Практикум по численным методам в задачах оптимального управления

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Author: К.Г. Григорьев | И.С. Григорьев | М.П. Заплетин

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@ $N ':#%e !; N$':#%e #7 2( ##.I6 ); @lN$':#%e !; N$':#%e #7 2( ##.I6 ); @ &x( #7;\ %" :M!N$':#%e !;L,D"[ e 2( ##.I6 - ?':#%e L,DE,D"7;bh= ) %" :M!N$':#%e b ,D"[ e x(·) L#7& 4 .REb" ,c; ON$':#%e ,DE,D"7;bh= %" :M!N$':#%e bW,D"[ e u(·) 9@ ! Q # # V V - #4 .RE2( 9 ,D.RE "+ , , (9" E2( G':;2( H1A !& kZ # G`NP4 E( E,D"#,D" K n .R#!; " ,c; +E,DD6O" *%!76 (#5kZ2,D"E ∆0 c1A$': # u(·) # .R # 515* V T M V Ym-1,D"7; "l,D [ Jp42515*'p( # ( 427& e ON$':#%e #7A B0(x(·), u(·), t0 , t1) V T #\(#5kZ2,D" Z _=D& # Jm':;2( J1A # ( *2,D%: J ,D ,D"2(. V V Z,G # * # !;!& ( V S #$': # R H" ( #7 #& 4 ( " *2,D% ( V V : R × Rn × Rr → R, 0 ≤ ν ≤ m + k, (fν 1+n+r ψν : R × Rn × R × Rn → R, 0 ≤ ν ≤ m + k, (ψν 2n + 2 ϕ : R × R n × Rr → Rn (ϕ n 1+n+r fν

2 +9C:Z#, )M, ,.$$9C: (%)*+!/ : ;(%$K%Y/: >, )M,.$Y !#"%$(%N5)*J!+!+!/*-, ." $/:

-F

* ,D"#E,D" !" ( #7 #./( L 5 ! \ 4 ( " *2,D% ( I 7& # * # !;( " V !- V Y- " ( " (I\*"l42515* (9%!,D ( 42e N$':#%e #7AC I,D 21A " ,c; %^ , , (9"7& ) → E2( Jq42515*Bd0(x(·), 42 ( #u(·), Jq4 t#0 ,t%!1^ -1 N$':#%e #7E(IPP,D%:& (./( d42515* V T M V YZ;;b" ,c; '!( #". N$':#%e #7A @p %" :M!N$':#%e #5*7 #.RJ t0 B0 (x(·), u(·), t0 , t1 ) K%: # *#.RJ t (E( #". 2( # :$-1Ax(·) A Du(·) " ,c;*"= _= # 42515* V T M1 V Y ,D':h=2,D" ' "5 515*' V T M V Y,N$':#%e #7A ( j eP(5kH#Z,D 2,D" H% 42515*$,PN$':#%e #7A ( L+J !, 1ADA42 ( #'K 2( ##.I6C Z

t t0

ED

fi (t, x(t), u(t))dt ≡ xn+1+i (t), 0 ≤ i ≤ m + k.

':* "E( ,--1A':bh= !6m 4 " J 42 ( #.p 2( ##.I6m * :1A#.I6 #,D"C x (t ) = 0 x (t ) = R t f (t, x(t), u(t))dt 427& 15* V T M n+1+i V Y,HN$0 ':#%e #n+1+i 7A ( 1j et\i J:1A "LL42515*'`, N$':#%e #7A ( L+J C 1

0

T0 (˜ x(·), u(·), t0 , t1 ) →

;

x˜˙ (t) − ϕ(t, ˜ x˜(t), u(t)) = 0 ∀t ∈ ∆0 ; u(t) ∈ U (t) ⊆ Rr

Ti (˜ x(·), u(·), t0 , t1 ) xn+1+i (t0 )

c1A

= 0, = 0,

∀t ∈ ∆; 1 ≤ i ≤ m; 0 ≤ i ≤ m + k;

Tm+j (˜ x(·), u(·), t0 , t1 ) ≤ 0,

1 ≤ j ≤ k;

(1.00 ) (1.10 ) (1.20 ) (1.30 ) (1.40 )

Tν (˜ x(·), u(·), t0 , t1 ) ≡ xn+1+ν (t1 ) + ψν (t0 , x(t0 ); t1 , x(t1 )), 0≤ν ≤m+k

x ˜(·) ≡ (x1 (·), ·, ·, ·, xn (·); 0, 0, ·, ·, ·, 0)T + +(0, ·, ·, ·, 0; xn+1 (·), ·, ·, ·, xn+1+m+k (·))T ;

ϕ˜ ≡ (ϕ1 (·), ·, ·, ·, ϕn (·); 0, 0, ·, ·, ·, 0)T +

VS

+(0, ·, ·, ·, 0; f0 , f1 , ·, ·, ·, fm+k )T .

( " (I7*"$ Z D6E21A "o42515* V T M V Y%Z42515* V T 0 M V Y 0 H %" :M!N$':#%e !; ': # J E,D"7 " ,c; # 42( ## J O %" :M!N$':#%e !; NP4 .I6d 2( #u(·) #.I6m': D * E "O42( !& #E,D"n# m + k + 1 I515* m " (97 # ': # !; V T M V Y/ V T 0 M V Y 0 % E7 #"#.P $21 % E7 #"#E,D" bq& # (9 " ,c;n,D 51A # =1; " !6O42515*O% .I6`42515*O #e (9%!,D ('!(9 , (I2# !kZ -D
! .R#!;b" ,c;\ # * # !; V !- V Y- u ,- \(#5kZ2,D" $1A 'E,D" (.I6K':;2(.I6H e2, ,D ?#o'E& ,D""G 4 # %! "42515*Z .j1AD # !;a 4 " G(#5kZ2,D"E j':;2( P e2, , 51AE,D"7;bh= ?( # (97 # 4 #5* # PN$':#%e #7' B <= 'E,D" (.RJ`':;20(.RJ8 e2, , ξb ≡ (bx(·), ub(·), bt , bt ) #7& 0 1 4 .RE " ,c; PA1AE,D"7;bh= ( N$':#%e #7' B (ξ) ,D ! #.RJ +( # ('!( ! P0 ! -52,- ,D':h=2,D" ' " "7%: 5*"1;ZE,c;!& %: K1A 'E,D" ( L':;2( G e2, δ, > 0ξ ≡ (x(·), u(·), t0 , t1) 1;L%: "

`D D

)-Y#,!J!J.'

GD

(%)*+!/-, :9C

(QJ.:9C D

!#"%$&' ,#:KQ9C! #"%$)- 9 N #Y#,!KQJ!J.!'#" %$GW!##"%KQ$!#"9C%$ +J.#"%$'! #"%J $:9C9; J -, (%5)*+!/# !#"%$9C )-Y#,!J!J.'

k(x(·), (t0 , t1 ))−(b x(·), (b t0 , b t1 ))kC(∆,Rn )×R2 ≡ q ≡ max( max (max |xi (t)−b xi (t)|), (t0 − b t0 )2 + (t1 − b t1 )2 < δ, 1≤i≤n t∈∆

.R#!; " ,c;m'E,- B0 (ξ)b ≤ B0(ξ) [ " (0 # (9# *" ? -1AD # H %!7 # J " (97 #E,D" H':;2( P e2,-& , , , (9" Eb" ,c;m1A 'E,D" (.R+':;2(.Ra e2, ,D. [ ξb4 % HaNP4 J+ 2( ## J x(·) O 2( # ( t t #L#ξ 0 1 ': # b u(·) 42515* V T M V Y#4 .RE " ,c;ZN$':#%e !;

(%$K%Y#,.N [&H)*$

L ≡ L(x(·), u(·), t0 , t1 , p(·), λ, λ0 ) ≡

Z

t1

Ldt + l. t0

V!

?':#%e !;

#4 .RE " ,c;

L

-&H)*$ #$'PC

L ≡ L(t, x, x, ˙ u, p, λ, λ0 ) ≡ Pm+k ≡ i=0 λi fi (t, x, u) + hp, x˙ − ϕ(t, x, u)i ≡ hp, xi ˙ − H;

N$':#%e !;

H

#4 .RE " ,c;

;(%$K%Y#,.NSA]$`)*/-&'$#7C

H ≡ H(t, x, u, p, λ, λ0 ) ≡ hp, ϕ(t, x, u)i −

N$':#%e !;

l

7P, ):$#$'PC

#4 .RE " ,c;

l ≡ l(t0 , x(t0 ); t1 , x(t1 )) ≡

m+k X i=0

m+k X

λi fi (t, x, u);

i=0

λi ψi (t0 , x(t0 ); t1 , x(t1 )).

Z:$ P, /: [

v$ ,- ∈ R * ,- J %" λ ≡ (λ1, ·, ·, ·, λm+k ) n %" :M $N ':#%e !λ; 0 p(·) 1 n∗ #4 .REb" ,c; 7 NP∈( KC " *2(∆, ,D%: Rn−) ( # ?E,D"#,D" Rn∗ ,D E;!& kZ ## /, Rn -1,D"7; "P,D [ JZE,D"#,D" n− ( #.I6Z %!& " =,D" %:$ "E(' p(·) ∈ Rn∗ @ n− ( #7;K %" :M!N$':#%e !; ,D" %! x(·) ∈ Rn @ n− ( #7; %" :M!N$':#%e !; ,D"[ e - P 1;\b[ .I6 n∗ n hp, xi ≡ ni=1 pi xi im17 # J_=2(n[ ' 1A':" ,Dp 4 ∈ ER"E,c;Kx ∈[ R4 #5* # !;C

&H)*$

%D

ϕ(t) b

ϕ bx (t)

b x (t) L

b x˙ (t) L

b H(t) b ∂ H(t) ∂x

V Y

≡ ϕ(t, xˆ(t), u ˆ(t)); ∂ϕ(t, x, u) ≡ x = xˆ(t) ; ∂x u=u ˆ(t) ∂L(t, x, x, ˙ u, p, λ, λ0 ) ≡ ∂x ∂L(t, x, x, ˙ u, p, λ, λ0 ) ≡ ∂ x˙

x=x ˆ(t) ... ˆ0 λ0 = λ x=x ˆ(t) ... ˆ0 λ0 = λ

ˆ λ ˆ 0 ); ≡ H(t, xˆ(t), u ˆ(t), pˆ(t), λ, ∂H(t, x, u, p, λ, λ0 ) ≡ x = xˆ(t) ; ∂x ... ˆ0 λ0 = λ

;

;

dLb dtk ∂ ˆl ∂x(tk ) ∂ ˆl ∂tk

Lfˆ"5(t) 1

≡

dL(x, u, t0 , t1 , p, λ, λ0 ) dtk

∂l(t0 , x(t0 ); t1 , x(t1 )) ≡ ∂x(tk ) ≡

∂l(t0 , x(t0 ); t1 , x(t1 )) ∂tk

≡ f (t, x ˆ(t), u ˆ(t)) ≡

x=x ˆ(t) ... ˆ0 λ0 = λ

,

t0 = tˆ0 ... x(t1 ) = x ˆ(tˆ1 )

k = 0, 1;

,

k = 0, 1;

, k= t0 = tˆ0 ... ˆ(tˆ1 ) 1) = x Pm+kx(t ˆ i fi (t, x λ ˆ(t), u ˆ(t)); i=0

0, 1;

V#

{ {

k f tfqp acom$knbDiHjfDrBaz dk[rDfDwxf df [zp ^ ] ixajvBlfDwxf jfB^ ]jvBlfDwxf c9a`l`a c9b co] b'l`^ `afBl]j] bDiHjfDrBaz d`[`a`l `a`d] co]B^ ixac9b co] r s ]Ap&]Mgk fBd`_Aa co]jvBlfDwxfyb'd`[]Drjkl`az { { O Q VACAEAY AEAV U E U

*

,+

"!$#&%

.-

('

,-0/1-

%

)

32

$ 'E,D" ξb ≡ (bx(·), ub(·), bt0, bt1) @q %!7 #? " (97 #.RJGZ,D ! #E( , (./,-H':;2(.RJ` e2, ,ZL42515* " (97 # \':D& # !;$ V T M V Y- 1AE,D"7;bh= JN$':#%e #7' B (ξ) ,D ! #.RJ=& %!7 #.RJZ( # ('!(I N$':#%e f (t, x, u) 0 ≤ i ≤0m+k - ϕ(t, x, u) O !68* ,D"#.RH 4 21A#.RZ i x # .R #.q+,D %':#E,D" 6E21;h= !6=?# !6H 2( ##.I6$ t x u 9?# %: " J %2,D"#& ,D" L " (97 # J\" %" #H(#5kZ2,D" n o V (t) ≡ (t, x) |x − xˆ(t)| < ε, ε > 0, t ∈ [tˆ0 , tˆ1 ]

! n& 1A': E(' # .R #. # ∈ U (t) t ∈ [tˆ0 , tˆ1 ] A1 %!∀u(t) " E(q 4 -1A # (#5kZ2,D" V (t) × U (t) t ∈ [tˆ , tˆ ] - 8N$':#%e ψ (t , x(t ), t , x(t )) 0 ≤ i ≤ m + k =# 0 .R 1# 1A NPN$ #e i' 20(. 0 %12,D"#1E,D" L" * % (bt0 , xb(bt0)) (bt1 , xb(bt1 )) t c1/#J:1A':" ,c;Z(#5kH "D =f? #!k λb λb ≡ {λb , ..., λb } $P #.R =1A# 02( ## 1L'2b=m+k ! pb(·) ∈ KC 1 ([t0 , t1 ], Rn∗ ) #5* j' 1A " E;bh= +'E,- b u s [ [ "': ,D& " " ,D" ':bhZ7;H .j1AD ##./(O[ ':%E ( -71;K%: " .I6 .R#!;" ,c; 'E,- !; M: - $ "E(`'E,- !; M: ( ':"$[ .R"P42 , #.d !& [ ? 9@r,/ ,D 4 E# 2(`N$':#%e J L L l [ Z @l,o ,D 4 E# 2(ON$':#%e J H Cl @ ,D !& ,D"2(9a': # # J J! 5M:f? #!k ,D E;!kZ ##7;x ,D ,D"D& (9 @

-S&H)*$$ `)*,!/-+&'N $#PJ.;K%D L)N ;, L), 7(QJ #+#,J.PY $# 1)-$J.I 2 +NG#, )M, ,.$$N

V

− −

ZA #!kZ J\N$ ( C −

db b x (t) = 0; Lx˙ (t) + L dt

#"E; #,D%: JLN$ ( C b ∂ H(t) ; pb˙ (t) = − ∂x

7(QJ #+/I`)*$#J!+, )MJ !#"%$J.?@

[

−

ZA #!kZ J\N$ ( C b x˙ (b L tk ) = (−1)k

∂b l , ∂x(tk )

#"E; #,D%: JLN$ ( C − pb(b tk ) = (−1)k

7(QJ #+#,:!#"%$J.E

−

A #!kZ JGN$ ( C

∂b l , ∂x(tk )

u(t)

k = 0, 1;

k = 0, 1;

@

b λ b0 ) u b(t) = arg abs min L(t, x b(t), x b˙ (t), u(t), pb(t), λ, u(t)∈U (t)

−

H #"E; #,D%: JLN$ ( C ∀t ∈ ∆0 ;

b λ b0 ) u b(t) = arg abs max H(t, x b(t), u(t), pb(t), λ, u(t)∈U (t)

∀t ∈ ∆0 ;

21A [ # 5;,D# # P, (./,-A [ 4 #5* # !;

b λ b0 ) arg abs min L(t, x b(t), x b˙ (t), u(t), pb(t), λ, u(t)∈U (t)

\ [ 4 #5* # !;

b λ b0 ) arg abs max H(t, x b(t), u(t), pb(t), λ, u(t)∈U (t)

U

, (Ij# !kZ8n4-1AD $E(9( #"7 J^%^'E,- !;( M: - ! VU

V

7(QJ #+/7J.PY$# )-$J.I

@ ZA #!kZ J\N$ ( C − d Lb = 0, d tk

tk

k = 0, 1;

#"E; #,D%: JLN$ ( C − ∂b l b b H( tk ) = (−1)k+1 , ∂tk

7(QJ #+/ '#$/ P,.NI$#, ,!J.K%J.Z@

1

−

k = 0, 1;

ED

ZA #!kZ JG \ #"E; #,D%: JLN$ (976G :1 " !6 'E,- JG21A #%: C bm+j · Bm+j (b λ x(·), u b(·), b t0 , b t1 ) = 0,

7(QJ #+/I$#,.`)-Y#P, #"%$J.=@

−

1 ≤ j ≤ k;

ED

ZA #!kZ JG \ #"E; #,D%: JLN$ (976G :1 " !6 'E,- JG21A #%: C b0 ≥ 0, λ

bm+j ≥ 0, λ

1 ≤ j ≤ k.

Q" EkZ1A # ,D# # J" 2(. =,D':h=2,D" E# H(#5kH "D& J f? #!k9' 1A " E;bh= !6 ,D %':#E,D" m'E,- J M: d'E,- b u s %"%:`#4 .RE " ,c; 1;42515* 8 " (97 # a': # !; V T M V Y- ! 1; " J 42515* 9 #76E21A " ,c;d #E(d,D " " ,D" L,$ [ h= (n #e E(df? #!k !

Z)-$Y' [5&H)-)*$$Y 'KQJ.:(EA]$`)*/-&'$# TD = ?

=9G J.YAX.UCAE q*WGHKHE AJWG -

+

-

/

-

E

1-

CAU *BJW ~> K ,/

J.)*/ ., $.J $9C)*:/ #,., $)M$, NE,.$J.$#9CJ.P:,. D

j$ 2( ##.R #4 .REb" ,c; Z ,D ,D"2(9=': #p(t) # J pb˙ (t) = −∂ H(t)/∂x @ !i 4 #':"E( :1A "7Z,D b,D"2(9Z # (9 "N$ (' # J!& # Jn pb(t) ,D ,D"2(.l [ .R%# ##.I6+1A NPN$ #e 7 #.I6O':& # # JC

:N V2X

pb˙ (t) + hb p(t), ϕ bx (t)i = fbx (t)

∀t ∈ ∆0 ,

c1A fbx (t) ≡ fx (t, x b(t), u b(t)) =

m+k X

bi fix (t, x λ b(t), u b(t)).

i , , (9" E2(.I63]$%" i=0 %'!(d42515*76] " (97 # ': # !;8(#5kZ2,D" U %: " E('a #51DkH "L': # ` "2 * G "+21;bh= a[ _= #,D"Ea[ "5E,D;h= #!& u #.I6 #e '^(9%!,D ('!(9I(5kZ "n42 ,D "n " t C u(t) ∈ U (t) , (I'E,- V V - "\#76E21A "G "7kZ # ZLN$ ('E& ∀t ∈ ∆ %: ,D# # JG" 2(. V S $21A [ # o [ "E(m, (I Y [$ N$ %!,D E##E( " *bh= j #:1A %!,D' % 'E,- o(5kH#ZZ" ( ##"x(tl k#)o %!b*"!E,D%: %'k ,D " "7& ,D" ':bh= j2('P'E,- "#,D , 7 #E,D" = "E(`,-':*I,-'EkH " 1;H -1AD # !;K(#5kH "D;Kf? #!ko x(t ) E%: " .RJ _= # L42515* G -1AD;"H# [5;42"D # k RQ ,- = " (97 #E,D" + G [ h=2( # E,D [ E( R,-'E& * 4 ; "I#J" P " (97 # ':u(t) # ub(t) ∀t ∈ [t0, t1] @ '!( #" N$':#%e ! N$':#%e -7 =%:& " E(\1AE,D"u(t) D " ,c; L(u(t)) o( # ('!(`N$':#%H(u(t)) e L(u(t)) ! ?(9%!,D ('!(ON$':#%e H(u(t)) -Eu ,- kZR': # ? 4 E ' ,

!;L " (97 #E,D" +I ! `,D E,D2(d#= -1AD; " ,c; u !ˆ (t)^ -1AD; " ,c;^# 21A# 4 #5*# "d#76E5kZ1A # uˆ(t) n"7%:E( ,-':*\(5kZ "+ " [ E"+ .I6E21;h= !6842+ (% n #!& e O(9%!,D ('!(9+1A # "D #.I6d , ,--1A E# Jj " " ,D" 'E& bh= PE,D [ E('H,-':*b)': # uˆ(t) #4 .RE " ,c; P idA #!kZ JHN$ ('E,- b ,D " " ,D" ' "?'E,- [& ,Db"# H( # ('!(9HN$':#%e L(u(t)) C

GD

D

J.' $9 N J. '

D

= ?

J.' $9 N

7J. 9C

b λ b0 ) = absmin L(t, x b(t), x b˙ (t), u(t), pb(t), λ,

u(t)∈U (t)

b λ b0 ) = L(t, x b(t), x b˙ (t), u b(t), pb(t), λ,

!∀t ∈ ∆0

b λ b0 ) ≤ L(t, x b λ b0 ) L(t, x b(t), x b˙ (t), u b(t), pb(t), λ, b(t), x b˙ (t), u(t), pb(t), λ,

∀u(t) r #∈"EU; (t) #,D∀t%: ∈Jp∆N$0 (^@ 'E,- ][E,Db"# r(9%!,D ('!(9 V

N$':#%e

H(u(t))

C

b λ b0 ) = H(t, x b λ b0 ) absmax H(t, x b(t), u(t), pb(t), λ, b(t), u b(t), pb(t), λ,

u(t)∈U (t)

! ∀t ∈ ∆0

b λ b0 ) ≥ H(t, x b λ b0 ) H(t, x b(t), u b(t), pb(t), λ, b(t), u(t), pb(t), λ,

D

Q ∀u(t) ,- ∈ !;ZU((t) # ∀t('!∈(9∆ 0L u (9%!,D ('!(9 H u % E7 #"#.P E,D%: %'`A #!kH # L mN$':#%e !;m$ #"E; # H ,D;42#. ,D "# _= # 2(IC L(t, x, x, ˙ u, p, λ, λ0 ) = hp, xi ˙ − H(t, x, u, p, λ, λ0 ),

%: " E(0 +,-A D2( \ Jm* ,D" " #+42 ,D "5 8 "E(' L H %!%8N$':#%e O '!( #"7 u "2 u*b" ,c;` _= 4 #%:E(Ii^,D;4 L, " (

D

c1A

u ˆ(t) = arg abs min L(u(t)) ≡ arg abs max H(u(t)) u(t)∈U (t)

u(t)∈U (t)

∀t ∈ ∆0 ,

ˆ λ ˆ 0 ) ∀t ∈ ∆0 , L(u(t)) ≡ L(t, x ˆ(t), xˆ˙ (t), u(t), pˆ(t), λ, ˆ ˆ H(u(t)) ≡ H(t, x ˆ(t), u(t), pˆ(t), λ, λ0 ) ∀t ∈ ∆0 .

i8%!5*2,D" (5kH#/[ "N$':#%e 5,D21A Ekh= " %:"?,-AL(u(t)) D2(.R$H(u(t)) N$':#%e J ˆ λ ˆ0 ), L(t, x ˆ(t), xˆ˙ (t), u(t), p(t), ˆ λ,

ˆ λ ˆ0 ), H(t, x ˆ(t), u(t), pˆ(t), λ,

%: " .RH42 ,c;"\ " ? ('2. a'E,- !;!6` " (97 #E,D" -/ -1AD;bh= uˆu(t) I1b" 4 #5* # !;3 '!( #" (t) ∀t ∈ ∆0 N$':#%e J L(u(t)) H(u(t)) %: " .I61AE,D" D " ,c;m !6 H( # ('!(I,D " " ,D" ##(9%!,D ('!(d u(t) "H2,D"

$9 N

J.

arg abs min L(u(t)) = u(t)∈U (t)

= {ˆ u(t) ∈ U (t)|L(u(t)) ≥ L(ˆ u(t))

ST

∀u(t) ∈ U (t) ∀t ∈ ∆0 },

arg abs max H(u(t)) = u(t)∈U (t)

= {ˆ u(t) ∈ U (t)|H(u(t)) ≤ H(ˆ u(t))

∀u(t) ∈ U (t) ∀t ∈ ∆0 }.

Q ,- " (97 #E,D" 8 -1AD; " %!%ON$':#%e br1A'E&

!6G 2( ##.I6t%# ( 4$'E,- uˆ (t) !; ˆ λ ˆ0 ) u ˆ(t) = arg abs max H(t, xˆ(t), u(t), pˆ(t), λ, u(t)∈U (t)

∀t ∈ ∆0

N$':#%e !; Z [ h=2(,-':*o -1AD; " ,c;G%!%\N$':#%e !;\ D& ( ##.I6 t uˆ(t) C 515*/ xˆ(t)-1ApDˆ(t) # !λˆ; uˆλˆ(t)0 @ uˆ(t) "= u(t, [ xˆh=(t), pˆ (t), E;λ,ˆ542λˆ501)5*#D !& # J# (9( E# !;C

D

ˆ λ ˆ 0 ) → max, H(t, xˆ(t), u(t), pˆ(t), λ,

u(t) ∈ U (t) ∀t∈∆0 .

i3 4 '27"7"= = _= # !;` `%!7kZ1AE( 4 #5* # ` 2( ##.I6 t ˆ λˆ0 H# E,D [ E(,-':*$ -1AD; " ,c;aN$':#%e !; x ˆ(t) pˆ(t) λ V #& ˆ λ ˆ 0 ). u ˆ(t) = u(t, x ˆ(t), pˆ(t), λ, -1AD # uˆ(t) E,D [ E(l,-':* , (I .R_= L(5kZ " "D& [ E"L,D e 7 # \ , , ( " # !; , (I# !kZH V ! $ ( . -

j$ D6E21\ "'E,- !;\,D"7e ##E,D" G t =A #!kZ J k N$ ( C dL/dt ˆ k = 0 (k = 0, 1) 9%m'E,- bB,D"7e ##E,D" d a #"E; #,D%: JN$ ( C ˆ tˆk ) = (−1)k+1∂ˆl/∂tk (k = 0, 1) tk ,D.RE " ,c;Ge *%: J\ #,DH( "C

V

U

ˆ k = 0 ⇔ (−1)k+1 L( ˆ tˆk ) + dˆl/dtk = 0 ⇔ dL/dt P m+k ˆ ˆ ˆ ⇔ (−1)k+1 p(tˆk ), x ˆ˙ (tˆk ) − ϕ( ˆ tˆk )i + i=o λi fi (tk ) + hˆ

(1.1) +∂ ˆl/∂tk + ∂ ˆl/∂x(tk )x ˆ˙ (tˆk ) = 0 ⇔ P m+k ˆ i fˆi (tˆk ) + hˆ ⇔ (−1)k+1 i=0 λ p(tk ), 0i+ ) ˆ ˆ +∂ l/∂tk + ∂ l/∂x(tk )ϕ(t ˆ k) = 0 ⇔ P m+k ˆ ˆ ˆ k ˆ ˆ ⇔ (−1)k+1 i=0 λ ˆ(tk )ϕ( ˆ tˆk ) = 0 ⇔ i fi (tk ) + ∂ l/∂tk + (−1) p P m+k ˆ ˆ ˆ k+1 ˆ k+1 ˆ ˆ ⇔ ∂ l/∂tk = (−1) pˆ(t )ϕ( ˆ t ) − (−1) i=0 λi fi (tk ) ⇔ Pm+k ˆk ˆ k k+1 ˆ ⇔ pˆ(tˆk )ϕ( ˆ tˆk ) − i=0 λ ∂ l/∂tk ⇔ i fi (tˆk ) = (−1) ˆ tˆk ) = (−1)k+1 ∂ ˆl/∂tk . ⇔ H(

SV

0 4P'E,- !;1A #!;bh=m+j ,D" ^,--1A' " #,D" B (ˆx(·), uˆ(·), tˆ , tˆ ) = 0 'E,- G1A& 0 1 #!;bh= J)#DkZ2,D"%:E,D" 3m+j "E(B,-':*n ,D':h=2,D" ##/ ! ) 1A': J3" ( # %" #-$ λˆ = 0 'E,- `1A& #!;bh= J #DkZ2,D"%:E,D" ^ .R#!; " ,c;^"m+j 5kZ1A2,D" ## #2,D':h=D& ,D" ##R ! 3#2%" #- d _= # xˆ(·) uˆ(·) tˆ0 tˆ1 %7& JB42515* B λˆm+j = 0 # [56E21A (9q %! 'E,- !; j Bm+j (ˆx(·), uˆ(·), tˆ0 , tˆ1) = 0 @ ˆ ˆ) ≤ 0 Bm+j (ˆ 7 ( " (I2*"% E7;P42515*I #!& x%!(·), R'E,-uˆ(·),

!t0; , tλˆ1m+j ≥0 e K(9%!,D ('!(9H λˆm+j = 0 (5kZ "H %!42"E,c;L .R5kZ1A ## J L " [ E"Z L _= # K1A # "D #.I6G , ,--1A E# J /i 'E,- !;a# " e"D #E,D" C ˆ ˆ ≥ 0 A6E21;" / G6E21;h= / _=Z(#5kH "DZf? #!k λˆ0 \N$λ':0#%≥e 0 #λ7m+j 'E,- !;1A #!;bh= JP#DkZ2,D"%:E,D" =(#5kH "D ?f? #!k λˆ m+j 1 ≤ j ≤ k I L # * # !;!6 V Yj" # #,D" pDA7;`21A* %#':"LE,D [ ':bq4 #5* (E,D"\'E,- !; u s ,D ,D"2(G'E,- J #e `(9%!,D ('!(9 %!b* (3 + "',D !& ,D"2('G%!%\'E,- k -

D

5D

7D $#,.$(MNJ #/ 5K@+)*J./#,!+J. %P( \ , : 9 % \ (G2 )-9 K%$$Y +#,.)M$ , $ 9J.PKQJ. +!/:>( , )M,. J.$ Y #N!##"JK@%)*$ 9 ,!$W+( N (@ )*J. + J.P, :9 %\ ZJ.#J.P, :9 (@)*+$#,.$N ;N-, )*[&H)*$ J.)*/# ,.$$N J.#J.P, :9 75i #"E; #,D%: JN$ (I,D ,D"2(9/1A NPN$ #!&

2n

e 7 #.I6n': # # Jd% Jd42515* d ( "8 :1 %!# # *2,D%: J

D ( !7" # JG,D ,D"2(.PC x ˆ˙ (t) pˆ˙(t)

ˆ ∂ H(t) ≡ ϕ(t, xˆ(t), u ˆ(t)), ∂p ˆ ∂ H(t) =− , ∂x =

c1A ˆ ≡ H(t, xˆ(t), uˆ(t), pˆ(t), λ,ˆ λˆ0 ) uˆ(t) -1AD; " ,c;`'E,-& 2(mH(t) " (97 #E,D" aI%!%+N$':#%e !;L 2( ##.I6 t xˆ(t) pˆ(t) S S

C V #&-E*"= 4 ; "P'E,D"# " # -1AD ##E,D"9,D;42##':bp,G ,D':" ,D" 2(rN$':#%e uˆ(t) ,D ,D"2(o1A NPN$ #e 7 #.I6K': # # J\% J\42515* 4 & 4 2,D"#./( 3^% J342515*O;;b" ,c; #.I6nE,D"5;##.I6n #" E# !;d L,D ,D"2(. 1A 2n NPN$ #e 7& #.I6?': # # J2 2( # 22,- = # =#2N$ %!,D E#.P Z(#& kH "D Gf? #!k λˆ λˆ tˆ≡0 (tˆλˆ1 , ..., λˆ ) @rE,D 2n + m + k + 3 # 4 2,D"#.I62t%P%!%?0 N$':#%e !;P1 f? m+k #!k @^21A# 21A#7; !& J8,D" # oN$':#%e !;8(#5kH "D J p(·) λL λ " " O(#5kH !& "D n(5kH#L4251"+," *#E,D" br1Aa5kH0 "D # L,DE(#5kH !& "D; # ( E"K !6 -Au ,- `G 4 '27"7"=#7 42H'E,- J #e L(9%!,D ('!(9G' 1 " ,c;` %!42"*"a# (97 #.RJ`,-'E& *J λˆ0 = 0 # 42(5kZ #"a+,D " " ,D" ,'E,- 2(^ $# !& ( %'`(#5kH "D J8f? #!k\(5kH#\E,D':h=2,D" "5kH #./(%!%: J!& # [ ' 1A?' 1A [ # JK G _= # \42515* K5kH !& λ "ˆD0 # JL%: #,D"7#" ( %'K(#5kH "D Jaf? #!kZ(5kH# E,D':h=2,D" "+ O1A': ( d,DE,D [E ( , (I# !kZGjV ! $ ( . -7$E,- # ( % Z(#5kH "D JZf? #!k/o% J?42517& * [ ' 1A " 2n+m+k+2 # 4 2,D"#.I6D
SD

U

V

V

U

D

D

D

( , $# S!

K%9;(%+$J."G"'(Q#$J #N +J.N #J.P , : 7 $N E(Q,-J #j+ %!NWb*)- #$ !Y;'E#,- !KQ;=J.#: (( EJ. +%

?,D ,D"2(''E,- JZ #e =(9%!,D ('!(9$'E,- !; V V M V !- M 4 $[ ' 1A':"a-1,D"7;"+,D [ J8% ':b 42515*' % ':b 42515*' l# [56E21A (.I6+1;d #e 8(9%!,D ('!(9?, _= # !;L'E,- J u ,- L(#5kZ2,D" #$42 ,D "= "Z 2( # L,D !& ,D"2(9$1A NPN$ #e 7U (t) #.I6Z': # # JK% J4251U5(t) * = U 77"P2,D"?N$':#%e !; H ! HN$':#%e !; L #R42 ,D "o "o 2( # ; # ! A*"G"KkZ=, ( #=42 ,c;"Z; #K " t N$':#%e ϕ t f 0 ≤ i ≤ m + k -C ϕ ≡ ϕ(x(t), u(t)) f ≡ f (x(t), " ,D ,D"i 2(9`1A NPN$ #e 7 #.I6n': # # Jm%i Jmi 42515*u(t)) m ( " !!V U C

7'#$9C $# L)-'

$#

I+$':

#, )M+9 NI$P,!&H)*! = ? PD

ˆ λ ˆ0 ) = C, ˆ H(t) ≡ H(ˆ x(t), u ˆ(t), pˆ(t), λ,

4 #5*bh= JE,D"5;#,D" PN$':#%e #$ " (97 # JH" %"& E$ "E(+1;H#76E5kZ1A # !;ZE,D"H5;## J C 1AE,D"7" *#$ D& 1AD "O4 #5* # LN$':#%e H O%!%: J!& # [ ' 1A`21A# Jn" *%:L !& " (97 # J+" %" i * ,D"#E,D" A2,- a6E "2;a[ .321A #a 4Z(& ( #" a 2( # k = 0, 1 #2N$ %!,D E#8 ∂ ˆl/∂tk = w E,D"5;##7; C (t5kkZ "n [ .R"n -1AD #n 4`'E,- !; K,D"7e & ##E,D" 0 tk C H( ˆ tˆk ) = (−1)k+1 ∂ ˆl/∂tk ! dL/dt ˆ k = 0 - E,D%: %' C ≡ (−1)k+1∂ˆl/∂tk k = 0, 1 -$u ,- ] , , (9"7& E2(E( ,-':*]N$':#%e ψi 0 ≤ i ≤ m + k n#]42 !& ,c;"m; #m " C ψi ≡ ψi (x(t0 ), x(t1 )) ]6E "2;][ . 21A# tk #2N$ %!,D E#t0 o"t1 H(t) ?E,D%: %'33"77& % !6 42515*76r" ( #ˆ#" l=# 0 42∀t ∈,D ")[t0;, t#1]0 " tk l "E(' i042515*76+[ ./,D"21A J,D" !;O,H( # ( !& C ≡ (−1)k+1 ∂ ˆl/∂tk = 0 4 ' 2(./( N$':#%e #7E( B (x(·), u(·), t , t ) ≡ t − t ON$':#%!& e !;( ψ 1 ≤ i ≤ m + k -#0=42 ,c;h= 0( L1 ; #K1 " t0 t A" !& ( ##" li ( " :1C l ≡ λ (t − t ) + Pm+k λ ψ (x(t 0), x(t1 )) 0 1 0 i i 0 1 i=1 ,--1A E"D # C ≡ (−1)k+1 ˆ0 k = 0, 1 - H(t) ˆ0 ˆ ∂ ˆl/∂tk = λ =λ ∀t ∈ [t0 , t1 ] 7 * K`,D ,D"2(H1A NPN$ #e 7 #.I68': # # J% J 42515* + K #" 7AG 4 ; "H42 ( # "\21A#G 4?1A NPN$D& #e 7 #.I6': # # Jd,D ,D"2(.l ./(3 #" 7E( #-1A No& N$ #e 7 #./( ,D "# _= # 2( I +"2( , (./( # 4 "K#K-1A !& # e'L E;:1A %`,D ,D"2(.]1A NPN$ #e 7 #.I6L': # # J+% J SY

24 515* R$ ]* ,- ##E( _= # ]% J 42515* .RJ] #!& " 7(5kH#L ,D 4 E"H1;8%: #";8 21A (.I6+ .R* ,-& # J %: #" '; #n " (97 # J " %" ^E,D"5;#,D" N$':#%e ˆ -E$ H 4 2,D"# JZE,D"5;## J C @0 =1; -1AD& # !;\21A#H(t) JG 4P# 4 2,D"#.I6G,DE,D"7;bh= !6 αi ≡ pi(t0 ) %" ( " ,D"D% , (I# !kZ - u ,- @B "%.R" (#5kZ2,D" G Rr ':& ;bhZ7;n U (t) %" !∀t&xN$∈':#[t%0e, t !1; ] u(·) # .R #C u(·) ∈ C(∆, Rr ) # .R #.n u * ,D"#.Rj 4 21A#.R ϕ f 0 ≤ i ≤ m+k - "= , , (9" E2(97;H42515* V T M V Y[ ' 1Au " iu G #"E; #,D%: J+N$ ( !V U vP ,D"#./( ,-':*2( 42515* af?7&

#!ka[ ' 1A " \ #"E; #,D%: J8N$ ( -\%: " JL1A NPN$ #e 7 #7;`,D;4 V V P ( "` :1C ˙ = u(t) ( " (I*"``42515*76Of? #!& k8 j,--1A E"Dx(t)

#O42515*76d%!A , ,D *2,D%: 8E e ## ,D* ,- # !; ∆0 = [t0, t1] V V -/t%3%!% " !6 42515*76 U (t) @) "%.R" R(#5kZ2,D" o r N$':#%e !; @)# D& ∀t ∈ [t0 , t1 ] .R #9 N$':#%e ϕ f @ # R.R #. u 9u(t) "nN$':#%e @ # u.R #iu=1A NPN$ #e ' 2(. u u ,- L L(u) H(u) ! /1A NPN$ #e ' 2(9L "E(]N$':#%e !; 1AkH# * ,-ZL(u) 4 :"Z GH(u) #76E5kZ1A # K " (97 # =':u=Duˆ& # !; uˆ(t) ( ':"?[ .R"? ,D 4 E#.d# [56E21A (.RI 1AE,D"7" *#.R 'E,- !;m ./,D_= !6d E;:1A%: ` %!7 # O( # ('!(9 ! ^(9%!,D !& ('!(91;O%: # *#E( # J`1A NPN$ #e ' 2( JON$':#%e 1A No& N$ #e ' 2( JN$':#%e n(# !68# 42 ,D (.I68 2( ##.I6 - * ,D"#E,D" 'E,- !;L " H E;:1A%!i],-':* 2,- L(u) ! @ 1AE7kZ1A. # .R #=1A NPN$ #e ' 2(97;\N$':#%e !;\21& H(u) # J\# 42 ,D ( JG 2( ## J u @l#$ %" -!'E,- !;G " E;:1A%!I1; " JZN$':#%e = ( b"$ :1C ∂ L/∂u ˆ 2≥0 ˆ = 0 ∂ 2 L/∂u ! ∂ H/∂u A @ 2 ˆ ˆ = 0 ∂ 2 H/∂u ≤0

*,.N [&H)*$ = ? *IK-J!J.*,!J.K%&'E+ )-#Y$$&'E#J.#J -,.$/ TD

D

]D $ #&'E:$:( #WKQJ.:( $#,. J. P :9;,>$(Q 9;J #, (Q+J#/:#+KQ/ !#"#% KQ!#"%$&'E](:# (I$:*"(8 ,D " #S *2,DKQ%J. m:( # %e\ m(9%!,D ('!(9`5; !A,c; %!%n " "`#+" [ E# G%" % m, N$ ('2 E"+# [56E21A !& (.R`'E,- !;m %!7 # D %!,D"2('!(9dN$':#%e #7A`1;^42515* " (97 # \': # !;8, !:K%$(%9Cr(#5kZ2,D" E( ':D&

ˆ ˆ ∂ H/∂u = 0 ∂ H/∂u <0 2

# J

U

Rr

2

2 ˆ ˆ ∂ L/∂u = 0 ∂ 2 L/∂u > 0

9<=5; # !; #e n(9%!,D ('!(9O , , (9" !& S#

E7 ,D842515* m " (97 # O': # !;d@ 42515* df? #!k 842515* `%!A , ,D *2,D%: \E e ## K ,D* ,- # !;\%: " .I6 @ \(#5kZ2,D" n Rr Iip,D;4 ^, " ( 42515* !& U " (97 # Z': # !;L, 3(#5kZ2,D" E(': # J (5kH#3'E,- #]#4 E"]42515* ( A$ "E(' 2,- aK42515* V T M V YR(#5kZ2,D" U (t) ! P@ ? Rr "`42515* V T M V Y ∀t ∈ [t0 , t1 ] [ ' 1A "a42515* J U A E(I

U (t) ∀t ∈ [t0 , t1 ] $ V T MU V Y 5$ = _= # Z42515* a,-':*Z42 (%#':" L(#5kZ2,D"E U (t) ∀t ∈ [t , t ] " (97 # ': # uˆ(t) # [56E21A (#76E21A "n%!% 4 0#5*1 # 8 '!( #!& "7 u(t) N$':#%e L(u(t)) ! `N$':#%e H(u(t)) - +%: " E( N$':#%e !; # (9 "$# ( # _= ! HN$':#%e !; H(u(t)) @ # [ L(u(t)) _= 4 #5* # o u(t) # ](#5kZ2,D" , (I # !kZPV ! ! c515* G " (97 # Z': # !; -
K@)-9 , !:K%$(%9C ID (%)*+!-,.$/ %D !:K%$(%, J. J.P+,.$$ !#"%$&' J.P!:K%$(%$J., (QT" J #:J. +$# J.,:P+,!)-,.J.$P$#$+ S -,$J.!#"% $ &' (% )*+!#-,.$/ E,!`J.KP " *! J. J.P+,.$$ !#"%$&' (%)*+!-,.$/M\

P !:K%$(%' V :U (QJ'# +NPKQ#J.KQ:!(#"%$ NS$# 9;+,.P!#"J /%$+J.J / J. ++ ABK%(%)*K%$J.K%"($#, ,. W J.)-+$:K%Y(%9 $J."I(QJ #+N 7\"L 2(;`%!%` 4 #5*7 #7= V U ? #e !& E( (9%!,D ('!(9H#4 .RE7E,DZ _=K'E,- P " (97 #E,D" a

u #"E; #,D%: JLN$ ( C uˆ = arg abs maxH(u) u∈U i)E,D"7# %:Z ,c6E21A# J`42515* V T M V Y-\"7%!kZaE,D"77& # %!76`E,DD6842515* _= ##.I6O+%!5*2,D" ( ++4-1ADA76 V !+ S S $%" %'!(9" [ ' " ,c;n -1AD;"8" *#':b # !kH#bb

#N$':#%e #7A B0(ξ(·)) → inf #`1A 'E,D" (.I6m':;D& (.I6O e2, , 76 6E "2;O ! # H" [ & E"O -1AD # !ξ(·) ; [E≡,D(x(·), b"#u(·), O(t) ,#t1 )('!(9N$':#%e #7A8 !& , "C B (ξ(·)) → abs min ! %"%: B (ξ(·)) → min - " ,D;42#m0 ,`"2(IR*"m42 ,D B0(ξ(·)) → inf0 #8 ,D%!b* "m 4D& (5kH#E,D"\,D':h=2,D" E# !;a _= # !;+42515* +G :1A( # ( 4 'E& bh= JGE,--1A E"D #E,D" Z1A 'E,D" (.I6H':;2(.I6H e2, ,D #%: " J lim B0 (ξn(·)) = inf B0(ξ(·)) -1A' "521A#7& {ξn (·)} %:!42 ( " "*"H n→∞ E,DD6G , , (9" E2(.I6G42515*76G$%" %'E& (9G _= # 42515* +\ :1A( # ( 4 ':bh= JOE,--1A E"D #&

B1

S

D, " ξ (·) " ,D':" ,D" ' "5j842 ,D B (ξ(·)) → inf E(2,D"42 ,D n A,D56!#!; " ,c;\%!%\# 0%!7;L"51A e !;a42 ,D LE,D"77& B0 (ξ(·)) → min # % %!,D"2(97 #.I6G42515* $ _= # 42515* " (97 # )': # !; #)E,D# #e (9%!,D ('!(9

D Y!#"%$&'KQWJ.(%:)*(+!-,.$A]/ $` $#)* ,P/-#"&',!&' !$#T/ +(% /(Q J.KQ$#)M, , # " J.P$#,. N+!$/+9- ,. J. N #)-O$" $#$,.J. ;:(%$9;K%,ZY(QJ $##!-+/ #, J )M-,+ &' +PL)*, #"/ % $KQ #'KQ#(!%#"*%,.$$$N 9;, +G)M,' (!#"#"%% PP, ,!&'W)-,.$#,.$ / KQJ. `KQ)MJ.`, )M!, #!#"A]%$$9;`,7)*/-(%&')*$#+! /-, : 9;(%,7)*)-+!-,Y#.,!$J!J.9 / $#KQ!#"%$WNW+ )M, !#",.%$#$ J.!#"%"$J`J. E#J.+E ' #"+ 9;P J.,O + $WJ #(#%, *L)*/$#,P,. K%+ -, +E:$#9J. #(Q.J.J / #P9+P,!, NP$+ #)M ,. )* / J.' P/ 7$ $$ 9 NWI= V Y? I #!#"!##"%%$$ J.&'J. P(%=!!)* V+!T-,$? .92$ #/G(Q J + # +')-#$WN+#NE! !KQ:!, #" )M%,$$$#NT,BJ.)* # KQJ.',P$# CP$#97J KQ)M!, #" %J.+P$, )M( +&@'T)M#"Q,.J OP/ ,.$# $,.J.'/SJ #)MJ, $J.( P $+,.N $$ N] !# #"]%K%$$9CJ!J.+:,.P$"B$+S,.NP* )-J. !+:$, )- $# J.P)-,) P$9;)*, /(Q:J #9C:+ /SP$#, #J.'!#"#"% $( J. :S +$#,.S # J -,`:&H9;)*, ##,.$# $ 9C#: !; ,.# J !#:.I6L!K (NP "4 (I :./*(n"? 1;2G(42 5#1#5./*K( = !"? (9;7': ## E=,D%': !6G427&& 15* = !? 8 # J#& %E51A" *#.I6`42515*W= #! V V@? _= # 42515*

ξˆ ≡ (ˆ x(·), u ˆ(·), tˆ0 , tˆ1 ) u ˆ(·) tˆ0 tˆ1

x ˆ(·)

':* ## #\E,D# Z #e L(9%!,D ('!(91AE,D"7; "\N$':#%!& e #7'G[E,Db"#.RJL( # ('!( ! a(9%!,D ('!( - Qo(2,D"#] " ( " "$*"$#2, ( "E;0#^"' 1A#E,D" r % %!,D"2(97 Jr$ #"E; #^#^ %!7 #':b r[E,Db"#':b " !& (97 #E,D"5/"7%!kZ #/ " ,D':" ,D" 1;=[ _= #,D"EI#D # J#.I6 42515*$ " (97 # R': # !;1A # ( *2,D% ( =,D ,D"2(9 ( ?" & 2(m,D':h=2,D" E# !;G L-1A #,D" ##E,D" a _= # !;"2(d#?( # a"7%: J,D "'Ee O 21A (.RH#LE,D# #e +(9%!,D ('!(9 , ,--1A E# !;42515*O " (97 # +': # !; 4 ;b"5%!% ! ':*"?,D21A Ek"D #.RR 4 '27"7". 1ADA"? 4D& #.Ro1;\%" % a .R 21A.P $E,D%: %'GE,DZ21Dkh= =* ,- ##E('a _= # brK1##E( $%" %'!(H42515* 8 " (97 # L': # !;O,D21A Ek"\7&

D

SEU

ID

( "` + `#'2 E( 4 #5* # " K ( "1A 'E,D%!b"G#7& " *2,D%: _= # %: " = O* ,- ##E(^ _= # 8% J 42515* \ #e H(9%!,D ('!(9( "21AE(m,D"D [ .] ,D 4 ' " ,c;G %!5*2,D" ,c6E21A# o1;G':* # !;G _= # !;K42515* K G# #'2D& E(\4 #5* # ? ( " , (I 5# ( X -5"o 4 # %! "/# [56E21A (E,D"R ( " [ 4 e.#7 " *2,D%: / _= # !;=E,D# #.I6$" , , (9" E7& 2(.I6\H$%" %'!(P42515*jN$ ('2 ' 2(nK,D;4 L, " (

= ,.? ' )- ' ,.$/ )M,.OP,.$/P # D

)*!,.`)-(

9S CAUKHEMWGRCAJ?J.YAE . U.BU GHVAXE U"WIHYAGHF*G VACAUK WJ.YAE ,-

T ? (97 4 E"K42515*' A"L2,D"\ 2,D" 8 Z%` :1A' V T M V Y- V a E,D"7 "LN$':#%e L L l 2,- 842515*G _Z " ,c;`\A7&

#!kZ JaN$ ( ! `N$':#%e 2,- +42515*Z _Z7& " ,c;\ #"E; #,D%: JLN$ ( H l S L , "K# [56E21A (.RP'E,- !;a M:4 R " (97 #E,D" +& e2, , ξˆ≡ (ˆx(·), uˆ(·), tˆ0 , tˆ1) #':b ,D ,D"2('L'E,- JL #!& e )(9%!,D ('!(9]1; 42515* " (97 # 3': # !; V V M V Y -s9 , , ( " "G 42(5kH#E,D"K# (97 # H,-'E& *7; λˆ = 0 2,- λˆ 6= 0 5kH λˆ #./( -1A # !& 0 eG ! b[ JO1A': Jn05kH "D # J%: 0#,D"7#" *"`[ 'E& 1A "/,D " " ,D" E"j21A# J$ 494 # :1A#E,D" J$'E,- !;o# !& ( % O(#5kH "D Jf? #!k- [ 4 E"G'E,- !; V V M V !- M:4 \% ':br42515*'a #e L(9%!,D ('!(9 Q[ -1A "E,c;*"a* ,-L# 4 2,D"#.I68L% J`42515*H,D & 51 ",o* ,-E(d'E,- JK1;\ !6\ -1AD # !; !\$E#7 4 E % ':b 42515*' #J" 1A 'E,D" (.R ':;2(.R+ e2, ,D. ξˆ ≡ (ˆx(·), uˆ(·), tˆ0 , tˆ1) j;;bh= D& ,c;] _= # !;( )% J]42515* /$ 3#7 * 3 % J 42515*8'E,- J^1A #!;bh= J3#DkZ2,D"%:E,D" 1K1;3%!7k& 1A + 4 " !6n'E,- Jn# [56E21A (a8,D " " ,D" d,G'E,-& !;( d = , , ( " "821,-':* λˆm+j = 0 λˆm+j > 0 1 ≤ j ≤ k -\ *2( # _= # G':* ##E( ?# [56E21A ( "3 .R# # 'E,- J ˆ λ = 0

SD

SX

/

m+j

K#G _= # ':* ##E( 7 @0'E,- J λˆm+j ≥ 0 1 ≤ j ≤ k Ya -1A W#J:1A ##.I6l1A 'E,D" (.I6B':;2(.I6p e2, ,D #J" _= # 42515* V T M V Ya@ " (97 #.RJr& e2, ,51AE,D"7;bh= JZ %!7 #.RJZ ! K g [E7 #.RJZ( # ('!( N$':#%e #7' V T -! ! L %!42":*"H _= # !;L# "5 s9 , , ( " ( ( . ( # # !;p$ !A3% #7 " *D& ,D%:E('q _= # b 42515*r " (97 # 3': # !; 0"E( * !& ,- o42515*3%!A , ,D *2,D%: E e ## ,D* ,- # !;0 042515* f? #!kL1;r# %: " .I606:%" #.I6qN$':#%e #7 o" !& ( #7 #.I6 % .I6 ! d # *#.I6 - 4 ( " *2,D% !6n " ( #7 #& 4 ( " *2,D% !6 # * # J " d #,D" )# #,D" " ( " (Io*"^ ( . 21A [ #. "7%*" [ . ,D # "D #RE,D" J$# E,D-1,D" ## JP %: J$(5kH#R[ .I 2,D" H , ,--1A E# /[E,Db"# J " (97 #E,D" H':* ##.I6 _= # Ji3[ _= #,D" =42515*+"7%`E,D" "a, 1ADA"G#Z' 17& " ,c;iI,DP , , (9" E2(.RPH$%" %'!(?H%!5*2,D" $ ( 42515* O`,D " " ,D" n,H#4 E# 2(]$%" %'!(9a 21;" ,c;O% :1A'=42515* " (97 # $': # !;$ V T M V Y _Zb" ,c;H# E,D# \ #e (9%!,D ('!(98 -1A ##E('n .R_=a$ !' 7 " (' j _= # !;L42515*G " (97 # ': # !;!#2, ( "7& E;?#/"2*"4 #5* "D #7;P* ,D" " !6?42515*?(5kZ "/[ .R"/ _= # R[ 9E,D"E('o7 " (' " ( " (L"7%!kZ *"D6E "2;o%!7kZ17; 4/42515*H$%" %'!(9?(5kZ "P[ .R"? _= #o [ =,R ,D 4 E# !& 2(n" %:N$':#%e J ZA #!kZ JGN$ ( -: [ H,o ,-& 4 E# 2(n" %:HLN$L':#%le J H l K #"E; #,D%: JaN$ ( - H$%" %'!( %!%\ ! " L42515* \K( "21A *2,D% (m,D [& 7kZ # !;(+ _Zb" ,c;=$ [ !6=N$ (976P21A# 2( ## 7D& #R ! 2,- "G' 1A [ # _Zb" ,c;+\, ( _Z## JaA #!kZ 7M #"E; #,D%: JLN$ ( Bm+j (ˆ x(·), u ˆ(·), tˆ0 , tˆ1 ) ≤ 0 Bm+j (ˆ x(·), u ˆ(·), tˆ0 , tˆ1 ) = 0

D

`D

7D

D

S

{ {$Z\[`acok[m ]Bl]ja`_Aa`gk ix^ fDwxf [k hkl`az

l ] fDixlfDrDk d`[`a`l `a`d] co]B^ ixac9b co] t]B[]B^`_k[`lmt s ]Ap&]Mg ^j]Diqixa`gk ix^ fDwxf rD]B[`a] `afBl`lfDwxf aixg`aiHjkl`az s ]Ap&]Mg ]DwH[]Bl h] a s ]Ap&]Mg fBd`_Aaco]jvBlfDwxf b'd`[]Drjkl`az ~> > ~> TDU.BU AE MWU~E AJ GHF*G?KHUCAEAU AEAGHYAYAGHF*GE AE*WJ.YA E # AB)-J.P,.NOP!/I * K-J!J.*,!J.K%&' + )-#Y$$&' #J.#J -,.$/ *J ;KQJ.)-+$$9C:T$#*!#"%$9C 5K%$#,.

