❈♦♥tr♦❧❡ Ót✐♠♦ ❆♣❧✐❝❛❞♦ ❡♠ ✉♠ ▼♦❞❡❧♦ ❞❡
❈â♥❝❡r
❊❞✉❛r❞♦ ❘❛♠♦s
❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛✳ ❉r❛✳ ❈r✐st✐❛♥❡ ◆❡s♣♦❧✐ ▼♦r❡❧❛t♦ ❋r❛♥ç❛
Pr♦❣r❛♠❛✿ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧
❯◆■❱❊❘❙■❉❆❉❊ ❊❙❚❆❉❯❆▲ P❆❯▲■❙❚❆
❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❡ Pr❡s✐❞❡♥t❡ Pr✉❞❡♥t❡
Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧
❈♦♥tr♦❧❡ Ót✐♠♦ ❆♣❧✐❝❛❞♦ ❡♠ ✉♠ ▼♦❞❡❧♦ ❞❡
❈â♥❝❡r
❊❞✉❛r❞♦ ❘❛♠♦s
❖r✐❡♥t❛❞♦r❛✿ Pr♦❢❛✳ ❉r❛✳ ❈r✐st✐❛♥❡ ◆❡s♣♦❧✐ ▼♦r❡❧❛t♦ ❋r❛♥ç❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧ ❞❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ❈✐ê♥❝✐❛s ❡ ❚❡❝♥♦❧♦❣✐❛ ❞❛ ❯◆❊❙P ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tí✲ t✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛❝✐♦♥❛❧✳
FICHA CATALOGRÁFICA
Ramos, Eduardo.
R142c Controle Ótimo Aplicado em um Modelo de Câncer / Eduardo Ramos. - Presidente Prudente : [s.n], 2015
98 p. : il.
Orientador: Cristiane Nespoli Morelato França
Dissertação (mestrado) - Universidade Estadual Paulista, Faculdade de Ciências e Tecnologia
Inclui bibliografia
1. Controle Ótimo. 2. Modelo de câncer com quimioterapia. 3.
❆❣r❛❞❡❝✐♠❡♥t♦s
✏ ❙❡♠♣r❡ ♠❡ ♣❛r❡❝❡✉ ❡str❛♥❤♦ q✉❡ t♦❞♦s ❛q✉❡❧❡s q✉❡ ❡st✉❞❛♠ s❡r✐❛♠❡♥t❡ ❡st❛ ❝✐ê♥❝✐❛ ❛❝❛❜❛♠ t♦♠❛❞♦s ❞❡ ✉♠❛ ❡s♣é❝✐❡ ❞❡ ♣❛✐①ã♦ ♣❡❧❛ ♠❡s♠❛✳ ❊♠ ✈❡r❞❛❞❡✱ ♦ q✉❡ ♣r♦♣♦r❝✐♦♥❛ ♦ ♠á①✐♠♦ ❞❡ ♣r❛③❡r ♥ã♦ é ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ s✐♠ ❛ ❛♣r❡♥❞✐③❛❣❡♠✱ ♥ã♦ é ❛ ♣♦ss❡✱ ♠❛s ❛ ❛q✉✐s✐çã♦✱ ♥ã♦ é ❛ ♣r❡s❡♥ç❛✱ ♠❛s ♦ ❛t♦ ❞❡ ❛t✐♥❣✐r ❛ ♠❡t❛✳✑
❘❡s✉♠♦
◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ❛ t❡♦r✐❛ ❞❡ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♥♦s ❢❛♠✐❧✐❛r✐③❛r♠♦s ❝♦♠ ❛ t❡♦r✐❛ ❞❡ ❈♦♥tr♦❧❡ Ót✐♠♦✳ ❊st✉❞❛♠♦s t❛♠❜é♠ r❡s✉❧t❛❞♦s ❞❛ ❚❡♦r✐❛ ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s✱❝♦♠ ❢♦❝♦ ❡♠ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛✱❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❞❡♠♦♥str❛r✲ ♠♦s ♦ Pr✐♥❝í♣✐♦ ▼í♥✐♠♦ ❞❡ P♦♥tr②❛❣✐♥✱❝♦♥❞✐çã♦ ❡st❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ♦t✐♠❛❧✐❞❛❞❡ ❞❡ ✉♠ ❝♦♥tr♦❧❡✳ P♦r s✉❛ ✈❡③✱❝♦♥❞✐çõ❡s ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ❡ ❡①✐stê♥❝✐❛ ♣❛r❛ ❈♦♥tr♦❧❡s Ót✐♠♦s ▲✐♥❡✲ ❛r❡s t❛♠❜é♠ ❢♦r❛♠ ❡st✉❞❛❞❛s✱✉♠❛ ✈❡③ q✉❡ sã♦ ❡ss❡♥❝✐❛✐s ♣❛r❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ❝♦♥tr♦❧❡s ót✐♠♦s✳ P♦r ✜♠✱✉t✐❧✐③❛♠♦s ❛ t❡♦r✐❛ ❞❡ ❈♦♥tr♦❧❡ Ót✐♠♦ ❛❝✐♠❛ ✐♥❞✐❝❛❞❛ ♣❛r❛ ❛ ❛♥á❧✐s❡ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡ ❝â♥❝❡r ❝♦♠ q✉✐♠✐♦t❡r❛♣✐❛ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♠✐♥✐♠✐③❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ q✉✐♠✐♦t❡rá♣✐❝♦ ❛♣❧✐❝❛❞❛ ♥♦ tr❛t❛♠❡♥t♦✳
❆❜str❛❝t
■♥ t❤❡ ✇♦r❦❡❞ ❤❡r❡ ♣r❡s❡♥t❡❞ ✇❡ st✉❞② t❤❡ t❤❡♦r② ♦❢ ❖♣t✐♠❛❧ ❈♦♥tr♦❧✳ ❲❡ ❛❧s♦ st✉❞✐❡❞ r❡s✉❧ts ✐♥ ❚❤❡♦r② ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ❢♦❝✉s✐♥❣ ♦♥ ▼❡❛s✉r❡ ❚❤❡♦r②✱✇✐t❤ t❤❡ ❣♦❛❧ ♦❢ ♣r♦✈✐♥❣ t❤❡ P♦♥tr②❛❣✐♥ ▼✐♥✐♥✉♠ Pr✐♥❝✐♣❧❡✱❛ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥ ❢♦r ♦♣t✐♠❛❧✐t② ♦❢ ❛ ❝♦♥tr♦❧✳ ■♥ t✉r♥✱r❡❣✉❧❛r✐t② ❛♥❞ ❡①✐st❡♥❝❡ ❝♦♥❞✐t✐♦♥s ❢♦r ❖♣t✐♠❛❧ ▲✐♥❡❛r ❈♦♥tr♦❧s ✇❡r❡ ❛❧s♦ st✉❞✐❡❞✱s✐♥❝❡ t❤❡② ❛r❡ ❡ss❡♥t✐❛❧ ❢♦r ❞❡t❡r♠✐♥❛t✐♦♥ ♦❢ ♦♣t✐♠❛❧ ❝♦♥tr♦❧s✳ ❋✐♥❛❧❧②✱✇❡ ✉s❡❞ t❤❡ t❤❡♦r② ♦❢ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❛❜♦✈❡ ♠❡♥t✐♦♥❡❞ ❢♦r ❛♥ ❛♥❛❧②s✐s ♦❢ ❛ ♠♦❞❡❧ ♦❢ t✉♠♦r ✇✐t❤ ❝❤❡♠♦t❤❡r❛♣② ✇✐t❤ t❤❡ ❣♦❛❧ ♦❢ ♠✐♥✐♠✐③✐♥❣ t❤❡ q✉❛♥t✐t② ♦❢ ❝❤❡♠♦t❤❡r❛♣② ❛♣♣❧✐❡❞ ✐♥ t❤❡ tr❡❛t♠❡♥t✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✸✳✶ ■❧✉str❛çã♦ ❞❛ ❝♦♥str✉çã♦ ❞❡ u(t, ǫ)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✶ ■❧✉str❛çã♦ ♣❛r❛ ♦ ❚❡♦r❡♠❛ ✶✻✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✺✳✶ ❘❡tr❛t♦ ❞❡ ❋❛s❡ ❞♦ ❈â♥❝❡r s❡♠ ◗✉✐♠✐♦t❡r❛♣✐❛✱ ❝♦♠ x❡ y ♥♦r♠❛❧✐③❛❞♦s✳ ✳ ✳ ✻✵
✺✳✷ ❘❡tr❛t♦ ❞❡ ❋❛s❡ ❞♦ ❈â♥❝❡r ❝♦♠ ◗✉✐♠✐♦t❡r❛♣✐❛✱ ❝♦♠ x❡ y ♥♦r♠❛❧✐③❛❞♦s✳ ✳ ✻✶
✺✳✸ P♦♥t♦s ❋✐♥❛✐s ♣❛r❛ c= 0.05 ❡ d= 0.01✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✺✳✹ ❚r❛❥❡t♦s Ót✐♠♦s ◆ã♦✲❙✐♥❣✉❧❛r❡s ♣❛r❛ c= 0.05 ❡ d= 0.01✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✺✳✺ P♦♥t♦s s✐♥❣✉❧❛r❡s ❝♦♠ 0 ≤ u(t) ≤ 1✱ ❡ s✐♥❣✉❧❛r❡s ❝rít✐❝♦s ❡ s✉♣❡r❝rít✐❝♦s
♣❛r❛ c= 0.05❡ d= 0.01✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✺✳✻ ❚r❛❥❡t♦s Ót✐♠♦s ♣❛r❛ c= 0.05❡ d= 0.01✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✺✳✼ ❆r❝♦s ❈❤❛✈❡ ♣❛r❛ c= 0.05 ❡ d= 0.01✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✺✳✽ ❚r❛❥❡t♦ Ót✐♠♦ ♣❛r❛ (x0, y0) = (600,0.2)✱ ♣❛r❛ c= 0.05❡ d= 0.01✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✺✳✾ ❈♦♥tr♦❧❡ Ót✐♠♦ ♣❛r❛ (x0, y0) = (600,0.2)✱ ♣❛r❛c= 0.05 ❡d= 0.01✳ ✳ ✳ ✳ ✳ ✻✻
✺✳✶✵ ❚r❛❥❡t♦s Ót✐♠♦s ♣❛r❛ c= 0.001 ❡ d= 0.28✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✺✳✶✶ ■❧✉str❛çã♦ ♣❛r❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✼✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✺✳✶ ❚❛❜❡❧❛ ❞❛s ❝♦♥st❛♥t❡s ✉t✐❧✐③❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
▲✐st❛ ❞❡ ◆♦t❛çõ❡s
L1(X, Y)✿ ❋✉♥çõ❡s ▲❡❜❡s❣✉❡ ✐♥t❡❣rá✈❡✐s ♣❛rt✐♥❞♦ ❞❡ X ⊂Rn ❡ ❝❤❡❣❛♥❞♦ ❡♠ Y ⊂R✳ ˆ
C(X, Y)✿ ❋✉♥çõ❡s ❝♦♥tí♥✉❛s ♣♦r ♣❛rt❡s ♣❛rt✐♥❞♦ ❞❡ X ⊂Rn ❡ ❝❤❡❣❛♥❞♦ ❡♠ Y ⊂R✳
f é Cn✿ ■♥❞✐❝❛ f s❡r n ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ f(n) ❝♦♥tí♥✉❛ ♥♦ ❞♦♠í♥✐♦ ❞❡ f✳
P❱■✿ Pr♦❜❧❡♠❛ ❞❡ ❱❛❧♦r ■♥✐❝✐❛❧✳
U❛❞✿ ❈♦♥❥✉♥t♦ ❞♦s ❝♦♥tr♦❧❡s ❛❞♠✐ssí✈❡✐s✳
❙✉♠ár✐♦
❘❡s✉♠♦ ✺
❆❜str❛❝t ✼
▲✐st❛ ❞❡ ❋✐❣✉r❛s ✽
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ✾
▲✐st❛ ❞❡ ◆♦t❛çõ❡s ✶✸
❈❛♣ít✉❧♦s
✶ ■♥tr♦❞✉çã♦ ✶✼
✷ Pr❡❧✐♠✐♥❛r❡s ✷✶
✷✳✶ ❚ó♣✐❝♦s ❞❡ ❆♥á❧✐s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ♣❛r❛ ❢✉♥çõ❡s ❆❜s♦❧✉t❛♠❡♥t❡ ❈♦♥tí♥✉❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✸ ❈♦♥tr♦❧❡ Ót✐♠♦ ✷✼
