P❖◆❚■❋❮❈■❆ ❯◆■❱❊❘❙■❉❆❉❊ ❈❆❚Ó▲■❈❆ ❉❖ ❘■❖ ●❘❆◆❉❊ ❉❖ ❙❯▲ ❋❆❈❯▲❉❆❉❊ ❉❊ ❊◆●❊◆❍❆❘■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊◆●❊◆❍❆❘■❆ ❊▲➱❚❘■❈❆
●❯■▲❍❊❘▼❊ ❆❘❆❯❏❖ P■▼❊◆❚❊▲
❈❖◆❚❘❖▲❊ ❘❖❇❯❙❚❖ P❖❘ ❘❊❆▲■▼❊◆❚❆➬➹❖ ▲■◆❊❆❘■❩❆◆❚❊ P❆❘❈■❆▲ ❉❊ ❇■❖❘❊❆❚❖❘❊❙ ❊▼ ▼❖❉❖ ❉❊ ❖P❊❘❆➬➹❖
✏❋❊❉✲❇❆❚❈❍✑
●❯■▲❍❊❘▼❊ ❆❘❆❯❏❖ P■▼❊◆❚❊▲
❈❖◆❚❘❖▲❊ ❘❖❇❯❙❚❖ P❖❘ ❘❊❆▲■▼❊◆❚❆➬➹❖ ▲■◆❊❆❘■❩❆◆❚❊ P❆❘❈■❆▲ ❉❊ ❇■❖❘❊❆❚❖❘❊❙ ❊▼ ▼❖❉❖ ❉❊ ❖P❊❘❆➬➹❖
✏❋❊❉✲❇❆❚❈❍✑
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♣❡❧♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❊♥❣❡♥❤❛r✐❛ ❊❧étr✐❝❛ ❞❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ❊♥❣❡♥❤❛r✐❛ ❞❛ P♦♥t✐❢í❝✐❛ ❯♥✐✈❡rs✐❞❛❞❡ ❈❛tó❧✐❝❛ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ❙✉❧✳
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ❋❡rr❡✐r❛ ❈♦✉t✐♥❤♦
❈❖◆❚❘❖▲❊ ❘❖❇❯❙❚❖ P❖❘ ❘❊❆▲■▼❊◆❚❆➬➹❖ ▲■◆❊❆❘■❩❆◆❚❊ P❆❘❈■❆▲ ❉❊ ❇■❖❘❊❆❚❖❘❊❙ ❊▼ ▼❖❉❖ ❉❊ ❖P❊❘❆➬➹❖
✏❋❊❉✲❇❆❚❈❍✑
●❯■▲❍❊❘▼❊ ❆❘❆❯❏❖ P■▼❊◆❚❊▲
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ ❡♠ ❊♥❣❡♥❤❛r✐❛ ❊❧étr✐❝❛✱ ❞❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ❊♥❣❡♥❤❛r✐❛ ❞❛ P♦♥t✐❢í❝✐❛ ❯♥✐✈❡rs✐❞❛❞❡ ❈❛tó❧✐❝❛ ❞♦ ❘✐♦ ●r❛♥❞❡ ❞♦ ❙✉❧✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ à ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❊♥❣❡♥❤❛r✐❛ ❊❧étr✐❝❛✳
Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ❋❡rr❡✐r❛ ❈♦✉t✐♥❤♦ ❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ❘✉❜❡♠ ❉✉tr❛ ❘✐❜❡✐r♦ ❋❛❣✉♥❞❡s
❈♦♦r❞❡♥❛❞♦r ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❊♥❣❡♥❤❛r✐❛ ❊❧étr✐❝❛ ❇❛♥❝❛ ❊①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳ ❉r✳ ❋❧á✈✐♦ ❆♥tô♥✐♦ ❇❡❝♦♥ ▲❡♠♦s Pr❡s✐❞❡♥t❡ ✕ P❯❈❘❙
Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ❋❡rr❡✐r❛ ❈♦✉t✐♥❤♦ ❖r✐❡♥t❛❞♦r ✕ ❯❋❙❈
❆❣r❛❞❡❝✐♠❡♥t♦s
▼❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s✿
• ❛♦ ♣r♦❢❡ss♦r ❞♦✉t♦r ❉❛♥✐❡❧ ❋❡rr❡✐r❛ ❈♦✉t✐♥❤♦✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❡ ✐♥❝❡♥t✐✈♦❀ • ❛♦s ❝♦❧❡❣❛s ❞♦ ▼❡str❛❞♦ ❡♠ ❊♥❣❡♥❤❛r✐❛ ❊❧étr✐❝❛ ❞❛ P❯❈❘❙❀
❘❡s✉♠♦
❊st❛ ❞✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛ ✉♠ ♠♦❞❡❧♦ ❣❡r❛❧ q✉❡ ❞❡s❝r❡✈❡ ❛ ❞✐♥â♠✐❝❛ ❞♦ ❝r❡s❝✐♠❡♥t♦ t❛♥t♦ ❞❛ ❜❛❝tér✐❛ ❊s❝❤❡r✐❝❤✐❛ ❈♦❧✐ q✉❛♥t♦ ❞❛ ❧❡✈❡❞✉r❛ ❙❛❝❝❤❛r♦♠②❝❡s ❈❡r❡✈✐s✐❛❡ q✉❛♥❞♦ ♣r♦❞✉③✐❞❛s ❡♠ ❜✐♦r❡❛t♦r❡s ♦♣❡r❛♥❞♦ ♥♦ ♠♦❞♦ ❞❡s❝♦♥tí♥✉♦ ❝♦♠ ❛❧✐♠❡♥t❛çã♦ ❝♦♥tr♦❧❛❞❛ ✭✏❢❡❞✲❜❛t❝❤✑✮✳ ❊♠ ❣❡r❛❧✱ ♣r♦❝✉r❛✲s❡ ♠❛①✐♠✐③❛r ❛ ♣r♦❞✉çã♦ ❞❛ ❜✐♦♠❛ss❛ ✭♠✐❝r♦♦r❣❛♥✐s✲ ♠♦s✮ ❛tr❛✈és ❞♦ ❝♦♥tr♦❧❡ ❞♦ s✉❜str❛t♦ ✭❛❧✐♠❡♥t♦✮ ✐♥❥❡t❛❞♦ ❛♦ ❜✐♦r❡❛t♦r✳ P❡❧♦ ♣r✐♥❝í♣✐♦ ❞♦ ❜♦tt❧❡✲♥❡❝❦✱ ❛ ♠á①✐♠❛ ♣r♦❞✉t✐✈✐❞❛❞❡ é ♦❜t✐❞❛ q✉❛♥❞♦ ♦ ♥í✈❡❧ ❞❡ s✉❜str❛t♦ é ♠❛♥t✐❞♦ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✈❛❧♦r ❝rít✐❝♦ q✉❡ ❞❡♣❡♥❞❡ ❞❡ ✈❛r✐á✈❡✐s ❜✐♦❧ó❣✐❝❛s ❞♦ ♣r♦❝❡ss♦ ✭q✉❡ ✈❛r✐❛♠ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✮ ❡ ❞❡ ❝❡rt♦s ♣❛râ♠❡tr♦s ❝♦♠ ❡❧❡✈❛❞♦ ❣r❛✉ ❞❡ ✐♥❝❡rt❡③❛✳ ❯♠❛ ❛❧t❡r♥❛✲ t✐✈❛ ❛ ❡st❛ ❛❜♦r❞❛❣❡♠ é ❛tr❛✈és ❞♦ ❝♦♥tr♦❧❡ ❞♦ ♣r♦❞✉t♦ s❡❝✉♥❞ár✐♦ ✭❛❝❡t❛t♦ ♥♦ ❝❛s♦ ❞❛ ✏❊✳ ❈♦❧✐✑ ♦✉ ❡t❛♥♦❧ ♥♦ ❝❛s♦ ❞❛ ✏❙✳❈❡r✈✐s✐❛❡✑✮ ♦ q✉❛❧ ❞❡✈❡ s❡r ♠❛♥t✐❞♦ ❡♠ ♥í✈❡✐s ♣ró①✐♠♦s ❛ ③❡r♦ ❡ ❞❡st❛ ❢♦r♠❛ t♦❞♦ ♦ s✉❜str❛t♦ é ✉t✐❧✐③❛❞♦ ♥❛ ♣r♦❞✉çã♦ ❞❡ ❜✐♦♠❛ss❛✳ ❆ ♣❛rt✐r ❞♦ ♠♦❞❡❧♦ ♥ã♦ ❧✐♥❡❛r ❞❛ ❞✐♥â♠✐❝❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞♦ ♠✐❝r♦♦r❣❛♥✐s♠♦✱ ♣r♦♣õ❡✲s❡ ♥❡st❛ ❞✐ss❡r✲ t❛çã♦ ✉♠❛ ❧❡✐ ❞❡ ❝♦♥tr♦❧❡ r♦❜✉st❛ ❜❛s❡❛❞❛ ♥❛ ✐❞❡✐❛ ❞❡ r❡❛❧✐♠❡♥t❛çã♦ ❧✐♥❡❛r✐③❛♥t❡ ♣❛r❝✐❛❧ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❡✈✐t❛r ❛ ♠❡❞✐çã♦ ❞❡ ✉♠ ❡❧❡✈❛❞♦ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s ❜✐♦❧ó❣✐❝❛s ❞❡ ❞✐❢í❝✐❧ ✐♥str✉♠❡♥t❛çã♦✳ P❛r❛ ♠❡❧❤♦r❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞✐♥â♠✐❝♦ ❞♦ s✐st❡♠❛✱ t❛♠❜é♠ é ♣r♦♣♦st♦ ✉♠ ♠❡❝❛♥✐s♠♦ ❞❡ ❛❞❛♣t❛çã♦ ♣❛r❛♠étr✐❝❛ ♣❛r❛ ❛ ❡st✐♠❛çã♦ ♦♥❧✐♥❡ ❞❛ t❛①❛ ❞❡ r❡❛çã♦ ❞❛ ♦①✐❞❛çã♦ ❞❛ ❣❧✐❝♦s❡✳ ❯t✐❧✐③❛♥❞♦ ❛ ❞❡s❝r✐çã♦ ❞❛s ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❛tr❛✈és ❞❛ ❛❜♦r❞❛❣❡♠ q✉❛s✐✲▲P❱ ❡ ❛ ❢♦r♠✉❧❛çã♦ ❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♣♦r ❞❡s✐❣✉❛❧❞❛❞❡s ♠❛tr✐❝✐❛✐s ❧✐✲ ♥❡❛r❡s ✭▲▼■s✮✱ ♣r♦❥❡t❛✲s❡ ❛ ❞✐♥â♠✐❝❛ ❧✐♥❡❛r ❧✐✈r❡✱ r❡s✉❧t❛♥t❡ ❞❛ r❡❛❧✐♠❡♥t❛çã♦ ❧✐♥❡❛r✐③❛♥t❡✱ ❞❡ ♠❛♥❡✐r❛ ❛ ❣❛r❛♥t✐r ❛ ❡st❛❜✐❧✐❞❛❞❡ r♦❜✉st❛ ✭❡♠ r❡❧❛çã♦ ❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ♥ã♦ ❝❛♥❝❡❧❛✲ ❞❛s ❡ ✈❛r✐❛çõ❡s ♣❛r❛♠étr✐❝❛s✮ ❡♠ ♠❛❧❤❛ ❢❡❝❤❛❞❛ ❡ t❛♠❜é♠ ✉♠ ❝❡rt♦ ❞❡s❡♠♣❡♥❤♦✳ P❛r❛ ✈❡r✐✜❝❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ♠❡t♦❞♦❧♦❣✐❛ ♣r♦♣♦st❛ sã♦ r❡❛❧✐③❛❞♦s ✈ár✐♦s t❡st❡s ❡♠ s✐♠✉✲ ❧❛çõ❡s ✉t✐❧✐③❛♥❞♦ ❛ ♣❧❛t❛❢♦r♠❛ M atlab/Simulinkr✱ ♦♥❞❡ ❡st✉❞❛✲s❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛
❆❜str❛❝t
