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♦♠ ♠
ssr② ♦♥t♦♥s ♦r ♦♥str♥♦♥s♠♦♦t rt♦♥ ♣t♠ ♦♥tr♦
Pr♦♠s
P tss s♠tt t♦ t t② ♦ ♥ ♦ ❯♥rst② ♦ P♦rt♦
P♦rt ♥ ♠♥t ♦ t rqr♠♥ts ♦r t r ♦ ♦t♦r ♦
P♦s♦♣② ♥ t♠ts
♣rt♠♥t ♦ t♠tst② ♦ ♥ ❯♥rst② ♦ P♦rt♦ P♦rt
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Necessary
Conditions for
Constrained
Nonsmooth
Fractional Optimal
Control ProblemsHegagi Mohamed Ali AhmedDoutoramento em Matemática AplicadaDepartamento de Matemática, Universidade do Porto, Portugal.
2016
Orientador Prof. Fernando Manuel Ferreira Lobo Pereira
Faculty of Engineering of the University of Porto, Portugal.
Co-orientador Prof. Sílvio Marques de Almeida Gama
Faculty of Science of the University of Porto, Portugal.
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strt
♠♥ ♦t ♦ ts tss s t♦ ♣r♦ ♦♥trt♦♥ t♦ t ♦② ♦ rsts rr♥t②
♥ t ♦♣t♠♦♥tr♦ t♦r② ♦r ②♥♠ ♦♥tr♦ s②st♠s ♠♦ ② rt♦♥
rt ♦ t stt r t rs♣t t♦ t♠ r♦r t ♣r♥♣ ♦s ♦ ts
ssrtt♦♥ s ♦♥ ♦t♥♥ t s♠♦♦t ♥ ♥♦♥s♠♦♦t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t②
♥ t ♦r♠ ♦ ♠①♠♠ ♣r♥♣ ♦ t P♦♥tr②♥ t②♣ ♦r rt♦♥ ♦♣t♠♦♥tr♦
♣r♦♠s
rt♦♥ ♦♣t♠♦♥tr♦ s ♥r③t♦♥ ♦ t ♦rrs♣♦♥♥ ♥tr ♦♣t♠
♦♥tr♦ t♦rs ①t ♦♥tr♦ s②st♠ ②♥♠s t rtrr②♦rr rts
♥ ♥trs s② ♥ t♦t ♥② ♦ss ♦ ♥rt② ♦ ♥tr ♦rr
rst ♠ ♦ ts tss s t♦ s♣② ♠①♠♠ ♣r♥♣ ♦ t P♦♥tr②♥ t②♣ ♦r t
♥r③t♦♥ ♦ ♥♦♥♥r rt♦♥ ♦♣t♠♦♥tr♦ ♣r♦♠s ♥r ♣r♦♣r ss♠♣t♦♥s
t s♠♦♦t t ♥ ♣r♦♣♦s ♥ ♣♣r♦ ♦♥trt♥ t♦ ♦t♥ ♠①♠♠ ♣r♥♣
♦♥r♥♥ t r♦♠♠♥ ♣r♦♠ ♥♦t② ♦ ♦r ♣♣r♦ ♦♥ssts ♥ ♠♦r
♣rs ♥st ♥r♥t t♦ t s ♦ ♦♣t♠♦♥tr♦ rt♦♥ ♠t♦s ♥ t ♦r♥
♠♦♥ r♠♦r r qt st♥t r♦♠ t♦s ♦r♥t ♥ t s ♦
rt♦♥ r♠♦r rq♥t② s ♥ t trtr ♦r♦r ♣♣② t ♣r♦♣♦s
♥ssr② ♦♥t♦♥s t♦ strt ♥ ①♠♣ ♦ ♥r③t♦♥ ♦ rt♦♥ ♦♣t♠♦♥tr♦
♣r♦♠s s s♦ ② t ttr ♥t♦♥
♦r ♠♦♥ t♦ t ♠♥ ♦ ♦ ts tss t♦ r ♥♦♥s♠♦♦t ♠①♠♠ ♣r♥
♣ ♦r rt♦♥ ♦♣t♠♦♥tr♦ ♣r♦♠s t stt ♦♥str♥ts ①t♥ rt♦♥
♥trs t♦ ♠♦r ♥r ♦♥ ♠srs ♥ ts rsts ♦ ♥♣♥♥t ♥trst t②
♣② ♥ ♠♣♦rt♥t r♦ ♥ t rt♦♥ ♦ t ♠①♠♠ ♣r♥♣ t stt ♦♥str♥ts
s♥ t ss♦t ♦♥t ♠t♣r ♥♦♠♣ss s t②♣ ♦ ♠srs
❲ s♦ ♣rs♥t ♥♦♥s♠♦♦t ♥ssr② ♦♥t♦♥ ♦ rt♦♥ r♥t ♥s♦♥ t
stt ♦♥str♥ts ♥r ss♠♣t♦♥s ♥ ♥tr♦ ♠♣♦rt♥t ♦♥♣ts ♥ rsts
♦♥r♥♥ t ①st♥ ♥ ♦♠♣t♥ss ♦ sts ♦ t rt♦♥ trt♦rs
♥② rss t ♠♥ ♦ ♦ ts ssrtt♦♥ t♦ stt rt♦♥ ♦♣t♠
♦♥tr♦ ♦r♠t♦♥ rqrs r② ss♠♣t♦♥s ♦♥ t t ♦ t ♣r♦♠ ♥
①t ♠♣♦rt♥t t②♣s ♦ ♦♥str♥ts ♥♦t② ♦♥tr♦ ♦♥str♥ts ♥ stt ♦♥str♥ts
tt ♥♦t ♥ ②t ♦♥sr ♥ t rt♦♥ ♦♥t①t ♣♣r♦ t♦ t ♣r♦♦ ♦
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t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♥ t ♦r♠ ♦ ♠①♠♠ ♣r♥♣ ♦ P♦♥tr②♥
t②♣ s s ♦♥ ♣♥③t♦♥ t♥qs rt♦♥ ♣r♥♣s ♥♦♥s♠♦♦t ♥②ss ♥
rt♦♥ s rsts ♥ ①♠♣ strt♥ t ♣♣t♦♥ ♦ t r ♠①♠♠
♣r♥♣ s s♦ ♥
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s♠♦
♣r♥♣ ♦t♦ st ts é ♣r♦♣♦r♦♥r ♠ ♦♥trçã♦ ♣r ♦ ♦♥♥t♦ rs
t♦s t♠♥t ①st♥t t♦r ♦♥tr♦ ót♠♦ sst♠s ♦♥tr♦s ♥â♠♦s
♠♦♦s ♣♦r rs r♦♥árs ♥s rás st♦ ♠ rçã♦ ♦ t♠♣♦ st
s♥t♦ ♦ ♥♦q ♣r♥♣ st ssrtçã♦ é str ♦♥çõs ♥ssárs ♦t♠
ss ♥ã♦ss s♦ ♦r♠ ♠ ♣r♥í♣♦ ♠á①♠♦ ♦ t♣♦ P♦♥tr②♥
♣r ♣r♦♠s ♦♥tr♦♦ ó♣t♠♦ r♦♥ár♦s
♦♥tr♦ ót♠♦ r♦♥ár♦ é ♠ ♥r③çã♦ ♦rrs♣♦♥♥t t♦r ♦♥tr♦
ót♠♦ ♠ rs ♥trs ♦♥ ♥â♠ ♦ sst♠ é ♦♥tr♦ ♣♦r rs
♥trs ♦r♠ rtrár r♠♥t s♠ qqr ♣r ♥r ♦r♠
♥tr
♣r♠r♦ ♦t♦ st tr♦ é ♣rs♥tr ♠ ♣r♥í♣♦ ♦ ♠á①♠♦ ♦ t♣♦ P♦♥
tr②♥ q ♥r③ ♣r♦♠s ♦♥tr♦ ót♠♦ r♦♥ár♦s ♥ã♦♥rs s♦ ♣ótss
qs ♦♠ ♦s ss ♣r♦♣♦♠♦s ♠ ♦r♠ q ♦♥stró ♠ ♣r♥í♣♦ ♦
♠á①♠♦ ♥♦ ♥♦ss ♦r♠ ♦♥sst ♥♠ sã♦ ♠s ♣rs ♥r♥t ♦ s♦
♠ét♦♦s r♦♥s ♦♥tr♦♦ ót♠♦ ♥♦ qr♦ ♠♦çã♦ ♦r♥ q é ♠t♦
st♥t♦ ♦ q s ♦té♠ ♥♦ ♠♣♦ ♦ á♦ rçõs rq♥t♠♥t ♥♦♥tr
♦s ♥ trtr s ♥ s ♦♥çõs ♥ssárs q ♣r♦♣♦♠♦s sã♦ strs ♥♠
①♠♣♦ q ♥r③ ♠ ♦♥♥t♦ ♣r♦♠s ♦♥tr♦ ót♠♦ r♦♥ár♦s q s
rs♦♠ rçs ♦ s♦ ♥çã♦ ttr
♥ts ♣ssr ♦ ♦t♦ ♣r♥♣ st ts ♣♦r ♦trs ♣rs str ♠ ♣r♥í
♣♦ ♦ ♠á①♠♦ ♥ã♦s ♣r ♣r♦♠s ♦♥tr♦♦ ó♣t♠♦ r♦♥ár♦ ♦♠ rstrçõs
st♦ st♥♠♦s ♦s ♥trs r♦♥ár♦s ♦ s♦ ♠s r ♠s ♦♥ ♥♦
st rst♦ ♦ s ♥trss ♣ró♣r♦ s♠♣♥ ♠ ♣♣ ♠♣♦rt♥t ♥ rçã♦
♦ ♣r♥í♣♦ ♦ ♠á①♠♦ ♦♠ rstrçõs st♦ ♠ ③ q ♦ ♠t♣♦r ♥t♦
ss♦♦ ♥♦ ♠ ♠ st t♣♦
♠é♠ ♣rs♥t♠♦s s♦ ♣ótss rs ♠ ♦♥çã♦ ♥ssár ♥ã♦s ♥
sã♦ r♥ r♦♥ár ♦♠ rstrçõs st♦ ♥tr♦③♠♦s ♦♥t♦s ♠♣♦rt♥ts
rst♦s s♦r ①stê♥ ♦♠♣ ♦s ♦♥♥t♦s s trtórs r♦♥árs
♥♠♥t ♦r♠♦s ♦ ♦t♦ ♣r♥♣ st ssrtçã♦ st♦ é str ♠ ♦r
♠çã♦ ♦♥tr♦ ót♠♦ r♦♥ár♦ q rqr ♥♦s ♦s ♦ ♣r♦♠ ♣ótss
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♠t♦ rs ①♠ ♠♣♦rt♥ts t♣♦s rstrçõs ♥♦♠♠♥t rstrçõs ♦♥
tr♦♦ rstrçõs st♦ s q ♥ ♥ã♦ ♦r♠ ♦♥srs ♥♦ ♦♥t①t♦ r♦♥ár♦
♠♦♥strçã♦ s ♦♥çõs ♥ssárs ♦t♠ s♦ ♦r♠ ♠ ♣r♥í♣♦ ♦
♠á①♠♦ ♦ t♣♦ P♦♥tr②♥ s té♥s ♣♥③çã♦ ♣r♥í♣♦s r♦♥s ♥ás
♥ã♦s rst♦s á♦ r♦♥ár♦ ❯♠ ①♠♣♦ q str ♣çã♦ st
♣r♥í♣♦ ♦ ♠á①♠♦ q ♣rs♥t♠♦s stá t♠é♠ q ♥í♦
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♥♦♠♥ts
♣♣rt♦♥ ♥ t♥s t♦ ♦ ♥ ♣ ♠ t♦ ts tss ♥
t♦ t♦s ♦ s♦♠♦ ♦♥trt t♦ t ♦♠♣s♠♥t ♦ ts P ♦r
rst ♦ ♦ t♦ ①♣rss ♠② ♠♦st s♥r rtt t♦ ♠② s♣rs♦rs Pr♦
r♥♥♦ ♥ rrr ♦♦ Prr Pr♦ss♦r ♥ t ♣rt♠♥t ♦ tr
♥ ♦♠♣tr ♥♥r♥ ♥ ❨ t t t② ♦ ♥♥r♥ ❯P
❯♥rst② ♦ P♦rt♦ P♦rt ♥ Pr♦ í♦ rqs ♠ ♠ ss♦t Pr♦
ss♦r ♥ t ♣rt♠♥t ♦ t♠ts ♥ ❯P t t t② ♦ ♥ ❯P
♥ rt♦r ♦ t ♦t♦r ♦rs ♥ ♣♣ t♠ts ❯♥rst② ♦ P♦rt♦ P♦r
t ♥t t♦ t♥ t♠ ♣r♠r② ♦r ♣t♥ t♦ ♦♥t② s♣rs ts tss ♦r
tr ♥ s♣♣♦rt ♥♦r♠♥t sst② tr r♠rs ♥ ♦♠♠♥ts r♥
t ♠t♥s t♦tr tt ♣ ♠ t♦ ♥rst♥ t r♥t ♦♥♣ts ♦ t
♣r♦♠ ♥ tr r♥s♣ r♥ ♠② sts t t ❯♥rst② ♦ P♦rt♦ s tss
♦ ♥♦t ♥ ♣♦ss t♦t tr ①♣rts
♠ rt ♦r t ♥♥ s♣♣♦rt ♥ ② rs♠s ♥sst♦ ❯♥rst② tr♦
t r♥t rs♠s ♥s t♠ r ♦rs♣ Pr♦r♠ ♦t ♦t ♥
r r♥t r♠♥t ♥♦ s♦ ♠ t♥ t♦ t P♦rts ♦♦r♥t♦r
♥ P ♥ t♦ t rs♠s ♥s t♠ ♥ P♦rt♦
s t♦ ①♣rss ♠② t♥♥ss t♦ r r♦ rt♥s ♦♥s♦ ♦r s s♣♣♦rt ♥ sss
t♥ r♥ t st ♠♦♥t ♦ rt♥
t♥ s♦♠ ♦s r♦♠ t t♠ts ♣rt♠♥t ♦ t ❯♥rst② ♦ P♦rt♦
s♣② r♦ r♥ P♥sr ♠♠ ♥ rs ♥ r♦
♥ ♦♦s s♦♣♥s ♦r tr s♣♣♦rt ♥ ssst♥ ♦♥ ♠t♠t qst♦♥s ♥
♥sts
♣ t♥s ♦ t♦ t♦s r♥s tt ♠ t rt ♠♦tt♦♥ t t rt t♠
♥ ♠ ♠ tt ♠ ♥♦t ♦♥ ♥ ♣rtr ♠♦ r♦ ♦ss♠♥
♠ ♦♠ ♦♠ t♥② ♥ s r
♠ rt s t♦ ♠② r♥s r ♥ P♦rt♦ ♥ s♦ ♥ ②♣t ♦r tr s♣♣♦rt
ssst♥ r♥s♣ ♦r t ♣st ②rs ♥ ♦r t ♥trt♥♠♥t tt t② ♣r♦ t♦
♠ ♦ t♦ t♥ r②♦② ♦ s ♥ ♠♣♦rt♥t t♦ t sss r③t♦♥
♦ ts tss
s ♠ ♦rt s t t♦ ♠② ♣r♥ts ♦r tr ♥♦r♠♥t s♣♣♦rt ♥ ♣r②rs
①
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♦r ♠ tt ②s ♠ str♥t ♥ t t♠s r② t♥ ②♦ ♦r t ♠♦st
♠♣♦rt♥t s ②♦ ♥ ♠ ♥ ♦ t♦ t♥ ♠② r♦trs ♠② sstrs
♥ ♠② ♠②
st t ♥♦ ♠♥s st ♥♥♦t t♥ ♥♦ ♠② ♦ ♥ st r♥ s♠
♦r r ♦ ♥ ♥♦r♠♥t tt st♦♦ s ♠ t t♠s ♥ ♦ ♣s
♠ t r ♥r♥ t ♦ s♦ t♦ ♥♦ t ♠♦st ♠♣♦rt♥t
♣rs♦♥s ♥ ♠② ♠② ♦ s♦♥ ♦♠ ♥ ♠② tr ♥♥
♥② ①♣rss ♠② ♠ ♣♦♦s t♦ t♦s ♦♠ ♠t ♦r♦tt♥ t♦ t♥
①
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♦♥t♥ts
strt
s♠♦
♥♦♠♥ts ①
♥tr♦t♦♥
ts
♦tt♦♥
♦♥trt♦♥s
r♥③t♦♥
tt♦trt
rt♦♥ s
♣ ♥t♦♥s
rt♦♥ ♥trs ♥ rts
rt♦♥ ♥trt♦♥ ② ♣rts
r ♦♥ ♦♥♥t♦♥ ♣t♠ ♦♥tr♦ ♦r②
ssr② ♦♣t♠t② ♦♥t♦♥s
P♦♥tr②♥ ♠①♠♠ ♣r♥♣
♣t♠ ♦♥tr♦ ♣r♦♠s t ♦♥str♥ts
①♠♠ ♣r♥♣
r ♦ s♦♠ t♦s t♦ ♦ ♣t♠③t♦♥ Pr♦♠s t rt♦♥
r♥t qt♦♥s
♦r♠t♦♥ ♦ rt♦♥ ♣t♠ ♦♥tr♦ Pr♦♠s
♥tr♦t♦♥
♥r ♦r♠t♦♥ ♦ rt♦♥ ♣t♠ ♦♥tr♦ Pr♦♠s
①♠♠ Pr♥♣ ♦r t s rt♦♥ ♣t♠ ♦♥tr♦ Pr♦♠
♥tr♦t♦♥
tt♠♥t ♥ ss♠♣t♦♥s
①♠♠ Pr♥♣ ♦ ♣t♠t②
①
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strt ①♠♣
♦♥s♦♥
rt♦♥ ♥trt♦♥ ♥ sr ♦♥♣ts
♥tr♦t♦♥
srs
♥trt♦♥
rt♦♥ ♥trt♦♥ t rs♣t t♦ sr
♥♠♥t Pr♦♣rts
ssr② ♦♥t♦♥s ♦ ♣t♠t② ♦r rt♦♥ ♦♥s♠♦♦t r♥
t ♥s♦♥ Pr♦♠s t tt ♦♥str♥ts
♥tr♦t♦♥
①r② ♥ sts
ssr② ♦♥t♦♥s ♦ ♣t♠t②
①♠♠ Pr♥♣ ♦r rt♦♥ ♣t♠ ♦♥tr♦ Pr♦♠s t tt
♦♥str♥ts
♥tr♦t♦♥
Pr♦♠ tt♠♥t ♥ ss♠♣t♦♥s
①♠♠ Pr♥♣ ♦ ♣t♠t②
strt ①♠♣
♦♥s♦♥s ♥ Pr♦s♣t sr
♦♥s♦♥s
tr ❲♦rs
♣♣♥s
rt♦♥ s
st♦r
rt♦♥ ♣rt♦rs
♥t♦♥s ♦ rt♦♥ ♥trs
♥t♦♥s ♦ rt♦♥ rts
t♦♥ t♥ t rt♦♥ rts
s Pr♦♣rts ♦ rt♦♥ s
♥r③ ②♦rs ♦r♠
♦♠ Pr♦♣rts ♦r ♠r rt♦♥ rt ♥ ♥tr
❱rt♦♥ sts
①t P♥③t♦♥
♥ ♦r♠
♥r③ r♦♥ ♥qt②
①
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rt♦♥ ♦s②♠♦♥ ♥♠♥t ♠♠
♦♥s♠♦♦t ♥②ss
sr ♦r② ♥ ♥trt♦♥
r ♥ σ−r ♦ ts
srs
♥trt♦♥
❯s ♦♥♣ts
♦r♣②
①
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♣tr
♥tr♦t♦♥
ts
♦s ♦ ts tss r t ♦♦♥
♠♣r♦♠♥t ♦ t rt♦♥ ♦♣t♠♦♥tr♦ ♦r♠t♦♥ ♥ r♥♠♥t ♦
♠t♦s ♥ ♣♣r♦s rqr ♦r t ♦♣♠♥ts ♦ ♥ssr② ♦♥t♦♥s ♦
♦♣t♠t② ♥ t ♦r♠ ♦ ①♠♠ Pr♥♣
s ♦ rqrs ♥ ♥r ♦r♠t♦♥ ♦r rt♦♥ ♦♣t♠♦♥tr♦ ♣r♦♠s
Ps ♦r t ♦t ♥t♦♥ s ♥ ② ♥ ♥tr ♦ rt♦♥ ♦rr
t ♦t② st s ♦s t ♦ ♥ ♥t st ♦ ♣♦♥ts ♥ ♥ ♣♣r♦ t♦
♣r♦ ♥ssr② ♦♥t♦♥ ♦ ♦♣t♠t② ♠①♠♠ ♣r♥♣ ♦ t P♦♥tr②♥
t②♣ ♣r♦♣♦s ♣♣r♦ s strt ② ♥ ♣♣t♦♥ ①♠♣
♦r♠t♦♥ ♦ ♦♣t♠♦♥tr♦ ♣r♦♠s Ps t rt♦♥ r♥t ②♥♠
s ♦ ♥r♠♥t② ♥rs♥ ♦♠♣①t② ♥♦t② ♥ t ♣rs♥ ♦ tr ♦♥tr♦
♥♣♦♥t ♦r stt ♦♥str♥ts ♦s t trs ss♠♣t♦♥s r t♥ t♦s
s② ♦♥sr ♦r Ps ♥ ♠♦r ♥ ♥ t t ♦♥s rr♥t② ♦♣t ♦r
♣r♦♠s t ♥tr rts
s ♦ rqrs ♥♦t ♦♥② t ♦♥srt♦♥ ♦ ♣♣r♦♣rt s♦t♦♥ ♦♥♣ts t
s♦ s♦♠ ♥stt♦♥ ♦♥ t ss♠♣t♦♥s t♦ ♠♣♦s ♥ ♦rr t♦ ♥sr t
s♥ss ♦ t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t②
♥♠♥t ♦ ♠t♦s ♥ ♣♣r♦s ♥♦t② ♣♥③t♦♥ t♥qs ♥
rt♦♥ ♠t♦s rqr t♦ rss t sss rs♥ ♥ t♠ ♦ ♥
t ♦♥t①t ♦ rt♦♥ r♥t ♦♥tr♦ s②st♠s
s ♦ s ♥ ♥tr♠t st♣ rqr t♦ rss t r♠♥♥ ♠s t s r
tt ♥ ♣♣r♦s t rs♣t t♦ t ♦♥s ♦♥sr ♥ t ♣st ♦r Ps r
rqr ♥ ♦rr t♦ t t ♥s rs♥ ♥ t P ♦r♠t♦♥s t♦
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♦♥sr ♥♠② ♥ rt♦♥ ♣r♥♣s ♥ ♥ ♣♥③t♦♥ t♥qs
♥ tr♥ ♠♣② t ♥ ♦ ♥♦♥s♠♦♦t s
rt♦♥ s ①♣♦t ♥ ♦rr t♦ ♥stt t ♥ ♠t♦s
t♦ ♠♣♦② ♥ t ♣r♦♦s ♦ t rsts t♦ ♦t♥
tt♠♥t ♥ ♣r♦♦ ♦ ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♠①♠♠ ♣r♥♣
♦ t P♦♥tr②♥ t②♣ ♦r t s r rt♦♥ ②♥♠s t stt
trt♦rs sts②♥ stt ♦♥str♥ts
♦ ♣rs ts ♦ sr ♦♥♣ts ♦ s♦t♦♥s ♦r ♦♥♥t♦♥ Ps r ①
♠♥ ♥ ♥ ss♥t r♦ s ♣② ② t ♥stt♦♥ ♦ ♦ t s♣ts
♥tr♥s t♦ t rt♦♥ ♥tr ♦ t rts ♥ Ps t ts ♦♥♣ts
♦r♦r t s ①♣t tt s♦♠ s♣ts ♦ t ♣r♦♠ ♦r♠t♦♥ ♥ ♣rtr
t ss♠♣t♦♥s ♦♥ t t ♦ t ♣r♦♠ ♠t ♦♠ ♥t♦ ♣② ♥ rss♥
ts sss
♥r ♣♣r♦ t♦ ♣rs ts ♦ s t♦ s t ♣♣t♦♥ ♦ r♠ts r
♦r st② ♥ ♥♥t♠♥s♦♥ ♦♣t♠③t♦♥ ♣r♦♠ t ♦♥str♥ts
s rqrs t s ♦ s♣②♥ ♣♥③t♦♥ t♥qs ♦♣ t
rt♦♥ ♣r♥♣ s♦ ♥ ①♠♣ s ♥ t♦ strt t ♣♣t② ♦
ts ♣♣r♦
♦tt♦♥
Ps r ♥r③t♦♥ ♦ ss Ps ♦r tr t ②♥♠s ♦ t ♦♥tr♦
s②st♠ s sr ② rt♦♥ r♥t qt♦♥s ♦r t ♣r♦r♠♥ ♥① s ♥
② rt♦♥ ♥trt♦♥ ♦♣rt♦r rs♦♥ ♥ t s ♦ rt♦♥ rts s
♥ t t tt t② ♣r♦ ♠♦r rt sr♣t♦♥ ♦ t ♦r ♦ t ♦♥sr
②♥♠ s②st♠ ♥ ♦♥sttt ♥ ①♥t t♦♦ ♦r t rtr③t♦♥ ♦ ♠♠♦r② ♥
rtr② ♣r♦♣rts ♦ sr ②♥♠ ♣r♦sss s ♣♦♥tt♦ t ❬❪ s ❬❪
s t ❬❪ ♥ P♦♥② ❬❪
Ps ♦♥ ♣♣t♦♥s ♥ ♠♥② r♥t s ♥ r ♥♥ ♦♦② s
♥ ❬❪ ♥s ♥ ♦r♥ ❬❪ ♥ t ❬❪ ♥ ♥rt ♥ ❲♦r♠♥ ❬❪
♦♦② s ♦♥ ❬❪ ♥ r♦r t ❬❪ ♥♥r♥ s ♦r♦t ♥
♦tt ❬❪ ♥ ❩♥ ♥ ♦rs♦ ❬❪ ♦♥♦♠s s t ♥ ♦♠♣s♦♥
❬❪ ♥ ❩♥ ♥ ♦rs♦ ❬❪ ♥♥ s ♥ ♥ s♠ ❬❪ ♥ s
♥ ③♥ ❬❪ rs♦r ♦t♦♥ ♥ ♠♥♠♥t s r ❬❪ ♥ t ♥
♦♠♣s♦♥ ❬❪ ♠♥ s ♦ t ❬❪ ♥ ♥ ❬❪ ♥ s♦ ♦♥
♥ t st tr s s ttrt s♥♥t ♥trst t♦ ts ♠t♣
♣♣t♦♥s ♥ ♠♥② rs ♥ ♣rtr rt♦♥ s②st♠s ♥ s ♥ rs ♦
♣♣t♦♥ t♦ ♥t s②st♠s t ♦♥r♥ ♥trt♦♥s ♦r ♣♦r ♠♠♦r② s
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s♥ ♥ ❩ss② ❬❪ ♥ ♦ ♥ r♠♦ ❬❪ rt♦♥ s②st♠s r ♦t♥
♠♦r ♣♣r♦♣rt t♥ t s ♦♥s ♥tr♦rr s②st♠s ♥ r ♣♣t♦♥s s
s tr♦♠str② s t ❬❪ tr ♠trs rs♦ ❬❪ s♦st
♠trs s ② ♥ ♦r ❬ ❪ ♥ ♥r② ❬❪ rt ♥t♦rs s
❲st t ❬❪ ♥ r♥ t ❬❪ r♦♦ts s térr③ t ❬❪ ❱ér♦
♥ á ♦st ❬❪ ♦♦ tsss r s rt r♥ t s ♦r♥
t ❬❪ ♦②t t ❬❪ ♦②s t ❬❪ ♥ é t ❬❪ tr rts
s ♣♦♥tt♦ t ❬❪ ♥ Ptrs ❬❪ s♥ ♣r♦ss♥ s s♥ ♥
❬❪ ♥ ❱♥r t ❬❪ ♦♥tr♦ s②st♠s s ①t ♥ s ❬❪ ♣♦♥tt♦
t ❬❪ ♦♥ t ❬❪ ♥ P♦♥② ❬❪ ♥ s♦ ♦♥
♥ ts ②rs s♦♠ ♣♦♣r ♥ s♠♣ strtrs ♥ ♣r♦♣♦s ♦r rt♦♥♦rr
♦♥tr♦rs s s rt♦♥♦rr Pr♦♣♦rt♦♥♥trrt P ♦♥tr♦rs
r ② s ♥ t ♥str② s ♠♠ ❬❪ P♦♥② ❬❪ ♥ ③♦
❬❪ ♥ tr ssss rt♦♥♦rr P ♥ P ♦♥tr♦rs s ♥ t
❬❪ ♥ ♦ ♥ ♥ ❬❪ s s r♥t ♥rt♦♥s ♦ ♦♠♠♥ ♦st rr
♦♥ ♥tr ♦♥tr♦rs r♣rs♥t t rst r♠♦r ♦r ♥♦♥♥tr♦rr
s②st♠ ♣♣t♦♥ ♥ t t♦♠t♦♥tr♦ r s ♥ss t ❬❪ ♥ss ♥
tr ❬❪ ♥ st♦♣ t ❬❪ t ♦r♦r rt♦♥♦rr ♦♥tr♦rs
♥ ♠♣♦② ♥ ♠♥② s ♣♣t♦♥s s s ♦♥tr♦ ♦ rs r sr♦ s②st♠s
♦ t ❬❪ ♦♥tr♦ ♦ ♠♥t ♠♥ ♣r♦sss ❬❪ s♣♣rss♦♥ ♦ ♦s ♥
♦t tr rts ③♦ t ❬❪ ♦♥tr♦ ♦ ♣♦r tr♦♥ ♦♥rtrs
ró♥ t ❬❪ ♦♥tr♦ ♦ ♦♠♣♦st ②r ②♥rs ❩♦ t ❬❪ ♦♥tr♦
♦ rrt♦♥ ♥s t t ❬❪ ♥ ♠♥② ♦trs
r♥ ♦ ♦♠♣①t② ♥ ♥♥ ♥ s♥ s t♥ t ♥tr ♥
rt♦♥ ♦♥t①ts t♦tr t t ss♦t r♥s ♥ t ♦♠tr ♥tr♣rtt♦♥
♠② ①♣♥ ② t ♦♣♠♥t ①t♥ts ♦ t rs♣t ss♦t ♦s ♦ ♦♣t♠
♦♥tr♦ t♦r② r s♦ ♠ ♥ ♦r ♦♥ t rsr rr ♦t ♦♥ ♦♣t♠t②
♦♥t♦♥s ♥ ♠t♦s ♦r s♦♥ Ps rs tr♠♥♦s t♥ ♦t sss ♦
♣r♦♠s ♠r② ♣ ①sts ♦r t ①t♥t ♦ t ♦♣♠♥t ♦ ①st♥ ♦♣t♠
♦♥tr♦ t♦r② ♦r ♦♥♥t♦♥ Ps ♥ Ps ♦r ♦♥② ♣r♦r♠♥ ♥tr
♥trs ♥ ♦♥sr
trt♥ t t ♣♦♥r♥ ♦r ② P♦♥tr②♥ ♥ s t♠ ❬❪ ♥
r♥ ♦ Ps ♥ ♦♥sr ♥♦t② stt ♦♥str♥ts srtt♠ ♥ t
s♦♣stt ♦♥♣t ♦ r① s♦t♦♥ r② ♥ ①♣♦t ♦♥♥t♦♥ ♦♣t♠
♦♥tr♦ t♦r② ♥t ♦r t ②rs tr♦ ①tr♠②♦♠♣t ♦♣♠♥ts ♥
t ss♠♣t♦♥s ♦ t ♣r♦♠ r str♦♥② ♥ ❱r② rs ♦r♠t♦♥s
rt② ♦ ♦♥str♥ts ♥ s♦ ♥♦♥ t♠ ♦r③♦♥s ♥ t sss ♦
♣♦s♥ss s♥stt② ♥♦♥s♠♦♦t♥ss ♥ ♥♦♥♥r② ♥ ♦♥sr s t
s s② ttst ♥ t ♦rs ♦ ♠♥② rsrrs ♠♦♥ Prr t ❬❪
rt②♥♦ ❬❪ rt②♥♦ t ❬❪ ❱♥tr ❬❪ r ❬❪ ♥ r t ❬❪
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❲t♥ t ♦ts ♦ ts tss st♣s t♦rs ♦♥trt♦♥s t♦ ♦s ts ♣ r
♣rs s ♦rt rqrs t ♥stt♦♥ ♦ ♥ ♠t♦s tt r ♥tr ♦♥ t
♦♣♠♥t ♦ rt♦♥ rsts ♣♣r♦♣rt ♦r Ps s s ♦♥ t r♥♠♥t
♦ rqr t♦ ♠t t ♠♥s ♦ t ♥s t♦ rss
♦♥trt♦♥s
rst ♦♥trt♦♥ ♦ ts tss s t ♦♣♠♥t ♦ Ps ② s♥ ♥r
♦r♠t♦♥ r ♠♣♦② t rt♦♥ ♥tr ♦♣rt♦r ♥ t ♦st ♥t♦♥ ♥
sr t ②♥♠s ♦ t ♦♥tr♦ s②st♠ tr♦ t ♣t♦ rt♦♥ rt
♦r♦r ♥tr♦ ♥ ♣♣r♦ t♦ ♣r♦ ♥ssr② ♦♣t♠t② ♦♥t♦♥s ♥ t
♦r♠ ♦ P♦♥tr②♥s ♠①♠♠ ♣r♥♣ ♦r t ♥r ♦r♠t♦♥ ♦ rt♦♥ ♥♦♥♥r
P ♥r s♠♦♦t ss♠♣t♦♥s ♦♥ t t ♦ t ♣r♦♠ rtr♠♦r ♣r♦ ♥
①♠♣ ♦r ts ss ♦ P s t ♥r③t♦♥ ♦ t ttr ♥t♦♥
t♦ s♦ t ♥ ♦♠♣r ♦r rsts t t ss ♦♥s ♥ α = 1 t♦ strt t
t♥ss ♦ ts ♥♠♥t ♥♥s rsts ♦ ts ♣tr r ♣rs♥t s
♥ strt t t ♥tr♥t♦♥ t♥ P P♦rt♦ P♦rt
♦♠♣t rs♦♥ ♦ t rsts s ♥ ♣s ♥ ❬❪
♥t♦♥ ♦ rt♦♥ ♥tr t rs♣t t♦ ♥r ♦♥ rr ♦r ♠sr
s♥t ② rt♦♥ tts ♥tr s s♦ ♦♥ ♦ t ♦♥trt♦♥s ♦ ts tss
r ts ♠sr ♥ rtt♥ s ♦♠♣♦st♦♥ ♦ t♦♠ srt s♦t②
♦♥t♥♦s ♥ s♥r ♦♥t♥♦s ♠srs ❲ ♣rs♥t s♦♠ ♣r♦♣rts ♦r ts ♦♥♣t
t♦♦ s rsts r♥t ♥ t ♥①t t♦ ♣trs ♥ ♥stt t
♠♥♠③r ♦ t P t stt ♦♥str♥ts s rsts r s♠tt ♦r ♣t♦♥
♥♦tr ♦♥trt♦♥ ♦ ts tss s t ♦♥strt♦♥ ♦ t ♥ssr② ♦♥t♦♥s ♦r
♥♦♥s♠♦♦t Ps ♥ t ②♥♠ s②st♠ s rtr③ ② rt♦♥ r♥t
♥s♦♥ t stt ♦♥str♥ts ♥r s♦♠ ss♠♣t♦♥s s rsts
r♥t ♥ t ♥①t ♣tr ♥ ♥stt t ♠①♠♠ ♣r♥♣ ♦r ♥♦♥s♠♦♦t
Ps t stt ♦♥str♥ts
♥ ♦♥trt♦♥ s t ♥ ♣♣r♦ ♥ t rt♦♥ ♦♥t①t t♦ ♣r♦ t
♥♦♥s♠♦♦t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♥ t ♦r♠ ♦ ♥♦♥s♠♦♦t ♠①♠♠
♣r♥♣ ♦r P ♥r ss♠♣t♦♥s ♦♥ t t ♦ t ♣r♦♠ t stt
♦♥str♥ts r ♦♣t t ♠r rt♦♥ rt t♦ ①♣rss t ②♥♠s ♦ t
♦♥tr♦ s②st♠ t② r ♥ ♦r ♦♥t♥♦s ♥t♦♥ ♥♦t ♥ssr② r♥t
rtr♠♦r s♣② ♥ ①♠♣ ♦r ts t②♣ ♦ rt♦♥ ♦♣t♠♦♥tr♦ ♣r♦♠
t stt ♦♥str♥t t♦ s♦ t t♥ss ♦ ♦r rsts rsts ♦ ts ♣tr
r s♠tt ♦r ♣t♦♥
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r♥③t♦♥
s tss s ♦r♥③ s ♦♦s
♥ ♣tr t ♦t t stt♦trt ♣rs♥t r ♦r ♦ t rt♦♥
s ♦♥♥t♦♥ ♦♣t♠♦♥tr♦ t♦r② ♥ s♦♠ ♠t♦s t♦ s♦ ♦♣t♠③t♦♥
♣r♦♠s t rt♦♥ r♥t qt♦♥s
♥ ♣tr ♥tr♦ ♥ ♦r ♦ t ② rsts ♦♥ ♦♥♥t♦♥ ♦♣t♠♦♥tr♦
t♦r② t ♦♥s rss♥ ♣r♦♠s ♦s ②♥♠s s ♥ ② ♥tr r♥t
♦♥tr♦ s②st♠s s ♥s t ♣r♦♠ ♦r♠t♦♥ t ♥ssr② ♦♥t♦♥s ♦
♦♣t♠t② ♥♦t② t ♦♥s ♥ t ♦r♠ ♦ ♠①♠♠ ♣r♥♣ ♥ P t
♦♥str♥ts ♥ ♣rs t ♦♣♠♥ts ♦♥ t ♦♣t♠♦♥tr♦ ♦ rt♦♥
r♥t ♦♥tr♦ s②st♠s s ♥ t ♣r♦♠ ♦r♠t♦♥ ♥ssr② ♦♥t♦♥s
♦ ♦♣t♠t② ♥ ♥♠r t♥qs t♦ s♦ ts ♣r♦♠s
♥ ♣tr ♣rs♥t ♥ ♣♣r♦ t♦ ♣r♦ ♥ssr② ♦♥t♦♥s ♦r ♦♣t♠t② ♥ t
♦r♠ ♦ P♦♥tr②♥ ♠①♠♠ ♣r♥♣ ♦r ♥r ♦r♠t♦♥ ♦ rt♦♥ ♥♦♥♥r
Ps ♦s ♣r♦r♠♥ ♥① s ♥ t rt♦♥ ♥tr ♦r♠ ♥ ♦s ②♥♠s s
♥ ② st ♦ rt♦♥ r♥t qt♦♥s s ♥ t ♣t♦ s♥s ♦r♦r
s ♥r③t♦♥ ♦ t ttr ♥t♦♥ t♦ s♦ ♥ ①♠♣ ♦ ts ♥r
♦r♠t♦♥ ♦ P ♥ ♦rr t♦ strt t ♥② ♦ ♦r rst rsts ♦ ts
♣tr ♥ ♣t ♦r ♣t♦♥ ♥ ❬❪
♥ ♣tr ♣rs♥t s♦♠ ♠♣♦rt♥t ♦♥♣ts ♦ ♠sr ♥ ♥trt♦♥ t♦r②
♣ s ♥rst♥ t rt♦♥ tts ♥tr s♣② ♥ ♠♣ ♥ t
♥t♦♥ rtr♠♦r ♥ ♦r♠ ♦r t rt♦♥ ♥tr t rs♣t t♦ t
♠sr ♥ t ♠r rt♦♥ ♥tr s♥s ♥ ts ♦r♠ r ♥trst ♥ t
♥♦♥s♠♦♦t s r t ♠sr ♥ ♦♠♣♦s ♥t♦ ♥ t♦♠ ♥ ♥♦♥t♦♠
♠sr s♦ ♣rs♥t s♦♠ ♣r♦♣rts ♦ ts ♦r♠
♥ ♣tr ♥stt t ♣r♦♠ t ②♥♠s ♥ ② rt♦♥ r♥t
♥s♦♥ ♥ ② st ♠♣ ♦ t t②♣ (t, x) → F (t, x). ❲ ♣r♦ s♦♠
♠♣♦rt♥t rsts rt t♦ r♥t ♥s♦♥s ♥ t rt♦♥ ♦♥t①t s
s ①st♥ ♥ ♦♠♣t♥ss ♦ rt♦♥ trt♦rs ♥ ♦r♠t t ♥♦♥s♠♦♦t
♥ssr② ♦♥t♦♥s ♦r t rt♦♥ r♥t♥s♦♥ ♣r♦♠ t stt ♦♥str♥ts
♥r rt♥ ss♠♣t♦♥s
♥ ♣tr ♥tr♦ ♥ ♣r♦r ♥ t rt♦♥ ♦♥t①t t♦ ♣r♦ ♥ssr②
♦♥t♦♥s ♥ t ♦r♠ ♦ P♦♥tr②♥ ♠①♠♠ ♣r♥♣ ♦r t P ♦s ②♥♠s
s ♥ ② t ♠r rt♦♥ rt st t♦ stt ♦♥str♥ts ♥ ♥r
ss♠♣t♦♥s ♦♥ t t ♦ t ♣r♦♠ ♦r♦r ♣rs♥t ♥ ①♠♣ t♦ strt
t t♥ss ♦ ♦r rsts
♥② ♥ ♣tr ♦s ts tss ② ♥ s♠♠r② ♦ t ♣rs♥t ♦♥trt♦♥s
r♦♠ ♦r ♦rs ♥ r♦♠♠♥t♦♥s ♦r tr ♦rs
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♣tr
tt♦trt
♥ ts ♣tr ♥tr♦ r r ♦ ② ♦♥♣ts ♥ rsts ♦ r♦s
♦♠♥s ♦ rt♦♥ s ♦♥♥t♦♥ ♦♣t♠♦♥tr♦ t♦r② ♥ s♦♠ ♠t♦s
t♦ s♦ ♦♣t♠③t♦♥ ♣r♦♠s t s r r♥t ♦r t rsr ♦♣
♥ t tss ♦r♦r ts ♦r s♦ ♠♦st ♣ t♦ ♣♣rt t
♦ ts tss t rs♣t t♦ t stt ♦ t rt
rt♦♥ s
rt♦♥ s s ♦ ♠t♠ts tt s t ♥trs ♥ rts
♦s ♦rr ♠② ♥ rtrr② r ♦r ♦♠♣① ♥♠r ts ♥r③♥ t ♥tr
♦rr r♥tt♦♥ ♥ ♥trt♦♥ s ♠② ♦♥sr ♦ ♥ ②t qt
②♦♥ ♦♥ t s ♥ ♦ t♦♣ s ts ♥♥♥ ♥ tr t♦ ♥③s ttr
t♦ ô♣t ♥ ♥ t ♥♦tt♦♥ ♦r r♥tt♦♥ ♦ ♥♦♥♥tr ♦rr 12 s
sss ♥ t♥ rt♦♥ s s ♥ ♦♣ r② ♥ ♥♦ ♦♥ ♦
t str♦♥② rsr rs ♦ t ♠t♠t ♥②ss s ttst ② t ♥♠r ♦
♣t♦♥s
♠♦tt♦♥ ♦r ts s ♥ t ♥rs♥ r♥ ♦ ♣♣t♦♥s rqr♥ t s ♦
rt♦♥ r♥tt♦♥ ♥ ♥trt♦♥ ♦♣rt♦rs ♥ r♦s s ♥♦t② ♥ ♣r ♥
♣♣ ♠t♠ts ♣②ss ♠ ♦♦ ♣r♦sss ♥♥r♥ ♦♥♦♠s ♥
♦♥tr♦ t♦r② ♠♦♥ ♦trs s r t ❬❪ ♦ ♥ ② ❬❪ ♥r ❬❪
rs t ❬❪ ♥ ♦ss♥ ♥ t♦ ❬❪ s t ♥ ♦♥sr ♥♦
t♦♣ s
r r ♠♥② ♥t♦♥s ♥ sr r♥t ♣♣r♦s ♥ rt♦♥ rts ♥
♥trs s r ♥ ♥rr♦ ❬❪ s t ❬❪ ♥ ♠♦ t ❬❪
r ♥tr♦ r ♦ s♦♠ s♣ ♥t♦♥s tt r s ♥ s ♣ t♦♦s
♥ s♦♠ ♥t♦♥s ♦r rt♦♥ ♥trs ♥ rts ♦ ♦♥sr ♦♥② t
♠♦st s ♦♥s ♦r ♦r ♣r♣♦ss
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♣ ♥t♦♥s
r ♣rs♥t ♥ ♦r ♦ s♦♠ ♥t♦♥s ♦r s♣ ♥t♦♥s tt r s ♦♥
ts tss
❼ ♠♠ ♥t♦♥
♥ ♠♣rt ♥t♦♥ ♦ t rt♦♥ s r♣rs♥ts ♦♥t♥♦s ①t♥s♦♥
♦ t t♦r ♥t♦♥ t s t ♠♠ ♥t♦♥ ♥r③s t t♦r
♥t♦♥ t♦ ♥♦♥♥tr ♥t ♥ ♦♠♣① r♠♥ts s♦ t r
♠♠ ♥t♦♥ Γ(z) s ♥ ②
Γ(z) =
∫ ∞
0e−ttz−1dt,
ts ♥tr ♥ ♦♥r♥t ♦r ♦♠♣① z ∈ C,ℜ(z) > 0.
❼ ttr ♥t♦♥ ♥ ♥r③t♦♥
ttr ♥t♦♥ ♣②s r② ♠♣♦rt♥t r♦ ♥ rt♦♥ s t s
rst ♥tr♦ ♥ ② t s ♠t♠t♥ öst ttr tt
r ❬❪ ♥ s ♥ ②
Eα(z) =∞∑
n=0
zn
Γ[nα+ 1],
r z s ♦♠♣① r α > 0 ♥ Γ(·) s t ♠♠ ♥t♦♥ t s
t ♦♥♣r♠tr ttr ♥t♦♥ s tr s s♦ ttr ♥t♦♥
t t♦ ♣r♠trs ♥ ♦♦♥ ♦r♠
Eα,β(z) =∞∑
n=0
zn
Γ[nα+ β]α, β > 0.
♥ t t t♦♣r♠tr ttr s ♥tr♦ ② r ♥
rs♦♥ ② ♥♦ t t♦♣r♠tr ttr s ♥♦t t r ♥t♦♥
t s♠♣② t ttr ♥t♦♥ s s r t t s♠ ♥♦tt♦♥ s
♦r t ♦♥♣r♠tr ttr ♥t♦♥ s s ❬❪ P♦♥② ❬❪
β = 1, Eα,1 = Eα t ♦r♥ ♦♥♣r♠tr ttr ♥t♦♥
♥r③t♦♥ ♦ t ttr ♥t♦♥ s ♦t♥ ② rt♥ t r♠♥t
♥ t ♦r♠ tα. t s r② ♠♣♦rt♥t t♦ s♦♥ s ♥ ♥ s ♦♦s t
A ∈ Rn×n α > 0, β > 0. ♥ t ♥r③t♦♥ ♦ t t♦♣r♠tr tt
r ♥t♦♥ s
Eα,β(Atα) =
∞∑
n=0
An tnα
Γ[nα+ β].
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β = 1 ♦t♥ t ♥r③t♦♥ ♦ t ♦♥♣r♠tr ttr ♥t♦♥ s
Eα(Atα) =
∞∑
n=0
An tnα
Γ[nα+ 1].