✸✳✶ ❈♦♥tr♦❧❡ Ót✐♠♦ ❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ P♦♥tr②❛❣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷ Pr♦❜❧❡♠❛ ❞❡ ❈♦♥tr♦❧❡ Ót✐♠♦ ♥❛ ❢♦r♠❛ ▼❛②❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✸ ❉❡♠♦♥str❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ P♦♥tr②❛❣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✹ Pr♦❜❧❡♠❛s ❞❡ ❈♦♥tr♦❧❡ Ót✐♠♦ ▲✐♥❡❛r❡s ✸✾
✹✳✶ ❆♣❧✐❝❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞♦ ▼í♥✐♠♦ ❞❡ P♦♥tr②❛❣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✷ ❚❡♦r❡♠❛ ❞❡ ❘❡❣✉❧❛r✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✸ ❆❧❣♦r✐t♠♦ ❞❡ ❉❡t❡r♠✐♥❛çã♦ ❞♦s ❈♦♥tr♦❧❡s Ót✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✹ ❊①✐stê♥❝✐❛ ❞❡ ❈♦♥tr♦❧❡ Ót✐♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✺ ❈♦♥❞✐çã♦ ❆❞✐❝✐♦♥❛❧ ♣❛r❛ P♦♥t♦ ❋✐♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✺ ▼♦❞❡❧♦ ❞❡ ❈â♥❝❡r ❞❡ ❙t❡♣❛♥♦✈❛ ✺✾
✺✳✶ ▼♦❞❡❧❛❣❡♠ ♣♦r ❈♦♥tr♦❧❡ Ót✐♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✺✳✷ ❆♥á❧✐s❡ ❞♦ Pr♦❜❧❡♠❛ ♣❡❧❛ ❚❡♦r✐❛ ❞❡ ❈♦♥tr♦❧❡ Ót✐♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺✳✸ ❊①✐stê♥❝✐❛ ❞♦ ❈♦♥tr♦❧❡ Ót✐♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✻ ❈♦♥❝❧✉sõ❡s ❡ Pr♦♣♦st❛s ❋✉t✉r❛s ✼✶
❘❡❢❡rê♥❝✐❛s ✼✶
✼ ❆♣ê♥❞✐❝❡ ❆ ✼✺ ✼✳✶ ❘❡❣r❛ ❞❛ ❈❛❞❡✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✼✳✷ ❉❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻
✽ ❆♣ê♥❞✐❝❡ ❇ ✽✶
❈❛♣ít✉❧♦
✶
■♥tr♦❞✉çã♦
❙❛❜❡♠♦s q✉❡ ♥❛s ár❡❛s ♠❛t❡♠át✐❝❛s ❞❡ ❈á❧❝✉❧♦ ❡ ❆♥á❧✐s❡ ♥♦ Rn ❛ q✉❡stã♦ ❞❡ ♠✐✲
♥✐♠✐③❛çã♦ ❞❡ ❢✉♥çõ❡s é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ❡ ❡♥❝♦♥tr❛ ❛♣❧✐❝❛çã♦ ❡♠ ❞✐✈❡rs❛s ár❡❛s✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ❛ ♠✐♥✐♠✐③❛çã♦ ❞❡ ❢✉♥❝✐♦♥❛✐s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❛s ✐♥t❡❣r❛✐s✱ t❡♠ t❛♠❜é♠ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ❡st❛ t❡♠ s✉❛ t❡♦r✐❛ ❜❛s❡❛❞❛ ❡♠ ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛ ❝♦♠♦ ♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧ ❡ ♦ ❈♦♥tr♦❧❡ Ót✐♠♦✳ ❊st❛ ✐♠♣♦rtâ♥❝✐❛ é ♠❛✐♦r ❛✐♥❞❛ ❡♠ ❝❡rt❛s ár❡❛s ❝♦♠♦ ❛ ♠❡❞✐❝✐♥❛ ♦♥❞❡ ❛ ♦t✐♠✐③❛çã♦ ♥♦ ✉s♦ ❞❡ ♠❡❞✐❝❛çõ❡s ❡♠ ❞♦❡♥ç❛s ❝♦♠♦ ♦ ❝â♥❝❡r sã♦ ❝r✉❝✐❛✐s ♣❛r❛ q✉❡ ❡st❡ ♥ã♦ ❝❛✉s❡ t❛♥t♦s ❞❛♥♦s✳
◆❡st❡ ❝♦♥t❡①t♦✱ ♣❛r❛ ❛ ár❡❛ ❞❡ ❝♦♥tr♦❧❡ ót✐♠♦✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞❛s ❡♥tã♦ ❞✐✈❡rs❛s t❡♦r✐❛s ♣❛r❛ r❡s♦❧✈❡r s❡✉s ♣r♦❜❧❡♠❛s✱ ❞❡♥tr❡ ♦s q✉❛✐s ❞❡st❛❝❛♠♦s ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ♦t✐♠❛❧✐❞❛❞❡ ❝❤❛♠❛❞❛ ❞❡ Pr✐♥❝í♣✐♦ ▼í♥✐♠♦ ❞❡ P♦♥tr②❛❣✐♥✳
❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ♥♦ ❡st✉❞♦ ❞❛ t❡♦r✐❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛♥á❧✐s❡ ❞❡ ♠♦❞❡❧♦s ❞❡ tr❛t❛♠❡♥t♦ ❞❡ ❝â♥❝❡r ❝♦♠ q✉✐♠✐♦t❡r❛♣✐❛✱ ✈✐st♦s ❝♦♠♦ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥tr♦❧❡ ót✐♠♦✳ ◆❡st❡ s❡♥t✐❞♦✱ ❛ ❞✐ss❡rt❛çã♦ ❡stá ❡str✉t✉r❛❞❛ ❝♦♠♦ s❡❣✉❡✳
◆♦ ❈❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ ❛♥á❧✐s❡ r❡❛❧ ❝♦♠♦ ❢✉♥çã♦ ❛❜s♦❧✉✲ t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡ t❡♦r❡♠❛s r❡❧❛❝✐♦♥❛❞♦s ❛ ❡st❛ ❞❡✜♥✐çã♦✳ ❚❛♠❜é♠ ❛♣r❡s❡♥t❛♠♦s ✉♠ r❡s✉♠♦ ❡♥✈♦❧✈❡♥❞♦ ❛ t❡♦r✐❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❛ ❢✉♥çõ❡s ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥t✐✲ ♥✉❛s✱ t❡♦r✐❛ ❡st❛ q✉❡ s❡rá ❡①t❡♥s❛♠❡♥t❡ ✉t✐❧✐③❛❞❛ ♥♦ ❞❡❝♦rr❡r ❞❡st❡ tr❛❜❛❧❤♦✳
◆♦ ❈❛♣ít✉❧♦ ✸ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ ✉♠ Pr♦❜❧❡♠❛ ❞❡ ❈♦♥tr♦❧❡ Ót✐♠♦✱ ❢♦❝❛♥❞♦ ♥♦s ♣r♦❜❧❡♠❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
• ❉❛❞♦s Ω = Ω1× · · · ×Ωm ⊂Rm ❡D⊂Rn✱ ♣❛r❛ ❝❛❞❛ ❢✉♥çã♦ u∈L1([0, T],Ω)✱ q✉❡ ❝❤❛♠❛♠♦s ❞❡ ❝♦♥tr♦❧❡✱ t♦♠❛♠♦s ✉♠❛ ❢✉♥çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ x : [a, b] → D✱ q✉❡
❝❤❛♠❛♠♦s ❞❡ tr❛❥❡tór✐❛✱ s❡♥❞♦ ❡st❛ ❞❡✜♥✐❞❛ ♣❡❧♦ P❱■ ❡♠ [a, b]
x′(t) = f(x(t), u(t)), x(0) =x0.
◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ♦ ❝♦♥tr♦❧❡ u✱ q✉❡ s❡rá ❝❤❛♠❛❞♦ ❞❡ ❝♦♥tr♦❧❡ ót✐♠♦✱ ❡ ♦ t❡♠♣♦
✜♥❛❧ T q✉❡ ♠✐♥✐♠✐③❡♠ ♦ ❝✉st♦
J(x, u) =φ(x(T)) + T
0
L(x(t), u(t))dt,
♦♥❞❡ φ ❡L sã♦ ❢✉♥çõ❡s ❞❡ Rn ❡Rn×Ω✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s♦❜r❡ R✳
P❛r❛ ♦ ✜♠ ❞❡ ❡♥❝♦♥tr❛r t❛❧ ❝♦♥tr♦❧❡ ót✐♠♦ u✱ ❛♣r❡s❡♥t❛♠♦s ❡ ♣r♦✈❛♠♦s ♦ Pr✐♥❝í♣✐♦
▼í♥✐♠♦ ❞❡ P♦♥tr②❛❣✐♥ ♣❛r❛ ❡st❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛✱ ♦ q✉❛❧ ❞✐③ q✉❡✱ ♣❛r❛ ✉♠ ❝♦♥tr♦❧❡
✶✳ ■♥tr♦❞✉çã♦ ✶✽ ót✐♠♦(x(t), u(t))✱ ❞❛❞♦H :D×Rn×Ω❞❡✜♥✐❞♦ ♣♦rH(x, λ, u) = L(x, u) +λf(x, u)✱ ❞❡✈❡
❡①✐st✐r λ: [0, T]→Rn t❛❧ q✉❡ s❡❥❛ s❛t✐s❢❡✐t♦ ♦ P❱■ ❡♠ [0, T]
λ′(t) =−Hx(x(t), λ(t), u(t)), λ(T) =φx(x(T))
❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ♠✐♥✐♠❛❧✐❞❛❞❡
H(x(t), λ(t), u(t)) = min
v∈Ω H(x(t), λ(t), v) q✳❝✳ ❡♠ [0, T]
P❛r❛ ♣r♦✈❛r ❡st❡ t❡♦r❡♠❛ s✉♣♦♠♦s ❛♣❡♥❛s q✉❡ L✱ f ❡φ s❛t✐s❢❛ç❛♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐✲
çõ❡s✿
• L(x, u)✱ f(x, u) ❡ φx(x, u)s❡❥❛♠ ♠❡♥s✉rá✈❡✐s ❡♠ Ω ♣❛r❛ ❝❛❞❛ x∈D ✜①❛❞♦❀
• Lx(x, u)✱ fx(x, u) ❡ φxx(x, u)❡①✐st❛♠ ❡ s❡❥❛♠ ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛s ❡♠ D×Ω✳ ❈♦♥s✐❞❡r❛♠♦s ❡st❛s ❝♦♥❞✐çõ❡s ♣❛r❛ ♦ Pr✐♥❝í♣✐♦ ❞❡ P♦♥tr②❛❣✐♥ ❜❡♠ ❢r❛❝❛s ❡♠ ❝♦♠♣❛r❛✲ çã♦ ❝♦♠ ❛s ❞❛ ❧✐t❡r❛t✉r❛ ❛t✉❛❧✱ ❛s q✉❛✐s ❝♦st✉♠❛♠ ❡①✐❣✐r ♣♦r ❡①❡♠♣❧♦ q✉❡ L(x, u)✱f(x, u) ❡φx(x, u) s❡❥❛♠ ❝♦♥tí♥✉❛s ❡♠u✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ Pr✐♥❝í♣✐♦ ▼í♥✐♠♦ ❞❡ P♦♥tr②❛❣✐♥✱ ❝♦♠ ❡st❛s ❝♦♥❞✐çõ❡s✱ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ r❡❝❡♥t❡ ❧✐t❡r❛t✉r❛✱ ✈✐❞❡ ❬✷✵❪✳
