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■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

◆♦♥❧✐♥❡❛r ▼♦❞❡❧ ❖r❞❡r ❘❡❞✉❝t✐♦♥ ✉s✐♥❣ P❖❉✴❉❊■▼ ❢♦r ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ♦❢ ❇✉r❣❡rs✬ ❊q✉❛t✐♦♥

▼❛♥✉❡❧ ▼✳ ❇❛✉♠❛♥♥ ❏✉❧② ✶✺✱ ✷✵✶✸

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❖✉t❧✐♥❡

✶ ❲❤❛t ✐s ▼♦❞❡❧ ❖r❞❡r ❘❡❞✉❝t✐♦♥ ✭▼❖❘✮ ❄ ✷ ▼♦❞❡❧ ❖r❞❡r ❘❡❞✉❝t✐♦♥ ✉s✐♥❣ P❖❉✲❉❊■▼

Pr♦♣❡r ❖rt❤♦❣♦♥❛❧ ❉❡❝♦♠♣♦s✐t✐♦♥ ✭P❖❉✮ ❉✐s❝r❡t❡ ❊♠♣✐r✐❝❛❧ ■♥t❡r♣♦❧❛t✐♦♥ ▼❡t❤♦❞ ✭❉❊■▼✮ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

✸ P❉❊✲❝♦♥str❛✐♥❡❞ ❖♣t✐♠✐③❛t✐♦♥

❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❋✐rst✲♦r❞❡r ♠❡t❤♦❞s✿ ❇❋●❙ ❛♥❞ ❙P●

✹ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❢♦r t❤❡ r❡❞✉❝❡❞✲♦r❞❡r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ✺ ❙✉♠♠❛r② ❛♥❞ ❢✉t✉r❡ r❡s❡❛r❝❤

✶ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

◆♦♥❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ s②st❡♠s

❈♦♥s✐❞❡r t❤❡ ♥♦♥❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ s②st❡♠ ˙ ②(t) = ❆②(t) + ❋(t, ②(t)), ②(t) ∈ R◆ ②(✵) = ②✵ ✭✶✮ ❛r✐s❡s ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✱ ❡✳❣✳ ♠❡❝❤❛♥✐❝❛❧ s②st❡♠s✱ ✢✉✐❞ ❞②♥❛♠✐❝s✱ ♥❡✉r♦♥ ♠♦❞❡❧✐♥❣✱ ✳✳✳ t❤❡ ♠❛tr✐① ❆ r❡♣r❡s❡♥ts t❤❡ ❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ ❜❡❤❛✈✐♦r ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ❋ r❡♣r❡s❡♥ts ♥♦♥❧✐♥❡❛r ❞②♥❛♠✐❝s ♦❢t❡♥ ❧❛r❣❡ ❞✐♠❡♥s✐♦♥ ♦❢ ✭✶✮ ❧❡❛❞s t♦ ❤✉❣❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✇♦r❦

✷ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

◆♦♥❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ s②st❡♠s

❈♦♥s✐❞❡r t❤❡ ♥♦♥❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ s②st❡♠ ˙ ②(t) = ❆②(t) + ❋(t, ②(t)), ②(t) ∈ R◆ ②(✵) = ②✵ ✭✶✮ ❛r✐s❡s ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✱ ❡✳❣✳ ♠❡❝❤❛♥✐❝❛❧ s②st❡♠s✱ ✢✉✐❞ ❞②♥❛♠✐❝s✱ ♥❡✉r♦♥ ♠♦❞❡❧✐♥❣✱ ✳✳✳ t❤❡ ♠❛tr✐① ❆ r❡♣r❡s❡♥ts t❤❡ ❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ ❜❡❤❛✈✐♦r ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ❋ r❡♣r❡s❡♥ts ♥♦♥❧✐♥❡❛r ❞②♥❛♠✐❝s ♦❢t❡♥ ❧❛r❣❡ ❞✐♠❡♥s✐♦♥ ♦❢ ✭✶✮ ❧❡❛❞s t♦ ❤✉❣❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✇♦r❦

