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Commande adaptative floue des systèmes non linéaires par la méthode de backstepping utilisant le mode glissant

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Ω = ≤ ≥ ≥ 24 DG3 ε σ . 24 %E3 24 %J3 ≥ >ε & 1 24 %E3 σ ≥σ S S W W 2 2 8 3 2 33 2 2 8 3 2 33 -= θ − + θ − 24 DJ3 24 ;F3 C S S W W W W 2 3 2 2 8 3 2 8 33 2 2 8 3 2 8 33 ( ( θ θ θ θ -= Λ − + − + − + 24 DN3 W W 24 %E3 24 %H3 24 DN3 2 3 2 3 2 3 ( ( - ( φ ξ φ ξ = Λ − + + + 24 DF3

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&# S S 2 3 φ =θ −θ φ =θ θ− ξ : + % ; % % % ; ; ; + ) φ φ φ φ γ γ = + + 24 E&3 % ; γ ξ + % % ; ; % % 2 3 2 2 33 ; % 2 2 3 3 + * )( )( - )( )( φ θ γ ξ γ φ θ γ ξ γ = − − + + + + + 24 E%3 24 ;G3 φ =θ φ =θ 24 D%3 2 3>& 5 2 3 )( ≥& % )( 2 3 θ = −γ ξ 24 E;3 ; )( 2 3 θ = −γ ξ 24 ED3 2E%3 % ; + ≤ − * + )( - 24 EE3 ( )( -1 -=& = 9 9 W W 6 + & < % ; - -1 W W 24 %J3 C W W . 1 . W2 8θS3 W2 8θS3 θ θ

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&% 5 (6O E H 6 ! :&; 5 2 3 2 3 + 2 3 % % + 2 3 24 EF3 2 3 % % + / 2 3 0 24 H&3 / 0 ℜ =% ; , =% ; , 5 , =, =& 24 EF3 24 H&3 ! ; ! 2 3 # " / 8 W2 8θ 3 W2 8θ 3 24 %E3 24 %H3 5 # " 8 • = -0 % ' ' % % ' − ' + + + 9 *2 × 3 0 24 ;G3 $;E' & )> 0 ε + • 5 - 8 0 " µ ! %" =% ; " % ; = / 24 EF3 24 H&3 0 W2 8θ 3 W2 8θ 3 % ; " ×" × ×" % ; = W2 8θ 3 W2 8θ 3 + 2% ; 3 % % % % 2% ; 3 + W 2 8θ 3 ! 24 H%3

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(' 2% ; 3 % % % % 2% ; 3 + W 2 8θ 3 1 24 H;3 % ; " = =% ; !2% ; 3 12% ; 3 ℜ + 1 4O 24 H%3 24 H;3 4O 24 EF3 24 H&3 2% ; 3 ! 12% ; 3 0 M 2% ; 3 ! 12% ; 3 !2% ; 3 2% ; 3 1 = Ω Ω < # 24 EF3 24 H&3 0( 2% ; 3 % % % % % % 2 3 2 3 2 3 " " µ ξ µ = = = = =

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ξ2 3 " 9 % ; % ; " = = 2 3 2 3 % % ! 1 µ µ C ξ2 3 θ 2&3 θ 2&3 W2 8θ 3 W2 8θ 3 + W 2 8θ 3=θ ξ2 3 24 HE3 W 2 8θ 3=θ ξ2 3 24 HH3 • 5 - * $ % 0 6 2 /3 24 ;N3 24 %N3 24 ;;3 24 D%3 W2 8θ 3 W2 8θ 3 24 HE3 24 HH3 07 ! θ +

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(& ; 2 3 2 3 & & 2 3 & )( )( )( γ ξ ξ θ ξ − < = ≥ 24 HN3 2 3 ξ %" 2 3 ξ • 6

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; ; ; 2 3 2 3 2 3 ) γ )(ξ γ )(ξ γ )( θ θ ξ θ − = − + 24 G&3 # " + :&; ( 24 %N3 24 ;N3 θ θ ! 24 HG3924 G&3 % ; + % θ 2 3 ≤ θ 2 3 ≤ θ ≥ε %8 ; ; 2 3 " + λ ≤ + 24 G%3 %8 ; 2 3 %8 ; 2 3 % ; 2 3 % % ; 2 3 2 3 2 3 " " , + ' + ' ε λ ε λ ≤ + + + + + + + + 24 G;3 < ≥& λ ) 2 2 %33 " " " " − = ; ; ; & & 2 3τ τ ≤ +( -2 3τ τ $4 GD3 < ≥& ( -24 DJ3 D * ! = 9 9 ; & 2 3 -∞ < ∞ 2 3 & →∞ = & / # 8 5 # E 7