❚❤✐s ❞✐ss❡rt❛t✐♦♥ ❛✐♠s t♦ ✉s❡ ❛ ❣❡♥❡r❛❧ ♠♦❞❡❧ t♦ ❞❡s❝r✐❜❡ t❤❡ ❣r♦✇t❤ ♦❢ ❜♦t❤ ❜❛❝t❡r✐❛ ✏❊s❝❤❡r✐❝❤✐❛ ❈♦❧✐✑ ❛s ②❡❛st ✏❙❛❝❝❤❛r♦♠②❝❡s ❈❡r❡✈✐s✐❛❡✑ ✐♥ ❛ ❜✐♦r❡❛❝t♦r ♦♥ ❢❡❞✲❜❛t❝❤ ♠♦❞❡✳ ■♥ ❣❡♥❡r❛❧✱ t♦ ♠❛①✐♠✐③❡ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ ❜✐♦♠❛ss ✭♠✐❝r♦♦r❣❛♥✐s♠s✮ ❜② ❝♦♥tr♦❧❧✐♥❣ t❤❡ s✉❜str❛t❡ ✭❢♦♦❞✮ ✐♥❥❡❝t❡❞ ✐♥t♦ t❤❡ ❜✐♦r❡❛❝t♦r✳ ❇② t❤❡ ♣r✐♥❝✐♣❧❡ ♦❢ ❜♦tt❧❡✲♥❡❝❦✱ t❤❡ ♠❛①✐♠✉♠ ②✐❡❧❞ ✐s ♦❜t❛✐♥❡❞ ✇❤❡♥ t❤❡ s✉❜str❛t❡ ❧❡✈❡❧ ✐s ♠❛✐♥t❛✐♥❡❞ ❛t ❛ ❝❡rt❛✐♥ ❝r✐t✐❝❛❧ ✈❛❧✉❡ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ ❜✐♦❧♦❣✐❝❛❧ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ ♣r♦❝❡ss ✭✇❤✐❝❤ ✈❛r② ✐♥ t✐♠❡✮ ❛♥❞ ❝❡rt❛✐♥ ♣❛r❛♠❡t❡rs ✇✐t❤ ❤✐❣❤ ❞❡❣r❡❡ ♦❢ ✉♥❝❡rt❛✐♥t②✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤ ✐s t❤❡ ❝♦♥tr♦❧ ♦❢ t❤❡ ❜②✲♣r♦❞✉❝t ✭❛❝❡t❛t❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ ✏❊✳❈♦❧✐✑ ♦r ❡t❤❛♥♦❧ ✐♥ t❤❡ ❝❛s❡ ♦❢ ✏❙✳❈❡r✈✐s✐❛❡✑✮ ✇❤✐❝❤ s❤♦✉❧❞ ❜❡ ♠❛✐♥t❛✐♥❡❞ ❛t ❧❡✈❡❧s ❝❧♦s❡ t♦ ③❡r♦ ❛♥❞ t❤✉s ❡♥t✐r❡ s✉❜str❛t❡ ✐s ✉s❡❞ ✐♥ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ ❜✐♦♠❛ss✳ ❚❤r♦✉❣❤ t❤❡ ♥♦♥❧✐♥❡❛r ♠♦❞❡❧ ♦❢ t❤❡ ❞②♥❛♠✐❝ ❣r♦✇t❤ ♦❢ t❤❡ ♠✐❝r♦♦r❣❛♥✐s♠✱ ✐t ✐s ♣r♦♣♦s❡❞ ✐♥ t❤✐s ❞✐ss❡rt❛t✐♦♥ ❛ r♦❜✉st ❝♦♥tr♦❧ ❧❛✇ ❜❛s❡❞ ♦♥ t❤❡ ✐❞❡❛ ♣❛rt✐❛❧ ❢❡❡❞❜❛❝❦ ❧✐♥❡❛r✐③❛t✐♦♥✱ ✐♥ ♦r❞❡r t♦ ❛✈♦✐❞ ♠❡❛s✉r❡ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❜✐♦❧♦❣✐❝❛❧ ✈❛r✐❛❜❧❡s ❞✐✣❝✉❧t ✐♥str✉♠❡♥t❛t✐♦♥✳ ❚♦ ✐♠♣r♦✈❡ t❤❡ ❞②♥❛♠✐❝ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ s②st❡♠✱ ✐s ❛❧s♦ ♣r♦♣♦s❡❞ ❛ ♠❡❝❤❛♥✐s♠ ❢♦r ♦♥❧✐♥❡ ❡st✐♠❛t✐♦♥ ❛♥❞ ♣❛r❛♠❡tr✐❝ ❛❞❛♣t❛t✐♦♥ ♦❢ t❤❡ r❡❛❝t✐♦♥ r❛t❡ ♦❢ ❣❧✉❝♦s❡ ♦①✐❞❛t✐♦♥✳ ❯s✐♥❣ t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ♥♦♥❧✐♥❡❛r✐t✐❡s ✇✐t❤ t❤❡ q✉❛s✐✲▲P❱ ❛♣♣r♦❛❝❤ ❛♥❞ t❤❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ st❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s t❤r♦✉❣❤ ❧✐♥❡❛r ♠❛tr✐① ✐♥❡q✉❛❧✐t✐❡s ✭▲▼■s✮✱ ✐s ❞❡s✐❣♥❡❞ ❛ ❢r❡❡ ❧✐♥❡❛r ❞②♥❛♠✐❝✱ ②✐❡❧❞ ♦❢ ❢❡❡❞❜❛❝❦ ❧✐♥❡❛r✐③❛t✐♦♥✱ t♦ ❡♥s✉r❡ t❤❡ r♦❜✉st st❛❜✐❧✐t② ✭r❡❧❛t❡❞ t♦ ♥♦♥❧✐♥❡❛r✐t✐❡s ♥♦t ❝❛♥❝❡❧❡❞ ❛♥❞ ♣❛r❛♠❡tr✐❝ ✈❛r✐❛t✐♦♥s✮✐♥ ❝❧♦s❡❞ ❧♦♦♣ ❛♥❞ ❛❧s♦ ❛ ❝❡rt❛✐♥ ♣❡r❢♦r♠❛♥❝❡✳ ❚♦ ✈❡r✐❢② t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞♦❧♦❣② ❛r❡ ❝♦♥❞✉❝t❡❞ s❡✈❡r❛❧ t❡sts ♦♥ s✐♠✉❧❛t✐♦♥s ✉s✐♥❣ t❤❡ ♣❧❛t❢♦r♠ M atlab/Simulinkr✱ ✇❤❡r❡ ✐s ♣♦ss✐❜❧❡ t♦ st✉❞②
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
▲✐st❛ ❞❡ ❙í♠❜♦❧♦s ❡ ❙✐❣❧❛s
✶ ■♥tr♦❞✉çã♦ ✶✻
✶✳✶ ❊str✉t✉r❛ ❞❛ ❉✐ss❡rt❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✷✵
✷✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❙✐st❡♠❛s ◆ã♦✲❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✶ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❙✐st❡♠❛s ◆ã♦✲▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷✳✷ ❚❡♦r✐❛ ❞❛ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ▲②❛♣✉♥♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸ ❙✐st❡♠❛s ■♥❝❡rt♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✸✳✶ ❉❡s❝r✐çã♦ ❞❡ ■♥❝❡rt❡③❛s P❛r❛♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸✳✷ ❘❡♣r❡s❡♥t❛çã♦ ▲P❱ ❡ q✉❛s✐✲▲P❱ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✹ ❈♦♥tr♦❧❡ r♦❜✉st♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹✳✶ ❉❡s✐❣✉❛❧❞❛❞❡s ▼❛tr✐❝✐❛✐s ▲✐♥❡❛r❡s ✕ ▲▼■s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✺ ❘❡❛❧✐♠❡♥t❛çã♦ ▲✐♥❡❛r✐③❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✻ ❙í♥t❡s❡ ❞♦ ❈❛♣ít✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✸ ❇✐♦r❡❛t♦r❡s ✹✷
✸✳✷ ❈❛r❛❝t❡r✐③❛çã♦ ❞♦s ❇✐♦r❡❛t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✷✳✶ ❇✐♦r❡❛t♦r ❡♠ ♠♦❞♦ ❞❡s❝♦♥tí♥✉♦ ❝♦♠ ❛❧✐♠❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✸ ▼♦❞❡❧❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✸✳✶ ❈♦♥❝❡✐t♦s ❞❛ ▼♦❞❡❧❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✸✳✸✳✷ ▼♦❞❡❧❛♥❞♦ ❛ ❞✐♥â♠✐❝❛ ❞❡ ✉♠ ❇✐♦r❡❛t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✸✳✸ ❈❛r❛❝t❡r✐③❛♥❞♦ ❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❡s♣❡❝í✜❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✸✳✹ ❈♦♥st❛♥t❡s Ps❡✉❞♦✲❊st❡q✉✐♦♠étr✐❝❛s ✭ki✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✸✳✸✳✺ ❘❡❛çõ❡s ❡①✐st❡♥t❡s ♥❛ ❝✉❧t✉r❛ ❞❡ ♠✐❝r♦♦r❣❛♥✐s♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✸✳✻ ❉✐✜❝✉❧❞❛❞❡s ♥❛ ♦❜t❡♥çã♦ ❞♦ ♠♦❞❡❧♦ ❞✐♥â♠✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✹ ❖ ♣r✐♥❝í♣✐♦ ❞♦ ❜♦tt❧❡✲♥❡❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✸✳✺ ❙í♥t❡s❡ ❞♦ ❈❛♣ít✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷
✹ Pr♦❥❡t♦ ❞♦ ❈♦♥tr♦❧❡ ❡ ❘❡s✉❧t❛❞♦s ✻✸
✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✹✳✷ ❘❡♣r❡s❡♥t❛çõ❡s ♣♦r ❱❛r✐á✈❡✐s ❞❡ ❊st❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✹✳✸ P❛râ♠❡tr♦s ❞❡ ❙✐♠✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✹ ❉✐❛❣r❛♠❛s ❞❡ ❙✐♠✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✹✳✺ ▲❡✐ ❞❡ ❈♦♥tr♦❧❡ ▲✐♥❡❛r✐③❛♥t❡ P❛r❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✹✳✺✳✶ ❈♦♥tr♦❧❛❞♦r ❘♦❜✉st♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✹✳✺✳✷ ❆♥á❧✐s❡ ❞❛s ❙✐♠✉❧❛çõ❡s ❞♦ ❈♦♥tr♦❧❡ ❘♦❜✉st♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✹✳✻ ❊st✐♠❛çã♦ ❞♦ P❛râ♠❡tr♦ ❱❛r✐❛♥t❡ ♥♦ ❚❡♠♣♦ σ(t) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺
✹✳✻✳✶ ❆♥á❧✐s❡ ❞❛s ❙✐♠✉❧❛çõ❡s ❝♦♠ ˇr1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼
✹✳✼ ❙í♥t❡s❡ ❞♦ ❈❛♣ít✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾
✺ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✽✵
✺✳✶ P❡rs♣❡❝t✐✈❛ ❞❡ tr❛❜❛❧❤♦s ❢✉t✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷
❆♣ê♥❞✐❝❡ ❆ ✲✲ ❈r♦♥♦❧♦❣✐❛ ❞❛ ❇✐♦t❡❝♥♦❧♦❣✐❛ ✽✼
❆♣ê♥❞✐❝❡ ❇ ✲✲ ❈ó❞✐❣♦ ❋♦♥t❡ ▼❛t❧❛❜✿ ❈á❧❝✉❧♦ ❛tr❛✈és ❞❡ ▲▼■s ❞❡ λ ✽✾
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶ ❊st❛❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷ ❘❡♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ♣♦❧✐t♦♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸ ❙✐st❡♠❛ ❝♦♠ ♣❡rt✉r❜❛çõ❡s ❡①t❡r♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹ ❉✐❛❣r❛♠❛ ❞❡ ❜❧♦❝♦s ❞♦ ❝♦♥tr♦❧❡ ❧✐♥❡❛r✐③❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✺ ❊sq✉❡♠❛ ❞❡ ✉♠ ❇✐♦r❡❛t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✻ ▼♦❞♦s ❞❡ ♦♣❡r❛çõ❡s ❞❡ ✉♠ ❇✐♦r❡❛t♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✼ ❆❞✐çã♦ ❞❡ s✉❜str❛t♦ ❡♠ ❞❡❣r❛✉ ❡ ❝♦♥tí♥✉♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✽ Pr✐♥❝í♣✐♦ ❜♦tt❧❡✲♥❡❝❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✾ ▼♦❞❡❧♦s ❞❡ ❙✐♠✉❧❛çã♦ ♥❛ P❧❛t❛❢♦r♠❛ ▼❛t❧❛❜✴❙✐♠✉❧✐♥❦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✶✵ ❈r❡s❝✐♠❡♥t♦ ❞❛ ❇✐♦♠❛ss❛ ❡ ❞♦ Pr♦❞✉t♦ ✜♥❛❧ ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✶✶ ❙✳❈❡r❡✈✐s✐❛❡ ✲ ❈r❡s❝✐♠❡♥t♦ ❞❛ ❇✐♦♠❛ss❛ ❡ ❞♦ Pr♦❞✉t♦ ✜♥❛❧ ❝♦♠ ❝♦♥tr♦❧❡
❧✐♥❡❛r✐③❛♥t❡ r♦❜✉st♦ ❝♦♠ r¯1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻
✶✷ ▼♦❞❡❧♦ ❞♦ ❙✐♠✉❧✐♥❦ ♣❛r❛ ❡q✉❛çã♦ rˇ1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼
✶✸ ❉✐♥â♠✐❝❛s ❞❡r1 ❡ rˇ1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
▲✐st❛ ❞❡ ❙í♠❜♦❧♦s ❡ ❙✐❣❧❛s
A′ ✕ ♠❛tr✐③ A tr❛♥s♣♦st❛
ka(t)k2,[0,T] ✕ ♥♦r♠❛✲2 ❞❡a(t)♥♦ ❤♦r✐③♦♥t❡ ❞❡ t❡♠♣♦ ❡♥tr❡ 0 ❡ T
¯
a ✕ ✈❛❧♦r ♠é❞✐♦ ✭♦✉ ♥♦♠✐♥❛❧✮ ❞❡ a
˙
V(x) ✕ ❞❡r✐✈❛❞❛ ♣r✐♠❡✐r❛ ❞❡ V(x)
A′ =A ✕ ♠❛tr✐③ s✐♠étr✐❝❛
A >0 ✕ ♠❛tr✐③ A é s✐♠étr✐❝❛ ❡ ❞❡✜♥✐❞❛ ♣♦s✐t✐✈❛ C ✕ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ❞✐ó①✐❞♦ ❞❡ ❝❛r❜♦♥♦
Csat ✕ s❛t✉r❛çã♦ ❞✐ó①✐❞♦ ❞❡ ❝❛r❜♦♥♦
CO2 ✕ ❞✐ó①✐❞♦ ❞❡ ❝❛r❜♦♥♦
D ✕ ❞✐❧✉✐çã♦ ❞♦ s✉❜str❛t♦ ❞❡ ❡♥tr❛❞❛✱ ✐❣✉❛❧ ❛ Fin/υ
δ ✕ ✐♥❝❡rt❡③❛ ♣❛r❛♠étr✐❝❛
δ ✕ ✈❛❧♦r ♠á①✐♠♦ ❞❛ ✐♥❝❡rt❡③❛δ δ ✕ ✈❛❧♦r ♠í♥✐♠♦ ❞❡δ
∆ ✕ ♣♦❧✐t♦♣♦ ❛❞♠✐ssí✈❡❧ ❞❡ ✐♥❝❡rt❡③❛s
Fin ✕ ✈❛③ã♦ ❞❡ ❡♥tr❛❞❛ ♥♦ ❜✐♦r❡❛t♦r
H∞ ✕ ♥♦r♠❛ ❍✲✐♥✜♥✐t♦ ❞❡ s✐st❡♠❛s KiP ✕ ❝♦♥st❛♥t❡ ❞❡ ✐♥✐❜✐çã♦
koa ✕ r❡❧❛çã♦ ❡♥tr❡ ♣r♦❞✉t♦ ✜♥❛❧ ❡ ♦ ❝♦♥s✉♠♦ ❞❡ ♦①✐❣ê♥✐♦
kos ✕ r❡❧❛çã♦ ❡♥tr❡ ♦ ❝♦♥s✉♠♦ ❞❡ ♦①✐❣ê♥✐♦ ❡ s✉❜str❛t♦
Ki ✕ ❝♦♥st❛♥t❡ ❞❡ s❛t✉r❛çã♦ ❞❡ i ♣❛r❛ i=X, S, O, P✳
kLa ✕ ❝♦❡✜❝✐❡♥t❡ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ✈♦❧✉♠étr✐❝❛
▲❋❘ ✕ r❡♣r❡s❡♥t❛çã♦ ❡♠ ❢r❛çõ❡s ❧✐♥❡❛r❡s
▲▼■ ✕ ❞❡s✐❣✉❛❧❞❛❞❡ ♠❛tr✐❝✐❛❧ ❧✐♥❡❛r ✭♦✉ ❧✐♥❡❛r ♠❛tr✐① ✐♥❡q✉❛❧✐t②✮ ▲P❱ ✕ s✐st❡♠❛s ❧✐♥❡❛r ❝♦♠ ♣❛râ♠❡tr♦s ✈❛r✐❛♥t❡s ♥♦ t❡♠♣♦ ▲❚■ ✕ s✐st❡♠❛s ❧✐♥❡❛r ✐♥✈❛r✐❛♥t❡ ♥♦ t❡♠♣♦
µi ✕ ✈❛❧♦r❡s ♠á①✐♠♦s ❞❛ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❡s♣❡❝í✜❝♦
O ✕ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ❖①✐❣ê♥✐♦ Osat ✕ s❛t✉r❛çã♦ ♦①✐❣ê♥✐♦
O2 ✕ ♦①✐❣ê♥✐♦
P ✕ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ♣r♦❞✉t♦ s❡❝✉♥❞ár✐♦ ✭❜②✲♣r♦❞✉❝t✮
Qgas ✕ ✈❛③ã♦ ❞❡ s❛í❞❛ ❞❛ ♠❛ss❛ ❞♦ ❝♦♠♣♦♥❡♥t❡ξi ❞❛ r❡❛çã♦ ❡♠ ❢♦r♠❛ ❣❛s♦s❛
q✉❛s✐✲▲P❱ ✕ s✐st❡♠❛ ❧✐♥❡❛r ❝♦♠ ♣❛râ♠❡tr♦s ❞❡♣❡♥❞❡♥t❡s ❞♦s ❡st❛❞♦s
ˇ
r1 ✕ ✈❛❧♦r ❡st✐♠❛❞♦ ❞❛ ✈❛r✐á✈❡❧ r1
Rn ✕ ❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s r❡❛✐s ❞❡ ❞✐♠❡♥sã♦n
rScrit ✕ ❝♦♥s✉♠♦ ❝rít✐❝♦ ❞♦ s✉❜str❛t♦
r1 ✕ t❛①❛ ❡s♣❡❝í✜❝❛ ❞❛ r❡❛çã♦ ❞❡ ♦①✐❞❛çã♦ ❞❛ ❣❧✐❝♦s❡
r2 ✕ t❛①❛ ❡s♣❡❝í✜❝❛ ❞❛ r❡❛çã♦ ❞❛ ❢❡r♠❡♥t❛çã♦ ❞❛ ❣❧✐❝♦s❡
r3 ✕ t❛①❛ ❡s♣❡❝í✜❝❛ ❞❛ r❡❛çã♦ ❞❡ ♦①✐❞❛çã♦ ❞♦ ♣r♦❞✉t♦ s❡❝✉♥❞ár✐♦
rO ✕ t❛①❛ ❞❡ ♦①✐❞❛çã♦ ❞♦ s✉❜str❛t♦ ❡♠ ❡①❝❡ss♦ ✭❢❡r♠❡♥t❛çã♦✮
rP ✕ t❛①❛ ❞❡ ♦①✐❞❛çã♦ ❞♦ ♣r♦❞✉t♦ s❡❝✉♥❞ár✐♦
rS ✕ t❛①❛ ❞❡ ❝♦♥s✉♠♦ ❞♦ s✉❜str❛t♦ ❛tr❛✈és ❞❛ ♦①✐❞❛çã♦ ❞❛ ❣❧✐❝♦s❡
S ✕ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s✉❜str❛t♦
Sin ✕ ❝♦♥❝❡♥tr❛çã♦ ❞♦ s✉❜str❛t♦ ❞❡ ❡♥tr❛❞❛
u ✕ ❡♥tr❛❞❛ ❞❡ ❝♦♥tr♦❧❡ ❞♦ s✐st❡♠❛
ulinearizante ✕ s✐♥❛❧ ❞❡ ❝♦♥tr♦❧❡ ♣❛r❛ ❧✐♥❡❛r✐③❛r ♦ s✐st❡♠❛
V(x) ✕ ❢✉♥çã♦ ❞❡ ▲②❛♣✉♥♦✈
ϕ ✕ t❛①❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞♦ ♠♦❞❡❧♦ w ✕ ❡♥tr❛❞❛ ❞❡ ❞✐stúr❜✐♦
✶✻
✶ ■♥tr♦❞✉çã♦
❆ ✐♥❞ústr✐❛ ❜✐♦t❡❝♥♦❧ó❣✐❝❛ ✈❡♠ ❝r❡s❝❡♥❞♦ r❛♣✐❞❛♠❡♥t❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧♦s ❛✈❛♥✲ ç♦s ♥❛ ❝♦♠♣r❡❡♥sã♦ s♦❜r❡ ♣r♦❝❡ss♦s ❜✐♦❧ó❣✐❝♦s ❝♦♠♣❧❡①♦s ❡ t❛♠❜é♠ ♣❡❧❛ ❛❧t❛ ♣r♦❝✉r❛ ❞♦s ♣r♦❞✉t♦s ❜✐♦❧♦❣✐❝❛♠❡♥t❡ ♠❛♥✉❢❛t✉r❛❞♦s ❝♦♠♦ ❛❧✐♠❡♥t♦s✱ ❜❡❜✐❞❛s✱ ❢❛r♠❛❝ê✉t✐❝♦s✱ ❝♦♠♠♦✲ ❞✐t✐❡s ❡ ♣r♦❞✉t♦s q✉í♠✐❝♦s ❡s♣❡❝í✜❝♦s✳ ❖ ✐♠♣❛❝t♦ ❞❛ ✐♥❞ústr✐❛ ❜✐♦t❡❝♥♦❧ó❣✐❝❛ ♥❛ ❡❝♦♥♦♠✐❛ ♠✉♥❞✐❛❧ é s✉❜st❛♥❝✐❛❧ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ♣r♦❞✉çã♦ ❡♠ ❣r❛♥❞❡ ❡s❝❛❧❛ ❞♦ ❡t❛♥♦❧ ❝♦♠♦ ❝♦♠❜✉stí✈❡❧ r❡♥♦✈á✈❡❧ ❬✶❪✳