✷ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

◆♦♥❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ s②st❡♠s

❈♦♥s✐❞❡r t❤❡ ♥♦♥❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ s②st❡♠ ˙ ②(t) = ❆②(t) + ❋(t, ②(t)), ②(t) ∈ R◆ ②(✵) = ②✵ ✭✶✮ ❛r✐s❡s ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✱ ❡✳❣✳ ♠❡❝❤❛♥✐❝❛❧ s②st❡♠s✱ ✢✉✐❞ ❞②♥❛♠✐❝s✱ ♥❡✉r♦♥ ♠♦❞❡❧✐♥❣✱ ✳✳✳ t❤❡ ♠❛tr✐① ❆ r❡♣r❡s❡♥ts t❤❡ ❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ ❜❡❤❛✈✐♦r ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ❋ r❡♣r❡s❡♥ts ♥♦♥❧✐♥❡❛r ❞②♥❛♠✐❝s ♦❢t❡♥ ❧❛r❣❡ ❞✐♠❡♥s✐♦♥ ♦❢ ✭✶✮ ❧❡❛❞s t♦ ❤✉❣❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✇♦r❦

✷ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

◆♦♥❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ s②st❡♠s

❈♦♥s✐❞❡r t❤❡ ♥♦♥❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ s②st❡♠ ˙ ②(t) = ❆②(t) + ❋(t, ②(t)), ②(t) ∈ R◆ ②(✵) = ②✵ ✭✶✮ ❛r✐s❡s ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✱ ❡✳❣✳ ♠❡❝❤❛♥✐❝❛❧ s②st❡♠s✱ ✢✉✐❞ ❞②♥❛♠✐❝s✱ ♥❡✉r♦♥ ♠♦❞❡❧✐♥❣✱ ✳✳✳ t❤❡ ♠❛tr✐① ❆ r❡♣r❡s❡♥ts t❤❡ ❧✐♥❡❛r ❞②♥❛♠✐❝❛❧ ❜❡❤❛✈✐♦r ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ❋ r❡♣r❡s❡♥ts ♥♦♥❧✐♥❡❛r ❞②♥❛♠✐❝s ♦❢t❡♥ ❧❛r❣❡ ❞✐♠❡♥s✐♦♥ ♦❢ ✭✶✮ ❧❡❛❞s t♦ ❤✉❣❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✇♦r❦

✷ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❚❤❡ ✐❞❡❛ ♦❢ ♠♦❞❡❧ ♦r❞❡r r❡❞✉❝t✐♦♥

❆♣♣r♦①✐♠❛t❡ t❤❡ st❛t❡ ✈✐❛ ②(t) ≈ ❯ℓ˜ ②(t), ❯ℓ ∈ R◆×ℓ, ˜ ② ∈ Rℓ, ✇❤❡r❡ t❤❡ ♠❛tr✐① ❯ℓ ❤❛s ♦rt❤♦♥♦r♠❛❧ ❝♦❧✉♠♥s✱ t❤❡ s♦✲❝❛❧❧❡❞ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ♦❢ ②✱ ❛♥❞ ℓ ≪ ◆✳

• ❛❧❡r❦✐♥ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❢✉❧❧✲♦r❞❡r s②st❡♠ ❧❡❛❞s t♦ ❛

r❡❞✉❝❡❞ s②st❡♠ ♦❢ ℓ ❡q✉❛t✐♦♥s✿ ❯❚

ℓ

• ❯ℓ ˙

˜ ② − ❆❯ℓ˜ ② − ❋(t, ❯ℓ˜ ②)

• = ✵

⇒ ˙ ˜ ② = ❯❚

ℓ ❆❯ℓ =:˜ ❆

˜ ② + ❯❚

ℓ ❋(t, ❯ℓ˜

②)

✸ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❚❤❡ ✐❞❡❛ ♦❢ ♠♦❞❡❧ ♦r❞❡r r❡❞✉❝t✐♦♥

❆♣♣r♦①✐♠❛t❡ t❤❡ st❛t❡ ✈✐❛ ②(t) ≈ ❯ℓ˜ ②(t), ❯ℓ ∈ R◆×ℓ, ˜ ② ∈ Rℓ, ✇❤❡r❡ t❤❡ ♠❛tr✐① ❯ℓ ❤❛s ♦rt❤♦♥♦r♠❛❧ ❝♦❧✉♠♥s✱ t❤❡ s♦✲❝❛❧❧❡❞ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ♦❢ ②✱ ❛♥❞ ℓ ≪ ◆✳

• ❛❧❡r❦✐♥ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❢✉❧❧✲♦r❞❡r s②st❡♠ ❧❡❛❞s t♦ ❛

r❡❞✉❝❡❞ s②st❡♠ ♦❢ ℓ ❡q✉❛t✐♦♥s✿ ❯❚

ℓ

• ❯ℓ ˙

˜ ② − ❆❯ℓ˜ ② − ❋(t, ❯ℓ˜ ②)