+

$9C $#*,.$/: 2 +N7#, )M, ,.$$NE7+ )M, ,.$-\

I(x(·)) ≡

Z

1 0

(x˙ 2 (t) − 12tx(t)) dt → inf,

!"

T ? (97 42e !;C! [ 4 #5* (

x(t) ˙ = u(t)

x(0) = x(1) = 0.

" c1

V 0 V U x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1]; V X u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 1]; V x(0) = 0, x(1) = 0. # * # !; V Y " Z# #,D"Z42515*o " ,D':" ,D" ':b"5 V ?':#%e !;Lf? #!kC B0 (x(·), u(·)) ≡

L≡

Z

Z

1

(u2 (t) − 12tx(t)) dt → inf;

1

(λ0 (u2 (t)−12tx(t))+p(t)(x(t)−u(t))) ˙ dt+ 0

+λ1 x(0)+λ2 x(1) ≡

!T

Z

1

L(t) dt+l; 0

A #!kH #C L ≡ λ0 (u2 − 12tx) + p(x˙ − u) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C

H ≡ pu − λ0 (u2 − 12tx);

" ( ##"C

l ≡ λ1 x(0) + λ2 x(1);

@ * ,- .R$ aN$':#%e #7 #.RJ\(#& kHλ0 "Dλ 1Lλf?2 p≡#!kp(t) S a ,D"2(90'E,- J #e r(9%!,D ('!(9)q42515* V M V -C K': # # J! 5M:f? #!kC

]1

−

d ˆ ˆ x (t) = 0 Lx˙ (t) + L dt ˆ ∂ H(t) ⇔ pˆ˙ (t) = − ∂x ˙ ˆ 0 t; ⇒ pˆ(t) = −12λ

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = 1 ˆ x˙ (tk ) = (−1)k L

= 0 t0 = 0 k = 1 ∂ ˆl ∂x(tk )

⇔ pˆ(tk ) = (−1)k

∂ ˆl ∂x(tk )

ˆ1 , pˆ(1) = −λ ˆ2 ; ⇒ pˆ(0) = λ

K'E,- o " (97 #E,D" L u C t%a%!% L(u) = λˆ0 u2 − pˆu H(u) = −L(u) !" ˆ 0 u2 − pˆ(t)u), u ˆ(t) = arg abs min L(u) ≡ arg abs min (λ u∈U

u∈(−∞,+∞)

!V

! ˆ0 u2 ). u ˆ(t) = arg abs max H(u) ≡ arg abs max (ˆ p(t)u − λ u∈U

u∈(−∞,+∞)

t% %!% 9" !& 1 = 0 [ =#$,D':h=U2,D"≡ ' R "= ≡" (−∞, (97 +∞) # ': # λˆ0uˆ(t) : [ #` .R#!; " ,c; 'E,- u s , (Ij# !kZ -I " (97 # ': # uˆ(t) (5kZ " [ .R" λ ˆ0 -6=1AD0 #? 4/'E,- J ∂ L/∂u ˆ 2 > 0 ! ˆ = 0 ∂ 2 L/∂u C 2 ˆ ˆ ∂ H/∂u = 0 ∂ 2 H/∂u <0 ˆ0u ˆ0 > 0 2λ ˆ(t) − pˆ(t) = 0, λ

! pˆ(t) − 2λˆ0uˆ(t) = 0, λˆ0 > 0). u ,- λˆ = 0 " V pˆ0(t) 6≡ 0 t ∈ [0, 1] N$':#%e !; H(u) ≡ pˆu # J# : \E,D%: %' : " (97 # 1 +∞) ':u # uˆ(t) #u,D':∈h=R2,D"≡ ' (−∞, "C uˆ(t) ; " ,c; t 'E∈,-[0, 1] S pˆ(t) ≡ 0 t ∈ [0, 1] #+ .R=#!±∞ u s E,D%: %'K "E( ,-':*P 4?'E,- J\[R,-D& 1A' " λˆ = 0 λˆ = 0 2 P "E('oE,D (#5kH "D ?f? #!& kL [ 1hZb" ,c2;Oa#C λˆ = 0 pˆ(t) ≡ 0 t ∈ [0, 1] 0 ˆ1 = 0 λ ˆ2 = 0 λ

K'E,- !;,D"7e ##E,D" Z t t $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t t @ N$ %!,D 0 E#1 . t = 0 t = 1 1G'E,- !;1A 0#!;1bh= JG#DkZ2,D"%:E,D" G 0" ,D':" ,D" 1':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ0 ≥ 0 k G'E,- ?# #,D"EH#'2brE,DD6a(#5kH "D J`f? #!& kZ21A# 2( ## 'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !\t%8%!%O# (97 #.RJ8,-':*J λˆ = 0 L , , (9" E2( J 42515*\# 42(5kZ # , (I .R_=a0#7 4\'E,- !;d - "O (

D

!S

'E,- b3 ˆ i^%!5*2,D" ˆ (5kH# 4D;"b[ $& 5kH "D # λZ0*> ,-0L* ,D"#E,D"λ 0 λˆ0 = 1/2 'E,- Z# !& ( % -E,--1,D" Z* G\'E,- + $21& ,D"7 uˆ(t) = pˆ(t) +': # # V U $ λˆuˆ0(t)= =1/2pˆ(t)+':& # # +-9 [ 4 ' 2(0'E,- !; V U - - V ZO% ':b 42515*' % ':b342515*'H #e H(9%!,D ('!(9-C

D

V V T

x ˆ˙ (t) = pˆ(t), pˆ˙ (t) = −6t;

x ˆ(0) = 0,

x ˆ(1) = 0.

i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D& E# !;+ ,D ,D"2(.31A NPN$ #e 7 #.I6a': # # JA 1;O !68 -1AD # !;8 ( " ,c;`1AEL% .I6+'E,- !;Q ,-& !;\[-C pˆ(0) = λˆ pˆ(1) = −λˆ !% ':b342515*'H#P %!b/& *b" ,c;"7%%!%1 # K,-'Ek"21;H -1AD # !;H(#5kH "D J %: " .R9 #E,c;" ,c;o/42515*'( "21AE(G 9 _= # !; λ #ˆ1λˆ 2_= # P% J\42515* L#? !;b"5 L "E('\ !6\ D& 1AD;"#P" [ ' " ,c;i^ 4 '27"7"$ _= # !;\% J\42517& * -1AD " ,c;9' 1A " E;bhZ7;d# [56E21A (./()'E,- !& ;( K E;:1A%! %!7 # JL " (97 #E,D" 'E,- !;( #e =(9%!,D ('!(9-: ,D%:E(97;K=42515* V M V -71A 'E& ,D" (97; %!,D"2(97 %!,D"2(978$ #"E; #-s _= "' % ':b]42515*'C

D

8D

ID

pˆ(t) = −3t2 + c1 , xˆ(t) = −t3 + c1 t + c2 ; x ˆ(0) = 0 ⇒ c2 = 0; x ˆ(1) = 0 ⇒ c1 = 1;

D

8D

':* (^1A 'E,D" (':b %!,D"2(97 %!,D"2(97`$ #"E; !& #-C xˆ(t) = −t3 + t ∀t ∈ [0, 1] Y\P, ,--1A' 2( ':* ##':b %!,D"2(97 xˆ(·) # [E,Db"#':b " (97 #E,D" E,D-1,D" ## J %: J %!7kZ2(I *" 1AE,D"7; "\N$':#%e #7'a[E,Db"#.RJ8( # ('!(I
7D

D

'E,- JG4 #%\ h= # !;aN$':#%e #7AC ∆I

≡ I(ˆ x(·) + x(·)) − I(ˆ x(·)) ≡ R1 R1 2 ≡ 0 [(x ˆ˙ + x) ˙ 2 − 12t(ˆ x + x)] dt − 0 (x ˆ˙ − 12tˆ x) dt ≡ R1 R1 2 ≡ 0 (2x ˆ˙ x˙ − 12tx) dt + 0 x˙ dt ≥ R1 R1 ≥ 0 2x ˆ˙ dx − 0 12tx dt ≡ 1 R R1 1 ¨ ≡ 2x ˆ˙ (t)x(t) − 0 2x ˆx dt − 0 12tx dt ≡ 0 R1 R1 ≡ 2x ˆ˙ (1) · 0 − 2x ˆ˙ (0) · 0 + 0 12tx dt − 0 12tx dt ≡ 0.

8D

t%0%!%) )b[ .I6 1A 'E,D" (.I6) 42(':h= # !;!6 D& _= # !; %!,D"2(97 xˆ(·) 8':* ## " & J )#x(·) WE,D#& # [56E21A (.I6 'E,- J E;:1A%! %!7 #&

%!,D"2('!(9 N$':#%e #7A% 'E,- J #e (9%!& # ':[ .RE7& ,D ('!(9 $ #"E; #-]N$':#%e #7 "C ∆I ≥ 0 W" %!,D"2(97 xˆI(x(·)) (t) = −t3 + t A 1 E D , 7 " ; n " 2 ( ' E [ D , b " # .RJ^( # ('!(I/$ ∀t =∈ [0, 1] "E( inf I(x(·)) ≡ abs min I(x(·)) ≡ I(ˆx(·)) = −0, 8

D

D

D

2$ /7 J+ $#NG, ;#, )MKQ,J.)-#,.$+$M$N $9C*:7 K-\K%J!$#J.,.*$,!9CJ.:K%S&'7 $#+ *)-,.#$Y/:$S$+ )M&', I#,.$J. #J -,.

T

x(T )

I(x(·), T ) ≡ x(0) = 0,

Z

T 0

x˙ 2 (t) dt → inf,

!"

T + x(T ) + 1 = 0,

T ? (97 42e !;C! [ 4 #5* (

x(t) ˙ = u(t)

B0 (x(·), u(·), T ) ≡

Z

T > 0.

" c1

T 0

u2 (t) dt → inf;

x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, T ];

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, T ]; x(0) = 0, T + x(T ) + 1 = 0;

!Y

−T < 0.

V V V V V S V V! V V Y V V #&

V ? ':#%e !;Lf? #!kC L≡

Z

T

(λ0 u2 (t)+p(t)(x(t)−u(t))) ˙ dt+

0

+λ1 x(0)+λ(T +x(T )+1)+µ(−T ) ≡

A #!kH #C

Z

T

L(t) dt+l; 0

L ≡ λ0 u2 + p(x˙ − u) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C

H ≡ pu − λ0 u2 ;

" ( ##"C

l ≡ λ1 x(0) + λ(T + x(T ) + 1) + µ(−T );

@ * ,- .RL $N ':#%e #7 #.RJ (λ0#5kHλ 1 "D λLf?µ p≡#!kp(t) S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V V V M V V #&-C K': # # J! 5M:f? #!kC

]1

−

ˆ d ˆ ˆ x (t) = 0 ⇔ pˆ˙ (t) = − ∂ H(t) ⇒ pˆ˙ (t) = 0; Lx˙ (t) + L dt ∂x

[K'E,- !;d"#,D , 7 #E,D" k -C tˆ = Tˆ 1

ˆ x˙ (tˆk ) = (−1)k L

⇔ pˆ(tˆk ) = (−1)k

= 0 tˆ0 = 0 k = 1

∂ ˆl ∂x(tk )

∂ ˆl ˆ 1 , pˆ(Tˆ) = −λ; ˆ ⇒ pˆ(0) = λ ∂x(tk )

K'E,- o " (97 #E,D" L u C t%a%!% L(u) = λˆ0 u2 − pˆu H(u) = −L(u) !"

ˆ 0 u2 − pˆ(t)u), u ˆ(t) = arg abs min L(u) ≡ arg abs min (λ u∈U

u∈(−∞,+∞)

!#

! ˆ0 u2 ). u ˆ(t) = arg abs max H(u) ≡ arg abs max (ˆ p(t)u − λ u∈U

u∈(−∞,+∞)

t%3%!% m , , (9" E2( J 42515*8,-':*J # 42(5kZ # , (I#7 4K'E,- !;$`$ ( λˆ`0 V =- 0 I"d " (97 # `': # U ≡ R1 ≡ (−∞, +∞) ( 5 Z k \ "

[ R . " \

A 1 D #\ 4Z'E,- J ∂ L/∂u ˆ u ˆ(t) =0 ! C 2 2 ˆ ˆ ˆ ∂ 2 L/∂u >0 ∂ H/∂u = 0 ∂ 2 H/∂u <0 ˆ0u ˆ0 > 0 2λ ˆ(t) − pˆ(t) = 0, λ

! pˆ(t) − 2λˆ uˆ(t) = 0, λˆ > 0); 0 0

K'E,- G,D"7e ##E,D" t +42515*K " ,D':" ,D" ' "5 "7%+%!% t @WN$ %!,D E# t0 = 0 'E,- ?,D"7e & ##E,D" L0 T ,$':* "E(d'E,- 0 J V V S -![ -C (

!

ˆ dLˆ ˆ Tˆ) + dl = 0 ⇔ = 0 ⇔ L( dT dT (1.12) ˆ 0 xˆ˙ (Tˆ) + pˆ(Tˆ) · 0 + λ(1 ˆ +x λ ˆ˙ (Tˆ)) − µ ˆ=0 ⇔ ) ˆ0u ˆ + λˆ ˆ u(Tˆ) − µ λ ˆ2 (Tˆ) + λ ˆ=0⇔ ˆ0u ˆ−µ λ ˆ2 (Tˆ) − pˆ(Tˆ)ˆ u(Tˆ) + λ ˆ = 0, ˆ ˆ0 u ˆ−µ ˆ Tˆ) = ∂ l ⇔ pˆ(Tˆ)ˆ H( u(Tˆ) − λ ˆ2 (Tˆ) = λ ˆ; ∂T

1G'E,- /1A #!;bh= JL#DkZ2,D"%:E,D" C µˆ · (−Tˆ) = 0 K'E,- !;G# " e"D #E,D" C λˆ0 ≥ 0 µˆ ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !\t%H%!%H?'E,- b V V #& (−Tˆ) < 0 "? 4/'E,- !;=1 ,-D& 1A' " µˆ = 0 E,D%: %'`# (97 #.RJO,-':*J λˆ0 = 0 , , (9" E2( J842515*H# 42(5kZ # , (I'E,- H -" !&

+'E,- b ˆ `%!5*2,D" ˆ (5kH#+ 4D;"Lb/&

[ 5kH "D λ#0 o>* 0,-:Z* ,D"#E,D"λ 0 λˆ0 = 1/2 'E,- # ( % -iI,--1,D" " aa'E,- 8 uˆ(t) = pˆ(t) O': # # V V S Z $21,D"7 'E,- ? -uˆ(t) [ =pˆ4 (t) ' 2(d'E,- !; V V S - V V Y-λˆ0-=[-1/2 A / % ':b]42515*' % ':b342515*'H #e H(9%!,D ('!(9-C

D

D

  xˆ˙ (t) = pˆ(t), pˆ˙(t) = 0; x ˆ(0) = 0, Tˆ + x ˆ(Tˆ) + 1 = 0,  2 ˆ ˆ ˆ ˆ pˆ(T ) = −λ; pˆ (T ) = 2λ.

V V

i " J0% J)42515*d# 4 2,D"#. 1A dE,D"5;##.Rd #!& " E# !;+ Z,D ,D"2(.)1A NPN$ #e 7 #.I6L': # # J 2(; ˆ Z(#5kH "D ˆ Yo# 4 2,D"#.I6 -5 P1;= !6? -1AD& # !;+ T( " ,c;`YG% λ.I6L'E,- !;Q ,- ?[-C pˆ(0) = λˆ =% ':b 42515*'Z#o %!b* " ,c;:"7%G%!%K #Z,-'EkH "P1;1 -1AD # !;G(#5kH "D; λˆ :%: " .RJK#? _= # o% J 42515* G#P !; "5 \ "E(1'G -1AD;"#P" [ ' " ,c; s _= "'G% ':b]42515*'C

PD

%D

pˆ(t) = c, x ˆ(t) = ct + c1 ; x ˆ(0) = 0 ⇒ c1 = 0; ˆ ⇒ c = −λ, ˆ x ˆ pˆ(Tˆ) = −λ ˆ(t) = −λt; 2 ˆ 2 ˆ⇒λ ˆ = 2λ, ˆ λ( ˆ λ ˆ − 2) = 0 : pˆ (T ) = 2λ ˆ ˆ 1) λ = 0, x ˆ(t) ≡ 0, T + 0 + 1 = 0, Tˆ = −1 Tˆ > 0; ˆ ˆ 2) λ = 2, x ˆ(t) = −2t, T − 2Tˆ + 1 = 0 ⇒ Tˆ = 1;

@l " * "Z'E,- b

D

8D

':* ( 2(; Tˆ = 1 L1A 'E,D" (':b %!,D"2(97 %!,D"D& (97H$ #"E; # xˆ(t) = −2t ∀t ∈ [0, 1] Y\P, ,--1A' 2(G':* ## _= # xˆ(·) Tˆ #/[E,Db"#':bn !& " (97 #E,D" E,D-1,D" ## J8 %: J8 %!7kZ2(I*" 1AE,D"7;b"+N$':#%e #7'8[E,Db"#.RJ( # ('!(I x
GD

E,D%: %' xˆ(0) = 0 xˆ(T ) = −2T " x(0) = 0 -1AD (3,K':* "E(]':* ##.I6O'E,- J 4 #%\ h= # !;LN$':#%e #7AC

T >0 x(T ) = T − 1 ∆I

≡ I(ˆ x(·) + x(·), T ) − I(ˆ x(·), Tˆ) ≡ RT R ˆ T ≡ 0 (x ˆ˙ + x) ˙ 2 dt − 0 x ˆ˙ 2 dt ≡ RT 2 RT RT R1 ≡ 0 xˆ˙ dt + 0 2x ˆ˙ x˙ dt + 0 x˙ 2 dt − 0 4 dt ≥ RT RT ≥ 0 4 dt + 0 (−4)x˙ dt − 4 ≡ T ≡ 4T − 4x(t) − 4 ≡ 4T − 4x(T ) + 4x(0) − 4 ≡ 0

8D

≡ 4T − 4(T − 1) + 0 − 4 ≡ 0.

t% %!% b[ .I6 1A 'E,D" (.I6 42(':h= # !;!6 _=D& # !; %!,D"2(97 xˆ(·) 2( # Tˆ -Z':* ## & .I6 #WE,D# l# [56E21A (.I6 'E,- J W E;:1A%!B& %!7 # %!,D"2('!(9 N$':#%e #7A 'E,- JW #e (9%!,D ('!(9$ #"E; #- N$':#%e #7 #+':[ .I& T) E "C ∆I ≥ 0 ]" 2(; Tˆ = I(x(·), ! % D , " 2(97 1 A 1 E D , 7 " ; b " 2 ( ' E [ D , b " # R . ^ J ( # ( ! ' I ( R$ x ˆ"(t)E( =inf−2t I(x(·), T ) ≡ abs min I(x(·), T ) ≡ I(ˆ x(·), Tˆ) = 4

D

D

D $:T/ 5J. IK%J.$#P,. )-$O9C.#:: M $# )-**,.7$ K+-/:$J! J.9C:+ *)M ,!, J. K%;,.&'$7KQ J. + 2)-)- # +Y+$$N59C$:#$,)M&'I,$##*,.J.$!#"$#%J $N -,9 . ,!,:)- + $9 -\

R1 I(x(·)) ≡ 0 x 2 (t) dt → inf, x(0) = x(0) ˙ = x¨(0) = 0, x(1) = 1,

!"

x(1) ˙ = 4,

T ? (97 42e ! ; CP [ 4 #5* ( x(t) " c1 x ¨(t) = x3 (t) x (t) = u(t) B0 (x1 (·), x2 (·), x3 (·), u(·)) ≡

! X

Z

= x1 (t) 1 0

x¨(1) = 12.

x(t) ˙ = x2 (t)

u2 (t) dt → inf;

x˙ 1 (t) − x2 (t) = 0, x˙ 2 (t) − x3 (t) = 0, x˙ 3 (t) − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1];

V V U V V2X

V V V ST x1 (0) = 0, x2 (0) = 0, x3 (0) = 0, x1 (1) − 1 = 0, x2 (1) − 4 = 0, x3 (1) − 12 = 0. # * # !; V Y " Z# #,D"Z42515*o " ,D':" ,D" ':b"5 V ?':#%e !;Lf? #!kC u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 1];

L≡

A #!kH #C

Z

1

L(t) dt + l; 0

L ≡ λ0 u2 + p1 (x˙ 1 − x2 ) + p2 (x˙ 2 − x3 ) + p3 (x˙ 3 − u) ≡ ≡ p1 x˙ 1 + p2 x˙ 2 + p3 x˙ 3 − H;

N$':#%e !;L$ #"E; #C " ( ##"C l

H ≡ p 1 x2 + p 2 x3 + p 3 u − λ 0 u 2 ;

≡ λ10 x1 (0) + λ20 x2 (0) + λ30 x3 (0)+ + λ11 (x1 (1) − 1) + λ21 (x2 (1) − 4) + λ31 (x3 (1) − 12);

j@p* ,- .RP +N$':#%!& e λ 0 #λ7i0 #λ.Ri1$(p#i5≡kH p"i(t) D Lf?i = 1,#!2,k3 S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V V U M V ST -C K': # # !; J! 5M:f? #!k (i = 1, 2, 3) C

I1

d ˆ ˆ xi (t) = 0 − dt Lx˙ i (t) + L ⇔ ˙pˆ1 (t) = 0, pˆ˙ 2 (t) = −ˆ p1 (t),

ˆ ∂ H(t) ˙ ˆp(t) = − i ∂xi pˆ˙ 3 (t) = −ˆ p2 (t);

[K'E,- !;d"#,D , 7 #E,D" k -C t = 1 i = 1, 2, 3

⇒

= 0 t0 = 0 k = 1

1

ˆ x˙ i (tk ) = (−1)k L

∂ ˆl ∂ ˆl ⇔ pˆi (tk ) = (−1)k ⇒ ∂xi (tk ) ∂xi (tk )

ˆ i0 , ⇒ pˆi (0) = λ

ˆ i1 ; pˆi (1) = −λ

!&

K'E,- o " (97 #E,D" L u C t%a%!% L(u) = λˆ u2 − pˆ u H(u) = −L(u) !" 0

3

ˆ 0 u2 − pˆ3 (t)u), u ˆ(t) = arg abs min L(u) ≡ arg abs min (λ u∈U

!

u∈(−∞,+∞)

ˆ0 u2 ). u ˆ(t) = arg abs max H(u) ≡ arg abs max (ˆ p3 (t)u − λ u∈U

u∈(−∞,+∞)

t%3%!% m , , (9" E2( J 42515*8,-':*J # 42(5kZ # , (I#7 4K'E,- !;$`$ ( λˆ`0 V =- 0 I"d " (97 # `': # U ≡ R1 ≡ (−∞, +∞) ( 5 Z k \ "

[ R . " \

A 1 D #\ 4Z'E,- J ∂ L/∂u ˆ u ˆ(t) =0 ! C 2 2 ˆ ˆ ˆ ∂ 2 L/∂u >0 ∂ H/∂u = 0 ∂ 2 H/∂u <0 ˆ0 u ˆ0 > 0 2λ ˆ(t) − pˆ3 (t) = 0, λ

! pˆ (t) − 2λˆ uˆ(t) = 0, λˆ > 0); 3 0 0

K'E,- !;,D"7e ##E,D" Z t0 t1 $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t t @ N$ %!,D E#. t = 0 t = 1 1G'E,- !;1A 0#!;1bh= JG#DkZ2,D"%:E,D" G 0" ,D':" ,D" 1':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ0 ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !\t%0%!%)# (97 #.RJ0,-':*J λˆ = 0 ^ , , (9" ED& ( J 42515*a# 42(5kZ # , (Ij'E,- 0 L - "nn'E,- b $5kH 'E,- =# ( % -& uˆ(t) = pˆ (t) a': # # ':*λˆ 0(I>C uˆ0(t) = pˆ3(t) λˆ$021=,D"71/2 V V2X-7 [ 4 ' 2(+'E,- !; V V2X- - V S3T P% ':b 427& (

YT

15*' % ':b)42515*'H #e H(9%!,D ('!(9-C                         

ID

x ˆ˙ 1 (t) = x ˆ2 (t), x ˆ˙ 2 (t) = x ˆ3 (t), ˙x ˆ3 (t) = pˆ3 (t); pˆ˙ 1 (t) = 0, pˆ˙ 2 (t) = −ˆ p1 (t), pˆ˙ 3 (t) = −ˆ p2 (t); x ˆ1 (0) = 0, x ˆ2 (0) = 0, x ˆ1 (1) = 1, x ˆ2 (1) = 4,

V S V xˆ3 (0) = 0; xˆ3 (1) = 12.

i " J?% Jo42515*9# 4 2,D"#.O_=2,D"/E,D"5;##.I6o #"D& E# !;+ ,D ,D"2(.31A NPN$ #e 7 #.I6a': # # JA 1;P !6$ -1AD # !;P ( " ,c;?_=2,D"/% .I6o'E,- JDQ ,-& !;O[-C i(0) = λˆi0 pˆi (1) = −λˆi i = 1, 2, 3 -a% ':b 42515*'n#pˆ` %!b* " ,c; "7% %!% # ^,-'Ek"81; -1AD& # !;+(#5kH "D J λˆ λˆ A%: " .RP#H _= # =% J 42515* `# !;b"5 Oi0 "Ei1('+ !6` -1AD;"a#" [ ' " ,c; s _= ( "'K% ':b342515*' t%L%!% pˆ1(t) = c1 " pˆ2 (t) = −c1 t + c2 pˆ3 (t) = c1 t2 /2 − c2 t + c3 x ˆ3 (t) = c1 t3 /6 − c2 t2 /2 + c3 t + c4 x ˆ2 (t) = c1 t4 /24 − c2 t3 /6 + c3 t2 /2 + c4 t + c5 5 4 3 2 x $ˆ1(t) =.Rc1'Et,-/120

!; − cV 2 StT /24 A +4 c3;tb/6"I+ c4-t1A/2 D +"Rc5tE+,D"c56;##.R -C - & ci i = 1, 6 c1 = 0 c2 = −24 cj = 0 j = 3, 4, 5, 6 ,D%: %' xˆ(t) = xˆ1 (t) A"K ,D%:E(97; %!,D"2(97 %!,D"2(97 $ #"E; # ( " :1C xˆ(t) = t4 ∀t ∈ [0, 1] Y\P, ,--1A' 2( ':* ##':b %!,D"2(97 xˆ(·) # [E,Db"#':b " (97 #E,D" E,D-1,D" ## J %: J %!7kZ2(I *" 1AE,D"7; "\N$':#%e #7'a[E,Db"#.RJ8( # ('!(I
WD

7D

D

ED

8D

N$ %!,D E#.PC

¨ x ˆ(1) + x ¨(1) = 12

E,D%: %'aL'E,- b 42515* 8%: #e.

¨ ¨ x ˆ(0) = xˆ˙ (0) = x ˆ(0) = 0, x ˆ(1) = 1, x ˆ˙ (1) = 4, x ˆ(1) = 12,

" -1AD& ˙ (Lx(0) ,':*= "x(0) E˙ (\=':x¨*(0) ##=.I06o'Ex(1) ,- = Jox(1) 4 #%P=x¨ (1)=h= 0# !;PN$':#%!& e #7AC ∆I

≡ I(ˆ x(·) + x(·)) − I(ˆ x(·)) ≡ R1 R1 2 2 ≡ 0 (x ˆ + x ) dt − 0 x ˆ dt ≡ R1 R1 2 R1 ≡ 0 2x ˆ x dt + 0 x ˆ dt ≥ 0 2 x ˆ x dt ≡ R1 R1 ≡ 0 2 · 24t x dt ≡ 0 48t d(¨ x) ≡ 1 R 1 ≡ 48t¨ x(t) − 0 48¨ x dt ≡ 0 R1 ≡ 48 · 1 · 0 − 48 · 0 · 0 − 0 48 d(x) ˙ ≡ 1 R 1 1 ≡ −48x(t) ˙ + 0 48x˙ dt = 0 + 48x(t) = 0.

0

D

0

t%)%!%) 3b[ .I6 1A 'E,D" (.I63 42(':h= # !;!6 & ':* ## J %!,D"2(97 xˆ(·) N$':#%e #7 I #q':x(·) [ .RE "C "aN$':#%e !; 1AE,D"7; "G2(' 4 ∆I ≥ 0 [E,Db"#.RJL( # ('!(IxˆA(t) $ = t"E( ∀t ∈ [0, 1]

SD

inf I(x(·)) ≡ abs min I(x(·)) ≡ I(ˆ x(·)) = 192.

$ +/ NJ ;#, )MKQ,J.)-,.$+$# $NGM$9C :+ *)M 7, $#K,.-*$!#"J!J.% $*9C,! J.K%# ,&'K%)-7$#+,.,.)-`$#)-9CYZ* ,!$J.$#K%$* &',.I$(Q#J/J.#:# +J#2-,, . # )*+,.$#J.P+%\

I(x(·)) ≡ x(0) = 1,

YS

Z

1 0

(x˙ 2 (t) + 24t2 x(t)) dt → inf,

x(1) = 5,

Z

1

tx(t) dt = 1. 0

!"

T ? (97 42e !;C! [ 4 #5* ( B0 (x(·), u(·)) ≡

Z

x(t) ˙ = u(t)

" c1

1 0

(u2 (t) + 24t2 x(t)) dt → inf;

x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1];

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 1]; Z 1 x(0) − 1 = 0, x(1) − 5 = 0, tx(t) dt − 1 = 0.

V S S V S ! V S Y V S #&

0

# * # !; V Y " Z# #,D"Z42515*o " ,D':" ,D" ':b"5 V ?':#%e !;Lf? #!kC L≡

A #!kH #C

Z

1

L(t) dt + l; 0

L ≡ λ0 (u2 + 24t2 x) + p(x˙ − u) + λtx ≡ px˙ − H;

N$':#%e !;L$ #"E; #C " ( ##"C

H ≡ pu − λ0 (u2 + 24t2 x) − λtx; l ≡ λ1 (x(0) − 1) + λ2 (x(1) − 5) − λ;

@ * ,- .RK dN$':#%e #7 #.RJ (λ0#5kHλ 1 "Dλ 2Lf?λ p≡#!kp(t) S a ,D"2(90'E,- J #e r(9%!,D ('!(9)q42515* V T M V !-C K': # # J! 5M:f? #!kC

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 ⇔ pˆ˙ (t) = − Lx˙ (t) + L ⇒ dt ∂x ˆ 0 t2 + λt; ˆ pˆ˙ (t) = 24λ

Y!

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = 1 ˆ x˙ (tk ) = (−1)k L

= 0 t0 = 0 k = 1

∂ ˆl ∂ ˆl ⇔ pˆ(tk ) = (−1)k ∂x(tk ) ∂x(tk )

ˆ1 , ⇒ pˆ(0) = λ

ˆ2 ; pˆ(1) = −λ

K'E,- o " (97 #E,D" L u C t%a%!% L(u) = λˆ u2 − pˆu H(u) = −L(u) !" 0

ˆ 0 u2 − pˆ(t)u), u ˆ(t) = arg abs min L(u) ≡ arg abs min (λ u∈U

!

u∈(−∞,+∞)

ˆ0 u2 ). u ˆ(t) = arg abs max H(u) ≡ arg abs max (ˆ p(t)u − λ u∈U

u∈(−∞,+∞)

t%3%!% m , , (9" E2( J 42515*8,-':*J # 42(5kZ # , (I #7 4L'E,- !;d=$ ( λˆ0=V = 0 I"d " (97 # `': # U ≡ R1 ≡ (−∞, +∞) ( 5 Z k \ "

[ R . " \

A 1 D #\ 4Z'E,- J ∂ L/∂u ˆ u ˆ(t) =0 ! C 2 2 ˆ ˆ ˆ ∂ 2 L/∂u >0 ∂ H/∂u = 0 ∂ 2 H/∂u <0 ˆ0u ˆ0 > 0 2λ ˆ(t) − pˆ(t) = 0, λ

! pˆ(t) − 2λˆ uˆ(t) = 0, λˆ > 0); 0 0

K'E,- !;,D"7e ##E,D" Z t0 t1 $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t0 t1 @ N$ %!,D E#. t0 = 0 t1 = 1 1G'E,- !;1A #!;bh= JG#DkZ2,D"%:E,D" G " ,D':" ,D" ':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ0 ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !\t%8%!%O# (97 #.RJ8,-':*J λˆ = 0 L , , (9" E2( J 42515*9# 42(5kZ # , (I2 .R_= 'E,-0 -2"/'E,- bn ˆ > 0 Z$%!5*2,D" λ ˆ (5kH#$ 4D;"b[ I5kH "D # λ (

Y Y

0

0

* ,-DR* ,D"#E,D" λˆ0 = 1/2 'E,- 9# ( % -DE,--1& ,D" P* H'E,- \ (t) = pˆ(t) A$21,D"7 uˆ(t) = pˆ(t) ': # # V V j λˆ0 =uˆ1/2 H': # # P- [ 4 ' 2( 'E,- !; V V -- V !j% ':b)42515*' % ':b)42515*' #e H(9%!,D ('!(9-C

x ˆ˙ (t) = pˆ(t), ˆ pˆ˙ (t) = 12t2 + λt;

R1

V S i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D&

E# !; L,D ,D"2(. 1A NPN$ #e 7 #.I6O': # # Jn (#5kH "D I 1;^ !6^ -1AD # !; ( " ,c; 1AE%D& .I6P Z 4 λ ( " *2,D%: j'E,- !;2Q ,- !;?[-C pˆ(0) = λˆ 1 ˆ 2 n% ':b 42515*'d#` %!b*b" ,c;j"7% %!% pˆ(1) = −λ # ,-'Ek"`1; -1AD # !; (#5kH "D J %: "& .R #E,c;" ,c;`\42515*'+( "21AE(^ _= #λˆ !1;λˆ#2\ _=D& # /% J42515* # !;b"5E H "E('Z !6 -1AD;"?# " [ ' " ,c;!s _= "'G% ':b342515*'C

D

x ˆ(0) = 1, xˆ(1) = 5;

0

tˆ x(t) dt = 1.

PD

ˆ 2 /2 + c1 , x ˆ 3 /6 + c1 t + c2 ; pˆ(t) = 4t3 + λt ˆ(t) = t4 + λt x ˆ(0) = 1 ⇒ c2 = 1; Z

1 0

x ˆ(1) = 5 ⇒

ˆ + c1 = 3 (∗); ⇒ λ/6

ˆ + 10c1 = 10 (∗∗); tˆ x(t) dt = 1 ⇒ λ

D

8D

ˆ = 30, c1 = −2, (∗)(∗∗) ⇒ λ

':* (^1A 'E,D" (':b %!,D"2(97 %!,D"2(97`$ #"E; !& #-C xˆ(t) = t4 + 5t3 − 2t + 1 ∀t ∈ [0, 1] Y\P, ,--1A' 2( ':* ##':b %!,D"2(97 xˆ(·) # [E,Db"#':b " (97 #E,D" E,D-1,D" ## J %: J %!7kZ2(I *" 1AE,D"7; "\N$':#%e #7'a[E,Db"#.RJ8( # ('!(I
D

D

E,D%: %'PP'E,-& b V ! " x(0) = 0 -1AD (O,R':* "E(O':* ##.I6 'E,- JG4 #%\ h= # !;aN$':#%e #7AC

R1 x(t)+x(t)) dt=1 x ˆ(1) + x(1) = 5 0 t(ˆ R1 xˆ(0) = 1 x ˆ(1) = 5 0 tˆ x(t) dt = 1 R1 x(1) = 0 0 tx(t) dt = 0

∆I ≡ I(ˆ x(·) + x(·)) − I(ˆ x(·)) ≡ Z 1 Z 1 2 2 ˙ ≡ [(x ˆ + x) ˙ + 24t (ˆ x + x)] dt − (x ˆ˙ 2 + 24t2 x ˆ) dt ≡ ≡ ≥ ≡

Z

Z

Z

0

1

2x ˆ˙ x˙ dt +

0 1

2x ˆ˙ x˙ dt +

0 1

2x ˆ˙ dx +

0

Z

Z

Z

0

1

1 0 1

1 Z ≡ 2x ˆ˙ (t)x(t) − 0

x˙ 2 dt +

0

≡ −60 ·

8D

24t2 x dt ≥

0

24t2 x dt ≡ 1

¨ 2x ˆx dt +

0

1 0

0

1

24t2 x dt ≡

≡ 2x ˆ˙ (1) · 0 − 2x ˆ˙ (0) · 0 − − Z

Z

Z

Z 1

0

1 0

24t2 x dt ≡

¨ (2x ˆ − 24t2 )x dt ≡

tx dt ≡ 0.

t%0%!%) )b[ .I6 1A 'E,D" (.I6) 42(':h= # !;!6 D& _= # !; %!,D"2(97 xˆ(·) N$':#%e #7 I #B':x(·) [ .RE "C ^" N$':#%e !; xˆ(t) = t4 + 5t3 − 2t + 1 ∆I ≥ 0 A 1 E,D"7; "2('=[E,Db"#.RJ( # ('!(I $ "E( ∀t ∈ [0, 1] 5 inf I(x(·)) ≡ abs min I(x(·)) ≡ I(ˆ x(·)) = 66

]D

7

( *# $ 4 ( " *2,D%: `'E,- R 1 tx(t) dt = 1 _= # 42515* H(5kH#P':*2,D"? H #./(I "2 *0 #./(O "P , , ( " ## $ .I& Rt _= ,DE,D [ E(I, 1ADAa42 ( #'8 2( ##.I6C 9i 4 '27"7""7%: JK42 ( #.^42515*?,/ 4 (y(t) " =*2,D%0 tx(t) (n'E,-dt !& 2(0 [ 4 ' " ,c;n842515*'n, % E7 #"#./( d2('n'E,- !;( C

SD

Y

y(t) ˙ = tx(t) y(0) = 0 y(1) = 1

5D

i " JL42515* C

L ≡ λ0 (u2 + 24t2 x) + p(x˙ − u) + q(y˙ − tx) ≡ px˙ + q y˙ − H; H ≡ pu + qtx − λ0 (u2 + 24t2 x); l ≡ λ1 (x(0) − 1) + λ2 (x(1) − 5) + µ1 y(0) + µ2 (y(1) − 1);

@ * ,- .RP +N$':#%e #7& # λ.R1 ?λ(2#5kHµ1 "Dµ 2Lf?p ≡ p(t) #!kq ≡ q(t) %':#E,D"L'E,- J M: $ #e `(9%!,D ('!(9++ [& 4 E## JG42515*o ( " :1C [

ˆ ˆ ∂ H(t) ˆ 0 t2 − qˆ(t)t, qˆ˙(t) = − ∂ H(t) ≡ 0; pˆ˙(t) = − ≡ 24λ ∂x ∂y ˆl ∂ ˆl ∂ ˆ 1 , pˆ(1) = − ˆ2 ; pˆ(0) = ≡λ ≡ −λ ∂x(0) ∂x(1) ∂ ˆl ∂ ˆl ≡µ ˆ1 , qˆ(1) = − ≡µ ˆ2 ; qˆ(0) = ∂y(0) ∂y(1) ˆ0u pˆ(t) − 2λ ˆ(t) = 0; ˆ0 ≥ 0. ; ; λ

" ,D':" ,D" ' " 1j " ,D':" ,D" ' " t%GkZ :%!%G GZ42515*o, 4 ( " *2,D% (d'E,- 2(IE& %!4 .RE " ,c; *"dn [ 4 E## Jm42515* λˆ0 6= 0 j$5kH 'E,- R# ( % - ':* ( uˆ(t) = pˆ(t) E K% E7; λ 42ˆ501=5*1/2 = ( "H :1C  ˙ x ˆ(t)    ˙ yˆ(t)  pˆ˙ (t)   ˙ qˆ(t)

WD

= pˆ(t), = tˆ x(t), = 12t2 − qˆ(t)t, = 0; x ˆ(0) = 1,

x ˆ(1) = 5,

yˆ(0) = 0,

yˆ(1) = 1.

s _= # 2( " JL% JG42515* \[ ' 1A ",D ,D"2(9HN$':#%e JC xˆ(t) = t4 + 5t3 − 2t + 1,

SD

pˆ(t) = 4t3 + 15t2 − 2,

1 6 t + t5 − 23 t3 + 12 t2 ; 6 qˆ(t) = −30. yˆ(t) =

D

$ E7;LN$':#%e !; " Ja,D ,D"2(.),D 51 "Z, %!,D"2(97 b0$ #"7& E ; # xˆ(t) ':* ## J+ .R_=?G 4 '27"7"= _= # !;+ ,c6E21A# J 42515* L,o 4 ( " *2,D% (m'E,- 2(I YU

$, / # )* *+,. #$# J.9;PJ.M+` )- * ;7,.N#KKQJ.-PJ.+)-J!J./G+*J$,!J.$K%9C :&'#7 , )-+$# )-*#,.!`#"Y%)-$9C*$,!$J.K% &'IK%#$#(QJ.,.J ##$J 9C+-,. $#*,.$/: 2 +N7#, )M, ,.$$N-\

Z

T (x(·), T ) ≡ T → inf, T

x˙ 2 (t) dt = 1,

0

x(0) = 0,

!"

T ? (97 42e !;C! [ 4 #5* (

x(T ) = 1,

x(t) ˙ = u(t)

T > 0.

" c1

B0 (x(·), u(·), T ) ≡ T → inf,

T > 0;

x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, T ];

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, T ];

V SEU V S X V S

V ! T 0# * # !; V Y " =# #,D"==42515* " ,D':" ,D" ':b"C 'E,- T > 0 =N$ (o,D" P# #,D"E?,--1A' "=,D* !& "7"K *#./(I"7%a%!%a H %!b* # =K # * # !; V Y K,D !'\,D " " ,D" ':bh= Z2('G'E,- !;K1A #!;bh= J #DkZ2,D"%:E,D" #n _= # 842515* ^#8 !; " , (II .R_= $ ( S \# !kZ?$ ( $- V ?':#%e !;Lf? #!kC Z

T

u2 (t) dt − 1 = 0,

L≡

Z

T

A #!kH #C 0

x(0) = 0,

x(T ) − 1 = 0.

p(t)(x(t) ˙ − u(t)) dt + λu2 ≡

Z

T

L(t) dt + l; 0

L ≡ p(x˙ − u) + λu2 ≡ px˙ − H;

N$':#%e !;L$ #"E; #C YEX

H ≡ pu − λu2 ;

" ( ##"C l ≡ λ0 T + λ1 x(0) + λ2 (x(T ) − 1);

@ * ,- .R$ aN$':#%e #7 #.RJ\(#& k λ0 "Dλ 1Lλf?2 p≡#!kp(t) H S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V SEU M V ! T -C K': # # J! 5M:f? #!kC

]1

−

ˆ ∂ H(t) ⇔ pˆ˙ (t) = − ⇒ ∂x

d ˆ ˆ x (t) = 0 Lx˙ (t) + L dt

pˆ˙ (t) = 0;

[K'E,- !;d"#,D , 7 #E,D" k -C tˆ1 = Tˆ ∂ ˆl ∂x(tk )

ˆ x˙ (tˆk ) = (−1)k L

ˆ1 , pˆ(0) = λ

K'E,- o " (97 #E,D" L "7%L%!% L(u) = λu ˆ 2 − pˆu

= 0 tˆ0 = 0 k = 1

⇔ pˆ(tˆk ) = (−1)k ˆ2 ; pˆ(Tˆ) = −λ

C

u H(u) = −L(u)

∂ ˆl ⇒ ∂x(tk )

!"

ˆ 2 − pˆ(t)u), u ˆ(t) = arg abs min L(u) ≡ arg abs min (λu

!

u∈U

u∈(−∞,+∞)

ˆ 2 ). u ˆ(t) = arg abs max H(u) ≡ arg abs max (ˆ p(t)u − λu u∈U

u∈(−∞,+∞)

$E,D%: %'H , , (9" E2( JG42515*

U ≡ R1 ≡ (−∞, +∞),

"n " (97 # a': # [ .R" λˆ-1A6=D0 # 4P'E,- J ! (

u ˆ(t)

2 ˆ ˆ ∂ L/∂u = 0, ∂ 2 L/∂u >0 2 2 ˆ ˆ ∂ H/∂u = 0, ∂ H/∂u < 0) :

(5kZ "

Y

ˆ u(t) − pˆ(t) = 0, λ ˆ>0 2λˆ ˆ ˆ > 0), pˆ(t) − 2λˆ u(t) = 0, λ (

!

V !V

\ " (97 # Z': # [ ' 1A "\427&

,D "Pλˆ "$= 0 -1A # !;HN$':#%e pˆ(t) E Puˆ(t) -1AD # " [ ' "H,D e 7 # #7 42 , (I# !kZ

K'E,- G,D"7e ##E,D" n t " ,D':" ,D" ':b"59"7%n%!% @WN$ %!,D E# t0 = 0 'E0,- Z,D"7e ##E,D" t0 T C ˆ dLˆ ˆ Tˆ) = ∂ l ⇒ =0 ⇔ H( dT ∂T 2 ˆ ˆ ˆ0 ; ˆ ˆ pˆ(T )ˆ u(T ) − λˆ u (T ) = λ

1G'E,- !;1A #!;bh= JG#DkZ2,D"%:E,D" G " ,D':" ,D" ':b"5"7% %!%LK42515*$# "H'E,- JL" H# #,D"'E,- 4H,D ,D"2(.r'E,- J V Y/ ,D%!b* #a O,D* "77& Tˆ > 0 " ,c;L *#./( K'E,- o# " e"D #E,D" C λˆ0 ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !GQ ,- !; V S X- V ! T -A M: - R-1,D"7;b"G,D [ JL%D& ':b342515*' % ':b342515*'H #e H(9%!,D ('!(9-C

'E,- u s !'E,- $# ( %

V ! S i " J0% J)42515*d# 4 2,D"#. 1A dE,D"5;##.Rd #!& " E# !;+ Z,D ,D"2(.)1A NPN$ #e 7 #.I6L': # # J 2(; 8(#5kH "D 8f? #!k #K# 4 2,D"#.I6 - H1;\ !Tˆ6G -1AD # !;\ ( " ,c; !='E,-λˆ 0 !λˆ; V ! T -!,--1,D" 'E,- !;H,D"7e ##E,D" K 9 K'E,- /# ( % #P'E,-& J -Q ,- !;\[-C pˆ(0) = λˆ1 pˆ(Tˆ) = −λˆ2 H% ':b342517& *'+# %!b*b" ,c;"7%8%!%8 # O,-'Ek"K1;O -1AD # !;

D

#T

 ˆ 2 ), x ˆ˙ (t) = u ˆ(t), u ˆ(t) = arg abs max (ˆ p(t)u − λu    u∈(−∞,+∞)     pˆ˙ (t) = 0; ˆ u2 (Tˆ) = λ ˆ0 ; x ˆ(0) = 0, x ˆ(Tˆ) = 1; pˆ(Tˆ)ˆ u(Tˆ) − λˆ   R ˆ  T   u ˆ2 (t) dt = 1; Tˆ > 0;   0 .