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(( 5 W W 24 %J3 ! σ & 8 • 5 -C # σ • 5 - 8 0 W2 8θ 3 W2 8θ 3 = 9 9 ; % % ; % % W 2 8 3 σ θ σ = = = = − − = − −

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(33)

() < 2 3 2 3 ! , , µ µ C 2 3 / µ µ 24 EF3 24 H&3

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/ µ µ 2&3 2 2&33 C 2 3 0 µ µ 24 DH3 24 DG3 • 5 - * $ % 9 W 2 8θ 3 θ ∂ ∂ + % W ( ( = ∂ = ∂ 24 GN3 ; % W ;2 3 W 2 3 ( ( σ = − − − ∂ = ∂ 24 GF3 ; D % W ;2 3 W 2 3 ( ( σ σ = − − − ∂ = ∂ 24 J&3 ; % ( σ = − =

− 24 J%3 24 GN3924 J&3 W 24 GE3 0 W 2 8θ 3 θ ∂ ∂ C 24 GN3924 J&3 > 0 /24 ;N3 24 %N3 24 ;;3 24 D%3 W2 8θ 3 W2 8θ 3 24 GE3 24 GH3 0 ! θ + • σ =σ % W W & W & & )( )( )( γ σ σ σ σ ∂ ∂ − < ∂ ∂ = ∂ ≥ ∂ 24 J;3 • 6

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( % ; % % W W 2 3 2 Z &3 W W W 2 Z &3 )( )( )( )( )( γ θ θ θ θ θ θ θ θ γ γ θ θ θ θ θ θ ∂ ∂ − < = ≥ ∂ ∂ = ∂ ∂ ∂ − + = < ∂ ∂ ∂ 24 JD3 0 ! θ + • =ε ; W W & W & & )( )( )( γ ∂ ∂ − < ∂ ∂ = ∂ ∂ 24 JE3 • σ =σ ; W W & W & & )( )( )( γ σ σ σ σ ∂ ∂ − < ∂ ∂ = ∂ ∂ 24 JH3 • 6 ; ; ; ; W W 2 3 2 Z &3 W W W 2 Z &3 )( )( )( )( )( γ θ θ θ σ θ θ θ θ γ γ θ θ θ θ θ θ ∂ ∂ − < = ≥ ∂ ∂ = ∂ ∂ ∂ − + = < ∂ ∂ ∂ 24 JG3 # & + :&; (6O > 9 + 0 θ ≤ σ θ ≥σ θ ≤ θ ≥σ 24 G%3 24 G;3 0 ; ; & & 2 3τ τ ≤ +( 2 3τ τ 24 JJ3 & ≥ ( 24 EN3 0 2 3 & →∞ = & ) [ 24 EN3 - ο θ

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(" , 2 4 24 JN3 24 %N3 # 5 2 3 2 3 D& " π = 2 3 < # , ; ; % % ; % ; ; ; % ; ; % ; % & &;H 2 3 FN % % 2 3 %H JN & &DG 2 3 ; & &H 2 3 E 2 3 D % % D " " " " " " − + + = ≤ = + = − − + 24 JF3 % ; % ; % ; % % 2 3 % 2 3 % EG 2 3 ; & &H 2 3 E % %2 3 2 3 D % % D " " " " " + = ≤ = = − − + 24 N&3 1 % ≤2 8 G3π % ; % ; ; 2 8 G3 % 2 3 % %; 2 3

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($ 2E3 E D % % ; ; % ; + 2 3 % 2 E3+ 24 NN3 2H3 H D % % ; ; % ; + 2 3 % 2 N3+ 24 NF3 2 % ; % ; H3 $ $ = = 2S3 Q 2S3 R %G 2 8 G3π =N N 2 8%;3π =E , % ; 2 3 % ; U 24 NH3924 NF3 ; = % ; H $ $= % → 2 % ;3 24 F&3 O 2&3= −2 π 8 G& &3 #

