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Lyapunov stability 2

  • 1. ! # $! " % & ' () *+ ' ' '+ , + $ D { x :V x 0, x * '0} -, .+ / 0 +, ! '+ ) ' + , " $D . + ' + * 0xe / + 0 - .+ , */ ' ( * ', '+ + " ** 00 0 ./ ' + + -, .+/ ) , ( )t ) , ' ' + ' + " 0 , 1 + ,1 +* * 0 ', '+ , + +, 1 '+ ( + - .+ " , 2 /) -) /) /) * , 1 '+ + x1 x1 2 x1 x 2 2 x2 3x x 1 2 4 x2 1 2 1 2 )# 4 3- ! #V " x x1 x2 ( +) V 0 0 'V x 0, x 0 2 2 * + ' V x x1 x1 x2 x2 2 2 x1 x1 2 x1 x 2 2 x 2 3 x1 x 2 4 x2 2 2 2 3 2x 1 8x 2 4 x x2 1 6x x 1 2 * 4x12 x2 , 1 * 4 * 2 x12 x2 2 0 4 x14 4 x12 x 2 2 x2 0 + 4 x12 x 2 4 x14 2 x2 ' * 4x1 x 23 , 1 * 6 * x1 3 3x2 2 0 x12 3 6 x1 x 2 6 9 x2 0 + 3 6 x1 x 2 x12 6 9 x2 ( ' * ' V x 2 x12 2 8x2 4 x14 2 x2 x12 6 9 x2 x12 2 7 x2 4 x14 6 9 x2
  • 2. x12 1 4 x12 2 4 x2 7 9 x2 $V x !# 4 + % 3- & ' D { x1 , x 2 : 4 x12 4 1 and 9 x 2 7} (* ., )0 -, .+ , $ + "D ‡ (. .5( + + "() ) ./ + " ( + 0! R n ( ( ) // . + 0 ' , x 6 / , ' t,x *' * ., ) t "0 7 8 "(+ x () ', '+' " +* ! , , ' 6 ) 9 !! " % 4 + * " 0 ,* 0 ' + '+' * 0 0 + ( +) 8 !* 0 %0 $* ' 0 8 . /V / :D !# 4 / 0: % 3- 4 ) * ' V x !% # 4 3- V x ) x * , 1 +, * + ; # 4 3- ! # " 2 x1 x2 2 V x 2 x1 x2 1 x1 x2 ,1 * x 8- 4<= x 2 x12 2 x2 !> 1 8 % 1? x + x1 0 * ' 2 x2 V x 2 2 x2 ) x2 1 x2 1= x + x2 0 * ' x12 V x x12 ) x1 1 x12 1> x + x1 x2 * ' 4 x12 V x 1 ) x1 1 4 x12
  • 3. 7 + , ?@ 0 ! # 4 3- ! # " +, + ; ‡ /# 4 3- ! #V " x $ ./ V x + ; =A V x ! # 4 + % 3- * 0 0 - .+ , , 4 " * -, .+ 8 . . + , 4 " * & ' + B7 - * ) ? ' = + ?> *' , ' ' + ' " +, + *( 1 + . 7 Vx / V x0 )x0 () ', '+ ' + , 4* * " " /) * ,1 '+ - + , x1 x2 x1 x12 2 x2 x2 x1 x2 x12 2 x2 1 2 )# - 34 ! # " V x x1 2 x2 ( + , C + 2 * + ' V x x1 x1 x2 x2 x1 x2 x1 x12 2 x2 x2 x1 x2 x12 2 x2 2 x12 2 x2 V2 0, x 2 $V x !# 4 + % 3- . + 0 x 2 ' 34 / ( # - ! #V x " + ; 8* + ?@ 0 * 0 ' + ! 0 + - .+ , ‡ * . ,x 0 ! * 0 0 %0 - .+ , * 0 '+ 0 + * ' * 0 8 1 '+ * 0 * + 0 / 0 , 0) * 0 ! x % , ' , x * ( "x ' t 0* " . ,$* * 0 ' + 1 ' - .+ , 8/ ! ' + / * 0 * % + 0 ' -, .+ ) , 1* 0 '+ * 0 + /x 0 * 0 0 x1 , x 2 x1e , x 2 e +