❯♠ ♣r♦❝❡ss♦ ❜✐♦t❡❝♥♦❧ó❣✐❝♦ tí♣✐❝♦ ❝♦♥s✐st❡ ❡♠ ❞✉❛s ❡t❛♣❛s ♣r✐♥❝✐♣❛✐s✳ ❆ ♣r✐♠❡✐r❛ ❡t❛♣❛ é r❡❧❛❝✐♦♥❛❞❛ ❛ r❡❛çõ❡s ♣r♦❞✉③✐❞❛s ♣♦r ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ❝é❧✉❧❛s q✉❡ ♣r♦❞✉③✐rã♦ ♦ ♣r♦❞✉t♦ ❞❡s❡❥❛❞♦ ❡ ♦ ♣r♦❞✉t♦ ❡①❝❡❞❡♥t❡✳ ◆❛ s❡❣✉♥❞❛ ❡t❛♣❛ é s❡♣❛r❛❞♦ ♦ ♣r♦❞✉t♦ ✜♥❛❧ ❞❡s❡❥❛❞♦ ❞♦ ♣r♦❞✉t♦ ❡①❝❡❞❡♥t❡✱ q✉❡ ♣♦❞❡rá s❡r ❞❡s❝❛rt❛❞♦✳ ❯♠❛ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦ ♣r♦❝❡ss♦ é ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ t✐♣♦s ❞❡ ❝é❧✉❧❛s ✭♦✉ ❝❡♣❛s✮ q✉❡ ♥ã♦ t❡♥❤❛♠ ✉♠ ❝✉st♦ ❡❧❡✈❛❞♦ ❡ q✉❡✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ t❡♥❤❛♠ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❣❡r❛r ♦ ♣r♦❞✉t♦ ✜♥❛❧ ❞❡s❡❥❛❞♦✳ ◆❛ ♠❛✐♦r✐❛ ❞♦s ❝❛s♦s✱ ❛ ✉t✐❧✐③❛çã♦ ❞❛ té❝♥✐❝❛ ❞❡ r❡❝♦♠❜✐♥❛çã♦ ❞❡ ♠✐❝r♦♦r❣❛♥✐s♠♦s é ❛ ♠❛✐s ✈✐á✈❡❧ ❡❝♦♥♦♠✐❝❛♠❡♥t❡ ♣❛r❛ ❛ ♣r♦❞✉çã♦ ❞❡ ❡♥③✐♠❛s ❬✷❪✳
❈♦♠ ♦ ❛✈❛♥ç♦ ❞❛ t❡❝♥♦❧♦❣✐❛ ❞❡ r❡❝♦♠❜✐♥❛çõ❡s ❞❡ ❉◆❆✱ ❡stá s❡ t♦r♥❛♥❞♦ ❝❛❞❛ ✈❡③ ♠❛✐s ❝♦♠✉♠ ❛ ❝r✐❛çã♦ ❞❡ ❝❡♣❛s ❡s♣❡❝✐❛✐s ♣❛r❛ ❛ ♣r♦❞✉çã♦ ❞❡ ♣r♦❞✉t♦s ❞❡s❡❥❛❞♦s ❬✸❪✳ ❚❡♥❞♦ ❡♠ ✈✐st❛ q✉❡ ❝❛❞❛ ❝é❧✉❧❛ ♣r♦❞✉③ ✉♠❛ q✉❛♥t✐❞❛❞❡ ♠✉✐t♦ ♣❡q✉❡♥❛ ❞♦ ♣r♦❞✉t♦ ❞❡s❡❥❛❞♦✱ é ♥❡❝❡ssár✐♦ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❝é❧✉❧❛s✱ s❡♥❞♦ ❛ss✐♠✱ ❛ ❝❡♣❛ ❡s❝♦❧❤✐❞❛ ❞❡✈❡ t❡r ✉♠ ❜❛✐①♦ ❝✉st♦✱ t♦r♥❛♥❞♦ ✈✐á✈❡❧ ❛ ♣r♦❞✉çã♦ ✐♥❞✉str✐❛❧ ❡♠ ❣r❛♥❞❡ ❡s❝❛❧❛✳
✶✼
♣r♦❞✉çã♦ ❞❡ ✐♥s✉❧✐♥❛ ♣❛r❛ ❞✐❛❜❡t❡s✱ ♣r♦❞✉çã♦ ❞❡ ✈❛❝✐♥❛s✱ ❡♥tr❡ ♦✉tr♦s ♣r♦❞✉t♦s✳ ❚❛♠✲ ❜é♠✱ é ✉♠❛s ❞❛s ♠❛✐s ♣♦♣✉❧❛r❡s ♥❛ ✐♥❞ústr✐❛ ❞❡ ♠✐❝r♦♦r❣❛♥✐s♠♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ♦❜t❡r s❡✉ ❝r❡s❝✐♠❡♥t♦ ❝♦♠ ✉♠ ♣r♦❝❡ss♦ r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s ❡ ❜❛r❛t♦ ❬✻❪✱ ❬✼❪✳ ❏á ❛ ✏❊✳❈♦❧✐✑ é ✉♠❛ ❞❛s ❜❛❝tér✐❛s ♠❛✐s ✉t✐❧✐③❛❞❛s ♥❛ ✐♥❞ústr✐❛ ❜✐♦❢❛r♠❛❝ê✉t✐❝❛ ❬✽❪✳
▲❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ❛✉♠❡♥t♦ ❞❛ ❝♦♠♣❡t✐çã♦ ❞♦ ♠❡r❝❛❞♦ ♠✉♥❞✐❛❧✱ ❛ r❡❞✉çã♦ ❞❡ ❝✉st♦s ❡ s❡✈❡r❛s ♥♦r♠❛s ❞❡ s❡❣✉r❛♥ç❛ ❞♦ ♠❡✐♦✲❛♠❜✐❡♥t❡✱ ❡①✐st❡ ✉♠ ♥❡❝❡ss✐❞❛❞❡ ❡♠ ♠❡❧❤♦r✐❛s ❞❡ ♣r♦❝❡ss♦s ❞❡ ❝♦♥tr♦❧❡ ❡ ♦t✐♠✐③❛çã♦ ❜❛s❡❛❞❛s ❡♠ ♠ét♦❞♦s ❛♥❛❧ít✐❝♦s ❛✈❛♥ç❛✲ ❞♦s ♣❛r❛ ♦ ❝♦♥tr♦❧❡ ❞♦ s✉❜str❛t♦ ❞♦s ♣r♦❞✉t♦s ❡ ❡st❛❞♦ ❞❛s ❝é❧✉❧❛s ❬✾❪✳ ❈♦♠ ❡st❡ ♦❜❥❡t✐✈♦✱ ❞✐✈❡rs♦s tr❛❜❛❧❤♦s ❢♦r❛♠ ♣r♦♣♦st♦s ♥❛ ❧✐t❡r❛t✉r❛ ♣❛r❛ ♦ ❝♦♥tr♦❧❡ ❞❡ ❜✐♦r❡❛t♦r❡s ♦♣❡r❛♥❞♦ ❡♠ ♠♦❞♦ ❞❡s❝♦♥tí♥✉♦ ❝♦♠ ❛❧✐♠❡♥t❛çã♦✳ ❊♠ ❬✶✵❪✱ ❬✶✶❪✱ ❬✶✷❪✱ ❬✶✸❪✱ ❬✶✹❪✱ ❬✶✺❪✱ ❬✹❪ sã♦ ❛♣r❡✲ s❡♥t❛❞♦s ❞✐✈❡rs♦s ❝♦♥❝❡✐t♦s q✉❡ ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ❝♦♠♦ ❜❛s❡ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳ ❈♦♠♦ ❛♣r❡s❡♥t❛❞♦ ♥❛s ❝✐t❛çõ❡s ❛❝✐♠❛✱ ❞✐✈❡rs❛s ❡str❛té❣✐❛ ❞❡ ❝♦♥tr♦❧❡ ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞❛s ♣❛r❛ ❞❡t❡r♠✐♥❛ ✉♠ ♣❡r✜❧ ót✐♠♦ ❞❡ ❛❧✐♠❡♥t❛çã♦ ♥♦ ❜✐♦r❡❛t♦r✱ ❣❡r❛❧♠❡♥t❡ ❝♦♥s✐❞❡r❛❞❛ ❛ ú♥✐❝❛ ✈❛r✐á✈❡❧ ♠❛♥✐♣✉❧❛❞❛ ♣❛r❛ ♦ ❝♦♥tr♦❧❡✳ ❈♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♠❛♥t❡r ♦ s✐st❡♠❛ ❡♠ ót✐♠❛s ❝♦♥❞✐çõ❡s ❡♠ r❡❧❛çã♦ ❛♦ ❝r❡s❝✐♠❡♥t♦ ❞❛ ❜✐♦♠❛ss❛✱ é ♣♦ssí✈❡❧ ❝♦♥tr♦❧❛r ♦ ❝♦❡✜❝✐❡♥t❡ r❡s♣✐r❛tór✐♦ ❞❛s ❝é❧✉❧❛s ✉t✐❧✐③❛♥❞♦ ✉♠ ❛♥❛❧✐s❛❞♦r ❞❡ ❣❛s❡s ✭♣❛r❛ ♠❛♥t❡r ♦ s✐st❡♠❛ ❡♠ ✉♠ ♥í✈❡❧ ❝rít✐❝♦ ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❞❡ s✉❜str❛t♦✮ ♦✉ ❝♦♥tr♦❧❛r ❛ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ❜②✲♣r♦❞✉❝t ♣❛r❛ q✉❡ ♣❡r♠❛♥❡ç❛ ❡♠ ✉♠ ♣❡q✉❡♥❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈♦❧✉♠❡✳
❚❡♥❞♦ ❝♦♠♦ ❛❧✈♦ ♦s ♠✐❝r♦♦r❣❛♥✐s♠♦s ❡♠ ❡st✉❞♦✱ ♣♦❞❡✲s❡ ❝✐t❛r ♣❛r❛ ♣❛r❛ ❛ ♠á①✐♠❛ ♣r♦❞✉çã♦ ❞❡ ❜✐♦♠❛ss❛ ✉t✐❧✐③❛♥❞♦ ✏❊✳❈♦❧✐✑ ♦s tr❛❜❛❧❤♦s ❞❡ ❍❛✜❞✐ ❡t ❛❧✳ ♦♥❞❡ ❡♠ ❬✽❪ é ✉t✐❧✐③❛❞♦ ✉♠ ❝♦♥tr♦❧❡ ♣r❡❞✐t✐✈♦ r♦❜✉st♦ ❡ ❡♠ ❬✶✻❪ é ✉t✐❧✐③❛❞♦ ✉♠ ❝♦♥tr♦❧❡ ♣r❡❞✐t✐✈♦ ♥ã♦ r♦❜✉st♦❀ ❡♠ ❘♦❡✈❛ ❡t ❛❧✳ ❬✷❪ ♦ ♣r♦❜❧❡♠❛ é tr❛t❛❞♦ ❛tr❛✈és ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ♣❛r❛ ♦ ❝♦♥tr♦❧❡ ót✐♠♦ ❞❛ t❛①❛ ❞❡ ❛❧✐♠❡♥t❛çã♦❀ ❡♠ ❏❡♥③❝❤ ❡t ❛❧✳ ❬✶✼❪ é tr❛t❛❞♦ ❝♦♠♦ ❝♦♥tr♦❧❡ ❛tr❛✈és ❞❡ ✉♠ ♠♦❞❡❧♦ ❣❡♥ér✐❝♦ ❡ ❡♠ ❘♦❝❤❛ ❡t ❛❧✳ ❬✶✽❪ é ❛♣❧✐❝❛❞♦ ✉♠ ❝♦♥tr♦❧❡ ❧✐♥❡❛r✐③❛♥t❡✳
P❛r❛ ❛ ✏❙✳❈❡r❡✈✐s✐❛❡✑✱ ♣♦❞❡✲s❡ ❞❡st❛❝❛r ❡♠ ❬✼❪ ❡ ❬✶✾❪ ❞♦✐s ❛❧❣♦r✐t♠♦s ❞❡ ♦t✐♠✐③❛çã♦ ❡♠ t❡♠♣♦ r❡❛❧✱ q✉❡ r❡❣✉❧❛♠ ♦ ♥í✈❡❧ ❞❡ s✉❜str❛t♦ ❛♦ r❡❞♦r ❞♦ ♣♦♥t♦ ❝rít✐❝♦✳ ❊♥tr❡t❛♥t♦✱ ❡st❛ té❝♥✐❝❛ só é ♣♦ssí✈❡❧ s❡r ❛♣❧✐❝❛❞❛ q✉❛♥❞♦ ♦ ✈❛❧♦r ❝rít✐❝♦ é ♠❡❞✐❞♦✱ s❡♥❞♦ ✐ss♦ ❣❡r❛❧♠❡♥t❡ ❞✐❢í❝✐❧ ❞❡ s❡ ♦❜t❡r ♣❡❧♦s ❜❛✐①♦s ♥í✈❡✐s ❞❡ ❝♦♥❝❡♥tr❛çã♦ ❡♥❝♦♥tr❛❞♦s ♥♦s ❞♦✐s ♠✐❝r♦♦r❣❛♥✐s✲ ♠♦s ✳
P♦❞❡✲s❡ ❞❡st❛❝❛r ❡♥tr❡ ❛s té❝♥✐❝❛ ♠❛✐s ♣♦♣✉❧❛r❡s ❞❡ ❝♦♥tr♦❧❡ ♦ ❝♦♥tr♦❧❡ ❧✐♥❡❛r✐③❛♥t❡ ❬✹❪✱ q✉❡ ✈❡♠ s❡♥❞♦ ❛♣❧✐❝❛❞❛ ❝♦♠ s✉❝❡ss♦ ❡♠ ❞✐✈❡rs♦s ❡st✉❞♦s ❬✶✵❪✱ ❬✷✵❪ ❡ ❬✻❪✳
✶✽
♠❡✐r❛♠❡♥t❡✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠ ♠♦❞❡❧♦ ❣❡r❛❧ q✉❡ ❞❡s❝r❡✈❡ ❛ ❞✐♥â♠✐❝❛ ❞♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ss❡s ❞♦✐s ♠✐❝r♦♦r❣❛♥✐s♠♦s ❡♠ ❜✐♦r❡❛t♦r❡s ♦♣❡r❛♥❞♦ ❡♠ ♠♦❞♦ ❞❡s❝♦♥tí♥✉♦ ❝♦♠ ❛❧✐♠❡♥t❛çã♦✶✳
❉❡✈✐❞♦ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ ♠♦❞❡❧♦ ✭❞❡s❝r✐t♦ ♣♦r ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♥ã♦ ❧✐♥❡❛r❡s ❡ s✉❥❡✐✲ t❛s ❛ ❣r❛♥❞❡s ✈❛r✐❛çõ❡s ♣❛r❛♠étr✐❝❛s✮✱ ♣r♦♣õ❡✲s❡ ✉♠❛ ❧❡✐ ❞❡ ❝♦♥tr♦❧❡ ❞♦ t✐♣♦ r❡❛❧✐♠❡♥t❛çã♦ ❧✐♥❡❛r✐③❛♥t❡ ♣❛r❝✐❛❧✳ ❯t✐❧✐③❛✲s❡✱ t❛♠❜é♠✱ ✉♠❛ ❧✐♥❡❛r✐③❛çã♦ ♣❛r❝✐❛❧ ✈✐s❛♥❞♦ ❞✐♠✐♥✉✐r ♦ ♥ú✲ ♠❡r♦ ❞❡ ♠❡❞✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❛ ❧❡✐ ❞❡ ❝♦♥tr♦❧❡✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❞✐♥â♠✐❝❛ ❧✐♥❡❛r ❛ s❡r ✐♠♣♦st❛ ♣❡❧♦ ❝♦♥tr♦❧❡ ❧✐♥❡❛r✐③❛♥t❡ é ♣r♦❥❡t❛ ❞❡ ♠❛♥❡✐r❛ ❛ ❣❛r❛♥t✐r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛ ❢r❡♥t❡ ❛ ✈❛r✐❛çõ❡s ♣❛r❛♠étr✐❝❛s✳ ❊♠ s❡❣✉✐❞❛✱ ❡st✉❞❛✲s❡ ✉♠ ♠❡✲ ❝❛♥✐s♠♦ ❞❡ ❡st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ♣❛r❛ ♠❡❧❤♦r❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦ s✐st❡♠❛ ❡♠ r❡❧❛çã♦ ❛ ♠❛①✐♠✐③❛çã♦ ❞❛ ♣r♦❞✉çã♦ ❞❡ ❜✐♦♠❛ss❛✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✐♠♣❧❡♠❡♥t❛✲s❡ ✉♠ ♠❡❝❛♥✐s♠♦ ❞❡ ❛❞❛♣t❛çã♦ ♣❛r❛ ❡st✐♠❛r ❛ t❛①❛ ❞❛ r❡❛çã♦ ❞❡ ♦①✐❞❛çã♦ ❞❛ ❣❧✐❝♦s❡ q✉❡ é ♦ ♣r✐♥❝✐♣❛❧ ♣❛râ♠❡tr♦ ✉t✐❧✐③❛❞♦ ♥❛ ❧❡✐ ❞❡ ❝♦♥tr♦❧❡ ❧✐♥❡❛r✐③❛♥t❡✳