♥r③t♦♥ ♦ t ttr ♥t♦♥ stss s♦♠ ♥trst♥ ♣r♦♣rts
s ♦③②rs ♥ ♦rrs ❬❪ Pr♣t ❬❪
Pr♦♣♦st♦♥ t α > 0 ♥ t ∈ [a, b] ♥ t rt♦♥ rt ♦ t
♥r③ ttr ♥t♦♥ ♦②s
CaD
αt Eα(A(t− a)α) = AEα(A(t− a)α),
r CaD
αt s t ♣t♦ rt♦♥ ♦♣rt♦r s ♥ ♥ t ♥①t
st♦♥
rt♦♥ ♥trs ♥ rts
r r ♠♥② ♥t♦♥s ♥ sr r♥t ♣♣r♦s ♦r rt♦♥ rts ♥
♥trs s ♣♣♥① r ♥tr♦ t rt♦♥ rts ♥ ♥trs
♠♣♦rt♥t ♦r ♦r ♦r s s t ♣t♦ ♠♥♥♦ ♥ ♠r ♦♥s
♥t♦♥ t f(·) ♥ ♥tr ♥t♦♥ ♥ ♥tr [a, b] ♦r t ∈ [a, b] ♥
α > 0 t t ♥ rt ♠♥♥♦ rt♦♥ ♥trs r rs♣t② ♥ ②
aIαt f(t) :=
1
Γ(α)
∫ t
a(t− τ)α−1f(τ)dτ,
tIαb f(t) :=
1
Γ(α)
∫ b
t(τ − t)α−1f(τ)dτ,
r Γ(·) s t r ♠♠ ♥t♦♥
♥t♦♥ t f(·) ♥ s♦t② ♦♥t♥♦s ♥t♦♥ ♥ t ♥tr [a, b] ♦r
α > 0, ♥ t ∈ [a, b], t t ♥ rt ♠♥♥♦ rt♦♥ rts r
rs♣t② ♥ ②
aDαt f(t) :=
dn
dtn
(
aIn−αt f(t)
)
=1
Γ(n− α)
(
d
dt
)n ∫ t
a(t− τ)n−α−1f(τ)dτ,
tDαb f(t) := (− d
dt)n
(
tIn−αb f(t)
)
=1
Γ(n− α)
(
− d
dt
)n∫ b
t(τ − t)n−α−1f(τ)dτ,
r n ∈ N s s tt n− 1 < α ≤ n.
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♥t♦♥ t f(·) ∈ ACn[a, b] ♦r t ∈ [a, b] ♥ α > 0 t t ♥ t rt
♣t♦ rt♦♥ rts r rs♣t② ♥ ②
CaD
αt f(t) := aI
n−αt
dn
dtnf(t) =
1
Γ(n− α)
∫ t
a(t− τ)n−α−1f (n)(τ)dτ,
CtD
αb f(t) := tI
n−αb
(
− d
dt
)n
f(t) =(−1)n
Γ(n− α)
∫ b
t(τ − t)n−α−1f (n)(τ)dτ,
r n ∈ N s s tt n− 1 < α ≤ n
α = n ∈ N0 t♥ t ♣t♦ ♥ ♠♥♥♦ rt♦♥ rts ♦♥ t
t ♦r♥r② rtdnf(t)
dtn ♦r s♦♠ ♣r♦♣rts ♥ rt♦♥s t♥ t ♣t♦ ♥
♠♥♥♦ rts s ♣♣♥①
Ps ♠ s ♦ r♥t t②♣s ♦ rt♦♥ rts ♠♦st ♣♦♣r ♠♦♥
t♠ r t ♣t♦ ♥ ♠♥♥♦ rt♦♥ rts t ♦t s♦♠
s♥ts ♦r ①♠♣ t ♣t♦ rt♦♥ rt ♦s ♥♦t ♣♣② ♥
t ♥t♦♥s r ♥♦t r♥t t ❬❪ ♥ t ♠♥♥♦ rt♦♥
rt ♦ ♦♥st♥t s ♥♦t q ③r♦
♦ ♦r♦♠ ts ss tr r rsts ♣r♦♣♦s ② ♠r ♥♦ st②
♠♦ ♥t♦♥ ♦ t rt♦♥ rt ♦ t ♠♥♥♦ rt ♠r
❬❪ ♥ ♦rr t♦ ♠♥t t s♥ts ♦ t ♠♥♥♦ ♥ ♣t♦
rt♦♥ rts ♠r rt♦♥ rt ♦ ♦♥st♥t s q t♦ ③r♦ ♥
t s ♥ ♦r ♦♥t♥♦s ♥♦t ♥ssr② r♥t ♥t♦♥
♥t♦♥ t f : [a, b] → R ♦♥t♥♦s ♥t♦♥ 0 < α < 1 ♥ t ∈ [a, b].
♥ t t ♥ rt ♠r rt♦♥ rts f(α)L (t) ♥ f
(α)R (t) r rs♣t②
♥ ②
f(α)L (t) := (aD
αt [f(·)− f(a)])(t)
=1
Γ(1− α)
d
dt
∫ t
a
f(τ)− f(a)
(t− τ)αdτ,
f(α)R (t) := (tD
αb [f(b)− f(·)])(t)
= − 1
Γ(1− α)
d
dt
∫ b
t
f(b)− f(τ)
(τ − t)αdτ.
α ≥ 1, ♥
f (α)(t) :=(
f (α−n)(t))(n)
, n ≤ α < n+ 1, n ≥ 1.
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♦r t s♣ s 0 < α < 1
f (α)(t) = (f (α−1)(t))′.
♠r f(a) = 0, t♥ t t ♠r rt♦♥ rt ♦♥s t t
♠♥♥♦ rt♦♥ rt
♥t♦♥ t f : [a, b] → R ♦♥t♥♦s ♥t♦♥ ♥ α > 0 r ♥♠r ♥
t ♥tr [a, b] ♠r rt♦♥ ♥tr s ♥ ②
aJαt f(t) =
1
Γ(α+ 1)
∫ t
af(τ)(dτ)α, 0 < α ≤ 1.
♦r♦r ♠r ♥tr♦ rt♦♥ ♥tr ♥♦tt♦♥ (dt)α ♥ ②
∫ t
af(τ)(dτ)α = α
∫ t
a(t− τ)α−1f(τ)dτ.
♠♣♦rt♥t ♣r♦♣rts ♦ t ♠r rt♦♥ rt ♥ ♥tr r s♦♥ ♥
♣♣♥①
♥t♦♥ t n − 1 < α ≤ n. ♥t♦♥ f(·) s s t♦ ♥ α−s♦t②
♦♥t♥♦s stss
f(t) =
n−1∑
k=0
f (k)(a)
Γ(k + 1)(t− a)k + aI
αt g(t), t ∈ [a, b],
r g(t) = aDαt f(t), t ∈ [a, b].
♦r ♠♦r ts s r ❬❪ ♦r s t ❬❪
rt♦♥ ♥trt♦♥ ② ♣rts
♥trt♦♥ ② ♣rts ♣②s ♥ ♠♣♦rt♥t r♦ ♥ r♥ t ♥r③ rr♥
qt♦♥s ♦r rt♦♥ rt♦♥ ♣r♦♠s ♥ ♥ ♣r♦♥ ♥ssr② ♦♣t♠t② ♦♥
t♦♥s ♦r rt♦♥ ♦♣t♠♦♥tr♦ ♣r♦♠s s r ❬❪ s t ❬❪ ♥
♠♦ t ❬❪
❼ t ϕ(t) ∈ Lp([a, b]) ψ(t) ∈ Lq([a, b]) 1 ≤ p ≤ ∞ 1 ≤ q ≤ ∞ ♥ 1p +
1q ≤ 1+α
ss♠ tt p > 1 ♥ q > 1, ♥ 1p + 1
q = 1 + α) t♥
∫ b
aϕ(t)(aI
αt ψ)(t)dt =
∫ b
aψ(t)(tI
αb ϕ)(t)dt.
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❼ t f(t) ∈ aIαt (L
p) g(t) ∈ tIαb (L
q) r aIαt (L
p) tIαb (L
q) ♥♦t t r♥s ♦ t
♦♣rt♦rs aIαt tI
αb ♦♥ Lp Lq rs♣t② ♥
∫ b
af(t)(aD
αt g)(t)dt =
∫ b
ag(t)(tD
αb f)(t)dt,
♥ aIαt (L
p) tIαb (L
q) ♦r ♥② 1 ≤ p ≤ ∞ 1 ≤ q ≤ ∞ α > 0 r ♥ ②
aIαt (L
p) := f : f = aIαt ϕ, ϕ ∈ Lp(a, b),
tIαb (L
q) := g : g = tIαb ψ, ψ ∈ Lq(a, b).
r s ♦r♠ ♦r rt♦♥ ♥trt♦♥ ② ♣rts ♦r t ♣t♦ rt♦♥ rts
s tt
∫ b
af(t)(CaD
αt g)(t)dt =
∫ b
ag(t)(tD
αb f)(t)dt+
n−1∑
i=0
(tDα+i−nb f(t))g(n−1−i)(t) |t=b
t=a,
∫ b
af(t)(CtD
αb g)(t)dt =
∫ b
ag(t)(aD
αt f)(t)dt+
n−1∑
i=0
(−1)n+i(aDα+i−nt f(t))g(n−1−i)(t) |t=b
t=a,
♥ ♥ ♣rtr ♦r α ∈ (0, 1)
∫ b
af(t)(CaD
αt g)(t)dt =
∫ b
ag(t)(tD
αb f)(t)dt+ (tI
1−αb f(t))g(t) |t=b
t=a,
∫ b
af(t)(CtD
αb g)(t)dt =
∫ b
ag(t)(aD
αt f)(t)dt− (aI
1−αt f(t))g(t) |t=b
t=a .
♥ t ♣r♦s qt♦♥s ♥ α → 1, t ss ♦r♠ ♦ ♥trt♦♥ ②
♣rts ∫ ba f(t)g
′(t)dt = f(t)g(t) |t=bt=a −
∫ ba g(t)f
′(t)dt) s CaD
αt = d
dt CtD
αb = − d
dt
aDαt = d
dt tDαb = − d
dt ♥ aI1−αt tI
1−αb r t ♥tt② ♦♣rt♦rs
rtr♠♦r ♠r ♥tr♦ ♦r♠ ♦ ♥trt♦♥ ② ♣rts s ♦♦s
∫ b
af (α)(t)g(t)(dt)α =
∫ b
a(f(t)g(t))(α)(dt)α −
∫ b
af(t)g(α)(t)(dt)α
= Γ(α+ 1)[f(t)g(t)]ba −∫ b
af(t)g(α)(t)(dt)α.
r ♦♥ ♦♥♥t♦♥ ♣t♠ ♦♥tr♦ ♦r②
ss Ps rs ♥tr② ♥ ♠♥② r♦s s ♥ ♥ sss ♦r ♦♥
t♠ tr♦r ♦t ♦ ♦r ①sts ♥ t r ♦ ♦♣t♠♦♥tr♦ ♦ ♥tr♦rr ②♥♠
s②st♠s ♥ ♥♥r♥ s♥ ♦♥♦♠s ♥ ♠♥② ♦tr s s t s ♥♦t sr♣rs♥
tt rst② ♦ Ps ♥ ♦♥sr r♥ ♦ sss ♥ ♣r♦♠s
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♥ ♠t♣ t②♣s ♦ t ♦♥sr ♦♥tr♦ rt♦♥s r rt t♦ t
r♦s t②♣s ♦ ♠♥♠♠ t②♣s ♦ ♦♥str♥ts stt ♦♥tr♦ ♠① s♦♣r♠tr
♥♣♦♥t ♥ ♥tr♠t stt ♦♥str♥ts ♥t ♦r ♥♥t t♠♦r③♦♥s sts
♦ ss♠♣t♦♥s t♦ ♦ sr t②♣s ♦ ♥♦♥♥rs ♦ t ♦♥t♦♥s s♥stt②
rsts r♦st♥ss t♦ ♣rtrt♦♥s ♥ t♦ ♥♥♦♥ ♠♦ ♣r♠trs r♥t
t②♣s ♦ ♠t♣rs ♥ ① ♦♥tr♦ ♠srs ♠♣s ♦♥tr♦ ♠♦♥ ♦trs s
rt②♥♦ ❬❪ rt②♥♦ t ❬❪ rt②♥♦ ♥ Prr ❬❪ rt②♥♦ t
❬❪ rt②♥♦ t ❬❪ r②s♦♥ ❬❪ r ❬❪ r t ❬❪ r ♥ Prr
❬❪ ♠r③ ❬❪ ♦ts ♥ ②t♥ ❬❪ r♦r② ♥ ♥ ❬❪ st♥s ❬❪
r♠③♥ t ❬ ❪ stt ❬❪ Prr ♥ ❬❪ P♦♥tr②♥ t
❬❪ ❱♥tr ♥ Prr ❬❪ Prr t ❬❪ ♥ ❱♥tr ❬❪ ♥ ♥r t
s♣t♦♥ ♦ ♥ P rqrs t ♦♦♥ t♠s stt ♥ ♦♥tr♦ s♣s
♣r♦r♠♥ ♥① ♦r ♦st ♥t♦♥ s② ♣♥♥ ♦♥ t stt ♥ ♦♥tr♦ rs
s② ♥♦t ② x(·) ♥ u(·) rs♣t② t♦ ♠♥♠③ ♦r t st ♦ ♠ss
♦♥tr♦ ♣r♦sss ②♥♠ ♦♥str♥ts tt sts t rt♦♥ t♥ t stt
trt♦r② t t ♦♥tr♦ ♥♣t ♥ ♥ ♥♣♦♥t stt r ♥ ♣♦ss②
♦♥ ♦r ♠♦r t②♣s ♦ ♦♥str♥ts s s ♣♦♥ts ♦♥tr♦ ♥♦r stt ♦♥str♥ts ♥
♦♥t ♦♥tr♦ ♥ stt ♦♥str♥ts ♦ ② ♦♥tr♦ ♣r♦ss ♠♥ ♥② ♣r (x;u) tt
stss t ②♥♠ ♦♥str♥ts ♦♥tr♦ ♣r♦ss s s ♦r t stss t
♦♥str♥ts ♥ t s ♦♣t♠ ♣r♦s ♦st ♦r t♥ tt ss♦t t ♥② ♦tr
s ♦♥tr♦ ♣r♦ss
♠♥ ♦t ♦ Ps s t♦ tr♠♥ t ♦♣♥ ♦r ♦s♦♦♣ ♦♥tr♦ strt②
tt ♦♣t♠③s ♠♥♠③s ♦r ♠①♠③s ♥ ♦♣t♠t② rtr♦♥ ♦r ♣r♦r♠♥ ♥①
s② ♥♦t ② J(·) ♣r♦r♠♥ ♥① ♠② r② ♥r t ♠② s♠♣②
♥t♦♥ ♣♥♥ ♦♥ t stt ♥♦r t♠ ♦r rt♠ ♣r♦♠s ♥♣♦♥ts ♦r s♦
♥♦ ♥ ♥tr ♦s ♥tr♥ ♠② ♥t♦♥ ♦ t s ♦ ♦t t stt ♥
t ♦♥tr♦ rs
♠r tt s♥ minJ(·) = −max−J(·) t s ♥r♥t t♦ ♦♥sr tr
♠①♠③t♦♥ ♠♦r ♦t♥ s ♥ ♦♥♦♠s ♦r ♠♥♠③t♦♥ ♠♦r ♦t♥ s ♥
♥♥r♥ ❲ ♦♣t t ttr
t J [x, u] ♣r♦r♠♥ ♥① x ∈ X st ♦ stt rs t♥ s ♥ Rn
♦♥tr♦ r u s ♦r ♠sr ♥t♦♥ U ♦s st ♦ ♦♥tr♦ rs s
tt u ∈ U t♥ s ♦♥ s♦♠ ♦s st Ω ⊂ Rm t ∈ [a, b] ♥ L(·) f(·) ♥ g(·)
♦♥t♥♦s② r♥t ♥t♦♥s ♥ tr ♦r♠t♦♥s ♠r tt f(t, x,Ω) s
♥ ♦♣♥ st ♦r (t, x), s ♦ rt♦♥s ♣r♦♠ ♥ t ♦♥r♥s t
♦st ♥t♦♥ tr r tr ♠♥ t②♣s ♦ Ps
♦③ ♦r♠t♦♥
J [x, u] = g(a, x(a), b, x(b)) +
∫ b
aL(t, x(t), u(t))dt,
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r♥ ♦r♠t♦♥
J [x, u] =
∫ b
aL(t, x(t), u(t))dt,
②r ♦r♠t♦♥
J [x, u] = g(a, x(a), b, x(b)).
t t♦
x(t) = f(t, x(t), u(t)), x(a) = x0.
♦r♠ ♦rs t ❬❪ r r ♦♥t♦♥s ♥r ts tr
♦♣t♠♦♥tr♦ ♣r♦♠s r q♥t
ssr② ♦♣t♠t② ♦♥t♦♥s
♦♥sr t s♠♣st ♥♦♥♥r P tt ♥ ♦r♠t s ♦♦s
Minimize J [x, u] =
∫ b
aL(t, x(t), u(t))dt,
subject to x(t) = f(t, x(t), u(t)),
x(a) = x0.
♥ ts P x(t) ∈ Rn s t stt r u ∈ U s t ♦♥tr♦ r L : [a, b] ×
Rn ×R
m → R s t ♥tr♥ f : [a, b]×Rn ×R
m → Rn s ♥t♦♥ ♥♥ t
②♥♠s ♦ t s②st♠ ♦r t s r t rst♦rr rt ♦ t stt r x s
♦♥sr ♥ t ♣♦♥ts a ♥ b r t ♥t ♥ ♥ t♠ ♣♦♥ts rs♣t②
♣r (x;u) s s♥t s ♦♥tr♦ ♣r♦ss ②♣♦tss t♦ sts ② L(·) ♥f(·) t♦ s tt ♥♠r ♦ ♣r♣♦ss ♥ ♦♥sr ♥ t ♥stt♦♥ ♦
t P s♠♣st sst ♦ ♣r♣♦ss ♦♥r♥s t ♣r♦♣r ♥t♦♥ ♦ t ♣r♦♠
①st♥ ♥ ♥q♥ss ♦ t s♦t♦♥ ♦ t r♥t qt♦♥ ♦r ♥ ♥t
stt ♥ ♦♥tr♦ ♥t♦♥ ♥ t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② r ♥♦r♠t
♥ t s♥s tt t② ♥ t sss rt♦♥ ♦ t ♥♠r ♦ ♥ts t♦ t
s♦t♦♥ ♦ t ♣r♦♠
❲♥ Ω = Rm r Ω s t st ♦ s t♥ ② t ♦♥tr♦ ♥t♦♥ t ♥ssr②
♦♥t♦♥s ♦ ♦♣t♠t② ♥ rtt♥ ♦♥ ② s♥ r♥ ♠t♣r s ♦♦s
x(t) =∂H
∂λ(t, x(t), u(t), λ(t)),
λ(t) = −∂H∂x
(t, x(t), u(t), λ(t)),
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0 =∂H
∂u(t, x(t), u(t), λ(t)),
r λ(·) s r♥ ♠t♣r ♥ H(t, x, u, λ) s t P♦♥tr②♥ ♥t♦♥ ♥ ②
H(t, x, u, λ) = L(t, x, u) + λT f(t, x, u).
P♦♥tr②♥ ♠①♠♠ ♣r♥♣
♠①♠♠ ♣r♥♣ ♦ ♦♣t♠♦♥tr♦ ♣r♦s ♥ssr② ♦♥t♦♥s t♦ sts ②
t ♦♣t♠♦♥tr♦ ♣r♦ss t ♣r (x;u) t s ♥ ♥ ♠♣♦rt♥t t♦♦ ♥ t ♠♥②
rs ♥ ♦♣t♠♦♥tr♦ ♣②s r♦ ♥♦♥ P♦♥tr②♥ ①♠♠ Pr♥♣
s ♦♣ ♥ t ♠ s ♥ t ♦t ❯♥♦♥ ② t ss♥ ♠t♠t♥
♠♥♦ P♦♥tr②♥ ♥ s ♦s ❬❪
P♦♥tr②♥ ♠①♠♠ ♣r♥♣ s stt s ♦♦s t (x∗;u∗) ♥ ♦♣t♠ s♦t♦♥
♦ t ♦♥tr♦ ♣r♦♠
Minimize J(x, u) =
∫ b
aL(t, x(t), u(t))dt,
subject to x(t) = f(t, x(t), u(t)),
r x(a) = x0 u ∈ U ⊂ Rm, ♥ t ∈ [a, b]. P♦♥tr②♥ ♥t♦♥ H(t, x, u, λ) s
♥ s ♦♦s
H(t, x, u, λ) = L(t, x, u) + λT f(t, x, u),
r t ♥t♦♥s f : [a, b] × Rn × R
m → Rn ♥ L : [a, b] × R
n × Rm → R r
rs♣t② t stt r ②♥♠s ♥ t ♥tr♥ ♦ t ♦st ♥t♦♥ ♦t
r r♥t t rs♣t t♦ rt x s♠sr rt t ♥ ♦r
♠sr rt u. P♦♥tr②♥s ♠①♠♠ ♣r♥♣ stts tt t ♦♣t♠ ♥♣t u∗(·)♠①♠③s H(t, x(t), u(t), λ(t)) ♠♦♥ ♠ss ♥♣ts u(·), tt s
H(t, x∗(t), u(t), λ∗(t)) ≤ H(t, x∗(t), u∗(t), λ∗(t)),
♦r ♠♦st t ∈ [a, b] r u∗(·) ∈ U s t ♦♣t♠♦♥tr♦ ♦r t ♣r♦♠
λ∗(·) s t ♦♣t♠ ♦stt trt♦r② ♥ x∗(·) s t ♦♣t♠ stt trt♦r② sts②♥
rs♣t② t ♦♥t qt♦♥ λ∗(t) = − ∂∂xH(t, x∗(t), u∗(t), λ∗(t)) ♥ t stt
qt♦♥s x∗(t) = ∂∂λH(t, x∗(t), u∗(t), λ∗(t)) t t ♦♥r② ♦♥t♦♥s x∗(a) = x0 ♥
λ∗(b) = 0.
❲♥ t ♥ t♠ s ① ♥ t P♦♥tr②♥ ♥t♦♥ ♦s ♥♦t ♣♥ ①♣t② ♦♥
t♠ t♥
H(x∗(t), u∗(t), λ∗(t)) = constant,
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♥ t ♥ t♠ s r t♥
H(x∗(t), u∗(t), λ∗(t)) = 0.
♣t♠ ♦♥tr♦ ♣r♦♠s t ♦♥str♥ts
♦♥str♥ts ♣♣r ♥ r♥t ②s ♥ Ps s ♦♥str♥ts rstrt t r♥ ♦
s ♦ ♦t t stt ♥ t ♦♥tr♦ rs
t ♦♥str♥ts r ♠♣♦s ♦♥ t ♦♥tr♦ u(·) ♦ t Ps t② r ♦♥tr♦
♦♥str♥ts
t ♦♥str♥ts r ♠♣♦s ♦♥② ♦♥ t stt trt♦rs ♦ t Ps x(·), t② r
♣r stt ♦♥str♥ts
♥② t ♦♥str♥ts r ♠♣♦s ♦♥ ♦t t stt ♥ t ♦♥tr♦ rs t② r
♠① stt ♦♥str♥ts
t s s♦ s♦♠ t②♣s ♦ t ♦r♠♥t♦♥ ♦♥str♥ts ♠♣♦s ♥ t Ps
♦♥tr♦ ♦♥str♥ts
♦♥tr♦ u ∈ U s ♦♥tr♦ ♦♥str♥t r u(t) ts s ♥ ♦s st
Ω(t) ♦r ♠♦st r② t ∈ [a, b] ♥ U : [a, b] → Ω(t) ⊂ Rm s ♠t♥t♦♥ t♥
♦♥ ♦s s
♥♣♦♥t ♦♥str♥ts
♥♣♦♥t ♦♥str♥ts ♥ ♠♣♦s t t ♥t ♥ ♦r tr♠♥ ♣♦♥ts ♦
① t♠ ♥tr [a, b] ♥ t ♠♦st ♥r ② t♦ rt t♠ s
(x(a), x(b)) ∈ C,
r C s ♦s st
tt ♦♥str♥ts
♥s♦♥ stt ♦♥str♥ts
t X : [a, b] → Rn ♠t♥t♦♥ t♥ ♦♥ ♦s s ♥ t
♥s♦♥ stt ♦♥str♥t s ♥ ②
x(t) ∈ X(t), ∀t ∈ [a, b].
♥qt② stt ♦♥str♥ts
t h : [a, b]×Rn → R ♥ ♥t♦♥ ♥ ♥ ♥qt② stt ♦♥str♥t
s ♥ ②
h(t, x(t)) ≤ 0, ∀t ∈ [a, b].
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qt② stt ♦♥str♥ts
t h : [a, b]× Rn → R ♥ ♥t♦♥ ♥ ♥ qt② stt ♦♥str♥t
s ♥ ②
h(t, x(t)) = 0, ∀t ∈ [a, b].
① stt ♦♥str♥ts
♥qt② ♠① stt ♦♥str♥ts
t h : [a, b] × Rn × R
k → Rm ♥ ♥t♦♥ ♥ ♥ ♥qt② ♠①
stt ♦♥str♥t s ♥ ②
h(t, x(t), u(t)) ≤ 0, t ∈ [a, b].
qt② ♠① stt ♦♥str♥ts
t h : [a, b] × Rn × R
k → Rm ♥ ♥t♦♥ ♥ ♥ qt② ♠①
stt ♦♥str♥t s ♥ ②
h(t, x(t), u(t)) = 0, t ∈ [a, b].
①♠♠ ♣r♥♣
r ♥ ♦r t ♥♦♥s♠♦♦t ♠①♠♠ ♣r♥♣ ♦r Ps t ♥ t♦t
stt ♦♥str♥ts ♥ t s r♥s r ♥r③ t ♦♥① sr♥ts
♦ ♦r t♦ ♦r ♣st③♦♥t♥♦s ♥t♦♥s ♥ t♦ s♦♠ ①t♥t ♦r s♠
♦♥t♥♦s ♥t♦♥s r ❬❪ ♣♣ ♥♦♥s♠♦♦t ♥②ss t♦ ♦♣t♠③t♦♥
♥ ♦♣t♠♦♥tr♦ t♦r② s♦ ♥ t s ♦r♦ ♣r♦♣♦s t ♦
♠t♥ sr♥ts ♥ ♠♦♥strt ♦ tr♥srst② ♦♥t♦♥s ♥ t ♥♦♥s♠♦♦t
♠①♠♠ ♣r♥♣ ♦ ♠♣r♦ ts ♠♥ t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t②
♠♦r ♣rs
❲ ♦♥sr t ♥♦♥s♠♦♦t P t stt ♦♥str♥ts s t ♦♦♥
(P )
Minimize g(x(a), x(b))
subject to x(t) = f(t, x(t), u(t)), t ∈ [a, b],
h(t, x(t)) ≤ 0, ∀t ∈ [a, b],
u ∈ U , t ∈ [a, b],
(x(a), x(b)) ∈ C.
♣r♦♠ (P ) stss t ♦♦♥ ②♣♦tss
t ♥t♦♥ (t, u) → f(t, x.u) s L × B−♠sr
t ♥t♦♥ f(t, ., u) s ♣st③ t ♥t♦♥ Kf ♥ L1 ♦r (t, u) ∈ (t,Ω(t)) :t ∈ [a, b];
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t r♣ ♦ Ω(t) s L×B−♠sr r Gr(Ω) s t r♣ ♦ t ♠t♥t♦♥
U : [a, b] → Rm ♥ ②
Gr(Ω) := (t, u) ∈ [a, b]× Rm : u ∈ Ω(t) ;
t ♥t♦♥ g s ♣st③ ♦ r♥ Kg;
t ♥t♦♥ h s ♣♣r s♠♦♥t♥♦s ♥ ♦r t ∈ [a, b] t ♥t♦♥ h(t, ·) s
♣st③ t ♦♥st♥t kh.
P♦♥tr②♥ ♥t♦♥ H : [a, b]× Rn × R
n × Rm → R s ♥ ②
H(t, x, p, u) = 〈p, f(t, x, u)〉 .
♦r♠ ♦♥s♠♦♦t ①♠♠ Pr♥♣ ♦r ♣t♠ ♦♥tr♦ Pr♦♠ (P )
t tt ♦♥str♥ts ♦r♠ r ❬❪
t (x∗, u∗) str♦♥ ♦ ♠♥♠③r ♦r t ♣r♦♠ (P ) ♥ ss♠ t ♣r♦s
②♣♦tss r sts ♥ tr ①st ♥ arc p, sr λ ≥ 0, ♥♦♥
♥t ♦♥ ♠sr µ(·) ♦♥ [a, b], ♥ ♠sr ♥t♦♥ γ(·) s tt t ♦♦♥
①♣rss♦♥s r sts
♦♥trt② ♦♥t♦♥
‖p‖+ ‖µ‖+ λ > 0,
t ♦♥t qt♦♥
−p(t) ∈ ∂xH(t, x∗(t), q(t), u∗(t), λ),
t ♠①♠♠ ♦♥t♦♥
H(t, x∗(t), q(t), u∗(t), λ) = max H(t, x∗(t), q(t), w, λ) : w ∈ U(t) ,
t tr♥srst② ♦♥t♦♥
(p(a),−q(b)) ∈ λ∂g(x∗(a), x∗(b)) +NC(x∗(a), x∗(b)),
γ(t) ∈ ∂>x h(t, x∗(t)) ♥ µ s s♣♣♦rt ♦♥ t st
t : h(t, x∗(t)) = 0.
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r q(·) rrr t♦ ♥ ♦♥t♦♥s ♥ s
q(t) =
p(t) +∫
[a,t) γ(s)µ(ds), t ∈ [a, b),
p(t) +∫
[a,b] γ(s)µ(ds), t = b,
♥ ∂>x (·) s rt♥ ♥r③ r♥t tt ts s s rt♥ ssts ♦ t
♥♦♥ rs ♥r③ r♥t ♥ ②
∂>x h(t, x) := coγ = limi→∞
γi : γi ∈ ∂xh(ti, xi), (ti, xi) → (t, x), h(ti, xi) > 0 ∀i.
s t s r r♦♠ ts rst tr r ♥♠r ♦ ♦tr ♦ts ♥r♥t t♦ t ♥r③
♥♦♥s♠♦♦t s ♦s ♥rst♥♥ rqrs ♥ ♦r ♦ sr s ♦♥♣ts
❲ ♣r♦ ♥ ♦r ♦ ts ♥ t ♦♦s ♥①t
♥t♦♥ r ❬❪ ♥ ❩♥ ❬❪ t ♥t♦♥ f : X → R s ♣st③
♥r ♥ ♣♦♥t x. rs ♥r③ rt♦♥ rt ♦ f(·) t t ♣♦♥t
x ∈ X ♥ t rt♦♥ d ♥ ②
f(x; d) := limλ→0
supy→x
f(y + λd)− f(y)
λ,
r y s t♦r ♥ X ♥ λ s ♣♦st sr
rs sr♥t ♦r ♥r③ r♥t ♦ t ♥t♦♥ f(·) t t ♣♦♥t x ∈ X,
♥♦t ② ∂f(x) s sst ♦ X∗
∂f(x) := x∗ ∈ X∗ : f(x; d) ≥ 〈x∗, d〉, ∀d ∈ X,
r X∗ s s♣ ♦ X.
♠r t stt ♦♥str♥ts r s♥t t♥ tr γ(·) = 0 ♦r t ♠sr
dµ(·) = 0 ♥ t ♦♥t♦♥s ♦♠ s♠♣r
t s s② t♦ s tt ♥r s♦♠ r♠st♥s ts ♦♥t♦♥s ♠② ♥rt ♦r
①♠♣ C = x0 × Rn h s s♠♦♦t t h(t, x0) = 0 ♥ h(t, x(t)) < 0 ♦r
s x(·) t t > t0 t♥ s ♠♠t t♦ s tt t ♠t♣r λ = 0 γ(0) =
∇h(t0, x0) dµ = δt0(·) ♥ p(0) = −∇h(t0, x0) stss t ♦♥t♦♥ ♦ t ♠①♠♠
♣r♥♣ ♦ P♦♥tr②♥ ♥ ♣rtr s ♥♦♥tr ♥ t t s♠ t♠ ♦s ♥♦t
♥② ♥♦r♠t♦♥ t♦ st t ①tr♠s ♦ t P
s ♣r♦♠ s ♥ rss ② sr t♦rs t r♦s ♣♣r♦s ♥ t
♠♦r s♥♥t ♦♥s ♦♥ ♥ s rt②♥♦ t ❬❪ ♥ ♦ts ♥
②t♥ ❬❪
r r t♦ ② ♣♣r♦s t♦ rss ts ♥ tr t♦ ♠♣♦s t♦♥
♦♥t♦♥s ♦♥ t t ♦ t ♣r♦♠ ♥ s ❱♥tr ❬❪ ♥ rt②♥♦ t
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❬❪ ♦r rs♦rt t♦ r ♦rr ♥♦r♠t♦♥ ♥ ♦rr t♦ ♠ sr tt t ♠t♣r
s♣ ② t ♦♥t♦♥s ♦ t ♠①♠♠ ♣r♥♣ ♦s ♥♦t ② ♥♦♥♥♦r♠t
♠t♣rs ♥ ♦r♦ ❬❪
r ♦ s♦♠ t♦s t♦ ♦ ♣t♠③t♦♥ Pr♦
♠s t rt♦♥ r♥t qt♦♥s
♥ ts st♦♥ r s♦♠ r♥t ♣♣rs ♥ s♦♠ ♦♣t♠③t♦♥ ♣r♦♠s t
rt♦♥ r♥t ♦♥str♥ts P ♦ Ps r s♣ s r
s♦
❲ ♥tr♦rr Ps ♥ sss ♦r ♦♥ t♠ ♥ r ♦② ♦ t♦r②
♥ ♥♠r t♥qs s ♥ ♦♣ t♦ s♦ t♠ t Ps ♦♥sttt ♥
r t ♠t ♥♠r ♦ ♣t♦♥s ♥ ♠♥② ♦♣♥ sss ♥r ♦r♠t♦♥
♥ s♦t♦♥ s♠ ♦r P r rst ♥tr♦ ② r ❬❪ r t P
♦r♠t♦♥ s ①♣rss s♥ t rt♦♥ rt♦♥ ♣r♥♣ ♥ t r♥
♠t♣r t♥q ♥ t rt♦♥ ②♥♠s ♦ t P s ♥ ♥ tr♠s ♦ t
♠♥♥♦ rt♦♥ rts ♦♥sr t stt ♥ t ♦♥tr♦ rs
s ♥r ♦♠♥t♦♥s ♦ tst ♥t♦♥s ♥ t t t ♥r qrt ♦♣t♠♦♥tr♦
♣r♦♠ s ♦♦s t t ♥t♦♥s q(t) ≥ 0 ♥ r(t) > 0 t♥
(P )Minimize J(u) =1
2
∫ 1
0[q(t)x2(t) + r(t)u2(t)]dt,
subject to 0Dαt x(t) = a(t)x(t) + b(t)u(t),
r x(0) = x0 ♥ α ∈ (0, 1). ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♦r P ♥ t
rr♥ ♦r♠ t♦ s②st♠ ♦ qt♦♥s ♦♠♣♦s ② ♥
0 = r(t)u(t) + b(t)λ(t),
tDα1 λ(t) = q(t)x(t) + a(t)λ(t).
♦♥tr♦ r u(t) s ♦t♥ ② s♥ s ♥t♦♥ ♦ t ♥♥♦♥ ♦stt
r λ(·) ♥ ♦rr t♦ tr♠♥ ♦t t stt ♥ ♦stt ♥t♦♥s ts qt♦♥
t♦tr t qt♦♥ ②s
0Dαt x(t) = a(t)x(t)− r−1(t)b2(t)λ(t).
s t ♦♥tr♦ ♥t♦♥ s tr♠♥ ② ♦♥t② s♦♥ ♥ st t♦ t
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tr♠♥ ♦♥t♦♥s x(0) = x0 λ(1) = 0 r s ♥ ♣♣r♦①♠t ♥♠r ♠t♦
t♦ ♥ x(t) ♥ λ(t) ② s♥ t st ♥r ♣♦②♥♦♠s
Pj(t) = (−1)jj
∑
k=0
(
j
k
)(
j + k
k
)
(−t)k,
tt sts② t ♦♦♥ ♦rt♦♥♦r♠t② ♦♥t♦♥s
∫ 1
0Pj(t)Pk(t)dt = δjk =
0, j = k,
1, j 6= k,
r δjk s t r♦♥r t ♥t♦♥ tr s♦♠ t♦♥s ♥ s♠♣t♦♥s
♦t♥ t ♦♦♥ s②st♠ ♦ 2m+ 2 qt♦♥s ♥ 2m+ 2 ♥♥♦♥s
0 =m∑
k=1
[F1(j, k)− F2(j, k)]ck +m∑
k=1
F3(j, k)dk + Pj(0)µ1,
0 = −m∑
k=1
[F0x(j, k)ck +
m∑
k=1
[F4(j, k)− F2(j, k)]dk + Pj(1)µ2,
x0 =
m∑
k=1
Pk(0)ck,
0 =
m∑
k=1
Pk(1)dk,
r µ1 ♥ µ2 r t r♥ ♠t♣rs ss♦t t t tr♠♥ ♦♥t♦♥s ♥
F0x(j, k) ♥ F1(j, k) t♦ F4(j, k) r ♥ s
F0x(j, k) =
∫ 1
0q(t)Pj(t)Pk(t)dt,
F1(j, k) =
∫ 1
0Pj(t)0D
αt Pk(t)dt,
F2(j, k) =
∫ 1
0a(t)Pj(t)Pk(t)dt,
F3(j, k) =
∫ 1
0r−1(t)b2(t)Pj(t)Pk(t)dt,
F4(j, k) =
∫ 1
0Pj(t)tD
α1Pk(t)dt.
♥ ♣♣r♦①♠t s♦t♦♥ t♦ ts ♣r♦♠ s ♦t♥ ② ♥r ♦♠♥t♦♥s ♦ t st
♥r ♣♦②♥♦♠s
rt ♥♠r t♥q t♦ s♦ P s s ② r ♥ ♥ ❬❪ r
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t t♦rs ♦♥sr ♠t♦♥♥ ♦r♠t♦♥ ② ♦♥sr t ♦♦♥ P ♥
t ♦♣t♠♦♥tr♦ u(·) tt ♠♥♠③s t ♣r♦r♠♥ ♥①
J(u) =
∫ 1
0f(x(t), u(t), t)dt,
st t♦ t s②st♠ ②♥♠ ♦♥str♥ts
0Dαt x(t) = g(x(t), u(t), t),
t t ♥t ♦♥t♦♥ x(0) = x0 r x(t) s t stt r f(·) ♥ g(·) r t♦♥ ♥t♦♥s ♥ 0 < α < 1 t λ(·) t r♥ ♠t♣r ♠t♦♥♥ ♦
t s②st♠ s ♥ ②
H(x(t), u(t), λ(t), t) = f(x(t), u(t), t) + λ(t)g(x(t), u(t), t).
♥ t ♥ssr② ♦♥t♦♥s ♥ tr♠s ♦ ♠t♦♥♥ ♦r t P r ♥ ②
tDα1 λ(t) =
∂H
∂x,
0 =∂H
∂u,
0Dαt x(t) =
∂H
∂λ,
st t♦ t ♥♣♦♥t ♦♥t♦♥s x(0) = x0 ♥ λ(1) = 0 t♦rs ♦s ♦♥ t
♣r♦♠s t qrt ♣r♦r♠♥ ♥① ♥ qt♦♥s t♦ ♥ t② s
rt ♥♠r ♠t♦ t♦ ♦♠♣t x(·) ♥ λ(·) ② ♦♣t♥ t rü♥t♥♦
♥t♦♥ ♥ ts ♠t♦ t ♥tr t♠ ♦♠♥ s ♦r♥③ ♥t♦ N q ♦♠♥s
② 0, 1, . . . , N ♥ t t♠ t ♥♦ j s ♥ ② tj = jh r h = 1N ②
s♥ t rü♥t♥♦ ♦♥♣t qt♦♥ t ♥♦ i ♥ ♣♣r♦①♠t s
s s t ❬❪ ♥ P♦♥② ❬❪
1
hα
i∑
j=0
w(α)j xi−j = a(ih)xi − r−1(ih)b2(ih)λi, i = 1, . . . , N,
r xi ♥ λi r t ♥♠r ♣♣r♦①♠t♦♥s ♦ x(·) ♥ λ(·) t ♥♦ i ♥ w(α)j
j = 0, . . . , i, r t ♦♥ts ♥ ②
w(α)j = (−1)j
(
α
j
)
.
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♠r② qt♦♥ t ♥♦ i ♥ ♣♣r♦①♠t s
1
hα
N∑
j=i
w(α)j−iλj = q(ih)xi + a(ih)λi, i = 0, . . . , N − 1.
s qt♦♥s ♣r♦ s②st♠ ♦ 2N qt♦♥s ♥ 2N ♥♥♦♥s tt ♥ s♦ ②
s♥ r♦s s♠s s s rt ss♥ ♠♥t♦♥ r♦♠ ts s②st♠ ♥ t
x(t) ♥ λ(t) ♥ ssttt ♥ t♦ ♦t♥ u(t)
r t ❬❪ s t ♦♠♥t♦♥ ♠t♦♥♥ ♦r♠t♦♥s ♦r P
♥ rt♦♥ rt rü♥t♥♦ ♦♥♣t t♦ s♦ t ♥♦♥ t
stt ♥ ♦♥tr♦ rs ♦r t② ♦♥sr P t t♦r stt
♥ ♦♥tr♦ rs ② stt t P s ♦♦s
Minimize J(u) =1
2
∫ 1
0[q(t)x2(t) + r(t)u2(t)]dt,
subject to 0Dαt x(t) = a(t)x(t) + b(t)u(t),
t ♥♣♦♥t ♦♥t♦♥s x(a) = c ♥ x(b) = d r t stt ♥ ♦♥tr♦ rs
x(t) ♥ u(t) r rs♣t② nx ♥ nu t♦rs f ♥ g r rs♣t② sr ♥
nx t♦r ♥t♦♥s ♥ c ♥ d r ♥ t♦rs ♠♥s♦♥s nx ♥ nu sts② t
rt♦♥ nu ≤ nx
♥♦tr st② r♥t rt♦♥ ♦ t ♣r♦s ♠t♦ s ♥tr♦ ② ♥ t
❬❪ t♦rs ♦♥sr t s♠ ♣r♦♠ t s ♠♦ rü♥t♥♦
♣♣r♦①♠t♦♥s ♦r t ♥ rt rt♦♥ rts
0Dαt x(ti− 1
2
) ∼= 1
hα
i∑
j=0
w(α)j xi−j , i = 1, . . . , n,
tDα1 x(ti+ 1
2
) ∼= 1
hα
n−i∑
j=0
w(α)j xi+j , i = n− 1, n− 2, . . . , 0,
r w(α)j r t ♦♥ts sts②♥ w
(α)0 = 1 ♥ w
(α)j =
(
1− α+1j
)
w(α)j−1, j = 1, . . . , n.
s ♣♣r♦①♠t♦♥s r rr ♦t t t ♥tr ♣♦♥ts ♦ rt♥ srt③t♦♥ ♦ t
t♠ ♦r③♦♥ ♥ t t♠ ♦♠♥ [0, 1] s ♥t♦ n q ♣rts ♥ t rt♦♥
rts 0Dαt x ♥ tD
α1 λ r ♣♣r♦①♠t t t ♥tr ♦ s♠♥t x(ti− 1
2
) ♥
♥ s t r ♦ t t♦ ♥ s ♦ t s♠♥t
x(
ti− 1
2
)
=xi−1 + xi
2.
♠r ♣♣r♦①♠t♦♥s r ♦♥sr ♦r x(ti+ 1
2
), λ(ti− 1
2
) ♥ λ(ti+ 1
2
).
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② ssttt♥ ts ♣♣r♦①♠t♦♥s ♥ ♥ t ♦♦♥ s②st♠ ♦ qt♦♥s
s ♦t♥
1
hα
i∑
j=0
w(α)j xi−j =
1
2a(i1h)(xi−1 + xi)−
1
2r−1(i1h)b
2(i1h)(λi−1 + λi), i = 1, . . . , n,
1
hα
n−i∑
j=i
w(α)j λi+j =
1
2q(i2h)(xi+1 + xi) +
1
2a(i2h)(λi−1 + λi), i = n− 1, . . . , 0,
r i1 = i− 12 ♥ i2 = i+ 1
2 . s s②st♠ ♦ 2n ♥r qt♦♥s ♥ tr♠s ♦ 2n ♥♥♦♥s
♥ s♦ s♥ st♥r ♥r s♦r
♥♠r t♥q s ♦♥ t ♥r ♦rt♦♥♦r♠ ♣♦②♥♦♠ s t ss ♦r
s♦♥ P s sss ② ♦t t ❬❪ t♦rs ♦s ♦♥ Ps t t
qrt ♣r♦r♠♥ ♥① ♥ t rt♦♥ ②♥♠s s ♥ ♥ tr♠s ♦ t ♣t♦
rt♦♥ rts t s♦t♦♥ ♠t♦ ♥ s ♦♥ t ♥r ♦rt♦♥♦r♠
♣♦②♥♦♠ ♣♣r♦①♠t♦♥ t♦t s♥ ♠t♦♥♥ ♦♥t♦♥s ② ♦♥sr t
♦♦♥ P ♦r♠t♦♥
(P ) Minimize J(u) =1
2
∫ 1
0[q(t)x2(t) + r(t)u2(t)]dt,
subject to 0Dαt x(t) = a(t)x(t) + b(t)u(t),
x(0) = x0,
r q(t) > 0, r(t) > 0 ♥ b(t) 6= 0.
stt ♥ ♦♥tr♦ rs x(·) ♥ u(·), r ①♣♥ ② s♥ t ♥r ss
Ψ(·) s ♦♦s
x(t) ≃ (CT Iα + dT )Ψ(t),
u(t) ≃ UTΨ(t),
r Iα s t ♠tr① ♦♣rt♦r ♦ rt♦♥ ♥trt♦♥ ♦ ♦rr α, CT = [c0, . . . , cm] ,
UT = [u0, . . . , um] ♥ dT = [x0, 0, . . . , 0] .
tr s♦♠ t♦♥s t ♣r♦r♠♥ ♥① J ♥ ♣♣r♦①♠t ②
J ≃ J [C,U ]
=1
2
∫ 1
0[(QTΨ(t))((CT Iα + dT )Ψ(t)Ψ(t)T (CT Iα + dT )T ) + (RTΨ(t))(UTΨ(t)Ψ(t)TU)]dt,
♥ t ②♥♠ s②st♠ s
CTΨ(t)−ATΨ(t)Ψ(t)T (CT Iα + dT )T −BTΨ(t)Ψ(t)TU = 0.
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♥ t t♦rs s♦ tt t ②♥♠s②st♠ ♣♣r♦①♠t♦♥ ♥ ♦♥rt ♥t♦
♥r s②st♠ ♦ r qt♦♥s ♦ t ♦r♠
CT − (CT Iα + dT )T V − UT W = 0,
♦r s♦♠ ♦♣rt♦rs V ♥ W .
♦ ♥
J∗[C,U, λ] = J [C,U ] + [CT − (CT Iα + dT )T V − UT W ]λ,
r λT = [λ0, λ1, . . . , λm] , V = [vij ]1≤i, j≤m+1 ♥ W = [wij ]1≤i, j≤m+1 . ♥ t
♣♣t♦♥ ♦ t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ②s
∂J∗
∂C= 0,
∂J∗
∂U= 0,
∂J∗
∂λ= 0,
r ② ∂J∗
∂C = 0 t s②st♠ ∂J∗
∂Cj= 0, j = 0, . . . ,m s ♠♥t s qt♦♥s ♥
s♦ ♦r C, U ♥ λ ② s♥ t t♦♥ trt ♠t♦
tr ♦♠♣t♥ C ♥ U, t ♣♣r♦①♠t s ♦ u(t) ♥ x(t) r ♦t♥ t t
qt♦♥s ♦ t ♥r ①♣♥s♦♥ ♦♠ ♣r♦♣rts ♦ ♥r ♣♦②♥♦♠s ♥ t
♦♥r♥ r s♦ ♦♥sr ♥ ts ♦r s ♦t t ❬❪
♥♦tr st② r♥t ♣♣r♦ s sss ② ❨♦s t ❬❪ ♥ ts ♦r
t ♠t♦♥♥ ♦♥t♦♥s t ♠♥♥♦ rt♦♥ rt ♥ t ♥r
♠tt ♦♦t♦♥ ♠t♦ r s t♦ s♦ t P P s stt s
♦♦s
Minimize J(u) =
∫ tf
t0
f(x(t), u(t), t)dt,
subject to 0Dαt x(t) = g(x(t), u(t), t),
r x(t0) = x0 x(t) ∈ Rn u(t) ∈ R
m f ♥ g r rs♣t② sr ♥ n t♦r
♥t♦♥s ♥ t ♥ssr② ♦♥t♦♥s ♥ tr♠s ♦ P♦♥tr②♥ ♥t♦♥ ♦r t
P r ♥ ②
tDαtfλ(t) =
∂H
∂x,
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0 =∂H
∂u,
t0Dαt x(t) =
∂H
∂λ,
x(t0) = x0, λ(tf ) = 0.