◆♦ ❈❛♣ít✉❧♦ ✹ tr❛t❛♠♦s ❞❡ Pr♦❜❧❡♠❛❜ ❞❡ ❈♦♥tr♦❧❡❜ Ót✐♠♦❜ ▲✐♥❡❛r❡❜✱ ♦♥❞❡ é ❢❡✐t❛ ❛ ❛♥á❧✐s❡ ❞❡ s❡✉s ❝♦♥tr♦❧❡s ót✐♠♦s ❛tr❛✈és ❞♦ Pr✐♥❝í♣✐♦ ❞❡ P♦♥tr②❛❣✐♥✳ ■♥tr♦❞✉③✐♠♦s ❡ ♣r♦✈❛♠♦s ❡♥tã♦ ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ❡st❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛✱ q✉❡ é ♥♦✈❛ ❛té ♦♥❞❡ ✈❛✐ ♥♦ss♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ ❧✐t❡r❛t✉r❛ ❛t✉❛❧✳ ▼❛✐s ❡s♣❡❝✐✜❝❛❞❛♠❡♥t❡✱ ❞❡♠♦s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ✉♠ ❝♦♥tr♦❧❡ ót✐♠♦ u(t) s❡❥❛ r❡❣✉❧❛r ✭❝♦♠♦ ❞❡✜♥✐❞♦ ♥♦ ❞❡❝♦rr❡r ❞♦ ❝❛♣ít✉❧♦✮ ❡ t❛♠❜é♠ s❡❥❛♠ ❝♦♥tí♥✉♦ ♣♦r ♣❛rt❡s✱ s❡♥❞♦ ❡st❛ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡s❡❥á✈❡❧ ♣❛r❛ t❛❧ ❝♦♥tr♦❧❡✳
◆❛ ❧✐t❡r❛t✉r❛ ❛t✉❛❧✱ ❡♠ ❝♦♠♣❛r❛çã♦ ❝♦♠ ❡st❛ ❝♦♥❞✐çã♦ ❞❡ r❡❣✉❧❛r✐❞❛❞❡✱ ❙✉ss♠❛♥♥ ♣r♦✈♦✉ ❡♠ ❬✶✽❪ ❝♦♥❞✐çõ❡s ❣❡♥ér✐❝❛s ✭❝♦♠♦ ❞❡✜♥✐❞♦ ❡♠ ❬✶✽❪ ♣♦r ❡①❡♠♣❧♦✮ ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥tr♦❧❡s ót✐♠♦s ❧✐♥❡❛r❡s ❝♦♠ ❞✐♠❡♥sã♦ n = 2✱ ❡♥q✉❛♥t♦ ❙❝❤❛tt❧❡r ♣r♦✈♦✉ ❡♠ ❬✶✺❪ ❝♦♥❞✐çõ❡s ❣❡♥ér✐❝❛s ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥tr♦❧❡s ót✐♠♦s ❧✐♥❡❛r❡s ❝♦♠ ❞✐♠❡♥sã♦ n = 3✳ ❆s ❤✐♣ót❡s❡s ❞❡ ♥♦ss❛s ❝♦♥❞✐çõ❡s ♣♦r ♦✉tr♦ ❧❛❞♦ sã♦ ♠❛✐s ❢r❛❝❛s ❞♦ q✉❡ ❡st❛s ❞✉❛s✱ ❡ ❛❝r❡❞✐t❛♠♦s q✉❡ ❡❧❛s s❡❥❛♠ ❝♦♥❞✐çõ❡s ❣❡♥ér✐❝❛s ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥tr♦❧❡s ót✐♠♦s ❧✐♥❡❛r❡s ❝♦♠ ❞✐♠❡♥sã♦ n = 4✳ ❯t✐❧✐③❛♠♦s ❡♥tã♦ t❛❧ ❝♦♥❞✐çã♦ ❞❡ r❡❣✉❧❛r✐❞❛❞❡ ♣❛r❛ ❝r✐❛r ✉♠ ❛❧❣♦r✐t♠♦ q✉❡ ❞❡t❡r♠✐♥❡ ❞❡ ❢♦r♠❛ ❡①❛t❛ t♦❞♦s ♦s ♣♦ssí✈❡✐s ❝♦♥tr♦❧❡s ót✐♠♦s ❝✉❥❛s tr❛❥❡tór✐❛s t❡r♠✐♥❡♠ ♥✉♠ ♣♦♥t♦ ✜♥❛❧ z(T)✳ ❊st❡ ❛❧❣♦r✐t♠♦ é út✐❧ ♣♦r ❡①❡♠♣❧♦ ♣❛r❛ ❛♥á❧✐s❡ ❣rá✜❝❛ ❞♦s ❝♦♥tr♦❧❡s ót✐♠♦s✳ ❈r✐❛♠♦s t❛♠❜é♠ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❞❡t❡r♠✐♥❛çã♦ ❛♣r♦①✐♠❛❞❛ ❞❡ ✉♠ ♣♦ssí✈❡❧ ❝♦♥tr♦❧❡ ót✐♠♦ u(t) ♣❛r❛ ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧
z0 ❞❛❞♦✳ ❊st❡ ❛❧❣♦r✐t♠♦ ♣♦❞❡ ✈✐r ❛ s❡r ❡s♣❡❝✐❛❧♠❡♥t❡ út✐❧ ♣♦✐s ♥♦t❛♠♦s ✉♠❛ ❣r❛♥❞❡
✐♥s❛t✐s❢❛çã♦ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❛rt✐❣♦s ❝♦♥s✉❧t❛❞♦s✱ ♣♦r ♣❛rt❡ ❞♦s ❛✉t♦r❡s✱ q✉❡ ✉t✐❧✐③❛r❛♠ ❞❡ ♠ét♦❞♦s ♥✉♠ér✐❝♦s ❝❧áss✐❝♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞❡ ❝♦♥tr♦❧❡s ót✐♠♦s✳ ❉❡ ❢♦r♠❛ ❛ ❝♦♠♣❧❡♠❡♥t❛r ❛ ✉t✐❧✐③❛çã♦ ❞❡st❡s ❛❧❣♦r✐t♠♦s✱ ♣r♦✈❛♠♦s t❛♠❜é♠ ✉♠❛ r❡str✐çã♦ ♥♦✈❛ ♣❛r❛ ♦s ♣♦♥t♦s ✜♥❛✐s
z(T) ❞❛ tr❛❥❡tór✐❛ z(t)✳ ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ✉♠ t❡♦r❡♠❛ ❞❡ ❡①✐stê♥❝✐❛ ❝❧áss✐❝♦ ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥tr♦❧❡s ót✐♠♦s ❧✐♥❡❛r❡s ❡ ✐♥tr♦❞✉③✐♠♦s✱ ❡ ❞❡♠♦♥str❛♠♦s✱ ✉♠ r❡s✉❧t❛❞♦ q✉❡ ❝♦♥s✐❞❡r❛♠♦s út✐❧ ♣❛r❛ ❞❡♠♦♥str❛r ❛ ❧✐♠✐t❛çã♦ ❞❡ tr❛❥❡tór✐❛s✱ s❡♥❞♦ ❡st❛ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❡①✐❣✐❞❛ ♥❡st❡ t❡♦r❡♠❛ ❞❡ ❡①✐stê♥❝✐❛✳
◆♦ ❈❛♣ít✉❧♦ ✺ ❛♣r❡s❡♥t❛♠♦s ✉♠ ♠♦❞❡❧♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞❡ ❝â♥❝❡r ❝♦♠ q✉✐♠✐♦t❡r❛♣✐❛✱ ❞❡✈✐❞♦ ❛ ❙t❡♣❛♥♦✈❛ ❬✶✼❪✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s
x′ =µCxF(x)−γxy−kXu ❡
✶✳ ■♥tr♦❞✉çã♦ ✶✾ ♦♥❞❡x r❡♣r❡s❡♥t❛ ❛ ♣r♦♣♦rçã♦ ❞❡ ❝é❧✉❧❛s t✉♠♦r❛✐s ♥♦ ✐♥❞✐✈í❞✉♦✱ ❡♥q✉❛♥t♦ y r❡♣r❡s❡♥t❛ ❛
♣r♦♣♦rçã♦ ❞❡ ❝é❧✉❧❛s ✐♠✉♥♦❝♦♠♣❡t❡♥t❡s ❞❡st❡ ❡ ur❡♣r❡s❡♥t❛ ✉♠❛ ❢✉♥çã♦ q✉❡ ♠❡❞❡ ❛ ❞♦s❡
❞❡ q✉✐♠✐♦t❡r❛♣✐❛ ❛ s❡r ❛♣❧✐❝❛❞❛✳ ❊♠ r❡❧❛çã♦ ❛ ❡st❡ ♠♦❞❡❧♦✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥tr♦❧❡ ót✐♠♦ q✉❡ ❝♦♥s✐st❡ ❡♠ ♠✐♥✐♠✐③❛r ♦ ❝✉st♦
J(u) = ax(T)−by(T) + T
0
cu(t) +d dt,
❝♦♠ u∈ L1([0, T],[0,1]) ❡ ❝♦♠ ❛ tr❛❥❡tór✐❛ (x(t), y(t))❞❡✜♥✐❞❛ ♣❡❧♦ P❱■ ❛❝✐♠❛✳ ❯t✐❧✐③❛✲ ♠♦s ❡♥tã♦ ♦s r❡s✉❧t❛❞♦s t❡ór✐❝♦s ❞♦ ❈❛♣ít✉❧♦ ✹ ♣❛r❛ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝♦♥tr♦❧❡ ót✐♠♦u(t) r❡❣✉❧❛r ❡ ❝♦♥tí♥✉♦ ♣♦r ♣❛rt❡s ♣❛r❛ ❡st❡ ♣r♦❜❧❡♠❛✳ P♦r ✜♠✱ ✉t✐❧✐③❛♠♦s ♦s ❛❧❣♦✲ r✐t♠♦s ❛❜♦r❞❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✹✱ ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ✉♠ ♣r♦❣r❛♠❛ ♥♦ s♦❢t✇❛r❡ ▼❛t❧❛❜ ♣❛r❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s ❝♦♥tr♦❧❡s ót✐♠♦s ❛ ♣❛rt✐r ❞♦s ♣♦ssí✈❡✐s ♣♦♥t♦s ✜♥❛✐s z(T)❡ ♣❛r❛ ❛♥❛❧✐s❛r ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✳ ❆tr❛✈és ❞❡ t❛✐s ❛❧❣♦r✐t♠♦s✱ ❞❡s❡♥✈♦❧✈❡♠♦s t❛♠❜é♠ ✉♠ ♣r♦❣r❛♠❛ ♣❛r❛ ♦❜t❡r ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞♦ ❝♦♥tr♦❧❡ ót✐♠♦ u(t) ❛ ♣❛rt✐r ❞❡ ✉♠ ❞❛❞♦ ♣♦♥t♦ ✐♥✐❝✐❛❧
z(t)✳
❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❛ ❛❜♦r❞❛❣❡♠ ❞❡st❡ tr❛❜❛❧❤♦ ❡ ❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r ▲❡❞③❡✇✐❝③ ❡t✳ ❛❧ ❡♠ ❬✼❪ é ❛ ❞❡ q✉❡ ❛q✉✐ ❝♦♥s❡❣✉✐♠♦s ✉t✐❧✐③❛r ❛ t❡♦r✐❛ ♣❛r❛ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ❝♦♥tr♦❧❡ ót✐♠♦ r❡❣✉❧❛r ❡ ❝♦♥tí♥✉♦ ♣♦r ♣❛rt❡s✱ ❛♥❛❧✐s❛r ❣r❛✜❝❛♠❡♥t❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥tr♦❧❡s ót✐♠♦s ❡ ♦❜t❡r♠♦s ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ✉♠ ❝♦♥tr♦❧❡ ót✐♠♦ ❛ ♣❛rt✐r ❞❡ ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ (x0, y0)
❞❛❞♦✳ P♦r s✉❛ ✈❡③✱ ❡♠ ❬✼❪ ❛ t❡♦r✐❛ ❢♦✐ ✉t✐❧✐③❛❞❛ ✉♥✐❝❛♠❡♥t❡ ♣❛r❛ ❛♥á❧✐s❡ ❣rá✜❝❛ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥tr♦❧❡s ót✐♠♦s ♦❜t✐❞♦s ♥✉♠❡r✐❝❛♠❡♥t❡✱ s❡♠ ❥✉st✐✜❝❛t✐✈❛s ❞♦s ❛❧❣♦r✐t♠♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ t❛❧ ❞❡t❡r♠✐♥❛çã♦✱ ♦✉ ♣r♦✈❛ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ t❛✐s ❝♦♥tr♦❧❡s ót✐♠♦s✳
❈❛♣ít✉❧♦
✷
Pr❡❧✐♠✐♥❛r❡s
❊st❡ ❝❛♣ít✉❧♦ ❝♦♥té♠ r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛✲ ❜❛❧❤♦✳
✷✳✶ ❚ó♣✐❝♦s ❞❡ ❆♥á❧✐s❡
❉✉r❛♥t❡ ♦ tr❛❜❛❧❤♦ ❛q✉✐ ❛♣r❡s❡♥t❛❞♦✱f(t)dt ✐♥❞✐❝❛rá ❛ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡✳
❉❡✜♥✐çã♦ ✶ ❉❛❞♦ D ⊂ Rn✱ ❞✐③❡♠♦s q✉❡ f : [a, b] → D é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♣♦r
♣❛rt❡s✱ q✉❛♥❞♦ ❡①✐st❡♠ t1✱ ✳✳✳✱ tm✱ t❛✐s q✉❡ f é ❝♦♥tí♥✉❛ ❡♠ (ti, ti+1) ❡ ♦s ❧✐♠✐t❡s ❧❛t❡r❛✐s
limt→ti+f(t) ❡ limt→ti+1−f(t) ❡①✐st❡♠✱ ♣❛r❛ ❝❛❞❛ 1 ≤ i < m✳ ◆❡st❡ ❝❛s♦ ❞❡♥♦t❛♠♦s
f ∈Cˆ([a, b], D)✳
❉❡✜♥✐çã♦ ✷ ❉❛❞❛ f : [a, b] → Rn✱ ❞✐r❡♠♦s q✉❡ f(t) é ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠
✉♠ ✐♥t❡r✈❛❧♦ [a, b]✱ ❝❛s♦ ❡①✐st❛ g : [a, b] → Rn✱ s❡♥❞♦ ▲❡❜❡s❣✉❡ ♠❡♥s✉rá✈❡❧ ❡♠ [a, b]✱ t❛❧
q✉❡
f(t) = f(a) + t
a
g(s)ds ∀t ∈[a, b].