(#5kH "D J %: " .R=#G _= # Z% J+42515* # !;b"5 8λˆ1 "Eλˆ(2 '+ !6` -1AD;"L#" [ ' " ,c;s _= ( % ':b]42515*' t%a%!% pˆ˙(t) = 0 " pˆ(t) = c a 42(5kH#.^1AE21,-':*7;C c = 0 c 6= 0 u ,- c = 0 " pˆ(t) ≡ 0 t ∈ [0, T ] 7 H 4R'E,- !;Z[,--1A' "C ˆ u ,- d "E( ˆ "+ 4G'E,- !;O ˆ λ ,-1-1A=' 0" λˆλ02 == 00 i 4 '27"7"\E,DλL(=#05kH "D mf? #!k %!4 .REb" ,c;= #./( #'2b= "P " * "P'E,- b u s : G "E('H = 0 λˆ = 0 % E7;K42515*P #!& e `(9%!,D ('!(9` _= #c !;d #\ ( "59u ,- " 'E,- b " (97 #E,D" V !V λˆ >c 0= 0uˆ(t)λˆ ≡6= 00 ` "E()#L ( "8 _= # !;d% E7;n42517& t ∈ [0, T ] *P1; xˆ(t) C xˆ˙ (t) = uˆ(t) ≡ 0 xˆ(0) = 0 xˆ(Tˆ) = 1 !Z "E(' c = 0 λˆ 6= 0 #` ( "n _= # !;^ ^% E7; 42515* #e H(9%!,D ('!(9u ,- 6= 0 λˆ = 0 "H " (97 # ': # G#L,D':h=2,D" ' "C uˆc(t) 9u ,- = ±∞ ∀t ∈ [0, Tˆ] ˆ 9"8 4L'E,- !; V !V P " (97 #E,D" m u c 6= 0 λ ,--1A' "C λˆ6=>0 0 2λˆ $5kH 'E,- # ( % -':ˆ*u (t)(IC =uˆ(t)c = c $21,D"7λˆ = 1/2 ˆ 'E,- L,D"7e ##E,D" 9':* (IC c2 = 2λˆ >uˆ(T0) = c " 4 #5* "5-*"/ .R#!; " ,c;$'E,- 9# " e"0D #E,D" $ # (97 #.RJL,-':*J 42515*$# 42(5kZ #!s _= 17 =': # # xˆ˙ (t) =λˆ0 uˆ=(t)0 ≡ c ':* (IC xˆ(t) = ct + B de *%',--1,D" JC xˆ(0) = 0 ⇒ B = 0 xˆ(Tˆ) = 1 ⇒ R Tˆ ˆ2(t) dt = 1 ⇒ c2Tˆ = 1 Tˆ = 1 cTˆ = 1 c = 1/Tˆ 0 u i " o _= # !;G% JG42515* c=1 u #e ˆ(t) $(9=%!1,D (xˆ(t) '!(9o=t':*2(+ 2(; Tˆ = 1 %!,D"2(97 $ #"E; #C xˆ(t) = t ∀t ∈ [0, 1] Y\P, ,--1A' 2(G':* ## _= # xˆ(·) Tˆ #/[E,Db"#':bn !& " (97 #E,D" E,D-1,D" ## J8 %: J8 %!7kZ2(I*" 1AE,D"7;b"+N$':#%e #7'8[E,Db"#.RJ( # ('!(I x
D

D

D

D

PD

GD

E,D%: %' d'E,- b V ! T I" A$ [ 4 EZE,--1A# #,D" :':* (IC

x(·) ∈ C 1 ([0, T ]) x ˆ(0) + x(0) = 0 xˆ(T ) + x(T ) = 1 RT 2 ˙ (t) + x(t)) ( x ˆ ˙ dt = 1 T > 0 x ˆ˙ (t) = 1 0 x ˆ(T ) = T x ˆ(0) = 0 x(0) = 0 RT ˙ 2 dt = 1 x(T ) = 1 − T 0 (1 + x)

T R T T + 2x(t) + 0 x˙ 2 (t) dt = 1 ⇒ 0 RT T + 2(x(T ) − x(0)) + 0 x˙ 2 (t) dt = 1 ⇒ RT T + 2(1 − T − 0) + 0 x˙ 2 (t) dt = 1 ⇒ RT 2 1 − T + 0 x˙ (t) dt = 0.

V !&!

D

$E,D%: %' R T 2 dt ≥ 0 "8 #,D" V !&!

42(5kH# T_=>K0 1 0− xT˙ (t) "G2,D"G " 4 #5* "5E*"Z Gb[ .I61A ≤'E0,D" (.I6K 42(':h=T #≥ !;!16H _=D& # !; %!,D"2(97 xˆ(·) = 2( # Tˆ N$':#%e #7 T (x(·), T ) #$':[ .RE "C

8D

%D

WD

∆T ≡ T (x(·), T ) − T (ˆ x(·), Tˆ) ≡ T − Tˆ ≡ T − 1 ≥ 0.

D

$ "E('O 2(; %!,D"2(97 1AE,D"7;b"Z2('GTˆ[E,D=b1"#.RJa( # ('!(Ixˆ(t) $ = "t E∀t ( ∈ [0, 1] inf T (x(·), T ) ≡ abs min T (x(·), T ) ≡ T (ˆ x(·), Tˆ) ≡ Tˆ = 1.

$ ,./G$# , J ;I(%$K@ K%)* Y,! +#9C$#!#M'(Q J #B*+7'##"K, %-Y#J!4J.$##*, ;,!J.$#K%KQ,J.)*&'7)-++,.+$# )-J.$P#$+9CY% \ $K%$$#&',.I$#9CJ.#+J )M-,,..

B(x(·), T ) ≡

Z

T 0

x˙ 2 (t) dt+x2 (0) → inf,

!"

T ? (97 42e !;C! [ 4 #5* ( #S

B0 (x(·), u(·), T ) ≡

Z

T 0

x(T )+T +3 ≤ 0,

x(t) ˙ = u(t)

T > 0.

" c1

u2 (t) dt + x2 (0) → inf;

V !Y

V !#& V !& u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, T ]; V ! U x(T ) + T + 3 ≤ 0, −T < 0. # * # !; V ! " #,D"42515*o " ,D':" ,D" ':b"5 V ?':#%e !;Lf? #!kC x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, T ];

L≡

A #!kH #C

Z

T

L(t) dt + l; 0

L ≡ λ0 u2 + p(x˙ − u) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C

H ≡ pu − λ0 u2 ;

" ( ##"C

l ≡ λ0 x2 (0) + λ(x(T ) + T + 3) + µ(−T );

@ * ,- .RH nN$':#%e #7 #.RJ(#& k λ0 "Dλ Lµf?p ≡#!p(t) H k S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V !Y M V ! U -C K': # # J! 5M:f? #!kC

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x pˆ˙ (t) = 0;

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = T ˆ x˙ (tˆk ) = (−1)k L

= 0 t0 = 0 k = 1

∂ ˆl ∂ ˆl ⇔ pˆ(tˆk ) = (−1)k ⇒ ∂x(tk ) ∂x(tk )

ˆ 0 xˆ(0), pˆ(0) = 2λ

ˆ pˆ(Tˆ) = −λ;

#!

K'E,- o " (97 #E,D" L u C t%a%!% L(u) = λˆ0 u2 − pˆu H(u) = −L(u) !" ˆ 0 u2 − pˆ(t)u), u ˆ(t) = arg abs min L(u) ≡ arg abs min (λ u∈U

!

u∈(−∞,+∞)

ˆ0 u2 ). u ˆ(t) = arg abs max H(u) ≡ arg abs max (ˆ p(t)u − λ u∈U

u∈(−∞,+∞)

t%3%!% m , , (9" E2( J 42515*8,-':*J # 42(5kZ # , (I #7 4L'E,- !;d=$ ( λˆ0=V = 0 I"d " (97 # `': # U ≡ R1 ≡ (−∞, +∞) ( 5 Z k \ "

[ R . " \

A 1 D #\ 4Z'E,- J ∂ L/∂u ˆ u ˆ(t) =0 ! C 2 2 ˆ ˆ ˆ ∂ 2 L/∂u >0 ∂ H/∂u = 0 ∂ 2 H/∂u <0 ˆ0u ˆ0 > 0 2λ ˆ(t) − pˆ(t) = 0, λ

! pˆ(t) − 2λˆ0uˆ(t) = 0, λˆ0 > 0);

K'E,- G,D"7e ##E,D" t +42515*K " ,D':" ,D" ' "5 "7%+%!% t @WN$ %!,D E# t0 = 0 'E,- ?,D"7e & 0 ##E,D" L0 T C (

ˆ dLˆ ˆ Tˆ) = ∂ l ⇒ =0 ⇔ H( dT ∂T ˆ0 u ˆ−µ pˆ(Tˆ)ˆ u(Tˆ) − λ ˆ2 (Tˆ) = λ ˆ;

1G'E,- !;H1A #!;bh= JL#DkZ2,D"%:E,D" C ˆ · (ˆ λ x(Tˆ) + Tˆ + 3) = 0,

#7Y

µ ˆ · (−Tˆ) = 0;

K'E,- !;G# " e"D #E,D" C λˆ0 ≥ 0 λˆ ≥ 0 µˆ ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !\t%H%!%H?'E,- b V ! U (−Tˆ) < 0 "? 4/'E,- !;=1 ,-D& 1A' "C µˆ = 0 E,D%: %'d# (97 #.RJ ,-':*J λˆ0 = 0 m , , (9" E2( J 42515*`# 42(5kZ #^ ] 'E,- b

R"m %!5*2,D" (5kH#m 4D;"mb[ 85kH !& "D # P* ,-K* ,D"#E,Dλˆ"0 λˆ0 = 1/2 'E,- $# ( & % -iI,--1,D" " ZZ'E,- G uˆ(t) = pˆ(t) $21,D"7 o': # # V !#&-5 'E,- !;?[ = - u ˆ(t) =[ pˆ(t) 4 ' 2(8'E,- !; V !#&-!-:[λˆ-0:= M:1/2 Z% ':b]42515*' % ':b342515*'H #e H(9%!,D ('!(9-C ˆ0 > 0 λ

D

  x ˆ˙ (t) = pˆ(t),      pˆ˙ (t) = 0; ˆ pˆ(0) = x ˆ(0), pˆ(Tˆ) = −λ;   2 ˆ λ ˆ · (ˆ  pˆ (Tˆ) = 2λ; x(Tˆ) + Tˆ + 3) = 0;    ˆ ˆ λ ≥ 0; T > 0.

D

V ! X

i " J0% J)42515*d# 4 2,D"#. 1A dE,D"5;##.Rd #!& " E# !;+ Z,D ,D"2(.)1A NPN$ #e 7 #.I6L': # # J 2(; Tˆ +(#5kH "D λˆ YK# 4 2,D"#.I6 -A \1;+ !6L D& 1AD # !;] ( " ,c; S % .I6 'E,- !;R,--1,D" 'E,- !; ,D"7e ##E,D" 'E,- G1A #!;bh= J #DkZ2,D"%:E,D" Y 'E,- !; - t%%!%% E7;O42515*L,D21A EkH "a'E,- Z1A #!;bh= J #DkZ2,D"%:E,D" "R,D " " ,D" P, $ !E(G _= # !;P42515* , "7% ( 'E,- !;( , (I7 !$$ ! # [56E21A (/ , , ( "7& "=1AEZ21,-':*7;C λˆ = 0 λˆ > 0 $ λˆ = 0 % E7;G42515*= ( "H :1C

TD

xˆ˙ (t) = pˆ(t), pˆ˙(t) = 0; pˆ(0) = xˆ(0),

pˆ(Tˆ) = 0,

Tˆ > 0.

V !&

i " JG% JK42515*!?# 4 2,D"#.I6C 1A oE,D"5;##.Ro #!& " E# !;n a,D ,D"2(. 1A NPN$ #e 7 #.I6': # # J G 2(; Tˆ : K _=?1AE?'E,- !;1;\ !6K -1AD # !; %7& E7;Z42515*o %!427A ,D .R5kZ1A ## J -s _= ( "'=% ':b 42515*'C pˆ(t) = c1 xˆ(t) = c1 t + c2 pˆ(0) = xˆ(0) ⇒ c1 = c2
GD

ID

ID

"+# #,D" ':* (IC ˆ " " * "n" [ E# b Tˆ > 0 j ^T "+E(3' ≤0 λˆ = 0 % E7;G42515*= _= # !;\#P ( "5 4K a'E,- !;`11A #!;bh= J#DkZ2,D"7& $ %:E,D" mλˆ ,->-1A0' "C xˆ(Tˆ) + Tˆ+3=0 m% E7;n42515*a ( " :1C

x ˆ(t) ≡ 0

D

 ˙ xˆ(t) = pˆ(t),    ˙ pˆ(t) = 0;  pˆ(0) = x ˆ(0),   2 ˆ ˆ pˆ (T ) = 2λ;

ˆ pˆ(Tˆ) = −λ; ˆ ˆ xˆ(T ) + T + 3 = 0;

V Y T Tˆ > 0.

i " J`% J+42515*=* ".R# 4 2,D"#.I6C1A ZE,D"5;#!& # .R #" E# !;= I,D ,D"2(.O1A NPN$ #e 7 #.I6P':& # # J 2(; =(#5kH "D 2 P ( " ,c;?* ".R % .I6 'E,- !;K1;a !Tˆ6L -1AD # !;λˆs _= ( "'L% ':b)42515*'C pˆ(t) = A1 x = xˆ(0) ⇒ A1 = A2 ,D%!b* (ˆ#(t) 5kH ="DA1tλˆ+ A482 "pˆ (0) n " " m% .I6 'E,- J!':* ( 2 ⇔ pˆ(Tˆ)(ˆ p(Tˆ) + 2) = 0 * " 2,- pˆ(Tˆ) = 0 " pˆA1(Tˆ=) =0 =−2ˆAp(2Tˆ )xˆ(t) ≡ 0 t ∈ [0, Tˆ] " * "P1A '!(E,--1A# (n% ./( 'E,- !;(O42515* C 2 , " A1 = A2 = −2 0 + Tˆ + 3 = 0 Tˆ > 0 pˆ(Tˆ) = −2 x ˆ(t) = −2t − 2 x Rip ˆ"( T ˆ8) +T'ˆ:+*32=( 0 ⇒ 2(; −2TˆTˆ=− 21 + Tˆ1A+ 3'E,D=" !0& ⇒ Tˆ = 1 (':b %!,D"2(97 %!,D"2(97\$ #"E; # xˆ(t) = −2t − 2 ∀t ∈ [0, 1] Y\P, ,--1A' 2(q':* ## a _= # xˆ(·) #[E,Db"#':b !& " (97 #E,D" E,D-1,D" ## J8 %: J8 %!7kZ2(I*" 1AE,D"7;b"+N$':#%e #7'8[E,Db"#.RJ( # ('!(I x 0 %' xˆ(T ) = −2T −2ˆ(T7")+x(T −2T )+T 7 "E(' −2+x(T )+T +3 ≤ 0 A 1 D ( P , : ' * " E ( "

K # #,D"E x(T ) − T + 1 ≤ 0

D

D

8D

GD

TD

#

4 #%\ h= # !;LN$':#%e #7AC ∆I

≡ I(ˆ x(·) + x(·), T ) − I(ˆ x(·), Tˆ) ≡ RT R ˆ T ˙2 ≡ 0 (x ˆ˙ + x) ˙ 2 dt − 0 x ˆ dt+ 2 + (ˆ x(0) + x(0)) − x ˆ2 (0) ≡ RT 2 RT RT R1 ≡ 0 x ˆ˙ dt + 0 2x ˆ˙ x˙ dt + 0 x˙ 2 dt − 0 4 dt+ + 2ˆ x(0)x(0) + x2 (0) ≡ RT RT R1 ≡ 0 4 dt + 0 (−4)x ˙ dt − 0 4 dt + 2 · (−2)x(0)+ RT 2 + ˙ dt + x2 (0) ≥ 0 x T ≥ 4T − 4x(t) − 4 − 4x(0) ≡ 0

≡ 4T − 4x(T ) + 4x(0) − 4 − 4x(0) ≡ ≡ −4 · (x(T ) − T + 1) ≥ 0.

8D

t %]%!% ]b[ .I6 1A 'E,D" (.I6 42(':h= # !;!6^ _= # !; ` 2( # Tˆ N$':#%e #7 I(x(·), T ) # %!,D"2(97 ':[ .RE "C ∆I ≥xˆ(·)0 2" 2(; Tˆ=1 %!,D"2(97 xˆ(t)=−2t−2 A 1 E D , 7 " ; b ^ " 2 ( 3 ' E [ D , b"#.RJq( # ('!(IP$ ∀t ∈ [0, 1] "E( inf I(x(·), T ) ≡ abs min I(x(·), T ) ≡ I(ˆx(·), Tˆ) = 8

D

D $*,/ C+: K%)- L)-#N)M,.M\OP,.$*#7,`J.K(-PJ!,!J.J.P*+,!(QJ.K%,.&'7#+O )-"#+Y$L$)$!&'#"I%$#'J.#JJ #(-,%.

ˆ0 = 0 λ

I(x(·), u(·)) ≡

Z

1 0

(u(t) + x2 (t)) dt → inf;

x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1];

Z

1

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 1]; (u2 (t) − 2u(t)) dt + 1 = 0,

x(0) − 1 = 0.

V YV V Y S V Y! V Y Y

# * 0# !; V Y " # #,D"42515*$ " ,D':" ,D" ':b"5

!"

T ? (97 42e !;CIN$ (97 42e ^42515* ^#8" [ ' " ,c;I"7% %!%\42515*='EkZPN$ (97 4 E# #U

V ? ':#%e !;Lf? #!kC L≡

A #!kH #C

Z

1

L(t) dt + l; 0

L ≡ λ0 (u + x2 ) + p(x˙ − u) + λ(u2 − 2u) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C " ( ##"C

H ≡ pu − λ0 (u + x2 ) − λ(u2 − 2u); l ≡ λ1 (x(0) − 1);

@p* ,- .R= 8N$':#%e #7 #.RJ`(#& kHλ0 "Dλ 1Lf?λ p≡#!kp(t) S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V YV M V Y Y-C K': # # J! 5M:f? #!kC

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 ⇔ pˆ˙ (t) = − Lx˙ (t) + L ⇒ dt ∂x ˆ0 x pˆ˙(t) = 2λ ˆ;

[K'E,- !;d"#,D , 7 #E,D" k -C t =1

= 0 t0 = 0 k = 1

1

ˆ x˙ (tk ) = (−1)k L

∂ ˆl ∂x(tk )

⇔ pˆ(tk ) = (−1)k

ˆ1 , pˆ(0) = λ

pˆ(1) = 0;

K'E,- o " (97 #E,D" L u C t%%!% L(u) = λˆ0 u− pˆu+λ(u2 −2u) " #X

u ˆ(t) = arg abs min L(u) ≡ u∈U

∂ ˆl ⇒ ∂x(tk )

H(u) = −L(u)

ˆ 2 +(λ ˆ 0 − pˆ(t)−2λ)u), ˆ ≡ arg abs min (λu

!

u∈(−∞,+∞)

u ˆ(t) = arg abs max H(u) ≡ u∈U

ˆ 2 −(λ ˆ 0 − pˆ(t)−2λ)u); ˆ ≡ arg abs max (−λu u∈(−∞,+∞)

K'E,- !;,D"7e ##E,D" Z t t $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t t @ N$ %!,D 0 E#1 . t =0 t = 1 1G'E,- !;1A 0#!;1bh= JG#DkZ2,D"%:E,D" G 0" ,D':" ,D"1 ':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ ≥ 0 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !\$ [ 4 ' 2(O'E,- !; V Y S - V Y Y-! M:-! M:4 % ':b 42515*' % ':b342515*'H #e H(9%!,D ('!(9-C

D

 ˙ x ˆ(t) = u ˆ(t),     u ˆ (t) = arg abs min L(u) ≡ arg abs max H(u),    u∈U u∈U  ˆ0x pˆ˙ (t) = 2λ ˆ;  x ˆ(0) = 1, pˆ(1) = 0;   R1 2   ˆ0 ≥ 0;  u (t) − 2ˆ u(t)) dt + 1 = 0; λ   0 (ˆ .

V Y#&

'E,- u s !'E,- $# ( %

i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D& E# !; L,D ,D"2(. 1A NPN$ #e 7 #.I6O': # # Jn (#5kH "D YK# 4 2,D"#.I6 -A \1;+ !6a -1AD # !; ( " ,c;O1AE+λˆ%0λˆ .I6O'E,- !; 4 ( " *2,D%: \'E,-& / G'E,- R# ( % Y='E,- !; -E$ /'E,- /[-C ˆ L% ':b 42515*'a# %!b* " ,c;"7%8%!%8 # pˆ(0) = λ ,-'EkH "\11 ; -1AD # !;n(#5kH "D; λˆ %: " .RJO#aD& _= # /% JH42515* # !; "5E K "E1(' ? -1AD;" #$" [ ' " ,c;!s _= ( "'G% ':b342515*'

WD

#

u ,-

ˆ=0 λ

"

ˆ 0 − pˆ(t))u ≡ −H(u) L(u) = (λ

ˆ 0 − pˆ)u]. u ˆ = arg abs min [(λ

D

u∈U

$ " (97 # Z': # Z#H,D':h=2,D" 'E& 6= 0 "CjN$λˆ':0#%−ep !ˆ(t) ; L(u) "E( ,-':*a # J# j(#5kZD& ,D" U ≡ R1 ≡ (−∞, +∞) @ # # * #/u ) "E(' $21,-':*J λˆ0 − pˆ(t) = 0 # 42(& u ∈ [0, 1] kZˆ(t) #="7±∞ %a%!% ∀tpˆ(1) a 4? #,D"E pˆ(t) = λˆ0 ∀t ∈ [0, 1] =0 ,--1A' "C *"O " * " 'E,- b λˆ0u s = 0 pˆ(t) E,D%:= 0%'o∀tE,D∈([0, #5kH1] "D =f? #!kR [& hZb" ,c;] #C λˆ = 0 λˆ0 = 0 λˆ1 = 0 pˆ(t) ≡ 0 -Ii " `':*2(I *"m,-':*J λˆ = 0 # 4D& t ∈ [0, 1] (5kZ # u ,- λˆ 6= 0 !"H " (97 # $': # uˆ(t) (5kZ "[ .R" -1AD #m 48'E,- J ∂ L/∂u ˆ 2 > 0 ! ˆ = 0 ∂ 2 L/∂u C 2 ˆ ˆ ∂ H/∂u = 0 ∂ 2 H/∂u <0 ˆ u(t) + λ ˆ0 − pˆ(t) − 2λ ˆ = 0, 2λˆ

ˆ>0 λ

! − 2λˆ ˆ u(t) − λ ˆ 0 + pˆ(t) + 2λ ˆ = 0, $4 " !6\': # # JL,--1A' "C

D

(

u ˆ(t) = 1 +

ˆ0 pˆ(t) − λ , ˆ 2λ

ˆ > 0). λ

ˆ > 0. λ

$ 21,D"7 uˆ(t) H 4 ( " *2,D%: ?'E,- $ a [ 7& 4 E= H,$':* "E(n #,D"E 1 = R 1 4t3 dt C 0

0 =

≡

≡

T

R1

R01

(ˆ u2 (t) − 2ˆ u(t) + 4t3 ) dt ≡

ˆ0 2 p(t)− ˆ λ ) ˆ 0 ((1 + 2 λ R 1 p(t)− ˆ0 2 ˆ λ ( 2λˆ ) dt, 0

ˆ

ˆ λ0 − 2 − 2 p(t)− + 4t3 ) dt ≡ ˆ 2λ

':* (IC pˆ(t) = λˆ0 ∀t ∈ [0, 1] u ,- ˆ " pˆ(t) = λˆ0 6= 0 ∀t ∈ [0, 1] "H " & * "Hλ'E0,-6= 0 b pˆ(1) a "E(' λˆ = 0 A$ λˆ = 0 4 =0

D

0

0

:' # # !; pˆ˙(t) = 2λˆ0xˆ(t) = 0 'E,- !; pˆ(1) = 0 ':* (IC E,--1,D" ?* C pˆ(t) ≡ 0 t ∈ [0, 1] " c1 xˆ(t) = t + 1 ∀t ∈ [0, 1] E,D%:uˆ(t)

%' =xˆ˙1(t)∀t=∈uˆ[0,(t)1]≡ 1 xˆ(0) = 1 -IQ ,- u s "E(l "E(9" *2,D% .R#!; " ,c;C i0 "&

':*2(I *λˆ"Z0 -=1A 0#,D"pˆ (t) ##=./0(∀t _=∈ #[0, 21](O%λˆ6= 0JK42515* #e O(9%!,D ('!(9` ,--1A E"D # ,c6E21A# Jd42515* %!A , ,D *2,D%: IE e ## R ,D* ,- # !;?[ ' 1A " DC λˆ0 = 0 xˆ(t) = t + 1 uˆ(t) = 1 pˆ(t) = 0 $E,D%: %' λˆ = 0 N$':#%e #7`# " ∀t ∈ [0, 1] λ 0 _= # $#Pˆ 6=!;0 "5 Y\P, ,--1A' 2(':* ##':b %!,D"2(97H$ #"E; #

D

$L)!#"% SD

$,P)M,.OP,.$#,

D

x ˆ(t) = t + 1,

D

t ∈ [0, 1]

#a[E,Db"#':b " (97 #E,D"
D

0 R1 R1 3 (1 − 2 + 4t ) dt + 0 (2x(t) ˙ − 2x(t)) ˙ dt + 0 x˙ 2 (t) = 0 ⇔ 0 R1 2 0 + 0 + 0 x˙ (t) = 0 ⇒ x(t) ˙ ≡ 0 t ∈ [0, 1] x(0) = 0 x(t) ≡ 0 t ∈ [0, 1] x ˆ(t) = t + 1 ∀t ∈ [0, 1]

R1

/$4 " "5kZ1A2,D"E\,=':* "E(^'E,- !; ,--1A' "G"5kZ1A2,D" " 4 #5* "5E*"Z42515*2,D"Z"& %:H21A#?1A 'E,D" (97; %!,D"2(97C 91A 'E,D" (.R E e P 7,--1A E"D #-1A 'E,D" (.R E7& e aN$':#%e #7A# 42(5kH#. -A515*Z (D& "L-1A #,D" ## H _= # "L@ ∆I = 0 DC $ " E ( ˆ =0 x λ ˆ(t) = t + 1 ∀t ∈ [0, 1] 0

DD

D

D $L)!#"%$, )M,.OP,.$#,

inf I(x(·)) ≡ abs min I(x(·)) ≡ I(ˆ x(·)) ≡ Z 1 Z 1 1 2 ˙ ≡ (x ˆ(t)+ x ˆ (t)) dt ≡ (1+(t+1)2) dt ≡ 3 . 3 0 0

V

$J. / ,+ \ K% L)-N # $#M +J!*,.NG7K-J!J.*!#",!J.%K%$&'N7`+ )*)-#,.K%YL)-$$G&'(%I)*#J.+!-,#.J $-,#.,

I(x(·)) ≡

Z

1 0

(x(t) − 1)2 dt → inf;

x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1];

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 1]; x(0) − 1 = 0,

x(1) − 1 = 0.

V Y V Y U V YEX V Y

# * # !; V Y " # #,D"42515*$ " ,D':" ,D" ':b"5 !" "742515* "#E,D " ,c;=% .R5kZ1A ##./(a42515* (a%!A , ,D *2,D%:&

8E e ## ` ,D* ,- # !; #` ( "O * :1A# \ _= # C !$ %!7kZ2(I %!% "= _= # o':*7& x ˆ" (t),c;L≡#Z1 E,Duˆ#(t) o≡0 #t e∈ [0, H1](9%!,D ('!(9 T ? (97 42e !;C!42515*='EkZPN$ (97 4 E# V ?':#%e !;Lf? #!kC

1

D

A #!kH #C

L≡

Z

1

L(t) dt + l; 0

L ≡ λ0 (x − 1)2 + p(x˙ − u) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C " ( ##"C

H ≡ pu − λ0 (x − 1)2 ;

l ≡ λ1 (x(0) − 1) + λ2 (x(1) − 1);

S

@ * ,- .R$ aN$':#%e #7 #.RJ\(#& kHλ0 "Dλ 1Lλf?2 p≡#!kp(t) S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V Y M V Y-C

]1

K': # # J! 5M:f? #!kC −

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x ˆ 0 (ˆ pˆ˙(t) = 2λ x(t) − 1);

[K'E,- !;d"#,D , 7 #E,D" k -C t =1

= 0 t0 = 0 k = 1

1

ˆ x˙ (tk ) = (−1)k L

∂ ˆl ∂x(tk )

ˆ1 , pˆ(0) = λ

⇔ pˆ(tk ) = (−1)k

∂ ˆl ⇒ ∂x(tk )

ˆ2 ; pˆ(1) = −λ

K'E,- o " (97 #E,D" L u C t%a%!% L(u) = −ˆpu H(u) = −L(u) !" u ˆ(t) = arg abs min L(u) ≡ u∈U

≡ arg abs max H(u) ≡ arg abs max (ˆ p(t)u); u∈U

u∈(−∞,+∞)

K'E,- !;,D"7e ##E,D" Z t0 t1 $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t0 t1 @ N$ %!,D E#. t0 = 0 t1 = 1 1G'E,- !;1A #!;bh= JG#DkZ2,D"%:E,D" G " ,D':" ,D" ':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ0 ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k ! # (97 #.RJ?,-':*J λˆ = 0 /42515* # 42(5kZ #D"7%P%!% λˆ = 0 4o'E,- !;G0 ,--1A' " pˆ(t) = const ∀t ∈ [0, 1] 42(5kH0 #.^" L21,-':*7;C ) " (97 # ?':D& α pˆ(t) = const < 0 ∀t ∈ [0, 1] ⇒ # uˆ #P,D':h=2,D" ' "C uˆ = −∞ ∀t ∈ [0, 1] ) " (97 # ?':D& β pˆ(t) = const > 0 ∀t ∈ [0, 1] ⇒ # uˆ #P,D':h=2,D" ' "C uˆ = +∞ ∀t ∈ [0, 1] &!

E,D ) ( pˆ(t) #5kH= "const D \f?≡ 0#!tk∈? [0,[ 1]hZ⇒ b" λˆ,c1;H=Z#0λˆ 2 ⇒= #0o .I& #!; " ,c;L'E,- u s t%Z%!% λˆ0 6= 0 7"$$'E,- bm λˆ0 > 0 $ #!; λˆ0 = 1/2 'E,- $# ( % - [ 4 ' 2('E,- !; V Y U --- V Yoa% ':b 42515*' % ':b 42515*'` #e +(9%!& ,D ('!(9-C γ

D

  xˆ˙ (t) = u ˆ(t),

u ˆ(t) = arg abs max (ˆ p(t)u),

 pˆ˙(t) = xˆ(t) − 1;

u∈(−∞,+∞)

xˆ(0) = 1,

x ˆ(1) = 1.

i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D& E# !;+ ,D ,D"2(.31A NPN$ #e 7 #.I6a': # # JA 1;O !68 -1AD # !;8 ( " ,c;`1AEL% .I6+'E,- !;Q ,-& !;L[-C pˆ(1) = −λˆ K% ':b)42515*'G#? %!b/& *b" ,c;pˆ"7(0) %=%!%λˆ1 # K ,-'Ek"12;H -1AD # !;H(#5kH "D J : %

"

R . P # H _= # ?% JL42515* L#? !;b"5 λ ˆ 1 "Eλˆ(2'8 !6 -1AD;"+#G" [ ' " ,c;s _= ( "'O% ':b 42515*' t% %!% I"m " (97 # 6= 0 ': # u ∈uˆ(t)(−∞, #I,D+∞) ':h=2,D" ' "C uˆ(t)pˆ=(t)±∞ 7$ ∀t ∈ [0, 1]

" 9 ( 7

#

? : ' # = 4 'E,-& pˆ(t) ≡ 0 t ∈ [0, 1] !; " (97 #E,D" u -1AD "?#D 4D; uˆ(t) ': # "E(d,-':*#=E,D2( " 4 %: Z -1AD # b][ 'E& u 1Aˆ " P "\': # = `21A[0,# 1] 2( ##G _= # =427& 15* neDE( #J:1A2(ICV 21,D"7 pˆ(t) ≡ 0 pˆ˙(t) ≡ 0 d " L1A NPN$ #e 7 # +': # # `%D& t ∈ [0, 1] J`42515* 8 8 -1AD xˆ(t) ≡ 1 t ∈ [0, 1] S 21,D"7 H o1A NPN$ #e 7 # $': # # x %ˆ(t) ≡ 1J+42t51∈5*[0, +1] 8 -1AD uˆ(t) ≡ 0 t ∈ [0, 1] $ .R 'E,- !; xˆ(0) = 1 xˆ(1) = 1 #L':* ##E( _= # .I& #!;b" ,c; i " _= # 2( % J 42515* [ ' 1A':" N$':#%e C $ "E(3* ,-& x ˆ.R(t)(≡#51kH uˆ"(t)D O≡f?0 pˆ(t)#!k≡G0 t#∈ (9[0, b1]"L4 #5* # !;C λˆ1 = 1 ˆ 2 = 0 7u h=R4I21A* %#2(I*"P " (97 # j': # λ #ZE,D J\ " (97 # J\" %" xˆ(t) @lE,D [ u ˆ(t)

D

]7D J. 9C 21

D

Y

V # T

D

SD

Y * :1A#*" %!,D"2(97 xˆ(t) ≡ 1 t ∈ [0, 1] 1AE,D"7; " N$':#%e #7' I(x(·)) [E,Db"#.RJL( # ('!(IA$ "E( inf I(x(·)) ≡ abs min I(x(·)) ≡ I(ˆ x(·)) = 0.

$, / # $# *, )* #+9;,.$#J.MJ.`P)- + * 7,.N#;KJ.-PKQ+J!J.J./G)-*J,!+J.K%$ $&'#9C7, :)-+ )-$##,.`*Y)-!#"*%$$,!$J.9CK%&'I #(QJ.K%J ##$#J +-,,.. $9C $#*,.$/: 2 +N7#, )M, ,.$$N-\

T (x(·), T ) ≡ T → inf, T > 0; Z T 1 x(0) = x(T ) = 0; (x˙ 2 (t) − 4x(t)) dt ≤ − . 3 0

!"

T ? (97 42e !;C! [ 4 #5* (

x(t) ˙ = u(t)

" c1

B0 (x(·), u(·), T ) ≡ T → inf,

T > 0;

x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, T ];

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, T ]; x(0) = 0,

x(T ) = 0;

V #!V V # S V #! V #7Y

V #&#& #,D" T 0 > 0 ,D* "7 " ,c;p *#./(I"7%p%!% E,D%: %'q #0,D" O 0 %!b* # ^0 # * # !; V #&#&o,D !'=,D " " ,D" ':bh= /2('P'E,- !;$1A #!;bh= J #DkZ2,D"%:E,D" #n _= # 842515* ^#8 !; " , (II .R_= $ ( .qV ! V V ! V - V ?':#%e !;Lf? #!kC Z

A #!kH #C

T

(u2 (t) − 4x(t)) dt +

L≡

Z

1 ≤ 0. 3

T

L(t) dt + l; 0

L ≡ p(x˙ − u) + λ(u2 − 4x) ≡ px˙ − H;

#

N$':#%e !;L$ #"E; #C H ≡ pu − λ(u2 − 4x);

" ( ##"C

0 @ 1 * ,- .R2K dN$':#%e #7 #.RJ (#5kH "D Lf? #!k S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V #!V M V #&#&-C K': # # J! 5M:f? #!kC

l ≡ λ T + λ x(0) + λ x(T );

λ λ0 λ1 λ2 p ≡ p(t)

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x ˆ pˆ˙ (t) = −4λ;

[K'E,- !;d"#,D , 7 #E,D" k -C t =T

= 0 t0 = 0 k = 1

1

ˆ x˙ (tˆk ) = (−1)k L

∂ ˆl ∂ ˆl ⇔ pˆ(tˆk ) = (−1)k ⇒ ∂x(tk ) ∂x(tk )

ˆ1 , pˆ(0) = λ

ˆ2 ; pˆ(Tˆ) = −λ

K'E,- o " (97 #E,D" L u C t%a%!% L(u) = λu ˆ 2 − pˆu H(u) = −L(u) !"

ˆ 2 − pˆ(t)u), u ˆ(t) = arg abs min L(u) ≡ arg abs min (λu

! *"H"kZu∈U P, (

u∈(−∞,+∞)

ˆ 2 ); u ˆ(t) = arg abs max H(u) ≡ arg abs max (ˆ p(t)u − λu u∈U

u∈(−∞,+∞)

K'E,- G,D"7e ##E,D" t0 +42515*K " ,D':" ,D" ' "5 "7%+%!% @WN$ %!,D E# 'E,- ?,D"7e & ##E,D" Lt0 T ,$':* "E(d'E,-t 0 = !; 0xˆ(Tˆ) = 0 -C

&

ˆ dLˆ ˆ Tˆ) = ∂ l ⇒ =0 ⇔ H( dT ∂T 2 ˆ u (Tˆ) − 4 · 0) = λ ˆ0 ; pˆ(Tˆ)ˆ u(Tˆ) − λ(ˆ

1G'E,- /1A #!;bh= JL#DkZ2,D"%:E,D" C ˆ· λ

Z

Tˆ 0

(ˆ u2 (t) − 4ˆ x(t)) dt +

1 = 0; 3

K'E,- !;G# " e"D #E,D" C λˆ0 ≥ 0 λˆ ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !GQ ,- !; V # S M V #7Y- M:4 -1,D"7;b"L,D [ J`% ':b 42515*' ,c6E21A#':b]% ':b342515*'K #e H(9%!,D ('!(9-C                                 

x ˆ˙ (t) = u ˆ(t), ˙pˆ(t) = −4λ; ˆ ˆ 2 ), u ˆ(t) = arg abs max (ˆ p(t)u − λu u∈(−∞,+∞)

x ˆ(0) = 0, xˆ(Tˆ) = 0; ˆ1 , pˆ(Tˆ) = −λ ˆ2 ; pˆ(0) = λ 2 ˆ ˆ ˆ0 ; ˆ ˆ pˆ(T)ˆ u(T ) − λˆ u (T ) = λ R Tˆ 2 1 ˆ· λ (ˆ u (t) − 4ˆ x (t)) dt + 0 3 = 0; ˆ ˆ ˆ T > 0; λ0 ≥ 0, λ ≥ 0;

V #

E' ,- u s !'E,- o# ( % . i " J0% J)42515*d# 4 2,D"#. 1A dE,D"5;##.Rd #!& " E# !;+ Z,D ,D"2(.)1A NPN$ #e 7 #.I6L': # # J 2(; (#5kH "D f? #!k U # 4D& 2,D"#.I6Tˆ-Z 1; !6 -1AD # !;lλˆ (λˆ 0" ,c;λˆ1 1AEλˆ02 % .I6 'E,- !; ,c6E21A# J 42515* 91AEn,--1,D" !; 'E,- !; "#,-& , 7 #E,D" ,--1,D" d'E,- !;),D"7e ##E,D" ) T 'E,- 1A #!;bh= J#DkZ2,D"%:E,D" H H'E,- I# ( % (#5kH "D J f? #!k U 'E,- J - Q ,- !; pˆ(0) = λˆ1 ˆ 4o% JK42515* G(5kH#= ,D%!b* "!"7%G%!% pˆ(Tˆ) = −λ # ,-'Ek2"`1; -1AD # !; (#5kH "D J λˆ λˆ %: "& .R #E,c;" ,c;`\42515*'+( "21AE(^ _= # !1;#2\ _=D& # L% Jm42515* m#L !;b"5 "E('n !6m -1AD;" #K" [ ' " ,c;$E,-H ,D%!b* # !;n 4K% JO42515* " !6 'E,- J n# J^[ ' 1A "#O# 4 2,D"#.I6 #O'E,- Jm1;^ !6 -1AD # !; U

D

D

$ [ L" c1ZE,D?(#5kH "D Lf? #!& 0 k8λˆ =#.p #'2b= pˆ*(t)"≡# 0 42(5kH# [ pˆ(t) ≡ c 6= 0 " c1Z " (97 # $': # P#?,D':h=2,D" ' "C uˆ(t) = ±∞ $ "E('K,-':*J λˆ = 0 # 42(5kZ # ∀t ∈ [0, Tˆ] $ ':* ".RE7; P 42(5kH#E,D"j .R[ I(#5kH !& "D Jdλˆ f?6=0 #!k+,G" *λˆ#≥E,D0" bl1A85kH "D # `,DE(#& kH "D; 5kH ( λˆ = 1/2 .R[ 7 $'E,- # ( % - i "E( ,-':* ˙ = −2 λˆ0 = p2(Tˆ)/2 a& #!;p(t) " ,c;P "E(9" *2,D% ij -1A # "E(''E,- λˆ uˆ≥(t)0 = .Rp(t) 1A # "D # J 2( ## J y(t) 4 ; "$ 4 [E "E,c;Z " #" 7A` ,D 2,D" d ,c6E21A#':b % ':bp42515*'O% ,--1A':b/& h= JC

%D

ZD

                

x ˆ˙ (t) = pˆ(t), pˆ˙ (t) = −2, yˆ˙ (t) = pˆ2 (t) − 4ˆ x(t), x ˆ(0) = 0, x ˆ(Tˆ) = 0, yˆ(0) = 0, yˆ(Tˆ) = −1/3; Tˆ > 0.

V # U

s _= ()% ':bp42515*'C pˆ(t) = −2t + c xˆ(t) = −t2 + ct "7%K%!% x(0) = 0 -!$4 xˆ(Tˆ) = 0 Tˆ > 0 ,--1A' " c = Tˆ !$4 yˆ˙ (t) = (−2t + c)2 − 4(−t2 + ct) = 8t2 − 8ct + c2 y(0) = 0 ,--1A' " yˆ(t) = 8t3/3 − 4ct2 + c2 t 9Q ,- yˆ(Tˆ) = −1/3 42 (.R%! "a _= # \% J42515* 4 ;!; -1AD " E,D"7 _= 2,c;G# 4 2,D"#.R C 8c3/3 − 4c3 + c3 = −1/3 c3 = 1 c = Tˆ = 1 t% (q [ 4 E() -1AD #.pE,DL# 4 2,D"#.RL% Jd427& 15* p #e (9%!,D ('!(9C Tˆ = 1 xˆ(t) = −t2 + t ˆ1 = 1 λˆ2 = 1 λˆ0 = 1/2 pˆ(t) = −2t + 1 ∀t ∈ [0, 1] λ Y\P, ,--1A' 2(G':* ## _= # xˆ(·) Tˆ #/[E,Db"#':bn !& " (97 #E,D" E,D-1,D" ## J8 %: J8 %!7kZ2(I*" 1AE,D"7;b"+N$':#%e #7'8[E,Db"#.RJ( # ('!(I x
GD

X

x(·) ∈ C 1 ([0, T ]) x ˆ(0) + x(0) = 0 xˆ(T ) + x(T ) = 0 RT 2 ˙ (t) + x(t)) ( x ˆ ˙ − 4(ˆ x (t) + x(t)) dt ≤ −1/3 T > 0 0

~> > > TDU.BUAEoUF*CAUY?U # M *B[&H)*$ J +&'L)*/ KQ];KQJ.)-+$$9C:

$#*>!#", )M%,.$$9CY #!S#"K%%$$#NP,.J!$+!9C/* " $##*, )-,. $/:I+ )M, ,.$ 2 +N7#, )M, ,.$$N-\

I(x(·), u(·)) ≡

Z

1 0

(u2 (t) − 2x(t)) dt → inf, u(t) ∈ R1

x(t) ˙ + x(t) = u(t),

!"

∀t ∈ [0, 1],

x(0) = 1, x(1) + 1/e = 1 + e.

T ? (97 42e !;C B0 (x(·), u(·)) ≡

Z

1 0

(u2 (t) − 2x(t)) dt → inf;

V #X

V # V T u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 1]; V V x(0) − 1 = 0, x(1) + 1/e − 1 − e = 0. # * # !; V Y " Z# #,D"Z42515*o " ,D':" ,D" ':b"5 V ?':#%e !;Lf? #!kC x(t) ˙ + x(t) − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1];

A #!kH #C

L≡

Z

1

L(t) dt + l; 0

L ≡ λ0 (u2 − 2x) + p(x˙ + x − u) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C

H ≡ p(−x + u) − λ0 (u2 − 2x);

&

" ( ##"C l ≡ λ1 (x(0) − 1) + λ2 (x(1) + 1/e − 1 − e);

@ * ,- .R$ aN$':#%e #7 #.RJ\(#& kHλ0 "Dλ 1Lλf?2 p≡#!kp(t) S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V #X M V V -C K': # # J! 5M:f? #!kC

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x ˆ0 ; pˆ˙(t) = pˆ(t) − 2λ

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = 1 ˆ x˙ (tk ) = (−1)k L

= 0 t0 = 0 k = 1

∂ ˆl ∂ ˆl ⇔ pˆ(tk ) = (−1)k ⇒ ∂x(tk ) ∂x(tk )

ˆ1 , pˆ(1) = −λ ˆ2 ; pˆ(0) = λ

K'E,- o " (97 #E,D" L u C "7%L%!% L(u) = λˆ u2 − pˆu H(u) = −L(u) !" 0

ˆ 0 u2 − pˆ(t)u), u ˆ(t) = arg abs min L(u) ≡ arg abs min (λ

!

u∈U

u∈(−∞,+∞)

ˆ0 u2 ). u ˆ(t) = arg abs max H(u) ≡ arg abs max (ˆ p(t)u − λ u∈U

u∈(−∞,+∞)

t %3%!% m , , (9" E2( J 42515*8,-':*J # 42(5kZ # , (I#7 4='E,- !;aRG$ ( λˆ\0V$=1;0 42515*3%!A , ,D *2,D%: E e ## ,D* ,- # !; - I"d " (97 # `': # U ≡ R1 ≡ (−∞, +∞) ( 5 Z k \ "

[ R . " \

A 1 D #\ 4Z'E,- J ∂ L/∂u ˆ u ˆ(t) =0 ! C 2 2 ˆ ˆ ˆ ∂ 2 L/∂u >0 ∂ H/∂u = 0 ∂ 2 H/∂u <0 U7T

! (

ˆ0u ˆ0 > 0 2λ ˆ(t) − pˆ(t) = 0, λ

ˆ0u ˆ0 > 0); pˆ(t) − 2λ ˆ(t) = 0, λ

K'E,- !;,D"7e ##E,D" Z t t $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t0 t1 @ N$ %!,D 0 E#1 . t0 = 0 t1 = 1 1G'E,- !;1A #!;bh= JG#DkZ2,D"%:E,D" G " ,D':" ,D" ':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ ≥ 0 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !\t%8%!%O# (97 #.RJ8,-':*J λˆ = 0 L , , (9" E2( J 42515*# 42(5kZ # , (I 'E,- 0 $/'E,- bO λˆ > 0 "RR%!5*2,D" λˆ0 (5kH#R 4D;"Ib[ 5kH "D # 9*0 ,- K* ,D"#E,D" λˆ0 = 1/2 'E,- ?# ( % -iI,--1,D" " `O'E,- n $21,D"7 ': # # V #? λˆuˆ0(t)==1/2pˆ(t)O': # # Luˆ-(t)= [ pˆ(t)4 'E& 2(q'E,- !; V #- - V V =O% ':bB42515*' % ':b 42515*'H #e H(9%!,D ('!(9-C

D

 ˆ˙ (t) + x ˆ(t) = pˆ(t),  x pˆ˙ (t) − pˆ(t) = −1;  x ˆ(0) = 1, x ˆ(1) = 1 + e − 1/e.

D

V S

i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D& E# !;+ ,D ,D"2(.31A NPN$ #e 7 #.I6a': # # JA 1;O !68 -1AD # !;8 ( " ,c;`1AEL% .I6+'E,- !;Q ,-& !;\[-C pˆ(0) = λˆ1 pˆ(1) = −λˆ2 !% ':b342515*'H#P %!b/& *b" ,c;"7%%!% # K,-'Ek"1;H -1AD # !;H(#5kH "D J %: " .RG #E,c;" ,c;+42515*'O( "21AE(3 G _=D& λ #ˆ1 !;λˆ#2H _= # P% JL42515* \#? !;b"5 a "E('\ !6 -1AD;"#P" [ ' " ,c; s _= "'G% ':b]42515*'C

PD

%D

pˆ˙(t) = pˆ(t) − 1, dˆ p(t)/(ˆ p(t) − 1) = dt, ln |ˆ p(t) − 1| = t + A, pˆ(t) = 1 + c1 et (c1 = eA ); 1 xˆ˙ (t) + x ˆ(t) = 1 + c1 et , x ˆ(t) = 1 + c1 et + c2 e−t ; 2 xˆ(0) = 1, x ˆ(1) = 1 + e − 1/e ⇒ ⇒ c1 = 2, c2 = −1,

UV

D

8D

':* (^1A 'E,D" (':b %!,D"2(97 %!,D"2(97`$ #"E; !& #-C L " (97 # $':D& = 1 + et − e−t ∀t ∈ [0, 1] # C xˆuˆ(t) (t) = 1 + 2et ∀t ∈ [0, 1] Y\P, ,--1A' 2( ':* ## ^ _= # xˆ(·) uˆ(·) #q[E,Db"7& #':b " (97 #E,D" E,D-1,D" ## J3 %: J3 %!7& kZ2(Io*" xˆ(·) uˆ(·) 1AE,D"7;b"^N$':#%e #7')[E,Db"7& #.RJ ( # ('!(IZ
D

∆I

≡ I(ˆ x(·) + x(·), u ˆ(·) + u(·)) − I(ˆ x(·), u ˆ(·)) ≡ R1 R 1 ≡ 0 [(ˆ u + u)2 − 2(ˆ x + x)] dt − 0 (ˆ u2 − 2ˆ x) dt ≡ R1 R1 2 R1 ≡ 0 2ˆ uu dt + 0 u dt − 0 2x dt = R1 R1 R1 = 0 2ˆ u(x˙ + x) dt − 0 2x dt + 0 u2 dt ≥ R1 R1 R1 ≥ 0 2ˆ ux˙ dt + 0 2ˆ ux dt − 0 2x dt ≡ R1 R1 ≡ 0 2ˆ u dx + 0 2(ˆ u − 1)x dt ≡ 1 R R1 1 ˙ ≡ 2ˆ u(t)x(t) − 0 2u ˆx dt + 0 2(ˆ u − 1)x dt ≡ 0 R1 ≡ 2ˆ u(1) · 0 − 2ˆ u(0) · 0 + 0 2(ˆ u−u ˆ˙ − 1)x dt ≡ R1 ≡ 0 2 · 0 · x dt ≡ 0.

t%m%!%d db[ .I6O1A 'E,D" (.I6d 42(':h= # !;!6 ':* ## K _= # !; xˆ(·) uˆ(·) N$':#%e #7 I #x(·) Z':[ .Ru(·) E7& "C ∆I ≥ 0 "+N$':#%e xˆ(·) uˆ(·) 1AE,D"7;b"\2('8[E,D& b"#.RJa( # ('!(IA$ "E(

D

inf I(x(·), u(·)) ≡ abs min I(x(·), u(·)) ≡ I(ˆ x(·), u ˆ(·)) =

US

= 2e2 + 2e − 2/e − 3.

+ $#L *)-,. $&' /: LE)*/ + )MKQ# , M,.$ ;E*KQPJ.2[)- &H+)*+$$ N7$ 9C#:, )MIJ , $#,.$*$>!#", N-)M%\ $,.$9CY:# !#"7%$K%NG$#,.J!+!$/*9C :"

I(x(·), u(·)) ≡

Z

1 0

(u2 (t) + 2x(t)) dt → inf,

x ¨(t) − x(t) ˙ = u(t), u(t) ∈ R ∀t ∈ [0, 1], = 0. ! "x(1)

x(0) = 0,

T ? (97 42e !;CD [ 4 #5* ( B0 (x1 (·), x2 (·), u(·)) ≡

Z

x(t) = x1 (t) x(t) ˙ = x2 (t) 1

0

(u2 (t) + 2x1 (t)) dt → inf;

" c1 V &!