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0 • * . & ." 8 & ! . = 9 " )& 3 $ % . = . 4 1 = .+α., ./ ,--" / 3 ., /. α . 1 " & 3 . 4 .= 1+ . ,--"0/ 4 .= 1+α.+ . ,--":/ 1 & 1 . . 1 " . 1 . 1 , /=, /" ; & . α ." + α. , / 4 . . . ., /. . ., /. α = − − α α= ,--"</ = .= − . .+ 1 ,--">/ . . 1 .=,.? 1/ . ,--"@/ 1 .= . .= ., 1− . ./= − . . + . 1 ,--"A/ .= . 1 .= . 1 1 . . α

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: 3 .>2 " )& 3 . " B α. . & <2 . 1 & " ; 1" • 1 =α.+ 1 1 = 1−α.= + 1−α. ,--".2/ + & 1 & 1 α = − ,--"../ 3 1 α 1 1 = + 1−α.= +α α1− +. 1 ,--".1/ * . ., /. α α= . . . . . 1 . . . , / α α α α =∂ =∂ =∂ + ,--". / 1 1 1 . 1 1 . 1 1 . 1 = + = + ,--".0/ -. . 1 . 1 1 1 . . α β α ∂ ∂ = − = − ∂ ∂ . . 1 1 . 1 1 1 . . α β α ∂ ∂ = − = − ∂ ∂ 1 1 = − . . + 1 + 1, .+α1+β1/ ,--".:/ 1 1 1 . 1 1 1 1 1 . . 1 1 . 1 3 2 α = − − −β > = − − + ,--".</ . 1 1 . 1 1 . 1 . 1 1 . 1 1 α α α α α β ∂ = + − + = + − + ∂ = + + = − − + ,--".>/ + α11, . 1/" • = 4 0 α1 0 α1 = + = − = + − ,--".@/

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< + 0 & = & 0 0 = 0 −α ,--".A/ 3α = 0 α1 0 α α1 = + − = + + − ,--"12/ * α11, . 1/ 1 1 1 1 1 1 . 1 1 . 1 . 1 . 1 , / , / α α α α α α = ∂ = ∂ +∂ =∂ + +∂ + ∂ ∂ ∂ ∂ ∂ ,--"1./ = 1 1 1 . 1 = + = + ,--"11/ 1 1 1 1 . 1 1 . 1 . 1 α α β α α ∂ ∂ ∂ ∂ = − − = − − ∂ ∂ ∂ ∂ 1 1 . . 1 1 0 , 1 α β / = − − + + + + " ,--"1 / 1 α = − − −β 3 >2 ,--"10/ # 1 1 1 . . 1 1 0 = − − − + ,--"1:/ 0 2 0 0 0 0 α 2 0 0 α = + + = − = + + − ,--"1</ 1 1 1 1 0 1 0 1 . 1 . . 1 1 0 1 0 α α α α α α α α β ∂ ∂ ∂ ∂ = + − + = + − − − − + ∂ ∂ ∂ ∂ = + + = − − + ,--"1>/ B . 1 , / α =α " • = 0 4 0 2 0 0 0 0 2 0 0 α α = + + = − = + + − ,--"1@/ * 0 & C " + 0 α 0 0 = 2+ 0+ 0 0 = 0−α ,--"1A/ . 1 1 1 α . α 1 α

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> 7 & . 1 , / α =α . 1 . 1 1 . 1 0 . 1 , / , / , / α α α α α α α α ∂ ∂ ∂ = = + + ∂ ∂ ∂ ∂ ∂ ∂ = + + + + + ∂ ∂ ∂ ,--" 2/ * 0 1 0 0 0 0 0 . 1 = + = + ,--" ./ 0 0 . 1 0 0 1 0 . 1 . 1 α α α β α α α ∂ ∂ ∂ ∂ ∂ ∂ = − − − = − − − ∂ ∂ ∂ ∂ ∂ ∂ 1 1 1 0 = − . . − 1 1 − + 0, + 2+ 0 +β0/ ,--" 1/ 0 0 0 0 α = − − −β 3 0 >2 0 2 0 0 . ,α / 2 = − ≠ ,--" / # 0 = − . .1− 1 11− − 0 01 0 2 0 0 2 0 0 1 . . 1 0 . . . 1 1 2 0 0 0 0 α α α α α α α α β = + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + − + − − − − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + = − − ,--" 0/ 7 $ % " # & α " D & 4 2 . ,α / = − ,--" :/ ! " ! !# + & " E " ,θ θ / 3 θ E " * & , / "

