

+

)

(

)

(

)

=

(

+

)

(

)

(

)

=

(

+

)

(

)

7/31/2019 optimal Control : Wave equation

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7/31/2019 optimal Control : Wave equation

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Ω = [

]

=

= [

]

[

]

=

= Ω ( )

(

) =

(

) =

(

)

=

( ) = Ω

= = Ω

(

)

=

(

)

=

(

)

=

(

)

=

7/31/2019 optimal Control : Wave equation

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(

)

7/31/2019 optimal Control : Wave equation

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(

)

7/31/2019 optimal Control : Wave equation

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(

)

7/31/2019 optimal Control : Wave equation

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(

)

7/31/2019 optimal Control : Wave equation

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7/31/2019 optimal Control : Wave equation

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