♦♣t s♦t♦♥ ♠t♦ s s ♦♥ t ♣♣r♦①♠t♦♥ ♦ x(·), u(·) ♥ λ(·) ②
tr♥t srs ♦ ♥r ♠tts ♦r t ∈ [t0, tf ] s ♦♦s
x(t) ≃2k−1∑
i=0
M∑
j=0
(t− t0)cxijΨij(t) + x0,
u(t) ≃2k−1∑
i=0
M∑
j=0
cuijΨij(t),
λ(t) ≃2k−1∑
i=0
M∑
j=0
(t− tf )cλijΨij(t),
r ♦rn(tf−t0)
2k+ t0 ≤ t ≤ (n+1)(tf−t0)
2k+ t0
Ψnm(t) =√2m+ 1
2k2
√
tf − t0Pm
(
2k(t− t0)
tf − t0− n
)
,
Pm(·) ♥ st ♥r ♣♦②♥♦♠s tr s♥ t ♦♦t♦♥ ♣♦♥ts Pi s tt
1 ≤ i ≤ 2k(M + 1) s t r♦♦ts ♦ ②s ♣♦②♥♦♠s ♦ r 2k(M + 1) t
♦♦♥ r s②st♠ ♦ qt♦♥s s ♦t♥
F1(x(Pi), u(Pi), λ(Pi)) = 0,
F2(x(Pi), u(Pi), λ(Pi)) = 0,
F3(x(Pi), u(Pi), λ(Pi)) = 0,
r F1(x(t), u(t), λ(t)) F2(x(t), u(t), λ(t)) ♥ F3(x(t), u(t), λ(t)) ♦rrs♣♦♥ t♦ t
qt♦♥s ♦ t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② tr s♦♥ ts qt♦♥s t
♦♥ts ♦ t srs ② ♥r ♠tts tt ♣♣r♦①♠t x(t) u(t) ♥ λ(t) r
♦t♥ ♥♦♥ ②s ♣♦②♥♦♠s ♦♥ t ♥tr [t0, tf ] ♥ tr♠♥
② t ♦♦♥ rrr♥ ♦r♠
Tn+1(t) = 2(2t
tf − t0− t0 + tf
tf − t0)Tn(t)− Tn−1(t),
t T0(t) = 1 ♥ T1(t) =2t
tf−t0− t0+tf
tf−t0
♥♦tr ♦r s♦♥ P ♥tr♦ ② ♥ Ptr♦ ❬❪ ♦♥ssts
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♥ tr♥s♦r♠♥ t rt♦♥ ♣r♦♠ ♥t♦ ss ♥tr♦rr ♣r♦♠ ② s♥ ♥
①♣♥s♦♥ ♦r♠ ♦r rt♦♥ rts ♦r s s ♦♥ ♥ ♣♣r♦①♠t♦♥
♦r♠ ♦t♥ ② t♥♦ ♥ t♥♦ ❬❪ t♦rs stt t P s
♦♦s
Minimize J(u) =
∫ 1
0f(x(t), u(t), t)dt,
subject to x(t) + k 0Dαt x(t) = g(x(t), u(t), t),
r x(t0) = x0 ♥ k s ♥ ♦♥st♥t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♦r ts
♣r♦♠ r
x(t) = −k 0Dαt x(t) + g(x(t), u(t), t),
0 =∂f
∂u+ λ(t)
∂g
∂u,
λ(t) = k tDα1 λ(t)− λ(t)
∂g
∂x,
st t♦ x(0) = x0 ♥ λ(1) = 0 tr s♦♠ t♦♥s t ♣♣r♦①♠t♦♥
0Dαt x(t) ≈ A(α)t−αx(t) +
N∑
p=2
B(α, p)t1−α−pVp(t),
s s r
A(α) =1
Γ(1− α)− 1
Γ(1− α)Γ(2− α)×
N∑
p=2
Γ(p− 1 + α)
(p− 1)!,
B(α, p) = − 1
Γ(1− α)Γ(2− α)× Γ(p− 1 + α)
(p− 1)!,
Vn(x(p))(t) =
∫ t
0x(p)(τ)τndτ, n ∈ N, t ≥ 0,
♥♦t♥ ② Vn(x(p))(t), n ∈ N t nth ♠♦♠♥t ♦ t ♥t♦♥ x(p)(·) ♥ ② x(p)(·),
p ∈ N t pth rt ♦ x ♥ t ♣r♦♠ ♦♠s ss ♥tr♦rr ♣r♦♠
♥ t rt♦♥ s②st♠ s tr♥s♦r♠ ♥t♦ t ♦♦♥ ♦r♥r② s②st♠
Minimize J(u) =
∫ 1
0f(t, x(t), u(t))dt,
subject to
x(t) = −k(A(α)t−αx(t)−∑N
p=2B(α, p)t1−α−pVp(t))+ f(t, x(t), u(t)),
Vp(t) = (1− p)(t− 0)p−2x(t),
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t t ♥♣♦♥t ♦♥t♦♥s Vp(0) = 0 ♦r p = 2, . . . , N ♥ x(0) = x0 s ♣r♦♠
♥ s♦ ② ♠♥s ♦ ss ♦♣t♠♦♥tr♦ t♦r② s♥ rt♦♥ rts ♦
♥♦t ♣♣r ♥ ts ♦r♠t♦♥
♥♦tr s♠r ♣r♦r s sss ② P♦♦s t ❬❪ t♦rs ♦♣t t
♣t♦ rt♦♥ rt t♦ ♠♦ t ②♥♠ ♦♥str♥ts ♥ s ♥ ♣♣r♦①♠t♦♥
♦r♠ t♦ ♦♥rt t rt♦♥ ♣r♦♠ ♥t♦ ♥ ♥tr♦rr ♦♥ t r tr♠♥ t♠
P s stt s ♦♦s
Minimize J(u) =
∫ T
aL(t, x(t), u(t))dt+ φ(T, x(T )),
subject to Mx(t) +N CaD
αt x(t) = f(t, x(t), u(t)),
t t ♥♣♦♥t ♦♥t♦♥ x(a) = xa r M ♥ N r ♥♦♥③r♦ ♥ xa s ① r
♥♠r
♦♣t♠ tr♣t (x, u, T ) ♠st sts② t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② r
♥ ♥ t ♥①t t♦r♠
♦r♠ (x, u, T ) ♠♥♠③s t ♣r♦r♠♥ ♥① sts②♥ t ②♥♠
♦♥str♥ts t t ♦♥r② ♦♥t♦♥ t♥ tr ①sts ♥t♦♥ λ(·) ♦r t
♦♦♥ ♦♥t♦♥s ♦
❼ ♠t♦♥♥ s②st♠
Mx(t) +N CaD
αt x(t) =
∂H∂λ (t, x(t), u(t), λ(t)), t ∈ [a, T ],
Mλ(t)−N tDαTλ(t) = −∂H
∂x (t, x(t), u(t), λ(t)), t ∈ [a, T ].
❼ stt♦♥r② ♦♥t♦♥
∂H
∂u(t, x(t), u(t), λ(t)) = 0.
❼ tr♥srst② ♦♥t♦♥s
[H(t, x(t), u(t), λ(t))−Nλ(t)CaDαt x(t) +Nx(t) tI
1−αT λ(t) +
∂φ
∂t(t, x(t))]t=T = 0,
[Mλ(t) +N tI1−αT λ(t) +
∂φ
∂x(t, x(t))]t=T = 0,
r H(t, x(t), u(t), λ(t)) s t P♦♥tr②♥ ♥t♦♥ ♥ ②
H(t, x(t), u(t), λ(t)) = L(t, x(t), u(t)) + λ(t)f(t, x(t), u(t)).
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t♦rs s t ①♣♥s♦♥ ♦r♠ ♦r rt♦♥ rts ♥ t rt♦♥ t♥
♠♥♥♦ ♥ ♣t♦ rts t♦ tr♥s♦r♠ t ♥ P ♥t♦ ss
♥tr♦rr ♦♣t♠ ♦♣t♠③t♦♥ ♣r♦♠ s ♦♦s t ♠♥♥♦ r
t♦♥ rt s ♣♣r♦①♠t ②
aDαt x(t) ≈ A(α,N)(t− a)−αx(t) +B(α,N)(t− a)1−αx(t)−
N∑
p=2
C(α, p)(t− a)1−α−pVp(t),
r Vp(t) s t s♦t♦♥ ♦ t s②st♠ Vp(t) = (1− p)(t− a)p−2x(t) t Vp(a) = 0 ♥
A(α,N) =1
Γ(1− α)[1 +
N∑
p=2
Γ(p− 1 + α)
Γ(α)(p− 1)!],
B(α,N) =1
Γ(2− α)[1 +
N∑
p=1
Γ(p− 1 + α)
Γ(α− 1)p!],
C(α, p) =1
Γ(α− 1)Γ(2− α)× Γ(p− 1 + α)
(p− 1)!.
rt ♠♥♥♦ rt♦♥ rt s ♣♣r♦①♠t ②
tDαb x(t) ≈ A(α,N)(b− t)−αx(t)−B(α,N)(b− t)1−αx(t) +
N∑
p=2
C(α, p)(b− t)1−α−pWp(t),
r Wp(t) s t s♦t♦♥ ♦ t s②st♠ Wp(t) = −(1− p)(b− t)p−2x(t) t Wp(b) = 0
♥ A(α,N) B(α,N) ♥ C(α, p) r s ♦
t♦rs st② s♦♠ ♣rtr ss ♦r rstrt♦♥s r ♠♣♦s ♦♥ t ♥
t♠ T ♦r ♦♥ x(T )
♠r ♠t♦s r sss ② r ♥ ♥ ❬❪ ♥ ts ♦r t rt♦♥
r♥tt♦♥ ♦♣rt♦r s ♥ t P s ♣♣r♦①♠t s♥ st♦♣s ♣♣r♦①
♠t♦♥ ♥t♦ stt s♣ r③t♦♥ ♦r♠ ♥ t P s r♦r♠t ♥t♦ ♥ ♥tr
P ② s♥ t t♦♦♦① t♦ s♦ ts ♣r♦♠ ♣r♦♠ ♦♥sr
♥ ts ♦r s
Minimize J(u) = G(x(a), x(b)) +
∫ T
aL(t, x(t), u(t))dt,
subject to aDαt x(t) = F (t, x(t), u(t)),
t ♥t ♦♥t♦♥ x(a) = xa ♥ t t ♦♦♥ ♦♥str♥ts
umin(t) ≤ u(t) ≤ umax(t),
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xmin(a) ≤ x(a) ≤ xmax(a),
Lvti(t, x(t), u(t)) ≤ 0,
Gvei(x(a), x(b)) ≤ 0,
Gvee(x(a), x(b)) = 0,
r L(·), G(·) ♥ F (·) r rtrr② ♥ ♥♦♥♥r ♥t♦♥s ssr♣ts ti, ei,
♥ ee ♦♥ t ♥t♦♥s L(·), G(·) r trt♦r② ♦♥str♥t ♥♣♦♥t ♥qt② ♦♥str♥t
♥ ♥♣♦♥t qt② ♦♥str♥t rs♣t② s t♦ s t ♣♣r♦①♠t♦♥
sα =∏N
n=11+s/wz,n
1+s/wp,nt♦ tr♥s♦r♠ t rt♦♥ r♥tt♦♥ ♦♣rt♦r ♥t♦ ♥ ♥tr
♦♣rt♦r s ♦♦s
aDαt x(t) ≈
z(t) = Az(t) +Bu(t),
x(t) = Cz(t) +Du(t),
r A =
−bN−1 −bN−2 . . . −b1 −b01 0 · · · 0 0
0 1 · · · 0 0
0 0 · · · 1 0
, B =
1
0
0
0
, D = d, ♥
C =[
CN−1 CN−2 . . . C1 C0
]
.
♥ t P s ♦♥rt t♦ t ♦♦♥ ♥tr P
Minimize J(u) = G(Cz(a) +Du(a), Cz(b) +Du(b)) +
∫ b
aL(t, Cz(t) +Du(t), u(t))dt,
subject to aDαt x(t) = Az(t) +BF (t, CZ(t) +Du(t), u(t)),
t ♥t ♦♥t♦♥ z(a) = xawCw r w =
[
1 0 · · · 0]T
♥ t t ♦♦♥
♦♥str♥ts
umin(t) ≤ u(t) ≤ umax(t),
xmin(a) ≤ Cz(a) +Du(a) ≤ xmax(a),
Lvti(t, Cz(t) +Du(t), u(t)) ≤ 0,
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Gvei(Cz(a) +Du(a), Cz(b) +Du(b)) ≤ 0,
Gvee(Cz(a) +Du(a), Cz(b) +Du(b)) = 0.
stt x(t) ♦ t ♥t P ♥ rtr ♦r♠ x(t) = Cz(t) + Du(t).
rst♥ stt♥ s ♣♣r♦♣rt s ♥ ♥♣t ♦r s t ♦♦♦①
♠♦ ❬❪ ♣rs♥ts ♦♣t♠t② ♦♥t♦♥s ♦ t P♦♥tr②♥ t②♣ ♦r P ♥r
♦♥①t② ss♠♣t♦♥s ♦ t ♦t② st ♥ ♦st ♥t♦♥ ♥ t rt♦♥ ②♥♠
s②st♠ ♥♦s t ♠♥♥♦ rt ♦r♦r stt t ♦♣t♠t②
♦♥t♦♥s ♥ t♦ ss rst t t ♥t ♦♥t♦♥ x0 = 0 ♥ s♦♥② ♥ t s
r♥t r♦♠ ③r♦ (x0 6= 0). ♦♥sr t ♦♦♥ P
Minimize J [x, u] =
∫ b
af0(t, x(t), u(t))dt,
subject to aDαt x(t) = f(t, x(t), u(t)), t ∈ [a, b] ,
aI1−αt x(a) = x0,
u(t) ∈M ⊂ Rm, t ∈ [a, b],
r f : [a, b]× Rn ×M → R
n, f0 : [a, b]× Rn ×M → R, 0 < α < 1, ♥ x0 ∈ R
n \ 0.
s ♣r♦♠ s stt ♥r t ♦♦♥ ss♠♣t♦♥s ♦♥ t t
t ♥t♦♥ f ∈ C1 t rs♣t t♦ x ∈ Rn;
t ♥t♦♥ f0(·, x, u) s ♠sr ♦♥ [a, b] ♦r x ∈ Rn, u(t) ∈M ♥ f0(t, x, ·)
s ♦♥t♥♦s ♦♥ M ♦r t ∈ [a, b] ♥ x ∈ Rn;
t ♥t♦♥ f0 ∈ C1 t rs♣t t♦ x ∈ Rn, ♦r 1 < p < 1
1−α , α ∈ (0, 1) stss
|f0(t, x, u)| ≤ a1(t) + c1 |x|p ,
|(f0)x(t, x, u)| ≤ a2(t) + c2 |x|p−1 ,
♦r t ∈ [a, b] ♥ x ∈ Rn, r a2 ∈ Lp′([a, b],R+
0 ), (1p + 1p′ = 1),
a1 ∈ L1([a, b],R+0 ), ♥ c1, c2 ≥ 0;
t ♥t♦♥s fx(·, x, u), (f0)x(·, x, u) r ♠sr ♦♥ [a, b] ♦r x ∈ Rn, u ∈M ;
t ♥t♦♥s fx(t, x, ·), (f0)x(t, x, ·) r ♦♥t♥♦s ♦♥ M ♦r t ∈ [a, b] ♥
x ∈ Rn;
t st M s ♦♠♣t
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♦r t ∈ [a, b] ♥ x ∈ Rn t st
(f0(t, x, u), f(t, x, u)) ∈ Rn+1 : u ∈M
,
s ♦♥①
t UM t st ♠♣♣♥ ♥ ②
UM =
u(·) ∈ L1([a, b],Rm) : u(t) ∈M, t ∈ [a, b]
.
r♦r r t ♦♣t♠t② ♦♥t♦♥s s ♦♦s t ♣r
(x∗(·), u∗(·)) ∈(
aIαt (L
p) +
d
(t− a)1−α, d ∈ R
n
)
× UM ,
s ♦② ♦♣t♠ s♦t♦♥ ♦ t ♣r♦♠ stt ♦r t♥ tr ①sts ♥t♦♥
λ ∈ tIαb (L
p′) s tt
tDαb λ(t) = fTx (t, x
∗(t), u∗(t))λ(t)− (f0)x(t, x∗(t), u∗(t)) ♦r t ∈ [a, b],
tI1−αb λ(b) = 0,
♥
f0(t, x∗(t), u∗(t))− λ(t)f(t, x∗(t), u∗(t)) = min
u∈Mf0(t, x∗(t), u(t))− λ(t)f(t, x∗(t), u(t)) ,
♦r t ∈ [a, b]. rtr♠♦r r ♦♣t♠t② ♦♥t♦♥s ♦r ♥ ♥t ♦♥t♦♥
x0 = 0 s ♠♦ ❬❪
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♣tr
♦r♠t♦♥ ♦ rt♦♥ ♣t♠
♦♥tr♦ Pr♦♠s
♥tr♦t♦♥
② Ps ♥♦t ♥ Ps ♦r tr t ♣r♦r♠♥ ♥① ♥♦r t
②♥♠ s②st♠ s♣②s t st ♦♥ rt♦♥ ♦♣rt♦r
t♦ sr Ps ♦r♠t♦♥s r ♣♦ss ♥ ♥r ♦♥sr ♦♥ ♦r
t ♣r♦r♠♥ ♥① s t ♥tr ♦ ♥t♦♥ tt ♣♥s ♦♥ ♦t t stt ♥
t ♦♥tr♦ rs ♥ t ②♥♠ ♦♥str♥ts r sr ② s rs♦♥
t♦ s s t♦ sr ②♥♠ s②st♠s ♥ Ps s s ♦♥sr ♥st♥s ♥
rt♦♥ rts ♣r♦ sr♣t♦♥ ♦ t ♦r ♦ t ②♥♠ s②st♠
s ♠♦r rt t♥ t ♦♥ ♥ ② ♥tr rts ♦r s♣② ts
s ♣rtr② r♥t ♦r s②st♠s t ♦♥r♥ ♠♠♦r② ♥ ♥♦♥♦ ts Ps
♥ ♥ t rs♣t t♦ r♥t ♥t♦♥s ♦ rt♦♥ rts ♦r t
♦♥s ♥ t s♥s ♦ ♠♥♥♦ ♥ ♦ ♣t♦ ♥ s ♠♦r ② r
r s♦ ♥♠r ♦ s♣ ♥♠r t♥qs t♦ s♦ Ps
Ps s ♠ r ♣♣t♦♥ r♥ ♦ ②♥♠♦♥tr♦ ♣r♦♠s t rs♣t
t♦ rt♦♥ s ♦ rt♦♥s ❱s ♥ ts s♠♣st rs♦♥ s ♥ ②
♠♥♠③♥ ♦st
J [x(·)] =
∫b
a
L(t, x(t), aDα
t x(t))dt,
st t♦ ♦♥r② ♦♥t♦♥s
x(a) = xa, x(b) = xb.
♦r ♠♦r ts ♦♥ ❱s s r ❬❪ ♥♦s ♥ ♦rrs ❬❪ ♥
♥♦s t ❬❪
♠♥ ♦s ♦ ts tss s Ps r rstt② ♥ r r♥ ♦ ♣♣t♦♥s
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♦♥sttt ② ♥ts ♦r ❱s s t r t ♠t♦s ♣♣ ♦r Ps
r sst♥t② r♦♠ t♦s ♦r ❱s ♦r ①♠♣ Ps ♠♣♦②s t P♦♥tr②♥
♠①♠♠ ♣r♥♣ ♥ t ♥r ♠①♠♠ ♦♥t♦♥ maxuH(·) ♦s t
♦♥tr♦ r t♦ s♦♥t♥♦s ♠♣♥ t t ♦♥r② ♣♦♥t ♥ ts ♦rs t
♦♥srt♦♥ ♦ t s♣ ♦ s♦t② ♦♥t♥♦s ♥t♦♥s ♦r t stt r
♥r ♦r♠t♦♥ ♦ rt♦♥ ♣t♠ ♦♥tr♦
Pr♦♠s
r r sr ♥t♦♥s ♦r Ps s t rs t②♣s ♦ rt♦♥ rts
♠ t ♠♣♦ss t♦ ♦♥sr t②♣ ♣r♦♠ tt r♣rs♥ts ♣♦ssts ❲
♦♥sr P s ♦♦s
❲ s ♦♥r♥ t ♥ ♥tr [a, b] ⊆ R. ❲ r ♥ ♠t♥t♦♥ U
♠♣♣♥ [a, b] → Rm, ♥ ♦♥tr♦ s st♦♥ u(·) ♦r Ω ⊂ R
m ♥t♦♥ u(·) ♠②
tr ♠sr ♦♥t♥♦s ♥tr ♣s ♦♥t♥♦s ♦r ♥ ♦trs
♣♥♥ ♦♥ t ♣r♦♠ sts②♥ u(t) ∈ Ω ♥ t♠ t, t a ≤ t ≤ b.
❲ r ♥ ②♥♠ ♥t♦♥ F : [a, b]×Rn × R
m → Rn. rt♦♥ trt♦r② ♦r
stt x(·) ♦♥ [a, b] ♦rrs♣♦♥♥ t♦ t ♦♥tr♦ u(·), stss t rt♦♥ ②♥♠
s②st♠
aDα
t x(t) = F (t, x(t), u(t)),
r t ♣♦♥t x(a) s r t♦ ♦s♥ t♥ ♥ st C0. r ♠ s t♦ ♥ ♦♥tr♦
u(t) ♦r P tt ♠♥♠③s t ♦st ♥t♦♥
J [x, u] =
∫b
a
L(t, x(t), u(t))dt,
r J [x, u] s ♣r♦r♠♥ ♥① ♦r ♦st ♥t♦♥ ♥ L(·) s r♥♥♥ ♦st ♦r
r♥♥
♥ s♠♠r② P s ♥ ②
(P ) Minimize J [x, u] =
∫b
a
L(t, x(t), u(t))dt
subject to aDα
t x(t) = F (t, x(t), u(t)) t ∈ [a, b],
u ∈ U ,
x(a) = x0 ∈ Rn t ∈ [a, b].
r aDαt r♣rs♥ts rt♦♥ r♥t ♦♣rt♦r s s ♠♥♥♦ ♦♣rt♦r
♣t♦ ♦♣rt♦r ♠r ♦♣rt♦r ♦r ♥♦tr ♦♥ x(·) s t stt r t r♣rs♥ts
t♠ ♥ L : [a, b]×Rn×R
m → R ♥ F : [a, b]×Rn×R
m → Rn r t♦ ♥ ♠♣♣♥s
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♦r♦r t Ps ♥ t ♦st ♥t♦♥ s ♥ s ♥ ♣r♦♠ (P ) r ♥♦♥
s ♣r♦♠s ♥ t r♥ ♦r♠ ♦r r♥ Ps r r t♦ ♦tr Ps t
rst ♥ t ♦③ P ♥ tr s tr♠♥ ♦st g(a, x(a), b, x(b)) ♥ t♦♥
t♦ t r♥♥♥ ♦st ♥ t ♣r♦r♠♥ ♥① J [x, u]. s♦♥ ♦♥ s ②r P ♥
t ♣r♦r♠♥ ♥① ♦♥② ♦♥ssts ♥ t tr♠♥ ♦st ♥ tr s ♥♦ r♥♥♥
♦st L(·) = 0
rt♦♥ trt♦r② s s♦t♦♥ ♦ t rt♦♥ r♥t qt♦♥ t t
♦♥r② ♦♥t♦♥ ♥ ♦r ♥ ♦♥tr♦ ♥t♦♥ sts②
♥② ♣r (x;u) stss t rt♦♥ ②♥♠ s ♦♥tr♦ ♣r♦ss ♦♥tr♦ ♣r♦ss
♦s rt♦♥ trt♦r② r♠♥s ♥ t ♦♥r② ♦ t tt♥ st st ♦ stt
s♣ ♣♦♥ts tt ♥ r r♦♠ t ♥t stt t ♠ss ♦♥tr♦ strts
s ♦♥r② ♣r♦ss
♥♠♠ s s♦t♦♥ ♦ Ps ♥ ts ♠♥♠♠ ♥ ♦ ♦r ♦ ♦r ♥st♥
♦♥sr ♥t♦♥ f : Rn → R, D ⊆ Rn, t♥ ♣♦♥t x∗ ∈ D s ♦ ♠♥♠♠ ♦ f(·)
♦r D tr ①sts ε > 0 s tt ♦r x ∈ D sts②♥ |x− x∗| < ε,
f(x∗) ≤ f(x),
x∗ s ♦ ♠♥♠♠ ♥ s♦♠ r♦♥ t f(·) ♦s ♥♦t tt♥ s♠r
t♥ f(x∗). ♦s ♦r x ∈ D, t♥ t ♠♥♠♠ s ♦ ♦r D. ♦r
♥ ts tss ♦♥sr ♦ ♠♥♠♠ ♦ t P♦♥tr②♥ t②♣ stt tt t
♦♥tr♦ ♠t♦♥♥ ♠st t ♠♥♠♠ ♦r s ♦♥tr♦s ♥ t s st
♠①♠♠ ♣r♥♣ ♦r♥ s♦t♦♥s t♦ t ♣r♦♠ (P ) ♦t♥ ♥r
s♦♠ ss♠♣t♦♥s ♦♥ t t s s
♦st ♥t♦♥ s ♦r s♠♦♥t♥♦s ♥ ♣st③
Ω(t) s ♦♠♣t st ♠♣ ∀t ∈ [a, b], ♥ t → Ω(t) s B−♠sr
t ♥t♦♥ F s ♦♥t♥♦s
t ♥t♦♥ F (t, ·, u) s ♣st③ s tt
|F (t, x1, u)− F (t, x2, u)| ≤ k(t, u) |x1 − x2| .
♥ ts tss stt sss ♥ r ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♥♦t ♦♥② ♦r
t P (P ) t s♦ r♦♠ ts s♠ ♣r♦♠ t t♦♥ ♦♥str♥ts ♦♥ t stt
r ♦ t ♦r♠ h(t, x(t)) ≤ 0, ♦r t ∈ [a, b].
♥ ♠♣♦rt♥t ♦srt♦♥ s ♥ ♦rr ssr② ♦♥t♦♥s ♦ ♦♣t♠t② r ♦♥②
♠♥♥ t ①st♥ ♦ s♦t♦♥ s r♥t r r r♦s sts ♦ s♥t
♦♥t♦♥s ♦r t ①st♥ ♦ s♦t♦♥ t♦ Ps ♦r ♥st♥ t ♦st ♥t♦♥ s
t st ♦r s♠♦♥t♥♦s ♥ t t s♠ t♠ t st ♥ ② t ♦♥str♥ts ♦
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t ♣r♦♠ s ♦♠♣t ♥ s♥s ♥ ♥ ♦♥t①t ♦ ♦♠♣t t♦♣♦♦s t♥
s♦t♦♥ ①sts ♦r ♥ ts tss ss♠ tt s ♥ ♦♣t♠♦♥tr♦ ♣r♦ss
r② ①sts ♥ r ♦♥r♥ ♦♥② t t ss♠♣t♦♥s ♦♥ t t ♦ t ♣r♦♠
♥r t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♥ r
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♣tr
①♠♠ Pr♥♣ ♦r t s
rt♦♥ ♣t♠ ♦♥tr♦ Pr♦♠
♥tr♦t♦♥
♥ ts ♣tr ♣rs♥t ♥ ♥r ♦r♠t♦♥ ♦ Ps ♦♥sr P ♦r
t ♣r♦r♠♥ ♥① s ♥ ② ♥ ♥tr ♦ rt♦♥ ♦rr ♥ t ②♥♠s
s ♠♣♣♥ s♣②♥ t ♣t♦ rt♦♥ rt ♦ t stt r t rs♣t
t♦ t♠ rs♦♥s t♦ ♦♦s t ♣t♦ rt♦♥ rt s s t s t ♠♦st
♣♦♣r ♦♥ ♠♦♥ ♣②ssts ♥ s♥tsts ♥ s♦ t t tt t rt♦♥ rt
♦ ♦♥st♥ts r ③r♦ ♦r♦r t ss♠♣t♦♥s tt ♠♣♦s ♦♥ t t ♦ t ♣r♦♠
♥s ♥♦ ♣♣r♦ t♦ t ♣r♦♦ s ♦♥ ♥r③t♦♥ ♦ ②♦rs ①♣♥s♦♥ ♥
rt♦♥ ♠♥ t♦r♠ ♥♦tr ♦♥trt♦♥ ♦ ts ♣tr ♦♥ssts ♦♥ ♥ ♥②t
♠t♦ t♦ s♦ t rt♦♥ r♥t qt♦♥ s s strt ② ♥ ①♠♣ s
♦♥ ♥r③t♦♥ ♦ t ttr ♥t♦♥
r ♣♣r♦ ♦♥ssts ♥t♦ ♦♥rt♥ t P ♥t♦ ♥ q♥t P ♥ t♥
♦♥ t ♦t♥ ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♥t♦ t t ♦ t ♦r♥
♣r♦♠ t s t♦ ② ♥ts rt② t♦ t tr♥ts ♦t♥ ♦♣t ♥ t
trtr ♠♦r ♣rs ♥st ♥r♥t t♦ t s ♦ rt♦♥ ♠t♦s ♥ t ♦r♥
♠♦♥ r♠♦r ♥ ♠♦r rt ♣♣r♦①♠t♥ ♦♠♣tt♦♥ ♣r♦rs
② t ♠①♠♠ ♣r♥♣ ♦♥t♦♥s
t s♦ r♠r tt ♦r rst rs sst♥t② r♦♠ t ♦♥ ♣rs♥t ②
♠♦ ❬❪ r ② s♥ t qt r♥t s ♦ rt♦♥s ♣♣r♦ ♥ssr②
♦♥t♦♥s ♦ ♦♣t♠t② r r ♦r r♥t P tt rqrs t ♦t② st
t st ♦ t♠ rts ♦ t stt r t♦ ♦♥① s s r② str♦♥
ss♠♣t♦♥ ♥ ♦♥sttts ② r♥ r♦♠ ♦r rst ♦rs ②♥♠ ♦♥tr♦
s②st♠s ♦s ♦t② sts ♠t ♠r srt st ♦ ♣♦♥ts ♦r♦r ♦r ♣♣r♦
s ♠ ♠♦r ♥ ♥ t t rt ss ♦r ♦ P♦♥tr②♥ t ❬❪
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s ♣tr s ♦r♥③ s ♦♦s ♥ t ♥①t t♦♥ stt sss ♥ ♣r♦
♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♥ t ♦r♠ ♦ P♦♥tr②♥ ①♠♠ Pr♥♣ ♦r
♥♦♥♥r Ps ♥ t♦♥ s♠♣ strt ①♠♣ ♦ P s♦ ②
♠t♦ s ♦♥ t ttr ♥t♦♥ s ♣rs♥t ♥② ♥ t♦♥
♣rs♥t s♦♠ ♦♥s♦♥s ♦ ts rsr s s s♦♠ ♦♣♥ ♥s rsts ♦
ts ♣tr ♥ ♥♥♦♥ ♥ ❬❪
tt♠♥t ♥ ss♠♣t♦♥s
♥ ts st♦♥ sss t P ♦♥sr ♥ ts ♣tr stt t ss♦t
♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♥ ♣rs♥t ts ♣r♦♦ ss ♥ ♣♣r♦ tt
rs r♦♠ t ♦♥s s② ♦♣t ♥ t trtr ♦r ts ss ♦ Ps
t s ♦♥sr t s♠♣ ♥r ♣r♦♠ s ♦♦s
(P )Minimize t0IαtfL(t, x(t), u(t))
subject to Ct0Dα
t x(t) = f(t, x(t), u(t)), [t0, tf ] L −
x(t0) = x0 ∈ Rn,
u(t) ∈ U ,
r U = u : [t0, tf ] → Rm : u(t) ∈ Ω(t) Ω : [t0, tf ] → R
m s ♥ st
♠♣♣♥ L : [t0, tf ] × Rn × R
m → Rn ♥ f : [t0, tf ] × R
n × Rm → R
n r
♥ ♥t♦♥s ♥♥ rs♣t② t r♥♥♥ ♦st ♦r r♥♥ ♥t♦♥ ♥ t
rt♦♥ ②♥♠s t0Iαtf
s t ♠♥♥♦ rt♦♥ ♥tr ♥ Ct0Dα
t s t t
♣t♦ rt♦♥ rt ♦ ♦rr 0 < α ≤ 1 ♦ t stt r t rs♣t t♦ t♠
t s ♥♦t r t♦ s tt s♠♣ tr♥s♦r♠t♦♥ ♦s s t♦ ♦♥rt t ♣r♦♠ (P )
♥t♦ ♥ q♥t ♦♥ s♠♣② ② ♥♥ ♥ t♦♥ stt r ♦♠♣♦♥♥t y ②
Ct0Dα
t y(t) = L(t, x(t), u(t)),
sts②♥ t ♥t ♦♥t♦♥ y(t0) = 0 ♥ ♦♥ tt ♣r♦♠ (P ) s q♥t
t♦ t ♦♥ s ♦♦s
(P )Minimize g(x(tf ))
subject to Ct0Dα
t x(t) = f(t, x(t), u(t)), [t0, tf ] L −
x(t0) = x0 ∈ Rn,
u(t) ∈ U ,
r ♥♦ g(x(tf )) = y(tf ) t stt r x =
[
y
x
]
t ♥s y s rst
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♦♠♣♦♥♥t t ♥t t 0 ♥ t ♠♣♣♥ f =
[
L
f
]
, t s L s rst
♦♠♣♦♥♥t
r♦♠ ♥♦ ♦♥ ♦♥sr ts s t s P ♥ t ♥♦r♠ ♦r♠ ❲ r♠r tt
t ♦ ♣r♦♠ stt♠♥t s t s♠♣st ♦♥ tt ♥ ♦♥sr ♦♥t♥♥
t ♥r♥ts rqr ② ♦♥ P
♦ stt t ss♠♣t♦♥s ♦♥ t t ♦ t ♣r♦♠ ♥r ♦r rst
♣r♦
♥t♦♥ g s C1 ♥ Rn ♦♥t♥♦s② r♥t ♥ ts ♦♠♥
♥t♦♥ f s C1 ♥ ♣st③ ♦♥t♥♦s t ♦♥st♥t Kf ♥ x, ♦r
(t, u) ∈ (t,Ω(t)) : t ∈ [t0, tf ]
♥t♦♥ f s ♦♥t♥♦s ♥ (t, u) ♦r x ∈ Rn
st ♠♣ Ω : [t0, tf ] → Rm s ♦♠♣t
r s M > 0 s tt |f(t, x,Ω(t))| < M, ♦r (t, x) ∈ [t0, tf ]× Rn
s r ② ♥♦ ♠♥s t st ②♣♦tss ♥♥ t ♣r♦♦ ♦ t ♠①♠♠
♣r♥♣s ♦r Ps ♦r t② r ♦ ♥trst ♥ tt t② ♦s t ♣rtr②
s♠♣ ♣r♦♦ ♦♣t ♥ ts ♣tr
①st♥ ♦ ♦♣t♠ s♦t♦♥s ♦r t ♥r s s ♥ r♥t② sss ②
♠♦ ❬❪ ①st♥ ♦ s♦t♦♥s s ss♥t t♦ ♥sr t ♠♥♥ ♦ t ♥ssr②
♦♥t♦♥s ♦ ♦♣t♠t② t♦ ts ♣tr ♦s ♥♦t ♦♥r♥ ♦♥t♦♥s ♦r t
①st♥ ♦ s♦t♦♥ t♦ P s t ♠①♠♠ ♣r♥♣ ss♠s ♥ ♦♣t♠♦♥tr♦
♣r♦ss ♣r♦r t s ♥♦t t t♦ ♦♥ tt ts s t s ♥r t ss♠♣t♦♥s
♥ t ①st♥ ♦ s♦t♦♥ t♦ P s r♥t t ♦st ♥t♦♥ g s
t st ♦r s♠♦♥t♥♦s ♥ t st ♦ ♣♦♥ts ♦ t stt s♣ R(tf ; t0, x0)
tt ♥ r t t ♥ t♠ tf s ♦♠♣t sr tt ♦♥t♦♥ s
♠♣ ② ♦♥t♦♥ ♦♦s r♦♠ t t tt tf < ∞ ♠♣s tt
R(tf ; t0, x0) s ♦♥ ♥ ts t♦tr t ♥ ♥ t ♦♥①②♥ t
♦ t ♥trt♦♥ ♦ t ②♥♠s ♠♣s tt R(tf ; t0, x0) s ♦s
♦♥sr
H(t, x, p, u) := pT f(t, x, u),
t p ∈ Rn t♦ t P♦♥tr②♥ ♥t♦♥ ss♦t t♦ ♣r♦♠ (P )
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①♠♠ Pr♥♣ ♦ ♣t♠t②
♦r r ♦♥ t♦ t ♠♥ t♦r♠ ♥tr♦ s♦♠ ♥t♦♥s ♣ s
t♦ ♣r♦ ts t♦r♠
♥t♦♥ t 0 < α ≤ 1 ♥ αs ♣♦♥t ♦ ♥ ♥tr ♥t♦♥ f : R → R
s ♣♦♥t t0 ∈ R sts②♥
limε→0+
1
2εt0−εI
αt0+ε |f(t)− f(t0)|
2−α = 0.
② rt② s♥ t ♥t♦♥ αrt♦♥ ♥tr t s s② t♦ ♦♥ tt t st
♦ s ♣♦♥ts ♥s t♦s ♦ t ♥tr ♦rr ♥tr s r♣t②♥ts ♥
♥sr ❬❪ t s ♠♣♦rt♥t t♦ ♣♦♥t ♦t t ♥♦♥ t tt t sst ♦ s
♣♦♥ts ♦ ♥ ♥tr ♥t♦♥ f(·) ♦♥sttts s ♠sr sst s
②♦r ❬❪
♥ t ♦♦s t ♥t♦♥ ♦ rt♦♥ stt tr♥st♦♥ ♠tr① ♦r ♥r
t♠ r②♥ t♦r rt♦♥ r♥t qt♦♥ ♦ t t②♣
CaD
αt x(t) = A(t)x(t), x(a) = xa,
r A(·) : R → Rn×n ♥ x(·) ∈ R
n s rqr
♥t♦♥ ♠tr① ♠♣ Φα : R × R → Rn×n s t t♦ t ♥r
rt♦♥ r♥t qt♦♥ ♦♥ t ♥tr [a, t] t stss
Ca D
αt Φα(t, s) = A(t)Φα(t, s), Φα(t, t) = In, ♥ Φα(t, s) = 0n , t < s,
r In ♥ 0n r rs♣t② t ♥tt② ♥ ③r♦ ♠trs ♦ ♦rr n t s s♠♣
t♦ ♦♥ tt x(t) = Φα(t, a)xa s s♦t♦♥ t♦
♥t♦♥ ♣r (x∗, u∗) s ♥ ♦♣t♠♦♥tr♦ ♣r♦ss ♦r ♣r♦♠ P t ②s
♦st ♦r t♥ tt ss♦t t ♥② ♦tr s ♦♥tr♦ ♣r♦ss
♦s② ♥ t q♥ t♥ P ♥ P t s♠ ♥t♦♥ ♦s ♦r (x, u)
t rs♣t t♦ P
♦r♠ t (x∗, u∗) ♦♣t♠♦♥tr♦ ♣r♦ss ♦r (P )
♥ tr ①sts ♥t♦♥ p : [t0, tf ] → Rn sts②♥
❼ t ♦♥t qt♦♥
tDαtfpT (t) = pT (t)Dxf(t, x
∗(t), u∗(t)),
r t ♦♣rt♦r tDαtf
s rt ♠♥♥♦ rt♦♥ rt ♥
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❼ t tr♥srst② ♦♥t♦♥
− pT (tf ) = ∇xg(x∗(tf )),
❼ u∗ : [t0, tf ] → Rm s ♦♥tr♦ strt② s tt u∗(t) ♠①♠③s [t0, tf ] L t
♠♣
u → H(t, x∗(t), p(t), u),
♦♥ Ω(t)
Pr♦♦ rst ② s tt ♥② ♣rtrt♦♥ ♦ t ♦♣t♠♦♥tr♦ u∗(·) tt ts
t ♥ ♦ t stt trt♦r② ♥ ♥♦t strt② rs t ♦st s t ♣r♦♦
rs ♦♥ t ♦♠♣rs♦♥ t♥ t ♦♣t♠ trt♦r② x∗(·) ♥ trt♦rs x(·)
r ♦t♥ ② ♣rtr♥ t ♦♣t♠♦♥tr♦ u∗(·)
t τ s ♣♦♥t ♥ (t0, tf ) ♥ ε > 0 s♥t② s♠ s♦ tt τ − ε ≥ t0
s ♣♦♥t ♥ t rt♦♥ ♦♥t①t ♥ ♥ t ♥t♦♥
♦ t s ♦♥sr t ♣rtr ♦♥tr♦ strt② uτ,ε ♥ ②
uτ,ε(t) =
u(t), t ∈ [τ − ε, τ),
u∗(t), t ∈ [t0, tf ] \ [τ − ε, τ),
r u(·) ∈ Ω(t) ♦r t ∈ [τ − ε, τ) ♥ τ s ♣♦♥t ♦ t rr♥ ♦♣t♠
♦♥tr♦ strt② ♦t tt tr s ♥♦ ♦ss ♦ ♥rt② ♦ t ♦ ♦ τ t♦ t t
tt t st s ♣♦♥ts s ♦ s ♠sr
t xτ,ε(·) t trt♦r② ss♦t t uτ,ε(·) ♥ t xτ,ε(t0) = x0 r② ②
♥t♦♥ ♦ ♦♣t♠t② ♦ (x∗, u∗)
0 ≤ g(xτ,ε(tf ))− g(x∗(tf ))
= ∇xg(x∗(tf ))[xτ,ε(tf )− x∗(tf )] + o(ε),
r ∇xg(·) s t r♥t ♦ g(·) ♥ o(ε) s st ♦ ♥t♦♥ r sts②♥ limε→0
r(ε)
ε= 0.
sr tt xτ,ε(t) = x∗(t) ♦r t ∈ [t0, τ − ε)
♦r♦r t s r tt ♦r t ∈ [τ − ε, τ)
|xτ,ε(t)− x∗(t)| ≤ τ−εIατ |f(s, xτ,ε(s), u(s))− f(s, x∗(s), u∗(s))|
≤ τ−εIατ Kf |xτ,ε(s)− x∗(s)|+ 2M
εα
Γ(α+ 1)
≤Mεα
Γ(α+ 1),
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r
M = 2M
(
1 +Kf
∞∑
n=1
Γ(α)n−1
Γ(nα+ 1)εnα
)
.
t s ♥♦t t t♦ s♦ tt ts srs ♦♥rs ♥ ts M s s♦♠ ♥t ♣♦st
♥♠r st ♥qt② s ♦t♥ ② t ♥①t t♦r♠ ♥ ♥ ♣rtr ♦s
♦r t = τ
♦r♠ ♥r③ ♠♥r♦♥ ♥qt② s ♥ ❬❪
♣♣♦s α > 0 t ∈ [0, T ) ♥ t ♥t♦♥s a(t) b(t) ♥ w(t) r ♥♦♥♥t ♥
♦♥t♥♦s ♥t♦♥s ♦♥ 0 ≤ t < T t
w(t) ≤ a(t) + b(t)
∫ t
0(t− s)α−1w(s)ds,
r b(t) s ♦♥ ♥ ♠♦♥♦t♦♥ ♥rs♥ ♥t♦♥ ♦♥ [0, T ). ♥
w(t) ≤ a(t) +
∫ t
0
[
∞∑
n=1
(b(t)Γ(α))n
Γ(nα)(t− s)nα−1a(s)
]
ds, t ∈ [0, T ).
♦r t ♣r♦♦ ♥ ♠♦r ts ♦t ♥r③ ♠♥r♦♥ ♥qt② s
♣♣♥①
♦r ♣r♦♥ t t ♣r♦♦ ♥ t ♦♦♥ ①r② rst
♥ t ♦♦s t Φα(·, ·) ♥♦t t stt tr♥st♦♥ ♠tr① s ♥t♦♥ ♦r t
♥r rt♦♥ r♥t s②st♠
Ct0Dα
t ξ(t) = Dxf(t, x∗(t), u∗(t))ξ(t).
♠♠ ♦♥sr t t♠ ♥tr [a, b] ♥ t ♥t♦♥ F (t, x(t)) = f(t, x(t), u(t)),
r u(t) s ♥r s ♦♥tr♦ ♥t♦♥ ♦r♦r ♦♥sr x(·), y(·), ♥ xν(·) t♦
rs♣t② t s♦t♦♥s t♦ t ♦♦♥ rt♦♥ r♥t s②st♠s ♥ ♦♥ t
♥tr [a, b] :
❼CaD
αt x(t) = F (t, x(t)) t x(a) = xa
❼CaD
αt y(t) = DxF (t, x(t))y(t) t y(a) = y ♥
❼CaD
αt xν(t) = F (t, xν(t)) t xν(a) ∈ xa + ναy + o(να)Bn
1(0)
♥ ♦r ν ♣♦st ♥ s♥t② s♠ r ♥♠r tt xν(·) stss ♦♥
t t♠ ♥tr [a, b]
xν(t) ∈ x(t) + ναy(t) + o(να)Bn1(0).
r Bn1 (0) ♥♦ts t ♦s ♥t ♦ Rn ♥tr t 0
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Pr♦♦ t s ♦♥sr t rst♦rr ①♣♥s♦♥ ♦ t ♠♣ x → F (t, ·) r♦♥ x(t). ❲
F (t, xν(t))− F (t, x(t))−DxF (t, x(t))(xν(t)− x(t)) ∈ o(‖xν(t)− x(t)‖),
♦r t ∈ [a, b] ♥CaD
αt y(t) = DxF (t, x(t))y(t),
t ναy(a) ∈ xν(a)− x(a) + o(να)Bn1(0), tt
CaD
αt [xν(t)− x(t)− να y(t)] = ζ(t),
♦r s♦♠ ζ ∈ L1 sts②♥ ζ(t) ∈ o(να)Bn1(0) ♥ L1
② ♥trt♥
xν(t)− x(t)− ναy(t) = 1Γ(α)
∫ t
a(t− τ)α−1(F (τ, xν(τ))− F (τ, x(τ))
−ναDxF (τ, x(τ))y(τ))dτ + 1Γ(α)
∫ t
a(t− τ)α−1ζ(τ)dτ.
♦r ♦♥t♥♥ s♦ ♦sr tt t s s♠♣ ①rs t♦ ♦♥ tt t
ss♠♣t♦♥ (H2) ♠♣s tt ‖DxF (t, x(t))‖ ≤ Kf ♦t s♦ tt t s ♥♦t t t♦
s tt β(t) := 1Γ(α)
∫ t
a(t− τ)α−1ζ(τ)dτ ∈ o(να), ♦r t ∈ [a, b],
♦ ② ♣tt♥ z(t) = xν(t) − x(t) − ναy(t) s♥ t ♦ ♦srt♦♥s ♥ t
ss♠♣t♦♥ (H2) ♦t♥ t ♥qt②
‖z(t)‖ ≤ β(t) +Kf
Γ(α)
∫ t
a
(t− τ)α−1‖z(τ)‖dτ.
♦r♠ ♥r③ ♠♥r♦♥ ♥qt② ②s
‖z(t)‖ ≤ ‖β(t)‖+
∫ t
a
[
∞∑
n=1
Knf Γ(α)
n
Γ(nα)(t− τ)nα−1(τ − a)‖β(τ)‖
]
dτ.
t
β = supt∈[a,b]
‖β(t)‖.
♦s② tt β ∈ o(να) ② ♣r♦r♠♥ t ♥tr ♥ t rt ♥ s ♦
t ♦ ♥qt② tt ♦r t ∈ [a, b]
‖z(t)‖ ≤ β
(
1 +
∞∑
n=1
Knf Γ(α)
n
Γ(nα+ 1)(t− a)nα
)
,
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② s♥ t ♦♥ ♣r♠tr ttr ♥t♦♥ s ♣tr ♥ ①♣rss
②
‖z(t)‖ ≤ βEα,α(KfΓ(α)(t− a)α).
s ♠♠ s ♣r♦ ♦r t ∈ [a, b]
xν(t) ∈ x(t) + ναy(t) + o(να)Bn1(0).