❋r❡♥t❡ ❛ ❞✐s❝✉ssã♦ ❛❝✐♠❛✱ ♦s ♣r✐♥❝✐♣❛✐s ♦❜❥❡t✐✈♦s ❛ s❡r❡♠ ❜✉s❝❛❞♦s ♥❡st❛ ❞✐ss❡rt❛çã♦ sã♦✿
✶✳ ♦❜t❡r ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞✐♥â♠✐❝❛ q✉❡ ❞❡s❝r❡✈❛ ❞❡ ♠♦❞♦ ❣❡r❛❧ ♦ ❝r❡s❝✐♠❡♥t♦ ❞♦s ♠✐❝r♦♦r❣❛♥✐s♠♦s ✏❊✳ ❈♦❧✐✑ ❡ ✏❙✳ ❈❡r❡✈✐s✐❛❡✑❀
✷✳ ♣r♦❥❡t❛r ✉♠❛ ❧❡✐ ❞❡ ❝♦♥tr♦❧❡ r♦❜✉st❛ q✉❡ ✈✐s❡ ❛ ♠❛①✐♠✐③❛çã♦ ❞❛ r❡♣r♦❞✉çã♦ ❞♦s ♠✐❝r♦♦r❣❛♥✐s♠♦s ✭♣r♦❞✉çã♦ ❞❡ ❜✐♦♠❛ss❛✮ ❡ q✉❡ ✉t✐❧✐③❡ ✉♠ ♥ú♠❡r♦ r❡❞✉③✐❞♦ ❞❡ ♠❡✲ ❞✐çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❜✐♦❧ó❣✐❝❛s❀ ❡
✸✳ ❞❡s❡♥✈♦❧✈❡r ♠❡❝❛♥✐s♠♦s ❞❡ ❡st✐♠❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ♣❛r❛ ♠❡❧❤♦r❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞❛ ❧❡✐ ❞❡ ❝♦♥tr♦❧❡ r♦❜✉st❛✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❞❛ t❛①❛ ❞❡ ♦①✐❞❛çã♦ ❞❡ ❣❧✐❝♦s❡✳
✶✳✶ ❊str✉t✉r❛ ❞❛ ❉✐ss❡rt❛çã♦
P❛r❛ ❞❡s❝r❡✈❡r ❛ ♠❡t♦❞♦❧♦❣✐❛ ♣r♦♣♦st❛ ♣❛r❛ ❛t✐♥❣✐r ♦s ♦❜❥❡t✐✈♦s ❛❝✐♠❛ ❛♣r❡s❡♥t❛❞♦s✱ ❡st❛ ❞✐ss❡rt❛çã♦ é ❞✐✈✐❞✐❞❛ ♥♦s s❡❣✉✐♥t❡s ❝❛♣ít✉❧♦s✿
• ❈❛♣✐t✉❧♦ ✷ ✲ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s
◆❡st❡ ❝❛♣ít✉❧♦✱ ❛❜♦r❞❛♠✲s❡ ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ✉t✐❧✐③❛❞♦s ❛♦ ❧♦♥❣♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ❞❡s❝r❡✈❡♠✲s❡ ♦s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❡st❛✲ ❜✐❧✐❞❛❞❡ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s✱ ❜❡♠ ❝♦♠♦ ❢♦r♠❛s ❞❡ ❞❡s❝r✐çã♦ ❞❡ ✐♥❝❡rt❡③❛s✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛✲s❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ♣♦r ♣❛râ♠❡tr♦s ✈❛r✐❛♥t❡s
✶✾
♥♦ t❡♠♣♦ ✭r❡♣r❡s❡♥t❛çã♦ q✉❛s✐✲▲P❱✮ ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ r❡❛❧✐♠❡♥t❛çã♦ ❧✐♥❡❛r✐③❛♥t❡✳ ◆❛ s❡q✉ê♥❝✐❛✱ ✐♥tr♦❞✉③✲s❡ ❞❡ ❢♦r♠❛ ❤✐stór✐❝❛ ❡ ❝♦♥❝❡✐t✉❛❧ ❛ t❡♦r✐❛ ❞❡ ❝♦♥tr♦❧❡ r♦❜✉st♦✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♣♦r ▲②❛♣✉♥♦✈ ✭❡♠ ♣❛rt✐❝✉❧❛r✱ ❝♦♥s✐❞❡r❛♥❞♦ ❢✉♥çõ❡s ❞❡ ▲②❛♣✉♥♦✈ q✉❛❞rát✐❝❛s✮✱ ❛ ❢♦r♠✉❧❛çã♦ ♠❛t❡♠át✐❝❛ ❞❡ ♣r♦❜❧❡♠❛s ♣♦r ❞❡s✐❣✉❛❧❞❛❞❡s ♠❛tr✐❝✐❛✐s ❧✐♥❡❛r❡s ✭▲▼■s✮ ❡ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❡♥tr❛❞❛✲s❛í❞❛ ✭❣❛♥❤♦ L2✮✳
• ❈❛♣✐t✉❧♦ ✸ ✲ ❇✐♦r❡❛t♦r❡s
❊st❡ ❝❛♣ít✉❧♦ ❜✉s❝❛ ✐♥tr♦❞✉③✐r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ ❜✐♦r❡❛t♦r❡s ♥♦ ❝✉❧t✐✈♦ ❞❡ ♠✐❝r♦♦r✲ ❣❛♥✐s♠♦s✳ ❆♣r❡s❡♥t❛♠✲s❡ ♦s ✈ár✐♦s ♠♦❞♦s ❞❡ ♦♣❡r❛çã♦ ❞❡ ❜✐♦r❡❛t♦r❡s✱ ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ♦❜t❡♥çã♦ ❞❡ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s✱ ❛s ♣r✐♥❝✐♣❛✐s r❡❛çõ❡s ❜✐♦❧ó❣✐❝❛s ❡♥✈♦❧✈✐❞❛s ♥♦ ❝✉❧t✐✈♦ ❞❡ ♠✐❝r♦♦r❣❛♥✐s♠♦s✱ ❡✱ ♣♦r ú❧t✐♠♦✱ ✐♥tr♦❞✉③✲s❡ ♦ ♣r✐♥❝í♣✐♦ ❞♦ ❜♦tt❧❡✲♥❡❝❦ q✉❡ é ❡①tr❡♠❛♠❡♥t❡ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛s ❝♦♥❞✐çõ❡s ✐❞❡❛✐s ♣❛r❛ ❛ ♣r♦❞✉çã♦ ❞❡ ♠✐❝r♦♦r❣❛♥✐s♠♦s ❡♠ ❜✐♦r❡❛t♦r❡s✳
• ❈❛♣✐t✉❧♦ ✹ ✲ Pr♦❥❡t♦ ❞♦ ❈♦♥tr♦❧❡ ❡ ❘❡s✉❧t❛❞♦s
◆❡st❡ ❝❛♣ít✉❧♦✱ ♣r✐♠❡✐r❛♠❡♥t❡ é tr❛ç❛❞♦ ♦ ♦❜❥❡t✐✈♦ ❞♦ ❝♦♥tr♦❧❡ ❡ t❛♠❜é♠ sã♦ ❛♣r❡✲ s❡♥t❛❞♦s ♦s ♠♦❞❡❧♦s q✉❡ ❛ s❡r❡♠ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❛ ❧❡✐ ❞❡ ❝♦♥tr♦❧❡✳ ❊♠ s❡❣✉✐❞❛✱ ✐♠♣❧❡♠❡♥t❛✲s❡ ✉♠❛ ❧❡✐ ❞❡ ❝♦♥tr♦❧❡ r♦❜✉st❛ ✉t✐❧✐③❛♥❞♦ ❛ té❝♥✐❝❛ ❞❡ r❡❛❧✐✲ ♠❡♥t❛çã♦ ❧✐♥❡❛r✐③❛♥t❡ ♣❛r❝✐❛❧ ♦♥❞❡ s❡ ❞❡t❡r♠✐♥❛ ❛ ❞✐♥â♠✐❝❛ ❧✐♥❡❛r ✐♠♣♦st❛ ♣❡❧♦ ❝♦♥tr♦❧❛❞♦r ❧✐♥❡❛r✐③❛♥t❡✳ ❇✉s❝❛♥❞♦ ✉♠ ♠❡❧❤♦r ❞❡s❡♠♣❡♥❤♦ ❞♦ s✐st❡♠❛ ❡♠ ♠❛❧❤❛ ❢❡❝❤❛❞❛✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠❛ té❝♥✐❝❛ ❞❡ ❛❞❛♣t❛çã♦ ♣❛r❛♠étr✐❝❛ ✈✐s❛♥❞♦ ❡st✐♠❛r ❛ t❛①❛ ❞❡ ♦①✐❞❛çã♦ ❞❛ ❣❧✐❝♦s❡ ♣♦♥t♦ ❝❤❛✈❡ ♥♦ ❞❡s❡♠♣❡♥❤♦ ❞♦ ❝♦♥tr♦❧❛❞♦r ♣r♦♣♦st♦✳ P♦r ✜♠✱ sã♦ r❡❛❧✐③❛❞♦s ✈ár✐♦s t❡st❡s ❞❡ s✐♠✉❧❛çã♦ ♣❛r❛ ✈❡r✐✜❝❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❧❡✐ ❞❡ ❝♦♥tr♦❧❡ ♣r♦♣♦st❛✳
• ❈❛♣✐t✉❧♦ ✺ ✲ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s
✷✵
✷ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s
✷✳✶ ■♥tr♦❞✉çã♦
◆❡st❡ ❝❛♣ít✉❧♦ sã♦ ❛❜♦r❞❛❞♦s ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s ✉t✐❧✐③❛❞♦s ❛♦ ❧♦♥❣♦ ❞❡st❛ ❞✐ss❡rt❛çã♦✳ ❙ã♦ ❞❡s❝r✐t♦s ❜r❡✈❡♠❡♥t❡ ❝♦♥❝❡✐t♦s ❞❡ s✐st❡♠❛s ♥ã♦✲❧✐♥❡❛r❡s ✐♥❝❡rt♦s ❡ ❛ r❡♣r❡s❡♥t❛çã♦ ♣♦r ♣❛râ♠❡tr♦s ✈❛r✐❛♥t❡s ♥♦ t❡♠♣♦✳ ❆❧é♠ ❞✐ss♦✱ ✐♥tr♦❞✉③✲s❡ ❢♦r♠❛s ❞❡ ❞❡s❝r✐çã♦ ❞❡ ✐♥❝❡rt❡③❛s ❡ ♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s r❡❧❛❝✐♦♥❛❞♦s ❛ r❡❛❧✐♠❡♥t❛çã♦ ❧✐♥❡❛✲ r✐③❛♥t❡✳ ❚❛♠❜é♠✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠❛ ❜r❡✈❡ r❡tr♦s♣❡❝t✐✈❛ ❤✐stór✐❝❛ ❡ ❝♦♥❝❡✐t✉❛❧ ❞❛ t❡♦r✐❛ ❞❡ ❝♦♥tr♦❧❡ r♦❜✉st♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❝♦♥s✐❞❡r❛♥❞♦ ❢✉♥çõ❡s ❞❡ ▲②❛♣✉♥♦✈ q✉❛❞rát✐❝❛s✳ ❋✐♥❛❧✐✲ ③❛♥❞♦ ♦ ❝❛♣ít✉❧♦✱ ✐♥tr♦❞✉③✲s❡ ❛ ❢♦r♠✉❧❛çã♦ ♣♦r ❞❡s✐❣✉❛❧❞❛❞❡s ♠❛tr✐❝✐❛✐s ❧✐♥❡❛r❡s ✭▲▼■s✮ r❡❧❛❝✐♦♥❛❞❛ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❡♥tr❛❞❛✲s❛í❞❛ ❡ ❣❛♥❤♦ L2✳
✷✳✷ ❙✐st❡♠❛s ◆ã♦✲❧✐♥❡❛r❡s