❖s t❡♦r❡♠❛s s❡❣✉✐♥t❡s ♣♦❞❡♠ s❡r ✈✐st♦s ❝♦♠♦ ♦s ❛♥á❧♦❣♦s✱ ♣❛r❛ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡✱ ❞♦ Pr✐♠❡✐r♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦✱ ♣♦✐s tr❛t❛♠ ❞❛ ❞❡r✐✈❛çã♦ ❞❛ ✐♥t❡❣r❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦✳
❚❡♦r❡♠❛ ✶ ✭Pr✐♠❡✐r♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ♣❛r❛ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡✮ ❙❡❥❛♠ ❞❛❞❛s f : [a, b]→Rn ❡ g : [a, b]→Rn ❝♦♥tí♥✉❛✱ t❛✐s q✉❡
f(t) = f(a) + t
a
g(s)ds ∀t ∈[a, b].
❊♥tã♦✱ ♣❛r❛ ❝❛❞❛ t∈[a, b]✱ t❡♠♦s q✉❡ f′(t) ❡①✐st❡ ❝♦♠ f′(t) = g(t)
❚❡♦r❡♠❛ ✷ ✭❚❡♦r❡♠❛ ❞❡ ❉✐❢❡r❡♥❝✐❛çã♦ ❞❡ ▲❡❜❡s❣✉❡✮ ❙❡❥❛♠ f : [a, b] → Rn ❡ g : [a, b]→Rn ✉♠❛ ❢✉♥çã♦ ▲❡❜❡s❣✉❡ ✐♥t❡❣rá✈❡❧✱ t❛✐s q✉❡
f(t) = f(a) + t
a
g(s)ds ∀t ∈[a, b].
❊♥tã♦✱ q✉❛s❡ ❝❡rt❛♠❡♥t❡ ❡♠ [a, b]✱ t❡♠♦s q✉❡ f′(t) ❡①✐st❡ ❝♦♠ f′(t) =g(t)✳
✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✷ ❆ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r ♥♦s ❞á ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡st❡ t❡♦r❡♠❛✳
❉❡✜♥✐çã♦ ✸ ❉❛❞❛ ✉♠❛ ❢✉♥çã♦ g : [a, b]→R ✐♥t❡❣rá✈❡❧ ❡♠ [a, b]✱ ❞✐③❡♠♦s q✉❡ ✉♠ ♣♦♥t♦
p é r❡❣✉❧❛r ❞❡ g ❡♠ [a, b]✑ q✉❛♥❞♦ f(t) = t
ag(s)ds ❢♦r ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ r❡❧❛çã♦ ❛ t ♥♦ ♣♦♥t♦ p✱ ❝♦♠ f′(p) =g(p)✳
❆ss✐♠✱ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞✐r❡t❛ ❞♦ ❚❡♦r❡♠❛ ✷✱ t❡♠♦s ♦ ❝♦r♦❧❛r✐♦ ❛ s❡❣✉✐r✳
❈♦r♦❧ár✐♦ ✶ ❉❛❞❛ g : [a, b]→R ✐♥t❡❣rá✈❡❧ ❡♠ [a, b]✱ t❡♠♦s q✉❡ pé ♣♦♥t♦ r❡❣✉❧❛r ❞❡ g(t)
q✉❛s❡ ❝❡rt❛♠❡♥t❡ ❡♠ [a, b]✳
❖✉tr❛ ❛♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✷ é q✉❡✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❢✉♥çõ❡s ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛s
f : [a, b]→Rn tê♠ ❞❡r✐✈❛❞❛ f′(t)q✉❛s❡ ❝❡rt❛♠❡♥t❡ ❡♠ [a, b]✳
◗✉❛♥❞♦ ✉♠❛ ❢✉♥çã♦ f : [a, b]→Rn ♣♦ss✉✐r ❞❡r✐✈❛❞❛ f′(t)✱ q✉❛s❡ ❝❡rt❛♠❡♥t❡ ❡♠ [a, b]✱
❝♦♠♦ ❛s ❢✉♥çõ❡s ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛s✱ ❞❡✜♥✐r❡♠♦s ❛ ❢✉♥çã♦ f′ : [a, b]×Rn ❝♦♠♦ s❡♥❞♦ ❛ ❞❡r✐✈❛❞❛ ❞❡ f ❡♠ t✱ q✉❛♥❞♦ ❡st❛ ❡①✐st✐r✱ ❝❛s♦ ❝♦♥trár✐♦ s❡rá 0✳
❊♠ r❡❧❛çã♦ ❛ ✐st♦✱ t❡♠♦s ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✱ q✉❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ❛♥á❧♦❣♦✱ ♣❛r❛ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡✱ ❞♦ ❙❡❣✉♥❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦✱ ♣♦r tr❛t❛r ❞❛ ✐♥t❡✲ ❣r❛çã♦ ❞❛ ❞❡r✐✈❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦✳
❚❡♦r❡♠❛ ✸ ✭❙❡❣✉♥❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ♣❛r❛ ✐♥t❡❣r❛❧ ❞❡ ▲❡❜❡s❣✉❡✮ ❙❡♥❞♦f : [a, b]→Rn ✉♠❛ ❢✉♥çã♦ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ [a, b]✱ ❡♥tã♦f′(t)é ▲❡❜❡s❣✉❡
■♥t❡❣rá✈❡❧ ❡
f(t) = f(a) + t
a
f′(s)ds.
❖s t❡♦r❡♠❛s ❛♥t❡r✐♦r❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✶✸❪ ✭▼❝❙❤❛♥❡✱ ✶✾✹✼✱ ❚❡♦r✳ ✸✸✳✷✱ ❚❡♦r✳ ✸✸✳✸ ❡ ❚❡♦r✳ ✸✺✳✸✮✱ ♦✉ ❬✻❪ ✭❋♦❧❧❛♥❞✱ ✶✾✾✾✱ ❚❡♦r✳ ✸✳✷✶ ❡ ❚❡♦r✳ ✸✳✸✺✮✳
P♦r ✜♠✱ ✐♥tr♦❞✉③✐♠♦s ❛❜❛✐①♦ ❞✉❛s ❢♦r♠❛s ❞✐st✐♥t❛s ❞❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛✳ ❆ ♣r✐♠❡✐r❛ é ✉♠❛ ❢♦r♠❛ ❡❝♦♥ô♠✐❝❛ ❞❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛ ✉s✉❛❧ ♣❛r❛ ❢✉♥çõ❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❏á ❛ s❡❣✉♥❞❛ é ✉♠❛ ✈❡rsã♦ ❞❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛ ♣❛r❛ ❢✉♥çõ❡s ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛s✳
Pr♦♣♦s✐çã♦ ✶ ❙❡❥❛♠ f : Rn×Rm → R✱ x : [a, b] → Rn ❡ y : [a, b] → Rm ❢✉♥çõ❡s t❛✐s
q✉❡ x(t)❡ y(t)s❡❥❛♠ ❞✐❢❡r❡♥❝✐á✈❡✐s ♥✉♠ ♣♦♥t♦ t0 ∈[a, b]✱ t❛❧ q✉❡ fy(x(t0), y(t0))❡①✐st❛✱ ❡
fx(x, y) ❡①✐st❛ ♣❛r❛ t♦❞♦ x ❡ y ❡♠ [a, b] ❡ s❡❥❛ ❝♦♥tí♥✉❛ ♥♦ ♣♦♥t♦ (x(t0), y(t0))✳
❊♥tã♦ f(x(t), y(t)) é ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ♣♦♥t♦ t=t0 ❡ ✈❛❧❡ ❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛
f′(x(t0), y(t0)) =fx(x(t0), y(t0))x′(t0) +fy(x(t0), y(t0))y′(t0).
Pr♦♣♦s✐çã♦ ✷ ❉❛❞♦ D⊂Rn ❛❜❡rt♦✱ s❡❥❛♠ f :D→ R ❡ x : [a, b] →D ❢✉♥çõ❡s t❛✐s q✉❡
x(t) s❡❥❛ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ [a, b]✱ ❡ fx(x) ❡①✐st❛ ❡ s❡❥❛ ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠
D✳
❊♥tã♦ f(x(t)) é ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ [a, b] ❡ ✈❛❧❡ ❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛
f′(x(t)) = fx(x(t))x′(t).
✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✸
✷✳✷ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ♣❛r❛ ❢✉♥çõ❡s ❆❜s♦❧✉t❛♠❡♥t❡
❈♦♥tí♥✉❛s
◆❡st❛ s❡çã♦✱ tr❛t❛♠♦s ❞❛ t❡♦r✐❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥♦✐❛✐s ♣❛r❛ ❢✉♥çõ❡s ❛❜s♦❧✉t❛♠❡♥t❡ ♦♦♥tí♥✉❛s✳ P❛r❛ ❢❛♦✐❧✐t❛r ❛ ❡s♦r✐t❛✱ ❞❛❞♦ x: [a, b]→D✱ ♦♦♠ D ⊂Rn✱ ❞✐③❡r q✉❡ x s❛t✐s❢❛③
♦ P❱■ ❡♠ [a, b]
x′(t) =f(t, x(t)), x(τ) =ρ, ✭✷✳✶✮
s✐❣♥✐✜♦❛ q✉❡ x(t) é ❛❜s♦❧✉t❛♠❡♥t❡ ♦♦♥t✐♥✉❛ ❡♠ [a, b]✱ q✉❡ τ ∈ [a, b]✱ ρ ∈ D ❡ q✉❡ ❛
✐❣✉❛❧❞❛❞❡ ♥❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥♦✐❛❧ ❛♦✐♠❛ ♦♦♦rr❡ q✉❛s❡ ♦❡rt❛♠❡♥t❡ ❡♠ [a, b]✳ ❙❡❣✉♥❞♦ ♦s ❚❡♦r❡♠❛s ✷ ❡ ✸✱ ✐st♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r q✉❡
x(t) = ρ+ t
τ
f(s, x(s))ds ∀t∈[a, b], ♦♦♠ τ ∈[a, b] ❡ ρ∈D.