V Y V #& u(t) ∈ R1 ≡ U ≡ (−∞, +∞) ∀t ∈ [0, 1]; V & x1 (0) = 0, x1 (1) = 0. # * # !; V Y " Z# #,D"Z42515*o " ,D':" ,D" ':b"5 V ?':#%e !;Lf? #!kC x˙ 1 (t) − x2 (t) = 0, x˙ 2 (t) − x2 (t) − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1];

A #!kH #C

L≡

Z

1

L(t) dt + l; 0

L ≡ λ0 (u2 + 2x1 ) + p1 (x˙ 1 − x2 ) + p2 (x˙ 2 − x2 − u) ≡ p1 x˙ 1 + p2 x˙ 2 − H;

N$':#%e !;L$ #"E; #C

H ≡ p1 x2 + p2 (x2 + u) − λ0 (u2 + 2x1 );

" ( ##"C

l ≡ λ1 x1 (0) + λ2 x1 (1);

@ * ,- .Ro \N$':#%e & (t) p2 ≡ p2 (t) #λ70 λ#1.R$λ(2 #5pkH1 ≡"Dp 1a f? #!k U!

S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V &! M V &-C K': # # !; J! 5M:f? #!k i = 1, 2 -C

I1

−

d ˆ ˆ xi (t) = 0 Lx˙ (t) + L dt i ˆ0 ; pˆ˙1 (t) = 2λ

ˆ ∂ H(t) ⇔ pˆ˙ i (t) = − ⇒ ∂xi

pˆ˙ 2 (t) = −ˆ p1 (t) − pˆ2 (t);

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = 1 i = 1, 2 ˆ x˙ i (tk ) = (−1)k L

= 0 t0 = 0 k = 1

∂ ˆl ∂ ˆl ⇔ pˆi (tk ) = (−1)k ⇒ ∂xi (tk ) ∂xi (tk )

ˆ 1 , pˆ1 (1) = −λ ˆ2 ; pˆ1 (0) = λ

pˆ2 (0) = 0, pˆ2 (1) = 0;

K'E,- o " (97 #E,D" L u C "7%L%!% L(u) = λˆ0 u2 − pˆ2u H(u) = −L(u) "

ˆ 0 u2 − pˆ2 (t)u), u ˆ(t) = arg abs min L(u) ≡ arg abs min (λ u∈U

!

u∈(−∞,+∞)

ˆ0 u2 ). ˆ u ˆ(t) = arg abs max H(u) ≡ arg abs max (ˆ p2 (t)u − λ u∈U

u∈(−∞,+∞)

t%3%!% m , , (9" E2( J 42515*8,-':*J # 42(5kZ # , (I .R_=Za$ ( aV?1;O42515λˆ*`0 %!=A ,-0& ,D *2,D%: qE e ## 0 ,D* ,- # !;B#7 4]'E,-& !;r -$ U ≡ R ≡ (−∞, +∞) P"] " (97 # ': # uˆ(t) (5kZ "q[ .R"q -1AD #q 4]'E,-& J ∂ L/∂u ˆ 2 > 0 ! ∂ H/∂u ˆ ˆ = 0 ∂ 2 L/∂u = 0 C 2 ˆ ∂ 2 H/∂u <0 (

UY

!

ˆ0 u ˆ0 > 0 2λ ˆ(t) − pˆ2 (t) = 0, λ ˆ0u ˆ 0 > 0); pˆ2 (t) − 2λ ˆ(t) = 0, λ

K'E,- !;,D"7e ##E,D" Z t t $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t0 t1 @ N$ %!,D 0 E#1 . t0 = 0 t1 = 1 1G'E,- !;1A #!;bh= JG#DkZ2,D"%:E,D" G " ,D':" ,D" ':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ ≥ 0 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !\t%8%!%O# (97 #.RJ8,-':*J λˆ = 0 L , , (9" E2( J 42515*# 42(5kZ # , (I 'E,- 0 $/'E,- bO λˆ > 0 "RR%!5*2,D" λˆ0 (5kH#R 4D;"Ib[ 5kH "D # 9*0 ,- K* ,D"#E,D" λˆ0 = 1/2 'E,- ?# ( % -iI,--1,D" " G\'E,- + $21,D"7 ': # # !; V YI uλˆˆ(t)==1/2pˆ2(t)H': # # !;+-uˆ(t) = [ pˆ2(t) 4 ' 2( 'E,- !; V Y- V &-09- [=O% ':bp42515*' % ':b 42515*'H #e H(9%!,D ('!(9-C

D

WD

 ˙ x ˆ1 (t) = x ˆ2 (t),    ˙  ˆ2 (t) = x ˆ2 (t) + pˆ2 (t),  x pˆ˙ 1 (t) = 1,    p1 (t) − pˆ2 (t);  pˆ˙ 2 (t) = −ˆ  x ˆ1 (0) = 0, x ˆ1 (1) = 0; pˆ2 (0) = 0, pˆ2 (1) = 0.

V U

i " JK% JH42515*/# 4 2,D"#. * ".RE,D"5;##.R #!& " E# !;+ Z,D ,D"2(.)1A NPN$ #e 7 #.I6L': # # J 1;d !6n -1AD # !;m ( " ,c;d* ".RL% .I6'E,- !; Q ,- !;G[-C !% ':b]42515*'# %!b*b" ,c;p9ˆ1"7(0) %m=%!λˆ%m1 #pˆ 1(1) ,-'E=k−"+λˆ21; -1AD # !; (#& kH "D J λˆ1 λˆ2 %: " .RK#a _= # K% JO42515* O# !;b"5! \ "E('K !6G -1AD;"#$" [ ' " ,c;!s _= ( "' % ':b]42515*' t%K%!% pˆ˙1(t) = 1 E" pˆ1(t) = t + c1 pˆ˙2(t) + pˆ2(t) = −t − c1 [ h= (` _= # 2(O21A# 21A# o': # # !; [ ' 1A "nN$':#%e !; pˆ2 (t) = ce−t IvP ,D"# a pˆ˙_=2(t) # +`pˆ2#(t) 21A=#0& 21A# ': # # !;a[ ' 1A2(m ,D%!"KK :1A C pˆ2 (t) = At + B $21,D"7 pˆ2 (t) # 21A# 21A# ]': # # K':* (IC " c1 pˆ (t) = ce−t − t + 1 − c A = −1 B = 1 − c1 2 1 U#

D

@ [ h= 8 _= # # 21A# 21A# m': # # !; -1AD 4L'E,- J pˆ2(0) = 0 pˆ2(1) = 0 C c = −e/(e − 1) c c1 #J:1A2(mN$':#%e pˆ2(t) uˆ(t) C c1 = −1/(e − 1) e (1 − e−t ). e−1

pˆ2 (t) = u ˆ(t) = −t +

$21,D"7 G ,c6E21A# $1A NPN$ #e 7 # ?': # # 1; xˆ(t) uˆ'(t):* (IC V X e x¨ ˆ(t) − x ˆ˙ (t) = −t + (1 − e−t ). e−1 [ h= ( _= # 2( ': # # !; V X8[ ' 1A "q,D'!(9(90 [ h=D&

_= # !; 21A# 21A# 8': # # !; * ,D"# n _= # !; # 21A# 21A# [ h= (m _= # 2(m21A# 21A# Z': # # !; [ ' 1A "$N$':#%e !; t 2 x @¨ˆ(t)− 6:xˆ˙(t) %" = 0,D" *2,D% Jp(# xˆ*! (t)#l=':M +N # # !e; kk −k= =00 @p%: # +(# *! #-A$E,D%: %' k = 0 A"1G* ,D"7& k2 = 1 # ? _= # Z# 21A# 21A# ': # # !; xˆ 1(t) ,--1A' "G ,-& %!"P? :1A C xˆ (t) = P e−t + t(Lt + Q) $ 2NPN$ e #". P -1AD (I21,D"7 xˆ (t) H': # # V X-C L Q

%D

" c1

P =−

e , 2(e − 1)

x ˆ(t) = M + N et −

1 , 2

L=

Q=−

1 . e−1

e 1 1 e−t + t2 − t 2(e − 1) 2 e−1

@W [ h= = _= # # 21A# 21A# G': # # !; V X- D& 1AD M N 4L% .I6'E,- J xˆ(0) = 0 xˆ(1) = 0 ':* (n ,D%:E(':b %!,D"2(97$ %!,D"2(97H$ #"E; #-C

D

x ˆ(t) =

8D

e2 + e − 4 e−2 t e 1 1 − e − e−t + t2 − t. 2(e − 1)2 (e − 1)2 2(e − 1) 2 e−1

Y\P, ,--1A' 2( ':* ## ^ _= # xˆ(·) uˆ(·) #q[E,Db"7& #':b " (97 #E,D" E,D-1,D" ## J3 %: J3 %!7& kZ2(Io*" xˆ(·) uˆ(·) 1AE,D"7;b"^N$':#%e #7')[E,Db"7& #.RJ ( # ('!(IZ
D

U

E,D"77 ,D?1A 'E,D" (./( i , , (9" E2( JK427& 15*PN$':#%e xˆ(·) + x(·) uˆ(·) + u(·) [ ' 1A':"=1A 'E,D" (./( 2,- x(·) ∈ C 2 ([0, 1]) u(·) ∈ C([0, 1]) x¨ˆ+x¨ −(xˆ˙ +x)˙ = uˆ +u ˆ(1) + x(1) = 0 E,D%: %'KK'E,- b x =0 x 42ˆ5(0) 15*+ x(0) ˆ(1) = 0 x¨ˆ−xˆ˙ = uˆ 2" x(0) = 0 x(1) = 0 x ˆ(0) = 0 x 1AD (d,o':* "E(d':* ##.I6G'E,- JG4 #% x ¨− x˙=h= u# !;LN$':#%e #7AC u ˆ(·) + u(·)

∆I ≡ ≡ ≡ ≥ ≡ ≡ − ≡ ≡ ≡ ≡

I(ˆ x(·) + x(·), u ˆ(·) + u(·)) − I(ˆ x(·), u ˆ(·)) ≡ R1 R1 2 2 [(ˆ u + u) + 2(ˆ x + x)] dt − 0 (ˆ u + 2ˆ x) dt ≡ R1 2 R1 R01 2ˆ u u dt + u dt + 2x dt ≥ 0 R01 R1 0 2ˆ u (¨ x − x) ˙ dt + 2x dt ≡ 0 R R01 R1 1 2ˆ u d x ˙ − 2ˆ u dx + 0 0 2x dt ≡ 1 0 R 1 2ˆ u(t)x(t) ˙ − 0 2x˙ · u ˆ˙ dt− 0 1 R R1 1 2ˆ u(t)x(t) + 0 2xu ˆ˙ dt + 0 2x dt ≡ 0 R1 R1 R1 0 − 0 2u ˆ˙ dx − 0 + 0 2u ˆ˙ x dt + 0 2x dt ≡ 1 R R1 R1 1 ¨ −2u ˆ˙ (t)x(t) + 0 2u ˆx dt + 0 2u ˆ˙ x dt + 0 2x dt ≡ 0 R1 ¨ 0 + 0 2 (u ˆ+u ˆ˙ + 1)x dt ≡ R1 2 · 0 · x dt ≡ 0. 0

t%m%!%d db[ .I6O1A 'E,D" (.I6d 42(':h= # !;!6 ':* ## K _= # !; xˆ(·) uˆ(·) N$':#%e #7 I #x(·) Z':[ .Ru(·) E7& "C ∆I ≥ 0 "+N$':#%e xˆ(·) uˆ(·) 1AE,D"7;b"\2('8[E,D& b"#.RJa( # ('!(IA$ "E(

D

inf I(x(·), u(·)) ≡ abs min I(x(·), u(·)) ≡ I(ˆ x(·), u ˆ(·)) = = 3(2e − 3)/(e − 1)2 − 7.

;IK@KQ)* J.,!)-+9C+ $(Q$J 9C.##:+7#,M$# *!*#" %#$P[9C$#, &H)*)*G+$ K%,. $#$#J.,.P+$J %9C\; (%$ $#K%Y*,.$$#!/#:' I+ B)M', #"%,.Y#$

R2 B0 (x(·), u(·)) ≡ 1 u2 (t) dt + x2 (1) → inf, x(t) ˙ = 1t x(t) + u(t), u(t) ∈ R1 ≡ (−∞, +∞) ∀t ∈ [1, 2], x(2) ≤ 3.

U U

!"

D

i " Ja42515*?[E,Db"#.RJ+( # ('!(^N$':#%e #7A=1AE,D" D7& " ,c; * :1A#! xˆ(t) ≡ 0 uˆ(t) ≡ 0 t ∈ [1, 2] $ %!7kZ2(I!%!% "H _= # $':* " ,c;\#ZE,D# o #e H(9%!,D ('!(9 T ? (97 42e !;C

D

B0 (x(·), u(·)) ≡

Z

2 1

u2 (t) dt + x2 (1) → inf;

V &

V U7T V U V u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [1, 2]; V US x(2) − 3 ≤ 0. # * # !; V ! " #,D"42515*o " ,D':" ,D" ':b"5 V ?':#%e !;Lf? #!kC 1 x(t) ˙ − x(t) − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [1, 2]; t

A #!kH #C

L≡

Z

2

L(t) dt + l; 1

1 L ≡ λ0 u2 + p(x˙ − x − u) ≡ px˙ − H; t

N$':#%e !;L$ #"E; #C " ( ##"C

x H ≡ pu − λ0 u2 + p ; t l ≡ λ0 x2 (1) + λ(x(2) − 3);

@l* ,- .Ro LN$':#%e #7 #.RJ\(#5kH "D& λ\0 f?λ p≡#!kp(t) S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V & M V US -C UX

]1

K': # # J! 5M:f? #!kC −

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x 1 pˆ˙ (t) = − pˆ(t); t

[K'E,- !;d"#,D , 7 #E,D" k -C t =2

= 0 t0 = 1 k = 1

1

ˆ x˙ (tk ) = (−1)k L

∂ ˆl ∂ ˆl ⇔ pˆ(tk ) = (−1)k ⇒ ∂x(tk ) ∂x(tk )

ˆ 0 xˆ(1), pˆ(2) = −λ; ˆ pˆ(1) = 2λ

K'E,- o " (97 #E,D" L u C "7%L%!% L(u) = λˆ u2 − pˆu H(u) = −L(u) !" 0

ˆ 0 u2 − pˆ(t)u), u ˆ(t) = arg abs min L(u) ≡ arg abs min (λ

!

u∈U

u∈(−∞,+∞)

ˆ0 u2 ). u ˆ(t) = arg abs max H(u) ≡ arg abs max (ˆ p(t)u − λ u∈U

u∈(−∞,+∞)

t%\%!%\,-':*J # 42(5kZ # , (I#7 4o'E,-& !;Z$$ ( Pλˆ0V =1;042515*?%!A , ,D *2,D%: oE e !& ## ,D* ,- # !; I U ≡ R1 ≡ (−∞, +∞) "K " !& (97 # Z': # uˆ(t) (5kZ "\[ .R"\ -1AD #G 4 'E,- J ∂ L/∂u ˆ 2 > 0 ! ∂ H/∂u ˆ ˆ = 0 ∂ 2 L/∂u =0 C 2 ˆ ∂ 2 H/∂u <0 ˆ0u ˆ0 > 0 2λ ˆ(t) − pˆ(t) = 0, λ

! pˆ(t) − 2λˆ uˆ(t) = 0, λˆ > 0); 0 0

K'E,- !;,D"7e ##E,D" Z t t $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t0 t1 @ N$ %!,D 0 E#1 . t0 = 1 t1 = 2 1G'E,- /1A #!;bh= JL#DkZ2,D"%:E,D" C (

ˆ · (ˆ λ x(2) − 3) = 0;

U

K'E,- o# " e"D #E,D" C λˆ0 ≥ 0 λˆ ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !\t%G%!%KZ'E,- b^ # (97 #.RJK,-':*J λˆ = 0 = ,-& , (9" E2( J=42515* # 42(5kZ #2"$$'E,- 0bd λˆ0 > 0 $%!5*2,D" λˆ0 (5kH#$ 4D;"$b[ I5kH "D # I* ,- K* ,D"#E,D" 'E,- ?# ( % -iI,--1,D" " `O'E,- λˆ 0 n= 1/2 $21,D"7 Ja=[-pˆA(t)D& ': # # V U7T -A λˆuˆ0(t)= =1/2pˆ(t)G ? 4?'E,-uˆ (t) [ 4 ' 2(O'E,- !; V U7T --[-E1j% ':b342515*' %7& ':b342515*'H #e H(9%!,D ('!(9-C

D

 1 ˆ(t) + pˆ(t), u ˆ(t) = pˆ(t),  xˆ˙ (t) = t x pˆ˙(t) = − 1t pˆ(t);  ˆ λ ˆ · (ˆ pˆ(1) = x ˆ(1), pˆ(2) = −λ; x(2) − 3) = 0.

V U !

D

i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D& E# !; L,D ,D"2(. 1A NPN$ #e 7 #.I6O': # # Jn (#5kH "D E H1;G !6H -1AD # !;K ( " ,c;H1AE?% .I6 'E,- !;G L'E,-λˆ /1A #!;bh= J\#DkZ2,D"%:E,D" $ #" EH,D ,D"2('K1A NPN$ #e 7 #.I6\': # # J % JG42515* L,o':* "E(d'E,- !; pˆ(1) = xˆ(1) ':* (IC pˆ(t) = ct ; x ˆ˙ (t) = xˆ(t)+c ⇒x ˆ(t) = c1 t − c; t pˆ(1) = x ˆ(1) ⇒ c1 = 2c ⇒ xˆ(t) = 2ct − c.

0 λ $ λˆ = 0 4 " R% I'E,- !;$':* (IC pˆ(2) = 0 9$ "E( " c1 21Dkh=c =$0 xˆ (t) %:$≡# 0 uˆ (t) #,D"≡ 0xˆ(2)t ∈≤[1,3 2] * :1A# .I& #!; " ,c;

WD

XT

$ 4$'E,- !;1A #!;bh= J\#DkZ2,D"%:E,D" L,--1A' " #λˆ,D"> 0 xˆ(2) − 3 = 0 E 4 ;bh= R -1AD "=%: #,D"7#!& "' c C c = 1 i^ " $':*2(n _= # C x ˆ(t) = 2t − 1,

u ˆ(t) =

1 = pˆ(t) t

ID

∀t ∈ [1, 2].

# * # # "E() _= # d# .R#!; " ,c; K1/2 #?=#p/ˆ(2) ;;= " −λ ,c;H≤ _=0 # 2(O% JH42515* #e H(9%!,D ('!(9 Y * :1A#2*" %!,D"2(97 xˆ(t) ≡ 0 t ∈ [1, 2] =': # 1AE,D"7;b"GN$':#%e #7'a[E,Db"#.RJ u (ˆ(t) # ≡('!0(It$∈ [1, 2]"E(

D

D

inf I(x(·), u(·)) ≡ abs min I(x(·), u(·)) ≡ I(ˆ x(·), u ˆ(·)) = 0.

B$J 9C #$# ,)-*!#" %,.$`9C)-# * M,!J.$#K% , ;Z*GKQ(QJ.[J #)-&H+)*+#, $ $ $9C #K%; $#)* ,. +$,.*$#9C J.ZP 9;+$#J.`*;)-,.$KQ J.,./N#:)-J.P ++2$/ +N7#, )M, ,.$$N-\

T (x(·), u(·), T ) ≡ T → inf,

T > 0;

x(t) ˙ − x(t) + tu(t) = 0 ∀t ∈ ∆0 ≡ ∆ ≡ [0, T ]; u(t) ∈ U ≡ {u(t) | 0 < u(t) < 2} ∀t ∈ [0, T ]; Z T sin2 u(t) dt − 1 = 0, x(0) − 1 = 0.

V U Y V U #& V U V U U

# * # !;0V Y " # #,D"42515*$ " ,D':" ,D" ':b"5

!"

T ? (97 42e !;C!42515*='EkZPN$ (97 4 E# V ?:' #%e !;Lf? #!kC L≡

Z

T

L(t) dt + l; 0

XV

A #!kH #C L ≡ p(x˙ − x + tu) + λ · sin2 u ≡ px˙ − H;

N$':#%e !;L$ #"E; #C

H ≡ p(x − tu) − λ · sin2 u;

" ( ##"C

l ≡ λ0 T + λ1 (x(0) − 1);

@p* ,- .R= 8N$':#%e #7 #.RJ`(#& kHλ0 "Dλ 1Lf?λ p≡#!kp(t) S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V U Y M V U U -C K': # # J! 5M:f? #!kC

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x pˆ˙(t) = −ˆ p(t);

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = T ˆ x˙ (tˆk ) = (−1)k L

= 0 t0 = 0 k = 1

∂ ˆl ∂ ˆl ⇔ pˆ(tˆk ) = (−1)k ⇒ ∂x(tk ) ∂x(tk ) ˆ1 , pˆ(Tˆ) = 0; pˆ(0) = λ

K'E,- o " (97 #E,D" L u C t%a%!% L(u) = pˆtu + λˆ sin2 u !

H(u) = −L(u)

ˆ sin2 u + pˆ(t)tu) u ˆ(t) = arg abs min (λ u∈(0,2)

ˆ sin2 u − pˆ(t)tu); u ˆ(t) = arg abs max (−λ u∈(0,2)

XS

" V U X

K'E,- R,D"7e ##E,D" t " ,D':" ,D" ' "5"7%H%!% t @BN$ %!,D E# t0 = 1 A'E,-0 =,D"7e ##E,D" a0 C T ˆ Tˆ) = ∂ ˆl ) ⇒ = 0 (⇔ H( ∂T ˆ sin2 u ˆ0 ; pˆ(Tˆ)(ˆ x(Tˆ) − Tˆu ˆ(Tˆ)) − λ ˆ(Tˆ) = λ ˆ dL dT

1G'E,- !;1A #!;bh= JG#DkZ2,D"%:E,D" G " ,D':" ,D" ':b"5"7% %!%3^42515*O# " 'E,- J V Y\" # #,D" 'E,- Tˆ > 0 ,D ,D"2('d'E,- J V YZ#L %!b*7& " ,c;L a,D* "7 " ,c;\ *#./( K'E,- o# " e"D #E,D" C λˆ ≥ 0 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !GQ ,- !; V U #&-& V U U -2 M: -D M:4 -1,D"7;b"/,D [ J$%7& ':b342515*' % ':b342515*'H #e H(9%!,D ('!(9-C   x ˆ˙ (t) = x ˆ(t) − tˆ u(t),    ˙  p ˆ (t) = −ˆ p (t),    ˆ sin2 u + pˆ(t)tu);  u ˆ(t) = arg abs min (λ    u∈(0,2)    x ˆ(0) = 1; pˆ(Tˆ) = 0; ˆ sin2 u ˆ0 ;  pˆ(Tˆ)(ˆ x(Tˆ) − Tˆu ˆ(Tˆ)) − λ ˆ(Tˆ) = λ    R ˆ  T  2  ˆ(t) dt = 1;  0 sin u    ˆ 0 ≥ 0; ˆ  T > 0; λ   

V U

E' ,- u s \'E,- o# ( % . i " J0% J)42515*d# 4 2,D"#. 1A dE,D"5;##.Rd #!& " E# !;+ Z,D ,D"2(.)1A NPN$ #e 7 #.I6L': # # J 2(; (#5kH "D #`# 4 2,D"#.I6 - 1;m !6 -1ADTˆ # !; ( " ,c;3,-λˆ-10 ,D"λˆ O'E,- !; ,D"7e ##E,D" T 2 4 ( " *2,D%: 'E,- D ,c6E21A# 9% 9'E,-& ,--1,D" ?'E,- J\"#,D , 7 #E,D" L a'E,- $# !& ( % #?'E,- J -:$ 'E,- [-C :Z%7& ':b 42515*'?#/ %!b* " ,c;7"7%H%!%Z #?,-p'ˆ(0) EkH ="oλˆ11; D& 1AD # !;(#5kH "D; λˆ1 %: " .RJ=$% ':b 42515*'P#I6E& 1A "5#a _= # K% J842515* O#K !; "5 "E('8 -1AD;"#P" [ ' " ,c;!s _= (d% ':b342515*' X&!

D

t%d%!%

"

pˆ˙ (t) = −ˆ p(t) pˆ(Tˆ) = 0 pˆ(t) ≡ 0 t ∈ [0, Tˆ] 2 2 ˆ ˆ ˆ0 u ˆ(t) = arg abs min (λ sin u) −λ sin u ˆ(Tˆ) = λ

t%%!%O E,--1A#2(nu∈(0,2) #,D" λˆ0 ≥ 0 sin2 uˆ(Tˆ) ≥ 0 " λˆ ≤ 0 u ,- λˆ = 0 E" λˆ = 0 E,D%: %' pˆ(t) ≡ 0 t ∈ [0, Tˆ] E" 0 ˆ 1 = 0 ! L#P .R#!; " ,c;\'E,- u s λ u ,- λˆ < 0 " uˆ(t) = π/2 ∀t ∈ [0, Tˆ] −λˆ · 1 = λˆ0 E,--1& ,D" * λˆ > 0 7$5kH λˆ = 1 'E,- # ( % - ':* (IC R Tˆ0 1 dt = 1 Tˆ = 1 0s _= H17 Z42515*'L$ _= C 0 ˆ(0) = 1 ':* (IC xˆ(t) = et − πt2/4 x ˆ˙ (t) = xˆ(t) − tπ/2 x ∀t ∈ [0, Tˆ] i 4 '27"7") _= # !;W% JB42515* B #e (9%!& ,D ('!(9^':*2(ICo 2(; Tˆ=1 o': # uˆ(t) = π/2 %!,D"2(97 $ #"E; # xˆ(t) = et − πt2/4 ∀t ∈ [0, Tˆ] ∀t ∈ [0, Tˆ] Y\P, ,--1A' 2(+':* ## I _= # R#P[E,Db"#':b " (97& #E,D"$-1A5kH (IA*"a,D':h=2,D" ' " T 4 #5* # ZN$':#%!& e #7A- 1; %: " T ≤ Tˆ = 1 * :1A#n e #%!C RT RT / 4%: " J3,--1A' "C T ≡ 0 1 dt ≥ 0 sin2 u(t) dt = 1 E,D%: %'OO-1A5kZ # b T ≤ Tˆ = 1 " T ≥ 1 d,--1A E"D #` , , (9" E2( Jn42515* T = 1 = Tˆ l @ E [ D , b"#H( # (97 # $4 #5* # T Tˆ = 1 i " P':*2(IC

D

inf T (x(·), u(·), T ) ≡ abs min T (x(·), u(·), T ) ≡ ≡ T (ˆ x(·), u ˆ(·), Tˆ) ≡ Tˆ = 1, t 2 x ˆ(t) = e − πt /4, u ˆ(t) = π/2 ∀t ∈ [0, Tˆ].

J.P+ (Q,. #O" # + M$L)*!#"%[$'&H )*J $#( %*, + %K )- L)-N )M,.OP ,.$#,EJ.( P,.

B0 (x(·), u(·)) ≡

XY

Z

1 0

ˆ0 = 0 λ

(u2 (t) + u(t)) dt → inf;

x(t) ˙ − x(t) − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1];

V X T V XV

Z

1

V X S V X&!

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 1]; (x2 (t) + 2tx(t)) dt +

1 = 0, 3

x(1) + 1 = 0.

# * # !; V Y " # #,D"42515*$ " ,D':" ,D" ':b"5 0

!"

T ? (97 42e !;C!42515*='EkZPN$ (97 4 E# V ?:' #%e !;Lf? #!kC L≡

A #!kH #C

Z

1

L(t) dt + l; 0

L ≡ λ0 (u2 + u) + p(x˙ − x − u) + λ(x2 + 2tx) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C

H ≡ p(x + u) − λ0 (u2 + u) − λ(x2 + 2tx);

" ( ##"C

1 p @ * , .R= 8N$':#%e #7 #.RJ`(#& kHλ0 "Dλ 1Lf?λ p≡#!kp(t) S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V X T M V X&!-C K': # # J! 5M:f? #!kC

l ≡ λ (x(1) + 1);

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x ˆ x(t) + t); pˆ˙ (t) = −ˆ p(t) + 2λ(ˆ

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = 1 ˆ x˙ (tk ) = (−1)k L

= 0 t0 = 0 k = 1

∂ ˆl ∂ ˆl ⇔ pˆ(tk ) = (−1)k ⇒ ∂x(tk ) ∂x(tk )

ˆ1 ; pˆ(0) = 0, pˆ(1) = −λ

X#

K'E,- o " (97 #E,D" L u C "7%L%!% L(u) = λˆ0 (u2 + u) − pˆu

H(u) = −L(u)

!"

ˆ 0 (u2 + u) − pˆ(t)u), u ˆ(t) = arg abs min (λ u∈(−∞,+∞)

!

ˆ 0 (u2 + u)); u ˆ(t) = arg abs max (ˆ p(t)u − λ u∈(−∞,+∞)

K'E,- !;,D"7e ##E,D" Z t0 t1 $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t t @ N$ %!,D E#. t = 0 t = 1 1G'E,- 1A #0 !;1bh= JL#DkZ2,D"%:E,D" L 0" ,D':" ,D" 1':b"5A"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ0 ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !L _=2(n 4 ( " *2,D%: $'E,- oH :1A C 0= ≡

Z

1

0 Z 1 0

(ˆ x2 (t) + 2tˆ x(t)) dt +

1 3

≡

(ˆ x2 (t) + 2tˆ x(t) + t2 ) dt ≡

Z

1

(ˆ x(t) + t)2 dt,

0

? [ 4 ' 2(\'E,- !; V XV -5 M:-2 M:4 - V X&!% ':b 42515*' % ':b342515*'H #e H(9%!,D ('!(9-C                   

V XY

E' ,- u s !'E,- o# ( % i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D&

E# !; L,D ,D"2(. 1A NPN$ #e 7 #.I6O': # # Jn

D

X&

x ˆ˙ (t) = xˆ(t) + u ˆ(t), ˆ x(t) + t), pˆ˙ (t) = −ˆ p(t) + 2λ(ˆ ˆ0 (u2 + u)); u ˆ(t) = arg abs max (ˆ p(t)u − λ u∈U R1 x ˆ(1) = −1; pˆ(0) = 0; (ˆ x(t) + t)2 dt = 0; 0 ˆ 0 ≥ 0; λ

(#5kH "D YK# 4 2,D"#.I6 -A \1;+ !6a -1AD # !; ( " ,c;O1AE+λˆ%0λˆ .I6O'E,- !; 4 ( " *2,D%: \'E,-& / G'E,- /# ( % Y='E,- !; -Eij" 'E,- /[-C ˆ K% ':b)42515*'K#? %!b* " ,c;"7%a%!%L # pˆ(1) = −λ ,-'EkH "\1;1 -1AD # !;n(#5kH "D; λˆ1 %: " .RJO#aD& _= # /% JH42515* # !; "5E K "E(' ? -1AD;" #$" [ ' " ,c;!s _= ( "'G% ':b342515*' $4q6E21;h= lp% ':b 42515*'B 4 ( " *2,D%: 'E,- !;`,--1A' "C xˆ(t) = −t ∀t ∈ [0, 1] $ "E( % 'E,- "E(9" *2,D% 8 .R#!; " ,c;t%%!% xˆ(t) =xˆ(1)−t =∀t−1 ( 2(IC pˆ˙(t) = −ˆp(t) 2 pˆ(0) = 0 D" ∈ [0, 1] "

c

1 uˆ(t) = arg abs max (−λˆ (u2 +u)) pˆ(t) ≡ 0 t ∈ [0, 1] 0 u∈(−∞,+∞) ˆ ≥0 λ u 0,- λˆ > 0 " #!; λˆ = 1 'E,- G# ( % - 0 ':* (I0 C uˆ(t) = −1/2 ∀t ∈ [0, & x ˆ(t) − 1/2 ,D%: %' xˆ(1) = −1 " xˆ(t) = (11] −xˆ˙3e(t)t−1=)/2 "7/N$':#%e !;P# ' 1A " E; "I 4 ( " *∀t2,D∈%:E[0,('$1]'E,-& b=!$ "E('K,-':* λˆ0 > 0 _= # $% JK42515* G ,--1A E"D # ,c6E21A# JH42515* Hf? #!kP#$,D':h=2,D" ' "5 u ,- λˆ = 0 "EE,D%: %' pˆ(t) ≡ 0 t ∈ [0, 1] ': # 40'E,- !; " (97 #E,D" H u -1AD "?#D 4D; u ˆ(t) "E('$ λˆ0 = 0 "': # [ ' 1A " P5u o(5k& #\ -1AD "L 4': # # !; 21,D"7 HG# xˆ(t) = −t $xˆ˙E(t),-?=xˆ21(t),D"7+#uˆ (t) % Lt ∈':[0,* 1](IC A ( " (I*"H'E,- u s u ˆ"(t)E(]= t.R−1#!∀t; ∈" ,c[0, ;C 1]λˆ0 = 0 pˆ(t) = 0 ∀t ∈ [0, 1] λˆ1 = 0 @ b[ O* ,D"#E,D" 9*"O,--1A' "8 4G1A NPN$D& λ 0 ˆ #e 7 # H': # # !;\1λˆ; 6=pˆ(t) C i] " ':*2(I *"= _= # 2(O% JH425015=* H0 +2 λˆ#e· 0=(9%!,D !& ('!(9K ,--1A E"D # ,c6E21A# Ja42515* af? #!k[ ' 1A " DC λˆ0 = 0 xˆ(t) = −t ∀t ∈ [0, 1] , : *2( pˆ(t) ≡ 0 u ˆ(t) = t − 1 ∀t ∈ [0, 1] @ b

[

# =

#

P #'2b=$E,D%: %' ˆ1 = 0 λ ˆ t ∈ [0, 1] λ $ N : ' # % e

# 7 L# " H = _ # P # $ !; "5 ˆ0 = 0 λ Y\P, ,--1A' 2(3':* ##':b %!,D"2(97 %!,D"2(97O$ #"E;!& XU

WD

D

GD

%D

D

]J. 9C

D

9C$L )(%!)*#"%+!$-,., $)M#,., OP ,.$#,

D

D

J.

8D

7D

# #Z[E,Db"#':b] " (97 #E,D" 1]
\= −t 42(':∀t" ∈( [0, E,D-1,D" E(^N$':#%e "7% *" [ .pN$':#%e !; xˆ(·) +xˆ(·)x(·) E,D"77A ,D+1A 'E,D" ( Jx(·) il ,-& , (9" E2( J`42515*N$':#%e !; xˆ(·) + x(·) [ ' 1A "K1A 'E,D" !& ( J!2,- x(·) ∈ C 1([0, 1]) xˆ(1) + x(1) = −1 Z

t%L%!% Z

1

1

[(ˆ x(t) + x(t))2 + 2t(ˆ x(t) + x(t))] dt + 1/3 = 0.

0

xˆ(t) = −t x ˆ(1) = −1

2

(ˆ x (t)+2tˆ x(t)) dt+ 0

Z

!"

x(1) = 0

1

(2ˆ x(t)x(t)+2tx(t)) dt+ 0

]D

+

Z

1

x2 (t) dt + 1/3 = 0

0

R1 R1 ⇔ −1/3 + 0 + 0 x2 (t) dt + 1/3 = 0 ⇔ 0 x2 (t) dt = 0 ⇔ x(t) ≡ 0 t ∈ [0, 1] x ˆ(t) = −t ∀t ∈ [0, 1]

7D

$4 " \"5kZ1A2,D"EG,--1A' "5A*"\G427& 1 5*=2,D"L" %:G21A#1A 'E,D" (97; %!,D"2(97C \1A 'E,D" (.RaE e m ,--1A E"D # 1A 'E,D" (.RRE e HN$':#%e #7Ao# 42(5kH#.Pt%H%!% 7" u(t) = x(t)−x(t) x ˆ˙ (t)+ x(t)− ˙ xˆ(t)−x(t) = u ˆ(t)+u(t) ˙ ≡0 n

" E ( ' @ A 1 # D , "

! # & t ∈ [0, 1] #7;L1A 'E,D" (97;`':uˆ(t);b=hZt7−;`1N$':∀t#%∈e ![0, ;1]i3 " Z':*7& 2(I*"`42515*\ ( "+-1A #,D" ## K _= # # "E(3D& _= # Z1AE,D" D " ,c;K[E,Db"#.RJK( # ('!(nN$':#%e #7A "@ _= # ˆ0 = 0 λ $ " E ( x ˆ(t) = −t u ˆ(t) = t − 1 ∀t ∈ [0, 1]

D $L)!#"%$,

ED JBJ. 9C S(%D)*+!-,.$#,

inf B0 (x(·), u(·)) ≡ abs min B0 (x(·), u(·)) ≡ ≡ B0 (ˆ x(·), u ˆ(·)) = −1/6.

`)* ,.K% L)- /GJ. # , ) M *Z(%*[J.&H)*$KS JSJ. + 9CK% (%L)-)*N +!-,.$#, ]\ !#"%$#!/

X X

I(x(·), u(·)) ≡

Z

2 0

(tu(t) − 1)2 (2x(t) − t2 + 1)2 dt → inf;

V X#&

x(t) ˙ − tu(t) = 0 ∀t ∈ ∆0 ≡ ∆ ≡ [0, 2];

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 2]; x(0) +

1 = 0, 2

x(2) − 1 = 0.

# * # !; V Y " # #,D"42515*$ " ,D':" ,D" ':b"5

V X& V X U V X X

!"

T ? (97 42e !;C!42515*='EkZPN$ (97 4 E# V ?:' #%e !;Lf? #!kC L≡

A #!kH #C

Z

T

L(t) dt + l; 0

L ≡ λ0 (tu − 1)2 (2x − t2 + 1)2 + p(x˙ − tu) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C " ( ##"C

H ≡ ptu − λ0 (tu − 1)2 (2x − t2 + 1)2 ; 1 l ≡ λ1 (x(0) + ) + λ2 (x(2) − 1); 2

@ * ,- .R$ aN$':#%e #7 #.RJ\(#& k λ0 "Dλ 1Lλf?2 p≡#!kp(t) H S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V X#& M V X X-C K': # # J! 5M:f? #!kC

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x ˆ 0 (tˆ pˆ˙ (t) = 4λ u − 1)2 (2ˆ x(t) − t2 + 1);

X&

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = 2 ˆ x˙ (tk ) = (−1)k L

= 0 t0 = 0 k = 1

∂ ˆl ∂ ˆl ⇔ pˆ(tk ) = (−1)k ⇒ ∂x(tk ) ∂x(tk )

ˆ1 , pˆ(2) = −λ2 ; pˆ(0) = λ

K'E,- o " (97 #E,D" L u C "7%Z%!% H(u) = pˆtu− λˆ0 (tu−1)2 (2ˆx −t2 +1)2 = −L(u) " ˆ0 (tu − 1)2 (2ˆ u ˆ(t) = arg abs max [ˆ p(t)tu − λ x(t) − t2 + 1)2 ]; u∈U

K'E,- !;,D"7e ##E,D" Z t t $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t t @ N$ %!,D 0 E#1 . t = 1 t = 2 1G'E,- !;1A 0#!;1bh= JG#DkZ2,D"%:E,D" G 0" ,D':" ,D" 1':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ ≥ 0 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !GQ ,- !; V X&-& V X X- M:-- M:4 -1,D"7;b"R,D [ Jo%7& ':bq42515*' ,c6E21A#':bq% ':bq42515*'\ #e \(9%!,D !& ('!(9\,O#./(l#[ E( # [56E21A (.I6 1;] _= # !; 'E,- JC          

x ˆ˙ (t) = tˆ u(t), ˙pˆ(t) = 4λ ˆ 0 (tˆ u − 1)2 (2ˆ x(t) − t2 + 1), ˆ0 (tu − 1)2 (2ˆ u ˆ(t) = arg abs max [ˆ p(t)tu − λ x(t) − t2 + 1)2 ]; u∈U

  x ˆ(0) = −1/2,    ˆ0 ≥ 0;  λ   

x(2) = 1;

ˆ1 , pˆ(0) = λ

'E,- u s \'E,- o# ( %

ˆ2 ; pˆ(2) = −λ

V X& i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D&

E# !; L,D ,D"2(. 1A NPN$ #e 7 #.I6O': # # Jn

D

T

.

(#5kH "D 8f? #!k #K# 4 2,D"#.I6 - a1; ! 6? -1AD # !;= ( " ,c;?λˆ1A0 Eλˆ%1λˆ 2 .I6P'E,- !;= ,c6E21A# JP427& 15* 51AEP,--1,D" !;H'E,- J"#,D , 7 #E,D" H'E,- # ( % `(#5kH "D J8f? #!k #K'E,- J -AQ ,- !; (5kH#KG% ':bq42515*'\#= %!b/& pˆ(0) = λ *"9"7ˆ%1 %!pˆ(2) %m =# −,-λˆ'2Ek"+1; -1AD # !; (#5kH "D J f? #!k !%: " .Ro#= _= # $% JK42515* K# !;b"5 Oλˆ 1"Eλˆ(2'+ !68 -1AD;"L#H" [ ' " ,c;$E,- ,-& %!b* # !;` 4Z% Ja42515* " !6+'E,- JaG42515*=E,D"77& # " ,c;" K# 4 2,D"#.I6Z H" H'E,- !;?1;H !6 -1AD # !; s _= (d% ':b]42515*' $ % E7;B42515* _= # !; #q ( "5L"7% = 0 %!%8 λˆ 0"E(] ,-':*Z [ a# .R#!; " ,c;8'E,- u s -/ [ #m,D':h=2,D" 'E& [0, 2] "O "pˆ˙ (t)(97= # const L':≡0 # t ∈ uˆ(t) = ±∞ ∀t ∈ [0, 2] E $

5 H k pˆ˙ (t) = const 6= 0 t ∈ [0, 2] 'E,- # ( % -:42 _=2(λˆ#0 >[56E021A ( R'E,- λˆ 0 = 1/2 %!,-& "2('!(9HN$':#%e H(u) C

ID

D

D

ˆ ∂H = 0 ⇒ pˆ(t) − (tˆ u(t) − 1)(2ˆ x(t) − t2 + 1)2 = 0 ∀t ∈ [0, 2]. ∂u

V T
" 9 ( 7 # m': # 2 4 'E,- !; " (9−71 #E,D" V T #e r(9%!,D (u'!ˆ(t) (9 -1AD "3#D 4D;PQ # uˆ(t) "E( ,-':* P -1AD ( K21,D"7 -1AD # b [ ' 1A " a : '

# # V X&-$':* (IC uˆ(t) = 1 2 x − 1)/2 tˆ(t) %8%!=% (t xˆ(0) "L " (97 #7;8" %!& " !; xˆ(t) #5*=# −1/2 " ,c;L,oE,Dx ˆ[ (2) =6=':*1 ,D"%!C xˆ(t) = (t2 − 1)/2 u ,- tˆu(t) = 1 ="0 4 ': # # !; V X&O ':D& # uˆ(t) = 1/t % E( 'E,- xˆ(2) = 1 j':* (IC x ˆ(t) = t − 1 V

D

J. 9C

D

-1AD (d(E( #"

SD

,D".R%: % " !6\ _= # JC

t∗

(t∗ 2 − 1)/2 = t∗ − 1 ⇒ t∗ = 1.

i " P':* (IC

  1, t ∈ [0, 1], u ˆ(t) = 1  , t ∈ [1, 2]; t

SJ. 9C

  1 t2 − 1 , t ∈ [0, 1], 2 x ˆ(t) = 2  t − 1, t ∈ [1, 2].

u =h \4L21A* %#2(I*"O': # uˆ(t) [ ' 1A " P Y\$4P7 " (9ZE,D" # !;\ _= # !;a,--1A' "C

t ∈ [0, 1]

inf I(x(·), u(·)) ≡ abs min I(x(·), u(·)) ≡ I(ˆ x(·), u ˆ(·)) = 0.

~> > > TDU.BUAE\GHVAXE U"WIHYAGHF*G VACAUK~WJ.YAE # M * !#"%$&'`(%)*+!-,.$/ J; +#J / + )M, ,.$ &H)*$*,.$#, $#:#, )M+( )- + $( $#, ;KQJ.)-+$ ,-

/

$9C K%$#,.$9C $#*,.$#, 2 + N7#, )M, ,.$$ N-\ I(x(·)) ≡

Z

1 0

(x˙ 2 (t) − 4x(t)) dt → inf,

!"

t ≤ x(t) ˙ ≤ 2t ∀t ∈ [0, 1],

T ? (97 42e !;C! [ 4 #5* ( B0 (x(·), u(·)) ≡

Z

1 0

x(0) = 0.

x(t) ˙ = u(t)

" c1

(u2 (t) − 4x(t)) dt → inf;

V V

V S V &! u(t) ∈ U (t) ≡ {u(t)|t ≤ u(t) ≤ 2t} ∀t ∈ [0, 1]; V Y x(0) = 0. # * # !; V Y " Z# #,D"Z42515*o " ,D':" ,D" ':b"5 x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 ⊆ ∆ ≡ [0, 1];

S

V ? ':#%e !;Lf? #!kC L≡

A #!kH #C

Z

1

L(t) dt + l; 0

L ≡ λ0 (u2 − 4x) + p(x˙ − u) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C

H ≡ pu − λ0 (u2 − 4x);

" ( ##"C

@l* ,- .Ro LN$':#%e #7 #.RJ\(#5kH "D& λ\0 f?λ p≡#!kp(t) S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 42515* V V M V Y-C K': # # J! 5M:f? #!kC l ≡ λx(0);

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x ˆ0 ; pˆ˙ (t) = −4λ

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = 1 ˆ x˙ (tk ) = (−1)k L

∂ ˆl ∂x(tk )

ˆ pˆ(0) = λ,

= 0 t0 = 0 k = 1

⇔ pˆ(tk ) = (−1)k

∂ ˆl ⇒ ∂x(tk )

pˆ(1) = 0;

K'E,- o " (97 #E,D" L u C "7%L%!% L(u) = λˆ u2 − pˆu H(u) = −L(u) !" 0

u ˆ(t)

= arg abs min L(u) ≡ arg abs max H(u) ≡ u∈U

u∈U

ˆ 0 u2 − pˆ(t)u) ≡ ≡ arg abs min (λ t≤u≤2t

ˆ 0 u2 ). ≡ arg abs max (ˆ p(t)u − λ t≤u≤2t

&!

[ " ( # (9# n*" " (97 # W': # ) , , (9" E2( J 42515* #D 4D; -1AD;" u ˆ(t) 4 'E,- J ∂ L/∂u ˆ 2 > 0 ! ˆ = 0 ∂ 2 L/∂u "7%`%!%8(#5kZ2,D" U (t) 2 ˆ ˆ ∂ H/∂u = 0 ∂ 2 H/∂u <0 @ D a # [56E21A (\#76E21A "a ∀t ∈ [0, 1] %!% 4 #5* # ^ '!( #"7 u = %: " E( N$':#%e !; # (9 "H# ( # _= ! aN$':#%!& ˆ 2 L(u) ≡ λ e !; H(u)0 u≡−pˆpuˆu− λˆ0 u2 @ # [ _= ?4 #5* # a# 3(#5kZ2,D" " 4 %: t ≤ u ≤ 2t -

K'E,- !; ,D"7e ##E,D" m t t " ,D':" ,D" ':b"5 "7% %!% t0 t1 @ N$ %!,D E#. t00= 01 t1 = 1 1G'E,- !;1A #!;bh= JG#DkZ2,D"%:E,D" G " ,D':" ,D" ':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ0 ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k ! # (97 #.RJ ,-':*J λˆ = 0 42515*a# 42(5kZ #9"7% %!%) λˆ0 = 0 4d'E,-0 !;0+,--1A' "C pˆ(t) = const ˆ u s pˆ(1) = 0 ⇒ pˆ(t) ≡ 0 t ∈ [0, 1] ⇒ λ $ λˆ0 > 0 , (I9'E,- \ -95kH =λˆ00=;1 'E,- # ( % - [ 4 ' 2(3'E,- !; V S - V Y-9 M:? % ':b]42515*' % ':b342515*'H #e H(9%!,D ('!(9-C

!:K%$(%,

!:K%$(%'

  x ˆ˙ (t) = u ˆ(t),

D

 pˆ˙ (t) = −4;

Y

u ˆ(t) = arg abs max (ˆ p(t)u − u2 ), t≤u≤2t

xˆ(0) = 0;

pˆ(1) = 0.

V #&

i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D&

E# !;+ ,D ,D"2(.31A NPN$ #e 7 #.I6a': # # JA 1;P !6$ -1AD # !;P ( " ,c;o1AEI% .I6o'E,- !;DQ ,- [-C pˆ(0) = λˆ P% ':b 42515*'?#R %!b* " ,c;"7%H%!%Z # ,-'EkH "$1;H -1AD # !;G(#5kH "D; E%: " .RJH#P _=D& # /% J42515* # !; "5 K "E(λˆ'Z P -1AD;"?# " [ ' " ,c;!s _= ( "'G% ':b]42515*' t%P%!% " $ "Ep(Iˆ˙(t) #=−4 ,D Epˆ(1) `1=; 0# g;pˆ:(t) 1A#E=,D"4−4t ≥N$0 %^∀tN$∈':#[0, %e1]

5D

WD

H(u) ≡ pˆu − u2

u ˆ(t) =

≡ ≡

$21,D"7 x ˆ(t) =

u ˆ(t) Z

t 0

u ∈ [t, 2t]   2t, pˆ(t)/2,  t,   2t, 2 − 2t,  t,   2t, 2 − 2t,  t,

!':* (IC pˆ(t)/2 ≥ 2t t ≤ pˆ(t)/2 ≤ 2t ≡ pˆ(t)/2 ≤ t 2 − 2t ≥ 2t t ≤ 2 − 2t ≤ 2t ≡ 2 − 2t ≤ t 0 ≤ t ≤ 1/2 1/2 ≤ t ≤ 2/3 . 2/3 ≤ t ≤ 1

': # # V S -!':* (IC

 2 0 ≤ t ≤ 1/2  t + C1 , 2t − t2 + C2 , 1/2 ≤ t ≤ 2/3 . u ˆ(t) dt ≡  2 t /2 + C3 , 2/3 ≤ t ≤ 1

t%O%!% " LE,D%: %'+N$':#%e !; xˆ(t) =0 # .R xˆ#(0) $ xˆ=(·)0 ∈ KCC11([0, -A"K 4?'E,- JL# .R& #E,D" L t = 1/2 1/4 + 0 =1])1 − 1/4 + C2 I L t = 2/3 − 4/9 + C = 2/9 + C 9,--1A' "C C = −1/2 C = 1/6 iO4/3 4 '27"7" 2':* (+ 3" (97 # ':2 # uˆ(t)3 %!,-& "2(97H$ #"E; # xˆ(t) C

]D

 0 ≤ t ≤ 1/2  2t, 2 − 2t, 1/2 ≤ t ≤ 2/3 , u ˆ(t) =  t, 2/3 ≤ t ≤ 1  2 0 ≤ t ≤ 1/2  t , 2t − t2 − 1/2, 1/2 ≤ t ≤ 2/3 . x ˆ(t) =  2 t /2 + 1/6, 2/3 ≤ t ≤ 1

D

Y\P, ,--1A' 2( ':* ##':b %!,D"2(97 xˆ(·) # [E,Db"#':b " (97 #E,D" E,D-1,D" ## J %: J %!7kZ2(I *" 1AE,D"7; "\N$':#%e #7'a[E,Db"#.RJ8( # ('!(I
D

ED

E,D%: %'dn'E,-& ∈ [0, 1]

b042515* A" ∀tx(0) $ "E( # * # #Z 4 21A#':b] ,D51 " ,c;\#=Z0" L'E,- !;C

t ≤ x ˆ˙ (t) + x(t) ˙ ≤ 2t x ˆ(0) = 0

t ≤ 2t + x(t) ˙ ≤ 2t ∀t ∈ [0, 1/2] ⇔ −t ≤ x(t) ˙ ≤ 0 ∀t ∈ [0, 1/2]; t ≤ 2 − 2t + x(t) ˙ ≤ 2t ∀t ∈ [1/2, 2/3] ⇔ 3t − 2 ≤ x(t) ˙ ≤ 4t − 2 ∀t ∈ [1/2, 2/3]; t ≤ t + x(t) ˙ ≤ 2t ∀t ∈ [2/3, 1] ⇔ 0 ≤ x(t) ˙ ≤ t ∀t ∈ [2/3, 1].