♦ ② ♦♥sr♥ ♥ ♣♣②♥ ♠♠ ♦♥ t t♠ ♥tr [τ, tf ] t
F (t, x) = f(t, x, uτ,ε(t)), να = ε a = τ t = tf xν = xτ,ε ♥ x = x∗ ♦♥
tt
0 ≤ ∇xg(x∗(tf ))[xτ,ε(tf )− x∗(tf )] + o(ε)
≤ ε∇xg(x∗(tf ))y(tf ) + o(ε)
= ε∇xg(x∗(tf ))Φα(tf , τ)y(τ) + o(ε),
r Φα(tf , τ) s t rt♦♥ stt tr♥st♦♥ ♠tr① ss♦t t t ♥r s②st♠
CaD
αt y(t) = DxF (t, x∗(t))y(t),
♥ t ♥tr [τ, tf ] ② tt♥
−pT (tf ) = ∇xg(x∗(tf )),
♥
pT (t) = pT (tf )Φα(tf , t),
♦♥ tt t ♦♥t r p(·) stss t rt ♠♥♥♦ rt♦♥
♥r qt♦♥
tDαtfpT (t) = pT (t)DxF (t, x∗(t)),
p(·) stss t ♦♥t qt♦♥ ♦ ♦r ♠①♠♠ ♣r♥♣ s s t ss♦t
tr♥srst② ♦♥t♦♥
♥② ② ♣tt♥ t♦tr ♥ ♥ ② ♦♦s♥
y(τ) = f(τ, x∗(τ), u)− f(τ, x∗(τ), u∗(τ)),
r u = u(τ), ♦t♥
0 ≥ εpT (τ)[f(τ, x∗(τ), u)− f(τ, x∗(τ), u∗(τ))] + o(ε).
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② ♥ ♦t ss ♦ ts ♥qt② ② ε > 0 ♥ ② t♥ t ♠t ε → 0+,
♦♥ t ♥qt②
0 ≥ pT (τ)[f(τ, x∗(τ), u)− f(τ, x∗(τ), u∗(τ))],
r♦♠ t rtrr♥ss ♦ u ∈ Ω(t) ②s t ♠①♠♠ ♦♥t♦♥ t t♠ t = τ
H(τ, x∗(τ), p(τ), u∗(τ)) ≥ H(τ, x∗(τ), p(τ), u).
t tt τ s ♥ rtrr② s ♣♦♥t ♥ [t0, tf ] ♠♣s tt t ♠①♠♠ ♦♥t♦♥
♦ ♦r ♠♥ rst ♦s tt s u∗(t) ♠①♠③s ♦♥ Ω(t) t ♠♣ u → H(t, x∗(t), p(t), u)
[t0, tf ] L
r ♠♥ rst s ♣r♦
strt ①♠♣
P♦♥tr②♥ ♠①♠♠ ♣r♥♣ ♣r♦ ♥ t ♣r♦s st♦♥ s ♥♦ ♣♣② t♦ s♦
s♠♣ ♣r♦♠ ♦ rs♦rs ♠♥♠♥t tt ♥♦s ♠♥♠③♥ rt♥ rt♦♥
♥tr st t♦ ♥ ♦♥tr♦ s
❲ ♦♥sr t ♦♦♥ ♣r♦♠
Minimize J(u)
subject to C0D
αt x(t) = u(t)x(t), t ∈ [0, T ],
x(0) = x0,
u(t) ∈ [0, 1],
r J(u) = 0IαT (1 − u(t))x(t), t 0 < α < 1 ♥ T > Γ(α + 1)α
−1
. r 0IαT s
rt♦♥ ♥tr ♥ C0D
αt s t ♣t♦ rt♦♥ rt
r x r♣rs♥ts ♥tr rs♦r tt ts ♣♦st s ♥♦t tt x0 > 0
♥ssr② r♦s ♦r♥ t♦ t r t ♥t♦♥ u(·) s♥t ②
♦♥tr♦ r♣rs♥ts t rt♦♥ ♦ t rs♦r tt s s t♦ ♣r♦♠♦t rtr
r♦t
♦r ♦ s t♦ ♥ t ♦♥tr♦ strt② tt ♠①♠③s t ♠♦♥t ♦ ♠t
rs♦r ♦r t t♠ ♥tr [0, T ] ♥ ② t rt♦♥ ♥tr
rst ♦♥sr ♥ t♦♥ stt r ♦♠♣♦♥♥t y sts②♥
C0D
αt y(t) = (1− u(t))x(t), y(0) = 0,
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♥ ♦rr ♦t♥ t ♥♦♥ ♣r♦♠ stt♠♥t ♥ t ♦r♠ ♦♥sr ♥ ♦r ♠♥ rst
tt s
Minimize y(T )
subject to C0D
αt x(t) = u(t)x(t), x(0) = x0,
C0D
αt y(t) = (1− u(t))x(t), y(0) = 0,
u(t) ∈ [0, 1].
r♦♠ ♦r♠ t ♦♥t qt♦♥ ♥ t tr♥srst② ♦♥t♦♥ ♦r
ts ♣r♦♠ r
tDαT p1(t) = [p1u
∗(t) + p2(1− u∗(t))], p1(T ) = 0,
tDαT p2(t) = 0, p2(T ) = 1,
r tDαT s rt ♠♥♥♦ rt♦♥ rt ♦ ♦rr α s tt
p2(t) ≡ p2(T ) = 1 ♥ qt♦♥ ♦♠s
tDαT p1(t) = [(p1(t)− 1)u∗(t) + 1].
r♦♠ t ♠①♠♠ ♦♥t♦♥ ♥♦ tt u∗(t) ♠①♠③s L ♥ [0, 1] t ♠♣♣♥
v → pT (t)f(t, x∗(t), y∗(t), v) = [p1(t)v + p2(t)(1− v)]x∗(t).
♥ p2(t) = 1 ♥ x∗(t) > 0 ♦r t ∈ [0, T ] ts s ♦♥ r♦♠ t t tt x0 > 0
t ♠♣♣♥ t♦ ♠①♠③ ♥ s♠♣ t♦ v → (p1(t)− 1)v s ♥ tt t
s②st♠ s t♠ ♥r♥t tt
u∗(t) =
1, p1(t) > 1,
0, p1(t) < 1.
♥ p1(T ) = 0 ♥ p1(·) s ♦♥t♥♦s ∃ b > 0 st u∗(t) = 0 ∀t ∈ [T − b, T ] s r♦♠
tDαT p1(t) = 1 ♥ ② rs ♥trt♦♥ ♦t♥
p1(t) =(T − t)α
Γ(α+ 1).
♦s② tt ♦r t∗ = T − (Γ(α+1))1
α ♦t♥ p1(t∗) = 1 ♦ t s tr♠♥ t
♦♣t♠♦♥tr♦ ♦r t < t∗ ♥ ♥♣♥♥t② ♦ t ♦♥tr♦ p1(·) r♠♥s ♠♦♥♦t♦♥②
rs♥ ♦r t < t∗ u∗(t) = 1 ♥ ts
tDαt∗p1(t) = p1(t).
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s♦t♦♥ ♦ ts ♥r rt♦♥ r♥t qt♦♥ s ♥ ②
p1(t) = p1(t∗)Φα(t
∗, t),
r p1(t∗) = 1 ♥ Φα(t
∗, t) s t ♥ t sr tt ♥ ♦♠♣t
② t ttr ♥t♦♥ ♥ ♥ ♣tr
② stt♥ β = α A = [1] ♥ ② r♣♥ t ② t∗ − t = T − Γ(α+1)α−1
− t ♦♥
tt
p1(t) = (t∗ − t)α−1∞∑
k=0
(t∗ − t)kα
Γ((k + 1)α).
♦t tt α = 1 t♥ ss s♦t♦♥ eT−t−1 ♥ t ♦♣t♠
♦♥tr♦ u∗, ♥ s② ♦♠♣t t ♦♣t♠ trt♦r② stss x∗(0) = x0 ♥
C0D
αt x
∗(t) =
x∗(t), t ∈ [0, t∗],
0, t ∈ [t∗, T ].
❲ ♥ ♦♠♣t t ♦♣t♠ trt♦r② x∗(t) ② t ♥r③t♦♥ ttr ♥t♦♥
♦r t ∈ [0, t∗], x∗(0) = x0, ♦♥ tt
x∗(t) = Eα(atα)
= x0Eα(tα)
= x0
∞∑
k=0
tkα
Γ(kα+ 1).
♦t tt α = 1 t♥ ss s♦t♦♥ x0et
♦ ♦♠♣t t ♦♣t♠ trt♦r② x∗(t) ♥ t ♥tr [t∗, T ], u∗(t) = 0,
x∗(t) = x∗(T ), ♦♥ tt
x∗(t) = x∗(t∗)
= x0Eα((t∗)α)
= x0
∞∑
k=0
(T − (Γ(α+ 1))1
α )kα
Γ(kα+ 1).
♦s ♦♣t♠♠ ♦st s
y∗(T ) = x0Eα((T )α).
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♦t tt α = 1 t♥ ss s♦t♦♥ x0eT−1
♦♥s♦♥
s ♣tr ♦♥r♥s t rt♦♥ ♦ ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♥ t ♦r♠ ♦
P♦♥tr②♥ ♠①♠♠ ♣r♥♣ ♦r ♥♦♥♥r P ♦s r♥t qt♦♥ ♥♦s
t ♣t♦ rt ♦ t stt r t rs♣t t♦ t♠ ❯♥r ♠ ss♠♣t♦♥s ♦♥
t t ♦ t ♣r♦♠ t ♣r♦♦ ♥♦ t rt ♣♣t♦♥ ♦ rt♦♥ r♠♥ts
ts ♦♥ t ♦t♥ s r♠♥t ♦ ♦♥rt♥ t P ♥t♦ ♦♥♥t♦♥ ♦♥ ♥
t♥ ①♣rss t ♦♣t♠t② ♦♥t♦♥s ♦r ts ①r② ♣r♦♠ ♥ t rt♦♥
rt ♦♥t①t ♥♦tr ♥trst♥ ♥♦t② ♦♥ssts ♥ t t tt ♥ ♥ ♠♦st
P ♦r♠t♦♥s ♦♥sr t ♦st ♥t♦♥ ♥ ② rt♦♥ ♥tr ♦
♠♥♥♦ t②♣
s♠♣ ①♠♣ strt♥ t ♣♣t♦♥ ♦ ♦r ♠①♠♠ ♣r♥♣ s ♣rs♥t
♦♣t♠♦♥tr♦ strt② s ♦♠♣t ♥②t② ♥ t rt♦♥ r♥t ♦♥t
qt♦♥ s♦ ② s♥ t♥q s ♦♥ ♥r③t♦♥ ttr ♥t♦♥
♥tr sq ♦ ts ♣tr ♦♥r♥s t ♥♥ ♦ t ss♠♣t♦♥s ♦♥ t t ♦
t ♣r♦♠ ♦t② t ♠r ♠srt② ♣♥♥ ♦ t ②♥♠s t rs♣t t♦
t♠ ♥ t♦ t ♦♥tr♦ rs s rt♥② rqr ♠♦r s♦♣stt rt♦♥
r♠♥ts ♥ t s ♦ ♠t♦s ♥ rsts ♦ ♥♦♥s♠♦♦t ♥②ss ♥♦tr rt♦♥ ♦
rsr ♦♥ssts ♥ ♥rs♥ t strtr ♦ t P ② ♦♥sr♥ t♦♥ stt
♥♣♦♥t ♦♥str♥ts ♥ stt ♥♦r ♠① ♦♥str♥ts ♥ ts ♦r♠t♦♥ ♥ ts s
t♦♥ rrt② ss♠♣t♦♥s ♥ t♦ ♥sr tt t ♦t♥ ♥ssr②
♦♥t♦♥s ♦ ♦♣t♠t② ♦ ♥♦t ♥rt
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♣tr
rt♦♥ ♥trt♦♥ ♥ sr
♦♥♣ts
♥ ts ♣tr ♥tr♦ s♦♠ rsts tt ♦ r♥ ♥ t ♥①t
♣trs r ♦s ♦r tt♥t♦♥ ♦♥ t ♥t♦♥ ♦ rt♦♥ ♥tr t rs♣t
t♦ ♥r ♦♥ ♠srs tt s t rt♦♥ tts ♥tr t♦ t t tt
ts ♠srs ♣♣r s ♠t♣rs ss♦t t stt ♦♥str♥ts ♥ t ♠①♠♠
♣r♥♣ ♦r Ps ❲ ♦♣t t ♠r rt♦♥ ♥tr ♦♣rt♦r ♥ ts ♣r♦♣rts
t♦ ♦r ♣r♣♦ss ♣rrqsts ♦r ♥rst♥♥ ♦r ♠♥ rsts r ♣rs♥t
♥ t rst st♦♥ ♦ ts ♣tr s ts ♦♥♣ts r ♠♣♦rt♥t t♦ ♦♣ t ♠♥
rsts
♥tr♦t♦♥
♥ ts st♦♥ ♣rs♥t s♦♠ s ♦♥♣ts ♦ t t♦r② ♦ ♠sr ♥ ♥trt♦♥
rqr t♦ ♦♣ t ♥ssr② ①r② rsts
srs
♠sr t♦r② s ♥t♠t② ♦♥♥t t ♥trt♦♥ ♣② t ♦♥♣t ♦
♠sr ♥r③s t ♦♥♣t ♦ ♥t ♦ ♥ ♥tr r ♦ rt♥ ♦♠ ♦
♣r♣♣ t t♦r② ♦ ♠sr ♥ ♥trt♦♥ ♥ ♦♥ ♥ ♥♠r
♦ ♦♦s s rtr ♥ ❱♥ r♥t ❬❪ ♦♥ ❬❪ t♥s♦♥ ❬❪ ②♦r ❬❪
♦♠s♦♥ ❬❪ ♦♠s♦♥ t ❬❪ ♥ ❨ t ❬❪
♠s ♦ sts tt sr s t ♦♠♥s ♦ ♠srs s s r σ−r ♥
♦r σ−r r ♥ ♥ ♣♣♥①
t X st q♣♣ t σ−r Ω st ♥t♦♥ µ(·) s ♠sr ♦♥
Ω ♦r ♦♥ (X,Ω) t stss t ♦♦♥ ♦♥t♦♥s
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µ(∅) = 0, r ∅ s ♥ st
Sj ⊂ Ω s ♦♥t ♦t♦♥ ♦ s♦♥t sts t♥
µ
⋃
j
Sj
=∑
j
µ(Sj).
♣r (X,Ω) s ♠sr s♣ ♥ t sts ♥ Ω r ♠sr sts
♦r♦r µ(·) s ♠sr ♦♥ (X,Ω), t♥ tr♣ (X,Ω, µ) s ♠sr s♣
♦♠ ♦ t s ♣r♦♣rts ♦ ♠srs r s♠♠r③ ♥ ♣♣♥①
♠sr µ : B(X) → [0,∞] s ♦r ♠sr µ(K) < ∞ ♦r r② ♦♠♣t
st K ⊂ B(X), r B(X) s t ♦r σ−r ♥ X. ♦♠ ♠srs s s rr
♠sr ♦tr rr ♠sr ♥♥r rr ♠sr ♥ ♦♥ ♠sr r ♥ ♥
♣♣♥①
♥t♦♥ t s♥r ♠srs ♦ ♠srs µ ♥ ν, ♥ ♦♥ σ−r
♦ X, r ♠t s♥r ♠srs tr r s♦♥t sts A ♥ B s tt
X = A∪B, µ s ③r♦ ♥ ♠sr ssts ♦ A, ♥ ν s ③r♦ ♥ ♠sr
ssts ♦ B.
♦r t ♣rtr s r ν s t s ♠sr ♦♥ X, s♠♣② s② tt t
♠sr µ s s♥r
♥t♦♥ s♦t② ♦♥t♥♦s ♠srs ♠sr ν ♦♥ ♠sr s♣
(X,Ω) s s♦t② ♦♥t♥♦s t rs♣t t♦ t ♠sr µ ♦♥ t s♠ ♠sr s♣
rt ν ≪ µ ♦r ♥② ♠sr st E ∈ Ω,
µ(E) = 0 ⇒ ν(E) = 0.
♥t♦♥ t♦♠ ♠sr ♥ t♦♠ ♠sr ♦r srt ♠sr s ♠sr
tt ♦♥② ts ♥♦♥③r♦ s ♦♥ srt ssts
st A ∈ Ω s ♥ t♦♠ ♦ µ, µ(A) > 0, ♥ ♦r r② V ∈ Ω, t V ⊂ A,
♦♥ s µ(V ) = 0 ♦r µ(V ) = µ(A).
rr ♦r ♠sr ♥ rtt♥ s ♥ t ♥♦♥ ♦♠♣♦st♦♥ ♦ t s♠ ♦ ♥
s♦t② ♦♥t♥♦s ♠sr (µac), s♥r ♦♥t♥♦s ♠sr (µsc) ♥ ♥ t♦♠
♠sr (µa), tt s
µ = µac + µsc + µa.
♥trt♦♥
ss ♥t♦♥ ♦ ♥ ♥tr s rst② ♣r♦♣♦s ② ② ♥ tr ♦♣
② ♠♥♥ ♦ ♥ t ♥tr∫ b
af(x)dx s ♠t ♦ t s♦ ♠♥♥ s♠s
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t♥s♦♥ ❬❪ ts ♦♥ ♠♥♥ ♥trs r ♣rs♥t ♥ ♣♣♥①
♥t♦♥s ♦r t ♠♥♥ ♥tr ①sts r s t♦ ♥tr ♥ t ♠♥♥
s♥s ♥ ♦rr ♦r t ♥t♦♥ f(·) t♦ ♠♥♥♥tr t s ♥ssr② tt t
s♦ ♦♥ ♥ tt t ♦s ♥♦t ①t ♥② ♣♦♥t ♦ ♠t♦♥ ♦ ♥♦♥s♠♠
s♦♥t♥ts ♣r♦♦ ♦ ts stt♠♥t ♥ ♦♥ ♥ ♠♥② ♦♦s s rtr
♥ ❱♥ r♥t ❬❪ ♦♥ ❬❪ ②♦r ❬❪ ♥ ❨ t ❬❪ s ts s♥♥t
rstrtt② ♥r♥t t♦ t ♥t♦♥ ♦ ♠♥♥ ♥tr ♠♦tts t ♥tr♦t♦♥ ♦
t s ♥tr
s ♥tr s ♥r③t♦♥ ♦ t ♠♥♥ ♥tr ♦r ♥st♥ t♦ st②
t ♠♥♥ ♥tr ♦♥ ♥s t♦ s t ♥tr ♦ ♥trt♦♥ ♥t♦ ♥t ♥♠r
♦ s♥trs t ♥ t s ♥tr t ♥tr s s ♥t♦ ♠♦r ♥r
sts ♠sr sts ♥ t ♦tr ♥ t s ♥tr ♠s ♥♦ st♥t♦♥
t♥ ♦♥ ♥ ♥♦♥ st ♥ ♥trt♦♥ r r ♥♠r♦s ♦♥♥♥
r♠♥ts ♦r ♦♥sr♥ t s ♥trt♦♥ s rtr ♥ ❱♥ r♥t ❬❪
rr ❬❪ ♦♥ ❬❪ t♥s♦♥ ❬❪ ②♦r ❬❪ ♥ ❨ t ❬❪ ts
♦♥ s ♥tr ♦r s♠♣ ♥t♦♥ ♦♥ ♠sr ♥t♦♥s ♥ ♥♦♥♥t
♥t♦♥s r ①♣♦s ♥ ♣♣♥①
♦ r ♦♥ t♦ ♣rs♥t t ♥t♦♥ ♦ rtr ♥r③t♦♥ ♦ ♥trs t♦ t
s♦ t tts ♥tr
♥t♦♥ t ψ(·) ♥t♦♥ ♥ ♦♥ t ♥tr t ∈ [a, b], ♥ µ(·)
♦♥ ♠sr t s ♣rtt♦♥ t ♥tr [a, b] ② n ♣♦♥ts a < t1 < t2 < · · · < tn < b,
♥ ♣t t0 = a, tn+1 = b, ∆ti = ti+1 − ti, i = 0, 1, . . . , n, ♥
Di =
[ti, ti+1[, i = 0, · · · , n− 1,
[ti, ti+1], i = n.
② ♦♦s♥ ♣♦♥t ξi ♥ Di tt t ♠t
limmax∆ti→0
n∑
i=0
ψ(ξi)µ(Di),
s t tts ♥tr ♦ t ♥t♦♥ ψ(·) t rs♣t t♦ t ♠sr µ(·), ♥
s ♥♦t ②∫
[a,b] ψ(t)dµ(t). r µ(Di) = µ(t−i+1) − µ(t−i ), ♦r i = 0, · · · , n − 1, ♥
µ(Dn) = µ(b+)− µ(t−n ).
② ♦♥ ♠sr t s ♠♥t ♥ ♥♥r rr ♦r ♠sr s ♥t♦♥ ♥
♣♣♥① ♦r ♠♦r ts ♥ ♠♣♦rt♥t ♣r♦♣rts ♦ t tts ♥tr s
r② ❬❪ rtr ♥ ❱♥ r♥t ❬❪ t♥s♦♥ ❬❪ ♥ ♦♠s♦♥ ❬❪
s ♠♥t♦♥ ♦r ♦r ♦s s ♦♥ rt♦♥ ♥trs ♠♦♥ t ♠♥♥
♦ ♦♥ s t ♠♦st ♦♠♠♦♥ ♥ ♦rr t♦ ♥stt ♦r ♥ ①♣♦t t
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rt♦♥ t♥ t ♠♥♥♦ ♥ ♠r rt♦♥ ♥trs r② stt
♥ ♣tr ♥①t ♥t♦♥ s♣②s t ♠♥♥♦ rt♦♥ ♥tr ♦r
♥t♦♥ t rs♣t t♦ ♥♦tr ♥t♦♥
♥t♦♥ t h : [a, b] → R ♠♦♥♦t♦♥ ♥rs♥ ♥t♦♥ ♦♥ (a, b], ♥ h′ ts
♦♥t♥♦s rt ♦♥ (a, b). rt♦♥ ♥tr ♦ ♦rr α > 0 ♦ ♥t♦♥ f(·)
t rs♣t t♦ t ♥t♦♥ h(·) ♦♥ [a, b] s ♥ s
aIαt;hf(t) =
1
Γ(α)
∫ t
a
(h(t)− h(τ))α−1 f(τ)h′(τ)dτ.
♦t tt h(t) = t, t ∈ [a, b], t rt♦♥ ♥tr aIαt;hf(t) t s ♠♥♥
♦ rt♦♥ ♥trt♦♥ ♠♦ t ❬❪
♦ ♣rs t t ② ♣r♣♦ss ♦ ts ♣tr
rt♦♥ ♥trt♦♥ t rs♣t t♦ sr
♥ ts st♦♥ ♥ t ♥tr ♥ rt♦♥ ♦♥t①t t rs♣t t♦ ♥r
♦♥ ♠sr ♥ ♥tr♦ t ♥ t ♠r rt♦♥ ♥tr ♦r♠
♥t♦♥ t f(·) ♥t♦♥ ♥ ♦♥ t ♥tr t ∈ [a, b], ♥ µ(·)
♦♥ ♠sr t s ♣rtt♦♥ t ♥tr [a, b] ② n ♣♦♥ts a < t1 < t2 < · · · < tn < b,
♥ ♣t t0 = a, tn+1 = b, ∆ti = ti+1 − ti, i = 0, 1, . . . , n, ♥
Di =
[ti, ti+1[, i = 0, · · · , n− 1,
[ti, ti+1], i = n.
② ♦♦s♥ ♣♦♥t ξi ♥ Di t♦ t ♠t
1
Γ(α+ 1)lim
max∆ti→0
n∑
i=0
f(ξi)(µ(Di))α,
t rt♦♥ tts ♥tr ♦ ♦rr α ♦ t ♥t♦♥ f(·) t rs♣t t♦ t ♠sr
µ(·), ♥ ♥♦t ②1
Γ(α+ 1)
∫
[a,b]f(t)(dµ(t))α,
r α > 0. α = 1, t ss tts ♥tr s stt ♥ t
♥t♦♥
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Pr♦♣♦st♦♥ t µ(·) ♣♦st ♦r ♠sr ♦♥ R. ♥ t rt♦♥ ♥tr
♦r t ♥t♦♥ f : R → [0,∞] t rs♣t t♦ t ♠sr µ(·) s ♥ ②
aJαt;µf(t) =
1Γ(α+1)
∫
[0,t] f(t)(dµ(t))α, t ∈ Supp(µc),
1Γ(α+1)
∫
[0,t) f(t)(dµ(t))α + V, t ∈ Supp(µa),
r V = f(t)(µ(t))α
Γ(α+1) µ(t) = µ(t+) − µ(t−), µa s t♦♠ ♠sr µc = µac + µsc s
t♦♠ss ♠sr µac ♥ µsc r ♥ ♦r ♥ aJαt;µ s ♦♣rt♦r ♦ ♠r rt♦♥
♥tr t rs♣t t♦ t ♠sr µ(·).
Pr♦♦ ♦ t ♣r♦♦ s s♥ s♦♠ ♥ ♦ rs t♦ tr♥s♦r♠ t ♥tr
tts ♥tr t♦ s ♥tr ♥ t s ①♣rss♦♥ ♦ t rt♦♥ ♥tr
s ♦♥♦t♦♥ ♥ ② ♥ ♥tr ♥tr s rtt♥ ♦♥ ♥ tr♠s ♦ t ♦r♥
♣r♠tr③t♦♥
♦ ss t♦ ♦♥sr t rst ♦♥ ♥ t ♠sr t♦♠ss t ∈ Supp(µc),
♥ ♥ t♦♠ ♠sr t ∈ Supp(µa).
❲♥ t ∈ Supp(µc), µc = µac + µsc, t ♥tr tts ♥tr ♥ ②
∫
[0,t]f(t)dµ(t).
② s♥ t t♠ ♥ ♦ rs ♥ ♦r t ∈ [0, T ], tr s s ∈ [0, µ([0, T ])]
sts②♥
s(t) =
∫
[0,t]dµ(t), ds = dµ(t)).
♦r♦r ♦r ♥tr ♥t♦♥s f : [0, T ] → Rn, t = θ(s) ♥ s = σ(t), ①sts
f : [0, σ(T )] → Rn s tt
f(s) = (f θ)(s),
♦♥sq♥t② tr ①st s ♥tr tt s q♥t t♦ t tts ♥tr tt
♠♥t♦♥ ♥ s tt
∫
[0,t]f(t)dµ(t) =
∫
[0,s]f(s)ds.
♦ ♦♥sr t rt♦♥ ♥tr ♦ f(·) ♦r α ∈ (0, 1], ♥ ①♣rss ♥
t r♣r♠tr③ t♠ r s ♦♦s
1
Γ(α)
∫ s
0f(s)(s− s)α−1ds.
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② s♥ t ss♠♣t♦♥s stt ♦r t♥ stss
1
Γ(α)
∫
[0,s]f(σ(t))(σ(t)− σ(t))α−1dσ(t).
♥ t > t, tt
µ(t)− µ(t) =
µ([t, t)), t < T,
µ([t, T ]), t = T.
s ② s♥ ♥ ♦ rs ♥ ♦♥ tt
1
Γ(α)
∫
[0,t]f(t) (µ(t)− µ(t))α−1 dµ(t).
s qt♦♥ r♣rs♥t ♠♥♥♦ rt♦♥ ♥tr t rs♣t t♦ ♠sr µ(·)
♦r t s ♥ t ∈ Supp(µc), ② ♣♣②♥ t rt♦♥ t♥ ♠♥♥♦
♥ ♠r rt♦♥ ♥tr ♦♣rt♦r s ♥ ♣tr r t ♥trt♥ ♠sr
dµ(t) r♣s dτ. ♦♥sq♥t② t rt♦♥ ♥tr t rs♣t t♦ t ♠sr dµ(·)
♥ ♠r rt♦♥ ♥tr ♦r♠ s ♦r t ∈ Supp(µc) ♥ ②
1
Γ(α+ 1)
∫
[0,t]f(t)(dµ(t))α.
❲♥ t ∈ Supp(µa), t ♥t♦♥ f(·) s♦♥t♥t② t ♣♦♥t t ∈ [t−, t+], t♥
1
Γ(α)
∫ µ(t+)
µ(t−)(µ(t+)− s)α−1f(t)m(s)ds,
r m(s) s t ♠sr strt♦♥ ♥t♦♥ ♦ t ♠sr µ sts②
m(s) =
1, s ∈ [µ(t−), µ(t+)],
0, otherwise.
♦♥sq♥t② ♦♥ tt
f(t)
Γ(α+ 1)(µ(t+)− µ(t−))α.
♦ ♥ rt ts qt♦♥ s ♦♦s
f(t)
Γ(α+ 1)(µ(t))α ,
r µ(t) = µ(t+)− µ(t−). s t ♥r s ♥ t ∈ Supp(µa) s ♥ ②
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1
Γ(α+ 1)
∫
[0,t)f(t)(dµ(t))α +
1
Γ(α+ 1)f(t)(µ(t)α.
♠r ❲♥ t ∈ Supp(µc) ∪ Supp(µa), ♥ rt t ♠sr s t ♥♦♥
♦♠♣♦st♦♥ ♦ t♦♠ ♥ ♥♦♥t♦♠ ♠sr µ = µc + µa, s ♦♦s
1
Γ(α+ 1)
∫
[0,t]f(t)(dµc(t))
α +∑
ti≤t
f(ti)
Γ(α+ 1)(µ(ti))
α .
♥♠♥t Pr♦♣rts
r ♥tr♦ s♦♠ ♣r♦♣rts ♦r t rt♦♥ tts ♥tr r
♥♦♥ ♥ t trtr ♦r s rt♦♥ ♥tr s ♥ ♠r ♦♥t①t s
s t ♥ ♦ r st♥r ♦r♠ ♦ ♥ ♦ r ♦ rt♦♥ ♥tr
♥tr♦ ② ♠r ❬ ❪
∫
f(x)(dx)α =
∫
f(g(t))(g′(t))α(dt)α, α ∈ (0, 1),
r ♠ t ssttt♦♥ x = g(t) ♥ g(t) ♥♦♥rs♥ r♥t
♥t♦♥ ❲ r s ♦r♠ ♦r rt♦♥ tts ♥tr s ♦♦s rst
♦♥sr strt② ♥rs♥ ♥t♦♥ g : R → R ♦♥ t ♥tr I, ♦r t1, t2 ∈ I
t t1 < t2 t♥ g(t1) < g(t2). rtr♠♦r t t ♥t♦♥ g(·) ♦♥t♥♦s ♥
strt② ♥rs♥ ♦♥ t ♥tr I, t♥ ♥ s② tt g(I) ♥ ♥tr ♥ ②
g(I) = g(t) : t ∈ I.
Pr♦♣♦st♦♥ rt♦♥ tts ♥ ♦ r t I ♥② ♥tr ♥
g : R → R ♦♥t♥♦s ♥t♦♥ strt② ♥rs♥ ♦♥ t ♥tr I. t µ : R → R
♦♥ ♠sr ♥
∫
I
(f g)(s) (d(µ g)(s))α =
∫
g(I)f(t)(dµ(t))α,
r (f g)(s) ♥♦ts t ♦♠♣♦st♦♥ ♦ f(s), ♥ g(s) s ♥ ② (f g)(s) = f [g(s)] ,
♦r s ∈ I.
Pr♦♦ ♦r t s ♦ s♠♣② ss♠ tt t = g(s), µ(s) = (µg)(s), f(s) = (f g)(s)
♥ ds = (g−1)′(t)dt.
s ♥♦ t ♠sr ♥♦♥ ♦♠♣♦st♦♥ s tt ♦r µc = µac + µsc,
µ = µc + µa sts②♥
dµ = dµc + dµa.
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r♦r
dµ = dµc + dµa.
② ♥t♦♥ ♦ µ(·), ♦r A ⊂ I
∫
A
dµ(s) =
∫
g(A)∩Sc
dµc(t) +∑
ti∈g(A)∩Sa
µa(ti),
r Sc = Sac+Ssc ♥ Sa r t s♣♣♦rts ♦ t ♠srs µc ♥ µa, rs♣t② t
B t s♣♣♦rt ♦ t ♠sr µ s tt µ(B) = ‖µ‖TV , ♥ µ(Bc) = 0.
♦r♥ t♦ t rsts ♦ t♦♥ ♦r s ∈ [0, S], ♥ t ∈ [0, T ],
σ(s) =
∫
[0,s](dµc(s) + dµa(s))
=
∫ s
0dµc(s) +
∑
si∈[0,s]
µa(si).
♦♥sq♥t②
dσ(s) = dµc(s) + µa(s)δs,
r δs s r t ♠♣s rtr♠♦r ♦r s1 ≤ S ♦♥ tt
1
Γ(α+ 1)
∫
[0,s1]f(s)(dµ(s))α =
1
Γ(α)
∫
[0,s1]f(s)(σ(s1)− σ(s))α−1dσ(s).
② s♥ t ♥t♦♥ ♦ σ(·). ❲
1
Γ(α+ 1)
∫
[0,s1]
f(s)(dµ(s))α =1
Γ(α)
∫
[0,s1]
f(s)
∫ s1
s
dµc(s) +∑
si∈[s,s1]
µa(si)
α−1
dσ(s)
r s t ss♠♣t♦♥s ti = g(si), f(t) = (fg−1)(t), µa(ti) = µa(g−1(ti)),
♥ µc(t) = (µc g−1)(t) s tt dµc(t) = d
(
(µc g−1)(t)
)
(g−1)′(t). ♥ t rt
♥ s ♦ stss
1
Γ(α)
∫
[0,t1]f(t)
∫ t1
t
dµc(t) +∑
ti∈[t,t1]
µa(ti)
α−1
dµ(t),
r dµ(t) = dµc(t) + µa(t)δt. r♦r
1
Γ(α)
∫
[0,t1]f(t) (µ([t, t1]))
α−1 dµ(t) =1
Γ(α+ 1)
∫
[0,t1]f(t) (dµ(t))α .
Pr♦♣♦st♦♥ r ♣r♦
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♣tr
ssr② ♦♥t♦♥s ♦ ♣t♠t②
♦r rt♦♥ ♦♥s♠♦♦t
r♥t ♥s♦♥ Pr♦♠s t
tt ♦♥str♥ts
♥tr♦t♦♥
r♥t ♥s♦♥s ♦ rt♦♥ ♦rr r♥t② ♥ rss ② sr rsrr
♦r ♠♥② ♣r♦♠s ♥ sr rsts rt t♦ rt♦♥ r♥t ♥s♦♥s
♣♣r ♥ t trtr s ♠ ♥ t♦②s ❬❪ ♥♦r t ❬❪
♥♦r t ❬❪ r♥ ❬❪ ② ♥ r♠ ❬❪ ♥ ♠♦ ♥ ③②s
❬❪
♠♥ ♦♥trt♦♥ ♦ ts ♣tr s t ♦r♠t♦♥ ♦ ♥ssr② ♦♥t♦♥s ♥ t ♦r♠
♦ ①♠♠ Pr♥♣ ♦r ♥♦♥s♠♦♦t ♦♣t♠♦♥tr♦ ♣r♦♠ t stt ♦♥str♥ts ♥
t ②♥♠s ♥ ② rt♦♥ r♥t ♥s♦♥ r ♦♥sr t ♠r
rt♦♥ rt
x(α)(t) ∈ F (t, x(t)), L − ,
r x(α)(·) s rt♦♥ ♠r rt ♦ ♦rr α ∈ (0, 1), ♥ F (t, x(t)) s st
♠♣ ♠t♥t♦♥ ♥ ♦♥ [a, b]× Rn.
♦♥sr t ♦♦♥ rt♦♥ r♥t ♥s♦♥ ♣r♦♠
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(P )
Minimize g(x(b))
subject to x(α)(t) ∈ F (t, x(t)), t ∈ [a, b],
h(t, x(t)) ≤ 0, a ≤ t ≤ b,
x(a) ∈ C,
r t ♥t♦♥ g : Rn → R s ♥ ♦st ♥t♦♥ x(α) s t rt♦♥ ♠r
rt ♦ ♦rr α ∈ (0, 1], F (t, x) s ♥ ♠t♥t♦♥ ♥ ♦♥ [a, b] × Rn, t
♥t♦♥ h : [a, b] × Rn → R ♥s t ♥ ♥qt② stt ♦♥str♥t x(a) r ♥t
♣♦♥t ♥ C s ♥ sst ♦ Rn.
s trt♦r② ♦ t ♣r♦♠ (P ) s s♦t♦♥ ♦ rt♦♥ r♥t ♥s♦♥
sts②♥ t ♦♥str♥ts ♦♥ t ♣r♦♠ (P ). ❲ s② tt x∗ ♦ ♠♥♠♠ ♦
t ♣r♦♠ (P ), t ♠♥♠③s t ♦t ♥t♦♥ ♦r ♦tr s stts x ∈ Rn,
♥ s♦♠ ♥♦r♦♦ s tt |x− x∗| ≤ ε.
♥t♦♥ t x : [a, b] → Rn ♥ ♦♥t♥♦s ♥t♦♥ ❲ s② tt ts
♥t♦♥ s ♥ Ω ⊂ [a, b] × Rn, t ♣♦♥t (t, x(t)) s ♥ Ω, ♦r t ∈ [a, b]. t ε > 0
s♠ ♣♦st ♦♥st♥t ♥ ε− t ♦t x s ♥ ②
T (x; ε) = (t, x) ∈ [a, b]× Rn : a ≤ t ≤ b, |x− x(t)| ≤ ε.
t−st♦♥ ♦ Ω ⊂ [a, b]× Rn ♥ ②
Ωt = x ∈ Rn : (t, x) ∈ Ω, ∀t ∈ [a, b].
♠t♥t♦♥ F (t, ·) s ♣st③ ♦ r♥ k(t) ♦r x1 ♥ x2, ♦r
η1 ∈ F (t, x1) tr ①sts η2 ∈ F (t, x2) s tt
|η1 − η2| ≤ k(t) |x1 − x2| .
♠t♥t♦♥ F (·, x) s s ♠sr ♦r ♦♣♥ st C ♥ Rn t st
t ∈ [a, b] : F (t, x) ∩ C 6= 0 ,
s s ♠sr ♦r x ∈ Rn. ♠srt② ♦ F ♥ ♥ q♥t②
t st C s ♥ rtrr② ♦s st s r ❬❪ ♥ ❱♥tr ❬❪ ♦r♦r
♠t♥t♦♥ F (·, ·) s ♠sr② ♣st③ ♦♥ Ω ⊂ [a, b]×Rn, ♦r x ∈ R
n, t
♠t♥t♦♥ F (·, x) s ♠sr ♦♥ [a, b], ♥ ♦r t ∈ [a, b] t ♠t♥t♦♥
F (t, ·) s ♥♦♥♠♣t② ♥ ♣st③ ♦ r♥ k(t) ♦♥ Ωt.
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s ②♣♦tss
♥t♦♥ g(·) s ♣st③ ♦♥ Ωb ♦ r♥ Kg, s tt
|g (x1(b))− g (x2(b))| ≤ Kg |x1(b)− x2(b)| .
F (·, x) s ♦s ♥ ♦♥① ♦♥ Ω
F s α−♥tr② ♦♥ ♦♥ Ω, s tt tr s ♥ α−♥tr ♥t♦♥
φ : [a, b] → R, s tt ♥② ♠sr st♦♥ η(t) ♦r ♠t♥t♦♥ F (t, x)
stss
|η(t)| ≤ φ(t).
♠t♥t♦♥ (t, x) → F (t, x) s ♠sr② ♣st③ ♦♥ Ω ⊂ [a, b] × Rn,
tt s ♦r x ∈ Rn, ♠t♥t♦♥ F (·, x) s ♠sr ♥ t ∈ [a, b], ♥ ♦r
t ∈ [a, b], ♠t♥t♦♥ F (t, ·) s ♣st③ ♦ r♥ K.
♥t♦♥ h s ♣♣r s♠♦♥t♥♦s ♥ tr ①sts ♦♥st♥t Kh > 0 s
tt t ♥t♦♥ h(t, ·) s ♣st③ ♦ r♥ Kh ♦♥ Ω ♦r t ∈ [a, b], t stss
|h(t, x1)− h(t, x2)| ≤ Kh |x1 − x2| .
t t ♠t♦♥♥ H ♦r t ♣r♦♠ (P ) s ♥t♦♥ H : Ω× Rn → R ♥ ②
H(t, x, p) = max〈p, v〉 : v ∈ F (t, x),
s ♣tr s ♦r♥③ s ♦♦s ♥ t ♥①t t♦♥ ♥tr♦ ♥♦tt♦♥s
♥t♦♥s ♥ ♣r♠♥r② ts s tr ♥ ts ♣tr ♥ t♦♥ t
♥ssr② ♦♣t♠ ♦♥t♦♥s r stt ♥ ♣r♦ s ♦♥t♦♥s r s ♥ ♥
ss♥t ② ♥ t ♣r♦♦ ♦ t ①♠♠ Pr♥♣ ♦r ♥♦♥s♠♦♦t rt♦♥ ♦♣t♠
♦♥tr♦ ♣r♦♠s rss ♥ ♣tr
①r② ♥ sts
♥ ts st♦♥ ♣rs♥t ♥ sss ② rsts ♦♥ ①st♥ s♦t♦♥s ♦r rt♦♥
r♥t ♥s♦♥s s s ♦♥ t ♦♠♣t♥ss ♦ t st ♦ trt♦rs ♦r ts ss
♦ ②♥♠ s②st♠s s ♦♥ssts ♥ ♣t♥ t rsts ♣r♦ ♥ r ❬❪
r♦♠ t ♥tr t♦ t rt♦♥ ♦♥t①t
s♣r♠♠ ♥♦r♠ ‖·‖ s ♥ ②
‖x‖ := maxt∈[a,b]
|x(t)|,
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r |·| s ♥ ♥♦r♠
t ts ♣♦♥t s♦♠ s ♥t♦♥s ♥ ♣r♦♣rts ♦♥ t ②♥♠s r stt t
v(·) : [a, b] → Rn s ♠sr st♦♥ ♦ t st ♠♣ F (·, x(·)), tt
s
v(t) ∈ F (t, x(t)), L −
s♦t♦♥ x(t) ♦ rt♦♥ r♥t ♥s♦♥ stss t rt♦♥ r♥t
qt♦♥
x(α)(t) = v(t),
♦r s♦♠ v(τ) s ♠sr st♦♥ ♦ F (τ, x(τ)), ♥ ♥tr t rs♣t t♦
(t− τ)α−1dτ, ♦r t ∈ [a, b], s tt x(·) sts②♥
x(t) = x(a) +1
Γ(α+ 1)
∫ t
a
v(τ)(dτ)α,
s ♥ r∫ t
a(·)(dt)α s ♠r rt♦♥ ♥tr ♦♣rt♦r s ♣tr
❲t♦t ♦ss ♦ ♥rt② ♦♥sr α ∈ (0, 1).
♥t♦♥ t dF (·,·)(·) : Ω × Rn → [0,∞] t st♥ ♥t♦♥ ss♦t t♦
♠t♥t♦♥ F : Rm → Rn, ♥ ②
dF (t,x)(v) := infz∈F (t,x)
|v − z|,
♥ x(·) s ♥ α−s♦t② ♦♥t♥♦s ♥t♦♥ r♦♠ [a, b] → Rn ②♥ ♥ Ω. ② ♥t♦♥
dα(x, F ) =
1
Γ(α+ 1)
∫ b
a
dF (t,x(t))(x(α)(t))(dt)α.
♦s② dF (t,x)(v) = 0, ♥ ♦♥② v ∈ F (t, x).
Pr♦♣♦st♦♥ ♦♥sr t ♠t♥t♦♥ F : [a, b] × Rn → R
n. ♥ ♦r ①
t ∈ [a, b], t st♥ ♥t♦♥ dF (t,x)(v) s ♣st③ ♦ r♥ k(t) s tt
∣
∣dF (t,x1)(v1)− dF (t,x2)(v2)∣
∣ ≤ k(t) |x1 − x2|+ |v1 − v2| ,
r t ♠♣ t→ dF (t,x)(v) s s ♠sr ♦r ♥② x ∈ Rn ♥ ♥② v ∈ R
n.
Pr♦♦ ❲ ♥ rt t t♥ s ♦ s ♦♦s
∣
∣dF (t,x1)(v1)− dF (t,x2)(v2)∣
∣ ≤∣
∣dF (t,x1)(v1)− dF (t,x2)(v1)∣
∣+∣
∣dF (t,x2)(v1)− dF (t,x2)(v2)∣
∣ .
♥ F (t, ·) s k(·) ♣st③ s tt
F (t, x1) ⊂ F (t, x2) + k(t) |x1 − x2|B1(0), ∀x1, x2 ∈ Rn,
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♦♥ tt
dF (t,x1)(v1) ≤ dF (t,x2)(v1) + k(t) |x1 − x2| , ∀v1 ∈ F (t, x1).
t ♦♦s r♦♠ ts rt♦♥ ♥ ② ①♥♥ t r♦s ♦ x1 ♥ x2, tt
∣
∣dF (t,x1)(v1)− dF (t,x2)(v1)∣
∣ ≤ k(t) |x1 − x2| .
t ε > 0 rtrr② s♠ ♥ r♦♠ t ♥t♦♥ ♦ dF (t,·)(·), tr s z2 ∈ F (t, x2)
s tt
dF (t,x2)(v1) ≥ |v1 − z2| − ε
≥ |v1 − z2 − v2 + v2| − ε
≥ |v2 − z2| − |v1 − v2| − ε
≥ dF (t,x2)(v2)− |v1 − v2| − ε.
♥ ε s rtrr② s♠ ♥ ② ①♥♥ t r♦s ♦ v1 ♥ v2, ♦♥ tt
∣
∣dF (t,x2)(v1)− dF (t,x2)(v2)∣
∣ ≤ |v1 − v2| .
♥② ssttt♥ ♥ ♥ t rt♥ s ♦ t t ♥qt②
♦ s♦ t ♠srt② ♦ t→ dF (t,x)(v). t rtrr② ♦♣♥ st Ck ♥
② Ck = z∗+εkB, r B s ♥ ♦♣♥ ♥t ♥tr t ③r♦ ♥ F s L−♠sr
t♥ t st
t ∈ [a, b] : Ck ∩ F (t, x) 6= ∅,
s L−♠sr ♥ Wk(z, t) = |v − z| : z ∈ Ck ∩ F (t, x). t s r tt ♦r
z, Wk(z, ·) s L−♠sr ② ♥♥ gk(t) = maxzWk(z, t), s♦ ♠② ssrt tt
gk(·) s L−♠sr ♦ ♦♥ tt
dF (t,x)(v) = limk→∞
gk(t),
s L−♠sr
♦r♠ t x(·) ♥ α−s♦t② ♦♥t♥♦s ♥t♦♥ ♥ t ε−t T (x; ε) ⊆
Ω, ♦r s♦♠ ♦♥st♥t ε > 0, ss♠ tt t ♠t♥t♦♥ F : Ω → Rn s ♣st③ ♦
r♥ k(t) ♦♥ Ω, ♥ dα(x, F ) < ε/K, r K = Eα
(
1Γ(α+1)
∫ b
ak(t)(dt)α
)
. ♥ tr
①sts rt♦♥ trt♦r② y(·) ♦r t ♠t♥t♦♥ F ②♥ ♥ t t T (x; ε) sts②♥
y(a) = x(a), ♥
‖x− y‖ ≤1
Γ(α+ 1)
∫ b
a
∣
∣
∣x(α)(t)− y(α)(t)
∣
∣
∣(dt)α ≤ Kd
α(x, F ) < ε.
Pr♦♦ ♦ t ♣r♦♦ s t♦ ♦♥strt sq♥ ♦ ♣♣r♦①♠t rt♦♥
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s♦t② ♦♥t♥♦s ♥t♦♥s xn(t), ♥♥♥ t x0(t) ≡ x(t), ♥ ② ♦♦s♥
x(α)n+1(t) s t ♦sst ♣♦♥t t♦ x
(α)n (t) ♦ t st F (t, xn(t)), tt ♦♥r ♦r
α−trt♦r② ♦r F. t vn(t) ♠sr st♦♥ ♦ ♠t♥t♦♥ F (t, xn(t))
s tt∣
∣
∣vn(t)− x(α)n (t)
∣
∣
∣= dF (t,xn(t))(x
(α)n (t)) .
t v0(t) ∈ F (t, x(t)) s tt
∣
∣
∣v0(t)− x(α)(t)
∣
∣
∣= dF (t,x(t))(x
(α)(t)) .
r♦♠ t ♦ v0(t) s α−♥tr ♥ ts
x1(t) = x(a) +1
Γ(α+ 1)
∫ t
a
v0(τ)(dτ)α.
s ♥ ♦♥sq♥t②
x(α)1 (t) = v0(t).
r♦r∣
∣
∣x(α)1 (t)− x(α)(t)
∣
∣
∣= dF (t,x(t))(x
(α)(t)).
② ♣♣②♥ t rt♦♥ ♠r ♥tr ♦♣rt♦r s s t st♥r ♠♦s
♥tr ♥qt② ♦♥ ♦t ss ♦ t
|x1(t)− x(t)| ≤1
Γ(α+ 1)
∫ b
a
dF (t,x(t))(x(α)(t))(dt)α.
r♦♠ ♥t♦♥ ♥ ♦♥t♦♥ ♦ ts rst ♦♥ tt
|x1(t)− x(t)| ≤ dα(x, F ) < ε/K.
s x1(t) s ♥ t t T (x; ε) ♥ tr♦r ♠② ♦♦s v1(t) ∈ F (t, x1(t))
s tt
∣
∣
∣v1(t)− x
(α)1 (t)
∣
∣
∣= dF (t,x1(t))(x
(α)1 (t)) .
s ♦r v1(t) s α−♥tr ♥ ♠② ♥ x2 ②
x2(t) = x(a) +1
Γ(α+ 1)
∫ t
a
v1(τ)(dτ)α.