  • 4. 0 x2e x1e x12e 2 x2e * ' x1e x12e 2 x2e x 2e 2DE ' 0 x1e x 2e x12e 2 x2e * ' x 2 e x12e 2 x2e x1e 2FE , x1e , x 2e / 0,0 . 2DE / 2FE x1e x2e x12e 2 x 2e 0 2GE x 2e x1e ( + . 2GE ( + ( x1e , x2e . ) 0,0 )* + , H 8* 0 ' + * ( * . , 0 + 0 )0 * , 1 '+ - + , x1 x1 x 2 x2 sin x1 x13 u 7 ; ! #/( u " u x1 , x 2 $ ./ 0 0 * '+ + -, .+ 1 2 )# - 34 ! # " V x x1 2 x2 ( + , C + 2 * + ' V x x1 x1 x2 x2 x1 x1 x2 x2 sin x1 x13 u x12 x1 x 2 x 2 sin x1 x 2 x13 x2 u x12 x 2 x1 sin x1 x13 u / u u x1 , x 2 $ ./ 0 * 0 '+ + -, .+ () V 0 0 ,V x 0, x 2 {0} ) u u x1 , x 2 sin x1 x1 x13 x2 ./ , / * 4sin x1 , x13 , x1 !* V x x12 !# 4 + % 3-2 x2 'V x + ; 8* + ?@ 0! * 0 '+ 0 + - .+ , ‡ , ; '+ + 0 +( ( # 4 8 3- !" "# /) $ " "# $ ./ ' '+ * 0 ) + 0 ' + "+ + * ' x1 , x2 ' + ' ! ; 4 )' + ' " + * %" / + *
  • 5. ' 0 * . ,* ./ 1 8u x1 , x 2 0 2 5 1 + 0 . E 0 ' 5 ( ( ( '+ . ) + ' * 0 ) 9 !2 t 0 E ' '+' " 0 1 8 +* * 0 ' u x1 , x 2 0 2 5 1 + 0 . E * 5 ( ( ' + '"+ * +* (8 '+ * ,8 ' , : + (+ 0 + ( 8 $! , %:/ ++ 8 )* ) ( u sin x1 x1 x13 x2 7 4 * '+ " * ' + ! " % 8 x1 x1 x2 1 1 /) x Ax 7 x 2 'A x2 x1 x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
  • 6. ! ! () )/ 8% , ( * ! % ! .% )* ! - ( : %0 . , ' + /0 . 9 : 9 4 *. * ( , 4 ! - 8* / * % 6 (, 1 ; 4, : + 0,N / 8% 0 * ( + ": - , *, ! 5 ? '((# ) * &" + - , ' = ** +./ 0* ! 12 ! 33 3 ,* , 4% 5 ! . 6 + - 1' , 3 3 D 72 ! - ' +., 8:= <AB BB 8 J LM < 9=QR<SSNBRTI 9; > C EH E K 9N O P 9S9<IN?NGD < ? D F II O @ G AJ ; F R V<INJW NB 9< 9E R8 DH V; ?D> ^ H U SSNB 9XYZS :D? P P: NBI ]8 _ J = := [ 9 N 51 `a! " % 9; >@ C EH " ! $ 8:< <AB BB & $ 3 1 b c ' ) & ; = E Y ,,./& # =? DF G #% 1 1d ( * ; B N : -((1 [+ 0 2 , 8 W W E <"3 h b Z C8 9E H S< N d e 8 S< N fGD G_ , e 8 _ S @& d f Wg B ]WB R 8 B = Y 8 8 B = Y SR ;I i ^9 R 8 8 j i B ?X K_I B f ^I = Gi B # (& 6 & k dc1 Z T 1d k 5 7 4($ 4 1b 3 N ;