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@ $ ! % # 4 1 , / 2 θ θ θ θ θ θ θ + + − = + − = ,--" </ 3 E ,--"./" * E E E , F " & $ ! % * " 4 1 , θ /, θ/ θ θ − θ+ = + ,--" >/ 3 " E 4 . 1 1 . 1 . , / , / ϕ ϕ = = + = ,--" @/ # 4 1 1 . . . 1 . . , / , / , / , / , / 0 , / ϕ − + = − + ,--" A/ θ

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A . 1 1 . , / , / , / 0 , / ϕ = + − + ,--"02/ • ) & . ξ . . ξ = − = − ,--"0./ # 4 1 . 1 . . , /ξ = ξ & 3 . . . , /ξ =ξ ξ <2 . . . . 1 , /ξ =ξ , − /=ξ , − / ,--"01/ 1 = − . .ξ + ,--"0 / & 3 1 . . . . , /ξ =ξ ,− ξ + + /= − ξ ,--"00/ +3 . >2 " • + E 1 ξ 1 1 1 1 . . ξ = − = + ξ − ,--"0:/ # = 1 1 . 1 . 1 . . , / 1 1 ξ ξ = ξ + ξ ,--"0</ . 1 . . 1 1 ,ξ ξ /=ξ ξ ξ ξ+ ,--"0>/ # 4 1 1 . 1 . . 1 . . . 1 ,ξ ξ /= − ξ +ξ φ, , /+ω, /+ −ξ ξ + ξ − / ,--"0@/ * 1 1 1 . . . 1 , / , / ω = − ξ ϕ− − +ξ ξ − ξ + ,--"0A/ 1 1 . 1 . . 1 1 ,ξ ξ /= −, ξ + ξ / ,--":2/ E = = . 1 ,ξ ξ /<2" 3 1 " = ,5#6/" = ,./ ,1/ ( E ,ξ ξ. 1/ . . 1 1 1 . 1 . . ξ ξ ξ ξ ξ = − = − = + − = − + − ,--":./

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02 . . . 1 ξ = − ξ ξ+ ,--":1/ 7 4 1 1 1 1 . . 1 1 . . . 1 . 1 ξ ξ ξ ξ ξ ϕ ϕ = − = + − = − + − + + ,--": / G 4 . . . 1 1 1 . . . 1 . 1 ξ ξ ξ ξ = −= − ξ++ ξ − +ϕ ϕ+ ,--":0/ 7 & 4 . 1 1 . . , / ϕ ω = = + = ,--"::/ # 1 1 , / , / ω ω =ϕ = ϕ ,--":</ + ,--".0/ ,--"11/ 4 1 1 1 . . . 1 1 1 , / , / , / 2 ξ ϕ ξ ξ ξ ϕ ϕ − − − + − + = ≠ ,--":>/ ' ! & 1 . . 1 , / = − − + + ( , " --"1/ & ,--" @/ 4 . 1 . . , / H 1" , /I ϕ = − − ,--":@/ 1 1, / H.?, " /I ϕ = ,--":A/ ' & " & ' ! , /

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0. ( ! ! ) F J " + , / ; ,--" /=,--"@/ 4 .2 2": . 2". = = = = ." .2 . , / ,1 / ,1 / = = = + 1" ,--" / ,--"</ E = = & ,--"0 --">/ & " + F E " + & & 9 2". . 1" & ) * ( " &

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01 & + , # & - ) * ( " & . - , "--": "--"@/ & &

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0 & / , # ' 0 " E ( " " ! E E E " F " F E " " ( ( E " % 1 % , K / & & " 6 & & , " --"A/ H1> ><I" 2 ! ! & " ( H1@I" # * * 6 6 6 & 3 ( ! %

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00 . ,"/ ! ϕ θ 1,"/ ! ϕ θ ,"/ ! ϕ θ + + + + + + 1 . 4 . 1 . . 1 . . 1 . . . 1 . . , / " , / " , """ / " , /" , / " ! ! ! ! ϕ θ ϕ θ ϕ θ β ϕ θ − − − = + = + = + = + = ,--"<2/ 3 ϕ 4ℜ ℜ" θ∈ℜ" " β, / , / 2 β ≠ ∀ ∈ℜ & & ." # & (