♥
x(α)2 (t) = v1(t).
∣
∣
∣x(α)2 (t)− x
(α)1 (t)
∣
∣
∣= dF (t,x1(t))(x
(α)1 (t)).
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r♦♠ t ♣st③ ♦♥t♦♥
dF (t,x1(t))(x(α)1 (t)) ≤ dF (t,x(t))(x
(α)1 (t)) + k(t) |x1(t)− x(t)| .
♦s② r♦♠ ♦ x(α)1 (t) ∈ F (t, x(t)), tr♦r dF (t,x(t))
(
x(α)1 (t)
)
= 0, ♥ ts
♦♥ tt∣
∣
∣x(α)2 (t)− x
(α)1 (t)
∣
∣
∣≤ k(t) |x1(t)− x(t)| .
② ♥trt♥ ♦t ss ♥ ♥ s♥ ♦♥ tt
|x2(t)− x1(t)| ≤ dα(x, F )
1
Γ(α+ 1)
∫ t
a
k(τ)(dτ)α.
♦t tt ♥ rt
|x2(t)− x(t)| ≤ |x2(t)− x1(t)|+ |x1(t)− x(t)|
≤ dα(x, F ) 1
Γ(α+1)
∫ t
ak(τ)(dτ)α + d
α(x, F )
≤ dα(x, F )
[
1Γ(α+1)
∫ t
ak(τ)(dτ)α + 1
]
≤ dα(x, F )Eα
(
1Γ(α+1)
∫ t
ak(τ)(dτ)α
)
≤ Kdα(x, F ) < ε,
r t s♦♥ ♥ tr ♥qts r t♦ ♥ s x2 s ♥
T (x; ε) s t♦ rst st♣s r② s♦ ♦ t ♥t♦♥ ♣r♦ss ♥ ♦♥strt
s ♦ t t tt t sss ♦ts ♥ T (x; ε), ♠② ♥r② ♦♦s
vn(t) ∈ F (t, xn(t)) sts②♥ r vn(t) s α−♥tr sts②♥
x(α)n+1(t) = vn(t).
tMα
n (t) = aJαt
(
k(t1)aJαt1
(
k(t2) · · ·(
aJαtn−1
k(tn))))
= 1n!
(
1Γ(α+1)
∫ t
ak(τ)(dτ)α
)n
,
r aJαt(·)
(·) s ♠r rt♦♥ ♥tr ♦♣rt♦r s ♣tr
♦♥sq♥t② x(α)n+1(t) ∈ F (t, xn(t)) stss
∣
∣
∣x(α)n+1(t)− x(α)n (t)
∣
∣
∣≤ k(t) |xn(t)− xn−1(t)| .
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s t ♠t♠t ♥t♦♥ s t♦ t rrs rt♦♥
∣
∣
∣x(α)n+1(t)− x(α)n (t)
∣
∣
∣≤ d
α(x, F )k(t)Mαn−1(t), n = 1, 2, · · · .
r♦r
|xn+1(t)− xn(t)| ≤ dα(x, F )Mα
n (t), n = 0, 1, 2, · · · .
t st♣ xn(t) s ♥ t t T (x; ε). t ♦♦s tt
|xn(t)− x(t)| ≤ dα(x, F )
∑n−1j=0
(Mαn (b))j
j
≤ dα(x, F )Eα
(
1Γ(α+1)
∫ b
ak(τ)(dτ)α
)
≤ dα(x, F )K < ε.
r♦♠ t sq♥ x(α)n (t) s ② sq♥ ♥ Lα([a, b];Rn). t v(·) ∈
Lα([a, b];Rn) ♠t ♦ ts sq♥ r♦♠ tt xn
♦♥r♥s t♦ ♦♥t♥♦s ♥t♦♥ y t y(a) = x(a). t ♦♦s tt
v(t) ∈ F (t, y(t)) .
♦ r♦♠
xn(t) = x(a) +1
Γ(α+ 1)
∫ t
a
x(α)n (τ)(dτ)α, n = 1, 2, · · · ,
♦♥ tt
y(t) = x(a) +1
Γ(α+ 1)
∫ t
a
v(τ)(dτ)α.
r♦r
y(α)(t) = v(t).
♥ t rt♦♥ trt♦r② y(t) ♦r t ♠t♥t♦♥ F stss
‖y(t)− x(t)‖ ≤ Eα
(
1
Γ(α+ 1)
∫ b
a
k(t)(dt)α)
dα(x, F )
≤ Kdα(x, F ).
rst s ♣r♦
♦r♠ t Ω ⊂ [a, b]×Rn ♥ ♠t♥t♦♥ F : Ω → R
n ♥ ss♠ tt
F s L× B ♠sr t ♥♦♥ ♠♣t② ♦s ♦♥① s ♦♥ Ω ♥ tt F s tt
|F (·, x(·)| ≤ φ(·) r φ(·) s ♥ ss♥t② ♦♥ ♥ α− integrable ♥t♦♥ ss♠
s♦ tr s ♠t♥t♦♥ G : [a, b] → Rn ♥ ♣♦st ♥t♦♥ π(t) s tt
t ♦♦♥ ②♣♦tss r sts
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♦r t ∈ [a, b], G(t) + π(t)B ⊂ Ωt,
♦r t ∈ [a, b], x ∈ G(t)+π(t)B, t ♠t♥t♦♥ F (t, ·) s ♣♣r s♠♦♥s♦s
♦r (t, x) ♥ t ♥tr♦r ♦ Ω, t ♠t♥t♦♥ F (·, x) s ♠sr
t xi sq♥ ♦ α−r♥t ♥t♦♥s ♦♥ [a, b], ri sq♥ ♦
♠sr ♥t♦♥ ♦♥ [a, b] ♦♥r♥ t♦ 0, s i → +∞, ♥ Λi s sq♥ ♦
♠sr sst ♦ [a, b] s tt L − meas(Λi) → (b − a) s i → ∞. rtr♠♦r
s♣♣♦s tt
xi(t) ∈ G(t), ♥∣
∣
∣x
(α)i (t)
∣
∣
∣≤ φ(t), t ∈ [a, b]
t sq♥ xi(a) s ♦♥
x(α)i (t) ∈ F (t, xi(t)) + ri(t)B, ∀t ∈ Λi
♥ tr s ssq♥ ♦ xi ♦♥r♥ t♦ ♥ α−r♥t ♥t♦♥ x sts②♥
s
x(α)(t) ∈ F (t, x(t)).
Pr♦♦ s t♦r♠ ♦♥r♥s t ♦♠♣t♥ss ♥ s♦♠ s♥s ♦ t st ♦ s
trt♦rs s s♦t♦♥s t♦ t α−r♥t ♥s♦♥ ♦r t s α = 1,
r♦♠ t r③s♦ t♦r♠ tt ♦r ♥ ♥ s♣ sst s ♦♠♣t ♥
♦♥② ♥② q♦♥t♥♦s ♥ ♥♦r♠② ♦♥ sq♥ s ssq♥ ♦♥r♥
t♦ ♥ ♠♥t ♦ t st ♦r ♥ ♥r ts s ♥♦t t s α ∈ (0, 1). s t
♦♦s r♦♠ ♦r♠ ♦ ♥ r ❬❪ t s r tt ♥ ♥r Iα s ♥♦t
♦♠♣t ♦♣rt♦r ♥ ts t ♥♦rPtts ♣r♦♣rt② ② ♦♠♣t ♦♣rt♦rs
tr♥s♦r♠ ② ♦♠♣t sts r♦♠ ♥ s♣ ♥t♦ ♥♦r♠♦♠♣t sts ♦ ♥♦tr
♥ s♣ ♦♠♣t ♦♥t♥t② ♥ ♣rtr ♦r ♥tr ♥tr ♦♣rt♦rs ♠♣♣♥
t s♣ ♦ L1 ♥tr ♥t♦♥s ♥t♦ tt ♦ ♦♥t♥♦s ♥t♦♥s ♥♥♦t ♣♣
♦r s s ♦ α, r ♥♦t♦♥s ♦ ♦♣rt♦r ♦♠♣t♥ss r ♥stt ♥ ♠
♥ ♦ ❬❪ ♦r ts ♣r♣♦s t r ♥♦t♦♥ ♦ Ptts ♥trt② s ①♣♦t
♥ ts rt ♥t♦♥ f : T → X, ♥ T ♠sr s♣ ♥♦ t t
strtr (T,Σ, µ), s Ptts ♥tr ♦r A ∈ Σ ∃ t♦r e ∈ X s♦ tt
〈ψ, e〉 =
∫
A
〈ψ, f(t)〉 dµ(t)
♦r ♥t♦♥s ψ ∈ X∗, r X∗ s t ♦ X. r♦♠ ♦r♠ ♥ ♠ ♥
♦ ❬❪ t ♦♦s tt f : T → X s Ptts ♥tr t♥ Iαf s ♥ ♥
T, f s rt♦♥② ♥tr ♦♥ T, ♥ t♦♥② f s str♦♥② ♠sr f
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s q t♦ t ♠t ♦ sq♥ ♦ ♠sr ♦♥t② ♥t♦♥s t
T = [0, 1], t♥ Iαf : T → X s ♦♥ ② ♦♥t♥♦s ♥
sup‖φ‖≤1
∫ 1
0φIαf(t)dt
≤1
Γ(α+ 1)sup‖φ‖≤1
∫ 1
0φf(t)dt
.
♦r♦r t ♦♦s r♦♠ ♥ s② ①t♥s♦♥ ♦ t r♠rs ♦ t ♦ t♦r♠ tt ♦r
α ≤ 1, tt Iα : Lp([0, 1];Rn) → Lp([0, 1];Rn) s ♦♠♣t ♥ p > max1, 1α, tt
Iα : Lp([0, 1];Rn) → C([0, 1];Rn) s s♦ ♦♠♣t s ♠♥s tt ② t r③s♦
♦r♠ tt ♥ ♦♦s sq♥ ♦ ♦♥ ♥ q♦♥t♥♦s α−r♥t
♥t♦♥s xi t ssq♥ ♦♥r♥ t♦ s♦♠ ♦♥t♥♦s ♥t♦♥ x ♥ tt ②
♣♣②♥ Ptts rtr♦♥ ♥ ♦♥sr rtr ssq♥ ♦r t sq♥
x(α)i ♦♥rs ② t♦ ♠t γ ∈ Lα([a, b];Rn).
r♦♠ t ②♣♦tss ♥♦ tt xi(a) s ♦♥ sq♥ ♥ ts tr s
ssq♥ ♦ xi(a) ♦ ♥♦t r tt ♦♥rs t♦ x(a)). ♦r s ssq♥
xi(t) = xi(a) +1
Γ(α+ 1)
∫ t
a
x(α)i (τ)(dτ)α,
♥ ts tt
x(t) = x(a) +1
Γ(α+ 1)
∫ t
a
γ(τ)(dτ)α,
ts x(·) s rt♦♥ trt♦r② s tt x(α)(t) = γ(t)
♦♥sr ♥ rtrr② s ♠sr st M ⊂ [a, b] ♥ t ♠t♦♥♥ H(t, z, p)
♥ ②
H(t, z, p) = max 〈p, v〉 : v ∈ F (t, z) .
r♦♠ ts ♥t♦♥ t ♦r t ∈M,
v ∈ F (t, z) ⇒ H(t, z, p) ≥ 〈p, v〉 , ∀p ∈ Rn.
♦♥sq♥t② t ♥t♦♥ z(·) s ♥ α−trt♦r② ♦r F ♥ ♦♥②
H(t, z, p) ≥⟨
p, z(α)⟩
, ∀p ∈ Rn, ∀t ∈M.
s r♦♠ ②♣♦tss t ♦♦s tt
∫
M∩Λi
H(t, xi(t), p)dt ≥
∫
M∩Λi
⟨
p, x(α)i (t)
⟩
dt−
∫
M∩Λi
ri(t) |p| dt− φ(b− a− |Λi|),
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r |Λi| s t s ♠sr ♦ t st Λi ♥ φ s t ss♥t s♣r♠♠ ♦ t
♥t♦♥ φ(·). ② t♥ ♣♣r ♠t s i→ +∞,
lim supi→∞
∫
M∩ΛiH(t, xi(t), p)dt ≥ lim supi→∞
∫
M∩Λi
⟨
p, x(α)i (t)
⟩
dt
− lim supi→∞
∫
M∩Λi(ri(t)|p|+ φ(b− a− |Λi|))dt.
r♦♠ t ②♣♦tss tt t ♠♥ts ♦ t sq♥ x(α)i r α−♥tr②
♦♥ ‖ri‖ → 0, ♥ L −meas(Λi) → (b− a). ♥ ♦♥ tt
∫
M
lim supi→∞
H(t, xi(t), p)dt− lim supi→∞
∫
M
⟨
p, x(α)i (t)
⟩
dt ≥ 0.
r♦♠ t ♣♣r s♠♦♥t♥t② ♦ H,
lim supi→∞
H(t, xi(t), p) ≤ H(t, x(t), p).
♥ ♦♥ tt
∫
M
(
H(t, x(t), p)−⟨
p, x(α)(t)⟩)
dt ≥ 0.
♥ ♥ t ♣r♦s ♥qt② M s ♥ rtrr② s ♠sr st ♦♥
tt
H(t, x(t), p) ≥⟨
p, x(α)(t)⟩
, t ∈ [a, b] .
♥ H s ♦♥t♥♦s ♥ p, ts ♥qt② ♥ ♦t♥ ∀p ∈ Rn.
s t ♦♦ tt
x(α)(t) ∈ F (t, x(t)), t ∈ [a, b] .
♣r♦♦ s ♦♠♣t
ssr② ♦♥t♦♥s ♦ ♣t♠t②
♥ ts st♦♥ ♣rs♥t sss ♥ ♣r♦ ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♦ t
P♦♥tr②♥ t②♣ ♦r ♣r♦♠ (P ) stt ♥ t rst st♦♥ ♦ ts ♣tr s♦
♥ t rqr ss♠♣t♦♥s ♦♥ t t ♦ t ♣r♦♠
t s ♥♦♥ tt ♥ssr② ♦♥t♦♥s r r♥t t♦ ♣r♦ ♠♥♥ ♥♦r♠t♦♥
t s♦t♦♥ t♦ t ♣r♦♠ ①sts ♥ t ♦♥t♦♥s ♦ ♥♦t ♥rt ♥ ts tss
r ♥♦t ♦♥r♥ t tr t②♣ ♦ rsts s♥ t② ♦ ①t♥ t ♦♠ ♦
♦r ♦ ts tss
♦r ♥ ♥ t ❬❪ ♠ s♥t ♦♥t♦♥s ♦r t ①st♥ ♦ s♦t♦♥ t♦
rt♦♥ ♦♥r② ♣r♦♠ r ♥ ♥ ♦r♠ ♦rrs♣♦♥s t♦ ①♥
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t ♦♥tr♦ ♥t♦♥ ♥ t ②♥♠s ♦ t s②st♠ ♥ ts ♣tr r ♥ ♥
t ♦r♠ ♦ r♥t ♥s♦♥ s ♠♥t♦♥ rr ♥ ts ♣tr ①t♥ t
♠①♠♠ ♣r♥♣ ♥ ♣tr ♦ rs ❬❪ t♦ t rt♦♥ ♦♥t①t ♥ ♥ ♠♥
t rt♦♥ ♦ t ♠♥ rst ♦ ♣tr
♦r♠ t x ♠ss rt♦♥ trt♦r② tt s♦s t ♣r♦♠ (P ), ♥
t ss♠♣t♦♥s sts ♥ tr ①sts ♠t♣r [p, λ, ξ, µ, γ], s
tt λ+ ‖µ‖ > 0, r ‖µ‖ = 1Γ(α+1)
∫ b
a|µ(dt)α| ♥♦ts t ♦♥ ♠sr ♥♦r♠ λ > 0
s sr p : [a, b] → Rn s rt♦♥ ♦♥t ♥t♦♥ ξ s ♣♦♥t ♥ R
n, µ(·) s
♥♦♥♥t ♦♥ ♠sr s♣♣♦rt ♦♥ t st
S := t ∈ [a, b] : ∂>x h(t, x(t)) 6= φ,
♥ γ : [a, b] → Rn s ♠sr ♥t♦♥ ♥
ξ ∈ ∂g(x(b));
♦r ♠♦st t ∈ [a, b],
(
−p(α)(t), x(α)(t))
∈ ∂x,pH
(
t, x(t), p(t) +1
Γ(α+ 1)
∫
[a,t)γ(τ)(dµ(τ))α
)
,
r ∂H rrs t♦ t ♥r③ r♥t s ♣♣♥① ♦ t ♠t♦♥♥ t
rs♣t t♦ (x, p), ♦r ① t;
♦r s♦♠ r > 0,
p(a) ∈ r∂dC(x(a)),
♥
−p(b)−1
Γ(α+ 1)
∫
[a,b]γ(τ)(dµ(τ))α ∈ λξ,
r dC(·) s ♥ st♥ ♥t♦♥ ♦r C.
γ(t) ∈ ∂>x h(t, x(t)) µ− a.e.,
∂>x h(t, x(t)) s rt♥ sst ♦ t s rs ♥r③ r♥t ♦ h(t, x(t)) t
rs♣t t♦ x, ♦r ① t s ♣♣♥① ♥ ②
∂>x h(t, x(t)) = coγ = limi→∞
γi : γi ∈ ∂xh(ti, xi), (ti, xi) → (t, x), h(ti, xi) > 0.
Pr♦♦ ♣r♦♦ s ♦r♥③ ♥ sr st♣s rst t ♦r♥ ♣r♦♠ s ♠♦ ②
♥r♥ t ♦t② st ♥ ♦rr t♦ ♥sr t st② ♦ ♣rtr trt♦rs ♦♥
♦♥strt sq♥ ♦ ①r② ♣r♦♠s ♥ t r♦s t②♣s ♦ ♦♥str♥ts
r r♠♦ ♥ t♦♥ ♥♦♥s♠♦♦t tr♠s s♥ t♦ ♣r♦♣r② ♣♥③ t ♦t♦♥
♦ t r♠♦ ♦♥str♥ts r t♦ t ♦st ♥t♦♥ ♦♣t♠③t♦♥ ♣r♦♠s
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♦ t ♦t♥ sq♥ r s♠♣r t rqr t ♣♣t♦♥ ♦ ♥s rt♦♥
♣r♥♣ r ♦♠♣t t ♥r③ r♥ts ♦r t ♣rtr ♣r♦♠ ♥
♣♣t♦♥ ♦ r♠t ♣r♥♣ ♥② ①♣rss t rsts ♥ tr♠s ♦ t ♦r♥
t ♥ t ♠t♦♥♥ ♦r♠
t♣ t ♠t♥t♦♥ F (t, x) ♥ ♥ Ω, ♥ ♦r s♦♠ rtrr② δ > 0,
♦♥sr t δ t ♦t x s ♥ s ♦r ♥ ♥♦t ② T (x; δ). ♦ ♦r ♥② β,
♥ ♥ ♠t♥t♦♥ Fβ(t, x) ♥ t ♦sr t T (x; δ/2) ♦♥t♥ ♥ Ωδ s
tt Fβ(t, x) = F (t, x) + βB. r♦r ♥ β ♦s t♦ ③r♦ t♥ t ♥ ♠t♥t♦♥
Fβ(t, x) ♦s t♦ t ♦r♥ ♠t♥t♦♥ F (t, x). rtr♠♦r ♥ rt♦♥
trt♦r② y(·) sts②♥
y(α)(t) ∈ Fβ(t, y(t)),
y(a) ∈ C,
(t, y(t)) ∈ Ωδ, t ∈ [a, b].
♥♦t t st tt ♦♥t♥s s rt♦♥ trt♦rs y(t) ② Λβ . ♥ ♦r ♥② s♠
ε > 0 t ♥t♦♥ ψε(y) ♥ ②
ψε(y) = max
g(y(b))− g(x(b)) + ε2, θ(y)
,
r t ♥t♦♥ θ(y) s ♥ ②
θ(y) = maxa≤t≤b
h+(t, y(t))
,
t h+(t, y(t)) = max0, h(t, y(t)). ❲ ♦♥sr ♠tr ♥t♦♥ ∆α(·, ·) ♥ ②
∆α(y, z) =1
Γ(α+ 1)
∫ b
a
|y(t)− z(t)| (dt)α + |y(a)− z(a)| .
t♣ r ♣♣② ♣♥③t♦♥ t♥q t♦ ♥♦r♣♦rt t ②♥♠ ♥s♦♥ ♥
stt ♦♥str♥t s ♣♥t② tr♠s ♥ t ♦st ♥t♦♥ ♥ t s♦t♦♥ t♦ t ♣♥③
♣r♦♠ s ♥♦t ♥♦♥ ♥ t♦ ♣♣② rt♦♥ ♣r♥♣ ♥ ♦♦s t ♦♥
t♦ r ♥ ❬❪ ♥ ♦r ♥② ♣♦st β < ε, ψε(x) = ε2, t ♦♦s tt
ψε(x) ≤ infΛβ
ψε + ε2.
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s ♥s ♦r♠ ssrts tt tr ①sts ♥ ♠♥t z ∈ Λβ tt ♠♥♠③s
ψε(y) + ε∆α(y, z) ♦r y ∈ Λβ s tt
∆α(x, z) ≤ ε, ψε(z) ≤ ε2.
♦ ♦♥ ts st♣ ♥ t♦ s♦ t ♦♦♥ ♠♠
♠♠ ♦r s♦♠ δ > 0, ♠♦♥ y ∈ Λβ sts②♥ ‖y − z‖ < δ, tr s rt♦♥
trt♦r② z tt ♠♥♠③s
ψε(y) + ε∆α(y, z) +R1dC(y(a)) +R2dα(y, Fβ),
r dα(·, ·) s ♥ s ♦r R1 = (L1 + εL2), R2 = (L3 + εL4), ♥ L1, L2, L3, L4
r ♥ ②
L1 = max Kg,Kh [K lnα(K) + 1] ,
L2 =
(
(b−a)α
Γ(α+1) + 1)
max Kg,KhL1,
L3 = Kmax Kg,Kh ,
L4 =
(
(b− a)α
Γ(α+ 1)+ 1
)
K,
r K, Kg ♥ Kh r ♥ ♦r ♥ t ②♣♦tss ♥ rs♣t②
♠r tt ts ♠♠ stts tt tr s ♥ ♦♣t♠③t♦♥ ♣r♦♠ t♦t ♦♥str♥ts
tt ♥ rr s ♣rtrt♦♥ ♦ (P ) ♥ ♦r s♦t♦♥ z ∈ Λβ .
Pr♦♦ ♠♠ ♣♣♦s tt ts ♠♠ s s t♥ tr s sq♥ ♦ rt♦♥
trt♦rs yj ♦♥r♥ t♦ z ♦r t ①♣rss♦♥ ♥ t ♠♠ s ss t♥ ts
t z s ψε(z). t cj ∈ C s tt
dC(yj(a)) = |yj(a)− cj | ,
♥ t yj t rt♦♥ trt♦r② ♥ ②
yj(t) = yj(t) + cj − yj(a).
r♦♠ t ♣st③ ♦♥t♦♥ ♦r ss♦t ♥t♦♥ dF (t,·)(·) s Pr♦♣♦st♦♥ ♥
∣
∣
∣dFβ(t,yj(t))(y
(α)j (t))− dFβ(t,yj(t))(y
(α)j (t))
∣
∣
∣≤ k(t) |yj(t)− yj(t)|+
∣
∣
∣y(α)j (t)− y
(α)j (t)
∣
∣
∣.
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② s♥ ♥ t t tt y(α)(t) = y(α)(t), ♦♥ tt
dFβ(t,yj)(y(α)j (t)) ≤ dFβ(t,yj)(y
(α)j (t)) + k(t) |yj(t)− yj(t)| .
② ♣♣②♥ t rt♦♥ ♠r ♥tr ♦♣rt♦r ♦r tr♠s ♦ ts ♥qt②
1Γ(α+1)
∫ b
adFβ(t,yj(t))(y
(α)j (t))(dt)α ≤ 1
Γ(α+1)
∫ b
adFβ(t,yj(t))(y
(α)j (t))(dt)α
+ 1Γ(α+1)
∫ b
ak(t) |yj(t)− yj(t)| (dt)
α.
r♦♠ ♥ ♥t♦♥
dα(yj , Fβ) ≤ d
α(yj , Fβ) + dC(yj(a))1
Γ(α+ 1)
∫ b
a
k(t)(dt)α.
t K = Eα(1
Γ(α+1)
∫ b
ak(t)(dt)α), t♥ lnα(K) = 1
Γ(α+1)
∫ b
ak(t)(dt)α, r lnα(·) ♥♦ts
t ♥rs ♥t♦♥ ♦ t ttr ♥t♦♥ Eα(·), ♦r ♠♦r ♥♦r♠t♦♥ ♦t t
rt♦♥s t♥ ttr ♥t♦♥ Eα(·) ♥ t ♥rs lnα(·), s ♣♣♥①
♥
dα(yj , Fβ) ≤ d
α(yj , Fβ) + dC(yj(a)) lnα(K).
♦r j s s♥t② r t♥ ♦r♥ t♦ ♦r♠ tr ①sts rt♦♥ trt♦r②
zj ∈ Fβ s tt zj(a) = yj(a) = cj ∈ C, ♥
‖zj − yj‖ ≤1
Γ(α+ 1)
∫ b
a
∣
∣
∣z(α)j − y
(α)j
∣
∣
∣(dt)α ≤ Kd
α(yj , Fβ).
rtr♠♦r ♥ rt
‖zj − yj‖ ≤ ‖zj − yj‖+ ‖yj − yj‖ .
♦ r♦♠ ♥
‖zj − yj‖ ≤ Kdα(yj , Fβ) + dC(yj(a)),
♥ ② ♦♥ tt
‖zj − yj‖ ≤ [1 +K lnα(K)] dC(yj(a)) +Kdα(yj , Fβ).
♥ r♦♠ t tr♥ ♥qt② ♥ ② ♥ ♥ strt♥ t q♥tts ψε(yj),
♦t♥
ψε(zj) + ε∆α(zj , z) ≤ ψε(yj) + ε∆α(yj , z) + ε∆α(yj , zj) + ψε(zj)− ψε(yj),
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♥ ② s♥ t ♥t♦♥ ♦ ∆α(·, ·), t rtr③t♦♥ ♦ t ♥t♦♥ ψε(·), ♥
♣st③ ♣r♦♣rt② ♦r t ♥t♦♥s g(·), θ(.) ♦♥ tt
ψε(zj)− ψε(yj) ≤ max Kg,Kh ‖zj − yj‖ ,
ε∆α(yj , zj) =ε
Γ(α+ 1)
∫ b
a
|yi(t)− zi(t)| (dt)α + ε |yi(a)− zi(a)|
≤ε(b− a)α
Γ(α+ 1)‖zj − yj‖+ ε ‖zj − yj‖
≤ ε
(
(b− a)α
Γ(α+ 1)+ 1
)
‖zj − yj‖ .
② ssttt♥ ♥
ψε(zj)+ε∆α(zj , z) ≤ ψε(yj)+ε∆α(yj , z)+
[
ε
(
(b− a)α
Γ(α+ 1)+ 1
)
+max Kg,Kh
]
‖zj − yj‖ .
② s♥ ♦♥ tt
ψε(zj) + ε∆α(zj , z) ≤ ψε(yj) + ε∆α(yj , z) +R1dC(yj(a)) +R2dα(yj , Fβ)
< ψε(z),
r R1 = (L1 + εL2), R2 = (L3 + εL4), ♥ L1, L2, L3, L4 r ♥ ♦ ♥
ts s ♦♥trt♦♥ ♦ tt z s t ♦♣t♠ s♦t♦♥ ♦ ψε(·) + ε∆α(·, z).
♠♠ s ♣r♦
❲ ♥♦ ♦♠♣t t ♥r③ r♥t ♦r t ♥t♦♥s ♥ t ♠♠
t♣ ♥ ts st♣ t ♥r③ r♥ts ♦r t ♣rtr ♣r♦♠
♥t♦♥s ♥ ♠♠ y = 0 s ♦ ♠♥♠③r ♦ ψε(z + y) + ε∆α(z + y, z) +
R1dC(y(a) + z(a)) +R2dα(y + z, Fβ). s ② r♠ts ♣r♥♣ tt
0 ∈ ∂y ψε(z) + ε∆α(z, z) +R1dC(z(a)) +R2dα(z, Fβ) ,
♥ r♦♠ t s♠ ♥r③ s r tt
0 ∈ ∂yψε(z) + ε∂y∆α(z, z) +R1∂ydC(z(a)) +R2∂ydα(z, Fβ).
f(·) s ♥② ♣st③ ♥t♦♥ ♦♥ Rn, ♥② ♠♥t ξ ♦ t ♥r③ r♥t ♦
t ♠♣♣♥ y → f(y(a)) t y0 s r♣rs♥t ② ♥ ♠♥t ξ0 ∈ ∂f(y0(a)), s♦ tt
ξ(y) = 〈ξ0, y(a)〉 ♦r y r ❬❪ t ♦♦s tt t stt ♦♥str♥t s ♥t
tt s θ(z) < ψε(z), t ♥t♦♥ ψε(y) ♦♠s ψε(y) = g(y(b))− g(x(b)) + ε2, ♥ t
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♥r③ r♥t t ♠♣ y → g(z(b) + y(b)) t 0 s s♦♠ ♠♥t ξ1 ♦ ∂g(z(b)) ♥
ts
ξ(y) = 〈ξ1, y(b)〉 .
θ(z) > 0, t♥ ξ s ♥ ♠♥t ♦ ∂θ(z) ♥ ②
ξ(y) =1
Γ(α+ 1)
∫
[a,b]〈γ(t), y(t)〉 (dµ(t))α,
r µ(·) s ♦♥ ♠sr ♦♥ [a, b] s♣♣♦rt ♦♥ t ♣♦♥ts ♥ t♠ t t ♦♥str♥t
♦♠s t ♥ γ(t) ∈ ∂>x h(t, z(t)), µ ② ♣tt♥ t♦tr ♣♦ssts
tt t ♥r③ r♥t ξ ♦ t ♥t♦♥ ∂yψε(z) stss
ξ(y) = λ 〈ξ1, y(b)〉+(1− λ)
Γ(α+ 1)
∫
[a,b]〈γ(t), y(t)〉 (dµ(t))α.
♠r② ♥② ♠♥t ξ ♦ R1∂ydC(z(a)) s r♣rs♥t ② ♥ ♠♥t ξ0 ∈ R1∂dC(z(a)),
s♦ tt
ξ(y) = 〈ξ0, y(a)〉 .
♥② ♠♥t ξ ♦ R2∂ydα(z, Fβ) s r♣rs♥t ② ♥ ♠♥t (q, s) ∈ R2∂d
α(z, Fβ), s t
♦♦s r♦♠ t ♥r③ s r r ❬❪
ξ(y) =1
Γ(α+ 1)
∫ b
a
〈q(t), y(t)〉 (dt)α +1
Γ(α+ 1)
∫ b
a
⟨
s(t), y(α)(t)⟩
(dt)α.
♠r② ♥② ♠♥t ξ ♦ ε∂y∆α(·, ·) ♦rrs♣♦♥s t♦ t ♥t♦♥ r(t) t r(t) ∈ εB ♥
♣♦♥t r0 ∈ εB s tt
ξ(y) = 〈r0, y(a)〉+1
Γ(α+ 1)
∫ b
a
〈r, y〉 (dt)α.
♠♠ r♦♠ tr s ♥r③ sr♥t sts②♥
〈ξ0, y(a)〉+ λ 〈ξ1, y(b)〉+ 〈r0, y(a)〉
+ 1Γ(α+1)
∫ b
a〈q, y〉 (dt)α + (1−λ)
Γ(α+1)
∫
[a,b] 〈γ(t), y(t)〉 (dµ(t))α
+ 1Γ(α+1)
∫ b
a
⟨
s, y(α)⟩
(dt)α + 1Γ(α+1)
∫ b
a〈r, y〉 (dt)α = 0.
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♦ r ♦♥ t♦ rt ts qt② ♥ tr♠s ♦ t t ♦ t ♣r♦♠
〈ξ0 + r0, y(a)〉+ 〈λξ1, y(b)〉
+ 1Γ(α+1)
∫ b
a〈q + r + (1− λ)γ(t)µ, y〉 (dt)α
+ 1Γ(α+1)
∫ b
a
⟨
s, y(α)⟩
(dt)α = 0.
② s♥ t ♠r rt♦♥ ♥trt♦♥ ② ♣rts s ♣tr
〈ξ0 + r0, y(a)〉+ 〈λξ1, y(b)〉+1
Γ(α+1)
∫ b
a〈q + r + (1− λ)γ(t)µ, y〉 (dt)α
+y(b)s(b)− y(a)s(a) − 1Γ(α+1)
∫ b
a
⟨
s(α), y⟩
(dt)α = 0.
♦♥sq♥t②
〈ξ0 + r0 − s(a), y(a)〉+ 〈λξ1 + s(b), y(b)〉
+ 1Γ(α+1)
∫ b
a
⟨
q + r + (1− λ)γ(t)µ− s(α), y⟩
(dt)α = 0.
② ♣♣②♥ t rt♦♥ ♦s②♠♦♥ ♠♠ s ♣♣♥① ♦♥ tt
q + r + (1− λ)γ(t)µ− s(α) = 0,
q + r + (1− λ)γ(t)µ = s(α).
② ♥trt♥ ♦t♥
1
Γ(α+ 1)
∫ t
a
s(α)(dτ)α =1
Γ(α+ 1)
∫ t
a
(q + r)(dτ)α +(1− λ)
Γ(α+ 1)
∫
[a,t)γ(t)µ(dτ)α,
♥
s(t) = s(a) +1
Γ(α+ 1)
∫ t
a
(q + r)(dτ)α +(1− λ)
Γ(α+ 1)
∫
[a,t)γ(t)µ(dτ)α,
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r
s(a) = ξ0 + r0,
−s(b) = λξ1.
r♦r
s(t) = ξ0 + r0 +1
Γ(α+ 1)
∫ t
a
(q + r)(dτ)α +(1− λ)
Γ(α+ 1)
∫
[a,t)γ(τ)(dµ(τ))α.
♦ ♥ ♥t♦♥ p(·) s
p(t) = ξ0 + r0 +1
Γ(α+ 1)
∫ t
a
(q + r)(dτ)α,
t♥ p(t) stss
p(a) = s(a) = ξ0 + r0,
p(b) = s(b)−(1− λ)
Γ(α+ 1)
∫
[a,b]γ(τ)(dµ(τ))α
= −λξ1 −(1− λ)
Γ(α+ 1)
∫
[a,b]γ(τ)(dµ(τ))α.
♦♥sq♥t② ♦♥ tt
[p(α) − r, p+(1− λ)
Γ(α+ 1)
∫
[a,t)γ(τ)(dµ(τ))α] ∈ R2∂d
α(z, Fβ),
p(a) ∈ R1∂dC(z(a)) + εB,
− p(b)−(1− λ)
Γ(α+ 1)
∫
[a,b]γ(τ)(dµ(τ))α = λξ1.
t♣ ♥ ts st st♣ ♦ t ♣r♦♦ ①♣rss t ♦t♥ ♦♥t♦♥s ♥ tr♠s ♦
t ♠t♦♥♥ ♥t♦♥ ♦ ts ♥ s♠♣② ♣♣② t ♦♦♥ ♠♠
♠♠ t (q, p) ∈ ∂Kdα(y, Fβ) ♥ d
α(y, Fβ) = 0. ♥
(−q, v) ∈ ∂H(t, y, p) + βB.
ts ♣r♦♦ ♥ ♦♥ ♥ r ❬❪ ♠♠ P ♥ t ②s t ♦♦♥ ♦♥t♦♥
(−p(α)(t), z(α)(t)) ∈ ∂H
(
t, z(t), p(t) +(1− λ)
Γ(α+ 1)
∫
[a,t)γ(τ)(dµ(τ))α
)
+ 2εB.
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r s t ♦♥s |r| ≤ ε, β ≤ ε, ♥ ε s ♥② ♣♦st ♣r♠tr s♥t②
s♠ ♦s② tt ♥ r t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t②
♦ t ♣rtr ♣r♦♠ ♣♣r♦①♠t♥ t ♦r♥ ♦♥ ♦ t ♥ ♦rt ♦ t ♣r♦♦
♦♥ssts ♥ s♦♥ tt ts qt♦♥s ♦♥rs t♦ t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t②
♦ t ♦r♥ ♣r♦♠ s t ♣rtrt♦♥ ♦s t♦ ③r♦ ❲ ♥♦ r♦♠ t st♣ tt
∆α(z, x) ≤ ε, s♦ z ♦♥rs t♦ x s ε→ 0. ❲ ♠② st sq♥ ♦ t ♣rtrt♦♥s
ε ♦♥r♥ t♦ ③r♦ s tt λ ♦♥rs t♦ λ0 ∈ [0, 1].
② s♥ ♦♥♦②♠ ♦r♠ r ❬❪ t ♥ s♦♥ tt ♦r rtr
ssq♥ t ♠srs η ♥ ② dη = (1 − λ)γdµ ♦♥r ∗ t♦ ♠sr
η0 ♦ t ♦r♠ dη0 = γ0dµ0, r µ0 s t ∗ ♠t ♦ (1−λ)γ, ♥ γ0 s ♠sr
st♦♥ ♦ ∂>x h(t, x(t)). ♥ ♦♥sq♥ λ0 + ‖µ0‖ > 0.
t ♥ ♠t♥t♦♥ F ♥ ♠sr ♥t♦♥ yε s ♦♦s
F (t, x, p) =
(−v, u) : (u, v) ∈ ∂H
(
t, x, p+1
Γ(α+ 1)
∫
[a,t)γ0(dµ0)
α
)
,
yε =(1− λε)
Γ(α+ 1)
∫
[a,t)γ(dµ)α −
1
Γ(α+ 1)
∫
[a,t)γ0(dµ0)
α.
s ♦r ε,
(z(α)ε , p(α)ε ) ∈ F (t, zε, pε + yε) + 2εB,
♥ tt yε ♦♥rs t♦ ③r♦ s ε→ 0. ② s♥ ♦r♠ t♦ t ♦♥r♥
♦ (zε, pε) t♦ (x, p).
♦♥sq♥t② ♦t♥ t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♦r t ♦r♥ ♣r♦♠
♥ ts t ♣r♦♦ ♦ t ♦r♠ s ♦♠♣t
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♣tr
①♠♠ Pr♥♣ ♦r rt♦♥
♣t♠ ♦♥tr♦ Pr♦♠s t
tt ♦♥str♥ts
♥tr♦t♦♥
♥ ts ♣tr stt sss ♥ ♣r♦ P♦♥tr②♥ ♠①♠♠ ♣r♥♣ ♦r rt♦♥
♦♣t♠♦♥tr♦ ♣r♦♠s Ps t stt ♦♥str♥ts ♥ ♥r ss♠♣t♦♥s
♠♣♦s ♦♥ t t ♦ t ♣r♦♠ ❲ ♦♥sr P t t ②♥♠s ♥♦♥
♠r rt♦♥ rts t rs♣t t♦ t♠
t♥q s ♥ t ♣r♦♦ ♦ ts ♣tr s s ♥ t ♦♥strt♦♥ ♦ rt♦♥
②♥♠ ♦♥tr♦ s②st♠ ♥ s ② tt ♦r ①tr♠ ♦ t P tr
♦rrs♣♦♥s ♣r♦ss ♦ t s tt t ♥♣♦♥t ♦ ts rt♦♥ trt♦r② s
♦♥ t ♦♥r② ♦ t ♠ ♦ t tt♥ st ② rt♥ ♣st③♥ ♥t♦♥ ♦
t stt r ♦ t ♦r♥ s②st♠ ♦♣♠♥t ♦ t P ♠st s♦ s
tt t ♥s t rt♦♥ ♦ t ♥ssr② ♦♥t♦♥s ♦r ♦♣t♠t② ♥ t ♦r♠ ♦
P♦♥tr②♥ ♠①♠♠ ♣r♥♣ ♦r t P r♦♠ t rtr③t♦♥ ♦ ♦♥r② ♦ t
♦♥sr ♥t♦♥ ♦ t tt♥ st ♦
♥t♦♥ t t st C sst ♦ Rn, ♥ x(·) s ♥ ♠ss rt♦♥
trt♦r② ss♦t t♦ ♦♥tr♦ u(·). ss♠ tt [a, b] ♥ ♥tr r x(·) stss
t ♥t ♦♥t♦♥ x(a) ∈ C. st ♦ ♣♦♥ts x(b) ♦t♥ ② ♦♥sr♥ ♥②
♠ss ♦♥tr♦ ♥t♦♥ s tt♥ st r♦♠ C, ♥ s ♥♦t ② A[C].
♠s ♦ ts ♣tr s t♦ st② t P ♥ t stt ♦♥str♥ts r ♣rs♥t
♥ t ♥ssr② ♦♥t♦♥s ♦r ts ♣r♦♠
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Pr♦♠ tt♠♥t ♥ ss♠♣t♦♥s
♥ ts st♦♥ stt t ♦♣t♠♦♥tr♦ ♣r♦♠ ♦r t ②♥♠s ts t
♦r♠ ♦ ♦♥tr♦ ♠r rt♦♥ r♥t qt♦♥ t stt ♦♥str♥ts
❲ ♦♥sr t rt♦♥ ♦♣t♠♦♥tr♦ ♣r♦♠ (PC) s t ♦♦♥
(PC)Minimize g(x(b))
subject to x(α)(t) = f(t, x(t), u(t)), t ∈ [a, b] L − ,
x(a) ∈ C0, x(b) ∈ C1,
u ∈ U ,
h(t, x(t)) ≤ 0, ∀t ∈ [a, b],
r f : [a, b] × Rn × R
m → Rn s rt♦♥ ②♥♠s g : Rn → R s ♦st ♥t♦♥
h : [a, b] × Rn → R s stt ♦♥str♥t u : [a, b] → R
m s ♠sr ♦♥tr♦ stss
t ♦♥tr♦ ♦♥str♥ts u ∈ U , ♥ U t st ♦ ♠sr ♥t♦♥s t♥ s ♦♥
st Ω(t) ♦r t ∈ [a, b], ♥ Ω s st ♠♣ t♥ ♦♥ s ♦♥ ssts ♦ Rm,
C0 ♥ C1 r ♦s sts x(a) ♥ x(b) r ♥t ♥ tr♠♥ ♣♦♥t rs♣t② ♥
t ♦♣rt♦r (α) s ♠r rt♦♥ rt ♦ ♦rr α ∈ (0, 1] ♦ t stt r
t rs♣t t♦ t♠
♠①♠♠ ♣r♥♣ t♦ sts ② t s♦t♦♥s t♦ ♣r♦♠ (PC) r ♦t♥ ♥r
t ♦♦♥ ②♣♦tss
♠♣♣♥ (t, u) → f(t, x, u) s L × B−♠sr r L ♥ B ♥♦t
t s sst ♦ [a, b] ♥ t ♦r sst ♦ Rn, rs♣t② ♦r
t → f(t, x(t), u(t)) s α−♥tr ♦♥ ♥② s ♦♥tr♦ ♣r♦ss
♦r (t, u) ∈ Gr(Ω), tr ①sts L × B−♠sr k(t, u) : [a, b] × Rm → R
♥ ♥ t Gr(Ω) s tt ♥t♦♥ f(t, ·, u) s ♣st③ ♦ r♥ k(t, u) ♥
|f(t, x1, u)− f(t, x2, u)| ≤ k(t, u) |x1 − x2| .
Gr(Ω) s L×B−♠sr r Gr(Ω) s r♣ ♦ t ♠t♥t♦♥ Ω : [a, b] →
P(Rm) ♥ ②
Gr(Ω) := (t, u) ∈ [a, b]× Rm : u ∈ Ω(t) .
♥t♦♥ g(·) s ♣st③ ♦ r♥ Kg s tt
|g(x)− g(x)| ≤ Kg |x− x| .
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♥t♦♥ h(·, ·) s ♣♣r s♠♦♥t♥♦s ♥ h(t, ·) s ♣st③ ♦ r♥ Kh ♦r
t ∈ [a, b] s tt
|h(t, x)− h(t, x)| ≤ Kh |x− x| .
P♦♥tr②♥ ♥t♦♥ H : [a, b]× Rn × R
n × Rm → R ♥ ②
H(t, x, p, u) = 〈p, f(t, x, u)〉 .
♣r (x, u) ♦♠♣rs♥ ♥ α−s♦t② ♦♥t♥♦s ♥t♦♥ x t stt rt♦♥
trt♦r② ♥ ♠sr ♥t♦♥ u t ♦♥tr♦ s s ♣r♦ss ♦ t
♣r♦♠ (PC), stss t ♦♥str♥ts ♦ t ♣r♦♠ (PC).
♦r Ps ♦♥ ♠② t ♦ ♦ ♦r ♦ ♠♥♠♠ s② tt x∗ ♦ ♠♥♠♠
♦ (PC), t ♠♥♠③s t ♦t ♥t♦♥ ♦r ♦tr s stts x ∈ Rn, ♥
s② tt x∗ ♦ ♠♥♠♠ ♦ (PC), t ♠♥♠③s t ♦t ♥t♦♥ ♦r ♦tr
s stts x ∈ Rn, ♥ s♦♠ ♥♦r♦♦ s tt |x− x∗| ≤ ε. r rstrt ♦r
sss♦♥ t♦ ♠♥♠③rs ♥ t ♦♥t①t ♦ P♦♥tr②♥ t②♣ ♦ ♠♥♠♠ P♦♥tr②♥ t
❬❪ ♦♥t♦♥s ♦ t ♠①♠♠ ♣r♥♣ sts ♦♥② t ♦♥tr♦ ♣r♦sss tt
r ♥ts t♦ ♦ ♠①♠♠
①♠♠ Pr♥♣ ♦ ♣t♠t②
♦r♠ t t ♦♥tr♦ ♣r♦ss (x, u) s♦t♦♥ t♦ t ♣r♦♠ (PC), ♥ ss♠
tt t ss♠♣t♦♥s r sts ♥ tr ①sts rt♦♥ ♦♥t
♥t♦♥ p : [a, b] → Rn, sr λ ≥ 0, ♠sr ♥t♦♥ γ(·), ♣♦st ♦♥
♠sr µ(·) s♣♣♦rt ♦♥ t st
t ∈ [a, b] : h(t, x(t)) = 0,
♥ ♥t♦♥ q(·) ♥ ②
q(t) =
p(t) + 1Γ(α+1)
∫
[a,t) γ(τ)(dµ(τ))α, t ∈ [a, b),
p(t) + 1Γ(α+1)
∫
[a,b] γ(τ)(dµ(τ))α, t = b,
sts②♥
♦♥t qt♦♥
−p(α)(t) ∈ ∂xH(t, x(t), q(t), u(t));
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tr♥srst② ♦♥t♦♥
p(a) ∈ NC0(x(a)),
−q(b) ∈ λ∂g(x(b)) +NC1(x(b));
♦♥tr♦ strt② u : [a, b] → Rm ♠①♠③s ♥ Ω(t) t ♠♣♣♥
v → H (t, x(t), q(t), v) ;
γ(t) ∈ ∂>x h(t, x(t)) µ− a.e.; ♥
|p|+ ‖µ‖+ λ > 0.
r ∂x(·) rrs t♦ t ♥r③ r♥t ♥ t s♥s ♦ r t rs♣t t♦ x ♦r
① t, NC0(·) s t ♠t♥ ♥♦r♠ ♦♥ ♥ t s♥s ♦ ♦r♦ ❬❪ ∂>
x h(t, x(t))
s ♥r③ r♥t ♦r t stt ♦♥str♥t ♥t♦♥ ♥ ♥ ♦r♠ ♣tr
♥ p(α)(·) s ♠r rt♦♥ rt ♦ t ♦♥t r t rs♣t t♦ t ♦
♦rr 0 < α ≤ 1.