▲♦❣♦✱ ✉t✐❧✐③❛r❡♠♦s ❛♠❜❛s ❛s ❢♦r♠❛s ❞❡ ❡s♦r✐t❛ ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ♠❡s♠♦ ❢❛t♦✳
❖❜s❡r✈❛♠♦s q✉❡ ❛ ❡q✉❛çã♦ ✭✷✳✶✮ ♣♦❞❡ s❡r t❛♥t♦ ✉♠❛ ❡q✉❛çã♦ ❞❡ ♠❛tr✐③❡s q✉❛♥t♦ ❞❡ ✈❡t♦r❡s ❧✐♥❤❛s ♦✉ ✈❡t♦r❡s ♦♦❧✉♥❛s✳
P❛r❛ ♦s ♣ró①✐♠♦s t❡♦r❡♠❛s s❡rá út✐❧ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
❉❡✜♥✐çã♦ ✹ ❉❛❞♦ D ⊂ Rn✱ ❞✐r❡♠♦s q✉❡ f : D → Rn é ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ D
q✉❛♥❞♦✱ ♣❛r❛ ❝❛❞❛ y ∈ D✱ ❡①✐st✐r ✉♠ ❛❜❡rt♦ A ⊂ D✱ ❝♦♠ y ∈ A ❡ f ❧✐♠✐t❛❞♦ ❡♠ A✳
❆♥❛❧♦❣❛♠❡♥t❡✱ ❞❛❞♦ f : [a, b]×D → Rn✱ ❞✐r❡♠♦s q✉❡ f é ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✐❛♥❛ ❡♠ [a, b]×D ❡♠ r❡❧❛çã♦ ❛ x ❝❛s♦✱ ♣❛r❛ ❝❛❞❛ y ∈D✱ ❡①✐st✐r ✉♠ ❛❜❡rt♦ A ⊂D ❡ M > 0✱ ❝♦♠ y∈A✱ t❛❧ q✉❡ |f(t, x)−f(t, y)|< M(x−y) ∀t∈[a, b] ❡ ∀x, y ∈A✱
❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ♥♦s ❢❛❧❛ s♦❜r❡ ❛ ❡①✐stê♥♦✐❛ ❧♦♦❛❧ ❞❡ s♦❧✉çõ❡s ❞♦ P❱■ ✭✷✳✶✮✳
❚❡♦r❡♠❛ ✹ ✭❊①✐stê♥❝✐❛ ▲♦❝❛❧✮ ❉❛❞♦ D ❛❜❡rt♦ ❡♠ Rn✱ s❡❥❛ f : [a, b]×D → Rn ✉♠❛
❢✉♥çã♦ ❞❡ (t, x)∈[a, b]×Rn✱ t❛❧ q✉❡✿
• f(t, x) é ♠❡♥s✉rá✈❡❧ ❡♠ t ♣❛r❛ ❝❛❞❛ x∈D ✜①❛❞♦❀
• f(t, x) é ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ [a, b]×D✳
❊♥tã♦✱ ❡①✐st❡ [c, d]⊂[a, b]✱ τ ∈(c, d) ❡ x: [c, d]→D s❛t✐s❢❛③ ♦ P❱■ ❡♠ [c, d]
x′(t) = f(t, x(t)), x(τ) =ρ.
P♦r ✜♠✱ ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✱ ♥♦s ❢❛❧❛ s♦❜r❡ ❛ ✉♥✐♦✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s✳
❚❡♦r❡♠❛ ✺ ✭❯♥✐❝✐❞❛❞❡✮ ❉❛❞♦ D ❛❜❡rt♦ ❞❡ Rn✱ s❡❥❛ f : [a, b]×D → Rn ✉♠❛ ❢✉♥çã♦ ❞❡ (t, x)∈[a, b]×D✱ t❛❧ q✉❡✿
• f(t, x) é ♠❡♥s✉rá✈❡❧ ❡♠ t ♣❛r❛ ❝❛❞❛ x∈D ✜①❛❞♦❀
• f(t, x) é ❧♦❝❛❧♠❡♥t❡ ▲✐♣s❝❤✐t③✐❛♥❛ ❡♠ [a, b]×D ❡♠ r❡❧❛çã♦ ❛ x✳
❊♥tã♦✱ ❞❛❞♦ [c, d]⊂[a, b]✱ ❝❛s♦ ❡①✐st❛ ❛ s♦❧✉çã♦ x: [c, d]→D ❞♦ P❱■ ❡♠ [c, d]
x′(t) = f(t, x(t)), x(τ) =ρ,
✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✹ ❚❛✐s t❡♦r❡♠❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ♣♦r ❡①❡♠♣❧♦ ❡♠ ❬✶✶❪ ✭▲✉❦❡s✱ ✶✾✽✷✱ ❚❡♦r✳ ✾✳✷✳✶✱ ❚❡♦r✳ ✶✵✳✸✳✷ ❡ ❚❡♦r✳ ✾✳✷✳✸✮✳
◆♦t❡♠♦s q✉❡ ♦ ❚❡♦r❡♠❛ ✺ ❣❛r❛♥t❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✱ ρ ∈ Rn t❡r❡♠♦s ✭❝❛s♦ ❡①✐st❛✮✱ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ x❞♦ P❱■ ✭✷✳✶✮✳ P♦❞❡♠♦s ❡♥tã♦ ❞❡♥♦t❛r ❡st❛ s♦❧✉çã♦ ♣♦r x(t, ρ)✱ ♦✉ s❡❥❛✱ x❞❡♣❡♥❞❡ ❞❡ t ❡ ❞❡ s✉❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ ρ✳
❊♠ r❡❧❛çã♦ à ❝♦♥t✐♥✉✐❞❛❞❡ ❡ ❞✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞❡ x(t, ρ)❡♠ r❡❧❛çã♦ à ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧
ρt❡♠♦s ♦ ❚❡♦r❡♠❛ ✻ ❛❜❛✐①♦✱ q✉❡ t❛♠❜é♠ ♥♦s ❞✐③ s♦❜r❡ ❛ ❡①✐stê♥❝✐❛ ❣❧♦❜❛❧ ❞❡st❛s s♦❧✉çõ❡s✳
❚❡♦r❡♠❛ ✻ ❉❛❞♦ D ❛❜❡rt♦ ❡♠ Rn✱ s❡❥❛ f : [a, b]×D → Rn ✉♠❛ ❢✉♥çã♦ ❞❡ (t, x) ∈ [a, b]×Rn✱ t❛❧ q✉❡✿
• f(t, x) é ♠❡♥s✉rá✈❡❧ ❡♠ t ♣❛r❛ ❝❛❞❛ x∈D ✜①❛❞♦❀
• fx(t, x) ❡①✐st❡ ♣❛r❛ t♦❞♦ (t, x)∈[a, b]×D ❡ é ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ [a, b]×Rn✳ ❙✉♣♦♥❤❛ ❛✐♥❞❛ q✉❡ ❡①✐st❛ ✉♠❛ s♦❧✉çã♦ ψ : [a, b]×D→Rn ❞♦ P❱■ ❡♠ [a, b]
ψ′(t) = f(t, ψ(t)), ψ(τ) =ρ0.
❊♥tã♦✱ ❡①✐st❡ γ > 0 ❡ x: [a, b]×B(ρ0, γ) →Rn ❝♦♥tí♥✉❛ ❡♠ [a, b]×B(ρ0, γ) s❛t✐s❢❛✲
③❡♥❞♦ ♦ P❱■ ❡♠ [a, b]
x′(t, ρ) =f(t, x(t, ρ)), x(τ, ρ) = ρ.
❆❧é♠ ❞♦ ♠❛✐s✱ x(t, ρ) s❡rá ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ r❡❧❛çã♦ ❛ ρ ❡♠ ρ = ρ0✱ ❝♦♠
dx(t, ρ0)
dρ
✭♠❛tr✐③ ❏❛❝♦❜✐❛♥❛ n×n ❞❡ x(t, ρ) ❡♠ r❡❧❛çã♦ ❛ ρ✮ s❛t✐s❢❛③❡♥❞♦ ♦ P❱■ ❡♠ [a, b]
dx(t, ρ0)
dρ
′
=fx(t, x(t, ρ0))
dx(t, ρ0)
dρ ,
dx(τ, ρ0)
dρ =In.
❊st❡ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✶✸❪ ✭▼❝❙❤❛♥❡✱ ✶✾✹✼✱ ❙❡❝✳ ✻✾✳✹✮✳ ❯♠❛ ❞❡✲ ♠♦♥str❛çã♦ ❡ss❡♥❝✐❛❧♠❡♥t❡ ♠❛✐s ❝✉rt❛ ❞♦ ♠❡s♠♦ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ♣♦r ♥ós ♥♦ ❆♣ê♥❞✐❝❡ ❆✳
❖ ❝♦r♦❧ár✐♦ ❛ s❡❣✉✐r é ❛♣❡♥❛s ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡st❡ t❡♦r❡♠❛✳
❈♦r♦❧ár✐♦ ✷ ❙❡❥❛ f : [a, b]×Rn→Rn ✉♠❛ ❢✉♥çã♦ ❞❡ (t, x)∈[a, b]×Rn✱ t❛❧ q✉❡✿
• f(t, x) é ♠❡♥s✉rá✈❡❧ ❡♠ t ♣❛r❛ ❝❛❞❛ x∈Rn ✜①❛❞♦❀
• fx(t, x) ❡①✐st❡ ♣❛r❛ t♦❞♦ (t, x)∈[a, b]×D ❡ é ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ [a, b]×D✳ ❙❡❥❛ ❞❛❞❛ t❛♠❜é♠ ω : (c, d) → R ✉♠❛ ❢✉♥çã♦ ❞❡ ǫ ∈ (c, d) ⊂ R✱ ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠
ǫ0 ∈(c, d)✱ t❛❧ q✉❡ ❡①✐st❛ ✉♠❛ s♦❧✉çã♦ ψ : [a, b]×D →Rn ❞♦ P❱■ ❡♠ [a, b]
ψ′(t) =f(t, ψ(t)), ψ(τ) = ω(ǫ0).
❊♥tã♦✱ ❡①✐st❡ δ > 0 ❡ x : [a, b]×B(ǫ0, δ)→Rn✱ ❝♦♥tí♥✉❛ ❡♠ [a, b]×B(ǫ0, δ)✱ s❛t✐s❢❛✲
③❡♥❞♦ ♦ P❱■ ❡♠ [a, b]
x′(t, ǫ) = f(t, x(t, ǫ)), x(τ, ǫ) =ω(ǫ).
❆❧é♠ ❞♦ ♠❛✐s✱ x(t, ǫ) s❡rá ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ r❡❧❛çã♦ ❛ ǫ ❡♠ ǫ0 ♣❛r❛ t♦❞♦ t ∈[a, b] ❝♦♠
dx(t, ǫ0)
dǫ s❡♥❞♦ ❛ s♦❧✉çã♦ ❞♦ P❱■ ❡♠ [a, b]
dx(t, ǫ0)
dǫ
′
=fx(t, x(t, ǫ0))
dx(t, ǫ0)
dǫ ,
dx(τ, ǫ0)
dǫ =ω
′(ǫ
✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✺ ❉❡♠♦♥str❛çã♦✳ P❡❧♦ ❚❡♦r❡♠❛ ✻✱ t❡♠♦s q✉❡ ❡①✐st❡♠γ >0❡y: [a, b]×B(ω(eo), γ)→Rn✱ ❝♦♥tí♥✉❛✱ s❛t✐s❢❛③❡♥❞♦ ♦ P❱■
y(t, ρ) =ρ+ t
a
f(s, y(s, ρ))ds. ✭✷✳✷✮
▲♦❣♦✱ ❜❛st❛ t♦♠❛r♠♦s✱ ♣♦r ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ω ❡♠ ǫ0✱ δ > 0 t❛❧ q✉❡ ǫ ∈ B(ǫ0, δ)
✐♠♣❧✐q✉❡ ❡♠ω(ǫ)∈B(ω(ǫ0), γ)✳ ❚♦♠❛♠♦s ❡♥tã♦ x: [a, b]×B(ǫ0, δ) ❞❡✜♥✐❞❛ ♣♦r
x(t, ǫ) = y(t, ω(ǫ)).
P♦r ❝♦♠♣♦s✐çã♦✱ x(t, ǫ) s❡rá ❝♦♥t✐♥✉❛✱ ❡✱ ♣❡❧❛ ❡q✉❛çã♦ ✭✷✳✷✮✱ t❡r❡♠♦s q✉❡ x s❛t✐s❢❛③
x(t, ǫ) = ω(ǫ) + t
a
f(s, y(s, ω(ǫ)))ds=ω(ǫ) + t
a
f(s, x(s, ǫ))ds.