❆ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞♦s s✐st❡♠❛s ❢ís✐❝♦s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ s✐st❡♠❛s ♠❡❝â♥✐❝♦s ❡ q✉í✲ ♠✐❝♦s✱ tê♠ ❛ ❞✐♥â♠✐❝❛ s✉❥❡✐t❛ ❛ ❛❧❣✉♠❛ ❢♦r♠❛ ❞❡ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡✱ ❛ q✉❛❧ é ❞❡✜♥✐❞❛ ❝♦♠♦ ♦ ❞❡s✈✐♦ às ❝❛r❛❝t❡ríst✐❝❛s ❧✐♥❡❛r❡s ❞❡ ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ❬✷✶❪✳
✷✶
❆♣❡s❛r ❞❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ✉t✐❧✐③❛çã♦ ❞❛ t❡♦r✐❛ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ♣❛r❛ tr❛t❛r ❞✐✲ ♥â♠✐❝❛s ❝♦♥t❡♥❞♦ ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡s ✭✉t✐❧✐③❛♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ♦♣❡r❛çã♦✮✱ ❡①✐st❡♠ ❞✐✈❡rs❛s r❛③õ❡s ♣❛r❛ s❡ ✉t✐❧✐③❛r té❝♥✐❝❛s ❞❡ ❛♥á❧✐s❡ ❡ ❝♦♥tr♦❧❡ ♠❛✐s ❝♦♠♣❧❡①❛s ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✷✷❪✿ (i) ❛ ❢❛✐①❛ ❞❡ ♦♣❡r❛çã♦ ❞♦ s✐st❡♠❛ é
♠❛✐♦r❀(ii)é ♣♦ssí✈❡❧ ✉♠❛ ❛♥á❧✐s❡ ❞❡ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡s ❢♦rt❡s ✭❞✉r❛s✮ ❝♦♠♦ s❛t✉r❛çã♦✱ ③♦♥❛
♠♦rt❛ ❡ ❤✐st❡r❡s❡❀ ❡(iii)é ♣♦ssí✈❡❧ ❛t✐♥❣✐r ✉♠ ♠❡❧❤♦r ❞❡s❡♠♣❡♥❤♦ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♥tr♦❧❡✳
❊♠ ❣❡r❛❧✱ ♦s s✐st❡♠❛s ❞✐♥â♠✐❝♦s ♥ã♦✲❧✐♥❡❛r❡s sã♦ ♠♦❞❡❧❛❞♦s ♥♦ ❡s♣❛ç♦ ❞❡ ❡st❛❞♦s ❛tr❛✈és ❞❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♥❛ ❢♦r♠❛✿
˙
x1 = f1(t, x1, ...xn, u1, ..., up)
˙
x2 = f2(t, x1, ...xn, u1, ..., up)
✳✳✳
˙
xn = fn(t, x1, ...xn, u1, ..., up)
♦♥❞❡ x˙i r❡♣r❡s❡♥t❛ ❛ ❞❡r✐✈❛❞❛ ❞❡ xi ❝♦♠ r❡❧❛çã♦ ❛ ✈❛r✐á✈❡❧ ❞❡ t❡♠♣♦ t✱ ♣❛r❛ i= 1, . . . , n❀
x1, x2, ..., xn sã♦ ♦s ❡st❛❞♦s ❞♦ s✐st❡♠❛❀ ❡ u1, u2, ..., up sã♦ ❛s ✈❛r✐á✈❡✐s ❞❡ ❡♥tr❛❞❛✳ ●❡r❛❧✲
♠❡♥t❡✱ ✉t✐❧✐③❛✲s❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♠❛✐s ❝♦♠♣❛❝t❛ ✉t✐❧✐③❛♥❞♦ ✈❡t♦r❡s ❡ ❢✉♥çõ❡s ✈❡t♦r✐❛✐s ❝♦♠♦ ♠♦str❛❞♦ ❛ s❡❣✉✐r✿
˙
x=f(t, x, u) ✭✷✳✶✮
♦♥❞❡✿ x= x1 x2 ✳✳✳ xn
, u=
u1 u2 ✳✳✳ up
, f(t, x, u) =
f1(t, x, u)
f2(t, x, u)
✳✳✳
fn(t, x, u)
❋r❡q✉❡♥t❡♠❡♥t❡✱ ❞❡✜♥❡✲s❡ ♦✉tr❛ ❡q✉❛çã♦ ♣❛r❛ r❡♣r❡s❡♥t❛r ❛s ✈❛r✐á✈❡✐s ❞❡ s❛í❞❛ ❞♦ s✐st❡♠❛✿
y =h(t, x, u) ✭✷✳✷✮
♦♥❞❡ y∈ Rq é ♦ ✈❡t♦r ❞❡ s❛í❞❛ ❡ h(t, x, u) é ✉♠❛ ❢✉♥çã♦ ✈❡t♦r✐❛❧ ❡♠(t, x, u)✳ ❆ ❡q✉❛çã♦
✭✷✳✷✮ é ♥♦r♠❛❧♠❡♥t❡ ❞❡♥♦♠✐♥❛❞❛ ❞❡ ❡q✉❛çã♦ ❞❡ s❛í❞❛✳ ❖ ♠♦❞❡❧♦ ❞❡ ❡s♣❛ç♦ ❞❡ ❡st❛❞♦s ❞❡ ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ é r❡♣r❡s❡♥t❛❞♦ ❛ ♣❛rt✐r ❞❛s ❡q✉❛çõ❡s ✭✷✳✶✮ ❡ ✭✷✳✷✮ ❛❝✐♠❛ ❞❡✜♥✐❞❛s✳
✷✷
sã♦ ❛♣r❡s❡♥t❛❞❛s ❛ s❡❣✉✐r✿
• ♦ ❡st❛❞♦ ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ✐♥stá✈❡❧✱ t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦ q✉❛♥❞♦ ♦ t❡♠♣♦ t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦❀ ♦ ❡st❛❞♦ ❞❡ ✉♠ s✐st❡♠❛ ♥ã♦✲❧✐♥❡❛r✱ ♣♦❞❡ ✐r ♣❛r❛ ♦ ✐♥✜♥✐t♦ ❡♠ ✉♠ t❡♠♣♦ ✜♥✐t♦❀
• ✉♠ s✐st❡♠❛ ❧✐♥❡❛r t❡♠ s♦♠❡♥t❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ❛ss✐♠ s❡♥❞♦✱ só ♣♦❞❡ t❡r ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ q✉❡ ❛tr❛✐ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ s❡✉ ❡st❛❞♦ ✐♥✐❝✐❛❧✳ ❯♠ s✐st❡♠❛ ♥ã♦✲❧✐♥❡❛r ♣♦❞❡ t❡r ♠❛✐s ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ s❡♥❞♦ q✉❡ ❡st❡ ❞❡♣❡♥❞❡ ❞♦ ❡st❛❞♦ ✐♥✐❝✐❛❧ ❞♦ s✐st❡♠❛❀
• ♣❛r❛ q✉❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❡ ✐♥✈❛r✐❛♥t❡ ♥♦ t❡♠♣♦ ♦s❝✐❧❡✱ ❞❡✈❡✲s❡ t❡r ✉♠ ♣❛r ❞❡ ❛✉t♦✈❛❧♦r❡s s♦❜r❡ ♦ ❡✐①♦ ✐♠❛❣✐♥ár✐♦✱ ❝♦♥❞✐çã♦ ♥ã♦ r♦❜✉st❛✱ q✉❡ é q✉❛s❡ ✐♠♣♦ssí✈❡❧ ❞❡ s❡ ♠❛♥t❡r ❞❡ ❢♦r♠❛ s✉st❡♥t❛❞❛ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ♣❡q✉❡♥❛s ♣❡rt✉r❜❛çõ❡s✳ ▼❡s♠♦ ❛ss✐♠✱ ❛ ❛♠♣❧✐t✉❞❡ ❞❛ ♦s❝✐❧❛çã♦ ❞❡♣❡♥❞❡rá ❞♦ ❡st❛❞♦ ✐♥✐❝✐❛❧✳ ◆❛ ♣rát✐❝❛✱ ❡ss❛s ♦s❝✐❧❛çõ❡s s♦♠❡♥t❡ sã♦ ♣r♦❞✉③✐❞❛s ♣♦r s✐st❡♠❛ ♥ã♦✲❧✐♥❡❛r❡s✳ ❊①✐st❡♠ s✐st❡♠❛s ♥ã♦✲ ❧✐♥❡❛r❡s q✉❡ ❡♥tr❛♠ ❡♠ ♦s❝✐❧❛çã♦ ❝♦♠ ✉♠❛ ❛♠♣❧✐t✉❞❡ ✜①❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ❡st❛❞♦ ✐♥✐❝✐❛❧✳ ❊st❡ t✐♣♦ ❞❡ ♦s❝✐❧❛çã♦ é ❝❤❛♠❛❞♦ ❞❡ ❝✐❝❧♦ ❧✐♠✐t❡❀
• ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❡stá✈❡❧ ❝♦♠♦ ✉♠❛ ❡♥tr❛❞❛ ♣❡r✐ó❞✐❝❛ ♣r♦❞✉③ ✉♠❛ s❛í❞❛ ❝♦♠ ❛ ♠❡s♠❛ ❢r❡q✉ê♥❝✐❛✳ ❯♠ s✐st❡♠❛ ♥ã♦✲❧✐♥❡❛r ❝♦♠ ✉♠❛ ❡①❝✐t❛çã♦ ♣❡r✐ó❞✐❝❛ ♣♦❞❡ ♦s❝✐❧❛r ❝♦♠ ❢r❡q✉ê♥❝✐❛s q✉❡ sã♦ s✉❜ ♦✉ ♠ú❧t✐♣❧♦s ❞❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ❡♥tr❛❞❛✳ ■ss♦ ❛✐♥❞❛ ♣♦❞❡ ❣❡r❛r ✉♠❛ ♦s❝✐❧❛çã♦ q✉❛s✐✲♣❡r✐ó❞✐❝❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ s♦♠❛ ❞❡ ♦s❝✐❧❛çõ❡s ♣❡r✐ó❞✐❝❛s ❝♦♠ ❢r❡q✉ê♥❝✐❛s q✉❡ ♥ã♦ sã♦ ♠ú❧t✐♣❧❛s ❡♥tr❡ ❡❧❛s❀
• ✉♠ s✐st❡♠❛ ♥ã♦✲❧✐♥❡❛r ♣♦❞❡ t❡r ✉♠ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ❡st❛❞♦s ♠❛✐s ❝♦♠♣❧✐❝❛❞❛s✱ q✉❡ ♥ã♦ é ♦ ❡q✉✐❧í❜r✐♦✱ ♦s❝✐❧❛çã♦ ♣❡r✐ó❞✐❝❛ ♦✉ ♦s❝✐❧❛çã♦ q✉❛s✐✲♣❡r✐ó❞✐❝❛✳ ❊ss❡ ❝♦♠♣♦rt❛♠❡♥t♦ ♥♦r♠❛❧♠❡♥t❡ é r❡❢❡r✐❞♦ ❝♦♠♦ ❝❛♦s✳ ❆❧❣✉♥s ❞❡st❡s ♠♦✈✐♠❡♥✲ t♦s ❝❛ót✐❝♦s tê♠ ❝❛r❛❝t❡ríst✐❝❛s r❛♥❞ô♠✐❝❛s✱ ❛♣❡s❛r ❞❛ ♥❛t✉r❡③❛ ❞❡t❡r♠✐♥íst✐❝❛ ❞♦ s✐st❡♠❛❀
✷✸