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0< 1 1 . 1 . 1 . . . 1 1 1 ! θ −θ = + + Γ ,--"<</ L θ = −θ 1 & 1 1 . 1 . . 1 1 . 1 . . 1 1 . . . . . 1 1 1 . L H I L ! ! ! ! ϕ α ϕ θ θ θ α θ ϕ ϕ θ − − = − − ∂ − Γ − ∂ ∂ = − − + Γ Γ − − ∂ ,--"<>/ & θ ! . . 1 1 . L , L/ α θ = Γτ θ = Γ ϕ −∂ ϕ ∂ ,--"<@/ & . . . . . 1 . 1 . . 1 . , / L , / , / τ ϕ α τ θ τ ϕ = ∂ = − ∂ ,--"<A/ # 1 =2 " 1 1 1 = − . . − 1 1 ≤2 ,--">2/ 7 & .= →∞ ., /= " & ( ! % 1 L θ & " & " ( ! ! ) F J " + ; ,--".1=--".>/ 4 . . ! ! ϕ α ϕ ∂ − ∂ . . ! ! ϕ α ϕ ∂ − ∂ . . . 1 − − − Γ . 1 L θ θ θ τ1

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0A & . B O & " & & , "--".>/ & & " (, # ) * 6 4 . 1 . 1 . , /! ϕ θ = + = = = " ,--">./ & ,--"<./" 1, . 1 L/ α θ & ,--"< / ( " + & = $ % & 4 1, . 1 L/ α θ = − − ,--">1/ 7 ! ,--"<@/ & 1 4 . 1 . 1 1 . 1 L L , / , / L α θ τ θ θ ∂ = Γ ∂ ,--"> / 1 τ & " # ,--":A/ ,--">1/ ,--"<A/ ,--"> / 4 . . . . . 1 1 1 1 . 2 . L . , / L ! ! ϕ θ α ϕ α τ θ θ − = + ∂ + ∂ + Γ − ∂ ∂ ,--">0/ & 4 1 1 . . 1 . . 1 1 1 1 1 1 L L , / , / L ! α τ θ θ τ θ θ − ∂ = − − + + Γ − + − Γ ∂ ,--">:/ & & ,--">1/ 4

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:2 , / 1 1 1 1 1 . 1 . L L , / L ! ! α α α α ϕ θ θ ϕ θ θ ∂ ∂ ∂ ∂ = − + − − − − ∂ ∂ ∂ ∂ ,--"></ 1 1 1 1 . 1 . 1 . . . . . 1 1 1 1 1 ! θ −θ = + = + + + Γ ,--">>/ & 4 1 1 . . . 1 1 1 , / 1 1 1 1 1 1 1 . 1 . 1 1 . L , / L L L , / L L , /" ! ! α τ θ θ α α α α α ϕ θ θ θ α θ τ ϕ −θ ∂ = − + + Γ − ∂ ∂ ∂ ∂ ∂ ∂ + + − + − − − − − ∂ ∂ ∂ ∂ ∂ ∂ + − − Γ ∂ ,--">@/ &

θ

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:. % % & " & P H1A 2I" + & & # # "L = H><I" & , / & ,G " --".@/" & / ( !2 % & & & " & " % % % * & !

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& ,--"A2/ ,--"A@/

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:: & 4 1 1 1 1 1 1 . . . . 1 . 1 . 1 . . . 1 1 1 1 1 1 . . . . 1 . 1 . 1 . . . . L . L , / 1 0 . L . L , / 1 0 0 ! ! ! ! ! ! ε ε ε θ λ φ θ θ θ ε ε ε θ λ φ θ θ θ = − − − − − + + + − + − ≤ − − − − − + + + − + − ≤ − 1 . . . 1 . 1 . 1 . . L . L , / ,--".2</ 0 ! ! ! ! ε ε θ λ φ θ θ θ − + + + − + − 4 . . 1 . . 1 , !/! τ λ φ τ = + = ,--".2>/ & ,--".2</ & 1 . . . . 1 . . . . L . L 0 ! ! ε ε θ τ θ θ τ θ = − − + + − + − ,--".2@/ • ( 1 & & & 4 1 . 1 1 . 1 1 . ! ) ε ε = + + ,--".2A/ 7 1 . 1 1 1 . ε ε! = + − ,--"..2/ )& ,--"11/ 4 1 1 . . . 1 . . 1 1 . 1 1 . . 1 . 1 . . . 1 . L . L . 0 . . L . L , / 0 ! ! ! ! ! ! ε ε θ τ θ θ τ θ ε ε ε ε ε ε θ τ θ θ τ θ ≤ − − + + − + − + − ≤ − − + + − + − + − ,--".../ # ,--"A0/ ,--"A@/ , .+ 1/ 4 1 . . 1 . . . 1 . 1 . . . . . . . . . . . L , / , / L L , / L L L L θ α α θ θ α ξ α λ α α α θ α θ ξ λ θ θ = − − = + ∂ = + − + + − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ − + + − − − ∂ ∂ ∂ ∂ ∂ ∂ ,--"..1/ # . . . . . . . . . . . . L L L L α α α α α α α α ξ λ θ θ ξ λ θ θ = ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + − − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ,--".. / R ,--".20/α. ξ1 λ1 θL & 4