P♦r ✜♠✱ ❝♦♠♦ y(t, p) é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ r❡❧❛çã♦ ❛ ρ ❡♠ ρ =ω(ǫ0)✱ ✉t✐❧✐③❛♥❞♦ ❛ r❡❣r❛
❞❛ ❝❛❞❡✐❛ ✉s✉❛❧✱ t❡♠♦s q✉❡ x(t, ǫ) = y(t, ω(ǫ)) é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ r❡❧❛çã♦ ❛ ǫ ❡♠ ǫ= ǫ0 ❡
s✉❛ ❞✐❢❡r❡♥❝✐❛❧ s❛t✐s❢❛③
dx(t, ǫ0)
dǫ =
dy(t, ω(ǫ0))
dρ ω
′(ǫ
0)
=Inω′(ǫ0) +
t a
fx(s, y(s, ω(ǫ0)))
dy(s, ω(ǫ0))
dǫ ω
′(ǫ
0)ds
=ω′(ǫ0) +
t a
fx(s, x(s, ǫ0))
dx(s, ǫ0)
dǫ ds.
❆ss✐♠✱ ❛ ❞❡♠♦♥str❛çã♦ ❡stá ❝♦♠♣❧❡t❛✳
◆♦t❡♠♦s q✉❡ ♥❡st❡ ❝♦r♦❧ár✐♦ t❡♠♦s q✉❡ dx(s, ǫ0)
dǫ s❛t✐s❢❛③ ✉♠ P❱■ ❧✐♥❡❛r ❞❡ ❞✐♠❡♥sã♦ n✳ P❛r❛ r❡s♦❧✈❡r ❡st❡ t✐♣♦ ❞❡ ❡q✉❛çã♦ ❧✐♥❡❛r s❡rá ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✳
❚❡♦r❡♠❛ ✼ ❉❛❞♦ D ❛❜❡rt♦ ❞❡ R✱ s❡❥❛ A: [a, b]→Mn(D) ✉♠❛ ♠❛tr✐③ ❝♦♠ ❡♥tr❛❞❛s ❡♠
❢✉♥çã♦ ❞❡ t✱ ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ [a, b]✳
❊♥tã♦✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ W : [a, b]×[a, b]→Mn(D) ♣❛r❛ ♦ P❱■ ❡♠ [a, b]
dW(t, τ)
dt =A(t)W(t, τ), W(τ, τ) = In.
❆❧é♠ ❞♦ ♠❛✐s✱ W(t, τ)s❡rát❛♠❜é♠ ❛ ú♥✐❝❛ s♦❧✉çã♦ ❞♦ P❱■ ❡♠ [a, b]✭❛❣♦r❛ ❡♠ ❢✉♥çã♦ ❞❡ τ✮
dW(t, τ)
dτ =−W(t, τ)A(τ), W(τ, τ) =In.
❆ ♠❛tr✐③ W(t, τ) ❛❝✐♠❛ é ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ❡❧❡♠❡♥t❛r ❞❡ A(t)✱ ♣♦❞❡♥❞♦ s❡r ❞❡✲ ♥♦t❛❞❛ ♣♦r WA(t, τ)✳
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡st❡ t❡♦r❡♠❛ t❡♠♦s ♦ ❝♦r♦❧ár✐♦ ❛ s❡❣✉✐r✳
❈♦r♦❧ár✐♦ ✸ ❉❛❞♦D ❛❜❡rt♦ ❞❡ R✱ s❡❥❛A: [a, b]→Mn(D)✉♠❛ ♠❛tr✐③ ❝♦♠ ❡♥tr❛❞❛s ❡♠
❢✉♥çã♦ ❞❡ t✱ ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ [a, b]⊂Mn(D)✳
❊♥tã♦✱ ❞❛❞♦ (τ, ρ) ∈ [a, b]×D✱ ♦♥❞❡ ρ ∈ Rn é ✉♠ ✈❡t♦r ❝♦❧✉♥❛✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ x: [a, b]→D ♣❛r❛ ♦ P❱■
✷✳ Pr❡❧✐♠✐♥❛r❡s ✷✻ ◆♦ ❝❛s♦✱ x(t) é ❞❛❞❛ ♣♦r
x(t) = WA(t, τ)ρ.
❆❧é♠ ❞♦ ♠❛✐s✱ ❞❛❞♦ (τ, δ)∈[a, b]×D✱ ♦♥❞❡δ∈Rn é ✉♠ ✈❡t♦r ❧✐♥❤❛✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛
s♦❧✉çã♦ y: [a, b]→D ♣❛r❛ ♦ P❱■
y′(t) =−y(t)A(t), y(τ) = δ.
◆♦ ❝❛s♦✱ y(t) é ❞❛❞❛ ♣♦r
y(t) =δWA(τ, t).
❱❛❧❡ ♦❜s❡r✈❛r q✉❡ x(t) ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ✈❡t♦r ❝♦❧✉♥❛ ❞❡ ❞✐♠❡♥sã♦ n✱ ❡♥q✉❛♥t♦ y(t) ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ✈❡t♦r ❧✐♥❤❛✱ t❛♠❜é♠ ❞❡ ❞✐♠❡♥sã♦ n✳
❊st❡ t❡♦r❡♠❛ ♥♦s ✐♥❞✐❝❛ ❝♦♠♦ r❡s♦❧✈❡r ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s ❞❡ ✈❡t♦r❡s ❛tr❛✈és ❞❛ ♠❛tr✐③ ❡❧❡♠❡♥t❛r ❛♥t❡r✐♦r♠❡♥t❡ ❞❡✜♥✐❞❛✳
❈❛♣ít✉❧♦
✸
❈♦♥tr♦❧❡ Ót✐♠♦
✸✳✶ ❈♦♥tr♦❧❡ Ót✐♠♦ ❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ P♦♥tr②❛❣✐♥
❖s ♣r♦❜❧❡♠❛s ❞❡ ❝♦♥tr♦❧❡ ót✐♠♦ t✐✈❡r❛♠ ♦r✐❣❡♠ ♣♦r ✈♦❧t❛ ❞❡ ✶✾✻✺✱ ❛tr❛✈és ❞❛ t❡♦r✐❛ ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧✳ ❖s ♣r✐♥❝✐♣❛✐s ♣❡sq✉✐s❛❞♦r❡s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♦ ✐♥í❝✐♦ ❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ t❡♦r✐❛ ❢♦r❛♠ ❇❡❧❧♠❛♥ ❬✷❪ ❡ P♦♥tr②❛❣✐♥ ❬✶✹❪
❉❡✜♥✐r❡♠♦s✱ ❡♠ ❣❡r❛❧ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥tr♦❧❡ ót✐♠♦ ❝♦♠♦ ✉♠ ♣r♦❜❧❡♠❛ ❞❛ ❢♦r♠❛✿
Pr♦❜❧❡♠❛ ❜❡r❛❧ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
▼✐♥✐♠✐③❛r J(x, u) =φ(T, x(T)) +T
t0 L(t, x(t), u(t))dt
s✉❥❡✐t♦ ❛
u∈Uad([t0, T],Ω), Ω⊂Rm✱ ❝♦♠
x: [t0, T]→D, ♣❛r❛ D⊂Rn, s❛t✐s❢❛③❡♥❞♦ ♦ P❱■ ❡♠ [t0, T]
x′(t) = f(t, x(t), u(t)), x(t
0) =xt0
❆❝✐♠❛✱ ♦ t❡♠♣♦ ✜♥❛❧ T ♣♦❞❡ ❡st❛r ✜①♦ ♦✉ ❡st❛r ❧✐✈r❡ ✭❞❡✐①❛❞♦ ❛ s❡r ❞❡t❡r♠✐♥❛❞♦✮✳
❊st❛rá ✐♠♣❧í❝✐t♦ ♥❛ ❢♦r♠✉❧❛çã♦ ❛❝✐♠❛ q✉❡ L✱ f ❡ φ sã♦ ❢✉♥çõ❡s ❞❡ D×Ω s♦❜r❡ Rn✳
❆ ❢✉♥çã♦ u : [t0, T] → Ω r❡♣r❡s❡♥t❛ ♦ ❝♦♥tr♦❧❡ q✉❡ t❡♠♦s ❞♦ s✐st❡♠❛✳ ❆ ✈❛r✐á✈❡❧ x :
[t0, T] →D r❡♣r❡s❡♥t❛ ❛ tr❛❥❡tór✐❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛ ❝❛❞❛ ❝♦♥tr♦❧❡ u✳ ❏á Uad([t0, T],Ω) é
✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❢✉♥çõ❡s ❞❡ [t0, T] s♦❜r❡Ω⊂Rm✱ r❡♣r❡s❡♥t❛♥❞♦ ❛s r❡str✐çõ❡s ✐♠♣♦st❛s ❛♦
♥♦ss♦ ❝♦♥tr♦❧❡ u✱ ❛ q✉❛❧ t♦❞♦ ❝♦♥tr♦❧❡ ❛❞♠✐ssí✈❡❧ u❞❡✈❡ ♣❡rt❡♥❝❡r✳
P❛r❛ ✉♠ ❝♦♥tr♦❧❡ u ∈ Uad([t0, T],Ω)✱ ❝♦♠ ✉♠❛ tr❛❥❡tór✐❛ ❛❞♠✐ssí✈❡❧ x(t)✱ ❛♠❜♦s s❛✲
t✐s❢❛③❡♥❞♦ ❛s r❡str✐çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❛❝✐♠❛ ❡ ♠✐♥✐♠✐③❛♥❞♦ ♦ ❝✉st♦ J(x, u)✱ ❞❛♠♦s ♦ ♥♦♠❡ ❞❡ ❝♦♥tr♦❧❡ ót✐♠♦ ❞♦ ♣r♦❜❧❡♠❛✳
◆♦t❡♠♦s ❞❡s❞❡ ❥á q✉❡✱ ❝❛s♦ t❡♥❤❛♠♦s u(t) =w(t) q✉❛s❡ ❝❡rt❛♠❡♥t❡ ❡♠ [t0, T]✱ ❡♥tã♦
❛♠❜♦s ♣♦ss✉❡♠ ❛ ♠❡s♠❛ tr❛❥❡tór✐❛ x(t) ❡ ♠❡s♠♦ ❝✉st♦ J(x, u) =J(x, w)✱ ❞❡ ♠♦❞♦ q✉❡✱ s❡u(t) é ✉♠ ❝♦♥tr♦❧❡ ót✐♠♦ ❞♦ ♣r♦❜❧❡♠❛ ❛❝✐♠❛✱ ❡♥tã♦ w(t) t❛♠❜é♠ ♦ é✱ ❡ ✈✐❝❡✲✈❡rs❛✳
P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡✱ ♥❛ ❧✐t❡r❛t✉r❛ ❛t✉❛❧✱ é ❝♦♠✉♠ ✐♠♣♦r✲s❡✱ ❛❧é♠ ❞❛s ❝♦♥❞✐çõ❡s ❛❝✐♠❛✱ ❝♦♥❞✐çõ❡s ♣❛r❛ ♦ ♣♦♥t♦ ✜♥❛❧ x(T)✳ P♦r ❡①❡♠♣❧♦✱ ✜①❛r x(T) = w ♣❛r❛ ❛❧❣✉♠ w∈Rn ♦✉ t♦♠❛r ψ(x(T)) = 0✱ ❞❡ ❢♦r♠❛ ♠❛✐s ❣❡r❛❧✱ ♣❛r❛ ❛❧❣✉♠❛ ❢✉♥çã♦ ψ :Rn→Rs✱ s