-1AD ( ,=':* "E( ':* ##.I6+'E,- J`4 #%` h=D& # !;LN$':#%e #7AC ∆I

≡ I(ˆ x(·) + x(·)) − I(ˆ x(·)) ≡ R1 R1 2 2 ≡ 0 [(x ˆ˙ + x) ˙ − 4(ˆ x + x)] dt − 0 (x ˆ˙ − 4ˆ x) dt ≡ R1 R1 2 R1 ≡ 2 0 xˆ˙ x˙ dt + 0 x˙ dt − 4 0 x dt ≥ R1 R1 ≥ 2 0 xˆ˙ x˙ dt − 4 0 x dt ≡ R1 R1 ≡ 2 0 xˆ˙ x˙ dt − 4 0 x d(t − 1) ≡ 1 R1 R1 ≡ 2 0 xˆ˙ x˙ dt − 4x(t)(t − 1) + 4 0 (t − 1)x˙ dt = 0 R1 = 0 (2x ˆ˙ + 4t − 4)x˙ dt − 0 ≡ R 1/2 R 2/3 R1 ≡ 0 (8t − 4)x˙ dt + 1/2 0 · x˙ dt + 2/3 (6t − 4)x˙ dt.

t% %!% ˙ R≤1/20(8t −∀t 4)∈x˙ dt[0, ≥1/2]0 " (8t − 4)8tx˙ −≥4 0 ≤ ∀t0 ∈−t[0,≤1/2]x(t) 0 t% %!% 6t − 4 ≥ 0 0 ≤ x(t) G" ˙ ∀t ∈ [2/3, 1] R 1 − 4)x˙ dt ≥ 0 (6t − 4)x˙ ≥ 0 ∀t ∈ [2/3, 1] i " ∆I ≥ 0 " 4 #5* "5o2/3*"(6t ] 0b[ .I6]1A& 'E,D" (.I6q 42(':h= # !;!6 x(·) ':* ## ] _= # !; xˆ(·) N$':#%e #7 #P':[ .RE "5 +,--1A E"D #!# %!,D"2(97& xˆ(·) 1AE,D" I D " ,c;8 a[E,Db"#.RJ( # ('!(I$ "E( 2 inf I(x(·)) ≡ abs min I(x(·)) ≡ I(ˆ x(·)) = −2

D

&

PD GD

9

KQ J.+ )M , !#",.$%$N5&H# )*M#$ P**,.$ #, ,$# !(W+#"%&H$)*L)-&';( (%K%')* )-+!K@-,)- .+#$ + /$NZ( J4 Z H+ J.#" J /*W )-I&H)*$*,.$$NI+L)-N7)- + $N %\

OX

I(x(·)) ≡

Z

1

x(t) dt → sup,

0

!"

−6t ≤ x ¨(t) ≤ 1 ∀t ∈ [0, 1],

T ? (97 42e !;CP [ 4 #5* ( " c1 x ¨(t) = u(t)

x(0) = x(1) = 0.

x(t) = x1 (t)

B0 (x1 (·), x2 (·), u(·)) ≡ −

Z

0

x(t) ˙ = x2 (t)

1

x1 (t) dt → inf;

V &

V U V X u(t) ∈ U (t) ≡ {u(t)| − 6t ≤ u(t) ≤ 1} ∀t ∈ [0, 1]; V & x1 (0) = 0, x1 (1) = 0. # * # !; V Y " Z# #,D"Z42515*o " ,D':" ,D" ':b"5 V ?':#%e !;Lf? #!kC x˙ 1 (t) − x2 (t) = 0,

A #!kH #C

x˙ 2 (t) − u(t) = 0 ∀t ∈ ∆0 ⊆ ∆ ≡ [0, 1];

L≡

Z

1

L(t) dt + l; 0

L ≡ −λ0 x1 + p1 (x˙ 1 − x2 ) + p2 (x˙ 2 − u) ≡ p1 x˙ 1 + p2 x˙ 2 − H;

N$':#%e !;L$ #"E; #C " ( ##"C

H ≡ p 1 x2 + p 2 u + λ 0 x1 ; l ≡ λ1 x1 (0) + λ2 x1 (1);

@ * ,- .Ro \N$':#%e & (t) p2 ≡ p2 (t) #λ70 λ#1.R$λ(2 #5pkH1 ≡"Dp 1a f? #!k U

S a ,D"2(9 'E,- J] #e (9%!,D ('!(9 24 515* V & M V &-C G,D ,D"2(9': # # J J! 5M:f? #!k i = 1 2 - C

I1

−

d ˆ ˆ xi (t) = 0 Lx˙ (t) + L dt i

ˆ ∂ H(t) ⇔ pˆ˙ i (t) = − ⇒ ∂xi

ˆ0 , pˆ˙1 (t) = −λ

pˆ˙ 2 (t) = −ˆ p1 (t);

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = 1 i = 1 2 ˆ x˙ i (tk ) = (−1)k L ˆ1 , pˆ1 (0) = λ

∂ ˆl ∂x(tk )

⇔ pˆi (tk ) = (−1)k

ˆ2 , pˆ1 (1) = −λ

= 0 t0 = 0 k = 1

pˆ2 (0) = 0,

K'E,- o " (97 #E,D" L u C "7%L%!% L(u) = −ˆp u H(u) = −L(u) !"

∂ ˆl ⇒ ∂x(tk )

pˆ2 (1) = 0;

2

u ˆ(t) = arg abs min L(u) ≡ arg abs max H(u) = u∈U  u∈U pˆ2 (t) < 0,  −6t, 1, pˆ2 (t) > 0, =  ∀u ∈ U, pˆ2 (t) = 0.

D $#7Y#, #' `)M,' KQ,

$ " (97 # $': # #$ D& 1AD #pˆ2(t)u ,-= 0 "E( pˆ (t) = 0 _=auˆa(t)21A# J8" *!& %: E"=': # uˆ(t) (52kZ "P[ .R"=21AkZ #P "' " *%'Po# .R #E,D" P@),- E/ ! Z,DE u ,- =kZ D9"8"7%: Z':& pˆ2 (t) ≡ 0 # uˆ(t) (5kZ "+ " [ E"L1;d,D a -1ADD& # !;^ .I6E21;h= !6m42 (% ^ #e m(9%!,D ('!(981A& # "D #.I6` , ,--1A E# J ?':#%e !; -1AD& ; "Z" *% \ %!b* # !;L': # !; uˆpˆ(t)2(t) G;; "7& ,c;G " JG42515* u ˆ(t)

K'E,- !; ,D"7e ##E,D" m t t " ,D':" ,D" ':b"5 "7% %!% t0 t1 @ N$ %!,D E#. t00= 01 t1 = 1

PD

X

7J. ,

D

;(%$K%Y#,.NI#, )M,.K *,.$/ (%)*+!-,.$/

1G'E,- !;1A #!;bh= JG#DkZ2,D"%:E,D" G " ,D':" ,D" ':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ0 ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k ! # (97 #.RJ ,-':*J λˆ = 0 42515*a# 42(5kZ #9"7% %!%a λˆ0 = 0 4='E,- 0 !;+/,--1A' "C pˆ2(t) = −Ct + C1 pˆ2 (0) = 0 pˆ2 (1) = 0 ⇒ C1 = C = 0 ⇒ pˆ2 (t) ≡ 0 t ∈ [0, 1] ˆ u s ˆ =⇒ pˆ1 (t) ≡ 0 t ∈ [0, 1] =⇒ λ $ λˆ0 > 0 , (I9'E,- \ -91=50kH λ 2 λˆ=0 0=;2 'E,- # ( % - [ 4 ' 2(3'E,- !; V U - V &-9 M:? % ':b]42515*' % ':b342515*'H #e H(9%!,D ('!(9-C  ˙ xˆ1 (t)     xˆ˙ 2 (t)        u ˆ(t)      pˆ˙ 1 (t)    ˙    pˆ2 (t) x ˆ1 (0)

WD

= x ˆ2 (t), = u ˆ(t),  pˆ2 (t) < 0,  −6t, 1, pˆ2 (t) > 0, =  ∀u ∈ U, pˆ2 (t) = 0, = −2, = −ˆ p1 (t); = 0, x ˆ1 (1) = 0; pˆ2 (0) = 0,

V V T T i " JK% JH42515*/# 4 2,D"#. * ".RE,D"5;##.R #!& " E# !;+ Z,D ,D"2(.)1A NPN$ #e 7 #.I6L': # # J 1;d !6n -1AD # !;m ( " ,c;d* ".RL% .I6'E,- !; Q ,- !;^[-C pˆ (0) = λˆ pˆ (1) = −λˆ jm% ':b 42517& *'+# %!b*b1 " ,c;"7%81%!%81 # O,-'Ek2"K1;O -1AD # !; (#5kH "D J %: " .R=#G _= # Z% J+42515* # !;b"5 8λˆ1 "Eλˆ(2 '+ !6` -1AD;"L#" [ ' " ,c;s _= ( % ':b]42515*' j" pˆ (t) = t2 − αt + β R$ t% %!% +α "E(r 4`'Epˆ,-1(t)

=J p−2t 2 ,--1A' "C α = 1 ˆ2 (0) = 0 pˆ2 (1) = 0 "

c

1 a 4?'E,- !; 2 β=0 ∀t ∈ (0, 1) " (97 #E,D" Lpˆ2(t) ':*= (It C −uˆ(t)t <=0−6t "7%L%!% ∀t ∈ [0, 1] 7 " ¨ x ˆ(t) = −6t x ˆ(0) = 0 xˆ(1) = 0 x ˆ(t) = −t3 +t ∀t ∈ [0, 1] pˆ2 (1) = 0.

D

&

D

Y\P, ,--1A' 2( ':* ##':b %!,D"2(97 $ #"E; # xˆ(·) # [E,Db"#':bm " (97 #E,D" E,D-1,D" ## J= %: J %!7kZ2(I*" xˆ(·) 1AE,D"7; "+N$':#%e #7'O[E,Db"#.RJ (9%!,D ('!(Iij 42(':" (pN$':#%e b E,D-1,D" E( N$':#%!& e x(·) :"7%*" [ . N$':#%e !; xˆ(·)xˆ+(·)x(·) E,D"77A ,D1A 'E,D" !& ( J i , , (9" E2( Jd42515*LN$':#%e !; xˆ(·) + x(·) [ 'E& 1A "K1A 'E,D" ( J2,- x(·) ∈ KC 2([0, 1]) xˆ(0) + x(0) = 0 t Z % ! % Z % x ≤ x¨ ˆ(t)+ x¨(t) ≤ 1 ∀t ∈ [0, 1] 'Eˆ,-(1)+x(1) bd425=150* −6t ¨ˆ(t) = −6t ∀t ∈ [0, 1] x ˆ(0) = 0 x 0 x " x(0) = 0 x(1) = 0 ˆ0(1)≤ =x¨(t) -1AD ( ,=':* "E( ':* ##.I6+'E,-≤ 1 +J`4 6t#%`∀t∈ [0,h=1]D& # !;LN$':#%e #7AC ∆I

≡ I(ˆ x(·) + x(·)) − I(ˆ x(·)) ≡ R1 R1 ≡ 0 x(t) dt ≡ 0 x(t) d(t − 21 ) ≡ 1 R1 ≡ x(t)(t − 12 ) − 0 (t − 12 )x(t) ˙ dt ≡ 0 2 R1 ≡ 0 − 0 x(t) ˙ d( t2 − 2t ) ≡ 1 R1 x(t) dt ≡ ≡ −x(t) ˙ 12 (t2 − t) + 12 0 (t2 − t)¨ 0 R 1 x(t) dt. ≡ 0 + 21 0 (t2 − t)¨

D

t %a%!% t2 − t ≤ 0 x¨(t) ≥ 0 ∀t ∈ [0, 1] " (t2 − t)¨x(t) ≤ 0 "^ 4 #5* "5R*" 3b[ .I6 1A∀t ∈'E,D"[0, (1].I6L ∆I 42(':≤h= #0 !;!6 x(·) ':* ## K _= # !; xˆ(·) N$':#%e #7 #P':[ .RE "5 +,--1A E"D #!# %!,D"2(97& xˆ(t) = −t3I+ t 1AE,D" D " ,c;a K[E,Db"#.RJa(9%!,D ('!(I $ "E( sup I(x(·)) ≡ abs max I(x(·)) ≡ I(ˆx(·)) = 0, 25

PD

SD (QJ # +#, #.#P$#,M)* +,.*$# J.P+%\ !#"%$&' (%)*+!-,.$/ JGK@)*,!+9C

I(x(·)) ≡

Z

1 0

(x˙ 2 (t) + 4x(t)) dt → inf,

−1 ≤ x(t) ˙ ≤ 0 ∀t ∈ [0, 1],

V T T

x(0) = 0,

x(1) + 3/4 ≤ 0.

!"

T ? (97 42e !;C! [ 4 #5* ( Z

B0 (x(·), u(·)) ≡

1 0

x(t) ˙ = u(t)

" c1

(u2 (t) + 4x(t)) dt → inf;

x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 ⊆ ∆ ≡ [0, 1]; u(t) ∈ U (t) ≡ {u(t)| − 1 ≤ u(t) ≤ 0} ∀t ∈ [0, 1]; x(0) = 0;

V ? ':#%e !;Lf? #!kC

x(1) + 3/4 ≤ 0.

L≡

A #!kH #C

Z

V V T V V V TES V V T ! V V T Y V V T #&

1

L(t) dt + l; 0

L ≡ λ0 (u2 + 4x) + p(x˙ − u) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C

H ≡ pu − λ0 (u2 + 4x);

" ( ##"C

l ≡ λ1 x(0) + λ(x(1) + 3/4);

@p* ,- .R= 8N$':#%e #7 #.RJ`(#& kHλ0 "Dλ 1Lf?λ p≡#!kp(t) S a ,D"2(9`'E,- Jn #e 8(9%!,D ('!(9`842515* V V T V M V V T #&-C K': # # J! 5M:f? #!kC

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x ˆ0 ; pˆ˙ (t) = 4λ

VTV

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = 1 ˆ x˙ (tk ) = (−1)k L

∂ ˆl ∂x(tk )

⇔ pˆ(tk ) = (−1)k

ˆ1 , pˆ(0) = λ

= 0 t0 = 0 k = 1

∂ ˆl ⇒ ∂x(tk )

ˆ pˆ(1) = −λ;

K'E,- o " (97 #E,D" L u C "7%L%!% L(u) = λˆ0 u2 − pˆu H(u) = −L(u) !" u ˆ(t) = arg abs min L(u) ≡ arg abs max H(u) ≡ u∈U

u∈U

ˆ0 u2 ). ≡ arg abs max (ˆ p(t)u − λ −1≤u≤0

!:K%$(%,

t%3%!%](#5kZ2,D" @ D " " (97 # ':U(t) #∀t ∈# [0, [56E1]21A ( #76E21A " %!% 4 #5* # ^ '!( #"7 u = %: " E( N$':#%e !; ˆ0 u2 # (9 "n# [ _= \4 #5* # H(u) ≡ pˆu − λ # @l " 4 %: −1 ≤ u ≤ 0

K'E,- !; ,D"7e ##E,D" m t t " ,D':" ,D" ':b"5 "7% %!% t t @ N$ %!,D E#. t 0=0 1 t = 1 1G'E,- 0 /1A1 #!;bh= JL#DkZ2,D"0%:E,D" C1

!:K%$(%':$ ,!J.P+,

ˆ · (ˆ λ x(1) + 15/16) = 0

K'E,- !;G# " e"D #E,D" C λˆ0 ≥ 0 λˆ ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !Gs9 , , ( " () 42(5kH#E,D"`842515*\# (97 # `,-':*7; 4o'E,- JK-:[-C ˙ ˆ ˆ 0 = 0 !$ λ ˆ −λ λ ,--1A' " pˆ(t) =0 =−λˆ0 ∀t ∈ [0, 1] 9 mO pˆ"(t) E=(0'n'Epˆ,-(1) = bp 42(5kH#. S ,-':*7;C λˆ = 0 λˆ > 0 u ,- λˆ = 0 5" pˆ(t) = 0 ∀t ∈ [0, 1] Zo E('$'E,- bm[ @0E,D/(#5kH "D Hf? #!ko [ " ! ,D$P# λ ˆ"1K=0 " * "K'E,- b u s + "E('\ λˆ = 0 42515*= _= # !;\#P ( "5

D

V TES

%D

u ,- " 0 'E,-λˆ > b] -C

ˆ ∀t ∈ [0, 1] H(u) = pˆu ≡ −λu ˆ pˆ(t) = −λ

ˆ ≡ −1. u ˆ(t) = arg abs max (−λu) −1≤u≤0

D

" c1G 4'E,- J V V TES - V V T Y-C ':* (IC xˆ(t) = −t ∀t ∈ [0, 1] "7=xˆ˙ (t) N$':#=%e−1 !;G#xˆo(0) ' 1A =0D& " E; "H,--1A':bh=2('G λˆ > 0 4?'E,- !;K1A #!;bh= J #DkZ2,D"%:E,D" O'E,- b xˆ(1) + 3/4 = 0 i0 " K':*2(I *"=,-':*J λˆ0 = 0 P , , (9" E2( J42515*R# 42(5kZ # $ λˆ0 > 0 , (I2'E,- -25kH λˆ0 = 1 'E,- # !& ( % - [ 4 ' 2(+'E,- !;$ V V TES - V V T Y- M:-519 % ':b]42515*' % ':b342515*'H #e H(9%!,D ('!(9-C        

x ˆ˙ = u ˆ, pˆ˙ = 4, u ˆ = arg abs max (ˆ pu − u2 );

−1≤u≤0    ˆ  x ˆ(0) = 0, pˆ(1) = −λ;    ˆ λ ≥ 0.

D

V V T

ˆ · (ˆ λ x(1) + 3/4) = 0;

i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D& E# !; L,D ,D"2(. 1A NPN$ #e 7 #.I6O': # # Jn (#5kH "D E H1;G !6H -1AD # !;K ( " ,c;H1AE?% .I6 'E,- !;Z H'E,-λˆ 1A #!;bh= JH#DkZ2,D"%:E,D" Q ,- /[-C ˆ L% ':b 42515*'a# %!b* " ,c;"7%8%!%8 # pˆ(0) = λ ,-'EkH "\11 ; -1AD # !;n(#5kH "D; λˆ1 %: " .RJO#aD& _= # /% JH42515* # !; "5E K "E(' ? -1AD;" #$" [ ' " ,c;!s _= ( "'G% ':b342515*' t%%!% " i ,D " p" ˆ˙(t) ,D" = a4,Pp$ˆ(1) = !−Eλˆ(n _=pˆ (t) # !;a=424t−4− 515*\ λˆ" ∀t(97∈ [0,# 1] ': # !;=, 'E,- 2(G1A #!;bh= J=#DkZ2,D"%:E,D" =# [56E& 1A ( , , ( " "Z1AE=21,-':*7;C λˆ = 0 λˆ > 0 $ 'E,- =1A #!;bh= J8#DkZ2,D"%:E,D" .R#!;!& " ,c;3λˆ ="E(90 " *2,D% 1A#%: "E(p,-':*E,-O D& 1AD # !; xˆ(·) n,D " " ,D" ,a$ !E(0" [ ' " ,c;m& %! ,c6E21A# n# #,D"E xˆ(1) + 3/4 ≤ 0 ':* "E( VT!

WD

TD

-1AD (

4?'E,- !;\ " (97 #E,D" a

C

pˆ(t) = 4t − 4 ≤ 0 ∀t ∈ [0, 1] u u ˆ(t) −1, u ˆ(t) = p ˆ(t)/2, −1, ≡ 2t − 2, −1, ≡ 2t − 2,

" c1

pˆ(t)/2 ≤ −1 ≡ −1 ≤ pˆ(t)/2 ≤ 0 2t − 2 ≤ −1 ≡ −1 ≤ 2t − 2 ≤ 0 0 ≤ t ≤ 1/2, 1/2 ≤ t ≤ 1.

x ˆ(t) =

Z

t

u ˆ(t) dt ≡

−t + C1 , 0 ≤ t ≤ 1/2, t2 − 2t + C2 , 1/2 ≤ t ≤ 1.

t %O%!% " C1 = 0 LE,D%: %'+N$':#%e !; xˆ(t) # .R xˆ#(0) x=ˆ(·)0 ∈ KC , (II .R_=8V j V -j" 1 4O'E,- !;^# .R #E,D"([0, "1]) J3N$':#%e ] t = 1/2 C I,--1A' "C i 4 '27"7" −1/2 = 1/4 − 1 + C2 λˆ = 0 %!,D"2(97 b0$ #"E; C#2Z=[ ' 1A1/4 "HN$':#%e !; V V T:U −t, 0 ≤ t ≤ 1/2, xˆ(t) = t2 − 2t + 1/4, 1/2 ≤ t ≤ 1. $ %!a#8':* ## J %!,D"2(97 m ,c6E21A# `# #!& ,D"E xˆ(1) + 3/4 ≤ 0 %!4 .RE "5:*" # .R#!; " ,c; $ λˆ > 0 4R'E,- !;P1A #!;bh= J#DkZ2,D"%:E,D" ,--1A' "C !t%\%!% :" ˆ x ˆ(1) G#+73/4

= a0,DG,-':*2( pˆλˆ(t)==0 4t 4=−'E4,-− λ !≤;L0 ∀t " (9∈7[0, #1]E,D" u ':* (IC 0

D

D

WD

u ˆ(t)

" c1 xˆ(t) =

VTY

−1, pˆ(t)/2 ≤ −1 ≡ pˆ(t)/2, −1 ≤ pˆ(t)/2 ≤ 0 ˆ ≤ −1 −1, 2t − 2 − λ/2 ≡ ˆ ˆ ≤0 ≡ 2t − 2 − λ/2, −1 ≤ 2t − 2 − λ/2 ˆ −1, 0 ≤ t ≤ 1/2 + λ/4, ≡ ˆ ˆ ˆ 2t − 2 − λ/2, 1/2 + λ/4 ≤ t ≤ 1 + λ/4.

=

ˆ −t + A1 , 0 ≤ t ≤ 1/2 + λ/4, 2 ˆ ˆ ≤ t ≤ 1 + λ/4. ˆ t − 2t − λt/2 + A2 , 1/2 + λ/4

t% %!% " A = 0 "7%m%!% xˆ(1) = −3/4 = 0 1 " 1 − 2 −xˆ(0) ˆ + A2 = −3/4 A2 = λ/2 ˆ + 1/4 A$4='E,- !; λ/2 ,--1A' "C # .R #E,D" LN$':#%e xˆ(t) t = 1/2 ˆ + λ/4 ˆ = −1/2 − λ/4 ˆ 2 −2(1/2+λ/4)−( ˆ ˆ ˆ ˆ = (1/2+λ/4) λ/2)·(1/2+ λ/4)+ λ/2+1/4,

ˆ λ ˆ − 4) = 0 λ( ˆ = 0 λ ˆ 1/2 + λ/4 ≡ 3/2 > 1 x ˆ(1) = −3/4 ˆ=4 λ

':*J , , ( " # .R_= Lu ,- λˆ = 4 L" t% %!% 'E,- xˆ (t) "E(d=#$−t .R0#≤!;t " ≤,c;1!"K,-':*J # 42(5kZ # Y\P, ,--1A' 2(q':* ##':b %!,D"2(97 xˆ(·) V V T:U Z#[E,D& b"#':b0 " (97 #E,D" E,D-1,D" ## J\ %: J\& %!7kZ2(I5*" xˆ(·) 1AE,D"7; "$N$':#%e #7'?[E,Db"#.RJZ( !& # ('!(II
D

D

D

D

−1 ≤ −1 + x(t) ˙ ≤ 0 ∀t ∈ [0, 1/2] ⇔ 0 ≤ x(t) ˙ ≤ 1 ∀t ∈ [0, 1/2]; −1 ≤ 2t − 2 + x(t) ˙ ≤ 0 ∀t ∈ [1/2, 1] ⇔ 1 − 2t ≤ x(t) ˙ ≤ 2 − 2t ∀t ∈ [1/2, 1].

-1AD ( ,=':* "E( ':* ##.I6+'E,- J`4 #%` h=D&

VT#

# !;LN$':#%e #7AC ∆I

≡ ≡ ≡ ≥ ≡ ≡ ≡ ≡ ≡ ≡

t%r%!%

I(ˆ x(·) + x(·)) − I(ˆ x(·)) ≡ R1 R1 2 2 ˙ ˆ + x) ˙ + 4(ˆ x + x)] dt − 0 (x ˆ˙ + 4ˆ x) dt ≡ 0R[(x R R 1 ˙ 1 2 1 2 0 x ˆx˙ dt + 0 x˙ dt + 4 0 x dt ≥ R1 R1 2 0 x ˆ˙ x˙ dt + 4 0 x(t) dt ≡ R1 R1 2 0 x ˆ˙ x˙ dt + 4 0 x(t) d(t − 1) ≡ 1 R1 R1 ˙ 2 0 x ˆx˙ dt + 4x(t) · (t − 1) − 4 0 (t − 1)x˙ dt ≡ 0 R1 R1 2x ˆ˙ x˙ dt + 0 − 0 4(t − 1)x˙ dt ≡ 0 R1 (2x ˆ˙ − 4t + 4)x˙ dt ≡ R01/2 R1 (2 − 4t)x˙ dt + 1/2 0 · x˙ dt ≡ 0 R 1/2 (2 − 4t)x˙ dt. 0

o" "0 4 #5* "5?*"0 b[ .I601A& "E(' 'E,D" (.I6q 42(':h= # !;!6 ':* ## ] _= # !; N$':#%e #7 #P':[ .RE "5 +,--1A E"D #!# %!,D"2(97& 1AE,D" D " ,c;8 a[E,Db"#.RJ( # ('!(I$ "E(

D

2 − 4t ≥ 0 0 ≤ x(t) ˙ ≤ 1 ∀t ∈ [0, 1/2] R 1/2 (2 − 4t)x(t) ˙ ≥ 0 ∀t ∈ [0, 1/2] 0 (2 − 4t)x˙ dt ≥ 0 ∆I ≥ 0 x(·) xˆ(·) I x ˆ(·) inf I(x(·)) ≡ abs min I(x(·)) ≡ I(ˆ x(·)) = −7/6

PD GD # M * !#"%$&' (%)*+!-,.$/ * 9;J.`)- ,.N#J.P+/EJ #, )-,.`)-*,!J.K% (QJ #+#, # )*+,.$ $J.P9C+ $#J ;*,.$KQJ./)-: +2$ $ 9C+N7$##*, !)M#", %$,.9C$ $EN-\ $#, ;KQJ.)-+$$9C K%$#,.

V V T X V V T V V V T u(t) ∈ U ≡ {u(t) | − 1 ≤ u(t) ≤ 2} ∀t ∈ [0, T ]; Z T V V V V cos2 u(t) dt − 1 = 0, x(0) − 1 = 0.

T (x(·), u(·), T ) ≡ T → inf, T > 0; x(t) ˙ + x(t) − tu(t) = 0 ∀t ∈ ∆0 ⊆ ∆ ≡ [0, T ];

# * # V Y " # #,D"42515*o " ,D':" ,D" ' "5 0

VT

!"

T ? (97 42e !;C!42515*='EkZPN$ (97 4 E#

V ? ':#%e !;Lf? #!kC L≡

A #!kH #C

Z

T

L(t) dt + l; 0

L ≡ p(x˙ + x − tu) + λ · cos2 u ≡ px˙ − H;

N$':#%e !;L$ #"E; #C " ( ##"C

H ≡ p(−x + tu) − λ · cos2 u;

@p0* ,- 1.R= 8N$':#%e #7 #.RJ`(#& λ0 λ1 λ p ≡ p(t) kH "D Lf? #!k S a ,D"2(9`'E,- Jn #e 8(9%!,D ('!(9`842515* V V T X M V V V V -C K': # # J! 5M:f? #!kC l ≡ λ T + λ (x(0) − 1);

]1

−

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 ⇔ pˆ˙ (t) = − Lx˙ (t) + L ⇒ dt ∂x pˆ˙ (t) = pˆ(t);

[K'E,- !;d"#,D , 7 #E,D" k -C tˆ = Tˆ

= 0 tˆ0 = 0 k = 1

1

ˆ x˙ (tk ) = (−1)k L

∂ ˆl ∂ ˆl ⇔ pˆ(tk ) = (−1)k ⇒ ∂x(tk ) ∂x(tk ) ˆ1 , pˆ(Tˆ) = 0; pˆ(0) = λ

K'E,- o " (97 #E,D" L u C "7%L%!% L(u) = −ˆptu + λˆ cos2 u

H(u) = −L(u)

"

u ˆ(t) = arg abs max H(u) = arg abs min L(u) = u∈U

u∈U

ˆ cos2 u − pˆ(t)tu), = arg abs min (λ u∈[−1,2]

ˆ cos2 u), = arg abs max (ˆ p(t)tu − λ u∈[−1,2]

V T:U

K'E,- G,D"7e ##E,D" n t " ,D':" ,D" ':b"59"7%n%!% @WN$ %!,D E# t0 = 1 'E0,- Z,D"7e ##E,D" t0 T C ˆ dLˆ ˆ Tˆ) = ∂ l ) ⇒ = 0 (⇔ H( dt ∂T ˆ cos2 u ˆ0 ; pˆ(Tˆ)(−ˆ x(Tˆ) + Tˆu ˆ(Tˆ) − λ ˆ(Tˆ)) = λ

1G'E,- !;1A #!;bh= JG#DkZ2,D"%:E,D" G " ,D':" ,D" ':b"5"7% %!%3^42515*O# " 'E,- J V Y\" # #,D" 'E,- Tˆ > 0 4o,D ,D"2(.^'E,- J V Y9 ,D%!b* #= ,D* "7 " ,c;L *#./( K'E,- o# " e"D #E,D" C λˆ ≥ 0 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !GQ ,- !; V V T - V V V V - M: - M:4 P-1,D"7;b"8,D [ J % ':b]42515*' % ':b342515*'H #e H(9%!,D ('!(9-C               

u∈[−1,2]

ˆ cos2 u ˆ0 , pˆ(Tˆ)(−ˆ x(Tˆ) + Tˆu ˆ(Tˆ)) − λ ˆ(Tˆ) = λ R Tˆ cos2 u ˆ(t) dt = 1, 0 x ˆ(0) = 1, pˆ(Tˆ) = 0; ˆ0 ≥ 0, Tˆ > 0; λ .

V V V S

E' ,- u s 'E,- o# ( % i " J0% J)42515*d# 4 2,D"#. 1A dE,D"5;##.Rd #!& " E# !;+ Z,D ,D"2(.)1A NPN$ #e 7 #.I6L': # # J 2(; 8(#5kH "D 8f? #!k #K# 4 2,D"#.I6 - L1;+ !Tˆ6+ -1AD # !;+ ( " ,c;`,--1λˆ,D0" λ ˆZ'E,- !;a,D"7e & ##E,D" T 4 ( " *2,D%: R'E,- 51AE$% .I6 'E,- !;+ `'E,- =# ( % #K'E,- J -$ ='E,-& j[-C pˆ(0) = λˆ1 o% ':b 42515*'P#I %!b* " ,c;7"7%Z%!% #\,-'EkH "K1;` -1AD # !;O(#5kH "D; %: " .RJ`# _= # % J842515* `#H !; "5 O "Eλˆ(1'+ L -1AD& ;"#P" [ ' " ,c;!s _= (d% ':b342515*'

D

VTX

             

x ˆ˙ = −ˆ x(t) + tˆ u(t), ˙pˆ(t) = pˆ(t), ˆ cos2 u − pˆ(t)tu); u ˆ(t) = arg abs min (λ

t%8%!%

"

" c1 t%8%!%` u∈[−1,2] E,--1A#2(n #,D" λˆ0 ≥ 0 cos2 uˆ(Tˆ) ≥ 0 " λˆ ≤ 0 u ,- λˆ = 0 " λˆ0 = 0 E,D%: %' pˆ(t) ≡ 0 t ∈ [0, Tˆ] ˆ 1 = 0 !"H#P .R#!; " ,c;\'E,- u s λ u ,- λˆ < 0 Z" uˆ(t) = arg abs min (λˆ cos2 u) = 0 ⇒ ˆ = λˆ0 >u∈[−1,2] ˆ 0 = 1 'E,- P# !& u ˆ(t) ≡ 0 t ∈ [0, Tˆ] −λ 0 ⇒ λ ( % R Tˆ 1 · dt = 1 ⇒ Tˆ = 1 xˆ˙ (t) = −ˆx(t) xˆ(0) = 1 0 ⇒ x ˆ(t) = e−t ∀t ∈ [0, 1] id 4 '27"7"R _= # !;H% J42515* #e =(9%!,D ('E& (9P':*2(IC 2(; E': # %!,D"2(97H$ #"E; Tˆ#= xˆ1(t) = e−t ∀t ∈uˆ[0,(t)1]≡ 0 t ∈ [0, 1] Y\P, ,--1A' 2(+':* ## I _= # R#P[E,Db"#':b " (97& #E,D"$-1A5kH (IA*"a,D':h=2,D" ' " T 4 #5* # ZN$':#%!& e #7A- 1; %: " T ≤ Tˆ = 1 * :1A#n e #%!C RT RT $E,D%: %'\\-1A& T ≡ 0 1 · dt ≥ 0 cos2 u(t) dt = 1 5kZ # b T ≤ 1 = Tˆ \\ e #%: T ≥ 1 " T = 1 = Tˆ "` 4 #5* "5*"`` , , (9" E2( J42515* Tˆ = 1 @ [E,Db"#( # (97 # $4 #5* # T i " P':*2(IC pˆ˙ = pˆ pˆ(Tˆ) = 0 pˆ(t) ≡ 0 t ∈ [0, Tˆ] 2 ˆ ˆ cos2 u ˆ0 u ˆ(t) = arg abs min (λ cos u) −λ ˆ(Tˆ) = λ

D

D

inf T (x(·), u(·), T ) u ˆ(t) x ˆ(t)

≡ ≡ ≡ =

abs min T (x(·), u(·), T ) ≡ T (ˆ x(·), u ˆ(·), Tˆ) ≡ Tˆ = 1, 0, t ∈ [0, ˆ 1], e−t ∀t ∈ [0, 1].

)-N7 )M ,.OP\ ,.$ #,S#J.( P,!MJ.P +(Q*,. #O" !+G#"%$$L&') !(%#"%)*$'+!-,.$J #(%/*G,+ %K )-

ˆ0 = 0 λ

T (x(·)) ≡ x(1) → inf, t − 1 ≤ x(t) ˙ ≤ t + 1 ∀t ∈ [0, 1], Z 1 (x2 (t) − 2tx(t)) dt = −1/3. 0

x(0) = 0,

VT

!"

" c1 T ? (97 42e !;Cr [ 4 #5* ( x(t) ˙ − t = u(t) n c , E 6 2 A 1 # 7 ; 2 4 5 1 5 * L [ 4 'E& −1 ≤ u(t) ≤ 1 ∀t ∈ [0, 1] " ,c;\42515*'K " (97 # Z': # !;C V V V! T (x(·), u(·)) ≡ x(1) → inf; V V V Y x(t) ˙ − u(t) − t = 0 ∀t ∈ ∆0 ⊆ ∆ ≡ [0, 1]; V V V #& u(t) ∈ U (t) ≡ {u(t)| − 1 ≤ u(t) ≤ 1} ∀t ∈ [0, 1]; Z 1 V V V x(0) = 0, (x2 (t) − 2tx(t)) dt + 1/3 = 0. 0

# * # !; V Y " Z# #,D"Z42515*o " ,D':" ,D" ':b"5 V ?':#%e !;Lf? #!kC L≡

A #!kH #C

Z

1

L(t) dt + l; 0

L ≡ p(x˙ − u − t) + λ(x2 − 2tx) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C " ( ##"C

H ≡ p(u + t) − λ(x2 − 2tx); l ≡ λ0 x(1) + λ1 x(0);

@p* ,- .R= 8N$':#%e #7 #.RJ`(#& kHλ0 "Dλ Lλf?1 p≡#!kp(t) S a ,D"2(9`'E,- Jn #e 8(9%!,D ('!(9`842515* V V V! M V V V-C K': # # J! 5M:f? #!kC

]1

−

V V T

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x ˆ x(t) − t); pˆ˙ = 2λ(ˆ

[K'E,- !;d"#,D , 7 #E,D" k -C t1 = 1 ˆ x˙ (tk ) = (−1)k L

∂ ˆl ∂x(tk )

ˆ1 , pˆ(0) = λ

= 0 t0 = 0 k = 1

⇔ pˆ(tk ) = (−1)k ˆ0 ; pˆ(1) = −λ

∂ ˆl ⇒ ∂x(tk )

K'E,- o " (97 #E,D" L u C "7%L%!% L(u) = −ˆpu H(u) = −L(u) !" u ˆ(t) = arg abs min L(u) ≡ arg abs max H(u) ≡ u∈U u∈U  pˆ(t) < 0,  −1, +1, pˆ(t) > 0, ≡ arg abs max (ˆ p(t)u) ≡  −1≤u≤1 ∀u ∈ U, pˆ(t) = 0.

u ,- pˆ(t) = 0 t ∈ [0, 1] d 4 E##.I6m" *%!76 `* ,D"#E,D" 821A# J" *%: "8': # (& kZ "K[ .R"K21AkZ # " +" *% " *%' RuKˆ(t)# D& .R #E,D" ]@ ,- Em ! 0,DERu ,- 3kZ pˆ(t) = 0 #^eD.I63 " 4 %!76 " 4 %: -C pˆ(t) ≡ 0 t ∈ [0, 1] 9" t ∈ [τ1i , τ2i ] τ1i < τ2i 1 ≤ i ≤ k 1 < k < ∞ " (97 # L': # L 4L'E,- !;m " (97 #E,D" -1AD "?#D 4D; Z1;H -1AD # !;H"7%: ': # !;+( ':"K " [ E"E,c;\ .I6E21;h= P42H (& % + #e G(9%!,D ('!(91A # "D #.R? , ,--1A E7& # !;

K'E,- !; ,D"7e ##E,D" m t t " ,D':" ,D" ':b"5 "7% %!% t t @ N$ %!,D E#. t 0= 01 t = 1 1G'E,- 0 !;1A1 #!;bh= JG#DkZ2,D"0%:E,D" G 1" ,D':" ,D" ':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ ≥ 0 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k !L _=2(n 4 ( " *2,D%: $'E,- oH :1A C

D

PJ. &'

0=

Z

1 0

1 (ˆ x (t) − 2tˆ x(t)) dt + ≡ 3 2

Z

1 0

(ˆ x(t) − t)2 dt,

V V V

+ [ 4 ' 2(d'E,- !; V V V Y- V V V- M:- M:4 RK%7& ':b342515*' % ':b342515*'H #e H(9%!,D ('!(9-C  ˙ x ˆ(t)     pˆ˙ (t)      u ˆ(t)        x ˆ(0)   ˆ0 λ

= u ˆ(t) + t, ˆ x(t) − t), = 2 λ(ˆ pˆ(t) < 0,  −1, +1, pˆ(t) > 0, =  ∀u ∈ U, pˆ(t) = 0; R1 ˆ0 ; = 0; pˆ(1) = −λ x(t) − t)2 dt = 0; 0 (ˆ ≥ 0;

'E,- u s !'E,- $# ( % . V V V U i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D&

E# !; L,D ,D"2(. 1A NPN$ #e 7 #.I6O': # # Jn (#5kH "D YK# 4 2,D"#.I6 -A \1;+ !6a -1AD # !; ( " ,c;O1AE+λˆ%λˆ0 .I6O'E,- !; 4 ( " *2,D%: \'E,-& $ a'E,- o# ( % YH'E,- !; -!Q ,- pˆ(0) = λˆ1 =% ':b 42515*'Z#o %!b* " ,c;:"7%G%!%K #Z,-'EkH "P1; -1AD # !;G(#5kH "D; :%: " .RJK#? _= # o% J 42515* `# !; "5 O "Eλˆ(1'+ \ -1AD;"a#" [ ' " ,c; s _= ( "'K% ':b342515*' $4q6E21;h= lp% ':b 42515*'B 4 ( " *2,D%: 'E,- !;n,--1A' "C xˆ(t) − t = 0 ∀t ∈ [0, 1] $ "E()%7& \'E,- "E(9" *2,D% m .R#!; " ,c; t% = 0 %!%a xˆ(t) −xˆ(0) ( 2(IC pˆ˙(t) = 0 ∀t ∈ [0, 1] "L,?':*D& t=0 "E(3% L'E,- !; pˆ(1) = −λˆ0 ':*2(IC pˆ(t) = −λˆ0 ∀t ∈ [0, 1] u ,- λˆ0 > 0 " #!; λˆ0 = 1 'E,- G# ( % - ':* (IC $ "E(^ 4'E,- !; 1] " (97 #pˆE(t) ,D" m=,-−1-1A' < "0C uˆ∀t(t)∈ =[0,−1 9 m "E(' ∀t ∈ [0, 1] 7 " ^ % ! % % " 2 x ˆ˙ (t) = −1 + t "\ _= # Z#xˆZ(0)' 1A =0 " E;xˆ (t) "K 4 = t /2 (− "7t& ∀t ∈ [0, 1] *2,D%:E(''E,- b R 1 (x(t) − t)2 dt = 0 !$ "E('Z,-':* 0 ˆ 0 > 0 _= # H% J`42515* 8 ,--1A E"D # ,c6E21& λ # J\42515* \ " (97 # Z': # !;\#P,D':h=2,D" ' "5 u ,- λˆ0 = 0 9" pˆ(t) ≡ 0 t ∈ [0, 1] 9 m': # uˆ(t) 4G'E,- !; " (97 #E,D" d u -1AD "`#D 4D; "

D

WD

D

D

V V S

GD

%D

>1

5J.

9C

:' # L λˆ0 = 0 pˆ(t) ≡ 0 t ∈ [0, 1] 9[ ' 1A " Pu (5kH#` -1AD "O 4\'E,- !; = u 21,D"7 `8# xˆ(t) = t $E,-\21,D"7#xˆ˙ (t) % n ˆ(t) ':*+ (It C A ( " (I*"H'E,- u s u ˆ"(t)E(0= 1.R−t#!∀t; ∈" ,c[0, ;1]"7%m%!% λˆ0 = 0 pˆ(t) ≡ 0 t ∈ [0, 1] @pb[ AG* ,D"#E,D" 6= 0 A*"L,--1A' "K 4 1Aλˆ 1 NP=N$0 #λeˆ 7 # o': # # !;?1; λˆpˆ(t) C in "&

I':*2(I7*"$ _= # 2(`% J=42515* =0 =2 λˆ#e· 0 P(9%!& ,D ('!(9\ ,--1A E"D # ,c6E21A# J`42515* 8 " (97 # ': # !;a[ ' 1A " / _= # C λˆ = 0 xˆ(t) = t 0 ∀t ∈ [0, 1] u ˆ(t) = 1 − t ∀t ∈ [0, 1] *2( pˆ(t) = 0 ∀t ∈ [0, 1] λˆ = 0 λ @ b[ #? # #'2b=$E,D%: %' λˆ0 = 0 N$':1#%e #7`# "\ _= # =# !; "5 Y\P, ,--1A' 2( ':* ##':b %!,D"2(97" %!,D"2(97 $ #"7& E; # # [E,Db"#':b !& " (97 #E,Dxˆ"(t)]
D

JPJ. 9CG $%( L))*!+!#"-,%.$$#,, D

D

R1 0

(ˆ x2 (t) − 2tˆ x(t)) dt +

R1 0

8D

D

(2ˆ x(t)x(t) − 2tx(t)) dt+ R1 + 0 x2 (t) dt = −1/3

R1 R1 ⇔ −1/3 + 0 + 0 x2 (t) dt = −1/3 ⇔ 0 x2 (t) dt = 0 ⇔ x(t) ≡ 0 t ∈ [0, 1]

D

/$E,--1A# 8"5kZ1A2,D" 4 #5* "5 *"^ 42515*2,D"^" %: 21A#d1A 'E,D" (97; %!,D"2(97C 8 =1A 'E,D" (.RE e %!%` +1A& x ˆ(t) 'E,D" =(t.R\∀tE∈ [0,e1] mN$':#%e #7A# 42(5kH#.Pt%m%!% " u(t) = xˆ(t) ≡ 0 x ˆ˙ (t) + x(t) ˙ −u ˆ(t) − u(t) − t = 0 n

" E ( ' @ -1A #,D" #!& t ∈ [0, 1] #7;L1A 'E,D" (97;`':uˆ(t);b=hZ17−;`N$t ':∀t#%∈e ![0, ;1]i3 " Z':*7& 2(I*"842515*a ( "`-1A #,D" ## G _= # #+%: " E( 1AE,D" D " ,c; [E,Db"#.RJ ( # ('!( N$':#%e #7AP " V V!

D

$L)!#"%$,= _= # L,GJ. 9C (%)*+!-,.$#, jP$C 5D "E(

@

ˆ0 = 0 λ

x ˆ(t) = t ∀t ∈ [0, 1] u ˆ(t) = 1 − t ∀t ∈ [0, 1] inf T (x(·)) ≡ abs min T (x(·)) ≡ T (ˆ x(·)) = 1

-, .$ #, ]!\ #"% $# !/ #`)*M,.K% *LI)-/EJ. , ) !#"%$&'E(%*(%J.)*+!-,K .J$/J. +79C K%(%L)*)-+N

V V V2X 0 V V V x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 ⊆ ∆ ≡ [0, 2]; V V ST u(t) ∈ U (t) ≡ {u(t)| − 1 ≤ u(t) ≤ 1} ∀t ∈ [0, 2]; V V S V x(0) − 1 = 0, x(2) = 0. # * # !; V Y " # #,D"42515*$ " ,D':" ,D" ':b"5 I(x(·), u(·)) ≡

Z

2

x2 (t) dt → inf;

!"

T ? (97 42e !;C!42515*='EkZPN$ (97 4 E# V ?:' #%e !;Lf? #!kC A #!kH #C

L≡

Z

2

L(t) dt + l; 0

L ≡ λ0 x2 + p(x˙ − u) ≡ px˙ − H;

N$':#%e !;L$ #"E; #C " ( ##"C

H ≡ pu − λ0 x2 ; l ≡ λ1 (x(1) − 1) + λ2 x(2);

@ * ,- .R$ aN$':#%e #7 #.RJ\(#& kHλ0 "Dλ 1Lλf?2 p≡#!kp(t) S a ,D"2(9`'E,- Jn #e 8(9%!,D ('!(9`842515* V V V2X M V V S V -C V V Y

]1

K': # # J! 5M:f? #!kC −

ˆ d ˆ ∂ H(t) ˆ x (t) = 0 Lx˙ (t) + L ⇔ pˆ˙ (t) = − ⇒ dt ∂x ˆ 0 xˆ(t); pˆ˙ (t) = 2λ

[K'E,- !;d"#,D , 7 #E,D" k -C t =2

= 0 t0 = 0 k = 1

1

ˆ x˙ (tk ) = (−1)k L

∂ ˆl ∂x(tk )

ˆ1 , pˆ(0) = λ

⇔ pˆ(tk ) = (−1)k

∂ ˆl ⇒ ∂x(tk )

ˆ2 ; pˆ(2) = −λ

K'E,- o " (97 #E,D" L u C "7%L%!% L(u) = −ˆpu H(u) = −L(u) !" u ˆ(t)

= arg abs min L(u) ≡ arg abs max H(u) ≡ u∈U

u∈U

≡ arg abs max (ˆ p(t)u) ≡ −1≤u≤1  pˆ(t) < 0,  −1, +1, pˆ(t) > 0, ≡  ∀u ∈ [−1, 1], pˆ(t) = 0;

K'E,- !;,D"7e ##E,D" Z t t $42515*I " ,D':" ,D" 'E& b"5"7%\%!% t0 t1 @ N$ %!,D 0 E#1 . t0 =0 t1 = 2 1G'E,- !;1A #!;bh= JG#DkZ2,D"%:E,D" G " ,D':" ,D" ':b"5"7% %!%\42515*$# "'E,- J V Yj" # #,D" K'E,- o# " e"D #E,D" C λˆ0 ≥ 0 k G'E,- u s 4 K'E,- o# ( % a(#5kH "D JLf? #!k ! # (97 #.RJ?,-':*J λˆ = 0 /42515* # 42(5kZ #D"7%P%!% λˆ = 0 4$'E,- !;K0 ,--1A' " pˆ(t) = w ∀t ∈ [0, 2] 42(5kH0 #.^" L21,-':*7;C V V #

w

pˆ(t) =

α

(1.119)

)

#?,D':h=2,D" ' "

"

⇒ x ˆ(t) = −t + A ∀t ∈ [0, 2] x ˆ(2) = 0 A

w

pˆ(t) =

"7%m%!%

< 0 ∀t ∈ [0, 2] ⇒ u ˆ(t) = −1 ∀t ∈ [0, 2]

x ˆ(0) = 1

)

I"7%^%!% xˆ(0) = 1 #?,D':h=2,D" ' " @WE,D ) γ pˆ(t) = w ≡ 0 t ∈ [0, 2] ⇒ λ (#5kH "D Gf? #!k$ [ hZbˆ" 1,c;==0#λˆ2 =#o0 .R& #!; " ,c;L'E,- u s - t% %!% λˆ0 6= 0 ?"0)'E,- b λˆ0 > 0 Z$ #!; ˆ 0 = 1/2 'E,- # ( % -= [ 4 ' 2( 'E,- !; λ V V V- V V S V ---k \% ':bq42515*' % ':br427& 15*'H #e H(9%!,D ('!(9-C β

(1.119)

"

> 0 ∀t ∈ [0, 2] ⇒ u ˆ(t) = 1 ∀t ∈ [0, 2]

⇒ x ˆ(t) = t + B ∀t ∈ [0, 2] x ˆ(2) = 0 B

  pˆ(t) < 0  −1,     +1, pˆ(t) > 0 , ˆ˙ (t) = u ˆ(t) ≡  x  ∀u ∈ [−1, +1], pˆ(t) = 0   pˆ˙ (t) = x ˆ(t);    x ˆ(0) = 1, x ˆ(2) = 0.