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:< . . . . . . 1 1 1 1 L L α α α α α α ξ λ θ ξ λ θ ∂ ∂ ∂ ∂ ∂ = + + + + ∂ ∂ ∂ ∂ ∂ & ,--"..1/ . . . . . . 1 . 1 . 1 1 1 1 . . . . . . 1 . 1 1 1 1 L L L , / , / L L L L L L α α α α α θ θ ξ λ θ ξ λ θ α α α α α θ θ θ ξ λ θ ξ λ θ ∂ ∂ ∂ ∂ ∂ + = + − + + − − − − − − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + − + − − − − − − ∂ ∂ ∂ ∂ ∂ ,--"..0/ + ,--".21/ . . 1 . 1 . 1 1 1 1 1 1 . . . . 1 . . 1 1 1 L L L L L , / , L / L L ! ! α θ θ θ ξ λ θ λ θ θ θ α α α α ε φ θ φ θ ξ λ θ ξ λ θ ∂ + = + − + − − + + + + ∂ ∂ ∂ ∂ ∂ + + + − − − − ∂ ∂ ∂ ∂ ,--"..:/ M 1 1 ξ λ " # ! 1 & " + 1 L α = θ α1 4 . . 1 1 1 . 1 . 1 1 . 1 1 . . . . 1 1 1 1 1 1 1 1 1 L , , /L L / L ! α α α θ ξ λ φ θ θ α α α α ξ λ τ τ ξ λ θ ∂ ∂ = − − − − − − + + + ∂ ∂ ∂ ∂ ∂ ∂ − − − + − − ∂ ∂ ∂ ∂ ,--"..</ # 1 1 τ τ ( " & ,--"..:/ 4 . . 1 1 1 1 1 1 1 . 1 . . . 1 1 1 1 1 L , / , / L L L ! α θ λ θ θ ε φ θ α θ α τ τ α θ θ ∂ + = − + − + + + ∂ ∂ ∂ ∂ − + − − ∂ ∂ ∂ ,--"..>/ & 4 1 . . 1 . . 1 1 1 1 1 1 1 . . 1 . . 1 1 1 1 1 . . 1 1 1 1 . . . 1 1 1 1 . L , / L L 0 . . L . L L 0 ! ! ! ! ! α α ε ε θ λ θ θ ε φ θ θ θ α τ τ α ε ε θ τ θ θ τ θ θ α ε ε ∂ ∂ ≤ − − + − + − + + + − ∂ ∂ ∂ ∂ + − − − − + − + − ∂ ∂ ∂ ≤ − − − − − ∂ 1 . 1 1 1 . . 1 1 1 1 . . . 1 1 1 1 1 . . . L . L , / L . L . L ! ! ! ! α ε ε ε α τ θ θ α λ φ τ θ θ α τ θ θ τ θ ∂ + − − ∂ ∂ ∂ + − + − + + − ∂ ∂ ∂ − − + − + − ∂ ,--"..@/

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:> * & . 1 L α θ θ ∂ ∂ & 1 1 θ 4 . 1 1 1 . . . 1 1 1 . , !/! α τ λ φ τ α τ τ ∂ = − + + ∂ ∂ = − + ∂ ,--"..A/ & & 4 1 1 . 1 . . 1 1 1 1 . 1 1 1 1 1 . L L 0 . L . L . L ! ! α ε ε τ θ θ θ τ θ τ θ θ τ θ ∂ ≤ − − − − − ∂ + − − − + − ,--".12/ • ) ! 4 . . . 1 . 1 . 1 1 . 1 1 . . . . 1 1 1 1 1 1 1 1 1 . L L L , , / / L L ! α α θ ξ λ φ θ θ θ α ξ α λ α τ τ α ξ λ θ ∂ ∂ = − − − − − − + + + ∂ ∂ ∂ ∂ ∂ ∂ − − − + − − ∂ ∂ ∂ ∂ ,--".1./ . 1 1 . 1 . . 1 1 . 1 L , / L ! ! α θ τ λ φ α θ τ ∂ = = − + + ∂ ∂ = = − + ∂ ,--".11/ ) 4 1 1 1 1 . . 0 ! + + + ε ε = = ≤ − − ,--".1 / 7 = 1 < ∀2 , ε/≠2