♠♥ ♦ t ♣r♦♦ s t♦ ♦♥strt ♥ ①r② ②♥♠ ♦♥tr♦ s②st♠ ♥ s
② tt t♦ ♦♣t♠♦♥tr♦ ♣r♦ss t♦ (PC) tr ♦rrs♣♦♥s ♦♥r② ♦♥tr♦
♣r♦ss t♦ t ①r② s②st♠ s ♥srs t ♥♦♥trt② ♦ t ♠t♣r r♦r
r t♦ r t ♥ssr② ♦♥t♦♥ ♦r ♦♣t♠♦♥tr♦ ♣r♦♠ (PC) r♦♠ t
rtr③t♦♥ ♦ rt♥ ♥t♦♥ ♦ t tt♥ st ♦ t ①r② ②♥♠
s②st♠ ♦ ts t s = [s, s1, s0] ♥♦t ♣♦♥ts ♥ Rn × R
n × R,
C = C0 × C1 × [0,∞),
f(t, s(t), u(t)) = (f(t, s(t), u(t)), 0, 0) ,
θ(s) = (g(s), s1 − s) ,
h(t, s) = h(t, s).
rtr♠♦r s♣♣♦s tt x(t) = [x, x1, x0] ♠ss rt♦♥ trt♦r② sts②♥
x(α)(t) = f(t, x(t), u(t)),
x(a) ∈ C,
h(t, x(t)) ≤ 0,
r θ(x(b)) s ♣st③ ♥t♦♥ s ♥ t ♦♥r② ♦ θ(A[C]), ♥ A[C] s
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tt♥ st r♦♠ C t t = b s ♥t♦♥ ♦ ♦♥sr t ♦♦♥ ①r②
rsts
♠♠ t x(t) ♠ss rt♦♥ trt♦r② stss
x(α)(t) = f(t, x(t), u(t)),
x(a) ∈ C,
h(t, x(t)) ≤ 0,
θ(x(b)) ∈ ② θ(A[C]),
r ② ♥♦ts t ♦♥r② ♥ tr ①sts t♦r ζ, rt♦♥ ♦♥t ♥t♦♥
p : [a, b] → Rn, ♠sr ♥t♦♥ γ(·), ♣♦st ♦♥ ♠sr µ(·) s♣♣♦rt ♦♥ t
st
t ∈ [a, b] : h(t, x(t)) = 0,
♥ q(·) ♥ ②
q(t) =
p(t) + 1Γ(α+1)
∫
[a,t) γ(τ)(dµ(τ))α, t ∈ [a, b),
p(t) + 1Γ(α+1)
∫
[a,b] γ(τ)(dµ(τ))α, t = b,
sts②♥
♦♥t qt♦♥
−p(α)(t) ∈ ∂xH(t, x(t), q(t), u(t)),
t tr♥srst② ♦♥t♦♥
p(a) ∈ NC(x(a)),
q(b) ∈ ζ∂xθ(x(b)),
♠①♠♠ ♦♥t♦♥
maxv∈Ω(t)
H (t, x(t), q(t), v) = H (t, x(t), q(t), u) ,
γ(t) ∈ ∂>x h(t, x(t)) µ− a.e.,
‖µ‖+ |ζ| > 0
♦ s♦ tt t ♥ssr② ♦♥t♦♥s ♦r (PC) r ♦♥sq♥ ♦ t
♥ssr② ♦♥t♦♥s ♦ ♠♠
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t p = (p, p1, p0), ♥ ts q = (q, q1, q0), r t rt♦♥ t♥ p ♥ q r s
♠♥t♦♥ rr ♥ t P♦♥tr②♥ ♥t♦♥ stss
H(t, x(t), q(t), u(t)) =⟨
q(t), f(t, x(t), u(t))⟩
.
r♦♠ ♦♥t♦♥ ♦ t ♠♠ ♥ t ♥t♦♥s ♦ p(t), q(t), f(t, x(t), u(t)), ♥
H(t, x(t), q(t), u(t)),
(
−p(α)(t),−p(α)1 (t),−p
(α)0 (t)
)
∈ (q(t), q1(t), q0(t))
∂xf(t, x(t), u(t))
0
0
.
♦♥sq♥t② ♦t♥ tt
−p(α)(t) ∈ q(t)∂xf(t, x(t), u(t)),
−p(α)1 (t) = 0, ♥ − p
(α)0 (t) = 0.
② s♥ t ♥t♦♥ ♦ P♦♥tr②♥ ♥t♦♥
−p(α)(t) ∈ ∂xH(t, x(t), q(t), u(t)).
♥ ♦♥t♦♥ ♦ ♦r♠ r♦♠ t ♦r♦r② ♦ ♣♣♥① ♥
♦♥t♦♥ ♦ ♠♠ ♦♥ tt
p(a) ∈ NC0(x(a)), p1 ∈ NC1
(x(b)), ♥ p0 ≤ 0.
s♦ r♦♠ ♦♥t♦♥ ♦ t ♠♠ tt
q(b) ∈ ζ∂xθ(x(b)),
♥ tr♦r r♦♠ t ♥t♦♥s ♦ q(·), x(·), ♥ θ(·), ② ♦♥sr♥ ζ = (ζ1, ζ2),
(q, q1, q0)(b) ∈ (ζ1, ζ2)∂(x,x1,x0)
(
g(x)
x1 − x
)
(b)
∈ (ζ1, ζ2)
(
∂xg(x(b)) 0 0
−I I 0
)
∈ (ζ1∂xg(x(b))− ζ2, ζ2, 0) ,
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ts ♠♥s tt
q(b) ∈ ζ1∂xg(x(b))− ζ2,
q1 = ζ2.
♥ ② λ = −ζ1, ♦♥t♦♥ ♦ t ♦r♠ ❲ ♦t♥ t ♦♥t♦♥s
♥ ② rt ssttt♦♥ ♦ q(·), x(·) ♥ ζ ♥ t ♦♥t♦♥s ♥
♥ t ♥ssr② ♦♥t♦♥s ♦ t ♦r♠ r ♦♥sq♥ ♦ t ♥ssr②
♦♥t♦♥s ♥ ♠♠ ♥♦ ♦♥② ♥ t♦ ♣r♦ t ♠♠ t s ♥♦t t
t♦ s tt t ②♣♦tss ♦ (PC) ♥ ♦r♠ ♠rt t♦ t ♦♥sr ♥
♠♠ ♦ r♦♠ ♥♦ ♦♥ ss♠ tt ②♣♦tss r♠♥ ♥ ♦r ♦r
t t ♦ r♠♥♥ ♦ t ♣r♦♦ ♦♥ssts ♥ ♦r♠t♥ ♥ ①r② ♠②
♦ ♦♣t♠♦♥tr♦ ♣r♦♠s t ②♥♠s ♥ ② rt♦♥ r♥t ♥s♦♥s ♦s
♦rrs♣♦♥♥ sq♥ ♦ s♦t♦♥s ♦♥r t♦ ♦♥r② ♦♥tr♦ ♣r♦ss ♦r t
s ♦s s t♦ s t ♠①♠♠ ♣r♥♣ ♣r♦ ♥ ♣tr ♥ ♦rr t♦ tt
t ♣r♦♦ t♦ t♦♥ ②♣♦tss t♠♣♦rr② ♦♥sr
①tr ②♣♦tss
♦r t ∈ [a, b], t st Ω(t) s ♥t ♥♠r ♦ ♣♦♥ts
♥t♦♥ f(t, x(t), v) s ♦♥ ② α−♥tr ♥t♦♥ σ(·) ♦r v ∈ Ω(t),
t ∈ [a, b] s tt
∣
∣
∣f(t, x(t), v)
∣
∣
∣≤ σ(t), k(t, v) ≤ σ(t),
r k(·, ·) s t ♥t♦♥ ♦s ♠♥t♦♥ ♦r ♥ t ②♣♦tss
♥ t st st♣ ♦ t ♣r♦♦ s♦ tt t rsts r♠♥ ♥ t s♥ ♦
ts ①tr ②♣♦tss
Pr♦♦ ♠♠ ♦♦♥ st♣s r rqr t♦ ♣r♦ ts ♠♠
t♣ ♦♥strt♦♥ ♦ ♥ ①r② ♠② ♦ rt♦♥ ♦♣t♠♦♥tr♦ ♣r♦♠s
ss♦t t t ♦r♥
♦r ts ♣r♣♦s strt ② ♦♥strt♥ st♣ ♥ ♥s rt♦♥
♣r♥♣ ♥ ♣♣ ♥ ♦rr t♦ rtr③ t sq♥ ♦ ①r② ♦♥tr♦ ♣r♦sss
♣♣r♦①♠t♥ t ♦♥r② ♦♥tr♦ ♣r♦ss
♦ ts ♦ t ε > 0, s tt t ε−t T (x; 2ε) s ♦♥t♥ ♥ Ω, r
Ω ⊆ [a, b]× Rn, ♥ V t s♣ ♦ t s ♦♥tr♦s ♥ ♥t ♦♥t♦♥ t
st ♦ ♣rs (v, z) r z s ♣♦♥t ♥ C ♥ v : [a, b] → Rm s ♠sr ♦♥tr♦
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sts②♥ v ∈ U ♦r tr s ♥ ♠ss rt♦♥ trt♦r② y(·) sts②♥ t
rt♦♥ r♥t qt♦♥
y(α)(t) = f(t, y(t), v(t)), y(a) = z,
r (α) s ♠r rt♦♥ rt ♦♣rt♦r 0 < α ≤ 1, t ∈ [a, b], ♥ stss t
stt ♦♥str♥t
h(t, y(t)) ≤ 0 a.e..
s♦ ♥ ♦♠♣t ♠tr s♣ t♦ ts st♣ ♦r ts t ♥② ♦♥tr♦s
v1, v2 ∈ V ♥ ②
δ(v1, v2) = L −meast ∈ [a, b] : v1(t) 6= v2(t),
♦r t ♣♦♥ts (v1, z1) ♥ (v2, z2) ♥ t st V, ♣r♦ t st V t t ♠tr
♥t♦♥ ∆ s tt
∆((v1, z1), (v2, z2)) = δ(v1, v2) + |z1 − z2| .
t s s② t♦ r tt ∆ s ♠tr s♣ ♦♥ V, s♥ ∆((v1, z1), (v2, z2)) ≥ 0,
∆((v1, z1), (v2, z2)) = ∆((v2, z2), (v1, z1)), ♥ t♦ s♦ tt
∆((v1, z1), (v2, z2)) ≤ ∆((v1, z1), (v3, z3)) + ∆((v3, z3), (v2, z2)),
t v1, v2, v3 ∈ V s tt
t : v1 6= v2 ⊂ t : v1 6= v3 ∪ t : v3 6= v2,
meast : v1 6= v2 ≤ meast : v1 6= v3+meast : v3 6= v2.
r♦r
δ(v1, v2) ≤ δ(v1, v3) + δ(v3, v2).
♦ s♦ tt t s♣ (V,∆) s ♦♠♣t ♠tr s♣
♠♠ t t sq♥ (vi, zi) ∈ V ② sq♥ ♥ tr s ♥
♠♥t (v0, z0) ∈ V s tt (vi, zi) ♦♥rs t♦ ♥ ♠ss ♣r (v0, z0).
Pr♦♦ ♥ t sq♥ s ② t ss t♦ s♦ tt ssq♥ ♦♥rs t♦
(v0, z0) ♥ V. t ♦♦s r♦♠ ❬♥ ❬❪ ♠♠ ❪ ♥ ①trt ssq♥
sts②♥
∆((vi, zi) , (vi+1, zi+1)) ≤ 2−i.
♥∣
∣
∣
∣
∪i≥k
t : vi(t) 6= vi+1(t)
∣
∣
∣
∣
≤ 21−k,
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tt vk(t) ♦♥rs ♥ ♠sr t♦ v0(t) s tt
v0(t) = vk(t) ∀t /∈ ∪i≥k
t : vi(t) 6= vi+1(t).
♥ ♦♥tr♦ v0 ①sts s tt δ(vi, v0) → 0 ♦r ♠♦st r② t. ♥ Rn s ♦♠♣t
♥ C s ♦s t♥ zi ♦♥rs t♦ ♥ ♠♥t z0 ∈ C s tt |zi − zi+1| ≤ 2−i. t
r♠♥s t♦ s♦ tt (v0, z0) s ♥ V. ♦ ♦ ts t yi(·) t rt♦♥ trt♦r②
ss♦t t (vi, zi) , Γ(t, y) st ♠♣ ♥ ②
Γ(t, y) = f(t, y, v0(t)).
r♦♠ t ♦ t s r tt t st Ai, ♥ ②
Ai = t ∈ [a, b] : vi(t) = v0(t),
s s tt L −meas(Ai) → (b− a), s♥
y(α)i (t) = f(t, yi(t), vi(t)), ∀t ∈ [a, b].
r♦r
y(α)i (t) ∈ Γ(t, yi(t)), ∀t ∈ Ai.
♥ ② ♣♣②♥ ♦r♠ ♦♥ tt tr ①sts ♥ α−s♦t② ♦♥t♥♦s
♥t♦♥ y0(t) s tt y0(0) = z0, ♥ y(α)0 (t) = f(t, y0(t), v0(t)). s y0(t) s ♥
♠ss trt♦r② ♦rrs♣♦♥♥ t♦ (v0, z0).
♠♠ s ♣r♦
♦ ♣r♦ tt tr s rt♦♥ trt♦r② yi(t) ss♦t t sq♥ (vi, zi)
♦♥rs t♦ y0(t) ss♦t t (v0, z0).
♠♠ (vi, zi) ∈ V ♦♥rs t♦ (v0, z0) ∈ V, t♥ |yi(t)− y0(t)| ♦♥rs t♦
③r♦
Pr♦♦ ❲ y(α)0 (t)−f(t, y0(t), v0(t)) = 0 ♦♥ t stAi. ♥
∣
∣
∣y(α)0 (t)− f(t, y0(t), v0(t))
∣
∣
∣
s ♦♥ ♥ L−meas(Ai) → (b−a). ♦r♦r ♥♦ r♦♠ t rsts ♦ ♣tr
tt dΓ(t,·)(·) s ss♦t ♥t♦♥ t♦ ♠t♥t♦♥ Γ(t, ·), ♥ dΓ(t,yi)(y(α)i ) = 0,
♥ t ∈ Ai, t ♦♦s r♦♠ t ♠♥t♦♥ rsts ♦ t ♦r♠ ♥ ♣tr
♦r ♥② ♣♦st δi, ♦r i s♥t② r dα(yi,Γ) ≤ δi. ♥ tr ①sts
rt♦♥ trt♦r② yi(t) ♦r ♠t♥t♦♥ Γ(t, yi(t)) sts②♥ yi(a) = yi(a) = zi,
|yi(t)− yi(t)| ≤ Kδi ♦r t r K ♥ ♥ ♦r♠ ♥ ♣tr ♥
y(α)i (t) = f(t, yi(t), v0(t)) t♥
∣
∣
∣y(α)0 (t)− y
(α)i (t)
∣
∣
∣=
∣
∣
∣f(t, y0(t), v0(t))− f(t, yi(t), v0(t))
∣
∣
∣
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≤ σ(t) |y0(t)− yi(t)| ,
r σ(t) s ♥ ♦r ♥ t ①tr ②♣♦tss ♥ s♦
|y0(a)− yi(a)| = |z0 − zi| .
② ♣♣②♥ rt♦♥ r♦♥ ♥qt② s ♣♣♥① ♦♥ tt
|y0(t)− yi(t)| ≤ |z0 − zi|Eα (σ(t)Γ(α)(t− a)α) ,
r Eα(·) s ♥r③t♦♥ ttr ♥t♦♥ s ♣tr r♦r
♦♥ tt
|y0(t)− yi(t)| ≤ |y0(t)− yi(t)|+ |yi(t)− yi(t)|
≤ |z0 − zi|Eα(σ(t)Γ(α)(t− a)α) +Kδi.
♥ ♦r i r |yi(t)− y0(t)| → 0.
♠♠ s ♣r♦
♦ ♦r ♣♦st ♥tr i ♦♦s ♣♦♥t ξ ♥ θ(x(b)) + i−2B r B s ♦♣♥
♥tr t ③r♦ s tt ξ /∈ θ(A[C]), t
G(x(b)) = |ξ − θ((x(b)))| ,
r θ(·) ♥ t ♦♥r② ♦ θ(A[C]). ♥ t ♥t♦♥ y(t) → f(t, y(t), v(t)) s
♣st③ t ♦♥st♥t σ(t), ♥ y(b) s rt♦♥ trt♦r② ss♦t t (v, z)
♥ t ♦♥r② ♦ t tt♥ st A[C]. ♥
G(y(b)) = |ξ − θ(y(b))| .
② ♣♣②♥ ♥ ♦r♠ s ♣♣♥① t♥ ♦r G(y(b)) s ♥♦♥♥t ♥
(u(t), x(a)) ∈ V stss
G(x(b)) ≤ infV
G(y(b)) + i−2,
r x(b) s rt♦♥ trt♦r② ss♦t t (u(t), x(a)) ♥ t ♦♥r② ♦ t
tt♥ st A[C], ♦r s♦♠ i−2 > 0, tr ①sts ♣♦♥t (v, z) ∈ V s tt
∆((u, x(a)), (v, z)) ≤1
i,
♥
G(y(b)) ≤ G(x(b)),
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r y(b) s rt♦♥ trt♦r② ♦rrs♣♦♥♥ t♦ (v, z) ♥ t ♦♥r② ♦ t
tt♥ st A[C], ♦r (v, z) 6= (v, z) ♥ t st V t♥
G(y(b)) + i−1∆((v, z), (v, z)) ≥ G(y(b)).
♦ ♥ s♠♣② t rsts ♦ ts st♣ ♥ t ♦♦♥ ♠♠
♠♠ t (y(t), v(t)) ♠ss ♣r♦ss ♦♥ t ♥tr [a, b] sts②♥
y(a) ∈ C,
h(t, y(t)) ≤ 0,
|y(t)− x(t)| ≤ ε.
♥
G(y(b)) + i−1∆((v, z), (v, z)) ≥ G(y(b)),
♦r (v, z) 6= (v, z) ∈ V. r♦r
|ξ − θ(y(b))|+ i−1δ(v, v) + i−1 |y(a)− z| ≥ |ξ − θ(y(b))| .
♦ t s s t rsts ♦t♥ ♥ t♦♥ ♦ ♣tr ♥ t rr♥t ♦♥t①t
t♣ ♥ ts st♣ ♦♥strt rt♦♥ r♥t ♥s♦♥ ♥ ♦rr t♦ t
♥t ♦ t ♠①♠♠ ♣r♥♣ ♣r♦ ♥ t ♣r♦s st♦♥ ♦ ♦ tt t t
rt♦♥ stts Y (t) ♥ tr ♦♠♣♦♥♥ts s ♦♦s Y (t) = [y1(t), y2(t), y3(t)], ♥
t ♠t♥t♦♥ F (t, Y (t)) ♥ ②
F (t, Y (t)) := [v
1 + |v|, χt(v), f(t, y3, v)] : v ∈ V ,
r t rst ♦♠♣♦♥♥t ♦ t ♠t♥t♦♥ F (t, Y (t)) s tr♠ rs♣♦♥s ② t
♦♠♣tt♦♥ t ♥srs t ♦♥r♥ ♦ ♠t♥ sq♥s ♥ ♥ t
♦♥tr♦ ♦♥str♥t st ♥ ♦♠ ♥♦♥ t s♦♥ ♦♠♣♦♥♥t s tr♠ t♦ ♣♥③
t t♦♥ ♦ t ♦♥tr♦ t rs♣t t♦ t ♦♣t♠③♥ ♦♥ s χt(v) s ♥t♦r ♥t♦♥
♥ ②
χt(v) =
1, v 6= v,
0, ♦trs,
♥ t st ♦♠♣♦♥♥t ♦ t ♠t♥t♦♥ s t s ②♥♠s ♣♣♦s tt t st
C ♥ ②
C := [y1, y2, y3] : y3 ∈ C.
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s ♥② rt♦♥ trt♦r② Y (t) = [y1(t), y2(t), y3(t)] ♦r ♠t♥t♦♥ F (t, Y (t))
r♦♠ t st C ♥ ②
Y = [β1 +1
Γ(α+ 1)
∫ t
a
v
1 + |v|(dτ)α, β2 +
1
Γ(α+ 1)
∫ t
a
χτ (v)(dτ)α, y(t)],
r (y(t), v(t)) s ♥ ♠ss ♦♥tr♦ ♣r♦ss s tt y(0) ∈ C, β1, β2 r ♦♥st♥t
♥∫ t
a(·)(dτ)α s rt♦♥ ♠r ♥tr ♦♣rt♦r t α ∈ (0, 1]. ♦ ♥
t♦ ♥t♦♥s M1(·), M2(·) s tt t s♠ ♦ ts ♥t♦♥s s q♥t t♦ t ♦st
♥t♦♥ tt s ♠♥♠③ ② t ♦♥r② ♦♥tr♦ ♣r♦ss t M1(Y ) ♥ M2(Y )
♥ ②
M1(Y ) = i−1 |y3 − z| − i−1y2,
M2(Y ) = |ξ − θ(y3)|+ i−1y2.
r♦r r♦♠ t ♠♠ t rt♦♥ trt♦r② Y (t) = [y1(t), y2(t), y3(t)] ♥
② t v = v, y = y, β1 = 0, β2 = 0 ♠♥♠③s
M2(Y (b)) +M1(Y (a)),
♦r t rt♦♥ trt♦rs ♦r t ♠t♥t♦♥ F (t, Y (t)) ♥ t st C, ♥ sts②
|y3(t)− x(t)| ≤ ε,
h(t, y3(t)) ≤ 0.
t
M(Y (a), Y (b)) = M2(Y (b)) +M1(Y (a))
= |ξ − θ(y3(b))|+ i−1 |y3 − z| .
♦s② tt
M(Y (a), Y (b)) = |ξ − θ(y(b))| .
t t ♥t♦♥ Φ(Y ) ♥ ②
Φ(Y ) = maxh+(t, y3(t)),
r h+ = maxh, 0.
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♥ ♦♥ tt t rt♦♥ trt♦r② Y (t) ♠♥♠③s
max
M(Y (a), Y (b))−M(Y (a), Y (b)), Φ(Y )
,
♦r t rt♦♥ trt♦rs Y (t) ♦r t ♠t♥t♦♥ F (t, Y (t)) s tt
Y (a) ∈ C,
t t ♦♥str♥ts ♥ ♥ ♥ ♦r ❲♥ i s s♥t② r ♠♣s
tt ♥② y(t) ♥r y(t) t♦♠t② sts② ♦ Y (t) ♣r♦s str♦♥ ♦
♠♥♠♠ s ♣♣♥① ♦r t ♥t♦♥ ♥ st t♦ t ♦♥str♥t
t♣ ♦ ♥ t ♠t♦♥♥ H(·, ·, ·). ♦r ts ♣r♣♦s t t ♦♥t
rt♦♥ ♥t♦♥ P (·) tt s♦ s tr ♦♠♣♦♥♥ts [p1(·), p2(·), p3(·)], tr♦r
Q(·) = [q1(·), q2(·), q3(·)] r t rt♦♥ t♥ q(·) ♥ p(·) s ♥ ♦r ♥
t ♠t♦♥♥ ♥ ②
H(t, Y,Q) := maxW∈F (t,Y )
〈Q,W 〉 ,
r Y, F (t, Y ) ♥ ♦r r♦r H(·, ·, ·) s s ♦♦s
H(t, Y,Q) := maxv∈U(t)
〈q1, v〉
1 + |v|+ χt(v)q2 +
⟨
q3, f(t, y3, v)⟩
.
② ♣♣②♥ t rsts ♦t♥ ♥ t♦♥ ♦ ♣tr ♥ ② s♥ t ♥r③
r♥t s ♣♣♥① ♦r t ♠t♦♥♥ H(t, Y,Q),
[−p(α)3 (t), 0] ∈ co[Dq3(t), r] : D, q3(t), ♥ r r t ♠♥ts tt ♥ ♥ s
♦♦s
q3(t) = p3(t) +1
Γ(α+ 1)
∫
[a,t)γ(τ)(dµ(τ))α, D = lim
i→∞∂yf(t, yi, vi), r = lim
i→∞χt(vi).
r♦♠ t ♥t♦♥ ♦ t ♥t♦r ♥t♦♥ tr r sr s ♦ r t tt
③r♦ s ♦r r i, ♦♥② r♥t sq♥s vi tt sts② vi = v, ♥ s
♦♥sq♥ yi(t) ♦♥rs t♦ y(t). ♥ ② stt♥ p3(t) = p(t), ♥ q3(t) = q(t)
♦♥ tt
−p(α)(t) ∈ q(t)∂yf(t, y(t), v(t)),
② s♥ t ♥t♦♥ ♦ P♦♥tr②♥ ♥t♦♥
− p(α)(t) ∈ ∂yH(t, y(t), q(t), v(t)).
♦t tt ♥ ♦r♠ ♦ ♣tr t tr♥srst② ♦♥t♦♥s ♥ ♦tr ♦♠♣♦♥♥t
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♦ t r♥t ♥s♦♥ ♠♣② tt p1 = 0, ♥ p2 s ♦♥st♥t ♥ t s♠ ♣♣♥s
t♦ q1, ♥ q2, rs♣t②
s s t ♦r♠ ♥ ♣tr tr s ♥♦♥♥t λ > 0 s s
p3(a) ∈ r∂y3dC(Y (a)) + λ∂y3M1(Y (a)),
p3(a) ∈ r∂y3dC(y(a)) + i−1B.
s i → ∞, t♥ r♦♠ ♥ t ♥t♦♥ ♦ t ♠tr∆(·, ·), tt t ♠sr
♦ t st t : v(t) 6= u(t) ♦s t♦ ③r♦ ♥ |y(a)− x(a)| → 0. t ♦♦s r♦♠ ♠♠
tt y(·) ♦♥rs t♦ x(·). ♥ t st qt♦♥ t y(·) = x(·), v(·) = u(·), ♥
p3(·) = p(·) ♦♠s
p(a) ∈ r∂dC(x(a)).
r♦♠ Pr♦♣♦st♦♥ ♥ ♣♣♥①
p(a) ∈ NC(x(a)).
♠r②
−p3(b)−1
Γ(α+ 1)
∫
[a,b]γ(τ)(dµ(τ))α ∈ λ∂y3M2(Y (b)),
s♥ q3(b) = p3(b) +1
Γ(α+1)
∫
[a,b] γ(τ)(dµ(τ))α, tr♦r
−q3(b) ∈ λ∂y3M2(Y (b)),
−q3(b) ∈ λ∂y3G(y(b)),
r G(·) s ♥ ♦r ❲ ♥♦ r♦♠ ♦ ξ /∈ θ(A[C]), tr♦r ξ 6= θ(y(b)),
t♥ t st♥ G(y(b)) 6= 0. ♦ ② s♥ ♦r♠ ♥ ♣♣♥①
−q3(b) ∈ λ∂y3G(y(b))∂y3θ(y(b)).
r♦♠ Pr♦♣♦st♦♥ ♥ ♣♣♥①
ζ =−λ(θ(y(b))− ξ)
|θ(y(b))− ξ|.
♥ ‖µ‖+ λ > 0, t♥
‖µ‖+ |ζ| > 0.
r♦r ♦♥ tt
q(b) ∈ ζ∂yθ(y(b)).
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② s♥ t s♠ ♥♦tt♦♥ ♥ i → ∞ stt ♦r ♦r p(a),
q(b) ∈ ζ∂xθ(x(b)).
♥② s♥ t ♦♠♣♦♥♥ts ♦ t ♠t♦♥♥ ♥s♦♥ sts② q1(·) = 0, ♥ q2(·) s
♦♥st♥t t♥ t ♠①♠③t♦♥ ♥ ②
⟨
q3(t), f(t, y(t), v)⟩
≥ maxv∈Ω(t)
⟨
q3(t), f(t, y(t), v)⟩
.
② s♥ t ♥♦tt♦♥ ♥ i → ∞, ♥ y(·) = x(·), v(·) = u(·), ♥ q3(·) = q(·),
⟨
q(t), f(t, x(t), u)⟩
= maxv∈Ω(t)
⟨
q(t), f(t, x(t), v)⟩
.
♦ ② ♣♣②♥ t P♦♥tr②♥ ♦r♠ ♦♥ tt
H(t, x(t), q(t), u) = maxv∈Ω(t)
H(t, x(t), q(t), v).
t♣ ♥② ♥ ♦rr t♦ ♦♠♣t t ♣r♦♦ r♠♦ t ①tr ②♣♦tss
② s♦♥ tt t rsts ♦t♥ ♦ r t♦t t♠ t t
②♣♦tss ♥ ♦r t t ②♣♦tss s♥t ♦r ts t
η(t) = p(t)Eα(aJαt fx),
r aJαt (·) s ♠r rt♦♥ ♥tr ♦♣rt♦r s ♣tr ♥ Eα(·) s tt
r ♥t♦♥ s ♣tr ② r♥tt♥ ♦t ss ② rt♦♥ ♠r
rt
η(α)(t) = p(α)(t)Eα(aJαt fx) + p(t)fxEα(aJ
αt fx).
r♦♠ t ♦♥t qt♦♥ ♦ t ♠♠ ♦♥t♦♥
−p(α) = (p+ aJαt;µhx)fx,
tr♦r
η(α)(t) = −p(t)fxEα(aJαt fx)− (aJ
αt;µhx)fxEα(aJ
αt fx) + p(t)fxEα(aJ
αt fx)
= −(aJαt;µhx)fxEα(aJ
αt fx).
② ♥trt♥ ♦t ss ♦t♥
p(t) = η(a)Eα(−aJαt fx)− aJ
αt
(
(aJαt;µhx)fx
)
,
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r♦♠ t ♣st③ ♦♥t♦♥ ♦r t ♥t♦♥ f , h
∣
∣
∣fx
∣
∣
∣ ≤ k(t, u),
∣
∣
∣hx
∣
∣
∣ ≤ Kh, ♥
‖µ‖ =1
Γ(α+ 1)
∫ t
a
|µ(dτα)| ≤ 1.
r Khs ♣st③ ♦♥st♥t ♦r t ♥t♦♥ h(t, ·). t♦♥② |p(a)| = |η(a)| ≤ 1.
♥ ♦♥ tt
|p(t)| ≤ |η(a)|Eα(aJαt
∣
∣
∣fx
∣
∣
∣) + aJ
αt
(
(aJαt;|µ|
∣
∣
∣hx
∣
∣
∣)∣
∣
∣fx
∣
∣
∣
)
,
≤ Eα(aJαt k(t, u)) +K
h aJαt k(t, u).
rt♥ s r♦♠ ts ♥qt② ♥ ②
M := Eα
(
1
Γ(α+ 1)
∫ b
a
k(t, u)(dt)α)
+Kh
(
1
Γ(α+ 1)
∫ b
a
k(t, u)(dt)α)
.
♦♥sq♥t②
|p(t)| ≤ M.
t Sj ♥ ♥rs♥ ♠② ♦ ♥t sst ♦ MB s tt MB ⊂ Sj + j−1B ♦r
j. ♦r s ∈ Sj , st ♠sr ♥t♦♥ vs(·) s tt vs(t) ∈ Ω(t) ♥
H(t, x(t), s) ≤⟨
s, f(t, x(t), vs)⟩
+ j−1,
r
H(t, ·, s) := sup⟨
s, f(t, ·, v)⟩
: v ∈ Ω(t).
t
Ωj(t) = vs(t) : s ∈ Sj ∪ u(t).
♦ ♦♥sr ♥ ♣r♦♠ ♥ t ♠t♥t♦♥ Ω(t) s r♣ ② Ωj(t). ②
ss♠♥ tt t ②♣♦tss r sts ② t t ♦r t ♥ ♣r♦♠ ♦r rst
②s ♠t♣rs p, µ, γ, ♥ ζ t ♣r♦♣rts st ♥ ♠♠ ①♣t tt ♥♦ t
♠①♠③t♦♥ ♦ t ♠t♦♥♥ ♦♥t♦♥ ts t ♦r♠
⟨
q, x(α)⟩
= Hj(t, x, q),
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r Hj ♥ ②
Hj(t, x, s) = max⟨
s, f(t, x, v)⟩
: v ∈ Ωj(t).
♦r t t ♦♥t♦♥ ♦ t ♠♠ s ♦♦s s ∈ Sj tt
q = p+1
Γ(α+ 1)
∫
[a,t)γ(τ)µ(dτ)α ∈ s+ j−1B.
♥ ♦♥t♥ t♦ ss♠ ♦♥t♦♥ ♦ ①tr ②♣♦tss σ(t) s ♣st③ ♦♥st♥t
♦r H(t, x(t), ·) ♥ Hj(t, x(t), ·) s tt
|Hj(t, x(t), s)−Hj(t, x(t), q)| = σ(t) |s− q| .
r♦♠
|Hj(t, x(t), s)| − |Hj(t, x(t), q)| ≤ σ(t)∣
∣j−1∣
∣ ,
|Hj(t, x(t), s)| − σ(t)∣
∣j−1∣
∣ ≤ |Hj(t, x(t), q)| .
② s♥ t ♥t♦♥ ♦ Hj(t, x(t), ·),
⟨
s, f(t, x(t), vs)⟩
− σ(t)∣
∣j−1∣
∣ ≤ |Hj(t, x(t), q)| .
② s♥
|Hj(t, x(t), q)| ≥ |H(t, x(t), s)| − j−1 − σ(t)∣
∣j−1∣
∣ ,
♥ ② ♣♣②♥ ♣st③ ♦♥t♦♥ ♦r H(t, x(t), ·) ♦t♥
|Hj(t, x(t), q)| ≥ |H(t, x(t), q)| − j−1 − 2σ(t)∣
∣j−1∣
∣ .
r♦♠ ♦♥ tt
⟨
q, x(α)(t)⟩
≥ |H(t, x(t), q)| − j−1 − 2σ(t)∣
∣j−1∣
∣ .
♥ ∣
∣
∣
⟨
q, x(α)(t)⟩
−H(t, x(t), q)∣
∣
∣≤∣
∣j−1∣
∣ (2σ(t) + 1).
s j → ∞ ♥ ②s ♦♥t♦♥ ♦ ♠♠ ♥ t ♦♥t♦♥ ♦ t
♠♠ ♠♣s q(b) s ♦♥ ② ♣♣②♥ ♦r♠ ♦ ♣tr ♦ ts tss
s t♦ p(·), γ(·), µ(·), ♥ ζ sts②♥ t rqr ♦♥t♦♥ ♥ t ♠t s j → ∞.
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♥② s♦ t ①tr ②♣♦tss ♥ t ♦r j ♥
Ωj(t) =
v ∈ Ω(t) : k(t, v) ≤ k(t, u(t)) + j,∣
∣
∣f(t, x(t), v)∣
∣
∣ ≤∣
∣
∣x(α)(t)∣
∣
∣+ j
.
♦t tt Ωj(t) s ♥ ♥rs♥ sq♥ ♦ ♠t♥t♦♥s ♥ tt ♥② ♠♥t ♦
Ω(t) ♦♥s t♦ Ωj(t) ♦r j s♥t② r ♦ ♦♥sr t ♣r♦♠ ♦t♥ ②
r♣♥ Ω(t) ② Ωj(t), ♥ t ②♣♦tss r sts ♥ ss♠ t ①st♥t ♦
♠t♣rs p(·), µ(·), γ(·), ♥ ζ ♣♥♥ ♦♥ j t t ♣r♦♣rts st ♥ ♠♠
①♣t tt t ♠①♠③t♦♥ ♦ t ♠t♦♥♥ ♦♥t♦♥ ♥♦ ts t ♦r♠
⟨
q, x(α)⟩
≥⟨
q, f(t, x(t), v)⟩
, ♦r v ∈ Ωj(t) t ∈ [a, b].
② ♣♣②♥ ♦r♠ ♣tr ②s p(·), µ(·), γ(·), ♥ ζ ♦♥t♥ t♦ sts②
t ♦♥t♦♥s ♥ ♦ t ♠♠ ② s♥ t t ♦r ♥② t ∈ [a, b],
v(t) ∈ Ω(t) ♠♣s v(t) ∈ Ωj(t) ♦r j r ♥♦ s t t sts② t
♦♥t♦♥ ♦ t ♠♠ ♦r t ♠t♥ t
♥ sts t ssrt♦♥s ♦ ♠♠
strt ①♠♣
P t stt ♦♥str♥ts ♦♥sr ♥ ts ①♠♣ t♦ strt t ♣♣t♦♥
♦ t ♣r♦ ①♠♠ Pr♥♣ ♦ P♦♥tr②♥ ♥ stt s ♦♦s
Minimize −y(T )
subject to x(α) = u(t)x(t), x(0) = x0,
y(α) = (1− u(t))x(t), y(0) = 0,
u(t) ∈ [0, 1],
x(t) ≤ a+ btα,
r t rt♦♥s ♦ s ♥ [0, T ] ♥ t ♦♥st♥ts x0 a b ♥ T sts②
a > x0 > 0 T > 1 b > 0 ♥ tr r ♦♥st♥ts c1 ♥ c2 s tt
c2 ≥ c1, Eα(cα1 ) >
a
x0, c2 = T − (Γ(α+ 1))
1
α .
♦t tt r ♦♥sr♥ s♠♦♦t t ♥ ♦rr t♦ tt t ♥rst♥♥ ♦ t
sss ♥♦ ♥ stt ♦♥str♥ts r ♣rs♥t
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t s ♥♦t ♦♣t♠ ♦♥tr♦ ♣r♦ss (x, y, u) ♦♠t strss ♥ ②
u(t) =
1 t ∈ [0, t1)bΓ(α+ 1)
a+ btα t ∈ [t1, t2)
0 t ∈ [t2, T ],
x(t) =
Eα(tα)x0 t ∈ [0, t1)
a+ btα t ∈ [t1, t2)
a+ btα2 t ∈ [t2, T ],
y(t) =
0 t ∈ [0, t1)(
a
Γ(α+ 1)− b
)
(t− t1)α +
bΓ(α+ 1)
Γ(2α+ 1)(t− t1)
2α t ∈ [t1, t2)
y(t2) +a+ btα2Γ(α+ 1)
(t− t2)α t ∈ [t2, T ],
r t1 ♥ t2 rs♣t② t rst ♥ t st t♠s t ♦r x(t) = a+ btα ①st
t♦ t ♦♥sr ss♠♣t♦♥s ♦r ①♠♣ ② ♦♥sr♥ t1 = c1 ♥ t2 = c2 ♥
r ♦s♥ ♥ ♦rr t♦ ♠①♠③ t ♦ y(T )
①t s♦ tt t ts ♦♥tr♦ ♣r♦ss stss t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t②
r ♥ ts ♣tr tt s u(t) ∈ [0, 1] ♠①♠③s s ♦♥ [0, T ] t ♠♣
v → H(x(t), y(y), v, p(t), q(t)),
r t P♦♥tr②♥ ♥t♦♥ H s ♥ ②
H(x, y, u, p, q) = [(p+ γ)u+ q(1− u)]x,
♥ γ(t) =1
Γ(α+ 1)
∫
[0,t)(dµ)α ♥ ② t ② γ(t+) =
1
Γ(α+ 1)
∫
[0,t](dµ)α s t
rt♦♥ tts ♥tr ♦♥ t ♥tr [0, t) ♦ t ♣♦st ♦♥ ♦r ♠sr µ
s♣♣♦rt t st ♦ ♣♦♥ts ♥ [0, T ] ♦♥ t stt ♦♥str♥t s t x(t) =
a+ btα ♥ (p, q) r t ♦♥t rs sts②♥ t ♦♦♥ r♥t qt♦♥s
s ♥ [0, T ]
−q(α)(t) = 0, q(T ) = 1
−p(α)(t) = (p(t) + γ(t))u(t) + q(t)(1− u(t)), p(T ) + γ(T+) = 1.
rst t s ♠♠t t♦ ♦♥ tt q(t) = 1, ♦r t ∈ [0, T ]
♦ ♦r t ∈ [0, t1) u(t) = 1 ♥ s♥ x(t) < a + btα γ(t) = 0, ♥ r♦♠ t t
tt −p(α)(t) = p(t) p(t) = p(t1)Eα((t1 − t)α) r♦♠ t ♠①♠③t♦♥ ♦ t
P♦♥tr②♥ ♥t♦♥ t rsts tt p(t) ≥ 1, ♦r t ∈ [0, t1) ♥ ts p(t1) ≥ 1
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♦r t ∈ [t1, t2] u(t) =bΓ(α+ 1)
a+ btα t s ♥♦t t t♦ r② tt ♥r
t ♦ ss♠♣t♦♥s u(t) ∈ (0, 1) ♦r♦r r♦♠ t ♠①♠③t♦♥ ♦ t P♦♥tr②♥
♥t♦♥ ♦♥ ♠♠t② tt p(t)+γ(t) = 1 t tt −p(α)(t) = 1 ♠♣s
tt p(t) = p(t1)−1
Γ(α+ 1)(t− t1)
α ♥
γ(t) = p(t)− p(t1) +1
Γ(α+ 1)(t− t1)
α ≥ 0,
♦r t ∈ [t1, t2] ♠♣s ♥♦t ♦♥② tt p(t1) = 1, t s♦ tt dµ = dt
♥ t st t♠ s♥tr (t2, T ] tt u(t) = 0 ♥ r♦♠ −p(α)(t) = 1 t
♦♦s tt
p(t) = p(t2)−1
Γ(α+ 1)(t− t2)
α.
♥
p(t2) = 1−1
Γ(α+ 1)(t2 − t1)
α,
♦♥ tt
p(t) = 1−1
Γ(α+ 1)((t− t2)
α + (t2 − t1)α) .
② s♥ t t tt
p(T ) = −1
Γ(α+ 1)
∫
[t1,t2](dt)α = −
1
Γ(α+ 1)(t2 − t1)
α,
tt
t2 = T − (Γ(α+ 1))1
α .
r♦♠ t ♦♥t♥t② ♥ t♠ ♦ t ♠①♠③ P♦♥tr②♥ ♥t♦♥ t t = t1
Eα(tα1 )x0 = a+ btα1 ♥ t t tt
tα1 =a
b
(x0aEα(t
α1 )− 1
)
> 0,
♦♥ tt Eα(tα1 ) >
a
x0.
❲t s♦♠ st♥r ♦rt t s ♥♦t t t♦ s tt t s ♦ t1 ♥ t2 t t2 ≥ t1
tt ♠①♠③ t ♦ y(T ) t ♦♥s tt ② t ♠♥♠♠ ♦st r ♦♠♣t
t ss♠♣t♦♥s t c1 = t1 ♥ c2 = t2 ♠♣♦s ♦♥ t t ♦ t ♣r♦♠ s
s♦♥ tt t ♦♣t♠ ♦♥tr♦ ♣r♦ss ♦♥sr ♦♥ ♥ ♥tt ss stss
t ♠①♠♠ ♣r♥♣ ♦ P♦♥tr②♥ ♣r♦ ♥ ts ♣tr s t s ♥♦t t t♦
r♦♥strt t s♦t♦♥ t♦ t ♣r♦♠ ② ♦♦s♥ t ♦♥tr♦s tt ♥♦r t t②
♦ t ♦♣t♠t② ♦♥t♦♥s ♠♦♥ rs r♦♠ t ♥ t♠
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♣tr
♦♥s♦♥s ♥ Pr♦s♣t
sr
♦♥s♦♥s
♠♥ ♦♥trt♦♥ ♦ ts tss s ♥ ♥tr♦ ♥ ♣trs ♥
♦tr ♣trs ♥ ts ssrtt♦♥ t tt t t r♥ ♥ tr ♦♥ t②
s♦ ♣② ssr② r♦ ♥ t ♦r♠t♦♥ ♥ rt♦♥ ♦ t ♠♥ rsts
r ♦t ♦♥r♥s t ♦r♠t♦♥ ♦ ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♦r rt♦♥
♣r♦♠s r t rtr③t♦♥ ♦ t ♦♣t♠♦♥tr♦ ♣r♦♠ ♥ t rt♦♥
♦♥t①t s ♠♦r rt t♥ ♦r t ♥tr ♦♥tr♣rt rt♦♥ ♦♣t♠♦♥tr♦
♣r♦♠ ♥ s ♥r③t♦♥ ♦r rt♦♥ s ♦ t ♦♣t♠♦♥tr♦
♣r♦♠ ♥ t ♥tr s♥s
❲ ♥ ② ♦r♠t♥ t rt♦♥ ♦♣t♠♦♥tr♦ ♣r♦♠ t t sts②♥
rt② str♦♥ ss♠♣t♦♥s sq♥t② ♥rs t ♦♠♣①t② ② st②♥ t
♣r♦♠ ♥ t s♥ ♦ s♠♦♦t♥ss ♦♥ ts t s♦ t stt ♦♥str♥ts r ♠♣♦s
♦♥ t t ♦ t ♣r♦♠
♥ ♣tr ♥r s♦♠ s♠♦♦t♥ss ss♠♣t♦♥s r P♦♥tr②♥ ♠①♠♠
♣r♥♣ ♦r ♥r ♦r♠t♦♥ ♦ rt♦♥ ♦♣t♠♦♥tr♦ ♣r♦♠s ♦s ♦st
♥t♦♥ s ♥ t rt♦♥ ♥tr ♦r♠ ♥ ♦s ②♥♠s s rtr③ ② t
♣t♦ rt♦♥ rt s♦ ♣rs♥t ♥ t♥q t♦ ♦t♥ rt♦♥
♣r♦♠ ♠①♠♠ ♣r♥♣ r t ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② r r
t rt♦♥s ♦♥ t ♦r♥ ♣r♦♠ ♥ ♥♦t ② ♦♥rt♥ t rt♦♥ ♣r♦♠ t♦
t ss ♦♥ t ♥tr ♦rr s ♦♥ ♥ t rr trtr rtr♠♦r ♥
strt ①♠♣ s s♦ ② s♥ t ♦♥t♦♥s ♦ t ♠①♠♠ ♣r♥♣ t♦tr
t t ttr ♥t♦♥ t♦ s♦ t t♥ss ♦ t ♣r♦♣♦s ♣♣r♦
♥ ♣tr ♥ t rt♦♥ ♥tr t rs♣t t♦ ♥r ♦♥ ♠sr ♥
t ♠r s♥s ♥ ♦r♠t ts rt♦♥ ♥tr ♥ t♦ ss t ♥ t♦t
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t♦♠ ♠sr ♦♠♣♦♥♥t ss ♥ ♥ rsts ♥ ♠sr ♥ ♥trt♦♥ t♦r②
♦r t rt♦♥ ♦♥t①t ts rsts r s♣② r♥t ♦r ♣trs ♥
♥ ♣tr ♦r♠t ♥ ♣r♦ ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♦r rt♦♥
♦♣t♠♦♥tr♦ ♣r♦♠s t stt ♦♥str♥ts ♥ t ②♥♠ s②st♠ s ♠♦ ②
rt♦♥ r♥t ♥s♦♥ ♥ t ♠r s♥s ss ♦ t ♥trst ♥ tr
♦♥ t rsts ♦ ts ♣tr r ♣rtr② ♣ t♦ ♦t♥ t ♠①♠♠ ♣r♥♣
♦ ♣tr
♥ ♣tr ♠①♠♠ ♣r♥♣ ♦r rt♦♥ ♦♣t♠♦♥tr♦ ♣r♦♠ t stt
♦♥str♥ts ♥ t ss♠♣t♦♥s r ♣rs♥t ♥ ♣r♦ ♦♣t ♣♣r♦
♦♦s t ♦♥ ♥ r ❬❪ s s ♥ ts ♣tr t♦ t ♥t ♦ t rsts
♦t♥ ♥ ♣tr ♦r♦r t ♣r♦♣♦s ♣♣r♦ s strt ② ♥ ①♠♣
♦ s♠ ♣ t ♠♥ ♦♥s♦♥ ♦ ts tss ♦♥ssts ♥ t t tt ①t♥ ♥
♥♠r ♦ r② s♥♥t ②s t rr♥t t♦r② ♦♥ ♥ssr② ♦♥t♦♥s ♦ ♦♣t♠t②
s♦ r ♦♣ ♦r ♦♣t♠ rt♦♥ r♥t ♦♥tr♦ ♣r♦♠s s rtr③ ♥
t ♦♥t♥ts ♦ t r♦s ♣trs ♦ t tss ♥ ts ② ts tss ♦♥sttts
♦♥trt♦♥ t♦ ss♥ t rr♥t ①st♥ ♣ t♥ ♦t ♦s ♦ t♦rs
tr ❲♦rs
s t s r r♦♠ t t ♦ sss tt rs ♥ ♦♣t♠♦♥tr♦ t♦r② tr r
r ♥♠r ♦ sss tt r t ♥t♦ t♦ t s♦rt ♣r♦ ♦ t♠ ②rs
♥♥ t s♦r ♣rt tt r ♠ t♦ ♦t ♠② ♦rts t♦ t ♣r♦♣♦s
♥s ♥ ts ② s♦♠ sts ♥♦t ♥ ①♣♦r t r t ♦r tr ♦rs
r r ♠♥② ♣♦♥ts ♦rt ♦ tr ♥stt♦♥ ♠♦♥ ♦ t♦ s♥
♦t t ♦♦♥
♥ ♣trs ♥ ♦♥② stt ♦♥str♥ts ♥ ♦♥sr ♥ t ♣r♦♠
♥ tr ♦rs ♥ r ♠①♠♠ ♣r♥♣ t ♠① ♦♥str♥ts ♥r
♣♣r♦♣rt ss♠♣t♦♥s
t s ♥♦t t t♦ ♦♥strt ♥ ①♠♣ ♦r t ♣r♦ ♠①♠♠ ♣r♥♣s
♥rt ♥ ♠♣♦rt♥t ss tt s ♦ ♥trst ♦♥r♥s t t♦♥ ss♠♣
t♦♥s ♥r t ♠①♠♠ ♣r♥♣ ♦s ♥♦t ♥rt tt s t ♦♥t♦♥s
r♠♥ ♥♦r♠t
rsts ♦t♥ ♥ ♣tr ♦♣♥ t ♦♦r ♦r t ♦♥srt♦♥ ♦ rt♦♥
♠♣s ②♥♠ ♦♥tr♦ s②st♠s ❲ ♥t♣t tt t ♥r②♥ ♥r♥t t
♥ sss ①tr♠② ♥♥ ♦r ♥ t r♥ ♦ ♣♣t♦♥s
ts s ♥♦tr rt♦♥ ♦♥ t ♣ t♥ ♥tr ♥ rt♦♥ ♦♣t♠
♦♥tr♦ t♦rs ♦ ♦♠ s♠r
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❲ r tt ♥ ts tss ♦♥② sss t rt♦♥ rt t
rs♣t t♦ t♠ t. ♦r tr r ♥♠r ♦ ♣r♦♠s ♦s ②♥♠s ♥♦
♣rt r♥t qt♦♥s ♦r t rt♦♥ rt t rs♣t t♦ t
stt ♦♥str♥ts s♦ ♦♥sr
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♣♣♥s
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♣♣♥①
rt♦♥ s
♥ ts ♣♣♥① ♣r♦ r r ♦ s♦♠ ② ♦♥♣ts ♦ rt♦♥ rt
♥ s rt♦♥ ♦♣rt♦rs ♥ ♣♣♥① r ♦t ♦ t s♦♣ ♦ ts tss
r st ♦r t♦ t rr ♦tr t②♣s ♦ rt♦♥ ♦♣rt♦rs
st♦r
st♦r② ♦ rt♦♥ s strt ♥ ♦s♣t r♦t t♦ ♥③ ttr t
♣t♠r t s♥ ♠ t s t ♠♥♥ ♦ dnxdtn
, n = 12 rt♦♥ ♥
♥③s rs♣♦♥s s ♥ ♣♣r♥t ♣r♦① r♦♠ ♦♥ ② s ♦♥sq♥s
r♥
qst♦♥ rs ② ♥③ ♦r rt♦♥ rt s ♥ ♥ ♦♥♦♥ t♦♣ ♥
t st ②rs ♥ t♥ rt♦♥ s s ttrt t tt♥t♦♥ ♦ ♠♥②
♠♦s ♠t♠t♥s s s r r♥ ♣ ♦rr
♦ ♠♥♥ rü♥
t♥♦ s ❲② ré② ♥ ♠♥②
♦trs s r ♥ s♦r ❬❪ ♥ ♦r♥♦ ♥ ♥r ❬❪ ♦r ♦♥②
s♥ t ♥ts rt♦♥ s s ♥ t ♦t ♦ s♣③ ♦♥r♥s ♥
trtss rt ♦r t rst ♦♣♥ s♥t ♥t s t♦ ♦ss ♦ ♦r♥③
t rst ♦♥r♥ ♦♥ rt♦♥ s ♥ ts ♣♣t♦♥s t t ♥rst② ♦
♥ ♥ ♥ ♥ t ts ♣r♦♥s rst ♠♦♥♦r♣ ♦t t♦ rt♦♥
s s ♣s ♥ ② ♠ ♥ ♣♥r t rsss tr ♦♥t
♦♦rt♦♥ tt ♥ ♥ s ♦♦rt♦♥ t♥ ♠st ♠ ♥
♠t♠t♥ ♣♥r ♥ trt♥ ♣r♦♠s ♦ ♠ss ♥ t tr♥sr ♥ tr♠s ♦
t s♦ s♠rts ♥ s♠♥trs r② ♠♥sts t ♦r♥ ♦ ♥ r
♦r rt♦♥ s s ♦♥ ♦t ♣②s ♥tt♦♥ ♥ ♠t♠t rstt② ♥
t r ♦♦ ② ♠♦ s ♥ r rrr t♦ ♥♦ s ♥②♦♣
♦ rt♦♥ s ♣♣r rst ♥ ss♥ ♥ tr tr♥st ♥t♦ ♥s
♦②s s♦♠ srs ♦ ♦♦s ♦r♥s ♥ t①ts ♥ ♦t t♦ rt♦♥ s
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♥ ts ♣♣t♦♥s ♦ t ❬❪ s ♥♠r s ①♣t t♦ r♦ ♥ t
♦rt♦♠♥ ②rs
rt♦♥ ♣rt♦rs
♥t♦♥s ♦ rt♦♥ ♥trs
t α > 0.