D

i " J% JO42515*H# 4 2,D"#.q1A KE,D"5;##.RH #"D& E# !;+ ,D ,D"2(.31A NPN$ #e 7 #.I6a': # # JA 1;O !68 -1AD # !;8 ( " ,c;`1AEL% .I6+'E,- !;Q ,-& !;\[-C pˆ(0) = λˆ1 pˆ(2) = −λˆ2 !% ':b342515*'H#P %!b/& *b" ,c;"7%%!% # K,-'Ek"1;H -1AD # !;H(#5kH "D J %: " .RP#H _= # ?% JL42515* L#? !;b"5 λ ˆ1 "Eλˆ(2'8 !6 -1AD;"+#G" [ ' " ,c;s _= ( "'O% ':b 42515*' v$" [ . 4 [ Dk" #7 42r 42(5kH#.I6 E #" r _=D& # !; % J 42515* ': D512( _= # ,c6E21A# J 427& 15* V V V2X M V V S V " (97 # p': # !;L#p%:& " E( 1AE,D" D " ,c;p[E,Db"#.RJB( # ('!( N$':#%e #7A I %!7kZ2(I *"d #O;; " ,c; _= # 2( %D& I(x(·), u(·)) Ja42515* A +# E,D-1,D" ## J+ %: Ja':[ -1A (9,c;A*" # ': D51##E( a _= # 1AE,D" D " ,c;^[E,Db"#.RJ^( # !& ('!(mN$':#%e #7A91ADA2( "

U

7U

V V

V V S S

CV

D

CV

D

"':%"':N$':#%e #7A V V V2X? %!4 .RE "5*" [& D, b"#.RJn( # ('!(]1AE,D" D " ,c; x(t) ≡ 0 t ∈ [0, 2] ` "E(0#L .R#!; " ,c;d% G'E,- x(0) = 1 v$" [ .^ .R# " "% o'E,- ( $N$':#%!& e b xˆ(t) ≡ 0 t ∈ [0, 2] O%!5*2,D" L _= # !;m 4 E(2( N$':#%e b=C

TD

PD

U

x ˆ(t) =

CV

1 − t, t ∈ [0, 1]; 0, t ∈ [1, 2].

)H )+" # !"#%$& '( ) +*,-/.+01 3211 '%4,576%8 6%6 9;:=<1576%8 6;>6=:?;&0@A#B2 1C( 'D/0 1=4EF GH#21@=EI4EF 1 J!DK-L@;E 0MHN !CO*%1 M '%1 DK# '%G.C+O1 '% PJC+O1A( '% -CJ2 '%O1 1$0B;Q0B'%ODK*R!1 COS$&A( OTUGV JEW4E4EV8 ?':#%e !; 4 ; " -1AD " ' 1A " E;bh= 'E,- !;(%xˆ(t) J\42515* \N$':#%e uˆ(t) pˆ(t) $ #" E=': # # pˆ˙(t) = xˆ(t) ':* (IC pˆ(t) =

t − 21 t2 + a1 , t ∈ [0, 1], a2 , t ∈ [1, 2],

c1A a1 a2 @ E,D"5;##.R #" E# !; t% %!% N$':#%e !; pˆ(t) # .R #B t ∈ [0, 2] E,D%: %' -P"3 4m'E,- !;0# .R #E,D" " J pˆ(·) ∈ KC 1 ([0, 2]) N$':#%e l t = 1 ':* ( ,D;4 q(DkZ1A' a1 a2 C a2 = a1 + 1/2 $ \': # # \% Jm42515* d 4 ; "8 -1AD " ': # uˆ(t) C u ˆ(t) = x ˆ˙ (t) ≡

−1, t ∈ [0, 1], 0, t ∈ [1, 2].

D

$ E,D%: %'d 4`'E,- !; " (97 #E,D" d': # b u ,--1A' "5*" 42(5kH#a _=` 9+ , , (9" E2u(ˆ(t) E(^=,-':0* uˆ(t) ≡ 0 t ∈ [1, 2] pˆ(t) " p=ˆ(t)0 ≡ 0 E,D%: %'d': # t ∈ [1, 2] #D 4D;d -1AD "8 4L'E,- !;n " (9uˆ7(t) #≡E,D"0 dt ∈u [1, "72]% %!% pˆ(t) ≡ 0 t ∈ [1, 2] , (I9'E,- G -9"8': # V V U

GJ."7 %`,P%!(%%8)*N$+!':-,#.%$e# !,D;t%+%!#% .R # @

" !"

u ˆ(t) ≡ 0 t ∈ [1, 2] t ∈ [1, 2] a2 = 0 pˆ(t) t=1 a1 = −1/2 t − t2 /2 − 1/2, t ∈ [0, 1], pˆ(t) = 0, t ∈ [1, 2].

pˆ(t) ≡ 0

( " (I *"n,D E;!kZ # L " Jm N$ %!ON$':#%e pˆ(t) t = 1 @l gA51A%: )H)+" #U 1N ! 21 ;# @DL& 2 K Q 1'D;'D M F2 1 t = 1pˆ2(t) /.+0 &A( .% 1 1=E,2 4?&'D+01 ,5 =OC&04;: pˆ˙ = −(t − 1) = 0 8 i` " ':*2(I *"o _= # 2(L%t=1 J$42515* Pt=1 #e (9%!,D ('!(9 %!,D"2(97 b0$ #"E; # [ ' 1A "N$':#%e !;C

8D

1 − t, t ∈ [0, 1], 0, t ∈ [1, 2],

J:(%*J.K%' J. &' (%)*+!-,.$/I)- Y\P, ,--1A' 2( ':* ##':b D %!,D"2(97 # [E,Db"#':b " (97 #E,D" E,D-1,D" ## J %: J %!7kZ2(I *" 1AE,D"7; "\N$':#%e #7'a[E,Db"#.RJ8( # ('!(I
t ∈ [1, 2] x ˆ(·)

x ˆ(·)

x ˆ(·) x ˆ(·) + x(·)

x(·)

, , (9" E2( J)42515*mN$':#%e !; [ ' 1A " 1A& + x(·) 'E,D" ( JG2,- x(·) ∈ KC 1([0, 2])xˆ (·)xˆ(0) + x(0) = 1 A $ E D , : % & x + x(t) ˙ %ˆ(2) 'O+8'Ex(2) ,- = b 0 42−1 515≤* xˆ˙ x(t) ≤ 1 ∀t ∈ [0,"2] x(0) = 0 ˆ(0) = 1 xˆ(2) = 0 $$ "E(B # * # n#^ 4 21A#':b ,-& x(2) = 0 51 " ,c;0# 1AE^'E,- !;C 0 ≤ x(t) ˙ ≤ 2 ∀t ∈ [0, 1] −1 ≤ x(t) ˙ ≤ 1 ∀t ∈ [1, 2] -1AD ( ,=':* "E( ':* ##.I6+'E,- J`4 #%` h=D& # !;LN$':#%e #7AC

D

∆I

V V2X

≡ ≡ ≡ ≡

I(ˆ x(·) − x(·)) − I(ˆ x(·)) ≡ R2 R2 2 2 (ˆ x + x) dt − x ˆ dt ≡ 0R R 1 20 R2 2 2 0 x ˆx dt + 0 x dt ≥ 2 0 x ˆx dt ≡ R1 R2 R1 2 0 (1 − t)x dt + 2 1 0 · x dt ≡ 2 0 (1 − t)x dt.

t%%!%

0 ≤ x(t) " x(t) ≥ 0 ˙ ≤ 2 ∀t ∈ [0, 1] R1 E D , : %

% ' : " ∀t ∈ [0, 1] "E(' ∆I ≥ 0 1"−mt 4 ≥#50* "5j2*"0 (1− ^t)x(t) b[ dt .I6 ≥1A0& 'E,D" (.I6q 42(':h= # !;!6 x(·) ':* ## ] _= # !; xˆ(·) N$':#%e #7 #P':[ .RE "5 +,--1A E"D #!# %!,D"2(97& xˆ(·) 1AE,D" I D " ,c;8 a[E,Db"#.RJ( # ('!(I$ "E( inf I(x(·)) ≡ abs min I(x(·)) ≡ I(ˆ x(·)) = 1/3 x(0) = 0

D

PD GD

=9G J.YAX.UCAE CAU *BJW ~> -

-

,/

KQK

i 4-1ADV !`#O ( 76d %!42# aO42515*76n%!A , ,D *D& ,D%: E e ## Z ,D* ,- # !;42515*76Gf? #!k= L42515*76 " (97 # P': # !; #e G(9%!,D ('!(9$ 4 ; "$'E& *" %!,D"2(97 #.R _= # !; %!,D"2(97 $ #"E; #-,D 21; %!7kZ1A':b 4 " !6842515*`%O% J842515*?1;,D ,D"2(.r [ .R%#& ##.I6+1A NPN$ #e 7 #.I68': # # Js9 , , ( " #. 42515* , E,D# #./( L" ( +N$':#%e #7 $ N$':#%e #7A ( Lf? #!k +J Z j e-!42515* \,D4 #5* "D # J\* ,D" b3 42(5kH#.I6 E #" a1A NPN$ #e 7 #.I6n,D;4 J # * # Jd#8':& # % .I6O'E,- JO" a #,D"a n# #,D" 4 !& ( " *2,D% !6='E,- J?" o #,D" Z# #,D"5"7%!kZj42517& * E?%: " .I6 _= # o,D':h=2,D" ' "$ _=?Z# (97 #E(8,-':* -E K42515* =%: " .I6 _= # o,D21A EkH "?':* ,D" %K,/E,D& [ λˆ./0(n=':0 # 2(I

ED D

8D

V V

|yfBdfjlkl`ako^ .w j]DrDk #H " " % ) +) &)

{e{ atfDc9a`[fDr

$-1,D"7; " ,c;d 4 #./(q 4 #%:E( "8* "7"D; %" %'E& 9( 8,G%: #e e Jn #e 8f? #!ka .R ,D.RE# !;# [56E21A !& (.I6\'E,- J %!,D"2('!(9H42515*76G# %!,D"2('!(I* ,D"#E,D" 42515*76K " (97 # Z': # !; i^%" %'!(= , , (9" E " ,c;L,--1A':bhZ7;a42515*( # ( !& 42e "H"7ZkZ$42515*:*"H V T M V Y -C

8D

1A2,D

SD

PD

B0 (ξ) → min, Bi (ξ) ≤ 0, i = 1, . . . , m0 , Bi (ξ) = 0, i = m0 + 1, . . . , m, x(t) ˙ − ϕ(t, x(t), u(t)) = 0, u(t) ∈ U (t). Ξ = {ξ = (x(·), u(·), (t0 , t1 ))}

V V S !

@lE,D"#,D"

KC 1 (∆, Rn ) × KC(∆, Rr ) × R2 ,

D, E,D"5;h= 4 %'!,D *#7M:# .R # 1A NPN$ #e ' 2(.I6 n M ( #.I6LN$':#%e J A%'!,D *#7M:# .R #.I6 r M!( #.I6LN$':#%!& e J u(·) = (t , x(·) @]4251##.RJ?%: # *#.RJ= " 4 % 0 t1 ) t i ∈ R ∆ l @ D , 2 ( J ,D" 21(#5kZ2,D" 4 Rr t , t ∈ int ∆ {U (t)} 0

1

Rt Bi (ξ) = t01 fi (t, x(t), u(t)) dt + ψi (t0 , x(t0 ), t1 , x(t1 )), fi : R × Rn × Rr → R, ψi : R × R n × R × R n , ϕ : R × R n × Rr → Rn , i = 0, 1, . . . , m.

*,.N [

&H)*$u ,-

V V S !p#4 .REb" n ]" U (t) , , (9≡"R Eb" [E#76E E( pE,D"#,D" # .R # 1A NPN$D& Ξ1 = C 1 (∆, Rn ) × C(∆, Rr ) × R2 #e ' 2(.I6 n &x( #.I6 N$':#%e J x(·) # .R #.I6 r & ( #.I6 N$':#%e J u(·) (t0 , t1), ti ∈ R , # ( J 5 1 *' V V S !/#7& kξkΞ = max kx(·)|C + kx(·)kC ) + |t0 | + |t1 | 4 .REb"

2( #" # 4 R . E

" c , L ; A 1

E ' D , " ( / . ( \42515*?f? #!& ko ! Z42515*j "ξ (97 # o': # !; -52,- #?DkH "$$& ,D"#,D" Ξ1 ! ,D " " ,D" ## Ξ ' 1A " E; "8E,D2( V ST

;L), %T\C 1 *,.N !#"%$&' (%)*+!-,.$/ + $`)*/-&'$#J.K%N 1

1

WD +I *, %\ ],!J #TJ.( P,!J.P1A+(%& . J #" % $ 9 N : $ : ( PK%, E ( /+!-,.P +L)*&'/ P ,!&'(Q5J.$#, )*:+,.&'$#J.P '-,+.( ,. ,.$P# J # +J -,.$#J.9PN + :$:(P"5: %!7\ #5.R15JK*('+ f?# ('!(#!k? G+ 9E, ,D,-"'#$-1A##':,.,D$b"S"a $# , )*#

# * # !;(II: E;"5j*" D2( #" ,D"7; "

ξb = (b x(·), u b(·), b t0 , b t1 )

ε > 0 ξ = (x(·), u(·), t0 , t1 ) kx(·) − x b(·)kC([t0 ,t1 ])| + |t0 − b t0 | + |t1 − b t1 | < ε b B0 (ξ) − B0 (ξ) ≥ 0

1A b]f? #!k ./,D%!42##':b (O1;G%: # *#E( # JK gAΞ511 & %: Jd42515* d,G #,D"E ( dmV U U 21A' 9(5kH#` ( #!;"`1; 2,DE(9=_= %: Z%!A , , Z42515*G" V V S Y f0 (x) → min, fi (x) ≤ 0, 1 ≤ i ≤ m, F (x) = 0,

c1A ( " ,c; ( # ( 4 ' 2(.RJ N$':#%e #7L%: # *# r* ,- # #,D" [ h=L E; [ 2,D%: # *# L* ,-n #,D" F " [ 7k "/ "E(+ ,c6E21A# # J# jE,D"#,D" /1A'E&

P # J# ?E,D"#,D" Y -i * ,D"#E,D" A42515*Zf?X #!k "#E,D " ,c;H%H , , (9" E2(E('=%!A , ,D'=42515*E515*o " (97& # /': # !;?# "#E,D " ,c;?%?#2(' 2# %: #*"D #.RJP 4 '2& "7"H,D " " ,D" ' "Z :1A \f? #!k ,D" *2,D% /[ 48" *# N$ ('2 E# !;]'E,- JI %: " .I6K :1AD;Lf? #!k?E,D':h=2,D" (9: P(5kH#Z .R4 ""7%C

D

1 ,!J.J #J.P++9 "%P, ;(%P$K%,Y$# N[E&H#)*KQ$! #"% $9 NI:$:( + *, %\

L(x, λ) =

n X

λi fi (x) + hy 0 , F (x)i

& ,$# L) :$Z #$#P,.N, $-,.&'NZ [;(%&H$)*K%Y$ $# !- J.J.$# #)-J!, ,.$ ,.P)+,7, $#,.: ~ 20 - : ,G3(Q!J #"+ #,D:!$) :(56 KW\ AB )- * , ' $# (%*,!J." ,+P9;,`, : +$ J.P,.#/#P, #"'QJ.P$+9 B J!9 +!/* " $$#$,.9 ` )-JPY#$#, P)*, +#",.%$#$J.9WP7+!+:9 '#$##J ,.#$9 )*'+#$,.$#/ J. PP+,. NI$#, ,!J.K%J. - J:1;L E#,D:4 .RDD E2(2 (#.R".P(QJ' #1A +/ :" E ;!& bh= $':* ##./(n': # # !;(I!,--1A' "= " [ "Z" "5!4 #5* #

i=0

λ

0 ≤ i ≤ m

y0

λi Y L(x, λ) → min

λi fi (x) = 0 1 ≤ i ≤ m

N$':#%e #7A$#P%: " E(8( # (97 # H" c1P42515*o[ ' 1A "? ,-& ,--1A E#P1A%: #e 2,- *2(I! 4 2,D"#!*"42515*= ( " VSV

PD

_= # -E$ %!7kZ2(I%!% "? ( #!;"$1;H , ,--1A E# !;42517& * V V S !- ?':#%e !;Of? #!k\42515* V V S ! ( "a :1 "4 '!(D& " ,c;"7=kZPN$':#%e !;*"H .R , #=#,D"V!=%" %'!(9-C L(ξ, λ) =

1

Pm

i=0

R t1

λi Bi (ξ) +

8D

R t1 t0

p(t) · (x(t) ˙ − ϕ(t, x(t), u(t))dt =

= t0 L(t, x(t), x(t), ˙ u(t, )λ)dt + l(t0 , x(t0 ), t1 , x(t1 ), λ), Pm L(t, x, x, ˙ u, λ) = i=0 λi fi (t, x, u) + p(t) · (x˙ − ϕ(t, x, u)), Pm l(t0 , x(t0 ), t1 , x(t1 ), λ) = i=0 λi ψi (t0 , x(t0 ), t1 , x(t1 )).

2( #" ,DE,D" "? 4o"D6G,DE,D"7;bh= !6C : :1AD;af? ξ#!k=-1A ,D.RE "H , , ( " "Hx(·), " au(·) 42515* (t 0N$, t 1%!),D !& ';K1A $%:E( # #".] 4P"D6 -C x(·)); V V S #& L(x(·), u b(·), (b t0 , b t1 )) → min ( u(·)); V V S L(b x(·), u(·), (b t0 , b t1 )) → min ( V V SEU L(b x(·), u b(·)), (t0 , t1 )) → min ( (t0 , t1 )). 515* V V S #& H42515* V V S ?,-':* 2,- H# # "? 7& # * # J "#E,c;" ,c; % * ,-'m42515*mE e ## 8 u,D* ,- # !; [ 4? # * # J !6a#4 .REb" A42515* a" V V S w # * # 2( u(t) ∈ U (t) R(.3#4 .RE2( 42515* V V SEU @ R2 At KE,D"77E,DZ _=H .R !& , "P# [56E21A (.RI'E,- !; %!,D"2('!(9o1;H * ,- ##.I6"D6 42515* 3- 3(*-&~ :*5 ;~- ?"":.) 8:", "" xb(·) !)H ) 4-0;< )

S *!: B'#"%Y# Z)-J.P,.NO :K%# ,: *! : ,' &H)*$!#"*%,.$$&'N :(%)*+!-,.$/ & - D B(x(·)) =

Z

t1

L(t, x(t), x(t))dt ˙ + l(x(t0 ), x(t1 )) → min

`[ )*.R*$##J!+, )MJ7!#"(@%)*$+$#J.,.$ / ;N-, )* ,DE,D"5;"+ 4

"5 - V S S

t0

Z c(Q1AJ #$%!+%N

d b b x (t) = 0 Lx˙ (t) − L − dt ib b Lx˙ (ti ) = (−1) lx(ti ) , i = 0, 1 b x˙ (t) = L(t, x L b(t), x b˙ (t))dt, b lx(ti ) = lx(ti ) (b x(t0 ), x b(t1 ))

3- 3(*-&~ :*5 ;~- ?"":.) 8:", "" -*5620!7 !7!)H) 4&- "2+)3;<*- -: ) ;~!"

Z

t1

(QJ #+/S:$:(7C

ψ(t, u(t))dt → min,

,DE,D"5;"Z 4?,--1A':bh= t0

u b(·)

u(t) ∈ U (t)

min ψ(t, u(t)) = ψ(t, u b(t))

u∈U (t)

" *%!76G# .R #E,D" L%'E,D *#Z# .R # JLN$':#%e ub(t) 3- 3(*-&~ :*5 ;~- "":.) ) (bt0, bt1) ;~)~,.- 7 !)H) 4 *o- )+""4!""7 h(t0 , t1 ) → min ,DE,D"5;" 4 ∂h ∂h u ,- 3" ^ !5kH " "∂t 0(N$bt0 , bt1()'2=. 0, %342∂t51(5bt*0 , bt( 1) =V V 0S #& M V V SEU -"\':* (IA*"\# [56E21A (.RP'E,- !;aG42515* V V S ! ,DE,D"5;"Z 4 G': # # !; J! $ -C x(·)

(QJ #+N7J.PY$# )-$J.5 D

0

1

E1

−

db b x (t) = 0, Lx˙ (t) − L dt

G'E,- JG"#,D , 7 #E,D" G

x(·)

b x˙ (ti ) = (−1)ib L lx(ti ) ,

w G'E,- J\( # ('!(9$

u(·)

V V S X

C

i = 0, 1,

V V S

-C

b min L(t, x b(t), x b˙ (t), u) = L(t),

V V! T

u∈U (t)

G'E,- JG"#,D , 7 #E,D" G t C

∂L b (ξ, λ) = 0, i = 0, 1. ∂ti

V V!V

u -, = , ,--1A' " ,c;?42515*Rf? #!kI#,-A[ .RJ=( # ('!(I7"E(D& ,D" V V! T .R ,D.RE " ,c; VS!

c0

1

G': # # J! $

u(·)

-C

V V! S

b u (t) = 0. L

t K(5kH#H, N$ ('2 E"Z" *#.RJL 4 '27"7"5 3- 3) #e ?f? #!k$1;L42515* V V S ! - : LC* ξb 0B 2C4EWGV I=O1=EF '?;&0@ U ,576%8 6;>;: f, f , f , ϕ, 1 2 N 1GV'%G ' ;1 ϕ x , ϕu x u 8 C N0#4;+O 0B'%ODK1;M+OC$ GV QEWC( {(t, x 4Eb4(t), E ?;ub&0(t)), @ tU∈ [bt 0,bt1]}#I'%GV21 O1 Gξb &%1 ) DK 576%8 6;>9;:# 576%8 6; > :#W576%8 6 A> :V-576%8 6 D6=: 8 :LC* 0B 2C4EWGV =O1=EF ,'R?;&0@R576%8 6;> ;: 2C( E O*%1 F2 ξb'%O1 DK# f, f , ϕ, ϕ , 1 2 N 1GV'%GR'M ;1 EW1 Q;' {(t, xb(t), U (t)), t ∈x [bt , btx]} 8 C N0#D;+O ξb 0B'%ODK1; +O*%GV EW4E4E ?;&0@I576%8 6;> 0;:# 1 J'%GV21 O1 G &%1 ) DK 576%8 6;>9;:#W576%8 6; > :# 576%8 6 : 576%8 6 D6=: 8

!" !)H ) 4 Z< 7 +(. ,--1A' 2( -1A #E('m!A#'Cj 7M: .I6 ,DE,D"7;2( N$':#%e b f? #!k ?f-P r,DE,D"7 # ?f #m %!b*7& 2( ,-A D2(.R`mN$ %!,D E###./( # *#./(l'E,- !;(Ij [ ,D " " ,D" ':bh= ?(#5kH "D +f? #!kZ# #N$ (9" #.PAij7M " .I69 .R ,D.RE2(0': # # !; J! Q - 'E,- !;m"#,-& , 7 #E,D" Qot -I'E,- !; ( # ('!(9 Qo _Z2(q'E& * ##.RH,D "# _= # !;i M:" " !6 [E,D'EkZ12(I*"L#J:1A #\@W .R':%!E(O,-':* %: c1PN$':#%e #7H .R':%!.RJEP # * # !; # J#.P,D4 '\':* " ,c;*"\#J:1A #`[E,Db"#.RJ+( # ('!(I #%: # e .R ,D.RE2( " "5ij :1A'+" *"a(.rE,D"':2( ,D" *2,D% 9++42%!b* "D # Jd* ,D" n [E,D'EkZ12(]':* #!& #.RH _= # !;E,D c1\A D2( ?f λ0 = 1 $ D6E21A (^%OD& _= # !;(I # J(x(·)) = R01 (x˙ 2 − 12tx)dt → min x(0) = x(1) = 0 ?f ,D 51 " , J pQ 7iOC,D !'x¨$ .R=':%!−6tE,D" ?42 51[ h=5* 5N$': #_=% e# ! ; x(t, C1 , C2 ) = t3 +C1 t+C2 ' 1A " E;bhZ7;% ./(8'E,- !;(I@r[E,Db"#.RJK( # ('!(I " "C xb(t) = −t3 + t ∈ absmin RT 2 # T > 0J(x(·), - T ) = 0 x˙ dt → min x(0) = 0 T + x(T ) + 1 = 0 VSY

Z1

D

:1

:1

:1

?fHC R T 2 Q C [ h= _= # K , :' * "E(m0 %x˙dt + λ(T'E,-+ x(T !;\))# E(dx¨ %:= #0e -C x(t, C) = Ct AQot] C Qotr C 2 $4 " !681A 'E6 x 2x(T x(T ˙ ) = 0 ': #˙ #) J =λ':*2(q1ATEx˙ _=(T )#+ !;λC +x1 (t) E,D-x12,D(t) " =## 0J ij" j _= # I "51 "57 [ o" c1 T = −1= −2t %: JG':[ DkZ12(9,c;!*"H " "C xb(t) = −2t .# J(x(·)) = R01(¨x2dt → min x(0) = x(0) ˙ = x¨(0) = 0 ˙ = 4 x¨(1) = 12 ⇔ R 1 x2 dt → min x˙ = x x(1) = 1 x(1) 3 0 1 2 x˙ 2 = x3 x1 (0) = x2 (0) = x3 (0) = 0 x1 (1) = 1 x2 (1) = 4 x3 (1) = 12 ?fHC R01(x23 + p1(x˙ 1 − x2) + p2(x˙ 2 − x3))dt \Q C −p˙1 = 0 " ,Dbj1 ':*2(IC (6) −p˙ 2 − p1 = 0 2¨ ':* "E( % .I6qx'E3,-= pJq2 #] E( %: #e [ h= x _== # 0 C i ,D !'H .R':%!E,D" G42515* x(t, C1 , C2 , C3 ) = C1 t3 + C2 t4 + C3 t5 N$':#%e !;!' 1A " E;bhZ7;K% ./('E,- !;(I @l[E,Db"#.RJ ( # ('!(I " "C xb(t) = t4 ∈ absmin # J1(x(·)) = R01(x˙ 2 + 24t2x)dt → min J2(x(·)) = R01 txdt = 1 x(0) =R 1 x(1) = 5 ?fHC (x˙ 2 + 24t2x + λtx)dt ZQ C 2¨x + 24t + λ = 0 [ h= _= # C x(t, C1 , C2 , C3) = t4 + C1t3 + C2t + C3 /$48 # *!& #.I6 ^ 4 ( " *2,D%: d'E,- !; !6E21A (r% #,D"E (IC C3 = 1 C1 + C2 = 3 C1 /5 + C2 /3 = 1/3 ⇒ C1 = 5 \ j i

: A 1 '

R . : ' ! % E D , " W 2 4 5 1 5 * W $ N : ' # % e ! ; G ' A 1

"

& C2 = −2 E;bhZ7; % ./( 'E,- !;(Ij@ [E,Db"#.RJ]( # ('!(I " "C x b(t) = t4 + 5t3 − 2t + 1 ∈ absmin # T → min J(x(·)) = R0T x˙ 2dt = 1 x(0) = 0 x(T ) = 1 T > 0 - ?fHC R0T x˙ 2dt + T + λx(T ) Q C x¨ = 0 3':* "E( % G'E,-& !; #n E( %: #e8 [ h= ` _= # C x(t) = Ct RQot x(·) C !Qot C 2 ) + 1 + λx(T : "%' 1 ˙ ) = 1 2x(T $˙':)*=2−λ (d-1A #,D" #T# Px˙ (T _= # x(t)˙ =)t= a0 4 x(T ) =x(T ,--1A'E& "5*" T = 1 E,D-1,D" ##7;\ %!Z %!4 .RE "5!1*" " [E,Db"#.RJL( # ('!(I " "C xb(t) = t Tb = 1 ∈ absmin

D

:1

:1

:1

D

iotj !6EE(

VS#

@HWUKHU$>

!

,. SJ.`)M, #" 9 )-#J.`)M, #K% $#K !&

s9 , , ( " (^,D':h=2,D" ( P42515* G " (97 # ': # !;C

S T t S V x(t) ˙ − ϕ(t, x(t), u(t)) = 0 ∀t ∈ ∆0 ⊆ [t0 , t1 ]; S S u(t) ∈ U (t) ∀t ∈ [t0 , t1 ]; S ! x(t0 ) − x0 = 0, x(t1 ) − x1 = 0. 515* S T M S !G@ * ,D"#.RJ0,-':*J) ,c6E21A# J]42515* V T M @ N$ %!,D E#. "7& V Y-Ca# JO D * #. ,D':" ,D" ':b"Z # * # !;t0 V x(t Y "0 ) Zt1# x(t 1 )#,D" : " ,D':" ,D" ' "=" !& ( #7 #7;L,DE,D"7;bhZ7;GHN$':#%e #7 C I(x(·), u(·)) ≡

Z

t1

0

f0 (t, x(t), u(t)) dt → inf;

ψ0 (t0 , x(t0 ); t1 , x(t1 )) = 0,

" ,D':" ,D" ':b" #" 7 #.R?,DE,D"7;bh= o'E,- !;!6 V !-C fν (t, x(t), u(t)) = 0,

1 ≤ ν ≤ m + k,

\" ( #7 #.RK,DE,D"7;bh= Z V !o,D " " ,D" ':b"\% ./( 'E,- !;( S !-C ψ1 (t0 , x(t0 ); t1 , x(t1 )) ≡ x(t0 ) − x0 = 0, ψ2 (t0 , x(t0 ); t1 , x(t1 )) ≡ x(t1 ) − x1 = 0;

"H2,D"$ V ! VS

i = 1, 2, . . . , m = 2n

s _= # 42515* S T M S ! 42%!b* " ,c; -1AD # :' ;bh= J= %" :M!N$':#%e uˆ(t) ≡ (ˆu1(t), . . . , uˆr (t))T ,D "7& " ,D" ':bh= JO JO" %" xˆ(t) ≡ (ˆx1 (t), . . . , xˆn (t))T " *% x NP4 GE,D"#,D"E1A !kZ # H%: " JO ,D.RE " ,c;8,D ,D"D& ( Jn [ .R%# ##.I6+1A NPN$ #e 7 #.I68': # # J S V - n%:& " 7;42a 2(; t1 − t0 c1A t0 t1 @ 4251#.P1AkH#a J" 4aN$ %!,D E## a#5*7 # +5kZ # !; x(t ) = x ON$ %!& ,D E## \%: # *# \5kZ # x(t ) = x 0 'E,- 0 *" N$':#%e #7 S T / # (9 "L "E1(]( #1 (97 # Z4 #5* # $#:1A %!, T P4-1A2,DZ G17 $ 4 #5* "Z"#,D # E# j' 1A2(+ _Z"$42515*'P$ #"E; #,D%: JN$ ( ,j ,D 4 E7& # 2(+N$':#%e J H l - -1A';? !'$ _= # !;=42515*P " (97& # ': # !; , (IV S -!42 _=2(dN$':#%e b0$ #"E; #C

GD

BU V

H ≡ hp, ϕ(t, x, u)i − λ0 f0 (t, x, u),

" ( ##"C

l ≡ hλ1 , x(t0 ) − x0 i + hλ2 , x(t1 ) − x1 i,

\#':b),D ,D"2('G'E,- JG #e (9%!,D ('!(9 'E,- !;\5M:4 -C \,D ,D"2(9m': # # J J! 5M:f? #!k ,D E;!kZ ##7;3,D !& ,D"2(9-C

1

ˆ ∂ H(t) pˆ˙ (t) = − ≡ ∂x ˆ 0 f0x (t, x ≡ −hˆ p(t), ϕx (t, xˆ(t), u ˆ(t))i+ λ ˆ(t), u ˆ(t));

[G'E,- !;G"#,D , 7 #E,D" C pˆ(t0 ) =

∂ ˆl ˆ1 , =λ ∂x(t0 )

G'E,- o " (97 #E,D" \

pˆ(t1 ) = − u

C

∂ ˆl ˆ2 ; = −λ ∂x(t1 )

ˆ0 ) ≡ u ˆ0 ) u ˆ(t) = arg abs maxH(t, x ˆ(t), u(t), pˆ(t), λ ˆ(t, xˆ(t), pˆ(t), λ u(t)∈U (t)

∀t ∈ ∆0

V SEU

G'E,- !;=,D"7e ##E,D" = t t " ,D':" ,D" ':b"5"7%=%!% t 0 1 0 N$ %!,D E#. t1 1G'E,- !;\1A #!;bh= J`#DkZ2,D"%:E,D" 8 " ,D':" ,D" ':b"5"7%`%!% 42515*o# "'E,- J\" # #,D" G'E,- o# " e"D #E,D" C λˆ ≥ 0 0 k G'E,- u s 4 G'E,- o# ( % L(#5kH "D JLf? #!k $E,-\21,D"7# % n': # !; 4 'E,- !;LjK': # # !; S V j +Iuˆ'(t) :* =( ,Du(t, ,D"xˆ2((t),'L pˆ[ (t), .R%λˆ#0 ) #!& #.I6K1A NPN$ #e 7 #.I6K': # # JK1;aN$':#%e J xˆ(t) pˆ(t) C ˆ ∂ H(t) x ˆ˙ (t) = ∂p ˆ ∂ H(t) pˆ˙ (t) = − ∂x

ˆ0 )), ≡ ϕ(t, xˆ(t), u(t, x ˆ(t), pˆ(t), λ

S Y

ˆ0 ))i + ≡ −hˆ p(t), ϕx (t, x ˆ(t), u(t, x ˆ(t), pˆ(t), λ ˆ 0 f0x (t, x ˆ0 )). +λ ˆ(t), u(t, xˆ(t), pˆ(t), λ

s _= # 8,D ,D"2(.% S YZ1AkH#d' 1A " E;"n% ./( 'E,-& !;( S !-j'E,- !;( "#,D , 7 #E,D" [H ^'E,- !;( M:4 - t%+E,D':h=2,D"; " ,c;L D6E21+ "H ,c6E21A# JL42515* L " (97 # ': # !; S T M S !j%+% J\42515* % JL42515*P #!& e (9%!,D ('!(91;L,D ,D"2(. 2n [ .R%# ##.I6Z1A NPN$ #e !& 7 #.I6P': # # J? E;:1A%! S Y, % ./( ?'E,- !& ;( S !R +'E,- !;( a"#,D , 7 #E,D" a4n [/ L1A # "D& #.I6O'E,- !;!68 M:4 -iq':* ## J% JO42515*H# 4 2,D"#. E,D"5;##.I6G #" E# !;L,D ,D"2(. S Y + 1 (#5kH !& 2n "D Jmf? #!k λˆ λˆ λˆ E,D 4n + 1 # 4 2n 2,D"#.I6 -
#!k E,D 4n + 1 'E,- - "d2,D"n* ,-d# 4 2,D"#.I6 % JG42515*$ #Z* ,-'G'E,- JH1;L !6\ -1AD # !; ( " (I*"C V 'E,- !;o"#,D , 7 #E,D" $[A 41## JP% Jo42515* P(5k& # ,D%!b* "E,D%: %'H # L,-'Ek"=1;\ -1AD # !;a(#5kH !& "D J`f? #!k λˆ1 λˆ2 %: " .R?\,D ,D"2(' S YR#Z6E21;"5# VSX

%D

_= # I% JZ42515* =#R !;b"5 "E('? !6= -1AD;"$# " [ ' " ,c; S j(#5kH "Df? #!k H,D !'K'E,- !;K# " e"D #E,D" $ [ + ##'2b # λˆ(90 7 #.RJn,-':*J 42(5kH#E,D"a%: "& $" [ ' "P " 1AD # P#7 42-5 [ P[ _=I#'2; K,I':* "E( 'E,- !;O# ( % 4 P(5kZ "+[ .R"+%: #% " 4 E# (5kH# # ( #!;" λˆ0 = 1 , (I .R_=R ( . #7 " *2,D%: _= # !;L42515* - i) " E,-H ,D%!b* # !;O'E,- J`"#,D , 7 #E,D" 8[ % J842515* λˆ > 0 a* ,D"#E,D" λˆ = 1 oE,D"7 "7& ,c; 2n # 4 2,D"#.I6PE,D0"5;##.I6P #" E# !;?,D 0 ,D"2(. S Y-2 ( " ,c; 2n 'E,- J S !A1;? !6? -1AD # !; 2 λˆ = 0 # !& (97 #.RJn,-':*J o% E7;O42515*\" [ ' "\1A # "0D #.I6O ,-& ,--1A E# J , (I .R_=j [E,D'EkZ1A # " $,-':*7;Zo4-1AD $E(& ( #"7 J%'E,- !;(] M: - ( . #7 " *2,D%: aD& _= # !;L42515*G λˆ0 = 0 V !- $" e ##.RJ+ e2, ,?* ,- ## G _= # !; \% J 42515* #e n(9%!,D ('!(9d( "21AE( ,D"D [ .P n* ,D"#E,D" ':* ## Jd% Jn42515* 42%!b* " ,c;d8E,--1A E"D #E( _= # d,D n42515*n$ _=
` _= # !;d42515* n$ _= n21Ak " ,c;81A` [ h= # !;`# %" !&xN$':#%e D#4 .RE2( J , (I2# !& kZ - [E,D' 1A ( "?21A [ # R#P ( , , (9" E2( J42517& *
V 4D

N

%0 L)-'# )*!,.`)-+E)-#J.`)M, #K% + . , % K # )*!,.`)-9 )-#J.`)M, #K% +,.K%L)- ;(%$K%Y#,.NP$#,!+!/* K D :KQK% D

;U

,H#5*7 #./( 'E,- !;( 42515*' $ _= +1;O,D ,D"2(. S Y -1AAx˜(t D0) " =,c;x0*"ap˜(t 0_=) =# αH42515* $ _= ?,D':h=2,D" ' "- #J:1A2(K1;?4251## α N$':#%e x˜(t) p˜(t) 5,--1A E"D #D4 #5* # 2u ,- ? u ˜"(t)E( ∀t%∈ [t 0.R, =t1'E] ,- !; x˜(t1) = x1 ' 1A " x˜ (tE;1" ),c≡; x˜*1"a(97 D& 5;"# E,D%: %'d %" α 4251# %!%!& " - "n42515*8[ ' 1A " _= #u ,- ZkZR% .RI'E,- !;Z#R' 1A " E;" ,c; x˜ 6= x - 1 1 "H , , (9" E " ,c;G %" C [t0 , t1 ]

D

U V

Z+,.K%L)-'$#,!+!/* K D

X≡x ˜1 − x1 ⇔ X ≡ (X 1 , . . . , X n )T ≡ (˜ x11 − x11 , . . . , x ˜n1 − xn1 )T ,

#4 .RE2(.RJ

$E,D%: %'` %" 42 ,D " "$#5*7 #.I6=4 #5* # J α C x˜1 ≡ x˜1 (α) E"P " α [ ' 1A "$x˜421 ,D "$ %" K# ;4 % X :"Z2,D" " "= %" K[ ' 1A "= %" !&xN$':#%e JC S #& X ≡ X(α) ≡ (X 1 (α1 , . . . , αn ), . . . , X n (α1 , . . . , αn ))T . "7` %" !&xN$':#%e !;d#4 .RE " ,c; V #! U XV #:V
[ *2,D% !6H': # # JH1; n # 4 2,D"#.I6 αi 1 ≤ i ≤ n C n S X i (α) ≡ X i (α1 , . . . , αn ) = 0, 1 ≤ i ≤ n. #5* # !; %" !&xN$':#%e 1; %!7kZ1A nN$ %!,D E##&

α -1AD;" ,c;aH 4 '27"7X(α) "P* ,- ## K _= # !;L#H " 4 %: 42515* O$ _= `1;,D ,D"2(. [ .R%# ##.I6L1A NPN$ #e !& [t0 , t1 ] 7 #.I6+': # # J S Y E;:1A%! 2n , 2n #5*7 #./( `'E,- !& ;( x(t ) = x p(t ) = p ≡ α $42( #!;!; α "2(), (./(]#7& *7 #.RK0 'E,- 0 !; p(t00 ) = α0 # [56E21A ( ,D"D;" ,D"D& E"E,c; - _Z7;K42515*'$ _= Z1A="D6K E %!P#o 4 J:1A " 51# ,94251## JP" *#E,D" bd/%: # *# 94 #5* # x(t1 ) = x1 #7;#7 !;,DP,D"D [ JZ 4I ' 1A !;C ,D"D E7;,D721& [ 7;?': Z .I "7,D#E;:1/ 4I,D" A/ ' 1A !; ': Z .I "7R@37& ( " ,D"D% - # [56E21A (R1A [ "E,c;= 51# !;,D#E;:1 4251##':breD "7L#7 !;+5;,D#!; "G ,c6E5kZ1A # Z ,D& 4 ' 2( Z" ( # -
1

?

V

]D

+,.K%L)- ;(%$K%Y#,.NT$#,!+!/* K=

U

V U

-GD 6 ,. J.`)M, #" 9B< 68)-#J.`)M, #K% <

V! T

u ˆ(t) ≡ (ˆ u1 (t), . . . , u ˆr (t))T

-1AD; " ,c; 4+'E,- !; H(9%!,D ('!(9 u N$':#%e H(u) C j"n2,D"n 4 '27"7"+ _= # !; [ h=\ & u → H(. . . , u, . . . ) E;# %: " JL42515* L#D # J# H (9( E# !;A (D& " (IA*"GK , , (9" E2(.I6\K%" %'!(=42515*76%!% & *2(I 8L[ _= #,D" = %!A51A#.I6`42515*` " (97 # \':& # !;R _Z2(.I6 #dE,D# 8 #e (9%!,D ('!(9/,D"':%"': N$':#%e ?(#5kZ2,D"E (t) 4 ;b"5 %!%? !D D& 1AD "+N$':H(u) #%e b u(t) = u(t,Ux(t), a 4 '27"7"-1AE7& E;bh= * ,- ## $ _= # P42515p(t), * G#2λ,-0)5kH# #7 42 $E,D%: %' *",--1A' "= 4 .R_=2,D%!42## ( "21a,D"D [ . -1,D"7; "R,D [ J$ " e ##.RJ$ e2, , 4 '27"7"E(G%: " &

K1AkH#L[ .R"\ .R# # H,Z4251## J`" *#E,D" b #G# %:& " E( k M:E( _Z " a e2, , α = αk ≡ (αkε, . . . , αk ) ,D ,D"2(. 'E,- J S - # [56E21A ($ ( " 2,D" N$':1#%e b n "7& * !;8#\%!7kZ1AE( k M:E( _Z Z " e ## \ e2, , G [ !& kZ ## _= # !;a,D ,D"2(.3': # # J S j " P" *# _=D& # !;ir%!5*2,D" H"7%: JnN$':#%e d(5kZ "a[ .R"+ 4D;"7# ( ,D%!7;#7;LN$':#%e !;

PD

S(αk ) =

@

n X

S U

(κ i X i (αk ))2

J.( KQ+ )*+T$L):)-+$$9 $#,!+!/* K R ! ] #7;]# "7& e"D # -1AD ##7;I [ hZbhZ7;,c; #m" %:m i=1

N$':#%e !;RiB%!5*2,D" +# ( *#.I6^(#5kH "D J ( 5kH#K 4D;"# ( A D * #.PA-15kZ ##.R?s ?D& 1A #%: V2XV ?':#%e !; S(αk ) 4 ; "O42 , "n'E,- ,c6E21A (E,D" \ " e ## Z e2, , C S X S(αk+1 ) < S(αk ), a'E,- P %: #*# !;\ " e ## H e2, , $ 'E,- P %: #*7& # !;a,D* "7-C S S(αk ) < ε. $ P _= # ?%: #% "#.I6P42515*$'E,- 9 %: #*# !;P,D* "7(5kZ " [ .R"L42 , #L OL #E(I "2 *#E( " S -# % E7 #"#E( 2('G :1A !':* ".REbh=2(m,D e N$ %'K _Z2(.I6G42515*
5= ?

X(α) = 0 κi

D

=

?

%!%: J:ME [ j 4 (# 4 #5*#.I6o( "21A R ! ?( "21A Is ':# D&x$':"". ST
: ;Y)-+$$9 NT= ,. R]" $#?

X(α0 ) = (X 1 (α0 ), . . . , X n (α0 ))T ,

c1A X i(α0 ) ≡ X i(α0 , . . . , α0 ) 1 ≤ i ≤ n ij 4 E(2(m42"2( %" : c1A 1α1 ≡ (αn1 , . . . , α1 )T ∆α1 ≡ (∆α1 , . . . , ∆α1 )T α1 ≡ α0 + ∆α1 + "E(' α1 = α0 + ∆α1 11 ≤ j ≤nn * "77;A*"\1 h= #n !; (97.3 aj 42(j 5kH#j -1,D"7 # ? %" !&xN$':#%e X(α) ∆α1j :1A C ∂X X(α) ≡ X(α + ∆α ) ≈ X(α ) + · ∆α1 , ∂α α=α0 0

1

-1AD (] %" h= # J 2,D" 4P'E,- !;C

1

X(α0 ) +

0

∆α1

4K'E,- !;

X(α) = 0

"

∂X · ∆α1 = 0. ∂α α=α0

"\'E,- ?-1,D"7; "G,D [ J`(9" *#':br42 ,D\,D ,D"2(. # J#.I6n# 21A# 21A#.I67 [ *2,D% !6O': # # JO1;d Dn& 1AD # !; n ,DE,D"7;bh= !6K %" ∆α1 C S V T ∂X · ∆α1 = −X(α0 ). ∂α α=α0

i 4 #':"E(n :1AP42 ,D,D ,D"2(. S V T "7%: EC V! S

n X ∂X i (α0 ) j=1

∂αj

· ∆α1j = −X i (α0 ),

1 ≤ i ≤ n.

S V V

_`)-Y( :K%

[ 4 #5* n(9" e' ∂X ≡ ∂X∂α(α ) α=α ∂α E,D-1,D" E( X 0(α0 ) 42 _=2(m,D ,D"2(. S V T - S V V H :1A C S V S X 0 (α0 ) · ∆α1 = −X(α0 ). id 4 '27"7"/ _= # !;K,D ,D"2(. # J#.I6H7 [ *2,D% !6Z':& # # J S V T ! S V V -9 ! S V S = -1AD " ,c;m %" ∆α1 # 4 2,D"#.I6 ∆α1 1 ≤ j ≤ n 9 ,D"2(9L # J#.I67 [ *D& ,D% !6K': # # JG(j 5kZ "=[ .R"Z _= #? 4 2,D"#./( \( "21 ( (D& "21AE(r!5'E, , 9( "21AE(q "7kZ # J ( "21AE(q " 8 "5 RV! ? (97 #8 _= # a,D ,D"2(. S V S =(5kH#O42 , "8 :1A C 0

i

0

j

= ?

c1A

∆α1 = −(X 0 (α0 ))−1 · X(α0 ),

(9" e! [ "#7;L(9" e 0(α0 ) 0 0 −1 @ (X-(α1AD ))H %" ∆α1 ≡ (. . . , ∆α1 , . . . ) 1 X #J:1A2( %" α1 ≡ (. . . , α1 , . . . ) I c1A α1 ≡j α0 + ∆α≤1 j 1≤ ≤n j ≤ n s _= 17 n42515*'3j $ _= 1;q,Dj ,D"2(j. S Yaj ,#5*7 #./( 'E,- !;( x(t ) = x p(t ) = α1 R':* ( %" # ;4 % 0 0 0 a (m'E,- P %: #*# !; X(α1 ) ≡ (X 1 (α1 ), . . . , X n (α1 ))T ,D* "7 'E,- S I ! % E7 #"# P2('\'E,- -u ,- a # .R# " ,c;"H D * #. 1 [ ' 1A':" ,D%:E(./( L !& [ !kZ ##./( H4 #5* # !;( H%:α j# 1JK≤,D j,D"≤2(n. ': # # J X(α) = 0 =2,- \#$ .R# " ,c;:" [ 4 ' 2(O %" 2 1 2 \& " (O .R* ,- # !;Eid 4 '27"7"/':* ( α = α + ∆α 7C S V! αk+1 = αk + ∆αk+1 , k = 0, 1, 2, . . . . <= # LN$ ('2' S V!/* ,- ./( (#5kH "D2( γ ':* ( k V U C S V Y αk+1 = αk + γk · ∆αk+1 , 0 < γk ≤ 1, k = 0, 1, 2, . . . , ν. ij %" k+1 S V!- S V YN$ (97 #o(5kZ "o[ .R"o-1,D"7& #L :∆α 1A C S V #& ∆αk+1 = −(X 0 (αk ))−1 · X(αk ). i S V! M S V #&

ED

:(#( K-J!J.*,!J.K%&':,. R]" $# TP, )*Y$$( ;L)- = P, )*Y? $$( ;L):(#( : ;Y)-+$$&'],. ]R]" $#

αk ≡ (αk1 , . . . , αkn )T ,

X(αk ) ≡ (X 1 (αk ), . . . , X n (αk ))T ,

V!&!

X 0 (αk )

@l(9" e i &x7;L,D" %!Z%: " J\ # (X i (αk ))0 = (∂X i (αk )/∂αk1 , . . . , ∂X i (αk )/∂αkn ),

@ [ "#7;L(9" e $v ,- aN$ ('2 S V Y/ '2 ' "\ .R[ +_Z DK 8D& D6E21A? " & J+% k + 1 & J+ " e `,= ,D 4 E# 2(d'E,- !; S XK,c6E21A (E,D" ^ " e ## d e2, , IP( " ,c;^#2,D%: %: 4D *#.I6\,DE,D [ ? .R[ ? D * # γ i %!A , ,D *2,D%:E(d( "& 1A b" # γk = 1 iq( "21A b" #k5M:s9 NP,D #K D * # γk -1AD; " ,c;\%!%L( # ('!( N$':#%e a,D # # !; S U -C (X 0 (αk ))−1 ) γk k

γk = min S(αk−1 + γ · ∆αk ).

-1AD # Z( # ('!(9KE,D':h=2,D"; " ,c;aG 4 '27"7"? ( #D& # !;0%!%: 7ME [ ( "21^21A#E( # ,D%!o$ ) ,D 4 & E# # %!7kZ1A Jq " e .R[ " ,c;d 4a* ,-A D2( #" O%: # *# JmE,--1A E"D #E,Dγ" k *2( {1, 2−1 , 2−2 , . . . , 2−N } u ,- G'E,- S X9 .R#!; " ,c;E"?γ01AD=A1 " ,cγ;Gk ,-=-min{1, 1A':bhZ7;K2γ k−1 " }7& e !;92,- d#L .R#!; " ,c;"O(#5kH "D γ E,--1A E"D # '!( # _Z " ,c;m71A G1An .R# # !;m'E,- !; k S X- 42"2(01AD& A " ,c;a D6E21`%+,--1A':bh= J k + 1 & Ja " e Au ,- + .R[ " 216E21;h= #Z' 1 " ,c;"L,D* "7 " ,c;8*"L( "21#H,D _=DA,c; L " e ##γ.Rk JL e2, ,o42%!#* E " ,c; $ 4 #%:E(L' 15*# o %: #*# !;= " e ## e2, , .I& * ,- # J\;; " ,c;G .R# # $# ν & JG " e L'E,- !; S ! 1A': % E7 #"# $2(' 'E,- !; %: #*# !;H,D* "7EijD& * # ε > 0 K'E,- L %: #*# !;a,D* "7@p4251##7;L" *#E,D" .R# # !; " 'E,- !; a "21 b" #3#4 .REb"3"7%!kZ (D;L :1A'G 4 2,D"#':b) E( " *2,D%':b) #" "7e b u ,- #5*7 # L [ !kZ # α0 1AE,D"7" *#O[ 4 %:% 4 #5*D& # b %: #!;"L1;d(# !6O42515*O "E(3,-':*G( "21 b"& #,c6E21A " ,c;K * #[ ./,D"!i " #E(d,-':*$( "21 b" # (5kZ "L ,c6E21A "E,c;i0,D;4 , " ( .R[ 4251# 6E _= #5*7 # $ [ !kZ # !; α0 ;; " ,c;H21A# (8 4 -1AD;bh= !6 [E,D"5;"DE,D"Z'E,D _=# Z ( # # !;a( "21 b" # i [ _= #,D" G-1A D2(.I6O1;m _= # !;d42515*m$%" !& %'!(9 #5*7 # n [ !kZ # α0 -1AD; " ,c;q] 4 '27"7" V!Y γ

: ;KQY GD J ,!+ ^ $$#

D SD

PD

ID

,. 'XKQJ P, #"%$9 = ?