,--"@A/ & ,--"A./"

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:A . 2 1 . 2 % = − ( − %2 S T 9 , & 1 . 1 2 + + = /" • + 4 . 1 1 . L θ α = − = − − ,--". / 3 α. " 6 )∈ℜ1 1× 2 2 2 ! ! )> ) =) , )% +% )= −* 4 1 1 . . . . . . . 1 1 1 ! ! ) θ θ θ ε ε = + + + ,--". 0/ & . . . . . . . L . L . . L . L . , / ! ! ! ! θ θ θ θ ε ε θ θ θ θ ε ε = + − + − − = − + − + − − ,--". :/ . 1 L1 ε1 ζ1, / λ θ1, / 1, /θ ε1 = = = + = + + + ,--". </ . . 1 1 1 1 . . . . 1 1 . 1 . 1 . 1 . . L . L . , / . . . L L L , / ! ! ! ! ! ζ λ θ θ ε θ θ θ θ ε ε α ζ λ θ ε θ λ θ θ θ ε ε = + + + − + − + − − = + + + + + − + − − ,--". >/ + & 4

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(106)

./ % ! 6 " # # # # 1284" 1/=4 5 # % " " ; # $ . =. # $ " K . > # " . . . = =. " . > * " " % > . = =. ." % ; ! $ < ' ) % 8 /9: 8 5 " . ' E ) 1/74<1/-4 5 # : ' 7) . . = =. =. − = '5; 7C) &" .= =. =.' −7)=. .' ) ' ) . . " . . = =. " ' -< ) 5 # .' " " ) # " # .' " ' )" ' )) % 8 / ; 5 17=4" < 7" ' ) ." 7" ' ) ." 7" / . . . . . . α α − > = − > ≤ − ≤ ≤ '5; 78) : . / α >α >

(107)

.0 ' ) ' ) ' ) ' " " ) ' " " ) ' ' " ))" . . . ∂ ∂ Λ ⋅ = ⋅ + ⋅ ∂ ∂ = Λ Λ Λ < '5; -.) &" >. . '78)" # <7".<.. : . . .= Λ Λ +. ∂ = Λ Λ −. α ∂ ∂ ∂ '5; -7) &" ' )" . ' )" . / . . . . . . α α α − > = − ≤ ! # # '-< ) < : 7 ∂. ∂ # E - # +' " )

[

." .

] [ ]

⊂ −7"7 = '5; 78) . ! " # . . . α ∂ > Λ Λ ∂ . .<. ' # . . α ∂ > ∂ ) ! $ + > $ , , .. '( 5; 77) 3 # . .<. # . ..< # * . . . α > % $ . . . α < * ) ' " ). . . .

(108)

. - ! < ' 9 J !) " " # $ E " 7 . 7 . 7 " 7" " 7 ' " " ) ' " " ) φ β + = = − = + '5; --) & φ.' )"⋅ β.' )⋅ 'β.' )⋅ )" '5; --) " $ # # " 1/24 6 & " " " I 9 J ! % + 1/34 # " $ # ρ −7 # 'ρ # # )" # " $ " # " " # " # .=." 0 # $ # " $ # < 9 J ! # 1//4 " 1/34 0 " 9 J < ' ) % " # " # # # # # " " # " # % " # # # # 0 " + # " -< < % < " <$< " 7 7 7 7 7 7 7 7 " " 7 7 ' " " " " " " ) ' ' ' )) + ' ' )) ) ' ' )) γ γ γ ρ ρ γ γ ρ ρ γ φ θ β θ φ θ + + + + + + − + + − + + + + = = + = + + '5; -=)

(109)

.) &" =7" " "ρ 0 < " 7 7 ' ' γ − ρ = = " ' )= 7' ) " " γ ' ) ∈ℜ " 7" " θ = θ θ ∈ℜ " φγ+7' ' ))∈ℜ 9 " 7"7 7"7 7" 7" " " "7 "7 ' ' )" ) + " + " " + " + " β β β β β + + = + + ' ' )" )" β ∈ℜ × " " " "∈ℜ β ' # ' ' )" )" β " " θ L θ) * " β' ' )" )" M # " β , # 7 ' ) ' )" 7" " " = γ + = " , ' , ) 6 " $ ρ # " - * < < , $ < < " ' 7) " 7 1γ + = γ+ − − −αγ+ − '5; -3) 7" " " 7" "ρ = = & αγ + −7 $ γ + −7" '"−7) 7 − " " " = γ +7 ' % ) 6 " " <$< " αγ + $ γ +

[

7" "

]

1= 1 1 $ #

[

." " .

