hegagi mohamed ali ahmed necessary conditions for … · 2019-06-06 · in particular, mahmoud...

♦♠ ♠

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

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.

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♦♦ ♦

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♦ ♥

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

♠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♦

♥♦♠♥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

♦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

♦♥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

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

rt♦♥ ♦s②♠♦♥ ♥♠♥t ♠♠

♦♥s♠♦♦t ♥②ss

sr ♦r② ♥ ♥trt♦♥

r ♥ σ−r ♦ ts

srs

♥trt♦♥

❯s ♦♥♣ts

♦r♣②

①

♣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♦

♦♥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

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 ❬❪

❲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♦♥

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

♣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

♣ ♥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].

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.

♥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.

♦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.

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

♥ ♠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,

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)),

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,

♥ 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].

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];

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.

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

❬❪ ♦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

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

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

)

.

♠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

).

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.

♥ 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,

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

♥ 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),

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)).

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),

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,

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

♦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 ♠♦ ❬❪

♣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

♦♥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

♦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 ♦

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

♣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 ❬❪

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

♦♠♣♦♥♥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 )

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 ♥

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),

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

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

)

,

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).

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,

♥ ♦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).

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).

♦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

♣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

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

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

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♦♥

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.

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

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.

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♦

♣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♦♠

(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.

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)|,

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,

♦♥ 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♦♥

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)).

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)| .

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

♦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

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|),

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♥

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

♦ 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.

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)

∣

∣

∣.

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),

♥ 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

♥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.

♦ 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,

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.

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

♣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♦♠

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| .

♥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));

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

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 ♦ ♠♠

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) ,

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♦

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,

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))

∣

∣

∣

≤ 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)),

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.

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.

♥ ♦♥ 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

♦ 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)).

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

)

,

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),

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 → ∞.

♥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

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

♦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♠

♣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

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

❲ 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

♣♣♥s

♣♣♥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

♥ 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.

♥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).

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

.

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).

♠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).

♥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♥

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.

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),

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).

♣♣♥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.

♥ ♦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.

♥

(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).

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,

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,

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

♥♦ 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♦♥

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

♣♣♥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.

♥ 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

♦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

♣♣♥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

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♦♥

♥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,

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 ❬❪

❯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∗ ❲∗

♦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 ♥♦

❬❪ 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♥ ♥♦

❬❪ ♣♦♥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

❬❪ ♦ ♥ 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 ♥♦

❬❪ ❱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 ♦♥ ♠♥ ♥♦

❬❪ ♦ ❱ 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

❬❪ 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 ♥♦

❬❪ 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