♥❛t✉r❛❧✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ t❛✐s t✐♣♦s ❞❡ ♣r♦❜❧❡♠❛s✱ ❛ t❡♦r✐❛ ❡♥✈♦❧✈✐❞❛ ❡♠ s✉❛ ❛♥á❧✐s❡ s❡ t♦r♥❛ ❝♦♠♣❧❡①❛ ❡ ❡①t❡♥s❛✱ ❛❧é♠ ❞❡ q✉❡ t❛❧ ❣❡♥❡r❛❧✐❞❛❞❡ ♥ã♦ ♥♦s s❡rá út✐❧ ♥♦s ♠♦❞❡❧♦s ❛q✉✐ ❡st✉❞❛❞♦s✳ P❛r❛ r❡❢❡rê♥❝✐❛s ♥❡st❡ ❛ss✉♥t♦ ✈❡r✿ ❬✶✹❪ ✭P♦♥tr②❛❣✐♥✱ ✶✾✻✷✱ ❙❡❝✳ ✶✳✸✮✱ ❬✶✵❪ ✭▲❡✐tã♦✱ ✷✵✵✶✱ ❈❛♣✳ ✸✮ ❡ ❬✶✻❪ ✭❙❝❤❛tt❧❡r✱ ✷✵✶✷✱ ❙❡❝✳ ✷✳✷✮✳
❋♦❝❛r❡♠♦s ❞❛q✉✐ ❡♠ ❞✐❛♥t❡ ♥❛ ✈❡rsã♦ ❛✉tô♥♦♠❛ ❞♦ Pr♦❜❧❡♠❛ ●❡r❛❧✱ ❛♣r❡s❡♥t❛❞❛ ♥❛ s❡q✉ê♥❝✐❛✳
✸✳ ❈♦♥tr♦❧❡ Ót✐♠♦ ✷✽ Pr♦❜❧❡♠❛ ❜ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
▼✐♥✐♠✐③❛r J(x, u) = φ(x(T)) +T
0 L(x(t), u(t))dt
❝♦♠ T ❧✐✈r❡✱ s✉❥❡✐t♦ ❛ u∈L1([0, T],Ω), Ω = Ω
1× · · · ×Ωm ⊂Rm✱ ❝♦♠
x: [a, b]→D, ♣❛r❛ D⊂Rn✱ s❛t✐s❢❛③❡♥❞♦ ♦ P❱■ ❡♠ ❬✵✱❚❪
x′(t) =f(x(t), u(t)), x(0) =x
0
♦♥❞❡ ♦s ❝♦♥❥✉♥t♦s Ωi ⊂R, ❝♦♠ i= 1✱ ...✱ m✱ r❡♣r❡s❡♥t❛♠ ♦s ❝♦♥tr❛✲❞♦♠í♥✐♦s ❞❛ i✲és✐♠❛ ❝♦♦r❞❡♥❛❞❛ ❞♦ ❝♦♥tr♦❧❡ u✳ ❆ ❝♦♥❞✐çã♦ T ❧✐✈r❡ s✐❣♥✐✜❝❛ q✉❡ ❛ ♠✐♥✐♠✐③❛çã♦ ❛❝✐♠❛ ❞❡✈❡
♦❝♦rr❡r ♣❛r❛ ❝❛❞❛ T > 0✳ ❏á L1(A, B)✱ ♦♥❞❡ A ⊂ R ❡ B ⊂ Rn✱ ✐♥❞✐❝❛ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s f :A→B ▲❡❜❡s❣✉❡ ✐♥t❡❣rá✈❡✐s ❡♠ A✳
◆❛ ❧✐t❡r❛t✉r❛ ❛t✉❛❧✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ❡st❡ ♣r♦❜❧❡♠❛ ❝♦♠ r❡str✐çõ❡s ❞♦ ❝♦♥tr♦❧❡ ♥❛ ❢♦r♠❛ u ∈ Cˆ([0, T],Ω)✱ ❛♦ ✐♥✈és ❞❡ u ∈ L1([0, T],Ω)✳ ❙❡ ♣♦r ✉♠ ❧❛❞♦ ✐st♦ s✐♠♣❧✐✜❝❛
❡♠ ♠✉✐t♦ ❛ t❡♦r✐❛ ❡♥✈♦❧✈✐❞❛ ❡♠ s✉❛ ❛♥á❧✐s❡✱ ♣♦r ♦✉tr♦✱ ♥ã♦ sã♦ ❝♦♥❤❡❝✐❞♦s r❡s✉❧t❛❞♦s út❡✐s ❞❡ ❡①✐stê♥❝✐❛ ❝♦♠♣❛tí✈❡✐s ❝♦♠ ❡st❛ r❡str✐çã♦✳ P❛r❛ r❡❢❡rê♥❝✐❛s ♥❡st❡ ❛ss✉♥t♦ ✈❡r ❬✸❪ ✭❇♦❧t②❛♥s❦✐✱ ✶✾✾✵✱ ❙❡❝✳ ✶✳✽✮✱ ❬✶✾❪ ✭❚r♦✉t♠❛♥✱ ✶✾✽✸✱ ❈❛♣✳ ✶✵✮✳
❆ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ♦ ♣r✐♥❝í♣✐♦ ❞❡ P♦♥tr②❛❣✐♥ ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ✶✱ ♦ q✉❛❧ r❡♣r❡s❡♥t❛ ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ q✉❡ ✉♠ ❝♦♥tr♦❧❡ us❡❥❛ ót✐♠♦✳ ❊st❡ ♣r✐♥❝í♣✐♦ ❝♦♥st✐t✉✐ ✉♠❛
❞❛s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❝♦♥❞✐çõ❡s ♣❛r❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s ❝♦♥tr♦❧❡s ót✐♠♦s✳
❚❡♦r❡♠❛ ✽ ✭P♦♥tr②❛❣✐♥✮ ❉❛❞♦s D ❛❜❡rt♦ ❡♠ Rn ❡ Ω⊂Rm ❝♦♠♣❛❝t♦✱ s❡❥❛♠ ❞❛❞❛s ❛s
❢✉♥çõ❡s L:D×Ω→R✱ f :D×Ω→Rn ❡ φ :D →R t❛✐s q✉❡✿
• L(x, u) ❡ f(x, u) s❡❥❛♠ ♠❡♥s✉rá✈❡✐s ❡♠ Ω ♣❛r❛ ❝❛❞❛ x∈D❀
• Lx(x, u)✱ fx(x, u) ❡ φxx(x) ❡①✐st❛♠ ❡ s❡❥❛♠ ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛s ❡♠ s❡✉s ❞♦♠í♥✐♦s✳ ❙✉♣♦♥❤❛ ❛✐♥❞❛ q✉❡ (x, u)s❡❥❛ ✉♠ ♠í♥✐♠♦ ❞♦ Pr♦❜❧❡♠❛ ✶ ❡ q✉❡ ❛ ❢✉♥çã♦ H :D×Rn×
Ω→R ❡st❡❥❛ ❞❡✜♥✐❞❛ ♣♦r
H(x, λ, u) = L(x, u) +λf(x, u).
❊①✐st❡ ❡♥tã♦ ✉♠❛ ❢✉♥çã♦ λ:Rn→R✱ t❛❧ q✉❡ é s❛t✐s❢❡✐t♦ ♦ P❱■ ❡♠ [0, T]
λ′(t) = −Hx(x(t), λ(t), u(t)), λ(T) =φx(x(T)) ❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ♠✐♥✐♠❛❧✐❞❛❞❡
H(x(t), λ(t), u(t)) = min
v∈Ω H(x(t), λ(t), v) q✳❝✳ ❡♠ [0, T].
❆❞✐❝✐♦♥❛❧♠❡♥t❡✱
H(x(t), λ(t), u(t)) = 0 q✳❝✳ ❡♠ [0, T],
❝♦♠ ✐❣✉❛❧❞❛❞❡ ❣❛r❛♥t✐❞❛ ♣❛r❛ t✱ q✉❛♥❞♦ ♦❝♦rr❡r H(x(t), λ(t), u(t)) = min
v∈ΩH(x(t), λ(t), v)✳
◆♦ t❡♦r❡♠❛ ❛❝✐♠❛✱ ❝♦♥s✐❞❡r❡ ♦ ❝♦♥❥✉♥t♦ A ❞♦s ♣♦♥t♦s t∈[0, T] t❛✐s q✉❡
H(x(t), λ(t), u(t))= min
v∈Ω H(x(t), λ(t), v)
❡ H(x(t), λ(t), v) ❛❞♠✐t❛ ✉♠ ♠í♥✐♠♦ w ❡♠ Ω✳ P❡❧♦ t❡♦r❡♠❛ ❛❝✐♠❛✱ t❡♠♦s q✉❡ A t❡♠
♠❡❞✐❞❛ ♥✉❧❛✳
❉❡✜♥❛♠♦s ❡♥tã♦ ✉♠ ♥♦✈♦ ❝♦♥tr♦❧❡ u∗ : [0, T]→Ω✱ ❛ ♣❛rt✐r ❞❡ u(t)✱ t♦♠❛♥❞♦
u∗(t) =
u(t), ♣❛r❛ t∈Ac
✸✳ ❈♦♥tr♦❧❡ Ót✐♠♦ ✷✾ ❆ss✐♠✱ u∗(t) = u(t) q✉❛s❡ ❝❡rt❛♠❡♥t❡ ❡♠ [0, T]✱ ❡
H(x(t), λ(t), u∗(t)) = min
v∈Ω H(x(t), λ(t), v)
♣❛r❛ t♦❞♦ t t❛❧ q✉❡ H(x(t), λ(t), v) ❛❞♠✐t❛ ✉♠ ♠í♥✐♠♦ w∈Ω✳
◆♦t❡ ❛✐♥❞❛ q✉❡✱ ❞❛❞♦ u(t)r❡❣✉❧❛r✱ ❝❛s♦H(x(t), λ(t), u)s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ u✱ ❡♥tã♦ ❡st❛
s❡♠♣r❡ ❛❞♠✐t✐rá ✉♠ ♠í♥✐♠♦ w∈Ω✱ ❞❡ ♠♦❞♦ q✉❡ ✈❛❧❡
H(x(t), λ(t), u(t)) = min
v∈Ω H(x(t), λ(t), v) = 0 ∀t∈[0, T].
❊st❛ ♦❜s❡r✈❛çã♦ ♥♦s ❞✐③ q✉❡✱ ❝❛s♦ H(x(t), λ(t), u) s❡❥❛ ❝♦♥tí♥✉❛ ❡♠ u✱ ♣r❡❝✐s❛♠♦s
❛♣❡♥❛s ♣r♦❝✉r❛r ♣❡❧♦s ❝♦♥tr♦❧❡s ót✐♠♦s u(t) q✉❡ s❡♠♣r❡ ♠✐♥✐♠✐③❡♠ ♦ ❍❛♠✐❧t♦♥✐❛♥♦✱ ♣♦✐s t♦❞♦s ♦s ♦✉tr♦s ✈ã♦ ❞✐❢❡r✐r ❞❡st❡ ❡♠ ❛♣❡♥❛s ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♠❡❞✐❞❛ ♥✉❧❛✳
✸✳✷ Pr♦❜❧❡♠❛ ❞❡ ❈♦♥tr♦❧❡ Ót✐♠♦ ♥❛ ❢♦r♠❛ ▼❛②❡r
❈❤❛♠❛r❡♠♦s ❞❡ ✉♠ Pr♦❜❧❡♠❛ ❜❡ ❈♦♥tr♦❧❡ Ót✐♠♦ ♥❛ ❢♦r♠❛ ▼❛②❡r ✭❝♦♠T ❧✐✈r❡
❡ ♣♦♥t♦ ✜♥❛❧ ❧✐✈r❡✮ ❝♦♠♦ t♦❞♦ ♣r♦❜❧❡♠❛ ♥❛ ❢♦r♠❛
Pr♦❜❧❡♠❛ ❜ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
▼✐♥✐♠✐③❛r J(x, u) =xn(T) ❝♦♠ ❚ ❧✐✈r❡✱ s✉❥❡✐t♦ ❛
u∈L1[0, T],Ω), Ω = Ω
1×Ωm ⊂Rm✱ ❝♦♠
x: [a, b]→D, ♣❛r❛ D⊂Rn✱ s❛t✐s❢❛③❡♥❞♦ ♦ P❱■ ❡♠ ❬✵✱❚❪
x′(t) = f(x(s), u(s)), x(0) =x
0.