❉❡st❛ ❢♦r♠❛✱ ❡♠ ✈ár✐❛s s✐t✉❛çõ❡s✱ ♥ã♦ é s❡❣✉r♦✱ t❛♥t♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ❡st❛❜✐✲ ❧✐❞❛❞❡ q✉❛♥t♦ ❞❡ ❞❡s❡♠♣❡♥❤♦✱ ❝♦♥s✐❞❡r❛r ❛♣❡♥❛s ♦ ♠♦❞❡❧♦ ❧✐♥❡❛r✐③❛❞♦ s❡♥❞♦ ♥❡❝❡ssár✐♦ ✉♠ ❜♦♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ♠♦❞❡❧♦ ♦r✐❣✐♥❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ✉♠❛ r❡s♣♦st❛ ❞❡s❡❥❛❞❛✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡ ❛ ❧✐♥❡❛r✐③❛çã♦ tê♠ ✉♠❛ r❡str✐çã♦ ❢✉♥❞❛♠❡♥t❛❧✳ ◆❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✱ ❛ ❧✐♥❡❛r✐③❛çã♦ ✈✐s❛ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❛♦ r❡❞♦r ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣♦♥t♦ ❞❡ ♦♣❡r❛✲ çã♦✳ ❈♦♠ ✐ss♦ s♦♠❡♥t❡ é ♣♦ssí✈❡❧ ❛♣r♦①✐♠❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ s✐st❡♠❛ ❧♦❝❛❧♠❡♥t❡ ❡♠ r❡❧❛çã♦ ❛♦ ♣♦♥t♦ ♣ré✲❞❡t❡r♠✐♥❛❞♦✳ ❉❡st❛ ❢♦r♠❛✱ ♥ã♦ é ♣♦ssí✈❡❧ ♣r❡✈❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ s✐st❡♠❛ ♥ã♦✲❧✐♥❡❛r ❧♦♥❣❡ ❞❛ ✈✐③✐♥❤❛♥ç❛ ❞♦ ♣♦♥t♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦✳ ▼❡s♠♦ q✉❛♥❞♦ ❛ ❛♣r♦①✐♠❛çã♦ ❧✐♥❡❛r é ♠✉✐t♦ ♣ró①✐♠❛ ❛ ♥ã♦✲❧✐♥❡❛r ❧♦♥❣❡ ❞♦ ♣♦♥t♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦✱ ❡①✐st❡♠ ❛❧❣✉♥s ❢❡♥ô♠❡♥♦s✱ ❡ss❡♥❝✐❛❧♠❡♥t❡ ♥ã♦✲❧✐♥❡❛r❡s✱ q✉❡ ♣♦❞❡♠ ♠♦❞✐✜❝❛r ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ s✐st❡♠❛ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛s ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡s ❞✉r❛s✶ ✭♦✉ ♥ã♦ ❞✐❢❡✲
r❡♥❝✐á✈❡✐s✮ ❬✷✸❪✳ P♦r ❡①❡♠♣❧♦✱ s❛t✉r❛çã♦✱ ③♦♥❛ ♠♦rt❛✱ ❡ r❡❧é ✭✐❞❡❛❧ ❡ ❝♦♠ ❤✐st❡r❡s❡✮✳ ❚❛✐s ♥ã♦✲❧✐♥❡❛r✐❞❛❞❡s sã♦ ❝♦♠✉♥s ❡♠ s✐st❡♠❛s ❝♦♥t❡♥❞♦ ✈á❧✈✉❧❛s ❞❡ ❛❜❡rt✉r❛ ❣r❛❞✉❛❧❀ tr❛♥s✲ ❢♦r♠❛❞♦r❡s ❡ ♠♦t♦r❡s❀ ♠❛t❡r✐❛✐s ❢❡rr♦♠❛❣♥ét✐❝♦s❀ ❢♦❧❣❛ ❞❡ ❡♥❣r❡♥❛❣❡♥s❀ t❛♠❜♦r ❤✐❞rá✉❧✐❝♦ ❡ ♣♥❡✉♠át✐❝♦ ❝♦♠ ♠✉❞❛♥ç❛ ❞❡ ❞✐r❡çã♦❀ ❧✐♥❡❛r✐③❛çã♦ ❞❡ ❡❧❡♠❡♥t♦s ❡❧étr✐❝♦s ❡ ❡❧❡trô♥✐❝♦s✳
❉❡✈✐❞♦ ❛♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❧♦❝❛❧ ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡♠ s✐st❡♠❛s ♥ã♦✲❧✐♥❡❛r❡s ❡ ❛ ❞✐✈❡rs✐❞❛❞❡ ❞❡ ❢❡♥ô♠❡♥♦s ♥♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞✐♥â♠✐❝♦ ❞♦ s✐st❡♠❛✱ ❛♣r❡s❡♥t❛✲s❡ ❛ s❡✲ ❣✉✐r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❡✜♥✐çõ❡s ❡♥✈♦❧✈❡♥❞♦ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦✱ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ❡st❛❜✐❧✐❞❛❞❡✳
✷✳✷✳✶ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❙✐st❡♠❛s ◆ã♦✲▲✐♥❡❛r❡s
❆♥t❡s ❞❡ s❡ ✐♥tr♦❞✉③✐r ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ s✐st❡♠❛s ♥ã♦✲❧✐♥❡❛r❡s✱ ❛♣r❡s❡♥t❛✲s❡ ❛ s❡❣✉✐r ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❣❛r❛♥t✐r ❛ ✉♥✐❝✐❞❛❞❡ ❡ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞♦ ✈❛❧♦r ✐♥✐❝✐❛❧✿
˙
x=f(t, x) , x(t0) =x0 ✭✷✳✸✮
❖ ♣r♦❜❧❡♠❛ ❞♦ ✈❛❧♦r ✐♥✐❝✐❛❧ ❝♦♥s✐st❡ ❡♠ ✈❡r✐✜❝❛r s♦❜ q✉❛✐s ❝♦♥❞✐çõ❡s ♦ s✐st❡♠❛ ❛❝✐♠❛ ♣♦ss✉✐ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ψ(t) ♣❛r❛ t♦❞♦t≥t0✳ ❆ s♦❧✉çã♦ ❛ ❡st❡ ♣r♦❜❧❡♠❛ é ❜❛s❡❛❞❛ ♥❛
❝♦♥❞✐çã♦ ❞❡ ▲✐♣s❝❤✐t③ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦ ❛ s❡❣✉✐r ❬✷✸❪✿
❚❡♦r❡♠❛ ✷✳✷✳✶ ✭❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞❡ ❙♦❧✉çã♦✮ ❙✉♣♦♥❤❛ q✉❡ ❛ ❢✉♥çã♦f(t, x)
❡♠ ✭✷✳✸✮ s❛t✐s❢❛ç❛ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿
kf(t, x)−f(t, y)k ≤Lkx−yk ✭✷✳✹✮
✷✹
♣❛r❛ t♦❞♦ x, y ∈ B := {x ∈ Rn : kx−yk ≤ r , r > 0} ❡ t♦❞♦ t ∈ [t0, t1]✳ ❊♥tã♦✱ ❡①✐st❡
✉♠ ❡s❝❛❧❛r d ♣♦s✐t✐✈♦ t❛❧ q✉❡ ❛ ❡q✉❛çã♦ ❞❡ ❡st❛❞♦s ❡♠ ✭✷✳✸✮ t❡♥❤❛ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ♥♦
✐♥t❡r✈❛❧♦ [t0, t0+d]✳
❆ ❢✉♥çã♦ f(x, t)s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✭✷✳✹✮ é ❞✐t❛ s❡r ▲✐♣s❝❤✐t③ ❡♠ x ❡ ❛ ❝♦♥st❛♥t❡ Lé ❝❤❛♠❛❞❛ ❞❡ ❝♦♥st❛♥t❡ ❞❡ ▲✐♣s❝❤✐t③✳ ❆ ❝♦♥❞✐çã♦ ❞❡ ▲✐♣s❝❤✐t③ ♣♦❞❡ s❡r ❧♦❝❛❧ ♦✉ ❣❧♦❜❛❧
❞❡♣❡♥❞❡♥❞♦ ❞♦ ❞♦♠í♥✐♦ ❞❡ ✈❛❧✐❞❛❞❡ ❞❡st❛ ❝♦♥❞✐çã♦✳
❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠✲s❡ ✈ár✐♦s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❡st❛❜✐❧✐✲ ❞❛❞❡ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ♣❛r❛ s✐st❡♠❛s ♥ã♦ ❧✐♥❡❛r❡s✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ✐♥tr♦❞✉③✲s❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳
❉❡✜♥✐çã♦ ✷✳✷✳✶ ✭P♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✮ ❯♠ ✈❡t♦rxeq ∈Rn é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡
✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ s❡ ♣❛r❛ ✉♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧x(0) =xeq✱ t❡♠✲s❡ q✉❡x(t) =xeq ♣❛r❛
q✉❛❧q✉❡r t≥t0✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ xeq é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❞♦ s❡❣✉✐♥t❡ s✐st❡♠❛✿
˙
x=f(x(t)) ✭✷✳✺✮
s❡ ❡ s♦♠❡♥t❡ s❡ ❛ r❡❧❛çã♦ ❛ s❡❣✉✐r ❢♦r s❛t✐s❢❡✐t❛✿
f(xeq) = 0 ✭✷✳✻✮
❆tr❛✈és ❞❛ ❡q✉❛çã♦ ✭✷✳✺✮✱ ♣♦❞❡✲s❡ ❝♦♥❝❧✉✐r q✉❡ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ é ✉♠ ♣♦♥t♦ ❡s✲ t❛❝✐♦♥ár✐♦ ❞❛ ❢✉♥çã♦ f(·)✱ ❡ ♣♦rt❛♥t♦ ❛ ❞❡r✐✈❛❞❛ ❞❡ f(·) ♥❡st❡ ♣♦♥t♦ ❞❡✈❡ s❡r ♥✉❧❛✳ ■st♦
é✱ s❡ ♦ s✐st❡♠❛ t❡♠ ❝♦♠♦ ❡st❛❞♦ ✐♥✐❝✐❛❧ ♦ ❡st❛❞♦ xeq✱ ❡❧❡ ♣❡r♠❛♥❡❝❡rá ♥❡st❡ ❡st❛❞♦ ✐♥❞❡✲
✜♥✐❞❛♠❡♥t❡✳ ❈♦♠♦ ♠❡♥❝✐♦♥❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ s✐st❡♠❛s ♥ã♦✲❧✐♥❡❛r❡s ♣♦❞❡♠ ❛♣r❡s❡♥t❛♠ ♠❛✐s ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳
❚❡♥❞♦ ❝❧❛r♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣♦♥t♦ ❞❡ ❡q✉✐❧✐❜r♦✱ ❛ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠✲s❡ ❛❧❣✉♠❛s ❞❡✜♥✐✲ çõ❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♣❛r❛ s✐st❡♠❛s ♥ã♦✲❧✐♥❡❛r❡s✳ ❆ ❋✐❣✉r❛ ✶ ✈✐s❛ ❛✉①✐❧✐❛r ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡st❛s ❞❡✜♥✐çõ❡s✳