#7 " *2,D%: = _= # !;\ , , (9" E2( JG42515* K " (97 #&

Z': # !;K \#'2 E(4 #5* # K6E21;h= ?Z42515*'7& ( "$ ( "42515* β - 2( #".]6E21;h= JK " e ##.RJL e2, ,$(9" e.]& 4 21A#.I6 X 0(α) (9" e. %: [ /( ':"G .R* ,-;"E,c;`\ !& [ !kZ ##./(d%: # *#& 4 #E,D"#./(nN$ ('2A (IC

1

∂X i (αk ) ∂αj

≈ [X i (αk1 , . . . , αkj−1 , αkj + δαkj , αj+1 k , . . . , αkn )− −X i (αk1 , . . . , αkj−1 , αkj , αkj+1 , . . . , αkn )] ·

1 , δαkj

S V @ l 9 ( 7

$ = h # ? D , E D , 7 " ; b = h J δαkj αkj 1 ≤ j, k ≤ n
c1A

PD

=

?

V!#

{ { jwxfB[`a`_c g`aiHjkl`lfDwxf [k hkl`az cok _fqp&fDc ix_A[kqjv Dm ^`[]Dk rDf s ]Ap&]Mg`a d`[`a`l `a`d] co]B^ ixac9b co] pj`z s ]Ap&]Mg`a { {

3

V A51b" ,c;\ %" \#5*7 # Z [ !kZ # !; α0 ≡ (α0 , . . . , α0 ) )" *#E,D" ε .R# # !;3'E,- !;] %: #*# !;] " 1e ## n e2, , 'E,- !;m %: #*# !;m,D* "7- % E7 #"# 8'E,- b S ! a,D 51bh= Z,$# (I S $+#5*7 #./(3'E,- !;( x(t0 ) = x0 p(t0) = α0 42'E,D%! "7& ,c;= , ##.RJZ .R_=j " e ##.RJZ e2, , * ,- ## $ _= # !; ( "21AE(^,D"D [ .0#K " 4 %: [t0 , t1] % J+42515* + #e (9%!,D ('!(9A$" e ##.RJ+ e2, ,P21Ak " ,c;G1AG .R#D& # !;L'E,- !;G %: #*# !;L,D* "7i 4 '27"7"$ _= # !;L -1AD;!& b" ,c;C': # uˆ(·) aNP4 E7;K" %" !; xˆ(·) i 6E21;h= /,D ,D"2(' S YA': # [ K -1AD; " ,c;+#7 " *2,D% + 4='E,- uˆ (t) !;L≡-uˆ(t, xˆ[ (t), K .Rpˆ(t), * ,-λ;ˆ!0 )& " ,c;`#KE,D# " G'E,- !;a#K%!7kZ1AE( _Z = _= # !;`42515* $ _= Aij 42(5kH#E,D"G# (97 # H,-':*7; λˆ = 0 R , ,--1A' "7& ,c;H42# $ λˆ0 6= 0 (#5kH "D λˆ0 (5kZ "P[ 0.R"? ,D 4 E# 1;G# ( % \(#5kH "D J\f? #!k ( λˆ A D7& " ,c;\ #./(n-1A # eP ! L%!%: J!& # [ ' 1AP1A': JL5kH0 "D # J %: #,D"7#" 9u ,- # 42(5kH#E,D"8# (97 # O,-':*7; λˆ0 = 0 d 4 '27"7"`" " *2,D%: d#7 4281A %!42"n#`' 1 " ,c;j" % E7;`42515*K _Z " ,c;8G1A 'E6+E #"776C λˆ0 = 0 8 4251E2(E( ,=':* "E( 'E,- !;+# ( % u ,- 8%D& λ ˆ.R0 ?>425015* L " !6G1A 'E6LE #"776\' 1 " ,c;a _= ""G,D " "7& ,D" ':bh= " ( E #"7 ( _= # !;,D # Eb" ,c;+L4 #5* # !& ;( N$':#%e #7A a"G 4 " !6a _= # J1;a%: " H4 #5* # N$':#%e #7AR[ _= " ,D E " ,c;5$#./( Z,- E ( 2 21A " ,c; ,D e !;` _= # Ji3 " #E(^,-':* [ 4H#7 42KE #!& "7 λˆ0 = 0 R _= # =% Ja42515* a ,--1A E"D # _= # ,c6E21A# JG42515* G " (97 # ': # !;\[ ' 1A':"Z# #./( s9 , , ( " ##.RJP .R_=1;?* ,D"# ,-':*7;P42515* P " (97& # `': # !; S T M S !?( "21 ,D"D [ . * ,- ## ` _=D& # !;+% Ja42515* L #e G(9%!,D ('!(9KE,-?# %: " .I6G1A&

D

D

PD D

V!&

D

# # Jd(5kZ "+[ .R"+ ( # #d n* ,- ##E(3 _= # n%7& J^42515* ^ #e m(9%!,D ('!(9O1; [ h= d,-':*7; 42515* " (97 # =': # !; V T M V Y- ':h=2,D" ( "21Z,D"D& [ . D6E21Ad "]* ,D"# 3,-':*7;r%r [ h=2(' $4 '!( " ,c; ,D56!# " ,c;C# [56E21A (LN$ ( E"L %" O ( " L !& ,D"D% d n %" !&xN$':#%e b # ;4 % * ,- ##8,D" "8(9" !& e' %: [ _Z7;a1; " a,D b 42515*8$ _= _Z"L " 7& e ##./(r( "21AE(q,D ,D"2('m7 [ *2,D% !6n': # # Jn1; ,D& ,D"7;bh= !6 %" !&xN$':#%e # ;4 %] ]"5 1R<= # # !; D6E21A\ "8* ,D"# 8,-':*7;m%m [ h=2('O[ ' 1A':",D;42#.PO !& ':b * -1A,I 42( # # 2(8,D"':%"':. %" o ( " o !& ,D"D% ` a %" !&xN$':#%e +# ;4 %K"7%!kZ=,=N$ ( E# 2( +42515*768,H#2N$ %!,D E##./( t t 'E,- JO#5*7A\ nE,D"77& # E`,D* "79i [ h=2(),-':*\425150* 1V T M V YP8 %" d7& ( " ? ,D"D% α E( (?# 4 2,D"#.I6%:E( # #"P %" ( ':"K J" +# 4 2,D"#.R=(#5kH "D `f? #!& x(t0 ) α ≡ p(t0 ) k λ0 λ ≡ (λ1 , . . . , λm+k ) R ,D':" ,D" ':bh= n1A NPN$ #e !& 7 #.I6H': # # !;!6% JH42515* ,D [ R # (9# "E( ,--1A' "L [ " "L#L42515* O, # * # !;( O" L# #,D" , (I # !kZ -R -1A' "21A* %#':" *"n -1AD;bh= ( NP%"& E('E,D _=# = ( # # !;a( "21Z,D"D [ .^ G _= # \%7& JP42515* P;; " ,c;? .R[ 4251# #5*7 # I [ !kZ # !; 1;H %" $ ( " P ,D"D% α "?' 15*# $ .R[ o#7& *7 # P [ !kZ # !;1;K %" α = ':b * -1A 42 !& ,D "K,c6E21A (E,D"H( "21 b" #-$E,D%: %'K21Dkh= $* ,-& ##E('a _= # brH1##E( $%" %'!(Z42515* + " (97 # ': # !;O,D21A Ek"\ ( "8 O O#'2 E( 4 #5* # " ( "( ':"[ .R"n _= #.B#7 " *2,D% "d#7 " *D& ,D%: / _= # $(5kZ "?[ .R"= ,D 4 E#$ G':* # K#5*7& # ? [ !kZ # !;Z1;K %" P ( " ? ,D"D% G KD& _= # 3% J]42515* ],O# #'2 ./( 34 #5* # !;( 3 ( " $E,-I -1AD # !;Z %" α #'2 E(`4 #5* # Z ( " 42515* L1;` _= # !;+% J+ 42515* + `# #'2 .I6a4 #5* # !;!6 ( "K\$%" %'!(Z ( #!; " ,c;8( "21 X s9 , , ( " (r ( .B':* # !; #5*7 # 8 [ !kZ # !; 1;? %" α ? _= # ?%: #% "#.I6$42515*P ? ':"#/ _= ( # %: " .R$" .]42515*!#$ , , (9" E _= 2,c;G#

D

D

>D

OP,.$/IP# )*!,.`)-(= ?

GD

T)- ' ,.$/ )M,. V! U

{ {$Z\[`acok[m d`[`acoklkl`az cok_fAp&] ix_A[kqjv Dm d`[`a [k hkl`a`a l] fDixlfDrDk d`[`a`l `a`d]Rco]B^ixac9b co] s ]Ap&]Mg d { { > > ~> TDU.BU AE MWU~E AJ GHF*G?KHUCAEAU AEAGHYAYAGHF*GE AE*WJ.YA E # AB)-J.P,.NOP!/I * K-J!J.*,!J.K%&' + )-#Y$$&'

+

#$9CJ.#J -,$#.$*,./ $/: 2* +J N7;#KQ, )MJ., )-,.+$$$$NE9C7:T+ )M$#, *,.!$#"-%\ $9C 5K%$#,.

I(x(·)) ≡

Z

1 0

(x˙ 2 (t) − 12tx(t)) dt → inf,

x(0) = x(1) = 0.

$ E7;K42515*= #e H(9%!,D ('!(9Z ( "H :1 V V T -C

x ˆ˙ (t) = pˆ(t), pˆ˙ (t) = −6t;

x ˆ(0) = 0,

xˆ(1) = 0.

s _= # q42515* W$ _= p1; #5*7 #.I6p'E,- J ( " :1C p(0) = α x(t) = −t3 + αt,

?':#%e !;L# ;4 %

x(0) = 0

p(t) = −3t2 + α.

X(α) = −1 + α

# J#L α 7 [ *2,D%: Z': # # X(α) = 0 _Z "7& ,c;( "21AE( b" #\42L21A#'+ " e bl Ob[ E( #5*7 #E( [ !kZ # α0 C α = 1 V! X

x(t) = −t3 + t,

p(t) = −3t2 + 1.

2$ /7 J+ $#NG, ;#, )MKQ,J.)-#,.$+$M$N $9C*:7 K-\K%J!$#J.,.*$,!9CJ.:K%S&'7 $#+ *)-,.#$Y/:$S$+ )M&', I#,.$J. #J -,.

T

x(T )

I(x(·), T ) ≡ x(0) = 0,

Z

T 0

x˙ 2 (t) dt → inf,

T + x(T ) + 1 = 0,

T > 0.

$ E7;K42515*= #e H(9%!,D ('!(9Z ( "H :1 V V-C  ˆ˙ (t) = pˆ(t),  x pˆ˙ (t) = 0; x ˆ(0) = 0, Tˆ + xˆ(Tˆ) + 1 = 0,  ˆ ˆ ˆ Tˆ) = λ. pˆ(Tˆ) = −λ; H(

s _= # q42515* W$ _= p1; #5*7 #.I6p'E,- J ( " :1C p(0) = α

x(0) = 0

WD

x(t) = α1 t,

p(t) = α1 .

s _= # R% J=42515* ( "21AE(8,D"D [ . % E7 #"#$ _=D& # b0,D ,D"2(.]': # # JC X1 (α) = (α1 )2 /2 + α1 , X2 (α) = α2 + α1 α2 + 1.

$ H _= # K,D ,D"2(. ': # # JK( "21AE( b" #$ " e #!& #.RJL e2, ,

∆αi1 ∆αi2

=−

αi+1 1 αi+1 2

αi1 + 1 αi2

=

αi1 αi2

0 αi1 + 1

+

−1

∆αi1 ∆αi2

,

(αi1 /2 + 1)αi1 αi2 + αi1 αi2 + 1

=

V!&

=−

(αi1 /2 + 1)αi1 /(αi1 + 1) i i 2 (α2 ((α1 ) /2 + αi1 + 1) + αi1 + 1)/(αi1 + 1)2

7D

( 5kZ "\,D J" ,D\%821A#E('a 4?1A 'E6`,D':h=2,D" ':bh= !6+%: # J` 8,c6E& 1A (E,D"m( "21n% 21A#E(' 4 " !6^%: # J^42 ,D "n "n .R[ #5*7 # [ !kZ # !;C 0 e2, ,n,c6E21A " ,c;]% #-1A 'E,D" (E('O,D !'n'E,- !; αT1 >>0 −1 %: #b p(0) = α1 = 0 I α0 < −1 @ %]%: #b p(0) = α1 = −2 T = α2 = −1 : ; ; bh=2('E1,c;\ _= # 2(d42515* ! α0 = −1 (9"7& T = α2 = 1 e %: [ ` .R5kZ1A #A ( "21 b" #G %!4 .RE1 " ,c;a# !& ( # (./(I P%n " ( *7E,D8# .R[ O1A': Jd .R* ,- "D # Jm,c6ED& (.p(5kZ "`'2':*_= ",c6E21A (E,D"8( "218,D"D [ .P ( 2,- OL 4 '27"7"H#7 42G% J842515* ` #e a(9%!,D ('E& (9d [ E,D# E"9*" I"m42 ( #n J^%:E( # #". %" :M!N$':#%e \# ;pˆ4 (T ˆ%G) #6= 0 H(T )/p(T ) + 1 ≡ αi /2 + 1

21A "K%8,D ,D"2(=': # # JA-1A #,D1" ## = _= # =%: " J -1AD; " ,c;n( "21AE( b" #L#G[ H*2(]42\1A K " e 1;\b[ Z#5*7 # = [ !kZ # !; .#

:$ T/ 5J.IK%J.$#P,.)-$O9C:: M $# )-**,.7$ K+-/:$J! J.9C:+ *)M ,!, J. K%;,.&'$7KQ J. + 2)-)- # +Y+$$N59C$:#$,)M&'I,$##*,.J.$!#"$#%J $N -,9 . ,!,:)- + $9-\ I(x(·)) ≡ x(0) = x(0) ˙ = x¨(0) = 0,

Z

1 0

x 2 (t) dt → inf,

˙ = 12. $ E7;K42515*= #e x(1) H(9%!=,D 1,('!(9x(1) Z (= 4,"H :1x¨(1) V S V -C             

V Y T

           

x ˆ˙ 1 (t) = x ˆ2 (t), x ˆ˙ 2 (t) = x ˆ3 (t), x ˆ˙ 3 (t) = pˆ3 (t); pˆ˙ 1 (t) = 0, pˆ˙ 2 (t) = −ˆ p1 (t), pˆ˙ 3 (t) = −ˆ p2 (t); x ˆ1 (0) = 0, x ˆ2 (0) = 0, x ˆ1 (1) = 1, x ˆ2 (1) = 4,

x ˆ3 (0) = 0; x ˆ3 (1) = 12.

D

x1 (·)[α1 , α2 , α3 ], p1 (·)[α1 , α2 , α3 ],

x2 (·)[α1 , α2 , α3 ], p2 (·)[α1 , α2 , α3 ],

ij %" :M!N$':#%e !;\# ;4 %G ( "H :1C

x3 (·)[α1 , α2 , α3 ], p3 (·)[α1 , α2 , α3 ].

X(α) ≡ (x1 (1)[α] − 1, x2 (1)[α] − 4, x3 (1)[α] − 12).

s _= # ]42515* l$ _= 1; #5*7 #.I6 'E,- J x1 (0) = 0 ( " x2 (0) = 0 x3 (0) = 0 p1 (0) = α1 p2 (0) = α2 p3 (0) = α3 :1C x1 (·)[α] = α1 t5 /120 − α2 t4 /24 + α3 t3 /6, x2 (·)[α] = α1 t4 /24 − α2 t3 /6 + α3 t2 /2, x3 (·)[α] = α1 t3 /6 − α2 t2 /2 + α3 t, p1 (·)[α] = α1 , p2 (·)[α] = −α1 t + α2 , p3 (·)[α] = α1 t2 /2 − α2 t + α3 .

WD

s _= # R% J=42515* ( "21AE(8,D"D [ . % E7 #"#$ _=D& # b0,D ,D"2(.]': # # JK1;L %" :M!N$':#%e \# ;4 % X1 (α) = α1 /120 − α2 /24 + α3 /6 − 1, X2 (α) = α1 /24 − α2 /6 + α3 /2 − 4, X3 (α) = α1 /6 − α2 /2 + α3 − 12.

t%d%!%d,D ,D"2(9`': # # J # J#+ n# .R5kZ1A #" #L _Z " ,c;( "21AE( b" #L42L21A#'`α " e b Ob[ E( #5*7 #E( [ !kZ # α0 C α1 = 0 α2 = −24 α3 = 0 #

$ +/ NJ ;#, )MKQ,J.)-,.$+$$NGM$9C :+ *)M 7, $#K,.-*$!#"J!J.% $*9C,! J.K%# ,&'K%)-7$#+,.,.)-`$#)-9CYZ* ,!$J.$#K%$* &',.I$(Q#J/J.#:# +J#2-,, . # )*+,.$#J.P+%\ I(x(·)) ≡

Z

1 0

(x˙ 2 (t) + 24t2 x(t)) dt → inf,

V YV

x(0) = 1,

x(1) = 5,

Z

1

tx(t) dt = 1.

$ E7;K42515*= #e H(9%!,D ('!(9Z ( "H :1 V S -C  x ˆ˙ (t) = pˆ(t),    ˙ yˆ(t) = tˆ x(t), 2 ˙ ˆ  p ˆ (t) = 12t − λt,   x ˆ(0) = 1, x ˆ(1) = 5,

0

yˆ(0) = 0,

yˆ(1) = 1.

?< ; _= # !; % J 42515* E,D 4 ' 2(9,c; %!A , ,D *D& ,D%: J .R* ,- "D # J ,c6E2( J ( "21W,D"D [ .P+i %!5*2,D" %" ( " ,D"D% .R[ 2( 1A %:E( # #". /51 " ^4 #5* # !; %!% (ME [ [& α = (α1 , α2 ) ≡ (p(0), λ) 4 E( ` _= G42515*'L$ _= + " t = 0 1A t = 1 ':* ( ,D "7& " ,D" ':bh= $ (mN$':#%e

D

x(·)[α1 , α2 ],

y(·)[α1 , α2 ],

ij %" :M!N$':#%e !;\# ;4 %G ( "H :1C

p(·)[α1 , α2 ].

X(α) ≡ (x(1)[α] − 5, y(1)[α] − 1).

s _= # q42515* W$ _= p1; # 5*7 #.I6p'E,- J λ = α ( " :1C y(0) = 0 p(0) = α 1

x(0) = 1

2

x(t)[α] = t4 − α2 t3 /6 + α1 t + 1, y(t)[α] = t6 /6 − α2 t5 /30 + α1 t3 /3 + t2 /2, p(t)[α] = 4t3 − α2 t2 /2 + α1 .

WD

s _= # R% J=42515* ( "21AE(8,D"D [ . % E7 #"#$ _=D& # b0,D ,D"2(.]': # # JK1;L %" :M!N$':#%e \# ;4 % X1 (α) = (1 − α2 /6 + α1 + 1) − 5, X2 (α) = (1/6 − α2 /30 + α1 /3 + 1/2) − 1.

t%d%!%d,D ,D"2(9`': # # J # J#+ n# .R5kZ1A #" #L _Z " ,c;( "21AE( b" #L42L21A#'`α " e b Ob[ E( #5*7 #E( [ !kZ # α0 C α1 = −2 α2 = −30 #

$, / # )* *+,. $# J.9;PJ.M+` )- * ;7,.N#KKQJ.-PJ.+)-J!J./G+*J$,!J.$K%9C :&'#7 , )-+$# )-*#,.!`#"Y%)-$9C*$,!$J.K% &'IK%#$#(QJ.,.J ##$J 9C+-,.

V Y S

$#*,.$/: 2 +N7#, )M, ,.$$N-\ Z

T (x(·), T ) ≡ T → inf, T

x˙ 2 (t) dt = 1,

x(0) = 0,

x(T ) = 1,

T > 0.

$ E7;G42515* #e K(9%!,D ('!(9$ V ! S -,P':* "E(m'E,-& !;a# ( % λ = 1/2 a42 ( #.] #" 7AZ1A NPN$ #e 7& # JL,D;4 b=,D 21A " ,c;\%\ :1A'C 0

          

x ˆ˙ (t) = pˆ(t), yˆ˙ (t) = pˆ2 (t), pˆ˙ (t) = 0; x ˆ(0) = 0, x ˆ(Tˆ) = 1, Tˆ > 0.

yˆ(0) = 0,

yˆ(Tˆ) = 1;

D

x(·)[α1 , α2 ],

y(·)[α1 , α2 ],

ij %" :M!N$':#%e !;\# ;4 %G ( "H :1C

p(·)[α1 , α2 ].

X(α) ≡ (x(T )[α] − 1, y(T )[α] − 1).

s _= # 842515* $ _= 1;]#5*7 #.I6 'E,- J ( "H :1C y(0) = 0 p(0) = α1 x(t)[α] = α1 t, y(t)[α] = α21 t, p(t)[α] = α1 .

x(0) = 0

WD

s _= # R% J=42515* ( "21AE(8,D"D [ . % E7 #"#$ _=D& # b0,D ,D"2(.]': # # JK1;L %" :M!N$':#%e \# ;4 % X1 (α) = α1 α2 − 1 = 0, X2 (α) = α21 α2 − 1 = 0.

V Y!

$ H _= # K,D ,D"2(. ': # # JK( "21AE( b" #$ " e #!& #.RJL e2, ,o ( " :1C αk+1 = αk + ∆αk , 1 1 2 1 k ∆α1 = k − 1 , ∆αk2 = −αk2 + k − k 2 . k α2 α1 α1 (α1 )

( #= J\%:E( # #".] %" :M!N$':#%e G# ;4 %\# x2 (α2 )[α]/y(α2 )[α] − 1 = 0,

Z " J\#

y(α2 )[α]/x(α2 )[α] − 1 = 0

D $,./G$#, J ;I(%$K@K%)*Y,!+9C$#!#M'(Q J #B*+7'##"K, %-Y#J!4J.$##*, ;,!J.$#K%KQ,J.)*&'7)-++,.+$# )-J.$P#$+9CY% \ $K%$$#&',.I$#9CJ.#+J )M-,,..

4 #5* "D # '2':*_Z " ,c6E21A (E,D" ( "21 b" # $ ( "21 b" #? "E(d,-':*$,c6E21A " ,c;K%G _=D& α01 6= 0 α02 6= 0 # b α = 1 α = 1 42Z21A#'K " e b= 1 2 #

B(x(·), T ) ≡

Z

T 0

x˙ 2 (t) dt+x2 (0) → inf,

x(T )+T +3 ≤ 0,

T > 0.

$ E7;K42515*= #e H(9%!,D ('!(9 V ! X ( "H :1C   x ˆ˙ (t) = pˆ(t),      pˆ˙ (t) = 0; ˆ pˆ(0) = xˆ(0), pˆ(Tˆ) = −λ;   2 ˆ ˆ ˆ ˆ  pˆ (T ) = 2λ; λ · (ˆ x(T ) + Tˆ + 3) = 0;    ˆ ˆ λ ≥ 0; T > 0.

i ,D " " ,D" ,a7 " (E(q _= # !; 42515* m# [56E21A ( , , ( " "=1AE,-':*7;C λˆ = 0 λˆ > 0 i^,-':* λ = 0 % E7;G42515*= ( " :1 V !&-C V Y Y

x ˆ˙ (t) = pˆ(t), pˆ˙ (t) = 0; pˆ(0) = x ˆ(0),

pˆ(Tˆ) = 0,

Tˆ > 0.

P%]E,D c1m 21A [ #.I6],-':*7;!6R* ,- # 4 2,D"#.I6] "7%: J % Jn42515*G[ _=K* ,-A`'E,- JO1;d !6d -1AD # !;9i [ h=2(3,-':*H1;n _= # !;d% J42515* n(5kH#+ E,D 4 & E"E,c;%!% (ME [ P,DE,D [ E(+ '2; 42e K42515* Z@0': #!;" * ,-# 4 2,D"#.I6P ?* ,-'E,- Jo1;? !6P -1AD # !; !& ( AG , , (9" E2( JL42515*?# 4 2,D"#':bq D * #' (5k& #= .j1AD "Z=* ,-= ( " =42515* :$ K _= # G%T J 42515* GN$ %!,D E##E( T ( "21AE(O,D"D [ . ?%!5*2,D" R7& ( "a ,D"D% n .R[ 2( D * #' -1AD;bh=':bl#7& *7 #.RK'E,- !;`N$ ('2A (IC x(0) = αα p(0) = α s _= # 42515* L$ _= G L"7% !6\#5*7 #.I6K'E,- !;!6K ( " :1C

?':#%e !;L# ;4 %\ -1AD; " ,c;\'E,- 2(# E(%: #e C x(t) = α(t + 1),

p(t) = α.

D

X(α) ≡ p(T )[α] = 0.

a "21 b" #Z -1AD; "H%: # " # J# H': # # !; 42P21A#' " e b= #7 4o ( " *2,D%: JH42 ,D (E,D" α=0 %!4 .RE "5*"+ #G # J#L f (T ) = x(T ) + T + 3 = T + 3 L L 5 H k " D

# I . 6 E ' ,

#P .R#!; " ,c; T $ "E('O1A 'E,D" ( 8 _=T # !; % TE7+;d3425≤150*` λ = 0 # ( "5 i^,-':* λ > 0 % E7;G42515*= ( " :1 V Y T -C

%D

 ˙ xˆ(t) = pˆ(t),    ˙ pˆ(t) = 0;  pˆ(0) = x ˆ(0),   2 ˆ ˆ pˆ (T ) = 2λ;

D

ˆ pˆ(Tˆ) = −λ; ˆ ˆ xˆ(T ) + T + 3 = 0;

Tˆ > 0,

ˆ > 0. λ

P,D%!b* (] 4 " JO% J842515* O(#5kH "D O E,D 4 'E& 2(9,c;G1;a P _= # !;a%!A , ,D *2,D%: J\ .R* ,- "D λ# Ja,c6E2( Ja(D& "21Z,D"D [ .P!i %!5*2,D" o %" ? ( " Z ,D"D% \ .I& [ 2()1A L D * #. α = (α1 , α2) ≡ (p(0), T ) j51 " 4 #7& * # !;O%!% (ME [ \ [ 4 E( O _= L42515*'+$ _= ` " t = 0 1A ':* (m,D " " ,D" ':bh= o (mN$':#%e t = T ≡ α2 x(·)[α1 , α2 ],

GD

p(·)[α1 , α2 ].

V Y#

s _= # /42515* $ _= =1;H#5*7 #.I6='E,- J ( " :1C x(t)[α] = α1 (t + 1),

x(0) = p(0) = α1

p(t)[α] = α1 .

ij %" :M!N$':#%e !;G# ;4 %L(5kZ "Z ( "H :1C

X(α) ≡ (x(T )[α] + T [α] + 3, p2 (T ) + 2p(T )).

D

s _= # a% Jd42515* ( "21AE(q,D"D [ .B,\"7%: Jm .R* ,- !& "D # J\,c6E2( J % E7 #"#? _= # b),D ,D"2(.^': # # J1; %" :M!N$':#%e \# ;4 %C X1 (α) = α1 (α2 + 1) + α2 + 3 = 0, X2 (α) = α1 (α1 + 2) = 0.

$ = _= # ?':* ## JZ,D ,D"2(.n': # # J=( "21AE( b" # " e ##.RJ+ e2, ,=(5kZ "K,D J" ,DG%+21A#E('\ 4P1A 'E6L ( b/& h= !6:,c;`%: # JC α = 0 α = T = −3 #-1A 'E,D" (.RJO%: #\ ,D !'O#':_= # !;n1 *2# a'E,- !; T > 0 P ! α = −2 1 # $ _= # P42515* - α1 = T = 1 ( " (Ia*"l1A': r ,D%!b* # 4r42515* (#5kH "D; % E7 #"# 042 ( #0 " JW%:E( # #". %" :M λ > 0 N$':#%e L# ;4 %G#

D

p(T ) + 2 = 0,

4 #5* "D #o'2':*_Z "$,c6E21A (E,D"$( "21 b" # Z 4 ; " 1;O *" 8E,DD6`#5*7 #.I6a [ !kZ # J α0 6= −1 D& 1AD "L-1A #,D" ## P1A 'E,D" ( Z _= # Z42515* `1 #Z[ =*2( 42?1A $ " e #

$*,/C+:K%)- L)-N)M,.M\OP,.$*#7,`J.K(-PJ!,!J.J.P*+,!(QJ.K%,.&'7#+O )-"#+Y$L$)$!&'#"I%$#'J.#JJ #(-,%.

ˆ0 = 0 λ

I(x(·), u(·)) ≡

Z

1 0

(u(t) + x2 (t)) dt → inf;

x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1];

V Y

Z

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 1]; 1

0

(u2 (t) − 2u(t)) dt + 1 = 0,

x(0) − 1 = 0.

$ E7;G42515*= #e K(9%!,D ('!(9 V Y#&-,$':* "E(d42 (D& #.] #" 7A?1A NPN$ #e 7 # J\,D;4 b=! ( " :1C                     

x ˆ˙ (t) = u ˆ(t), yˆ˙ (t) = u ˆ2 (t) − 2ˆ u(t), ˙pˆ(t) = 2λ ˆ0x ˆ, ˆ

ˆ λ0 u ˆ(t) = 1 + p(t)− ; ˆ 2λ x ˆ(0) = 1, yˆ(0) = 0, ˆ > 0; λ ˆ0 ≥ 0; λ

pˆ(1) = 0,

yˆ(1) + 1 = 0;

E' ,- u s !'E,- $# ( % . 0 7 Z "E(' %!5*2,D" $'E,- !;G# ( % \ .R[ " ,c;G'E,- λˆ = 1
7D

x(t)[α] = (α1 /2 + 1)t + 1,

C

α2 > 0 x(t)[α] p(t)[α] y(t)[α]

α2 < 0

y(t)[α] = (α21 /4 − 1)t;

√ √ = C sh( α2 t) + ch( α2 t), √ √ √ √ = 2C α2 ch( α2 t) + 2 α2 sh( α2 t) + α2 − 2, √ √ √ √ α2 α2 2 (C + 1) sh(2 C ch(2 α2 t)+ = α t) + 2 4 2 √ √ α2 2 + 2 (C√ − 1)t − 2C sh( α2 t) − 2 ch( α2 t)+ α + 2 − 2 2 C;

√ √ C sin( 2 t), √ −α2 t)√+ cos( −α√ √ 2C −α2 cos( −α2 t) − 2 −α2 sin( −α2 t)+ α − 2, √2 √ −α2 y(t)[α] (C 2 − 1) sin(2 −α2 t)+ √4 √ −α2 cos(2 −α2 t) − α22 (C 2 + 1)t− √ 2 C√ √ 2 2C sin( −α2 t) − 2 cos( −α2 t) + 2 − −α 2 C, p C = (2 + α1 − α2 )/(2 |α2 |)

x(t)[α] p(t)[α]

c1A

C

p(t)[α] = α1 ,

= = + = + −

V Y U

i %" :M!N$':#%e b3# ;4 %G %!b* (n1A P,DE,D"7;bh= C !

X(α) ≡ (y(1)[α] + 1; p(1)[α]) = (0; 0),

 2 α1 /4,     √  √ α2  2   4√ (C + 1) sh(2 α2 )+  √  α   + 2 2 C ch(2 α2 ) + α22 (C 2 − 1)−  √ √    sh( α2 ) − 2 ch( α2 ) + 3−  −2C √ α − 2 2 C, X1 (α) =   √ √  −α2  2  −α2 )+  4 (C − 1) sin(2 √  √  −α2 α2 2   + 2 C√ cos(2 −α2 ) − √  2 (C + 1)−    −2C sin( −α2 ) − 2 cos( −α2 ) + 3−    √−α2 − 2 C,  α1 ,     √ √ √ √    2C α2 ch( α2 ) + 2 α2 sh( α2 )+ +α2 − 2, X2 (α) =   √ √   2C √−α2 cos(√ −α2 )−    −2 −α2 sin( −α2 ) + α2 − 2,

α2 = 0,

α2 > 0,

α < 0, 2 α = 0,

2

α2 > 0,

α2 < 0. $ 0 _= # r':* ## Jq,D ,D"2(. #D # J#.I6q7 [ !& *2,D% !6d': # # Jm(21A N$ e E##./(0( "21AE( b" #` "D& e ##.RJa e2, ,?,c6E21A " ,c;L%a-1A #,D" ##E('L,D':h=2,D" ':bh=2(' %: #b -1AD;bh=2('` _= # ,c6E21A# J`427& 15* α1 = 0 α2 = 0 #

$J. / ,+ \ K%L)-N $#M + J!*,.NG7K-J!J.*!#",!J.%K%$&'N7`+ )*)-#,.K%YL)-$$G&'(%I)*#J.+!-,#.J $-,#., I(x(·)) ≡

Z

1 0

(x(t) − 1)2 dt → inf;

x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1];

V YEX

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 1];

x(0) − 1 = 0,

x(1) − 1 = 0.

$ E7;K42515*= #e H(9%!,D ('!(9 V # T ( "H :1C   x ˆ˙ (t) = u ˆ(t),

u ˆ(t) = arg abs max (ˆ p(t)u), u∈(−∞,+∞)

D

 pˆ˙ (t) = x ˆ(t) − 1;

x ˆ(0) = 1,

x ˆ(1) = 1.

$ #7 * # %!,D"2(97 ':* ,D"%: pE,D [ ':D& # !;O ,D 4 E# Z%!A , ,D *2,D%: J8 .R* ,- "D # JO,c6E2(. ( "& 1G,D"D [ .) 2,D" `%`'E,DD6!'\#(5kZ " ,c6E2(9K #e 7& #?# ( # (9-:im , , (9" E2(E(8,-':*/ .R[ 4D *#.I6 #5*7 #.I6]'E,- J3[ 4':* "7 #76E5kZ1A # !;3# ':* ,D"%:nE,D [ &

`': # !; 21A "+%n# 42(5kH#E,D" 81;n *" dE,DD6O#7& *7 #.I6^'E,- J _= " 42515*'^$ _= 1;),D ,D"2(. 1A NPN$D& #e 7 #.I6+': # # J`% J+42515* "7%8%!%` p(t) 6= 0 -:$ .R"%!P':*2,D"?'E,- !;#76E5kZ1A # !;H#P':* ,D"%:/E,D& u = ±∞ [ +': # !;n 21A "`%n _= # bp42515* n[ 4G ,D 4 E7& # !; ( "21O,D"D [ . E6!#E,D"`E,D [ +': # !;m,DE,D"& " 4$-1A #,D" ## J\ 42(5kH# J %!,D"2(97 -*" \21ADA#Z V ! V v$ ,- ## qE,D" # %!,D"2(97 J ,q':* ,D"%! ( E,D [ ': # !;P / E;:1A%!R 42(5kH#/, ,D 4 E# 2(\,D e !& 7 #=21A [ ##.I6H .R* ,- "D #.I6\,c6E2( # %!A , ,D *2,D% !6K .I& * ,- "D #.I6H,c6E2(`( "21P,D"D [ . -E$ ( .m"7% !6Z _= # J 21;" ,c;!# ( XV2X #

D

D

= ? $, / # $#*, )* +9;,.$#J.MJ.`P)- + * 7,.N#;KJ.-PKQ+J!J.J./G)-*J,!+J.K%$ $&'#9C7, :)-+ )-$##,.`*Y)-!#"*%$$,!$J.9CK%&'I #(QJ.K%J ##$#J +-,,.. $9C $#*,.$/: 2 +N7#, )M, ,.$$N-\

T (x(·), T ) ≡ T → inf, x(0) = x(T ) = 0;

Z

T

T > 0;

1 (x˙ 2 (t) − 4x(t)) dt ≤ − . 3

i ,D " " ,D" ,a7 " 0(E(q _= # !; 42515* m# [56E21A ( , , ( " "H1AEK,-':*7;C #7 " *2,D%: = , ,-D& 1A E# o %!4 .RE "5:*"Hλˆ= λˆ0 = 0λˆ > _=0 # !;L42515* G# "5 V Y

$ E7;P42515*R #e o(9%!,D ('!(97, ':* "E(a'E,- !;?# !& ( % λˆ = 1/2 V # U -! ( "H :1C   x ˆ˙ (t) = pˆ(t),     yˆ˙ (t) = pˆ2 (t) − 4ˆ x(t),    ˙ pˆ(t) = −2,  x ˆ(0) = 0, x ˆ(Tˆ) = 0,     yˆ(0) = 0, yˆ(Tˆ) + 1/3 = 0;    ˆ T > 0.

Q -, Tˆ > 0 % JG42515*;; " ,c;L *#./(I
D

x(·)[α],

p(·)[α],

y(·)[α].

s _= # q42515* W$ _= p1; #5*7 #.I6p'E,- J ( "H :1C y(0) = 0 p(0) = α

x(0) = 0

1

x(t)[α] = −t2 + α1 t, p(t)[α] = −2t + α1 , 8 y(t)[α] = t3 − 4α1 t2 + α21 t. 3

ij %" :M!N$':#%e !;G# ;4 %\ ( " :1C

X(α) ≡ (x(T )[α], y(T )[α] + 1/3) = (0; 0).

D

s _= # a% Jd42515* ( "21AE(q,D"D [ .B,\"7%: Jm .R* ,- !& "D # J\,c6E2( J % E7 #"#? _= # b),D ,D"2(.^': # # J1; %" :M!N$':#%e \# ;4 %C X1 (α) = −α22 + α1 α2 , 8 X2 (α) = α32 − 4α1 α22 + α21 α2 + 1/3. 3

$ a _= # +':* ## J`,D ,D"2(.)': # # J`(21A N$ e & E##./( ( "21AE( b" #m " e ##.RJ] e2, ,O,c6E21A " ,c; % -1A #,D" ##E('G%: #b α1 = 1 α2 = T = 1 V#T

ED

[ " (+ # (9# 7*"P42 ( ## ;4 % = x(T )[α] = 0 # % E7 #"#':bl Jn`,D !'8'E,- !; T > X0 1(α) x(T )[α]/T [α] = 0 4 #5* "D #l'2':*_Z " ,c6E21A (E,D"l( "21 b" # ': # # P,D"7# " ,c;K # J#./( -

> > > TDU.BUAEoUF*CAUY?U # M *B[&H)*$ J +&'L)*/ KQ];KQJ.)-+$$9C:

$#*>!#", )M%,.$$9CY #!S#"K%%$$#NP,.J!$+!9C/* " $##*, )-,. $/:I+ )M, ,.$ 2 +N7#, )M, ,.$$N-\ I(x(·), u(·)) ≡

Z

1 0

(u2 (t) − 2x(t)) dt → inf,

x(t) ˙ + x(t) = u(t),

u(t) ∈ R1

∀t ∈ [0, 1],

x(0) = 1, x(1) + 1/e = 1 + e.

$ E7;K42515*= #e H(9%!,D ('!(9Z ( "H :1 V S -C  ˆ˙ (t) = −ˆ x(t) + pˆ(t),  x pˆ˙ (t) = pˆ(t) − 1;  x ˆ(0) = 1, xˆ(1) = 1 + e − 1/e.

.R* ,- "D # J?,c6E2( J?( "21/,D"D [ .P5i`%!5*2,D" ( " ,D"D% ^ .R[ 2(r D * #' R51 "n4 #5* # %!% (ME [ G [ 4 E( ` _= \425α15=*'ap(0) $ _= ` " t = 0 1A t = 1 ':* (m,D " " ,D" ':bh= PN$':#%e C x(·)[α],

D

p(·)[α].

s _= # q42515* W$ _= p1; #5*7 #.I6p'E,- J ( " :1C p(0) = α

x(0) = 0

α − 1 t α − 1 −t e − e , 2 2 p(t)[α] = 1 + (α − 1)et .

x(t)[α] = 1 +

?':#%e !;L# ;4 %G ( "H :1C

X(α) ≡ x(1)[α] − (1 + e − 1/e) = 0.

V #!V

D

s _= # a% Jd42515* ( "21AE(q,D"D [ .B,\"7%: Jm .R* ,- !& "D # J,c6E2( J % E7 #"#o _= # b ': # # !;P1;HN$':#%e # ;4 % C

TD D

X1 (α) =

α−1 α − 1 −1 1+ e− e 2 2

8D

− (1 + e − 1/e) = 0.

t% %!% "': # # \ # J# j( "21 b" # % E7& #"#.RJ\ "E(d,-':*o6E21;h=2('Z# ,DE,D"7 # JH* ,D" b)(D& "21A'L _= # !;a # J#.I6a': # # J I -1AD; "K _= # =%D& J\42515* G42Z21A#'K " e b=C α x(t)[α] p(t)[α]

= 3, = 1 + et − e−t , = 1 + 2et .

+ $#L *)-,. $&' /: LE)*/ + )MKQ# , M,.$ ;E*KQPJ.2[)- &H+)*+$$ N7$ 9C#:, )MIJ , $#,.$*$>!#", N-)M%\ $,.$9CY:# !#"7%$K%NG$#,.J!+!$/*9C :"

I(x(·), u(·)) ≡

Z

1 0

(u2 (t) + 2x(t)) dt → inf,

x ¨(t) − x(t) ˙ = u(t), u(t) ∈ R ∀t ∈ [0, 1], x(0) = 0,

x(1) = 0.

$ E7;K42515*= #e H(9%!,D ('!(9Z ( "H :1 V U -C  ˙ xˆ1 (t) = xˆ2 (t),      xˆ˙ 2 (t) = xˆ2 (t) + pˆ2 (t), pˆ˙1 (t) = 1,   ˙  p1 (t) − pˆ2 (t);   pˆ2 (t) = −ˆ xˆ1 (0) = 0, x ˆ1 (1) = 0; pˆ2 (0) = 0, pˆ2 (1) = 0.

7D

V#S

[ 4 E(q _= n42515*'m$ _= " ,D " " ,D" ':bh= $ (mN$':#%e C x1 (·)[α],

x2 (·)[α],

t = 0

p1 (·)[α],

1A

t = 1

p2 (·)[α].

s _= # ]42515* l$ _= 1; #5*7 #.I6 'E,- J ( " :1C x (0) = α p (0) = α p (0) = 0 2

1

1

2

':* (

x1 (0) = 0

2

α2 + 1 t α2 − 1 −t t2 e + e + + α2 t − α1 + 1, x1 (t)[α] = α1 − 2 2 2 α2 + 1 t α2 − 1 −t x2 (t)[α] = α1 − e − e + t + α2 , 2 2 p1 (t)[α] = t + α2 , p2 (t)[α] = (α2 − 1)e−t − t + 1 − α2 .

ij %" :M!N$':#%e !;G# ;4 %\ ( " :1C

X(α) ≡ (x1 (1)[α], p2 (1)[α]) = (0, 0).

D

s _= # a% Jd42515* ( "21AE(q,D"D [ .B,\"7%: Jm .R* ,- !& "D # J\,c6E2( J % E7 #"#? _= # b),D ,D"2(.^': # # J1; %:E( # #" %" :M!N$':#%e G# ;4 %C α2 + 1 α2 − 1 −1 1 e+ e + + α2 − α1 + 1 = 0, 2 2 2 X2 (α) = (α2 − 1)e−1 − α2 = 0. X1 (α) =

α1 −

:D

8D

t%G%!%G,D ,D"2(9Z': # # JH # J#:"( "21 b" # %!& E7 #"#.RJ "E(O,-':*I6E21;h=2('?P# ?,DE,D"7 # JZ* ,D" b ( "21A'Z _= # !;H,D ,D"2(. # J#.I6H7 [ *2,D% !6=': # # J -1AD; " _= # $% J\42515* G42Z21A#'K " e b=C α1

=

α2

=

x ˆ1 (t) = x ˆ2 (t) = pˆ1 (t) = pˆ2 (t) =

e2 − 5e + 6 , 2(e − 1)2 −1/(e − 1), α2 + 1 t α2 − 1 −t t2 α1 − e + e + + α2 t − α1 + 1, 2 2 2 α2 + 1 t α2 − 1 −t α1 − e − e + t + α2 , 2 2 t + α2 , (α2 − 1)e−t − t + 1 − α2 .

V #!

;IK@KQ)* J.,!)-+9C+ $(Q$J 9C.##:+7#,M$# *!*#" %#$P[9C$#, &H)*)*G+$ K%,. $#$#J.,.P+$J %9C\; (%$ $#K%Y*,.$$#!/#:' I+ B)M', #"%,.Y#$

R2 B0 (x(·), u(·)) ≡ 1 u2 (t) dt + x2 (1) → inf, x(t) ˙ = 1t x(t) + u(t), u(t) ∈ R1 ≡ (−∞, +∞) ∀t ∈ [1, 2], x(2) ≤ 3.

$ E7;K42515*= #e H(9%!,D ('!(9Z ( "H :1 V U !-C  ˆ˙ (t) = 1t xˆ(t) + pˆ(t), u ˆ(t) = pˆ(t),  x pˆ˙ (t) = − 1t pˆ(t);  ˆ λ ˆ · (ˆ pˆ(1) = x ˆ(1), pˆ(2) = −λ; x(2) − 3) = 0.

$ Z _= # Z% J=42515* ,--1A';$ !'P _= # !;Z%7& .I6?42515*Z,j'E,- 2(G1A #!;bh= JZ#DkZ2,D"%:E,D" , (I7!$$7& ! -# [56E21A (= , , ( " "=1AE,-':*7;C λˆ = 0 λˆ > 0 $ λˆ = 0 % E7;G42515*= # (9 "H :1C  x ˆ˙ (t) = 1t x ˆ(t) + pˆ(t),    ˙ pˆ(t) = − 1t pˆ(t);  ˆ(1), pˆ(2) = 0;  pˆ(1) = x  x ˆ(2) − 3 ≤ 0.

S V U

Q -, xˆ(2) − 3 ≤ 0 +% J842515* S V U /;; " ,c; D& *#./(I $ λˆ > 0 % E7;G42515*= # (9 "H :1C  ˆ˙ (t) = 1t x ˆ(t) + pˆ(t),  x pˆ˙ (t) = − 1t pˆ(t);  pˆ(1) = xˆ(1), x ˆ(2) − 3 = 0;

S V2X ˆ ≤ 0. pˆ(2) = −λ

`% Jn42515* S V2XP;; " ,c;n *!& Q ,- #./(I pˆ(2) ≤ 0
ID

V #7Y

x(·)[α],

p(·)[α].

s _= # I42515* =$ _= P1;Z"7% !6?#5*7 #.I6P'E,- J? ( "$ :1C

?':#%e L# ;4 %G

x(t)[α] = α(2t − 1), p(t)[α] = α/t.

D " !6K1A 'E6G,-':*7;!6G ( b"H :1C

X(α) ≡ p(2)[α] = 0, X(α) ≡ x(2)[α] − 3 = 0.

S V

TD

s _= # /% .I6Z42515*H( "21AE(,D"D [ . % E7 #"#PD& _= # b)': # # !;C X(α) = α/2 = 0

1;L% JG42515* S V U -

D 1

X(α) = α(2 · 2 − 1) − 3 = 0

8D

1;L% JG42515* S V2X- " ': # # !;Z # J#.m K( "21 b" # % E7 #"#.RJ "E(^,-':*=6E21;h=2('\G# \,DE,D"7 # Ja* ,D" br( "21A'a _=D& # !;K # J#.I6G7 [ *2,D% !6': # # J -1AD; "? !6K _=D& # !;L42Z21A#'K " e b=C α = 0, x(t) = 0, p(t) = 0

α x(t) p(t)

D

= 1, = 2t − 1, = 1/t.

$ %!G'E,- J #L':* ##.I68 _= # !& ;!6?E,D"7; "/" %:ox(2) 21A#' ≤ %!3,D"p(2) 2(97≤50 * :1A# 1AE,D"7;bh=':b [E,Db"#.RJL( # ('!( N$':#%e #7'H42515* [ " (0 # (9# 9*"81;d1## O ( 8 42(5kH#8 # E,D-1,D" ## _= # % JP42515* V U ![ 4 4-1AD # !; #,-':* λ = 0 λ > 0 $ L .R[ $N$':#%e \# ;4 % X(α) ≡ p(2)[α](x(2)[α] − 3) = 0

V #&#

,`':* "E( _= # !; 42515* $ _= S VH ,D 4 E# `( "21 ,D"D [ . 21A "%\%E51A"#E('K': # # b=C

D

c (3c − 3) = 0. 2

$ n _= # " +': # # !;n( "21AE( b" #+42 ,D (& ,D" + "H .R[ Z#5*7 # [ !kZ # !; -1AD; " ,c;a21A #L 4 1A 'E6G 42(5kH#.I6H%: # JC α = 0 α0 < 1/2 ! α = 1 -$ 0 4 21A#7;N$':#%e # ;4 % & α0 > 1/2 #G#'2b 8( "21 α =b"1/2 #K# ( # (I #7 *# %! 'E,- J x(2) ≤ 3 p(2) ≤ 0 #Z':* ##.I6G _= # !;!6GE,D"7; " " %:?21A#' %!,D"2(97 * :1A#51AE,D"7;bh=':b [E,Db"#.RJ ( # ('!( N$':#%e #7'H42515* #

D J$B9C #$#,)-*!#"%,.$`9C)- * M,!J.$#K% , ;Z*GKQ(QJ.[J #)-&H+)*+#, $ $ $9C #K%; $#)* ,. +$,.*$#9C J.ZP 9;+$#J.`*;)-,.$KQ J.,./N#:)-J.P ++2$/ +N7#, )M, ,.$$N-\ T (x(·), u(·), T ) ≡ T → inf,

T > 0;

x(t) ˙ − x(t) + tu(t) = 0 ∀t ∈ ∆0 ≡ ∆ ≡ [0, T ]; u(t) ∈ U ≡ {u(t) | 0 < u(t) < 2} ∀t ∈ [0, T ]; Z T sin2 u(t) dt − 1 = 0, x(0) − 1 = 0.

$ -1A2(80#7 4R'E,- !; " (97 #E,D" V U X- $ K#7 !& 4 ?42515* LHV ! S " ( *7E,D*" λˆ 6= 0 λˆ > 0 A$4?'E,- J ,D"7e ##E,D" −λˆ sin2 u(T ) = λˆ ,--1A' "5 *"0 −λˆ > 0 Eij.R[ %!5*2,D" $'E,- !;G# ( % −0 λˆ = 1 ':* (IC ∂L ∂H = 0 (⇔ = 0) ⇒ sin 2ˆ u(t) = p(t)t, ∂u ∂u

#?,D':h=2,D" ' "5! !

c1A @ ( # (97 #.RJL5kH "D #.RJ\%: #':& 1, 16556 # # !β; ≈tg(β/2) =β

u ˆ(t) =

V #

(π − arcsin(ˆ p(t)t))/2, sin(π − 4) < p(t)t ≤ sin β, sin β < pˆ(t)t pˆ(t)t ≤ sin(π − 4),

[ h= J :1m42 ,D (E,D" nN$':#%e d$ #"E; #+ "+':D& # !;H H4D *#.I6Z4 #5* # !;!6 p(t)t -1,D"7 ##P , S V ,D':#%:jkH #./(` .j1AD #.n42 ,D (E,D" ,D " " ,D" ':bh= j 7& # *#./(r,-':*7;(IC p(t)t = sin β # !kH#!;!; = p(t)t = sin(π − 4) E6!#!;!; -$_=" !6EM:':#%" E(B@ 42 ,D (E,D" ':#%" E(n@l42 ,D (E,D"Z p(t)t = 1 > sin β p(t)t = 0

s , S VZ ,D (E,D"GN$':#%e +$ #"E; #H "K': # !;a 4D *#.I6G4 #5* # !;!6 p(t)t ,D# # J8#-1AE,D"7" %"7%: a'E,- !;O * ,- ##E(3 _=D& # `42515* `,DE,D" "G\ G# # J+ -1AD ##E,D" <= -1AD& (d " (97 # $': # $H,--1A':bh=2('G42%: #'C  p(t)t > sin β,  0, sin(π − 4) < p(t)t ≤ sin β, u ˆ(t) = (π − arcsin(ˆ p(t)t))/2, p(t)t ≤ sin(π − 4).  2, S ST t%: 31A -1AD # % E7 #"# 42 ( #) ,c6E21A# JB "%.I& " Jq [A ,D" 31A 'E,D" ( ': # !;0# 42 (%#':"':b [A ,D" U (t) ≡ [0; 2] $ E7;G42515* #e K(9%!,D ('!(9$ V U -,P':* "E(m'E,-& V#U

D

!;G# ( % K1A -1AD # !;G " (97 # =':& # !; S ST -! ( λ ˆ"= :−1 1C x ˆ˙ (t) = xˆ(t) − tˆ u(t), yˆ˙ (t) = sin2 u ˆ(t), pˆ˙ (t) = −ˆ p(t);   0, (π − arcsin(ˆ p(t)t))/2, u ˆ(t) =  2, x ˆ(0) = 1, yˆ(0) = 0, yˆ(Tˆ) = 1,

p(t)t > sin β, sin(π − 4) < p(t)t ≤ sin β, p(t)t ≤ sin(π − 4); pˆ(Tˆ) = 0.