❆❝✐♠❛✱ xn r❡♣r❡s❡♥t❛ ❛ n✲és✐♠❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ x✱ s❡♥❞♦ q✉❡ xn t❛♠❜é♠ ❡stá ✐♥❝❧✉s♦ ♥❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞♦ P❱■ ❛❝✐♠❛✳
◆♦t❡♠♦s q✉❡ ❡st❡ Pr♦❜❧❡♠❛ é ❛♣❡♥❛s ✉♠ ❝❛s♦ ❡s♣❡❝í✜❝♦ ❞♦ Pr♦❜❧❡♠❛ ✶✳ ❱❡r❡♠♦s ❛❞✐❛♥t❡ t❛♠❜é♠ q✉❡✱ ❛♣❡s❛r ❞❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞♦ Pr♦❜❧❡♠❛ ♥❛ ❢♦r♠❛ ▼❛②❡r✱ t♦❞♦ ♣r♦❜❧❡♠❛ ♥❛ ❢♦r♠❛ ❞♦ Pr♦❜❧❡♠❛ ✶ ♣♦❞❡ s❡r ❝♦♥✈❡rt✐❞♦ ♥❛ ❢♦r♠❛ ❞♦ Pr♦❜❧❡♠❛ ✷✳
❆ s❡❣✉✐r t❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❡ P♦♥tr②❛❣✐♥ ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ✷✳
❚❡♦r❡♠❛ ✾ ✭P♦♥tr②❛❣✐♥✮ ❉❛❞♦sD❛❜❡rt♦ ❡♠Rn❡Ω⊂Rm❝♦♠♣❛❝t♦✱ s❡❥❛f :D×Ω→ Rn t❛❧ q✉❡✿
• f(x, u) é♠❡♥s✉rá✈❡❧ ❡♠ Ω ♣❛r❛ ❝❛❞❛ x∈D✱
• fx(x, u) ❡①✐st❛ ❡ s❡❥❛ ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛ ❡♠ D×Ω✳
❙✉♣♦♥❤❛ ❛✐♥❞❛ q✉❡ (x, u) s❡❥❛ ✉♠ ♠í♥✐♠♦ ❞♦ Pr♦❜❧❡♠❛ ✷ ❡ q✉❡ ❛ ❢✉♥çã♦ H :D×Rn× Ω→R ❡st❡❥❛ ❞❡✜♥✐❞❛ ♣♦r
H(x, λ, u) = λf(x, u).
❊①✐st❡ ❡♥tã♦ ✉♠❛ ❢✉♥çã♦ λ : [0, T]→Rn t❛❧ q✉❡ és❛t✐s❢❡✐t♦ ♦ P❱■ ❡♠ [0, T]
λ′(t) = −Hx(x(t), λ(t), u(t)), λ(T) = (0,0,· · · ,0,1) ❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ♠✐♥✐♠❛❧✐❞❛❞❡
H(x(t), λ(t), u(t)) = min
v∈ΩH(x(t), λ(t), v) q✳❝✳ ❡♠ [0, T].
❆❞✐❝✐♦♥❛❧♠❡♥t❡✱
H(x(t), λ(t), u(t)) = 0 q✳❝✳ ❡♠[0, T],
❝♦♠ ✐❣✉❛❧❞❛❞❡ ❣❛r❛♥t✐❞❛ ♣❛r❛ t✱ q✉❛♥❞♦ ♦❝♦rr❡r H(x(t), λ(t), u(t)) = min
✸✳ ❈♦♥tr♦❧❡ Ót✐♠♦ ✸✵ ◆♦t❡ q✉❡ ❡st❡ t❡♦r❡♠❛ é ❛♣❡♥❛s ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐r❡t❛ ❞♦ ❚❡♦r❡♠❛ ✽ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ♥❛ ❢♦r♠❛ ▼❛②❡r✳ ■♥t❡r❡ss❛♥t❡♠❡♥t❡✱ ❛ r❡❝í♣r♦❝❛ t❛♠❜é♠ é ✈❡r❞❛❞❡✐r❛✿
▲❡♠❛ ✶ ❖ ❚❡♦r❡♠❛ ✽ é ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ✾✳
❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ✈❡❥❛♠♦s ❝♦♠♦ tr❛♥s❢♦r♠❛r ♦ Pr♦❜❧❡♠❛ ✶ ♣❛r❛ ❛ ❢♦r♠❛ ▼❛②❡r✿ ❉❛❞♦s x✱ f✱ L ❡φ s❛t✐s❢❛③❡♥❞♦ ♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ❚❡♦r❡♠❛ ✽✱ ❝♦♥s✐❞❡r❡
g∗(x(t), u(t)) =L(x(t), u(t)) +φx(x(t))f(x(t), u(t)),
y∗(t) = φ(x0) +
T
0
g∗(x(t), u(t))dt
❡ ❞❡✜♥❛✿ g = (f1, ..., fn, g∗)✱ y = (x1, ..., xn, y∗) ❡ y0 = (x0, φ(x0))✳
❆ss✐♠✱ ♣❡❧❛s ❞❡✜♥✐çõ❡s ❛♥t❡r✐♦r❡s✱ t❡♠♦s q✉❡ y s❛t✐s❢❛③ ♦ P❱■
y(t)′ =g(y(t), u(t)), y(0) =y0. ✭✸✳✶✮
◆♦t❡ q✉❡✱ ♣♦r s✉❛ ❞❡✜♥✐çã♦✱ g(y(t), u(t)) ♥ã♦ ❞❡♣❡♥❞❡ r❡❛❧♠❡♥t❡ ❞❛ ✈❛r✐á✈❡❧ y∗✱ ❡
J(x, u) = φ(x(T)) + T
0
L(x(t), u(t))
=φ(x(0)) + T
0
φ′(x(t))dt+ T
0
L(x(t), u(t))dt
=φ(x0) +
T
0
(L(x(t), u(t)) +φx(x(T))f(x(t), u(t))) dt
=φ(x0) +
T
0
g∗(x(t), u(t))dt =y(T).
❖✉ s❡❥❛✱ ❡♥❝♦♥tr❛r ✉♠ ♠í♥✐♠♦ (x, u)❞♦ Pr♦❜❧❡♠❛ ✶ ❝♦rr❡s♣♦♥❞❡ ❛ ❡♥❝♦♥tr❛r ✉♠ ♠í✲ ♥✐♠♦(y, u)❞♦ Pr♦❜❧❡♠❛ ✷ ✭♣r♦❜❧❡♠❛ ♥❛ ❢♦r♠❛ ▼❛②❡r✮✱ ❝♦♠ y: [a, b]→D×Rs❛t✐s❢❛③❡♥❞♦
♦ P❱■ ✭✸✳✶✮✳
❆ss✐♠✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ❚❡♦r❡♠❛ ✾ ❛ ❡st❡ ♥♦✈♦ ♣r♦❜❧❡♠❛✱ ❞❡ ♠♦❞♦ q✉❡✱ s❡♥❞♦
W : (D × R) × Rn ×Ω → R ❞❡✜♥✐❞♦ ♣♦r W(y, µ, u) = µg(y, u) t❡r❡♠♦s q✉❡ ❡①✐st❡
µ:R→Rn+1 s❛t✐s❢❛③❡♥❞♦ ♦ P❱■
µ′(t) = −Wx(y(t), µ(t), u), µ(T) = (0,0,· · · ,0,1) ✭✸✳✷✮ ❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ♠✐♥✐♠❛❧✐❞❛❞❡
W(y(t), µ(t), u(t)) = min
v∈ΩW(y(t), µ(t), v) q✳❝✳ ❡♠ [0, T], ✭✸✳✸✮
❛❧é♠ ❞❡ q✉❡
W(y(t), µ(t), u(t)) = 0 q✳❝✳ ❡♠ [0, T].
Pr♦✈❛r❡♠♦s ❛❣♦r❛ ❛ ❝♦♥❞✐çã♦ ❞❡ ♠✐♥✐♠❛❧✐❞❛❞❡ ❞♦ ❍❛♠✐❧t♦♥✐❛♥♦✳ ✭✐✮ ❉❛ ❡q✉❛çã♦ ✭✸✳✷✮ s❡❣✉❡ q✉❡ µn+1(t) s❛t✐s❢❛③ ♦ P❱■ ❡♠ [0, T]
µ′n+1(t) =Hy∗(y(t), µ(t), u(t)), µ(t
0) = 1.
❉❡ ♠♦❞♦ q✉❡✱ s❡♥❞♦ Hy∗(y(t), µ(t), u(t)) = µgy∗(y(t), u(t)) = 0 ✭♣♦✐s g(y, u) ♥ã♦ ❞❡✲
♣❡♥❞❡ ❞❡ y∗✮ t❡♠♦s q✉❡ µ′
n+1(t) s❛t✐s❢❛③ ♦ P❱■ ❡♠ [0, T]
✸✳ ❈♦♥tr♦❧❡ Ót✐♠♦ ✸✶ ❖✉ s❡❥❛✱ µn+1(t) = 1 +
T
0 0dt= 1 ∀t ∈[0, T]✳
❉❡♥♦t❡ ❛❣♦r❛ ρ= (µ1, µ2, ..., µn)✳ ❊♥tã♦✱
W(y(t), µ(t), u(t)) =µ(t)g(y(t), u(t)) =µn+1(t)g∗(x(t), u(t)) +ρ(t)f(x(t), u(t))
=L(x(t), u(t)) +φx(x(t))f(x(t), u(t)) +ρ(t)f(x(t), u(t)) =L(x(t), u(t)) + (φx(x(t)) +ρ(t))f(x(t), u(t)).
❉❡✜♥❛λ(t) =φx(x(t)) +ρ(t)❡ s❡❥❛H(x, λ, u) =L(x, u) +λf(x, u)❝♦♠♦ ♥♦ ❡♥✉♥❝✐❛❞♦✳ ❙❡❣✉❡ ❡♥tã♦ q✉❡
H(x(t), λ(t), u(t)) =W(y(t), µ(t), u(t)), ✭✸✳✹✮
❞❡ ♦♥❞❡ s❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❡ ✭✸✳✸✮ q✉❡
H(x(t), λ(t), u(t)) = min
v∈ΩH(x(t), λ(t), v) q✳❝✳ ❡♠ [0, T]
❡ q✉❡
H(x(t), λ(t), u(t)) = 0 q✳❝✳ ❡♠ [0, T],
❝♦♠ ✐❣✉❛❧❞❛❞❡ ❣❛r❛♥t✐❞❛ ♣❛r❛ t q✉❛♥❞♦ ♦❝♦rr❡r H(x(t), λ(t), u(t)) = min
v∈ΩH(x(t), λ(t), v)✱
❝♦♠♦ q✉❡rí❛♠♦s✳
❆❣♦r❛ ♣r♦✈❛♠♦s ♦ P❱■ s❛t✐s❢❡✐t♦ ♣♦r λ(t)✳
✭✐✐✮ ❉❡gx(x(t), λ(t), u(t)) = (fx(x(t), u(t)), gx∗(x(t), λ(t), u(t))) ❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ρ✱ t❡♠♦s q✉❡ρ s❛t✐s❢❛③
ρ′(t) =−µ(t)gx(x(t), λ(t), u(t)) =−g∗x(x(t), u(t))−ρ(t)fx(x(t), u(t))
=−Lx(x(t), u(t))−φxx(x(t))f(x(t), u(t))−φx(x(t))fx(x(t), u(t))−ρ(t)fx(x(t), u(t)),
❡ρ(T) = 0✳
❆❧é♠ ❞♦ ♠❛✐s✱ ❝♦♠♦ φxx(x) ❡①✐st❡ ❡ é ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦ ♣♦r ❤✐♣ót❡s❡ ❡ x(t) é ❛❜✲ s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ [0, T]✱ s❡❣✉❡ ❞❛ ❘❡❣r❛ ❞❛ ❈❛❞❡✐❛ ✭Pr♦♣♦s✐çã♦ ✷✮ q✉❡ φx(x(t)) é ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ [0, T] ❝♦♠
φ′x(x(t)) =φxx(x(t))x′(t) = φxx(x(t))f(x(t), u(t)).
❆ss✐♠✱ ❞❡ λ(t) = φx(x(t)) +ρ(t)✱ t❡♠♦s q✉❡ λ(t)t❛♠❜é♠ é ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥tí♥✉❛✱ ❡
λ′(t) = φ′x(x(t)) +ρ′(t) =φxx(x(t))f(x(t), u(t)) +· · ·
...−Lx(x(t), u(t))−φxx(x(t))f(x(t), u(t))−φx(x(t))fx(x(t), u(t))−ρ(t)fx(x(t), u(t)) =−Lx(x(t), u(t))−(φx(x(t)) +ρ(t))fx(x(t), u(t))
=−Lx(x(t), u(t))−λ(t)fx(x(t), u(t)),
❝♦♠λ(T) = ρ(T) +φx(x(T)) =φx(x(T))✱ ♦✉ s❡❥❛✱ t❡♠♦s q✉❡ λ(T)❞❡ ❢❛t♦ s❛t✐s❢❛③ ♦ P❱■
λ′(T) =−Hx(x(t), λ(t), u(t)), λ(T) =φx(x(T)),