❉❡✜♥✐çã♦ ✷✳✷✳✷ ✭❊st❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✮ ❯♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦
xeq é ❡stá✈❡❧ s❡ ♣❛r❛ q✉❛❧q✉❡r R >0✱ ❡①✐st❛ ✉♠ r✱ ❝♦♠ 0< r < R✱ t❛❧ q✉❡ ♣❛r❛ t♦❞♦ x(0)
❞❡♥tr♦ ❞❛ r❡❣✐ã♦ S(xeq;r) := {x : kx(t)−xeqk ≤ r}✱ ❛ tr❛❥❡tór✐❛ x(t) ♣❡r♠❛♥❡❝❡ ❞❡♥tr♦
❞❛ r❡❣✐ã♦ S(xeq, R) ♣❛r❛ t♦❞♦ t >0✳
❉❡✜♥✐çã♦ ✷✳✷✳✸ ✭❊st❛❜✐❧✐❞❛❞❡ ❆ss✐♥tót✐❝❛✮ ❯♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦xeq é ❛ss✐♥t♦t✐❝❛✲
♠❡♥t❡ ❡stá✈❡❧ s❡ ❡❧❡ ❢♦r ❡stá✈❡❧ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❡①✐st✐r ✉♠ r > 0 t❛❧ q✉❡ s❡ x(0) ❡st✐✈❡r
✷✺
❋✐❣✉r❛ ✶✿ ❚r❛❥❡tór✐❛s ❞❡ ✉♠ s✐st❡♠❛ ♥ã♦✲❧✐♥❡❛r ❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❬✷✷❪✳
❉❡✜♥✐çã♦ ✷✳✷✳✹ ✭P♦♥t♦ ❞❡ ❊q✉✐❧í❜r✐♦ ■♥stá✈❡❧✮ ❯♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ xeq é ✐♥stá✲
✈❡❧ s❡ ❡❧❡ ♥ã♦ ❢♦r ❡stá✈❡❧✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ xeq é ✐♥stá✈❡❧ s❡ ♣❛r❛ ❛❧❣✉♠ R > 0 ❡ ♣❛r❛
q✉❛❧q✉❡r r > 0 ❡①✐st✐r ✉♠ ❡st❛❞♦ ✐♥✐❝✐❛❧ x(0) ∈ S(xeq, r) t❛❧ q✉❡ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ❡♠
❛❧❣✉♠ ♠♦♠❡♥t♦ s❛✐rá ❞❛ r❡❣✐ã♦ S(xeq, R)✳
❆ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛ ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s tr❛❥❡tór✐❛s ❞♦ s✐st❡♠❛ é ✉♠❛ t❛r❡❢❛ q✉❡ ❡①✐❣❡ ✉♠ ❣r❛♥❞❡ ❡s❢♦rç♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❡ ♥ã♦ é ♣rát✐❝❛ ♣❛r❛ s✐st❡♠❛s ♠❛✐s ❝♦♠♣❧❡①♦s✳ ❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛✲s❡ ❞❡ ❢♦r♠❛ r❡s✉♠✐❞❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ ❛tr❛✈és ❞❛ t❡♦r✐❛ ❞❡ ▲②❛♣✉♥♦✈✳
✷✳✷✳✷ ❚❡♦r✐❛ ❞❛ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ▲②❛♣✉♥♦✈
✷✻
t✐✈❛ ❡ ❞❡❝r❡s❝❡♥t❡ ❛♦t → ∞✳ P♦r ♠❛✐s ❞❡ ♠❡✐♦ sé❝✉❧♦✱ ♦ tr❛❜❛❧❤♦ ♣✐♦♥❡✐r♦ ❞❡ ▲②❛♣✉♥♦✈
s♦❜r❡ ❡st❛❜✐❧✐❞❛❞❡ r❡❝❡❜❡✉ ♠✉✐t♦ ♣♦✉❝♦ ♣r❡stí❣✐♦ ❢♦r❛ ❞❛ ❘úss✐❛✱ ❡♠❜♦r❛ t❡♥❤❛ s✐❞♦ tr❛✲ ❞✉③✐❞♦ ♣❛r❛ ♦ ❋r❛♥❝ês ❡♠ ✶✾✵✽✱ ♣♦r P♦✐♥❝❛ré ❬✸✵❪✱ ❡ ✐♠♣r❡ss♦ ♥♦✈❛♠❡♥t❡ ♣❡❧❛ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss ❡♠ ✶✾✹✼✳ ❈♦♠ ❛s ♣✉❜❧✐❝❛çõ❡s ❞♦s tr❛❜❛❧❤♦s ❞❡ ▲✉r✬❡ ❬✸✶❪✱ ▲❛ ❙❛❧❧❡ ❬✸✷❪ ❡ ▲❡❢s❝❤❡t③ ❬✸✵❪✱ ❛ t❡♦r✐❛ ❞❡ ▲②❛♣✉♥♦✈ ♣❛ss♦✉ ❛ r❡❝❡❜❡r ❣r❛♥❞❡ ❛t❡♥çã♦ ❞❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛ ❛ ♣❛rt✐r ❞♦ ✐♥í❝✐♦ ❞♦s ❛♥♦s ✻✵✳ ❉❡s❞❡ ❡♥tã♦✱ ✈ár✐❛s ♠❡❧❤♦r✐❛s ♥❛ ♠❡t♦❞♦❧♦❣✐❛ ♦r✐❣✐♥❛❧ ❞❡ ▲②❛♣✉♥♦✈ ❢♦r❛♠ ♣r♦♣♦st❛s✱ s❡♥❞♦ ❛té ✉♠ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦✳ ❆t✉❛❧♠❡♥t❡✱ ♦ ♠é✲ t♦❞♦ ✐♥❞✐r❡t♦ ❞❡ ▲②❛♣✉♥♦✈ t♦r♥♦✉✲s❡ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♥❛ ❥✉st✐✜❝❛t✐✈❛ t❡ór✐❝❛ ❞❡ s✐st❡♠❛s ❞❡ ❝♦♥tr♦❧❡ ❧✐♥❡❛r❡s✱ ❡♥q✉❛♥t♦ ♦ ♠ét♦❞♦ ❞✐r❡t♦ t♦r♥♦✉✲s❡ ❛ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❛♥á❧✐s❡ ❡ ♣r♦❥❡t♦ ❞❡ s✐st❡♠❛s ❞❡ ❝♦♥tr♦❧❡ ♥ã♦✲❧✐♥❡❛r❡s ❬✷✷✱ ✸✸✱ ✷✸❪✳
❖ ♠ét♦❞♦ ❞✐r❡t♦ ❞❡ ▲②❛♣✉♥♦✈ é ✉♠❛ ❡①t❡♥sã♦ ♠❛t❡♠át✐❝❛ ❞❡ ✉♠❛ ♦❜s❡r✈❛çã♦ ❢ís✐❝❛ ❢✉♥❞❛♠❡♥t❛❧✿ s❡ ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞❡ ✉♠ s✐st❡♠❛ ♠❡❝â♥✐❝♦ ♦✉ ❡❧étr✐❝♦ é ❝♦♥t✐♥✉❛♠❡♥t❡ ❞✐ss✐♣❛❞❛✱ ❡♥tã♦ ♦ s✐st❡♠❛✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ s❡r ❧✐♥❡❛r ♦✉ ♥ã♦✲❧✐♥❡❛r✱ ❡✈❡♥t✉❛❧♠❡♥t❡ ✐rá ❡♥tr❛r ❡♠ ❡q✉✐❧í❜r✐♦ ❡♠ ✉♠ ❝❡rt♦ ♣♦♥t♦✳ ❆ss✐♠✱ ♣♦❞❡✲s❡ ❞❡t❡r♠✐♥❛r ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠ s✐st❡♠❛ ❡①❛♠✐♥❛♥❞♦ ❛ ✈❛r✐❛çã♦ ❞❡ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦ ❡s❝❛❧❛r ✭❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ❡♥❡r❣✐❛ ♦✉ ❞❡ ▲②❛♣✉♥♦✈✮✳ ❈♦♠♣❛r❛♥❞♦ ❛s ❞❡✜♥✐çõ❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❡ ❡♥❡r❣✐❛ ♠❡❝â♥✐❝❛✱ ♣♦❞❡✲ s❡ ✈❡r ❛❧❣✉♠❛s r❡❧❛çõ❡s ❡♥tr❡ ❛ ❡♥❡r❣✐❛ ♠❡❝â♥✐❝❛ ❡ ❝♦♥❝❡✐t♦s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❞❡s❝r✐t♦s ❛♥t❡r✐♦r♠❡♥t❡✿
• ❡♥❡r❣✐❛ ③❡r♦ ❝♦rr❡s♣♦♥❞❡ ❛♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✭x= 0,x˙ = 0✮❀
• ❡st❛❜✐❧✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ✐♠♣❧✐❝❛ ♥❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❛ ❡♥❡r❣✐❛ ♠❡❝â♥✐❝❛ ♣❛r❛ ③❡r♦❀ • ✐♥st❛❜✐❧✐❞❛❞❡ é r❡❧❛❝✐♦♥❛❞❛ ❛♦ ❝r❡s❝✐♠❡♥t♦ ❞❛ ❡♥❡r❣✐❛ ♠❡❝â♥✐❝❛❀
❊ss❛s r❡❧❛çõ❡s ✐♥❞✐❝❛♠ q✉❡ ♦ ✈❛❧♦r ❞❛ q✉❛♥t✐❞❛❞❡ ❡s❝❛❧❛r ❞❛ ❡♥❡r❣✐❛ ♠❡❝â♥✐❝❛✱ ✐♥❞✐✲ r❡t❛♠❡♥t❡ r❡✢❡t❡ ❛ ♠❛❣♥✐t✉❞❡ ❞♦ ✈❡t♦r ❞❡ ❡st❛❞♦s✳ ❊ ♠❛✐s✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛ ♣♦❞❡ s❡r ❝❛r❛❝t❡r✐③❛❞❛ ♣❡❧❛ ✈❛r✐❛çã♦ ❞❛ ❡♥❡r❣✐❛ ❞♦ s✐st❡♠❛ ♠❡❝â♥✐❝♦✳ ❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠✲s❡ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❛ss♦❝✐❛❞♦s ❛♦ ♠ét♦❞♦ ❞✐r❡t♦ ❞❡ ▲②❛♣✉♥♦✈✳ P❛r❛ s✐♠♣❧✐✜❝❛r ❛ ❛♥á❧✐s❡ s❡rá ❝♦♥s✐❞❡r❛❞♦ q✉❡ ♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❡♠ ❛♥á❧✐s❡ é ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛✱ ✐st♦ é✱xeq = 0✳
❉❡✜♥✐çã♦ ✷✳✷✳✺ ✭❋✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦s✐t✐✈❛✮ ❯♠❛ ❢✉♥çã♦ ❡s❝❛❧❛rV(x)é ❞✐t❛ ❞❡✜♥✐❞❛
❧♦❝❛❧♠❡♥t❡ ♣♦s✐t✐✈❛✱ s❡ V(0) = 0 ❡ ♣❛r❛ t♦❞♦ x ♣❡rt❡♥❝❡♥t❡ ❛ ✉♠❛ ❜♦❧❛ S(0;R)✱ ✐st♦ é✱
x∈ {x:kxk ≤R , R >0}✱ ❛ r❡❧❛çã♦ ❛❜❛✐①♦ é s❛t✐s❢❡✐t❛✿