]

! % $ # αγ+7 7 γ α + =7" " " ' ' )" ' )) α N < < $ γ + " <$< " ' ) 7 7 " 1γ + + = γ + + − −αγ + '5; -2)

(110)

.* - 7 7 7 7 L L ' ) ' ) - -- 1 γ γ θ θ θ θ + − + = = + − Γ − '5; -/) 7 7 7 7 7 7 7 7 7 7 " 7 7 " L L L 1 1 2 1 γ γ γ γ γ γ γ γ γ γ γ γ α α α θ θ α α τ θ + − + − + + − + + + + + = = + + + − + − + − = ∂ ∂ = − − + Γ − + ∂ ∂ ∂ ∂ + + ∂ ∂ '5; -A) 7 7 2γ+ φγ + αγ + − φ = ∂ = − ∂ '5; -C) 7 1 2 γ γ γ γ τ + = Γ + + +τ + − '5; -8) 7 7 7 7 7 ' ) L ' ) L L 1 1 1 1 2 1 2 γ γ γ γ γ γ γ γ γ γ θ α α τ θ θ θ + + − + + + + + − + + + + = = − − + + ∂ ∂ + Γ − − ∂ ∂ '5; =.) " =7" " " =." "ρ α" γ =."τ. =

[

." " .

]

∈ℜ γ+ " # τγ + $ γ + 9 # # # # $ γ + ' # αγ =.) ! # $ γ " 7 7 " 1γ + = γ + − : " # # 7 γ α + ( " # < ' +7)0 < τ - ( " " $ < " " # γ + " # " $ " " 7 ' ) L ' )=β' ' )" )θ α− ' ' ))+ ∗ ' ) '5; =7) &" 7 7 7 7 7 7 7 7 -7 7 7 7 '7 ) ' ' )" ) '7 ) " ρ ρ ρ γ γ α α β β β α β α β + + − − = − − − = ∂ ∂ − − ∂ ∂ = ∂ ∂ − − ∂ ∂ " β β' ' ))" ' ) ' 7) ' ) 7" ' )" " " ' ) ρ ρ ∗ = "

(111)

.+ 7 7 7 7 7 7 7 7 ' ) ' ' )) ' ) ρ ρ γ γ α α β α α α β + + − = − = ∂ − = ∂ − * " '5; -2)<'5; -8) % " # $ 7 7 γ + − " αγ " 7 γ + + -7 7 7 7 ' ) "7 " 7 7 7 "- "7 1 ρ γ+ αγ+− γ+− γ+− = − − + = '5; =-) & 7 7 γ+− $ # # 7 7 1γ +− " " # " " 6 " 7 7 7 7 7 7 7 7 7 - 7 7 7 "7 7 3 -7 7 7 L ' ) ' ) L ' ) L L 1 1 1 2 1 2 γ γ γ γ γ γ ρ γ γ γ γ γ θ θ α α τ θ θ θ + + + + + + + + − − − − − − − − + − + − = = − − + + + − ∂ ∂ − − + Γ ∂ ∂ '5; ==) ! " : L L L L L ' " ) ' " ) ' " ) L L ' " " " " ) ' " " " ) 1 3 1 1 4 1 5 1 6 7 1 8 1 θ θ θ θ θ θ θ θ θ = + + + = + '5; =3) 9&" 7 7 7 -7" " 7" 7" " 7 " 7"7" " "7" 7"-" " "- " " " " " 1 1 1γ 1γ 1γ 3 7 8 4 5 +− ++ +− = = ∈ℜ 6 # 0 : @7 7 L 4 1' " )L 1 γ θ τ= = Γ θ '5; =2) & 7 7 γ τ +− $ ' −7)" #

[

." " .

]

γ τ = ∈ℜ " # '5; =3) $ " L L ' " ) ' " ) 1= 3 1θ 1 4 1+ θ θ +6 '5; =/) > '5; =3)" " 7 L L ' ' ) ' )) -- = 1 1+ −θ θ Γ −θ θ '5; =A) - '5; -2)< '5; =.)" θL Γ4 1' " )θL 1 :

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