♥t♦♥ ♠r rt♦♥ ♥tr
❼ t ♠r rt♦♥ ♥tr ♦ ♦rr α s
Ha I
αt f(t) =
1
Γ(α)
∫ t
a
(lnt
τ)α−1 f(τ)
τdτ, t ∈ [a, b].
❼ rt ♠r rt♦♥ ♥tr ♦ ♦rr α s
Ht I
αb f(t) =
1
Γ(α)
∫ b
t
(lnτ
t)α−1 f(τ)
τdτ, t ∈ [a, b].
♥t♦♥ ♥ rt♦♥ ♥tr
❼ t ♥ rt♦♥ ♥tr ♦ ♦rr α s
Iαc f(t) =1
Γ(α)
∫ t
c
(t− τ)α−1f(τ)dτ, t > c.
❼ rt ♥ rt♦♥ ♥tr ♦ ♦rr α s
Iαc f(t) =1
Γ(α)
∫ c
t
(τ − t)α−1f(τ)dτ, t < c.
♥t♦♥ ♦r rt♦♥ ♥tr
❼ t ♦r rt♦♥ ♥tr ♦ ♦rr α s
Iα1,ηf(t) =t−α−η
Γ(α)
∫ t
0(t− τ)α−1f(τ)dτ.
❼ rt ♦r rt♦♥ ♥tr ♦ ♦rr α s
Iα1,ηf(t) =tη
Γ(α)
∫ ∞
t
(τ − t)α−1f(τ)dτ.
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♥t♦♥ ré② rt♦♥ ♥tr
❼ t ré② rt♦♥ ♥tr ♦ ♦rr α s
Iασ,ηf(t) =σt−σ(α+η)
Γ(α)
∫ t
0(tσ − τσ)α−1τση+σ−1f(τ)dτ.
❼ rt ré② rt♦♥ ♥tr ♦ ♦rr α s
Iασ,ηf(t) =σtσα
Γ(α)
∫ ∞
t
(τσ − tσ)α−1τσ(1−α−η)−1f(τ)dτ.
♥t♦♥s ♦ rt♦♥ rts
t α > 0 ♥ n− 1 < α ≤ n, n ∈ N.
♥t♦♥ ♦ rt♦♥ rt
❼ t ♦ rt♦♥ rt ♦ ♦rr α s
Dα+f(t) =
1
Γ(n− α)
dn
dtn
∫ t
−∞
(t− τ)n−α−1f(τ)dτ, t > 0.
❼ rt ♦ rt♦♥ rt ♦ ♦rr α s
Dα−f(t) =
(−1)n
Γ(n− α)
dn
dtn
∫ +∞
t
(t− τ)n−α−1f(τ)dτ, t < +∞.
♥t♦♥ rü♥t♥♦ rt♦♥ rt
❼ t rü♥t♥♦ rt♦♥ rt ♦ ♦rr α s
GLa Dα
t f(t) = limh→0
1
hα
[ t−a
h]
∑
k=0
(−1)k(
α
k
)
f(t− kh).
❼ rt rü♥t♥♦ rt♦♥ rt ♦ ♦rr α s
GLt Dα
b f(t) = limh→0
1
hα
[ b−t
h]
∑
k=0
(−1)k(
α
k
)
f(t+ kh).
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r(
αk
)
s t ♥r③t♦♥ ♦ ♥♦♠ ♦♥ts t♦ r ♥♠rs ♥ ②
(
α
k
)
=Γ(α+ 1)
Γ(k + 1)Γ(α− k + 1).
♥t♦♥ ♠r rt♦♥ rt
❼ t ♠r rt♦♥ rt ♦ ♦rr α s
HaD
αt f(t) =
1
Γ(n− α)
(
td
dt
)n ∫ t
a
(lnt
τ)n−α−1 f(τ)
τdτ, t ∈ [a, b].
❼ rt ♠r rt♦♥ rt ♦ ♦rr α s
Ht D
αb f(t) =
1
Γ(n− α)
(
−td
dt
)n ∫ b
t
(lnτ
t)n−α−1 f(τ)
τdτ, t ∈ [a, b].
♥t♦♥ ♥ rt♦♥ rt
❼ t ♥ rt♦♥ rt ♦ ♦rr α s
Dαc f(t) =
1
Γ(1− α)
d
dt
∫ t
c
(t− τ)−αf(τ)dτ, t > c.
❼ rt ♥ rt♦♥ rt ♦ ♦rr α s
Dαc f(t) = −
1
Γ(1− α)
d
dt
∫ c
t
(τ − t)−αf(τ)dτ, t < c.
♥t♦♥ r rt♦♥ rt
❼ t r rt♦♥ rt ♦ ♦rr α s
Dα+f(t) =
α
Γ(1− α)
∫ ∞
0
f(t)− f(t− τ)
τα+1dτ.
❼ rt r rt♦♥ rt ♦ ♦rr α s
Dα−f(t) =
α
Γ(1− α)
∫ ∞
0
f(t)− f(t+ τ)
τα+1dτ.
♥t♦♥ s③ rt♦♥ rt
Dαt f(t) = −
1
2 cos(απ2 )
1
Γ(α)
dn
dtn
∫ t
−∞
(t− τ)n−α−1f(τ)dτ +
∫ ∞
t
(τ − t)n−α−1f(τ)dτ
.
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t♦♥ t♥ t rt♦♥ rts
r st r s♦♠ rt♦♥s s ♦r ♦r ♣r♣♦ss ♦r tr ♣r♦♦s s s
t ❬❪ ♥ P♦♥② ❬❪
♠♥♥♦ ♥ ♣t♦ rts r rt ♥ t ♦♦♥ ② t t > 0
α ∈ R ♥ n− 1 < α ≤ n ∈ N ♥
aDαt f(t) = C
aDαt f(t) +
n−1∑
k=0
f (k)(a)
Γ(k + 1− α)(t− a)k−α,
tDαb f(t) = C
tDαb f(t) +
n−1∑
k=0
f (k)(b)
Γ(k + 1− α)(b− t)k−α,
♥
CaD
αt f(t) = aD
αt
(
f(t)−n−1∑
k=0
(t− a)k
Γ(k + 1)f (k)(a)
)
,
CtD
αb f(t) = tD
αb
(
f(t)−n−1∑
k=0
(b− t)k
Γ(k + 1)f (k)(b)
)
.
s Pr♦♣rts ♦ rt♦♥ s
♠♦♥ t sr ♣r♦♣rts ♦ t ♦♣rt♦rs ♦ r♥tt♦♥ ♥ ♥trt♦♥ ♦ rtrr②
♦rr r ①♣rss s♦♠ ♦ t ♠♦st s ♦r ♦r ♣r♣♦ss ♥♦t② ♦r t ♠♥♥
♦ ♥ ♣t♦ rts
Pr♦♣♦st♦♥ ♦♥st♥t ♥t♦♥
♦r t ♠♥♥♦ rt♦♥ rt ♦r ♥② ♦♥st♥t k,
Dαk =k
Γ(1− α)t−α.
♥ t ♦♥trr② ♦r t ♣t♦ rt♦♥ rt ♦r ♥② ♦♥st♥t k,
CDαk = 0.
Pr♦♣♦st♦♥ ♥rt②
t n− 1 < α < n ∈ N, f(t) ♥ g(t) t♦ ♦♥t♥♦s ♥t♦♥s ♥ ♦♥ [a, b] s tt
aDαt f ♥ aD
αt g ①st ♠♦st r②r ♦r♦r t λ1, λ2 ∈ R. ♥ Dα
a (λ1f ± λ2g)
①sts ♠♦st r②r ♥ t ♠♥♥♦ rt ♦②s
aDαt [λ1f(t)± λ2g(t)] = λ1aD
αt f(t)± λ2aD
αt g(t).
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♠r② t ♣t♦ rt stss
CaD
αt [λ1f(t)± λ2g(t)] = λ1
CaD
αt f(t)± λ2
CaD
αt g(t).
Pr♦♣♦st♦♥ s♠r♦♣ ♣r♦♣rt② ♦ t ♠♥♥♦ ♥tr ♦♣rt♦r
t α, β > 0, t > 0, ♥ f(t) ∈ Lp(a, b), 1 ≤ p ≤ ∞. ♥
IαIβf(t) = IβIαf(t) = Iα+βf(t), t ∈ [a, b] a.e..
Pr♦♣♦st♦♥ t n − 1 < α ≤ n ∈ N, t ∈ [a, b] ♥ f(t) ∈ Lp(a, b), 1 ≤ p ≤ ∞.
♥
CaD
αt aI
αt f(t) = f(t),
aIαtCaD
αt f(t) = f(t)−
n−1∑
k=0
f (k)(a)
Γ(k + 1)(t− a)k.
s♠ rst ♦s ♥ s♥ t ♠♥♥♦ rt
Pr♦♣♦st♦♥ ♥tr♣♦t♦♥
t n − 1 < α ≤ n ∈ N, t ∈ [a, b], ♥ f(t) ♥t♦♥ s tt Dαf(t) ①sts ♥
t ♠♥♥♦ rt ♦②s
limα→n
aDαt f(t) = f (n)(t),
limα→n−1
aDαt f(t) = f (n−1)(t).
♦r t ♣t♦ rt t ♦rrs♣♦♥♥ ♥tr♣♦t♦♥ ♣r♦♣rt② rs
limα→n
CaD
αt f(t) = f (n)(t),
limα→n−1
CaD
αt f(t) = f (n−1)(t)− f (n−1)(a).
Pr♦♣♦st♦♥ ♥③
t α ∈ R, t > 0, n− 1 < α ≤ n ∈ N ♥ f(t), g(t) ♦♥t♥♦s ♥t♦♥s ♦♥ [a, b]; t♥
t ♥r③ ♥③ ♦r♠ ♦r t ♠♥♥♦ rt s ♥ s
Dαa+[f(t)g(t)] =
∞∑
k=0
(
α
k
)
(
Dα−kf(t))
Dk[g(t)],
r s t ♥♦♠ ♦♥t
(
α
k
)
=Γ(α+ 1)
Γ(k + 1)Γ(α− k + 1).
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♥③ ♦r♠ ♦r t ♣t♦ rt stss
CaD
αt [f(t)g(t)] =
∞∑
k=0
(
α
k
)
(
Dα−kf(t))
Dkg(t)−n−1∑
k=0
tk−α
Γ(k − α+ 1)Dk[g(t)f(t)](a).
♥r③ ②♦rs ♦r♠
②♦rs ♦r♠ s ♥ ♥r③ ② ♠♥② t♦rs t ♥ ❬❪
r st r ♦♥② t♦ ♦r♠s ♣t♦ ♥ ♠♥♥♦
❼ ♥r③t♦♥ ♦ ②♦rs ♦r♠ ♥♦♥ ♣t♦ rt♦♥ rts
t 0 < α ≤ 1 n ∈ N ♥ f(x) ♦♥t♥♦s ♥t♦♥ ♥ [a, b] t ♥
❬❪ ♥ ♦r x ∈ [a, b],
f(x) =n∑
k=0
(x− a)kα
Γ(kα+ 1)CaD
kαx f(a) +Rn(x, a),
r Rn(x, a) s t r♠♥r ♦ t ♥r③ ②♦rs srs ♥ ②
Rn(x, a) =CaD
(n+1)αx f(ξ)
(x− a)(n+1)α
Γ((n+ 1)α+ 1).
r a ≤ ξ ≤ x ♥ CaD
αx s t t ♣t♦ rt♦♥ rt ♦ ♦rr α
❼ ♥r③t♦♥ ♦ ②♦rs ♦r♠ ♥ t ♠♥♥♦ s♥s
t α > 0 n ∈ Z+ ♥ f(x) ∈ C
[α]+n+1([a, b]) ♥♠♠r ❬❪ ♥
f(x) =n−1∑
k=−n
(x− x0)k+α
Γ(k + α+ 1)Dk+α
a+ f(x0) +Rn(x),
♦r a ≤ x ≤ b r Rn(x) s t r♠♥r ♥ ②
Rn(x) = Iα+na+ Dα+n
a+ f(x).
r Dαa+ s t t ♠♥♥♦ rt♦♥ rt ♦ ♦rr α Iαa+ s t
t ♠♥♥♦ rt♦♥ ♥tr ♦ ♦rr α ♥ [α] s t ♥tr ♣rt ♦ α
♦♠ Pr♦♣rts ♦r ♠r rt♦♥ rt ♥
♥tr
♥t♦♥ ♠r rt♦♥ rt rt♦♥ r♥
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t f : R → R, x→ f(x) ♦♥t♥♦s ♥t♦♥ t ♥♦t ♥ssr② r♥t ♥
h > 0 ♥♦t ♦♥st♥t srt③t♦♥ s♣♥ ♦rr ♦♣rt♦r FW (h) s ♥ s
FW (h).f(x) := f(x+ h).
♦r α ∈ R ♥ 0 < α ≤ 1 t rt♦♥ r♥ ∆αf(x) s ♥ ②
∆αf(x) := (FW − 1)α.f(x) =
∞∑
k=0
(−1)k(
α
k
)
f(x+ (α− k)h),
♥ t ♠r rt♦♥ rt ♦ ♦rr α s
f (α)(x) = limh↓0
∆α[f(x)− f(0)]
hα.
♠r rt♦♥ rt s t ♦♦♥ ♣r♦♣rts
❼ αth rts ♦ ♦♥st♥t s ③r♦
❼ rt♦♥ rr♦s ♦r♠
∫ t
a
f (α)(τ)(dτ)α = Γ(α+ 1)(f(t)− f(a)).
❼ rt♦♥ rt ♦ ♦♠♣♦♥ ♥t♦♥s
dαf ∼= Γ(1 + α)df,
♦r ♥ tr♠ ♦ rt♦♥ r♥ ∆αf ∼= Γ(1 + α)∆f.
❼ rt♦♥ ♥③ r
(f(t)g(t))(α) = (f(t))(α)g(t) + (g(t))(α)f(t).
❼ ♥rs ♦ ttr ♥t♦♥ ♥ ♠r ♦r♠
∫ x
0
dαt
t= lnα x, x = Eα(lnα x).
❼ rt♦♥ rt r♥
f (α)(t) = limh↓0
∆αf(t)
hα= Γ(1 + α) lim
h↓0
∆f(t)
hα, 0 < α ≤ 1,
r ∆αf ∼= Γ(1 + α)∆f, ♥ ♦r t ♥r③t♦♥ ♦r♠ ♥ ②
f (α)(t) = Γ(1 + (α− n)) limh↓0
∆f (n)(t)
hα−n, n < α ≤ n+ 1.
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❼ rt♦♥ rt ♦ ♦♠♣♦st♦♥ ♥t♦♥ rt♦♥ ♥ r
f (α)[x(t)] =df(x)
dxx(α)(t),
= f (α)x (x)(dx(t)
dt)α = Γ(2− α)xα−1f (α)x (x)x(α)(t).
♦t tt ♥ t ♦r♠ ♦ t rt♦♥ rt ♦ ♦♠♣♦st♦♥ ♥t♦♥
x(t) s ♥♦♥r♥t ♥ t rst qt♦♥ ♥ r♥t ♥ t s♦♥ ♦♥
t f(x) s r♥t ♥ t rst qt♦♥ ♥ ♥♦♥r♥t ♥ t s♦♥
♦♥
❼ ♥r③t♦♥ ♦ ②♦rs ①♣♥s♦♥
Pr♦♣♦st♦♥ t f : R → R ♦♥t♥♦s ♥t♦♥ x→ f(x) rt♦♥
rt ♦ ♦rr kα, ♦r ♥② ♣♦st ♥tr k, ♥ 0 < α ≤ 1. ♥ t rt♦♥
②♦r srs s ♥ ②
f(x+ h) =
∞∑
k=0
hαk
Γ(1 + αk)f (αk)(x), 0 < α ≤ 1.
♦r t ♣r♦♦ ♥ rtr ts s ♠r ❬❪ ♦r♦r ts srs ♥
rtt♥ s
f(x+ h) = Eα(hαDα
x )f(x),
r Eα(·) ♥♦ts t ttr ♥t♦♥ ♥ ♥ ♣tr
♦t tt s rt♦♥ ②♦r srs ♦♥② ♣♣s ♦♥ t ♥♦♥r♥t
♥t♦♥s ♦ t ♦s ♥♦t ♦r t t st♥r ♠♥♥♦ rt
♠r ❬❪
♦r♦r② ss♠ tt n < α ≤ n + 1, n ∈ N − 0 ♥ tt f(x) s
rts ♦ ♦rr k ♥tr 1 ≤ k ≤ n, ♥ ss♠ tt f (n)(x) s rt♦♥
②♦rs srs ♦ ♦rr α− n =: β, ♣r♦ ② t ①♣rss♦♥
f (n)(x+ h) =∞∑
k=0
hk(α−n)
Γ(1 + k(α− n))Dk(α−n)f (n)(x), n < α ≤ n+ 1.
♥ ♥trt♥ ts srs t rs♣t t♦ h,
f(x+ h) =n∑
k=0
hk
k!f (k)(x) +
∞∑
k=1
h(kβ+n)
Γ(1 + (kβ + n+ 1))f (kβ+n)(x), β := α− n.
♦r♠ ♦ ♥ ②♥ ❬❪ ss♠ tt f(x) s ♦♥t♥♦s ♥t♦♥
♥ s rt♦♥ rt ♦ ♦rr α, t♥ ♦r 0 < α ≤ 1,
dα
dxαJαf(x) = f(x),
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Jα dα
dxαf(x) = f(x)− f(0).
Pr♦♦ ♦r♠ t ♥t♦♥ ♦ t rt♦♥ ♥trt♦♥
dα
dxαJαf(x) =
dα
dxα
(
1
Γ(α)
∫ x
0(x− τ)α−1f(τ)dτ
)
=dα
dxα
(
1
Γ(α)
1
α
∫ x
0f(τ)(dτ)α
)
=1
Γ(α+ 1)
dα
dxα
(∫ x
0f(τ)(dτ)α
)
=1
Γ(α+ 1)Γ(α+ 1)f(x)
= f(x).
♠r②
Jα dα
dxαf(x) =
1
Γ(α+ 1)
(∫ x
0
(
dα
dταf(τ)
)
(dτ)α)
=1
Γ(α+ 1)Γ(α+ 1)f(τ) |τ=x
τ=0
= f(x)− f(0).
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♣♣♥①
❱rt♦♥ sts
①t P♥③t♦♥
s t♦ tr♥s♦r♠ ♦♥str♥ ♣r♦♠s ♥t♦ ♥♦♥str♥ ♦♥s ② ♥ t♦ t
♦r♥ ♦t ♥t♦♥ tr♠ tt ♣♥③s ♥② ♦t♦♥ ♦ t ♦♥str♥t
♥①t t♦r♠ s t ♦♥t♦♥s ♥r ♠♥♠③r ♦r ♦♥str♥
♦♣t♠③t♦♥ ♣r♦♠ s s♦ ♠♥♠③r ♦r ♥ ♥♦♥str♥ ♣r♦♠ ♥ t t
r ♣st③ ♦♥t♦♥s
♦r♠ ①t P♥③t♦♥ ♦r♠ ❱♥tr ❬❪
t (X,M) ♠tr s♣ C ⊂ X st ♥ f : X → R ♥t♦♥ ♣st③
♦♥t♥♦s ♦♥ X t ♣st③ ♦♥st♥t K. ♣♣♦s tt t ♣♦♥t x s ♠♥♠③r ♦r
t ♦♥str♥ ♠♥♠③t♦♥ ♣r♦♠
Minimize f(x) ♦r x ∈ Rk,
satisfying x ∈ C.
♥ ♦r ♥② K ≥ K, t ♣♦♥t x s ♠♥♠③r s♦ ♦r t ♥♦♥str♥ ♣r♦♠
Minimize f(x) + KdC(x),
over points x ∈ Rk,
r dC(x) s st♥ ♥t♦♥ ♦♥ X ♥ s
dC(x) := infx′∈C
M(x, x′), ♦r x ∈ X.
♦t tt K > K ♥ C s ♦s st t♥ t ♦♥rs ssrt♦♥ s s♦ tr
♥② ♠♥♠③r x ♦r t ♥♦♥str♥ ♣r♦♠ s s♦ ♠♥♠③r ♦r t ♦♥str♥
♣r♦♠ ♥ ♣rtr x ∈ C.
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♥ ♦r♠
♦ ts t♦r♠ s tt ♣♦♥t u ♣♣r♦①♠t② ♠♥♠③s ♥t♦♥ f(·) t♥
s♦♠ ♥♦r♥ ♣♦♥t u ♦s t♦ u s ♠♥♠③r ♦r s♦♠ ♥ ♣rtr ♥t♦♥ f(·)
♦t♥ ② ♥ s♠ ♣rtrt♦♥ tr♠ t♦ t ♦r♥ ♥t♦♥ f(·).
♦r♠ ♥s ♦r♠ ♥ ❬❪
t (V,∆) ♦♠♣t ♠tr s♣ F : V → R ∪ +∞ ♦r s♠♦♥t♥♦s
♥t♦♥ ♦♥ r♦♠ ♦ ♥ u ∈ V ♣♦♥t u s ♠♦st ♠♥♠③r ♦r V
sts②♥
F (u) ≤ inf F + ε,
♦r s♦♠ ε ≻ 0, t♥ ♦r r② λ ≻ 0 tr ①sts ♥r② ♣♦♥t v ∈ V s ♥ t
♠♥♠③r ♦r st② ♣rtr ♥t♦♥ s tt
F (v) ≤ F (u),
∆(u, v) ≤ λ,
F (v) < F (w) + ελ∆(w, v), ∀w 6= v.
Pr♦♦ ♦r ♣r♦ ts t♦r♠ ♣rs♥t t ♥①t ♠♠
♠♠ t S ♦s sst ♦ V × R, s tt ♦r s♦♠ sr m, r②
♠♥t (v, r) ∈ S stss r ≥ m. ♥ ♦r r② (v1, r1) ∈ S, tr ①sts ♥ ♠♥t
(v, r) ∈ S sts②♥ (v1, r1) ≤α (v, r) s ♠①♠ ♥ S ♦r t ♣rt ♦rr ≤α .
♥t♦♥ r ♥ ♣rt ♦rr♥ s♦♣ ♥ P♣s ❬❪ ♦r ♥②
α > 0, t ♣rt ♦rr♥ ≤α ♦♥ V ×R s ♥ ②
(v1, r1) ≤α (v2, r2) ⇔ r2 − r1 + α∆(v1, v2) ≤ 0.
s rt♦♥ s r① ♥ts②♠♠tr ♥ tr♥st
♦ ♣r♦ tt ♣rt ♦rr♥ stss ts rt♦♥s
rst ≤α s r① t (v1, r1) ∈ V ×R, s tt (v1, r1) ≤α (v1, r1). ♥
r1 − r1 + α∆(v1, v1) = α∆(v1, v1) ≤ 0,
r ∆(v1, v1) = 0 s ∆ s st♥ ♥ (v1, r1) ≤α (v1, r1). rst rt♦♥
s ♣r♦♥
♦♥ ≤α s ♥ts②♠♠tr t (v1, r1), (v2, r2) ∈ V ×R ♥♦ tt
(v1, r1) ≤α (v2, r2) ⇔ r2 − r1 + α∆(v1, v2) ≤ 0,
(v2, r2) ≤α (v1, r1) ⇔ r1 − r2 + α∆(v2, v1) ≤ 0.
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♥
(r2 − r1 + α∆(v1, v2)) + (r1 − r2 + α∆(v2, v1)) ≤ 0.
s ♠♥s tt 2α∆(v1, v2) ≤ 0 ♥ r♦♠ t ♥t♦♥ α > 0, t♥ ∆(v1, v2) ≤ 0. ♦
∆(v1, v2) = 0 ⇔ v1 = v2.
tr ssttt♥ ∆(v1, v2) = 0
r2 − r1 ≤ 0 ⇔ r2 ≤ r1,
r1 − r2 ≤ 0 ⇔ r1 ≤ r2.
s ♠♥s tt r1 = r2, t♥ (v1, r1) = (v2, r2). s♦♥ rt♦♥ s ♣r♦♥
r ≤α s tr♥st t (v1, r1), (v2, r2), (v3, r3) ∈ V ×R, s tt
(v1, r1) ≤α (v2, r2) ⇔ r2 − r1 + α∆(v1, v2) ≤ 0,
(v2, r2) ≤α (v3, r3) ⇔ r3 − r2 + α∆(v2, v3) ≤ 0.
♦ ♥ t♦ ♣r♦ tt (v1, r1) ≤α (v3, r3). ♦♠♣t♥ r3 − r1 + α∆(v1, v3) ② ♥
♥ r♠♦♥ r2 ♥ s♥ t tr♥ ♥qt②
r3 − r1 + α∆(v1, v3) = r3 − r2 + r2 − r1 + α∆(v2, v3)
≤ r3 − r2 + r2 − r1 + α(∆(v1, v2) + ∆(v2, v3)),
tr♦r
(r3 − r2 + α∆(v2, v3)) + (r2 − r1 + α∆(v1, v2)) ≤ 0.
♥ (v1, r1) ≤α (v3, r3). tr rt♦♥ s ♣r♦♥
♦ ♦♠ t♦ t ♣r♦♦ ♦ t ♠♠ t S ♦s sst ♦ V × R, s
tt ♦r s♦♠ sr m, r② ♠♥t (v, r) ∈ S stss r ≥ m.
t (vn, rn) sq♥ ♥ S, (v1, r1) t rst ♠♥t ♥ ts sq♥ ♥ (vn, rn)
♥♦♥ ♥
Sn := (v, r) ∈ S : (vn, rn) ≤α (v, r),
mn := infr : (v, r) ∈ Sn ♦r s♦♠ v ∈ V .
② t ♠♠ ♦r ♠♥t ♦ S, r ≥ m, s♦ Sn ≤ S ♥ mn ≥ m.
t (vn+1, rn+1) ♥② ♣♦♥t ♥ Sn, s tt
rn − rn+1 ≥1
2(rn −mn).
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Sn r ♦s ♥ ♥st
sn = (v, r) ∈ S : r − rn + α∆(vn, v) ≤ 0,
= (v, r) ∈ S : ∆(vn, v) ≤rn − r
α.
♠r② ♥ ♥ Sn+1 s t ♦♦♥
sn+1 = (v, r) ∈ S : r − rn+1 + α∆(vn+1, v) ≤ 0,
= (v, r) ∈ S : ∆(vn+1, v) ≤rn+1 − r
α,
t (vn, rn) ≤α (vn+1, rn+1), s (vn+1, rn+1) ∈ Sn. ♥ ≤α s tr♥st s ♣r♦♥
♦r t♥ (vn, rn) ≤α (v, r) ⇒ (v, r) ∈ Sn ♥ (v, r) ∈ Sn+1, t♥ Sn+1 ⊂ Sn.
r♦♠ t ♥t♦♥ ♦ Sn ♦r (vn+1, rn+1) ∈ Sn, rn+1 ≤ rn ♥ r♦♠
|rn+1 −mn+1| = |rn+1 − rn + rn −mn +mn −mn+1|
≤
∣
∣
∣
∣
1
2(mn − rn) + rn −mn
∣
∣
∣
∣
=1
2|rn −mn|
≤1
2(1
2|rn−1 −mn−1)| ≤
1
22|rn−1 −mn−1)| ≤ · · ·
≤1
2n|r1 −m1)| ≤
1
2n|r1 −m)| .
♥ ♦r r② (v, r) ∈ Sn+1,
(vn+1, rn+1) ≤α (v, r) ⇔ r − rn+1 + α∆(vn+1, v) ≤ 0, mn+1 ≤ r.
r♦♠ t
|rn+1 − r| ≤ |rn+1 −mn+1| ≤1
2n|r1 −m| .
r r − rn+1 + α∆(vn+1, v) ≤ 0 ⇒ ∆(vn+1, v) ≤1α|rn+1 − r| t♥
|∆(vn+1, v)| = ∆(vn+1, v) ≤1
α|rn+1 − r| ≤
1
2nα|r1 −m| .
s
limn→∞
1
2nα|r1 −m| =
1
2∞α|r1 −m| = 0.
♦ 0 ≤ ∆(vn+1, v) ≤1
2nα |r1 −m| → 0. s s♦s tt t ♠tr ♦ Sn ♦s t♦ ③r♦ s
n→ ∞. ♥ V ×R s ♦♠♣t ♠tr t sts Sn ♦♥ ♣♦♥t (v, r) ♥ ♦♠♠♦♥
(v, r) =⋂
n≥1
Sn.
② t ♥t♦♥ ♦ ♣rt ♦rr (vn, rn) ≤α (v, r) ♦r r② n. ♣♣♦s tt (v, r) ∈ S,
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s tt (v, r) ≤α (v, r), ∀n ∈ N, t♥ ② t tr♥stt② ♦ ♣rt ♦rr♥
(vn, rn) ≤α (v, r), ∀n ∈ N, ♥ (v, r),∈⋂
n≥1 Sn, ♥ tr♦r (v, r) = (v, r). s
♠♥s tt t ♠♥t (v, r) s t ♠①♠ ♥ S.
♦ ♣r♦ t t♦r♠ t
S = epi(V ) = (v, F (v)) : v ∈ V, F (v) ∈ R,
♥ ♣♣② ♥ t ♣r♦s ♠♠ α = ελ♥ (v1, r1) = (u, F (u)). ♥ ♦r t ♠①♠
♠♥t (v, r) ∈ S sts②♥
(u, F (u)) ≤α (v, r),
s♥ (v, r) s ♥ S, (v, r) ≤α (v, F (v)) ⇒ r = f(v). ② t ♠①♠t② ♦ (v, r)
①♣rss♦♥ ♦♠s
(u, F (u)) ≤α (v, F (v)) ⇔ F (v)− F (u) + α∆(u, v) ≤ 0,
s♥ α∆(u, v) ≥ 0, t♥ F (v)− F (u) ≤ 0 ⇔ F (v) ≤ F (u), s t rst ♦♥t♦♥
♦ t t♦r♠ ♠①♠t② ♦ (v, F (v)) ∈ S ♠♣s tt ♦r ♥② w ∈ V s tt
w 6= v, F (w) s ♥t t♥ t rt♦♥ (v, F (v)) ≤α (w,F (w)) ♦s ♥♦t ♦ ♦
F (w)− F (v) + (ε
λ)∆(v, w) > 0 ⇔ F (w) + (
ε
λ)∆(v, w) > F (v),
♠♥s t tr ♦♥t♦♥ ♦ t t♦r♠ s ♣r♦♥
♥② s F (u) ≤ inf(F ) + ε, t♥ tr ①sts F (v) ≥ F (u)− ε ♥ ② ♦♠♥♥
ts rt♦♥ t
F (v)− F (u) + α∆(u, v) ≤ 0 ⇔ F (u)− ε− F (u) + α∆(u, v) ≤ 0
⇔ α∆(u, v) ≤ ε⇒ ∆(u, v) ≤ε
α= λ.
♥ t s♦♥ ♦♥t♦♥ ♦ t t♦r♠ s ♣r♦♥
♥r③ r♦♥ ♥qt②
r♦♥ ♥qt② s ♥ ♠♣♦rt♥t r♦ ♥ ♥♠r♦s r♥t ♥ ♥tr
qt♦♥s ss ♦r♠ ♦ ts ♥qt② s sr ② t ♦♦♥ t♦r♠
♦r♥♥ ❬❪
♦r♠ ♦r ♥② t ∈ [t0, T ] t a(t) b(t) ♥ w(t) ♦♥t♥♦s ♥t♦♥s t
b(t) ≥ 0. w(t) stss
w(t) ≤ a(t) +
∫ t
t0
b(τ)w(τ)dτ,
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r b(t) ≥ 0, t♥
w(t) ≤ a(t) +
∫ t
t0
a(τ)b(τ) exp
(∫ t
τ
b(s)ds
)
dτ.
♥ ♣rtr a(t) s ♥♦♥rs♥ t♥
w(t) ≤ a(t) exp
(∫ t
t0
b(τ)dτ
)
.
♦ ♣rs♥t ♥r③t♦♥ ♦ t r♦♥ ♥qt② ♥ s ♥
rt♦♥ r♥t qt♦♥ r r sr ♥r③t♦♥s ♦ t r♦♥♠♥
♥qts s ♥ ❬❪ ❨ t ❬❪ ♥ ❩♥ ❬❪ t s r t ♦♦♥
♦♥
♦r♠ t α > 0, a(t) ♥♦♥♥t ♥t♦♥ ♦② ♥tr ♦♥ t ∈ [0, T ]
r T ≤ +∞), ♥ b(t) ♥♦♥♥t ♥♦♥rs♥ ♦♥t♥♦s ♥t♦♥ ♥
♦♥ 0 ≤ t ≤ T, r b(t) s ♦♥ ② ♣♦st ♦♥st♥t K b(t) ≤ K). w(t) s
♥♦♥♥t ♥ ♦② ♥tr ♦♥ t ∈ [0, T ] ♥ stss
w(t) ≤ a(t) + b(t)
∫ t
0(t− τ)α−1w(τ)dτ,
t♥
w(t) ≤ a(t) +
∫ t
0
[
∞∑
n=1
(b(t)Γ(α))n
Γ(nα)(t− τ)nα−1a(τ)
]
dτ.
Pr♦♦ t θ(t) ♦② ♥tr ♥t♦♥ ♥ t s ♥ ♥ ♦♣rt♦r B ♦♥ θ s
♦♦s
Bθ(t) := b(t)
∫ t
0(t− τ)α−1θ(τ)dτ, t ≥ 0.
r♦♠ ♥qt②
w(t) ≤ a(t) +Bw(t),
ts ♠♣s
w(t) ≤n−1∑
k=0
Bka(t) +Bnw(t).
♥ ♦rr t♦ t t sr ♥qt② s♥ ♥ ♣r♦ tt
Bnw(t) ≤
∫ t
0
(b(t)Γ(α))n
Γ(nα)(t− τ)nα−1w(τ)dτ,
♥ Bnw(t) ♥ss s n ♥rss Bnw(t) → 0 s n→ +∞) ♦r t ∈ [0, T ).
❲ s t ♠t♠t ♥t♦♥ ♠t♦ t♦ r② t ♥qt② ♥ rst
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♥♦ tt t ♥qt② ♥ s tr ♦r n = 1. ♦♥ ss♠ tt t ♥qt②
♥ s tr ♦r n = k, t♥ ♣r♦ tt t s s♦ tr ♦r n = k + 1
Bk+1w(t) = B(Bkw(t)) ≤ b(t)
∫ t
0(t− τ)α−1
[
∫ τ
0
(b(s)Γ(α))k
Γ(kα)(t− s)kα−1w(s)ds
]
dτ.
♥ b(t) s ♥♦♥♥t ♥ ♥♦♥rs♥ ♥t♦♥ t ♦♦s tt
Bk+1w(t) ≤ bk+1(t)
∫ t
0(t− τ)α−1
[
∫ τ
0
(Γ(α))k
Γ(kα)(t− s)kα−1w(s)ds
]
dτ,
② ♥tr♥♥ t ♦rr ♦ ♥trt♦♥
Bk+1w(t) ≤ bk+1(t)
∫ t
0
[
∫ t
s
(Γ(α))k
Γ(kα)(t− τ)α−1(τ − s)kα−1dτ
]
w(s)ds.
♥ t ssttt♦♥ τ = s+ z(t− s) ♥ t ♣r♦s ♥tr ♥ s♥ t ♥t♦♥
♦ t ♥t♦♥ s P♦♥② ❬❪ ♦t♥
∫ t
s
(t− τ)α−1(τ − s)kα−1dτ = (t− s)kα+α−1
∫ 1
0(1− z)α−1zkα−1dz
= (t− s)(k+1)α−1B(kα, α)
=Γ(α)Γ(kα)
Γ((k + 1)α)(t− s)(k+1)α−1.
♥
Bk+1w(t) ≤
∫ t
0
(b(t)Γ(α))k+1
Γ((k + 1)α)(t− s)(k+1)α−1w(s)ds,
♥ t ♥qt② s ♣r♦
♥ Bnw(T ) ≤∫ t
0(KΓ(α))n
Γ(nα) (t − τ)nα−1w(τ)dτ → 0, s n → +∞, ♦r t ∈ [0, T ), t
♣r♦♦ s ♦♠♣t
♦r♦r② ♣♣♦s t ②♣♦tss ♣rs♥t ♥ ♦r♠ r sts ♥ t
a(t) ♥♦♥rs♥ ♦♥ t ∈ [0, T ]. ♥
w(t) ≤ a(t)
[
1 +
∫ t
0
∞∑
n=1
(b(t)Γ(α))n
Γ(nα)(t− τ)nα−1dτ
]
≤ a(t)Eα(b(t)Γ(α)tα),
r Eα(·) s t ♥r③ ttr ♥t♦♥ ♥ Γ(·) s t ♠♠ ♥t♦♥
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Pr♦♦ r♦♠ t Pr♦♦ ♦ ♦r♠
w(t) ≤ a(t) +
∫ t
0
[
∞∑
n=1
(b(t)Γ(α))n
Γ(nα)(t− τ)nα−1a(τ)
]
dτ.
♥ a(t) s ♥♦♥rs♥ ♥ rt
w(t) ≤ a(t)
[
1 +
∫ t
0
∞∑
n=1
(b(t)Γ(α))n
Γ(nα)(t− τ)nα−1dτ
]
≤ a(t)
∞∑
n=0
(b(t)Γ(α))n
Γ(nα+ 1)tnα
≤ a(t)Eα(b(t)Γ(α)tα).
♦r♦r② s ♣r♦
rt♦♥ ♦s②♠♦♥ ♥♠♥t ♠♠
r r sr ♦r♠s ♦ t rt♦♥ ♦s②♠♦♥ ♠♠ s ♠ ♥
♦rrs ❬❪ ♦r♥ ♥ ③ ❬❪ ♠♦ ❬❪ ♥ ③♦ ♥ ♦rrs ❬❪ r
♥tr♦ t ♦♥s ♠♦st ♠♣♦rt♥t ♦r ♦r ♦r
♠♠ ③♦ ♥ ♦rrs ❬❪ t h r♥t ♥t♦♥ ♥ t ♥tr
[a, b] t h(a) = 0, h(b) = 0, ♥ t f ∈ L1([a, b]) s tt tr s ♥♠r δ ∈ [a, b]
t |f(t)| ≤ k(x− a)β ♦r t ∈ [a, δ], r k > 0 ♥ β > −α r ♦♥st♥ts ♥
aIαb (f(t)aD
αt h(t)) = 0,
♥
f(t) = c,
r c s ♦♥st♥t Dα(·) s t ♠♥♥♦ rt♦♥ rt ♦♣rt♦r ♥
Iα s t rt♦♥ ♥tr ♦♣rt♦r
♠♠ ♠ ♥ ♦rrs ❬❪ t f ♦♥t♥♦s ♥t♦♥ sts②♥
∫ b
a
f(t)g(t)(dt)α = 0,
♦r r② ♦♥t♥♦s ♥t♦♥ g sts②♥ g(a) = g(b) = 0. ♥ f = 0. r t ♥tr∫ b
a(·)(dt)α s t ♠r rt♦♥ ♥tr ♦♣rt♦r
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♣♣♥①
♦♥s♠♦♦t ♥②ss
♦♥s♠♦♦t ♥②ss s ♥ ♠♣♦rt♥t t♦♦ ♥ ♦♣t♠♦♥tr♦ t♦r② t rst ♣♣r ♥
t ss ♥②ss t♦ st♠t ♣♣r♦①♠t♦♥s t♦ ♥♦♥r♥t ♥t♦♥s
♥ t♦ t sts t ♥♦♥r♥t ♦♥rs r tr r ♠♥② ♥t♦♥s
r ♦♥t♥♦s r②r t ♥♦t r♥t t s♦♠ ♣♦♥ts
rst t♦ ♦♣ ♥♦♥s♠♦♦t ♥②ss s t♥ r♦♠ t ♦♠tr rt♦♥s♣
t♥ t rt ♦ s♠♦♦t ♥t♦♥s r♥t ♥t♦♥s ♥ t r♣ ♦ ts
♥t♦♥s s ♦♦s
♥ ss s♠♦♦t ♥②ss t rts ♦ ♥t♦♥ g r rt t♦ t♦rs ♥♦r♠ t♦
t♥♥t ②♣r♣♥s ♦r ♥② r♥t ♥t♦♥ g t t♦r (g′(x),−1) s ♦♥r
♥♦r♠ t♦ t r♣ ♦ g t (x, g(x)). r t r♣ ♦ g s ♥ ②
Gr g = (x, r) ∈ Rn × R : r = g(x).
♥st ♦ ♦♥sr♥ rts s ♠♥ts ♦ ♥♦r♠ ss♣s t♦ s♠♦♦t sts
♥r③ rts r ♥ t♦ ♠♥ts ♦ ♥♦r♠ ♦♥s t♦ ♣♦ss② ♥♦♥s♠♦♦t
sts
♦ t ♦♣t♠♦♥tr♦ sss s t ♦♦♥ rsts
♥t♦♥ t X sst ♦ ♥ s♣ ψ ♥t♦♥ g : X → R s s t♦
sts② ♣st③ ♦♥t♦♥ ♦♥ X
|g(x1)− g(x2)| ≤ k ‖x1 − x2‖ ,
♦r ♣♦♥ts x1, x2 ∈ X, r k s ♣♦st ♦♥st♥t s♦ rrr t♦ s ♣st③
♦♥t♦♥ ♦ r♥ k. ❲ s② tt g s ♣st③ ♦ r♥ k ♥r x ♦r s♦♠ δ > 0, g
stss ♣st③ ♦♥t♦♥ ♦ r♥ k ♦♥ t st x+ δB, r B s t ♦♣♥ ♥t
Pr♦♣♦st♦♥ t C ♥♦♥♠♣t② sst ♦ X ♥ dC(·) : X → R st♥
♥t♦♥ ♥ ②
dC(x) = inf‖x− c‖ : c ∈ C.
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♥ t st♥ ♥t♦♥ dC(·) stss t ♣st③ ♦♥t♦♥ ♦♥ X s ♦♦s
|dC(x)− dC(y)| ≤ ‖x− y‖ .
♦r♠ r ❬❪ t h(·) ♣st③ ♥r x, s♣♣♦s S s ♥② st ♦ s
♠sr 0 ♥ Rn, ♥ t Ωh t st ♦ ♣♦♥ts r ♥ ♥t♦♥ h(·) s t♦
r♥t ♥ t ♥r③ r♥t s ♥ s
∂xh(x) := colim∇h(xi) : xi → x, xi /∈ S, xi /∈ Ωh.
♥t♦♥ t x ♣♦♥t ♥ t st C ⊆ X, ♥ λ ∈ X t♥♥t t♦r t♦ C t
x sts②♥ dC(x;λ) = 0. ♥ t st ♦ t t♥♥ts t♦ C t x s t t♥♥t
♦♥ TC(x) ♥ s ♥ ②
TC(x) := λ ∈ Rn : dC(x;λ) = 0,
r dC(x;λ) s t rt♦♥ rt ♦ t st♥ ♥t♦♥
♥t♦♥ t x ♣♦♥t ♥ C ⊆ X, ♥ λ ∈ X t♥♥t t♦r t♦ C t x.
♥ t ♥♦r♠ ♦♥ t♦ C t x NC(x) s ♥ s
NC(x) := ξ ∈ X∗ : 〈ξ, λ ≤ 0〉 ∀λ ∈ TC(x).
♠t♥ ♥♦r♠ ♦♥ s ♥tr♦ ② ♦r♦ ♥ ❬❪ s ♦♦s
t C ♥♦♥♠♣t② sst ♦ Rn, ♥ t
P (x,C) := z ∈ clC : ‖x− z‖ = d(x,C)
t st ♦ st ♣♣r♦①♠t♦♥s ♦ x ♥ clC t rs♣t t ♥ st♥ ♥t♦♥
d(x,C).