Q ,- !; ;;b" ,c;a *!& #./( T > 0 sin(π − 4) < p(t)t ≤ sin β
51O4 #5* # !; ( " n ,D"D% ^%!% (ME [ [ 7& 4 E(8 K _= ?42515*'Z$ _= H " t = 0 1A t = α2 E':* (O,D " "7& ,D" ':bh= PN$':#%e C x(·)[α],

y(·)[α],

p(·)[α],

u(·)[α].

s _= # o42515* G$ _= 1;LN$':#%e #$42 ,D "? "?1A'E&

!6dN$':#%e Jd d ( "8 :1 p(t)[α] = α ep(t) ,D (E,D" −t −t # J#+,D;42##7;O,HN$':#%e JO %!b1* # !; p(t)t -1,Dte"7& #K#G , S S " ( " ( ,--1A':bh= ,D J,D"EKN$':#%e ` D& %!b* # !; p(t)t C V o #\ #\#'2b # " e"D #\ α1 > 0 # 5kH "D #= α1 α<1 0= 0 S = α1 6= 0 N$':#%e !;m %!b* # !;d1AE,D" D " %!,D"2('!(9 α1 e−1 ! t ∈ [0; 1] t =t1∈ [1; ∞] @B':* ,D"% 8( # " ##E,D" 8N$':#%e 8D& %!b* # !; t% ( [ 4 E(I9# %!,D"2(97 ^(5kZ "[ .R"n#a[ \1A 'E6 (E( #" a %!b* # !; τ1 τ2 ;;bh= !6:,c;8%: #!;( ': #D& # !; p(t)t = sin β α1 > 0 p(t)t = sin(π − 4) α1 < 0 V #X

7D

D

$ -1A ##.RJd#7 4G'E,- !;n " (97 #E,D" S ST P 4 ; " -1AD "C y(t)[α] =

c1A

Z

∆ = [0; T ] \ [τ ; τ ] 1

1+

√

∆

1 − α1 s2 e−2s ds + A, 2

2

A = − min(0, sign α1 ) · (max(τ1 , min(τ2 , T )) − τ1 ) · sin2 2.

s _= # P,D ,D"2(.]#D # J#.I6\': # # JL( "21,D"D [ . X1 (α) ≡

R

1+

√

∆

1−α1 s2 e−2s ds 2

+ A − 1 = 0,

X2 (α) ≡ α1 exp(−α2 ) = 0

,Z ,D 4 E# 2(^(21A N$ e E## G( "21 b" #G 4 & ; "\ -1AD "L-1A #,D" ##.RJO,D':h=2,D" ':bh= J`%: # α1 = 0 -1AD;bh= J\ _= # P ,c6E21A# JG42515* α2 = 1

s , S S -1AD # P%: # J\': # # !;

J.P+ (Q,. #O" # + M$L)*!#"%[$'&H )*J $#( %*, + %K )- L)-N M) ,.OP\ ,.$#,EJ.( P,. p(t)t = 1

α1 > 0

B0 (x(·), u(·)) ≡

Z

1 0

ˆ0 = 0 λ

(u2 (t) + u(t)) dt → inf;

V #

x(t) ˙ − x(t) − u(t) = 0 ∀t ∈ ∆0 = ∆ ≡ [0, 1]; Z

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 1];

1

(x2 (t) + 2tx(t)) dt +

1 = 0, 3

x(1) + 1 = 0.

$ E7;K42515*= #e H(9%!,D ('!(9 V XY ( "H :1C 0

         

xˆ˙ (t) = x ˆ(t) + u ˆ(t), yˆ˙ (t) = xˆ(t)2 + 2tˆ x(t), ˆ x(t) + t), pˆ˙(t) = −ˆ p(t) + 2λ(ˆ ˆ 0 (u2 + u)); u ˆ(t) = arg abs max (ˆ p(t)u − λ    u∈U    yˆ(0) = 0, pˆ(0) = 0, xˆ(1) = −1, yˆ(1) = −1/3;   

ˆ 0 ≥ 0; λ 'E,- u s !'E,- o# ( % %!,D"2(97+ ':* " ,c;` 4 '27"7"\#7 " !& *2,D%: 8 _= # !; 4251λ50* =u 0,- λ0 = 0 p(t) 6≡ 0 "2,D" E,--1A E"D # 0 , τ1) ⊂ [0, 1] p(t) 6= 0 - ∃t ∈ [0, 1] p(t) 6= 0 ! \#= Z* ,D" " " (97 # ': # o#= " 4 %:∃(τ #H,D':h=2,D" ' "5 8 "E('a#H,D':h=2,D" ' "G[0, 1]_= # Z% J`42517& * u ,- akZ p(t) ≡ 0 A" p(t) t c12,- λ = 0 "`E,DG˙ (#5=kH0 "+D 2λ(x(t) nf? +#!t)k=L0 ∀t#. ∈ #[0,'2b1]= *"G# 42(5kH#H2,- " aE,D 'E,- !;G% J\42515* G λ.R6=0# #.Px(t) + t ≡ 0 ∀t ∈ [0, 1] $ λ 6= 0 4P'E,- !;G " (97 #E,D" L':* " ,c;C 0

1

u ˆ(t) =

pˆ(t) 1 − . ˆ0 2 2λ

ij.R[ ZH%!5*2,D" $'E,- !;\# ( % % ':b342515*'H%L :1A'C

ˆ0 = 1/2 λ

 xˆ˙ (t) = x ˆ(t) + pˆ(t) − 1/2,    ˙ yˆ(t) = xˆ(t)2 + 2tˆ x(t), ˙ ˆ  p ˆ (t) = −ˆ p (t) + 2 λ(ˆ x(t) + t),   yˆ(0) = 0, pˆ(0) = 0, x ˆ(1) = −1,

! [ 4 ' 2(

yˆ(1) = −1/3.

?< ;p _= # !;B':* ## Jp% J 42515* p E,D 4 ' 2(9,c; %!A , ,D *2,D%: Jd .R* ,- "D # Jm,c6E2( Jd( "21O,D"D [ .P9i %!7& *2,D" B %" ( " ,D"D% .R[ 2( D * #. V T

D

R51 " 4 #5* # !; %!% (ME [ [& 4 E( ` _= G42515*'L$ _= + " t = 0 1A t = 1 ':* ( ,D "7& " ,D" ':bh= PN$':#%e C

α ≡ (α1 , α2 ) = (x(0), λ)

x(·)[α],

y(·)[α],

p(·)[α].

s _= # ^42515* l$ _= q1;l#5*7 #.I6r'E,- J x(0) ( "H,--1A':bh= JL :1 y(0) = 0 p(0) = 0 λ = α2 $ α2 < − 1 C

= α1

2

1 − 2α2 α1 β + sin βt+ 2β 1 1 α1 β 2 + cos βt + 2α2 t − , 2 2 1 2α2 α2 α2 α2 2α1 α2 + 2 sin βt− 2 cos βt−2 2 t+ 2 . p(t)[α] = β β β β β 1 x(t)[α] = − 2α2 + 1

$ $

C

α2 = − 21

x(t)[α] = −

t3 t2 t − − +α1 (t+1), 6 4 2

α2 > − 21

p(t)[α] =

C

t3 t2 − −α1 t. 6 4

β2 − 2 α1 β + sh βt+ 2β 1 1 α1 β 2 − ch βt − 2α2 t + , 2 2 α2 1 p(t)[α] = 2 α1 β − sh βt + ch βt + 2t − 1 . 2α2 + 1 β 1 x(t)[α] = 2α2 + 1

y(t)[α] =

Z

t

(x2 (s)[α] + 2tx(t)[α])ds.

i -1A ##.I6LN$ ('2A76 β = p|2α2 + 1| ij %" :M!N$':#%e !;G# ;4 %\ ( " :1C 0

X(α) ≡ (x(1)[α] + 1, y(1)[α] + 1/3) = (0, 0).

VV

P ?%!% !6$#5*7 #.I6o'E,- !;!6$ " 7;$%:E( # #"7/N$':#%!& e a# ;4 %\H#H#P [ hZ " ,c;t%a%!%a,D ,D"2(9': # # J _= # !;G#$ ( "5!"H( "21 b" #=#$(5kZ "Z Z -1AD " \ .R"% L _= # !;\42515* L( "21AE(d,D"D [ .^%L'E,DD6!'H#$ !& 21;"5 #

`)*,.K%L)-/GJ. , ) M *Z(%*[J.&H)*$KSJ SJ. + 9CK% (%L)-)*N +!-,.$#, ]\ !#"%$#!/ I(x(·), u(·)) ≡

Z

2 0

(tu(t) − 1)2 (2x(t) − t2 + 1)2 dt → inf;

x(t) ˙ − tu(t) = 0 ∀t ∈ ∆0 ≡ ∆ ≡ [0, 2];

u(t) ∈ U ≡ R1 ≡ (−∞, +∞) ∀t ∈ [0, 2]; x(0) +

1 = 0, 2

x(2) − 1 = 0.

$ E7;G42515* #e K(9%!,D ('!(9$ V X&-,P':* "E(m'E,-& !;\# ( % λˆ = 1/2 ! ( " :1C 0

 x ˆ˙ (t) = tˆ u(t),    ˙ pˆ(t) = 2(tˆ u − 1)2 (2ˆ x(t) − t2 + 1),  pˆ(t) − (tˆ u(t) − 1)(2ˆ x(t) − t2 + 1)2 = 0;   x ˆ(0) = −1/2, x(2) = 1.

" ( " ( E,D [ ##E,D"42515* C ': # ( 5kZ "[ .R"o .R[ #/b[ ./(I o#7 "t *=2,D%:0 _= # I4251u(0) 5* 4 #5* # ': # !;H=21A# JK" *%:# !; "5!? K* ,- ##E( _= # L': # $ .R[ " ,c;G# .R #E,D" a,DE $#.RJ0#7 4d42515* ]1AE,D"7" *#],-5kZ #ooA , ,D *2,D%!7; .R* ,- "D #7;Z,c6E2(9o( "21o,D"D [ .n ZE,D" # %!,D"D& (97 JZ, ':* ,D"%! ( =E,D [ /': # !;?%Z'E,DD6!'$ #e 7 # 2,D" #G(5kZ "5$ "E('` _= # 42515* O , , ( " ( " %:Z,-':*J\ .R[ P .R* ,- "D # J\,c6E2(. ( "21Z,D"D [ . 1;KE,D" # !; %!,D"2(97 :,D21A Ekh= JDkH (OE,D [ $':& # !; , (I V ! S -$ .R[ .R* ,- "D # J,c6E2(.r-1& A D " ,c;A*" %!,D"2(97L,DE,D" "\ 4=1A 'E6+':* ,D"%: $ .RJ ':* ,D" %L,D " " ,D" ' "ZDkH ('GE,D [ =': # !;C

D

V S

D

%D

D

x(t) ≡ (t2 − 1)/2,

p(t) ≡ 0,

u(t) ≡ 1.

aE( #" %: #*# !; ':* ,D"%!)E,D [ )': # !; ;; " ,c; # 4 2,D"#τ./(d ( "E(n ,D"D% C τ = α.

? " E(O':* ,D"%:': # / -1AD; " ,c;K 4o'E,- !;K(9%!,D !& ('!(9PN$':#%e K$ #"E; #o H _= # R42515* H$ _= Z " t = τ 1A ( "H :1C t=2 p(t) ≡ 0,

u(t) = 1/t,

x(t) ˙ = t·

1 = 1, t

%D

x(t) ≡ (t−τ )+(τ 2 −1)/2.

Q # u(t) Z" *%: τ (5kZ "Z ( "4 .R$ "E('HE(D& ,D"21A# JG42515* L$ _= G " 1A _Zb" ,c;K1A $42515* $ _= C! " t = 0 1A t = τ L t" =t 0= τ 1At =t2= 2
D

X(α) ≡ x(2)[α] − 1 = (2 − α) + (α2 − 1)/2 − 1 = (α − 1)2 /2.

[ h= # G8#8N$':#%e m# ;4 % d4251 "`%E51A"# G':& # # H,=-1A #,D" ##./(^%: #2(I " ( " (I*"\':* ## Z\D& 4 '27"7"G _= # !;d42515* n': # uˆ(t) %!4 .RE " ,c;# D& .R #./(I s _= # /': # # !;K( "21AE( b" #$ 42(5kH##="7%H%!% %: #K%"#.RJA"G,D%: E,D"G,c6E21A (E,D" a( "21 b" #H %!7& 4 .RE " ,c;G# ./,D %: J V!

7= ?

> > > TDU.BUAE\GHVAXE U"WIHYAGHF*G VACAUK~WJ.YAE # M * !#"%$&'`(%)*+!-,.$/ J; +#J / + )M, ,.$ &H)*$*,.$#, $#:#, )M+( )- + $( $#, ;KQJ.)-+$ ,-

/

$9C K%$#,.$9C $#*,.$#, 2 + N7#, )M, ,.$$ N-\ I(x(·)) ≡

Z

1 0

(x˙ 2 (t) − 4x(t)) dt → inf,

t ≤ x(t) ˙ ≤ 2t ∀t ∈ [0, 1],

x(0) = 0.

V&!

$ E7;K42515*= #e H(9%!,D ('!(9Z ( "H :1 V #&-C  x ˆ˙ (t) = u ˆ(t),     ˙ (t) = −4;  p ˆ    pˆ(t) ≥ 4t  2t,  u ˆ (t) = p ˆ (t)/2, 2t ≤ pˆ(t) ≤ 4t ;      t, p ˆ (t) ≤ 2t   x ˆ(0) = 0, pˆ(1) = 0.

.R* ,- "D # J?,c6E2( J?( "21/,D"D [ .P5i`%!5*2,D" ( " ,D"D% ^ .R[ 2(r D * #' R51 "n4 #5* # %!% (ME [ G [ 4 E( ` _= \425α15=*'ap(0) $ _= ` " t = 0 1A t = 1 ':* (m,D " " ,D" ':bh= $2('GN$':#%e C

D

ID

x(·)[α],

p(·)[α].

D

$ Z _= # Z42515* Z$ _= =# [56E21A (':*2,D"oE,D [ ##E,D" " J 24 515* C# %!,D"2(97 d( ':"+[ .R"+" *% d %!b* # !; d# , (E()1ADa2,D" , (I V ! !-9"7% %!%mE7;m* ,D"n,D ,D"2(. 1A NPN$ #e 7 #.I6)': # # J0;; " ,c;r# .R # J$#]# # .R #7M 1A NPN$ #e ' 2( JaN$':#%e J` 2( # t *% 8D& %!b* # !;? ?* ,- ##E(\ _= # ?42515* $# [56E21A (I -1AD& "L,=(9%!,D (97 #G 42(5kH# Ja" *#E,D" b 21A [ # = [ "E( , (I X - ?':#%e !;L# ;4 %G ( "H :1C

D

= ?

X(α) ≡ p(1)[α] = 0.

s _= # 842515* $ _= 1;]#5*7 #.I6 'E,- J α ∈ (0, 6) ( " :1C p(0) = α

VY

x(0) = 0

p(t)[α] = α − 4t, t ≤ α/8,  2t, u ˆ(t)[α] = pˆ(t)/2, α/8 ≤ t ≤ α/6,  t, α/6  2≤ t; t ≤ α/8,   t , Rt α α2 2 t − t − , α/8 ≤ t ≤ α/6, x(t)[α] = 0 u(s)ds = 2 32   t2 α2 α/6 ≤ t ≤ 1. 2 + 96 ,

D

s _= # % JP42515* ?( "21AE(a,D"D [ ., "7%: JP .R* ,- !& "D # J,c6E2( J % E7 #"#o _= # b ': # # !;P1;HN$':#%e # ;4 % C

TD D

8D

X1 (α) = (α − 4) = 0.

t % %!% "': # # \ # J# j( "21 b" # % E7& #"#.RJ\ "E(d,-':*o6E21;h=2('Z# ,DE,D"7 # JH* ,D" b)(D& "21A'L _= # !;a # J#.I6a': # # J I -1AD; "K _= # =%D& J\42515* G42Z21A#'K " e b=C α = 4, pˆ(t) = 4 − 4t, 0 ≤ t ≤ 1/2,  2t, 2 − 2t, 1/2 ≤ t ≤ 2/3, u ˆ(t) =  2/3 ≤ t ≤ 1;  t,2 0 ≤ t ≤ 1/2,  t , 2t − t2 − 1/2, 1/2 ≤ t ≤ 2/3, x ˆ(t) =  2 t /2 + 1/6, 2/3 ≤ t ≤ 1.

KQ J.+ )M , !#",.$%$N5&H# )*M#$ P**,.$ #, ,$# !(W+#"%&H$)*L)-&';( (%K%')* )-+!K@-,)- .+#$ + /$NZ( J4 Z H+ J.#" J /*W )-I&H)*$*,.$$NI+L)-N7)- + $N %\

OX

I(x(·)) ≡

Z

1 0

x(t) dt → sup,

−6t ≤ x ¨(t) ≤ 1 ∀t ∈ [0, 1],

x(0) = x(1) = 0.

$ E7;K42515*= #e H(9%!,D ('!(9Z ( "H :1 V V T T -C                       

x ˆ˙ 1 (t) x ˆ˙ 2 (t)

= xˆ2 (t), = u ˆ(t),  pˆ2 (t) < 0,  −6t, u ˆ(t) = 1, pˆ2 (t) > 0,  ∀u ∈ U, pˆ2 (t) = 0, pˆ˙ 1 (t) = −2, pˆ˙ 2 (t) = −ˆ p1 (t); x ˆ1 (0) = 0, x ˆ1 (1) = 0; pˆ2 (0) = 0,

pˆ2 (1) = 0.

V#

sDkH ( E,D [ ': # !;^d , , (9" E2( J^42515*+ !& " (97 # H': # !;a# 42(5kZ #i], (E(n1AD 2,- +,D':h=D& ,D" ' " [τ1, τ2 ] ⊆ [0, 1] "7%: J9*" pˆ2(t) = 0 ∀t ∈ [τ1, τ2 ] "O pˆ1 (t) = 0 ∀t ∈ [τ1 , τ2 ] ?':#%e !;8 %!b* # !*; "Hpˆ2(t) " ; ; *" ,c ;8"(':# # *! # #b E(^pˆ˙1 8(t)= "E−2(' (5kZ " [ hZ"E,c;KH#" %:H%: # *#E(n* ,-P" * %" *!& # #$[ $*2(d=1A 'E6G" *%!76G " 4 %! [0; 1]
7D

D

x1 (·)[α],

x2 (·)[α],

p1 (·)[α],

p2 (·)[α].

D

$ Z _= # Z42515* Z$ _= =# [56E21A (':*2,D"oE,D [ ##E,D" " J 42515* C!# %!,D"2(97 L( ':"[ .R"" *% L %!b* # !; #, 7& (E()1ADL !6d#a[ ' 1A "5 , (I /V ! !-"7%m%!%mE7;d* ,D",D !& ,D"2(.n1A NPN$ #e 7 #.I6=': # # JZ#I;; " ,c;Z# .R # J N$':#%e Ja 2( # At *% + %!b* # !;a +* ,- ##E(m _=D& # 42515* O# [56E21A (\ -1AD "`,K(9%!,D (97 #L 42(5kH# J " *#E,D" b 21A [ # o [ "E(d, (I X - ?':#%e !;L# ;4 %G ( "H :1C

PD

= ?

X(α) ≡ (x1 (1)[α], p2 (1)[α]).

s _= # L42515* m$ _= 1; #5*7 #.I6n'E,- J x (0) = 0 ( "H,--1A':bh= JL :11C x (0) = α p (0) = α p (0) = 0 2

$ $ V&

1

α2 ≤ 0

1

2

p1 (t)[α] = α2 − 2t, p2 (t)[α] = t2 − α2 t.

C

u ˆ(t)[α] = 1, 0 < α2 < 1

2

C

x ˆ2 (t)[α] = t + α1 ,

u ˆ(t)[α] =

x ˆ1 (t)[α] = t2 /2 + α1 t.

−6t, t ∈ [0, α2 ), 1, t ∈ (α2 , 1],

xˆ2 (t)[α] = x ˆ1 (t)[α] =

$

α2 ≥ 1

C

−3t2 + α1 , t ∈ [0, α2 ), t − 3α22 − α2 + α1 , t ∈ (α2 , 1],

−t3 + α1 t, t ∈ [0, α2 ), t2 /2 + (α1 − 3α22 − α2 )t + (2α32 + α22 /2), t ∈ (α2 , 1].

u ˆ(t)[α] = −6t,

D

x ˆ2 (t)[α] = −3t2 + α1 ,

x ˆ1 (t)[α] = −t3 + α1 t.

s _= # % JP42515* ?( "21AE(a,D"D [ ., "7%: JP .R* ,- !& "D # J\,c6E2( J % E7 #"#? _= # b),D ,D"2(.^': # # J1; %" :M!N$':#%e \# ;4 %C  α2 ≤ 0,  1/2 + α1 , 0 < α2 < 1, X1 (α) = 1/2 + 2α32 − 25 α22 − α2 + α1 , α ≥ 1;  −1 + α , 1

2

X2 (α) = 1 − α2 .

-1AD # =-1A #,D" ## G,D':h=2,D" ':bh= K%: #!;`( "21AE( b" #Z" [ ' "Z#P[ /1A 'E6G " e J .#

(QJ #+#, #P$#,M)* +,.*$# J.P+%\ !#"%$&' (%)*+!-,.$/ JGK@)*,!+9C I(x(·)) ≡

Z

1 0

(x˙ 2 (t) + 4x(t)) dt → inf,

−1 ≤ x(t) ˙ ≤ 0 ∀t ∈ [0, 1],

x(0) = 0,

x(1) + 3/4 ≤ 0.

$ E7;K42515*= #e H(9%!,D ('!(9Z ( "H :1 V V T -C          

xˆ˙ = u ˆ, pˆ˙ = 4,  pˆ(t) ≤ −2,  −1, pˆ(t)/2, −2 ≤ pˆ(t) ≤ 0, u ˆ(t) =    0, pˆ(t) ≥ 0,     ˆ λ ˆ · (ˆ  x ˆ (0) = 0, p ˆ (1) = − λ; x(1) + 3/4) = 0;   ˆ λ ≥ 0.

i ,D " " ,D" ,a7 " (E(q _= # !; 42515* m# [56E21A ( , , ( " "=1AE,-':*7;C λˆ = 0 λˆ > 0 V U

.R* ,- "D # J?,c6E2( J?( "21/,D"D [ .P5i`%!5*2,D" ( " ,D"D% ^ .R[ 2(r D * #' R51 "n4 #5* # %!% (ME [ G [ 4 E( ` _= \425α15=*'ap(0) $ _= ` " t = 0 1A t = 1 ':* (m,D " " ,D" ':bh= $2('GN$':#%e C

D

x(·)[α],

p(·)[α].

$ B _= # W42515* B$ _= B# [56E21A (q':*2,D"lE,D [ ##E,D" D " J^42515* C # D %!,D"2(97 ( ':"n[ .R"n" *% %!b* # !; , (IKaV ! !-H"7%p%!%pE7; * ,D" ,D ,D"2(. 1A NPN$ #e !&

7 #.I6n': # # Jd;; " ,c;m# .R # J9#O#a# .R #7M 1A NPN$ #e ' 2( J0N$':#%e J0 2( # Pt *% q %!b* # !; 8* ,- ##E( _= # 842515* +# [56E21A (H -1AD "a,Z(9%!& ,D (97 # 42(5kH# JG" *#E,D" b 21A [ # $ [ "E(m, (I X - ?':#%e L# ;4 %G ( b"H :1C

D

= ?

X(α) ≡ x(1)[α] + 3/4

`% Jn42515* ,D " " ,D" ':bh= Jn'E,- !;(

λ≥0

x ˆ(1) + 3/4 = 0

X(α) ≡ p(1)[α]

`% Jn42515* ,D " " ,D" ':bh= Jn'E,- !;( xˆ(1) + 3/4 ≤ 0 λ=0 s _= # 842515* $ _= 1;]#5*7 #.I6 'E,- J x(0) = 0 ( " :1C p(0) = α $ $

C

C

α ≤ −6 u ˆ(t)[α] = −1 x ˆ(t)[α] = −t −6 < α ≤ −4 −1, 0 ≤ t ≤ −(α + 2)/4, u ˆ(t)[α] = 2t + α/2, −(α + 2)/4 ≤ t ≤ 1, xˆ(t)[α] =

V X

pˆ(t)[α] = 4t + α.

  −t,

α  t2 + t + 2

α+2 4

2

0 ≤ t ≤ −(α + 2)/4, , −(α + 2)/4 ≤ t ≤ 1.

$

C

−4 < α ≤ −2  0 ≤ t ≤ −(α + 2)/4,  −1, 2t + α/2, −(α + 2)/4 ≤ t ≤ −α/4, u ˆ(t)[α] =  0, −α/4 ≤ t ≤ 1,

 −t, 0 ≤ t ≤ −(α + 2)/4,   2  α α+2 x ˆ(t)[α] = t2 + t + , −(α + 2)/4 ≤ t ≤ −α/4,  2 4   (α + 1)/4, −α/4 ≤ t ≤ 1.

$

−2 < α ≤ 0

C

u ˆ(t)[α] = x ˆ(t)[α] =

2t + α/2, 0 ≤ t ≤ −α/4, 0, −α/4 ≤ t ≤ 1, t2 + αt/2, 0 ≤ t ≤ −α/4, −α2 /16, −α/4 ≤ t ≤ 1.

$ α > 0 C uˆ(t)[α] = 0 xˆ(t)[α] = 0 s _= # m% .I6042515*r( "21AE( ,D"D [ . ,m"7% ( r .I& * ,- "D #./( G,c6E2(9 ( % E7 #"#$ _= # b^': # # J=1; N$':#%e L# ;4 %C  α ≤ −6, −1,    2  − 6 < α ≤ −4, α α+2    1+ + , 2 4 X(α) = − 4 < α ≤ −2, S S V  (α + 1)/4,  − 2 < α ≤ 0,    −α2 /16,  0 < α  0, `% Jn42515* ,D " " ,D" ':bh= Jn'E,- !;( xˆ(1) + 3/4 = 0 λ≥0 S S S X(α) = 4 + α = 0 `% Jn42515* ,D " " ,D" ':bh= Jn'E,- !;( xˆ(1) + 3/4 ≤ 0 λ=0 t%%!%Z E7;N$':#%e !;# ;4 % S S V # .R #7M 1A NPN$D& #e ':2(9= \( # " ##E"Z(21A N$ e E##.RJG( "21 b"& #P -1AD; "P-1A #,D" ##.RJK,D':h=2,D" ':bh= JH%: # " P':& # # !; −6 < α0 < 0 t%%!%Z " 7;ZN$':#%e !;Z# ;4 % S S S V&

PD

:D

# J#!"Z%: #Z " =': # # !; α = −4 -1AD; " ,c;G42 21A#'K " e b0( "21 b" # $E,- -1AD # !; ,D%:E(.I68%: # JO+ [ !6`% .I6842517& *76 E;b" ,c;H" [ ' 2(.R/# #,D"E im,-':*/ !6H .R#D& # !;#J:1A ## I _= # I[ ' 1A "$ _= # 2(` ,c6E21A# J?42515* 7 #5* #J:1A ## $ _= # ?E,DE( D"D # JK% J\42515* \ _= # 2( ,c6E21A# JG42515* G#P[ ' 1A "5 #

;J.P9 J.+`)-J ;,.N#J.KQPJ.+)-/E+J M$$ 9C*# , )-$#*!,.#"`%)-$!9C#"* %,!$J.EK%&'$# , ;(%(QJ KQ#)*J.++!)--,#., +$$/ $9C# )*K% $#+,.,.*$ $9C $#*,.$/: 2 +N7#, )M, ,.$$N-\

T (x(·), u(·), T ) ≡ T → inf,

T > 0;

x(t) ˙ + x(t) − tu(t) = 0 ∀t ∈ ∆0 ⊆ ∆ ≡ [0, T ]; u(t) ∈ U ≡ {u(t) | − 1 ≤ u(t) ≤ 2} ∀t ∈ [0, T ]; Z T cos2 u(t) dt − 1 = 0, x(0) − 1 = 0. 0

D

i^%!5*2,D" $'E,- !;\# ( % L .R[ 2(n D * #' i "E( ,-':*?% E7;a42515* #e G(9%!,D ('!(9K (λˆ ="G−1 :1 V V V S -C          

x ˆ˙ = −ˆ x(t) + tˆ u(t), pˆ˙ (t) = pˆ(t), yˆ˙ (t) = cos2 u ˆ(t), u ˆ (t) = arg abs max (cos2 u + pˆ(t)tu);   u∈[−1,2]     x ˆ(0) = 1, yˆ(0) = 0, yˆ(Tˆ) = 1, pˆ(Tˆ) = 0;    ˆ T > 0.

5D

$ 4L,--1,D" !;d'E,- !;d,D"7e ##E,D" d -1AD; " ,c;d D * # 'E,- L# " e"D #E,D" λˆ0 ≥ 0 "E( 2 λ (Tˆ) ˆ.R0 =#cos !; " uˆ,c;L "E(9" *2,D%
D

4 E(n L _= 42515*'K$ _= \ " " ,D" ':bh= $ (mN$':#%e C x(·)[α],

t=0

y(·)[α],

1A

t = α2

':* (d,D "7&

p(·)[α].

D t" *Jm$% G42 851 5 * _= %! Cb## G* O D #42%! !5,D1;G"5*2 8(9 K7$ * _=,-( 8 # #':# "`E ([5[ 6E.R21A "_= O( "#G *': K%* m4225,D1"a5 *E H,D%! #b[ #*[5#6E #E21A,D !" !; & ( -1AD "o,j(9%!,D (97 #/ 42(5kH# J$" *#E,D" b 21A [ # [P D "E(m, (I = X? - ij %" :M!N$':#%e !;G# ;4 %\ ( " :1C X(α) ≡ (X1 (α), X2 (α)) ≡ (p(T )[α], y(T )[α] − 1).

$ d _= # \% Jn42515* d( "21AE(0,D"D [ .l -1AD;!& " ,c;n _= # G42515* $ _= 81;n#5*7 #.I68'E,- J x(0) = 0 $ "E( y(0) = 0 p(0) = α1

D

pˆ(t)[α] = α1 et .

$E( # #". %" :M!N$':#%e \# ;4 %\ ( b" :1C X1 (α) X2 (α)

= α1 eα2 = 0, = y(T )[α] − 1 = 0.

#7 " *2,D%!7; 42 ,DB " J %:E( # #". %" :M!N$':#%e # ;4 %K1AE,D"7" *#Z E( 4-1A%!Z \ "E('G#$ 21A " ,c; P,D 4 E# (21A N$ e E## Z( "21 b" #= 4 & ; "\ -1AD "L-1A #,D" ##.RJO,D':h=2,D" ':bh= J`%: # α = 0 1 !;;bh= J,c;\ _= # 2(d ,c6E21A# JK42515* α2 = 1 #

)-N7)M,.OP\ ,.$#,SJ.( P,!MJ.P +(Q*,. #O" !+G#"%$$L&') !(%#"%)*$'+!-,.$J #(%/ *G,+ %K )-

ˆ0 = 0 λ

T (x(·)) ≡ x(1) → inf,

t − 1 ≤ x(t) ˙ ≤ t + 1 ∀t ∈ [0, 1], Z 1 (x2 (t) − 2tx(t)) dt = −1/3.

x(0) = 0,

0

VUV

$ E7;K42515*= #e H(9%!,D ('!(9 V V V U ( " :1C                         

xˆ˙ (t) = u ˆ(t), yˆ˙ (t) = x ˆ2 (t) − 2tˆ x(t), ˙pˆ(t) = 2λ(ˆ ˆ x(t) − t),  pˆ(t) < 0,  t − 1, u ˆ(t) = t + 1, pˆ(t) > 0,  ∀u ∈ [t − 1; t + 1], pˆ(t) = 0; ˆ 0 , yˆ(1) = −1/3; x ˆ(0) = 0, yˆ(0) = 0, pˆ(1) = −λ ˆ 0 ≥ 0; λ

E' ,- u s 'E,- o# ( % . ,D " " ,D" ' "RD& #7 " *2,D%: _= # 42515* P kH ('GE,D [ Z': # !; pˆ(t) ≡ 0 Q* λ"H0 = 420(5kH#E,D" GDkH (9 E,D [ Z': # !;A,D [E,D" ##! a 21A "H%+#7 " *2,D%:E(' _= # b)42515* $ %!7kZ2(I!*"K .R"% aE,D" # !;\1A': JL .R* ,- "D # J ,c6E2(.]%L'E,DD6!'G#? 21;"5ij7M: .I6! ,D%!b* (m 4?% J "E(^':* !& 42515* `(#5kH "D Q ,- ".RE " ,c;G%!%G λ 0*# p(1) E; " =,c;G−λ "0 ≤%:Z0#=E,D" ## J %!,D"2(97 -Ea#5kH "D λ (5kZ "$ # (9"?21A#P 4R"D64 #7& * # JC &gV V` T jij.R[ .I6d1A 'E6 4 #5* # J^,D " " ,D" ' " 4251# b)1A 'E6a4D *#.I6a 42(5kH#.I6L# ( %+(#5kH "D J f? #!kid%!5*2,D" I ( "P ,D"D% K "E(8 .R[ 7& " ,c;O D * # α = p(0) ?':#%e !;O# ;4 % O -1AD; " ,c;8'E,-& 2(IC

GD

D

D

u ,-

X(α) ≡ y(1)[α] + p(t)[α] ≤ 0 ∀t ∈ [0; 1]

!"KN$':#%e

u(t)[α] = t − 1, 2 x(t)[α] = t2 − t,

?4 #5* # #!; " ,c;L

y(t)[α] =

X(α) = 53/60 λ=0 λ=1

t5 20

−

t4 2

+ t3

1

# 42 ,c;"R "/ D * #. α " .R& ! L λ = −1 α ≤ −5/3 C α<0

p(t)[α] = λ

V US

1 = 0. 3

t3 − 2t2 3

+ α.

u ,-

p(t)[α] ≥ 0 ∀t ∈ [0; 1]

!"KN$':#%e

u(t)[α] = t + 1, 2 x(t)[α] = t2 + t, t5 20

y(t)[α] =

−

-1

t3 3

L4 #5* # X(α) = 1/20 "7%!kZ$#P42 ,c;"Z " D * #. α " .R#!; " ,c;d λ = 0 λ = 1 α > 0 ! m λ = −1 C α ≥ 1/3 p(t)[α] = λ

t3 + α. 3

':*J # 42(5kZ #KZ,D !'H'E,- !; u s ! -, ':*J λ =λ1= α0 =α0=,D0 21A " ,c;+%`21A#E('a 4?1A 'E6+-1A.j1A':h= !6 d42 ,D (E,D" "n .R[ O D * #.W': # !; d#5*7 #.RJ (E( #" 2( # $ # %!,D"2(97 ( " ,c;H21A#o" *%! %!bλ* =# !−1 ; τ 0=<√α3α
D

3

 u(t)[α] = t + 1,    2   x(t)[α] = t + t, 2

 u(t)[α] = t − 1,    2   x(t)[α] = t − t + 2τ, 2

5

3

t  y(t)[α] = 20 − t3 ,    3  p(t)[α] = − t + α, 3

5

t ∈ [0; τ ],

4

t ατ 3 2 2  y(t)[α] = 20 − t2 (1 + 2τ  3 )t − 4τ t + 4τ t − 4α − 2 ,    p(t)[α] = − t3 + 2t2 − 2τ t + α, t ∈ [τ ; 1]; 3 √ α 11 √ 53 3 3 X(α) = −4α + 4( 3α)2 − + 3α + . 2 3 60

D

$ # %!,D"2(97 8(5kZ "\ %!42"& ,c;=#2,D%:λ %:=−1 " * %=−5/3 <%!bα*< #0 !;E ,D (E,D" u(t)[α] x(t)[α] "E(^ %!4 .REb" ,c;L1AE,D"7" *#G & y(t)[α] p(t)[α] ( 4-1A% ( " ( " X(α) (m _=*"GN$':#%e !;L# ;4 %L# a a%!%:E( .R[ α H#H#? [ hZ " ,c;! a "E('G _= "K42515*'G,P ,-& 4 E# 2("7%: J\ .R* ,- "D # JL,c6E2(. #$' 1 " ,c; VU!

GD

-, .$ #, ]!\ #"% $# !/ #`)*M,.K% *LI)-/EJ. , ) !#"%$&'E(%*(%J.)*+!-,K .J$/J. +79C K%(%L)*)-+N

I(x(·), u(·)) ≡

Z

2 0

x2 (t) dt → inf;

x(t) ˙ − u(t) = 0 ∀t ∈ ∆0 ⊆ ∆ ≡ [0, 2];

$ V V S S -C

u(t) ∈ U (t) ≡ {u(t)| − 1 ≤ u(t) ≤ 1} ∀t ∈ [0, 2]; x(0) − 1 = 0, x(2) = 0.

ˆ0 = 1 λ

% E7;G42515*? #e (9%!,D ('!(9Z ( " :1

  pˆ(t) < 0  −1,     +1, pˆ(t) > 0 , ˆ˙ (t) = u ˆ(t) ≡  x  ∀u ∈ [−1, +1], pˆ(t) = 0   pˆ˙ (t) = 2ˆ x(t);    x ˆ(0) = 1, xˆ(2) = 0.

s _= # j% J=42515* =,j ,D 4 E# 2(L%!A , ,D *2,D%: J? .I& * ,- "D # J],c6E2(. ( "21d,D"D [ .PI%!% ],--1A E7n5kH !& 1"A%`'E,DD6!'L#Z 21A "5i3, (E(m1AD A .R[ 2( \%!5*2,D" ( "G ,D"D% O D * #' α = p(0) GL%!5*2,D" ZN$':#%!& e L# ;4 %\ D * #' X(α) = x(2)[α].

51 α %!% (ME [ P [ 4 E(+ H _= P42515*'=$ _= " t = 0 1A !':* (m,D " " ,D" ':bh= PN$':#%e x(·)[α] p(·)[α] t=2 $ α < −1 C $

VUY

−1 < α < 0

C

x(t)[α] = −t + 1, p(t)[α] = −(t − 1)2 + 1 + α, u(t)[α] = −1.

x(t)[α] = −t + 1, p(t)[α] = −(t − 1)2 + 1 + α, u(t)[α] = −1,

t ∈ [0; 1 −

√

1 + α],

$

√ x(t)[α] = t + 2 1 + α − 1, √ p(t)[α] = (t + 1)2 − (2 − 1 + α)2 , u(t)[α] = 1, 0≤α

C

t ∈ [1 −

√ 1 + α; 2].

x(t)[α] = t + 1, p(t)[α] = (t + 1)2 − 1 + α, u(t)[α] = 1.

$ ,D':h=2,D" ' "\#2,D%: %:\ _= # JO42515* O$ _= # \#5* α#=b−1 " ,c;L21A# (n L"2(nkZP':* ,D"%:E(IC

D

x(t)[α] = −t + 1, p(t)[α] = −(t − 1)2 , u(t)[α] = −1, t ∈ [0; 1].

"2( %!,D"2(97 G4-1AD;b" ,c;K=42 ,D (E,D" K "=,DE,D [E$ .R[ & Z': # !;\ J\ %2,D"#E,D" L" *% t = 1 $ \ .R[ $4 #5* # !; u(1+) = 1 C x(t)[α] = t − 1, p(t)[α] = (t + 1)2 − 4, u(t)[α] = 1, t ∈ [1; 2].

$ \ .R[ $4 #5* # !;

u(1+ ) = −1

x(t)[α] = −t + 1, p(t)[α] = −(t − 1)2 , u(t)[α] = −1, t ∈ [1; 2].

$ .R[ 84 #5* # !; E,D [ =': # !;C

u(1+ ) = 0

/,D " " ,D" ':bh= dDkH ('

x(t)[α] = 0, p(t)[α] = 0, u(t)[α] = 0.

Q * ,D" %HE,D [ o': # !;K(5kZ "$[ .R"?42 _= #$b[ JK(& ( #" d`42 ,D (E,D" n "` .R[ L D * #.l':& # !; τ2u(τ∈2+[1;) 2) m,c6E21AG,G':* ,D"%!`':*b" ,c;O1A \4D *#.R %!,D"2(97 P,D 4 E# I%!A , ,D *2,D%: JH .R* ,- "D # JG,c6E2(.^( "21 ,D"D [ .q"7%: J8 42(5kH#E,D" `#H':* ".RE "5 O ( ##a "E(' VU#

D

%D

D

E,D" " ,D%:E(':b %!,D"2(97,o':* ,D"%:E(nE,D [ ?': # !; #P 4 ; "5
D

:D

α = (p(0), τ1 ),

c1A τ [ 4 #5* " (E( #"l .I6E21r# E,D [ 0(# [ 4 ij %" 1:M!N$':#%e !;\# ;4 %G ( "H :1C X(α) ≡ (X1 (α), X2 (α)) ≡ (x(τ1 )[α], p(τ1 )[α]) = (0, 0).

[ h= # an# %" :M!N$':#%e # ;4 % ≤ τ ≤ 2

D#" ' "0 42(5kH#E,D"021AkZ # !;l _= # !; 0#0 "1 4 %: E,D [ ./(':* ,D"%:E( ,--1A E"D #E .R# # o%D& t ∈ [τ1 , 2] Z'E,- !; x(2) = 0 ij.R[ ?1A': JH .R* ,- "D # JH,c6E2(. ( "21?,D"D [ . ,RD& _= # 2(+42515* Z$ _= =o [ "#E(a 2( # ,D 21A " _= # I%D& J$42515* o%P21A#E('/ # J#E('o': # # b= $ "E(G/%!5*2,D" ( " ,D"D% + .R[ " ,c;L D * # α = τ1 Ai](E( #" 4 2,D"#.)1AE\'E,- !;8#76E5kZ1A # !;`#\E,D [ E(^(# [ 4 τ1 -1AD Z'E,- !;\,c6E21=,oE,D [ =(# & x(τ1 ) = 0 p(τ1 ) = 0 [ 4 !; u(τ1−) = −1 L _= H42515*'G$ _= \ " t = τ1 1A t = 0 .R* ,- (mN$':#%e x(·)[α] p(·)[α] u(·)[α] C

:D

x(t)[α] = −t + α, p(t)[α] = −(t − α)2 , u(t)[α] = −1, t ∈ [0, α].

?':#%e !;L# ;4 % \ ( "H :1C

S S ! $ -1AD # 'E,- !; ,c6E21 , E,D [ (# [ 4 !; N$':#%e x(·)[α] p(·)[α] u(·)[α] ( b"H :1C u(τ ) = 1 X(α) ≡ x(0)[α] − 1 = 0.

1−

VU

x(t)[α] = t − α, p(t)[α] = (t − α)2 , u(t)[α] = 1, t ∈ [0, α].

?':#%e !;`# ;4 % S S !R#K"7% !6+ _= # !;!6a\#G#Z [ hZ7& " ,c; <=': / ( . ( # # !;\( "21?,D"D [ . K _= # 42515*\,$DkH (9 ( LE,D [ =': # !;L, (I X

= ?

]Ap&]Mg`a

$-1A D2(.Rj1;G, (E,D"5;"D # $ _= # !;K42515* K,D21A Ek" #2,D%: %:ZE #" ZE,D"7# % V R 10 (x˙ 2 + αx3 )dt → inf x(0) = 0 x(10) = 0 α ∈ [0, A] iI0 #".PC K[ 41A # "D # #.I6G # * # J [ |x|˙ ≤ 1 |¨x| ≤ 1

K'E,- x(10) = 0 42 ( #!; " ,c; # * # 2( O1A #!; " ,c;b[ ./() 4\'E,- J −B ≤ x(10) ≤ B M:- @ ( " 42515* A ∈ R+ B ∈ R+ @ # %: " .R %:α #,D"7#". 42515* S R 10 (x˙ 2 + αx˙ 3 )dt → inf x(0) = 0 x(10) = 0 α ∈ [0, A] iI0 #".PC K[ 41A # "D # #.I6G # * # J [ |x|˙ ≤ 1 |¨x| ≤ 1

K'E,- x(10) = 0 42 ( #!; " ,c; # * # 2( O1A #!; " ,c;b[ ./() 4\'E,- J −B ≤ x(10) ≤ B M:- @ ( " 42515* A ∈ R+ B ∈ R+ @ # %: " .R %:α #,D"7#". 42515* !

R4 0

x˙ 2 +

αx 1 + x˙ 2

dt → inf x(0) = 0 x(4) = 0 α ∈ [0, A]

V U U

iI #".PC K[ 41A # "D # #.I6G # * # J [ |x|˙ ≤ 1 |¨x| ≤ 1

K'E,- x(4) = 0 42 ( #!; " ,c; # * # 2( 1A #!; " ,c;mb[ ./( 4+'E,- J −B ≤ x(4) ≤ B M:- @ ( " 42515* A ∈ R B ∈ R @ # %: " .R + + %:α #,D"7#". 42515* Y R 4 x˙ 2 + x2 + αx dt → inf x(0) = 0 x(4) = 0 0 1 + x˙ 2 α ∈ [0, A] iI #".PC K[ 41A # "D # #.I6G # * # J [ |x|˙ ≤ 1 |¨x| ≤ 1

K'E,- x(10) = 0 42 ( #!; " ,c; # * # 2( 1A #!; " ,c;mb[ ./( 4+'E,- J −B ≤ x(4) ≤ B M:- @ ( " 42515* A ∈ R+ B ∈ R+ @ # %: " .R %:α #,D"7#". 42515* #!

VUX

RT

|u + α| + |u − α| xs + −α 0 2 x(0) = −1 x(T ) = 1

2 !

dt → inf

iI #". 1A NPN$ #e 7 # J\,D;4 C x˙ = u [ x¨ = u x¨ + x = u

x¨ − x = u = {1, 2, 3} T = {1, 10, 20} @ ( ". ': # 42α51=5* {0, u1}∈ Rs @l

o ,D"2(9?1A NPN$ #e 7 #.I6G': # # JC  x˙ = vx ,       y˙ = vy , vx + ux , v˙ x = −G(y, vx , vy ) 2 (v + vy2 )1/2  x   vy   + uy ,  v˙ y = −g − G(y, vx , vy ) 2 (vx + vy2 )1/2

y @ 17 #E,D"O ./,D "7 vx vy @ %:E( # #". %!& " x H ,D%: E,D" @ 'E,D%: # !; %:E( # #".3 %" ': # !; ux uy @l "7e ## 'E,D%: # ( "L42515* g @q,D " # /" (E, N$ .P G(y, vx , vy ) = α(vx2 + vy2 )e−µy @

( " ! % D ,

# # e 7 # J\(21AD L" (E, N$ .P µ≥0 @

( " n D ,

" # !; 1A !kZ # α≥ 0 [ 4P,D " # !; ⇒ #7 " *2,D%: $ _=α #= 0- $ .RK'E,- !; mN$':#%e #7 D "8#8(9%!,D (97& #':b 17 #E,D"-C

D

x(0) = 0, y(0) = 0, vx (0) = 0, vy (0) = 0, x(T ) → max, y(T ) = 0, vx (T ) = 0, vy (T ) = 0.

iI #".PC T > 0 @ ( " 42515* N$ %!,D E## \* ,-- # * # o#=': # !; u @ u2x + u2y ≤ u2max max ( "L42515* [ T > 0 @ ,D [ 21A#7; " ( 4 ' 2(97; n D * # RT q @ 6:%" ,D" *D& u2x + u2y dt ≤ V V > 0 0 ,D%!7;L,D%: E,D" ( "L42515* T > 0 @ ,D [ 21A#7; " ( 4 ' 2(97; n D * # RT 2 @ ( "L42515* ux + u2y dt ≤ C C > 0 0 U /$E,D"7# %!K42515* 8 -1AD; " ,c;Oa 4 '27"7"H .R[ \ 21A#E('Z 4o-1,D"7 ##.I6HE #" ZN$':#%e #7A71A No& N$ #e 7 # J`,D;4 # * # !;`#\': # !;%D& .I6K'E,- J\ 42(5kH# 1A # "D #.I6G # * # J VU

?':#%e #7C

i

<

V2X T

42515*=[ ./,D"21A J,D" !; - ( # ( 42e !;a42""Z': # !; - ( # ( 42e !;a42""Z': # !; - ! hZ51AZ21a N$ %:E( - " ( #7 # $4 #5* # $ D * #. - x(T ) → extr T → inf RT |u|dt → inf 0 RT 2 u dt → inf 0 RT xdt → inf 0

U

U

CV

4V

@W7& ( ".3%:E(E( , ,D# N$':#%e #7A <= NPN$ #e 7 #7;G,D;4 C x˙ = u ,D%: E,D"- [ x¨ = u 'E,D%: # - x¨ + x = u (97;"# % -

x¨ − x = u 1 x¨ + αx˙ + ωx = u x¨ + ωx|˙ x|˙ = u " # - Q # !;C > u ∈ R+ > > |u| ≤ 1 $ .Ro'E,- !;C ? %!,D E##.Ro4 #5* # !;\NP4 .I6H 2( ##.I6GZ#7& *7$ L%: #e \vP ,D" *#7M!,D [ 21A#.R?4 #5* # !;ONP4 .I6a 2( ##.I6 #5*7$ L%: #e <= # "D #.Ro # * # !;C V R T f (t, x, u)dt = C 0 S m ≤ R T f (t, x, u)dt ≤ M RT kT T + kxT x(T ) + 0 (ku1 |u| + ku2 u2 + kx x)dt → inf kT > 0 kxT > 0 ku1 > 0 ku2 > 0 kx > 0

0

c1A ,D " " ,D" ' "?%!%:E(' ME [ =N$':#%e #7'Z427& 15* f (t, #x,u) ( oi ! a j ! x cos t x sin t !)H)+""7

V-M:[ Y7M: -M:[ 5M! T = 10 u = 1 max ˙ = 2 x(T ) = 0 x(T U M M:7M:> > M x(0) = 5 x(0) ˙ )=0 R T u2dt ≤ 4 - U M M M S x(0) = 0 x(0) ˙ =0 0

V2XV

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