• = ✵

⇒ ˙ ˜ ② = ❯❚

ℓ ❆❯ℓ =:˜ ❆

˜ ② + ❯❚

ℓ ❋(t, ❯ℓ˜

②)

✸ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❚✇♦ q✉❡st✐♦♥s ❛r❡ ❧❡❢t✳✳✳

❈♦♥s✐❞❡r✐♥❣ t❤❡ r❡❞✉❝❡❞ ♠♦❞❡❧ ˙ ˜ ②(t) = ˜ ❆˜ ②(t) + ❯❚

ℓ ❋(t, ❯ℓ˜

②(t)), ˜ ②(t) ∈ Rℓ t✇♦ q✉❡st✐♦♥s ❛r❡ ❧❡❢t✿

✶ ❍♦✇ t♦ ♦❜t❛✐♥ t❤❡ ♠❛tr✐① ❯ℓ ♦❢ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ❄ ✷ ◆♦t❡ t❤❛t ❯ℓ˜

②(t) ∈ R◆ ✐s st✐❧❧ ❧❛r❣❡✳ ❍♦✇ ❞♦ ✇❡ ❡✈❛❧✉❛t❡ ❋(t, ❯ℓ˜ ②(t)) ❡✣❝✐❡♥t❧② ❄

✹ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❚✇♦ q✉❡st✐♦♥s ❛r❡ ❧❡❢t✳✳✳

❈♦♥s✐❞❡r✐♥❣ t❤❡ r❡❞✉❝❡❞ ♠♦❞❡❧ ˙ ˜ ②(t) = ˜ ❆˜ ②(t) + ❯❚

ℓ ❋(t, ❯ℓ˜

②(t)), ˜ ②(t) ∈ Rℓ t✇♦ q✉❡st✐♦♥s ❛r❡ ❧❡❢t✿

✶ ❍♦✇ t♦ ♦❜t❛✐♥ t❤❡ ♠❛tr✐① ❯ℓ ♦❢ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ❄ ✷ ◆♦t❡ t❤❛t ❯ℓ˜

②(t) ∈ R◆ ✐s st✐❧❧ ❧❛r❣❡✳ ❍♦✇ ❞♦ ✇❡ ❡✈❛❧✉❛t❡ ❋(t, ❯ℓ˜ ②(t)) ❡✣❝✐❡♥t❧② ❄

✹ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❚✇♦ q✉❡st✐♦♥s ❛r❡ ❧❡❢t✳✳✳

❈♦♥s✐❞❡r✐♥❣ t❤❡ r❡❞✉❝❡❞ ♠♦❞❡❧ ˙ ˜ ②(t) = ˜ ❆˜ ②(t) + ❯❚

ℓ ❋(t, ❯ℓ˜

②(t)), ˜ ②(t) ∈ Rℓ t✇♦ q✉❡st✐♦♥s ❛r❡ ❧❡❢t✿

✶ ❍♦✇ t♦ ♦❜t❛✐♥ t❤❡ ♠❛tr✐① ❯ℓ ♦❢ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ❄ ✷ ◆♦t❡ t❤❛t ❯ℓ˜

②(t) ∈ R◆ ✐s st✐❧❧ ❧❛r❣❡✳ ❍♦✇ ❞♦ ✇❡ ❡✈❛❧✉❛t❡ ❋(t, ❯ℓ˜ ②(t)) ❡✣❝✐❡♥t❧② ❄

✹ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ Pr♦♣❡r ❖rt❤♦❣♦♥❛❧ ❉❡❝♦♠♣♦s✐t✐♦♥ ✭P❖❉✮

❚❤❡ Pr♦♣❡r ❖rt❤♦❣♦♥❛❧ ❉❡❝♦♠♣♦s✐t✐♦♥ ✭P❖❉✮

❉✉r✐♥❣ t❤❡ ♥✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥✱ ❜✉✐❧❞ ✉♣ t❤❡ s♥❛♣s❤♦t ♠❛tr✐① ❨ := [②(t✶), ..., ②(t♥s)] ∈ R◆×♥s, ✇✐t❤ ♥s ❜❡✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ s♥❛♣s❤♦ts✳ P❡r❢♦r♠ ❛ ❙✐♥❣✉❧❛r ❱❛❧✉❡ ❉❡❝♦♠♣♦s✐t✐♦♥ ✭❙❱❉✮ ❨ = ❯Σ❱ ❚ ❛♥❞ ❧❡t ❯ℓ := ❯✭✿✱✶✿❧✮ ❝♦♥s✐st ♦❢ t❤♦s❡ ❧❡❢t s✐♥❣✉❧❛r ✈❡❝t♦rs ♦❢ ❨ t❤❛t ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ℓ ❧❛r❣❡st s✐♥❣✉❧❛r ✈❛❧✉❡s ✐♥ Σ✳