♥t♦♥ ♥ x ∈ clC t ♦♦♥ ♦s ♦♥
N(x, C) := lim supx→x
cone (x− P (x,C))
s t ♥♦r♠ ♦♥ t♦ t st C t ♣♦♥t x. x /∈ clC, ♣t N(x, C) = ∅.
Pr♦♣♦st♦♥ ♥♦r♠ ♦♥ NC(x) s t ♦s ♦♥① ♦♥ ♥rt ② ∂dC(x)
♥ stss
NC(x) = cl⋃
λ≥0
λ∂dC(x),
r cl s ∗ ♦sr
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♦r♦r② t X = X1 ×X2, r X1, X2 r ♥ s♣s C = C1 ×C2, r
C1 ♥ C2 r ssts ♦ X1 ♥ X2, rs♣t② ♣♣♦s tt x = (x1, x2) ∈ C. ♥
TC(x) = TC1(x1)× TC2
(x2),
NC(x) = NC1(x1)×NC2
(x2).
Pr♦♣♦st♦♥ r ❬❪ ♦♥ ❬❪ t C ⊆ Rn ♦s st ♥ ∇dC(x)
①st ♥ r♥t r♦♠ ③r♦ ♥ x s ♦ts C, t st C ①t② ♦♥t♥s
♥q ♦sst ♣♦♥t c t t ♠♥♠♠ st♥ t♦ x s tt♥ s tt
∇dC(x) =x− c
|x− c|.
♦r♠ r ❬❪ t f = g F r F : Rn → Rm s ♣st③ ♥r x, ♥
g : Rm → R s ♣st③ ♥r F (x). ♥ f s ♣st③ ♥r x ♥ stss
∂f(x) ⊂ co ∂g(F (x))∂F (x) .
g s strt② r♥t t F (x), t♥ qt② ♦s ♥ co s ♥ss
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♣♣♥①
sr ♦r② ♥ ♥trt♦♥
r ♥ σ−r ♦ ts
♥t♦♥ r t X ♥ rtrr② ♥♦♥♠♣t② st ♥ Ω(X) ♦t♦♥
♦ ssts ♦ X. ♥ Ω(X) s ♥ r stss
∅, X ∈ Ω(X) r ∅ s t ♠♣t② st
A ∈ Ω(X) =⇒ Ac ∈ Ω(X), r Ac s t ♦♠♣♠♥t ♦ A ♥ ②
Ac = a ∈ X | a /∈ A.
A,B ∈ Ω(X) =⇒ A ∪B ∈ Ω(X).
♥t♦♥ σ−r ♥ r Ω(X) s s t♦ σ−r s♠r
stss t t♦♥ ♦♥t♦♥
♦r ♥② sq♥ An ⊂ Ω(X) =⇒⋃
∞
n=1An ∈ Ω(X).
♦t tt t ♥trst♦♥s ♦ ts sq♥ s♦ ♦♥ t♦ Ω(X)
An ⊂ Ω(X) =⇒∞⋂
n=1
An ∈ Ω(X).
♥t♦♥ ♦r σ−r t X ♠tr s♣ ♦r σ−r ♦
X s ♥ t♦ σ−r ♥rt ② ♦♣♥ ssts ♦ X ♥ s ♥♦t ② B(X).
♠♥ts ♦ B(X) r s t♦ ♦r ♠sr st
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srs
❲ s② tt t ♠sr µ(·) s ♥t t ♦r ♥② ♠② S1, . . . , Sn ∈ Ω ♦ s♦♥t
sts
µ
n⋃
j=1
Sj
=
n∑
j=1
µ(Sj).
❲ s② tt µ(·) s ♥t st ♦r ♥② ♠② S1, . . . , Sn ∈ Ω ♦ s♦♥t sts
µ
n⋃
j=1
Sj
≤n∑
j=1
µ(Sj).
♠sr µ(·) s ♠♦♥♦t♦♥ ♦r ♥② S1, S2 ∈ Ω, t S1 ⊂ S2,
µ(S1) ≤ µ(S2),
♥ s ♦♥t stt② ♦r ♥② sq♥ Sn ⊂ Ω,
µ
(
⋃
n∈N
Sn
)
≤∑
n∈N
µ(Sn).
♠♠ ♠sr µ(·) ♦♥ σ−r Ω ♦ sst ♦ st X s ♥t t
♥t st ♠♦♥♦t♦♥ ♦♥t st ♥
µ(S2 \ S1) = µ(S2)− µ(S1),
♦r S1, S2 ∈ Ω, S1 ⊂ S2, t µ(S1) <∞.
♣r♦♦ ts ♠♠ ♥ ♦♥ ♦r ♥st♥ ♥ ❨ ❬❪
♥t♦♥ ♦♥sr t ♠sr µ : B(X) → [0,∞] ♥ t S ∈ B(X). ♠sr
µ(·) s ♥ ♦tr rr ♠sr
µ(S) = infµ(U) | S ⊆ U ♥ U s ♦♣♥,
♥ s ♥ ♥♥r rr ♠sr
µ(S) = supµ(K) | K ⊆ S ♥ K s ♦♠♣t.
♠sr s rr t s ♦t ♦tr ♥ ♥♥r rr ♦♥ ♠sr s ♥
♥♥r rr ♦r ♠sr
♥t♦♥ t (X,Ω) ♥ (Y, Ψ) t♦ ♠sr s♣s ♠♣ f : X → Y s
s t♦ ♠sr ♦r A ∈ Ψ, t st f−1(A) ∈ Ω. Y s ♠tr s♣ ♥
Ψ = B(Y ), f s ♦r ♥t♦♥ Y = R, f ♦r ♥t♦♥
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♥trt♦♥
♥t♦♥ ♠♥♥ ♥tr t f : [a, b] → R ♦♥ ♥t♦♥ ♥ ♦♥
♦s ♥tr [a, b], ♦♥sr ♥② ♣rtt♦♥ P : a = x0 < x1 < · · · < xn = b, t
δxi = xi − xi−1, ♥ ♥ t ♣♣r ♥ ♦r s♠ rs♣t② ss♦t t
♣rtt♦♥ P s ♦♦s
UP =n∑
i=1
Miδxi,
LP =n∑
i=1
miδxi,
r
Mi = supf(x) : xi−1 < x ≤ xi,
mi = inff(x) : xi−1 < x ≤ xi.
♥ ♥ t ♣♣r ♥ ♦r ♠♥♥ ♥tr ♦ f ♦r [a, b], rs♣t② s
♦♦♥
∫ b
a
f(x)dx = infPUP ,
∫ b
a
f(x)dx = supP
LP .
❲ s② tt f s ♠♥♥ ♥tr ♦♥ [a, b], ♥♦t ②∫ b
af(x)dx, t ♥♠♠ ♦ ♣♣r
s♠s tr♦ ♣rtt♦♥s ♦ [a, b] s q t♦ t s♣r♠♠ ♦ ♦r s♠s tr♦
♣rtt♦♥s ♦ [a, b],
∫ b
a
f(x)dx =
∫ b
a
f(x)dx =
∫ b
a
f(x)dx.
♥t♦♥ ♠♣ ♥t♦♥ ♥t♦♥ s s♠♣ ts r♥ s ♥t st t ψ
s s♠♣ ♥t♦♥ r♣rs♥t ②
ψ =n∑
i=1
aiχEi,
r ai r st♥t s ♦ ψ ♥ χEis ♠sr ♥t♦♥ t ♥t♦r
♥t♦♥ ♦ t st Ei ♥ ②
χEi(x) =
1, x ∈ Ei,
0, x /∈ Ei,
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s tt Ei = ψ−1(ai). ♦♥rs② ♥② ①♣rss♦♥ ♦ ts ♦r♠ r ai ♥ ♥♦t
st♥t ♥ Ei ♥♦t ♥ssr② ψ−1(ai) s♦ ♥s s♠♣ ♥t♦♥
♥t♦♥ t (X,µ) ♠sr s♣ µ(·) s s ♠sr ♥
s ♥tr ♦r D ⊂ X ♦ ♠sr s♠♣ ♥t♦♥ ψ s ♥ ②
∫
D
ψdµ =
∫
D
n∑
i=1
aiχEidµ =
n∑
i=1
aiµ(Ei).
q♥tt② ♦♥ t rt r♣rs♥ts t s♠ ♦ t rs ♦ t r♣ ♦ ψ(·).
♥t♦♥ t f(·) ♦♥ ♠sr ♥t♦♥ ♥ ♦♥ st D ♦ ♥t
♠sr s tt t ♣♣r ♥ ♦r s ♥tr rs♣t② ♥ ②
U = sup
∫
D
ψdµ | ψ s s♠♣ ♥t♦♥ ♥ ψ ≤ f
,
L = inf
∫
D
ψdµ | ψ s s♠♣ ♥t♦♥ ♥ ψ ≥ f
.
U = L tt f s s ♥tr ♥
L =
∫
D
fdµ = U
r∫
Dψdµ s stt ♦r
♥t♦♥ t (X,Ω, µ) ♠sr s♣ ♠sr ♥t♦♥ f : X → R
s s ♥tr ♦♥ Y ∈ Ω t rs♣t t♦ µ, t ♥♦♥♥t ♥t♦♥
|f | = f+ + f− stss∫
Y
|f | dµ <∞,
♥ ts ♥tr ♦♥ Y s ♥ s
∫
Y
fdµ =
∫
Y
f+dµ−
∫
Y
f−dµ
r f+, f− r s t♦ t ♣♦st ♥ ♥t ♣rts ♦ t ♥t♦♥ f, ♥ ②
f+(x) = max(f(x), 0),
f−(x) = max(−f(x), 0).
♦r ♠♦r ts ♥ ♣r♦♣rts ♦♥ s ♥tr s rtr ♥ ❱♥ r♥t ❬❪
♦♠s♦♥ ❬❪ ♥ ❨ t ❬❪
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❯s ♦♥♣ts
♥t♦♥ f(t), ♥ ♦♥ t ♦s ♥tr [a, b] s s t♦ ♥rs♥
f(t1) ≤ f(t2), ♦r t1 < t2,
strt② ♥rs♥
f(t1) < f(t2), ♦r t1 < t2,
rs♥
f(t1) ≥ f(t2), ♦r t1 < t2,
strt② rs♥
f(t1) > f(t2), ♦r t1 < t2,
♠♦♥♦t♦♥ ♦r ♠♦♥♦t♦♥ strt② ♠♦♥♦t♦♥ ♥rs♥ ♥ rs♥ strt②
♥rs♥ ♥ strt② rs♥ ♥t♦♥s
♥t♦♥ tr♦♥ ♦ ♠♥♠♠ ♥ ♠ss ♣r♦ss (x∗, u∗) s str♦♥ ♦
♠♥♠③r ♦r ♥ ♦♣t♠♦♥tr♦ ♣r♦♠ ♦r ε > 0, t ♠♥♠③s t ♦st ♦r ♦tr
♠ss ♣r♦sss ① s tt
|x(t)− x∗(t)| ≤ ε, ∀t ∈ [a, b].
♥t♦♥ ❲ ♦♥r♥ t X ♥♦r♠ ♥r t♦r s♣ ♥ X∗
t s♣ ♦ X. sq♥ xn s ♦♥r ② t♦ x ∈ X ♦r x∗ ∈ X∗,
〈xn, x∗〉 → 〈x, x∗〉 , ♥ rt xn → x ②
♥t♦♥ ❲∗ ♦♥r♥ t X ♥♦r♠ ♥r t♦r s♣ ♥ X∗
t s♣ ♦ X. sq♥ x∗n ♥ X∗ s ♦♥r ❲∗ str t♦
x∗ ∈ X∗ 〈x, x∗n〉 → 〈x, x∗〉 ♦r x ∈ X, ♥ rt x∗n → x∗ ❲∗
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♦r♣②
❬❪ ♦ P Pr♦s ♥ s♥ ♥♦♦ ♦ ♠ss t sts ❱♦ ♣r♥r
❬❪ P r ♦r♠t♦♥ ♦ rr♥ qt♦♥s ♦r rt♦♥ rt♦♥ ♣r♦♠s ♦r♥
♦ t♠t ♥②ss ♥ ♣♣t♦♥s ♥♦
❬❪ ♥r ♦r♠t♦♥ ♥ s♦t♦♥ s♠ ♦r rt♦♥ ♦♣t♠ ♦♥tr♦ ♣r♦♠s ♦♥♥r
②♥♠s ♥♦
❬❪ rt♦♥ rt♦♥ s ♥ tr♠s ♦ s③ rt♦♥ rts ♦r♥ ♦ P②ss
t♠t ♥ ♦rt ♥♦
❬❪ P r ♥ ♥ ♠t♦♥♥ ♦r♠t♦♥ ♥ rt ♥♠r s♠ ♦r
rt♦♥ ♦♣t♠ ♦♥tr♦ ♣r♦♠s ♦r♥ ♦ ❱rt♦♥ ♥ ♦♥tr♦ ♥♦
❬❪ P r tr ♥ ♥ rt♦♥ ♦♣t♠ ♦♥tr♦ ♣r♦♠s t sr stt
♥ ♦♥tr♦ rs ♦r♥ ♦ ❱rt♦♥ ♥ ♦♥tr♦ ♥♦
❬❪ ♠ ♥ t♦②s rt♦♥ r♥t ♥s♦♥s t rt♦♥ s♣rt ♦♥r②
♦♥t♦♥s rt♦♥ s ♥ ♣♣ ♥②ss ♥♦
❬❪ Prr ♥ ♠ ♥ ♣♣r♦ t♦ t P♦♥tr②♥ ♠①♠♠ ♣r♥♣
♦r ♥♦♥♥r rt♦♥ ♦♣t♠ ♦♥tr♦ ♣r♦♠s t♠t t♦s ♥ t ♣♣ ♥s
❬❪ ♠ ♥ ♦rrs rt♦♥ rt♦♥ s ♦r ♥♦♥r♥t ♥t♦♥s
♦♠♣trs t♠ts t ♣♣t♦♥s ♥♦
❬❪ P r♥ ♣♦♥tt♦ ♦rt♥ ♥ P♦rt♦ ♦♥♥r ♥♦♥♥tr ♦rr rts ♥ s②st♠s
♥ ♥tr♦t♦♥ ❱♦ ❲♦r ♥t
❬❪ rt②♥♦ ♣t♠t② ♦♥t♦♥s ♥♦r♠ ♥ ♥rt ♣r♦♠s ❱♦ ♣r♥r
❬❪ rt②♥♦ ❱ ②t ♥ Prr ssr② ♦♥t♦♥s ♦r ♠♣s ♥♦♥♥r ♦♣t♠
♦♥tr♦ ♣r♦♠s t♦t ♣r♦r ♥♦r♠t② ss♠♣t♦♥s ♦r♥ ♦ ♣t♠③t♦♥ ♦r② ♥
♣♣t♦♥s ♥♦
❬❪ ❱ rt②♥♦ ❨ r♠③♥ ♥ Prr ①♠♠ ♣r♥♣ ♥ ♣r♦♠s t ♠①
♦♥str♥ts ♥r ss♠♣t♦♥s ♦ rrt② ♣t♠③t♦♥ ♥♦
❬❪ ♠①♠♠ ♣r♥♣ ♦r ♦♣t♠ ♦♥tr♦ ♣r♦♠s t stt ♦♥str♥ts ② ❱
♠r③ rst ♦r♥ ♦ ♣t♠③t♦♥ ♦r② ♥ ♣♣t♦♥s ♥♦
❬❪ ❱ rt②♥♦ ❨ r♠③♥ ♥ Prr ♥ ♥r③t♦♥ ♦ t ♠♣s ♦♥tr♦
♦♥♣t ♦♥tr♦♥ s②st♠ ♠♣s srt ♦♥t♥ ②♥ ②st ♥♦
❬❪ ❱ rt②♥♦ ❨ r♠③♥ ♥ Prr ❱ ♠r③s ♠①♠♠ ♣r♥♣
♦r ♦♣t♠ ♦♥tr♦ ♣r♦♠s t ♦♥ ♣s ♦♦r♥ts ♥ ts rt♦♥ t♦ ♦tr ♦♣t♠t②
♦♥t♦♥s ♦② t♠ts ♥♦
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❬❪ tt ♦♥str♥ts ♥ ♠♣s ♦♥tr♦ ♣r♦♠s ♠r③ ♦♥t♦♥s ♦ ♦♣t♠t②
♦r♥ ♦ ♣t♠③t♦♥ ♦r② ♥ ♣♣t♦♥s ♥♦
❬❪ ♦♥t♦♥s ♦r t s♥ ♦ ♠♣s ♦ t s♦t♦♥ t♦ t ♦♥t s②st♠ ♦ t ♠①♠♠
♣r♥♣ ♦r ♦♣t♠ ♦♥tr♦ ♣r♦♠s t stt ♦♥str♥ts Pr♦♥s ♦ t t♦ ♥sttt ♦
t♠ts ♥♦
❬❪ ❱ rt②♥♦ ❨ r♠③♥ Prr ♥ ♥stt♦♥ ♦ rrt② ♦♥t♦♥s
♥ ♦♣t♠ ♦♥tr♦ ♣r♦♠s t ♦♠tr ♠① ♦♥str♥ts ♣t♠③t♦♥ ♥♦
❬❪ ❱ rt②♥♦ ♥ Prr ♦♥♦rr ♥ssr② ♦♥t♦♥s ♦r ♦♣t♠ ♠♣s ♦♥tr♦
Pr♦♥s ❱♦♠s P♣rs♥♥ ♣♣
❬❪ ♦♥♦rr ♥ssr② ♦♣t♠t② ♦♥t♦♥s ♦r ♣r♦♠s t♦t ♣r♦r ♥♦r♠t②
ss♠♣t♦♥s t♠ts ♦ ♣rt♦♥s sr ♥♦
❬❪ ❱ rt②♥♦ ❨ r♠③♥ ♥ Prr P♦♥tr②♥s ♠①♠♠ ♣r♥♣ ♦r ♦♥str♥
♠♣s ♦♥tr♦ ♣r♦♠s ♦♥♥r ♥②ss ♦r② t♦s ♣♣t♦♥s ♥♦
❬❪ t♥♦ ♥ t♥♦ ♥ ♥♠r s♠ ♦r s♦♥ r♥t qt♦♥s ♦
rt♦♥ ♦rr ♥s sr ♦♠♠♥t♦♥s ♥♦
❬❪ ①t ♥ s rt♦♥ s ♣♣t♦♥ ♥ ♦♥tr♦ s②st♠s Pr♦♥s ♦ t
t♦♥ r♦s♣ ♥ tr♦♥s ♦♥r♥ ♣♣
❬❪ ② ♥ ♦r rt♦♥ s r♥t ♣♣r♦ t♦ t ♥②ss ♦ s♦st②
♠♣ strtrs ♦r♥ ♥♦
❬❪ ② ♥ P ♦r t♦rt ss ♦r t ♣♣t♦♥ ♦ rt♦♥ s t♦
s♦stt② ♦r♥ ♦ ♦♦② ♥♦
❬❪ ♥ tr ♥ P r ♥tr r♥ ♥♠r s♠ ♦r rt♦♥
♦♣t♠ ♦♥tr♦ ♣r♦♠s ♦r♥ ♦ ❱rt♦♥ ♥ ♦♥tr♦ ♥♦
❬❪ ♥ ③♣♦r ♥ ♦♠♠ ♦♠ ①st♥ rsts ♦♥ ♥♦♥♥r rt♦♥
r♥t qt♦♥s P♦s♦♣ r♥st♦♥s ♦ t ♦② ♦t② ♦ ♦♥♦♥ t♠t
P②s ♥ ♥♥r♥ ♥s ♥♦
❬❪ ♥ ♣t♠ ♦♥tr♦ ♦ tr♠♥st ♣♠s ♣t♠ ♦♥tr♦ ♣♣t♦♥s ♥ t♦s
♥♦
❬❪ ♥♦r ♥ ♠♥ ♦♥r② ♣r♦♠s ♦r r♥t ♥s♦♥s t
♠♥♥♦ rt♦♥ rt ♦♥♥r st♦♥s ♥♦
❬❪ ♥♦r ♥rs♦♥ t♦②s ♥ ①st♥ rsts ♦r rt♦♥
♥t♦♥ r♥t ♥s♦♥s t ♥♥t ② ♥ ♣♣t♦♥s t♦ ♦♥tr♦ t♦r② rt♦♥
s ♥ ♣♣ ♥②ss ♥♦
❬❪ s♦♣ ♥ P♣s s♣♣♦rt ♥t♦♥s ♦ ♦♥① st Pr♦♥s ♦ ②♠♣♦s ♥
Pr t♠ts ♣♣
❬❪ ♦r♥ ♥ ③ rt♦♥ ♥♠♥t ♠♠ ♥ rt♦♥ ♥trt♦♥ ② ♣rts
♦r♠♣♣t♦♥s t♦ rt ♣♦♥ts ♦ ♦③ ♥t♦♥s ♥ t♦ ♥r ♦♥r② ♣r♦♠s
♥s ♥ r♥t qt♦♥s ♥♦
❬❪ r② ♠♥tr② ♣r♦♣rts ♦ t tts ♥tr ♥♥s ♦ t♠ts ♥♦
❬❪ r②s♦♥ ♣♣ ♦♣t♠ ♦♥tr♦ ♦♣t♠③t♦♥ st♠t♦♥ ♥ ♦♥tr♦ Prss
❬❪ ró♥ ❱♥r ♥ ❱ rt♦♥ ♦rr ♦♥tr♦ strts ♦r ♣♦r tr♦♥
♦♥rtrs ♥ Pr♦ss♥ ♥♦
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❬❪ ♣♦♥tt♦ ♦♥♦ ♦rt♥ ♥ Ptrá ♦♥ ♥ ♦♥tr♦ ♣♣t♦♥s ❲♦r
♥t
❬❪ rtr ♥ ❱♥ r♥t stts ♥tr ♣r♥r
❬❪ r♥ ♥ rt♦♥ r♥t ♥s♦♥ t ♦♥r② ♦♥t♦♥s t ❯♥rstts
s♦② t♠t
❬❪ P ♥ ♥ s♠ ♣t♠ ♦♥tr♦ ♠♦s ♥ ♥♥ ♣r♥r
❬❪ ❲ rt♠t ♦♦♥♦♠s ♦♣t♠ ♥♠♥t ♦ st♥ s♦rs ❲②
♥trs♥ ❨♦r
❬❪ r ♠①♠♠ ♣r♥♣ ♥ ♦♣t♠ ♦♥tr♦ t♥ ♥ ♥♦ ♦♥tr♦ ♥ ②r♥ts
❬❪ r ♣t♠③t♦♥ ♥ ♥♦♥s♠♦♦t ♥②ss ❲② ❨♦r
❬❪ r ❨ ② tr♥ ♥ P ❲♦♥s ♦♥s♠♦♦t ♥②ss ♥ ♦♥tr♦ t♦r②
♣r♥r
❬❪ ❨ ♦♥ ♣♣t♦♥s ♦ ♦♥tr♦ t♦r② ♥ ♦♦② Pr♦♥s ♦ t ②♠♣♦s♠ ♦♥ ♣t♠
♦♥tr♦ t♦r② t t tt ❯♥rst② ♦ ❨♦r ②rs ❨♦r st
❱♦ ♣r♥r ♥ s♥ss
❬❪ ♦r♥♥ Pr♥♣s ♦ r♥t ♥ ♥tr qt♦♥s ❱♦ ♠r♥ t♠t
♦t②
❬❪ r ♥ s♦r ♣♣t♦♥s ♦ rt♦♥ s ♣♣ t♠t ♥s
♥♦
❬❪ s ♥t♦♥ rt♦♥ s ♣r♥r ♥ s♥ss
❬❪ s ♥ ③♥ s♦t♦♥ ♦ ♥ ♦♣t♠ ♦♥tr♦ ♣r♦♠ ♥ ♥♥ ♠♦♥
♣rt♦♥s sr ♥♦
❬❪ rr sr t♦r② ♥ ♥trt♦♥ sr
❬❪ r ♥ ♥rr♦ r ♦ ♥t♦♥s ♦r rt♦♥ rts ♥ ♥trs
t♠t Pr♦♠s ♥ ♥♥r♥ ♥♦
❬❪ ♦r♥ r r ♥ ❱s② rt♦♥ ♦rr s♦stt② ♦ t ♦rt
s♣ ♥ tr♥t t♦ qs♥r s♦stt② ♦r♥ ♦ ♦♠♥ ♥♥r♥
♥♦
❬❪ ❨ ♦ts ♥ ②t♥ ①tr♠♠ ♣r♦♠s t ♦♥str♥ts ♦ t ♦
❬❪ rt♦♥ ♦rr s②st♠s ♥ ♥str t♦♠t♦♥ sr② r♥st♦♥s ♦♥
♥str ♥♦r♠ts ♥♦
❬❪ ♥ ♥ t rt♦♥ ♣r♥♣ ♦r♥ ♦ t♠t ♥②ss ♥ ♣♣t♦♥s
♥♦
❬❪ ② ♥ r♠ t ♥tr qt♦♥s ♦ rt♦♥♦rrs ♣♣
t♠ts ♥ ♦♠♣tt♦♥ ♥♦
❬❪ ❱ t Pr③ ♥ ♦r③ rt♦♥ r♦st ♦♥tr♦ ♦ ♠♥ rrt♦♥ ♥s
t r ②♥♠ ♣r♠trs ♦♥tr♦ ♥♥r♥ Prt ♥♦
❬❪ ♦♥ ♥②ss ♠♦r♥ t♥qs ♥ tr ♣♣t♦♥s ♦♥ ❲② ♦♥s
❬❪ r ♥ Prr ♠t♦♥♦♠♥ qt♦♥ ♥ s②♥tss ♦r ♠♣s
♦♥tr♦ t♦♠t ♦♥tr♦ r♥st♦♥s ♦♥ ♥♦
❬❪ ❱ ♠r③ ♣t♠ ♣r♦sss t ♦♥ ♣s ♦♦r♥ts ③ r
t
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❬❪ ♦ ♥ ②♥ ♥②t trt♠♥t ♦ r♥t qt♦♥s t rt♦♥ ♦♦r♥t
rts ♦♠♣trs t♠ts t ♣♣t♦♥s ♥♦
❬❪ ♦r♥♦ ♥ ♥r ss♥ts ♦ rt♦♥ s t♠t P②ss ♥ t♦sts
♥tr
❬❪ r♦r② ♥ ♥ ♦♥str♥ ♦♣t♠③t♦♥ ♥ t s ♦ rt♦♥s ♥ ♦♣t♠ ♦♥tr♦
t♦r② ♣r♥r
❬❪ ❱ r♦r ♦ ♥ rt♦♥♦ ♣t♠ ♦♥tr♦ ♦ ♣♦t♦♥ st♦ tr♦
♦♦ ♥trt♦♥ ♦ t ♠♥trr ♥ t stt st t♠t ♦r ② ♣♦♥s
❬❪ térr③ ♦sár♦ ♥ ♥rr♦ ♦ rt♦♥ ♦rr s s ♦♥♣ts
♥ ♥♥r♥ ♣♣t♦♥s t♠t Pr♦♠s ♥ ♥♥r♥
❬❪ ♠♠ ♥ ♦rt♠ ♦r st③t♦♥ ♦ rt♦♥♦rr t♠ ② s②st♠s s♥ rt♦♥
♦rr P ♦♥tr♦rs r♥st♦♥s ♦♥ t♦♠t ♦♥tr♦ ♥♦
❬❪ ♥s ♥ ❱ ♦r♥ ♦rt ♣♣r♦s t♦ ♦♦ ♦♥tr♦ ♠r ❯♥rst②
Prss
❬❪ st♥s s ♦ rt♦♥s ♥ ♦♣t♠ ♦♥tr♦ t♦r② ❲② ❨♦r
❬❪ r r♦ ♥tr♦t♦♥ t♦ rt♦♥ rts ♥♦♠♦s r♥s♣♦rt ♦♥t♦♥s ♥
♣♣t♦♥s
❬❪ r P t③r ❯ ❲st♣ ♦s ❲ ♥r ❩ss② ♦♥♥♠r
♠♥ ♥ ❲st ♣♣t♦♥s ♦ rt♦♥ s ♥ ♣②ss ❱♦ ❲♦r ♥t
❬❪ ♦②t st♥ ❩♥ P r P ♥t♥s ❱ ♦s♣ tr♥
♥s ♥ Prr ss stt② ♣r♦♣rts s ♦♠rrs ♦r ♣r♦stt ♥r ♥r
♦♠rrs ♥♦
❬❪ s ❨ ②♥ ♥ ♦♠ ♥ ♥♦ s♠t♦♥ ♦ ♥♦♥♥tr ♦rr tr♥sr
♥t♦♥s ♦r ♥②ss ♦ tr♦ ♣r♦sss ♦r♥ ♦ tr♦♥②t ♠str② ♥ ♥tr
tr♦♠str② ♥♦
❬❪ ❩ ♥ Ptr♦ ♣t♠t② ♦♥t♦♥s ♥ s♦t♦♥ s♠ ♦r rt♦♥ ♦♣t♠
♦♥tr♦ ♣r♦♠s trtr ♥ ts♣♥r② ♣t♠③t♦♥ ♥♦
❬❪ ❨ ♥ ❨ ♥ ♥ ❳ ♠♦♥st♥t r♦st ♥②ss ♦ rt♦♥ ♦rr ❬♣r♦♣♦rt♦♥
rt❪ ♦♥tr♦r ♦♥tr♦ ♦r② ♣♣t♦♥s ♥♦
❬❪ ♠r ♥ t r♣rs♥tt♦♥ ♦ rt♦♥ r♦♥♥ ♠♦t♦♥ s ♥ ♥tr t rs♣t t♦ (dt)α
♣♣ t♠ts ttrs ♥♦
❬❪ ♦ ♠♥♥♦ rt ♥ rt♦♥ ②♦r srs ♦ ♥♦♥r♥t
♥t♦♥s rtr rsts ♦♠♣trs t♠ts t ♣♣t♦♥s ♥♦
❬❪ ♦ s♦♠ s rt♦♥ s ♦r♠ r r♦♠ ♠♦ ♠♥♥♦
rt ♦r ♥♦♥r♥t ♥t♦♥s ♣♣ t♠ts ttrs ♥♦
❬❪ rt♦♥ r♥t s ♦r ♥♦♥r♥t ♥t♦♥s ♥s ♦♠tr②
st♦sts ♥♦r♠t♦♥ t♦r② ♣ ♠rt ♠
❬❪ ♥ ♥rt ♥ ❩ ♥ ♣t♠ ♦♥tr♦ ♦ trt♠♥ts ♥ t♦str♥ tr♦ss ♠♦
srt ♥ ♦♥t♥♦s ②♥♠ ②st♠s rs ♥♦
❬❪ ♠♦ ♥ t ①st♥ ♦ ♦♣t♠ s♦t♦♥s t♦ rt♦♥ ♦♣t♠ ♦♥tr♦ ♣r♦♠s ♣♣
t♠ts ♥ ♦♠♣tt♦♥
❬❪ P♦♥tr②♥ ♠①♠♠ ♣r♥♣ ♦r rt♦♥ ♦r♥r② ♦♣t♠ ♦♥tr♦ ♣r♦♠s t♠t
t♦s ♥ t ♣♣ ♥s ♥♦
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❬❪ ❱rt♦♥ ♠t♦s ♦r rt♦♥ rt ♣r♦♠ ♥♦♥ ♠r rt
t♠t Pr♦♠s ♥ ♥♥r♥
❬❪ ♠♦ ♥ ③②s ♥ rt♦♥ r♥t ♥s♦♥s t t ♠r rt
♦r♥ ♦ t♠t P②ss ♥♦
❬❪ ❨ r♠③♥ ❱ r Prr ♥ ♥ s♦♠ ①t♥s♦♥ ♦ ♦♣t♠ ♦♥tr♦
t♦r② r♦♣♥ ♦r♥ ♦ ♦♥tr♦ ♥♦
❬❪ Pr♠tr ♥rt♥t② ♥ stt ♦♥str♥ ♦♣t♠ ♦♥tr♦ ♣r♦♠s P❨
st♥ r② st
❬❪ r♣t②♥ts ♥ ♥sr s ♣♦♥ts ♦r rt♦♥ ♥tr ♥tr r♥s♦r♠s ♥
♣ ♥t♦♥s ♥♦
❬❪ s rst ♥ r♦ ♦r② ♥ ♣♣t♦♥s ♦ rt♦♥ r♥t
qt♦♥s ❱♦ sr ♥ ♠t
❬❪ ❨ ♦②s ❲t♥ ♦s ♠r ♥ ❱s♦st ♥ ♥♦♥♥r
r ♠♦♥ ♦r ♥ ♥srt♦♥ s♠t♦♥ ♦t ss ♦♠♥ ♦♥ ♦r ♦♠♣tr
ssst rr②
❬❪ ♦ ♥ ② ♣♣t♦♥ ♦ rt♦♥ rts t♦ ss♠ ♥②ss ♦ ss♦t
♠♦s rtq ♥♥r♥ trtr ②♥♠s ♥♦
❬❪ P ♥ss t ♥ P ♦r r♦♥ ♦♥tr♦ s②st♠ s♥ t♦♦♦① ♦r t ♦♥tr♦ ♥♥r♥
♦♠♠♥t② tt♦r ♥ s st② P♦s♦♣ r♥st♦♥s ♦ t ♦② ♦t② ♦ ♦♥♦♥
t♠t P②s ♥ ♥♥r♥ ♥s ♥♦
❬❪ P ♥ss ♥ tr P ♠♣♠♥tt♦♥ ♦ r♦♥ ♦♥tr♦r rt♦♥ s ♥
♣♣ ♥②ss ♥♦
❬❪ s♥ ♥ ❩ss② ♦♥♥r rt♦♥ ②♥♠s ♦♥ tt t ♦♥ r♥ ♥trt♦♥s
P②s ttst ♥s ♥ ts ♣♣t♦♥s ♥♦
❬❪ ③♦ ♥ ♦rrs ♦s②♠♦♥ ♥♠♥t ♠♠ ♦ t rt♦♥ s
♦ rt♦♥s ♥ ♥ rr♥ qt♦♥ ♥♦♥ ♦♥② rts ♦ ♣t♦ ♦r♥ ♦
♣t♠③t♦♥ ♦r② ♥ ♣♣t♦♥s ♥♦
❬❪ ♥rt ♥ ❲♦r♠♥ ♣t♠ ♦♥tr♦ ♣♣ t♦ ♦♦ ♠♦s Prss
❬❪ ❨ ♥ ♥r③ r♦♥ ♥qts ♥ tr ♣♣t♦♥s t♦ rt♦♥ r♥t qt♦♥s
♦r♥ ♦ ♥qts ♥ ♣♣t♦♥s ♥♦
❬❪ ❩ ♥ ❩♥ ①st♥ ♥ ♦♥tr♦t② ♦r rt♦♥ ♦t♦♥ ♥s♦♥s ♦ r
sr♥t t②♣ ♣♣ t♠ts ♥ ♦♠♣tt♦♥
❬❪ P ♦♥ ♣t♠ ♦♥tr♦ ♥♦♥s♠♦♦t ♥②ss ♠r♥ t♠t ♦t②
❬❪ ♦t ♥ ♥ ❨♦s ♥♠r t♥q ♦r s♦♥ rt♦♥ ♦♣t♠ ♦♥tr♦
♣r♦♠s ♦♠♣trs t♠ts t ♣♣t♦♥s ♥♦
❬❪ ♦ ♥ ❱ r♠♦ ♦♥r♥ ♥trt♦♥s st♦stt② ♥ rt♦♥ ②♥♠s t
t♦ ♦r ❩ss② ♣r♥r ♥ s♥ss
❬❪ ❨ ♦ ♥ ❨ ♥ t③♥ ♥ r♦st rt♦♥ ♦rr P ♦♥tr♦r s②♥tss ♦r rst ♦rr
♣s t♠ ② s②st♠s t♦♠t ♥♦
❬❪ ❨ ♦ ❩♥ ♥ ♥ ❨ ♥ rt♦♥♦rr ♣r♦♣♦rt♦♥ rt ♦♥tr♦r
s②♥tss ♥ ♠♣♠♥tt♦♥ ♦r rsr sr♦ s②st♠ r♥st♦♥s ♦♥ ♦♥tr♦
②st♠s ♥♦♦② ♥♦
❬❪ é ♦♥ ♦♥t♦ s ♥ ♥ ♥tr ♥ ♦ ♠♣♣♥ ♦ r♥ stt②
♥ s♠ ♥♠s s♥ sr ♠♥ r♥st♦♥s ♦♥ ♠♥ ♥♦
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❬❪ ♦ ❱ r②♦ ♥ ♥r ♥t st♦r② ♦ rt♦♥ s ♦♠♠♥t♦♥s
♥ ♦♥♥r ♥ ♥ ♠r ♠t♦♥ ♥♦
❬❪ ♥r rt♦♥ r①t♦♥♦st♦♥ ♥ rt♦♥ s♦♥ ♣♥♦♠♥ ♦s
♦t♦♥s rts ♥♦
❬❪ rs rs ♥ ♦♥st♥t♥♦ ②♥♠ ♥②ss ♦ s♦st ♠♣rs
♦r♥ ♦ ♥♥r♥ ♥s ♥♦
❬❪ ♥♦s ③③ ♥ ♦rrs ♥ ♠t♦s ♥ t rt♦♥ s ♦
rt♦♥s ♣r♥r
❬❪ ♥♦s ♥ ♦rrs ♥tr♦t♦♥ t♦ t rt♦♥ s ♦ rt♦♥s ❲♦r
♥t
❬❪ ttr r ♥♦ ♦♥t♦♥ Eα(x) Prs
❬❪ ♦♥ ❨ ♥ ❱♥r ❳ ♥ ❱ t rt♦♥♦rr s②st♠s ♥
♦♥tr♦s ♥♠♥ts ♥ ♣♣t♦♥s ♣r♥r ♥ s♥ss
❬❪ ♦r♦ ①♠♠ ♣r♥♣ ♥ t ♣r♦♠ ♦ t♠ ♦♣t♠ rs♣♦♥s t ♥♦♥s♠♦♦t
♦♥str♥ts ♦r♥ ♦ ♣♣ t♠ts ♥ ♥s ♥♦
❬❪ ♥r③ r♥t s ♦r ♥♦♥s♠♦♦t ♥ st ♠♣♣♥s ♦r♥ ♦
t♠t ♥②ss ♥ ♣♣t♦♥s ♥♦
❬❪ ♦③②rs ♥ ♦rrs ♦ ♦♣t♠ ♥r② ♥ ♥t ♠♠♦r② ♦ rt♦♥
♦♥t♥♦st♠ ♥r s②st♠s ♥ Pr♦ss♥ ♥♦
❬❪ ♥♠♠r ♠♥♥♦ rt♦♥ rts ♥ t ②♦r♠♥♥ srs ❯❯
Pr♦t ♣♦rt
❬❪ P t♥s♦♥ ♦r② ♦ ♥t♦♥s ♦ r r rr ❯♥r ❨♦r
❬❪ ❲ stt ♥ strt rt♦♥ t♦r② t ♣♣t♦♥s t♦ r♦ ss ♦ ♦♣t♠③t♦♥
♣r♦♠s ♣♣t♦♥s ♦r♥ ♦♥ ♦♥tr♦ ♥♦
❬❪ ❩ t ♥ ♥r③ ②♦rs ♦r♠ ♣♣ t♠ts ♥
♦♠♣tt♦♥ ♥♦
❬❪ ♥ ❲ r rtr ♦r tr♠♥♥ ♦♠♣t♥ss ♦ t ♥trt♦♥ ♦♣rt♦r
ss♦t t t♦r ♠sr tr t t P ♥♦
❬❪ st♦♣ ♦s P ♥ss P ♦r ❳ ♦r ♥ tr ♣r♦
♦rt ♦♣♠♥ts ♥ ♠♦r ♣♣t♦♥s rt♦♥ r♥tt♦♥ ♥ ts ♣♣t♦♥s
♥♦
❬❪ Prr ♥ ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♦r t♦r ♠♣s ♦♥tr♦
♣r♦♠s ②st♠s ♦♥tr♦ ttrs ♥♦
❬❪ tt② ♦r ♠♣s ♦♥tr♦ s②st♠s ②♥♠ ②st♠s ♥♦
❬❪ ♠①♠♠ ♣r♥♣ ♦r ♥♥t t♠ s②♠♣t♦t② st ♠♣s ②♥♠ ♦♥tr♦
s②st♠s Pr♦♥s ❱♦♠s P♣rs♥♥
❬❪ ssr② ♦♥t♦♥s ♦ ♦♣t♠t② ♦r stt ♦♥str♥ ♥♥t ♦r③♦♥ r♥t
♥s♦♥s t ♦♥r♥ ♦♥ s♦♥ ♥ ♦♥tr♦ ♥ r♦♣♥ ♦♥tr♦ ♦♥r♥
♣♣
❬❪ Prr ♥ ❱ r ♥r♥ ♦r ♠♣s ♦♥tr♦ s②st♠s t♦♠t♦♥ ♥
♠♦t ♦♥tr♦ ♥♦
❬❪ Ptrs rt♦♥♦rr ♥♦♥♥r s②st♠s ♠♦♥ ♥②ss ♥ s♠t♦♥ ♣r♥r ♥
s♥ss
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❬❪ P♦♥② rt♦♥ r♥t qt♦♥s ♥ ♥tr♦t♦♥ t♦ rt♦♥ rts rt♦♥
r♥t qt♦♥s t♦ ♠t♦s ♦ tr s♦t♦♥ ♥ s♦♠ ♦ tr ♣♣t♦♥s ❱♦ ♠
Prss
❬❪ rt♦♥♦rr s②st♠s ♥ PIλD
µ♦♥tr♦rs r♥st♦♥s ♦♥ t♦♠t ♦♥tr♦
♥♦
❬❪ P♦♥tr②♥ ❱ ♦t②♥s ❱ ♠r③ ♥ s♥♦ ♠t♠t
t♦r② ♦ ♦♣t♠ ♣r♦sss ❲② ♥trs♥ ❨♦r
❬❪ P♦♦s ♠ ♥ ♦rrs rt♦♥ ♦rr ♦♣t♠ ♦♥tr♦ ♣r♦♠s t r
tr♠♥ t♠ ♦r♥ ♦ ♥str ♥ ♥♠♥t ♣t♠③t♦♥ ♥♦
❬❪ Pr♣t rt♥ ♣r♦♣rts ♦ ttr ♥t♦♥ t r♠♥t xα α > 0 t♥
♦r♥ ♦ Pr ♥ ♣♣ t♠ts
❬❪ ♥r② ❲ rs ♥ ♦ t♠t ♣r♦♠s ♥ s♦stt② r♥
t♠t ♦t② ❨♦r
❬❪ ♦rs ♦♥tr♦ ♥ ♦rrs ♣t♠ ♦♥tr♦ ♥ ♥♠r s♦tr
♥ ♦r ♣r♣r♥t r❳
❬❪ ♦ss ♠♦ ♥ ♦ ♥t♦♥s tt ♥♦ rst ♦rr rt ♠t
rt♦♥ rts ♦ ♦rrs ss t♥ ♦♥ ♥②ss ①♥ ♥♦
❬❪ ❨ ♦ss♥ ♥ ❱ t♦ ♣♣t♦♥s ♦ rt♦♥ s t♦ ②♥♠ ♣r♦♠s ♦
♥r ♥ ♥♦♥♥r rtr② ♠♥s ♦ s♦s ♣♣ ♥s s ♥♦
❬❪ ♠ ♥ ♦ ♥ s♦t♦♥s ♦ rt♦♥ ♦rr ♦♥r② ♣r♦♠s t ♥tr
♦♥r② ♦♥t♦♥s ♥ ♥ s♣s ♦r♥ ♦ ♥t♦♥ ♣s ♥ ♣♣t♦♥s
❬❪ ♠♦ s ♥ r rt♦♥ ♥trs ♥ rts t♦r② ♥
♣♣t♦♥s ♦r♦♥ ♥ r ♠str♠ ♥ r♥s r♦♠ t ss♥ t♦♥
❬❪ P t ♥ ♦♠♣s♦♥ ♣t♠ ♦♥tr♦ t♦r② ♣♣t♦♥s t♦ ♠♥♠♥t s♥ ♥
♦♥♦♠s ♣r♥r
❬❪ ❲ ♥ ♣♣t♦♥s ♦ ♦♣t♠ ♦♥tr♦ t♦r② ♥ ♦♠♥ Prss
❬❪ ❱ rs♦ ❯♥rs tr♦♠♥t s ♥ tr ♦r♥ ♦ P②ss ♦♥♥s ttr
♥♦
❬❪ rt♦♥ ②♥♠s ♣♣t♦♥s ♦ rt♦♥ s t♦ ②♥♠s ♦ ♣rts s ♥
♠ ♣r♥r ♥ s♥ss
❬❪ ③♦ ♠ rs♣♦♥s ♥②ss ♦ rt♦♥♦rr ♦♥tr♦ s②st♠s sr② ♦♥ r♥t rsts
rt♦♥ s ♥ ♣♣ ♥②ss ♥♦
❬❪ ③♦ r r ♦♦ ♥ ♠ ♦♠ ♣♣t♦♥s ♦ rt♦♥ s
♥ s♣♣rss♦♥ ♦ ♦t ♦st♦♥s r♥st♦♥s ♦♥ ♥str ♥♦r♠ts ♥♦
❬❪ ②♦r sr t♦r② ♥ ♥trt♦♥ ♠r♥ t♠t ♦t②
❬❪ ♦♠s♦♥ ♦r② ♦ t ♥tr ss♥②s ♦♠
❬❪ ♦♠s♦♥ r♥r ♥ r♥r ♠♥tr② r ♥②ss ss♥②ss
♦♠
❬❪ ♦r♦t ♥ P ♦tt ②♥♠ ♣r♦r♠♠♥ ♥ ts ♣♣t♦♥ t♦ ♦♣t♠ ♦♥tr♦ ❱♦
sr
❬❪ r ♥ ❨ ♥ ♥ ♣♣r♦①♠t ♠t♦ ♦r ♥♠r② s♦♥ rt♦♥ ♦rr ♦♣t♠
♦♥tr♦ ♣r♦♠s ♦ ♥r ♦r♠ ♦♠♣trs t♠ts t ♣♣t♦♥s ♥♦
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❬❪ s♥ ♥ t ♥trt♦r s♥ s♥ rrs ♦♠r ♥trt♦♥ r ♥
rt♦♥ s♠♣ ② ♥ Pr♦ss♥ ♥♦
❬❪ ❱ér♦ ♥ á ♦st ♦♥♥tr ♦rr ♦♥tr♦ ♦ ① r♦♦t Pr♦♥s ♦ t
♦rs♦♣ ♦♥ rt♦♥ r♥tt♦♥ ♥ ts ♣♣t♦♥s ♦r① r♥ ♣♣
❬❪ ❱♥r ❨ ♥ ♥ Ptrá ♦ rt st♥ srt③t♦♥ ♠t♦s ♦r rt♦♥
♦rr r♥tt♦r♥trt♦r ♦r♥ ♦ t r♥♥ ♥sttt ♥♦
❬❪ ❱♥tr ♣t♠ ♦♥tr♦ ♣r♥r
❬❪ ❱♥tr ♥ Prr ♠①♠♠ ♣r♥♣ ♦r ♦♣t♠ ♣r♦sss t s♦♥t♥♦s
trt♦rs ♦r♥ ♦♥ ♦♥tr♦ ♥ ♦♣t♠③t♦♥ ♥♦
❬❪ ❲st ♦♦♥ ♥ P r♦♥ P②ss ♦ rt ♦♣rt♦rs ♣r♥r ♥ s♥ss
❬❪ ❨ ♦ ♥ ❨ ♥ ♥r③ r♦♥ ♥qt② ♥ ts ♣♣t♦♥ t♦ rt♦♥
r♥t qt♦♥ ♦r♥ ♦ t♠t ♥②ss ♥ ♣♣t♦♥s ♥♦
❬❪ ❨ ♥②ss t♦r② ♦ ♠sr ♥ ♥trt♦♥ ❲♦r ♥t
❬❪ ❨♦s ♦t ♥ ♥ s ♦ ♥r ♠tt ♦♦t♦♥ ♠t♦ ♦r
s♦♥ t rt♦♥ ♦♣t♠ ♦♥tr♦ ♣r♦♠s ♦r♥ ♦ ❱rt♦♥ ♥ ♦♥tr♦ ♥♦
❬❪ ❩♥ ♥ ❱ ♦rs♦ ♦r② ♦ ttr♥ ♦♥tr♦ t ♣♣t♦♥s t♦ str♦♥ts
r♦♦ts ♦♥♦♠s ♥ ♥♥r♥ ♣r♥r ♥ s♥ss
❬❪ ❩♥ ♦♠ ♥ r♦♥♠♥t②♣ ♥qts s ♦♥ t ♠♦ ♠♥♥♦
rt♦♥ rt ♦r♥ ♦ ♣♣ t♠ts