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APPENDIX
Proof of Theorem 1: The following Lyapunov-Krasovskii functions are well constructed
( )] (17)
( ) = ( ) + 1 ( ) + 2 ( ) + 1 ( ) + 2
where
( ) = ( ) ( )
∫
( ) 1 ( )
( ) = 2 ( − )
1
−
∫
−
( ) 2 ( )
( ) = 2 ( − )
2
−
∫ ∫
0
¤ ( ) 1 ¤( )
( ) = 2 ( − )
1
− +
∫ ∫
−
¤ ( ) 2 ¤( )
( ) = ( − ) 2 ( − )
2
− +
By differentiating the above designed functionals along with ( ), we can obtain
¤
( ) = 2 ( ) [ ( ) + ( − ( ))
− ( ℎ) + ( )] (18)
¤
1 ( ) = −2 1 ( ) + ( ) 1 ( )
− −2 (19)
( − ) 1 ( − )
¤ ( ) + −2
2 ( ) = −2 2 ( − ) 2 ( − )
− −2 (20)
( − ) 2 ( − )
¤ 2
1 ( ) = −2 1 ( ) + ¤ ( ) 1 ¤( )
∫
¤ ( ) 1 ¤( )
− 2 ( − ) (21)
−