✺ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❉✐s❝r❡t❡ ❊♠♣✐r✐❝❛❧ ■♥t❡r♣♦❧❛t✐♦♥ ▼❡t❤♦❞ ✭❉❊■▼✮

❚❤❡ ❉✐s❝r❡t❡ ❊♠♣✐r✐❝❛❧ ■♥t❡r♣♦❧❛t✐♦♥ ▼❡t❤♦❞ ✭❉❊■▼✮

❈♦♥s✐❞❡r t❤❡ ♥♦♥❧✐♥❡❛r✐t② ◆ := ❯❚

ℓ

• ℓ×◆

❋(t, ❯ℓ˜ ②(t))

• ◆×✶

❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❋ ≈ ❲ ❝, ❲ ∈ R◆×♠, ❝ ∈ R♠ ✐s ♦✈❡r✲❞❡t❡r♠✐♥❡❞✳ ❚❤❡r❡❢♦r❡✱ ✜♥❞ ♣r♦❥❡❝t♦r P s✉❝❤ t❤❛t✿ P❚❋ = (P❚❲ )❝ ⇒ ❋ ≈ ❲ ❝ = ❲ (P❚❲ )−✶P❚❋ ⇒ ◆ ≈ ❯❚

ℓ ❲ (P❚❲ ) ♠×♠ −✶P❚❋(t, ❯ℓ⑦

②(t))

✻ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❉✐s❝r❡t❡ ❊♠♣✐r✐❝❛❧ ■♥t❡r♣♦❧❛t✐♦♥ ▼❡t❤♦❞ ✭❉❊■▼✮

❚❤❡ ❉✐s❝r❡t❡ ❊♠♣✐r✐❝❛❧ ■♥t❡r♣♦❧❛t✐♦♥ ▼❡t❤♦❞ ✭❉❊■▼✮

❈♦♥s✐❞❡r t❤❡ ♥♦♥❧✐♥❡❛r✐t② ◆ := ❯❚

ℓ

• ℓ×◆

❋(t, ❯ℓ˜ ②(t))

• ◆×✶

❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❋ ≈ ❲ ❝, ❲ ∈ R◆×♠, ❝ ∈ R♠ ✐s ♦✈❡r✲❞❡t❡r♠✐♥❡❞✳ ❚❤❡r❡❢♦r❡✱ ✜♥❞ ♣r♦❥❡❝t♦r P s✉❝❤ t❤❛t✿ P❚❋ = (P❚❲ )❝ ⇒ ❋ ≈ ❲ ❝ = ❲ (P❚❲ )−✶P❚❋ ⇒ ◆ ≈ ❯❚

ℓ ❲ (P❚❲ ) ♠×♠ −✶P❚❋(t, ❯ℓ⑦

②(t))

✼ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❉✐s❝r❡t❡ ❊♠♣✐r✐❝❛❧ ■♥t❡r♣♦❧❛t✐♦♥ ▼❡t❤♦❞ ✭❉❊■▼✮

❆❧❣♦r✐t❤♠ ✶ ❚❤❡ ❉❊■▼ ❛❧❣♦r✐t❤♠ ❬❈❤❛t✉r❛♥t❛❜✉t✱ ❙♦r❡♥s❡♥✱ ✷✵✶✵❪

✶✿ ■◆P❯❚✿ {✇✐}♠

✐=✶ ⊂ R◆ ❧✐♥❡❛r ✐♥❞❡♣❡♥❞❡♥t

✷✿ ❖❯❚P❯❚✿

• ℘ = [℘✶, ..., ℘♠]❚ ∈ R♠, P ∈ R◆×♠

✸✿ [|ρ|, ℘✶] = ♠❛①{|✇✶|} ✹✿ ❲ = [✇✶], P = [❡℘✶],

℘ = [℘✶]

✺✿ ❢♦r ✐ = ✷ t♦ ♠ ❞♦ ✻✿

❙♦❧✈❡ (P❚❲ )❝ = P❚✇✐ ❢♦r ❝

✼✿

r = ✇✐ − ❲ ❝

✽✿

[|ρ|, ℘✐] = ♠❛①{|r|}

✾✿

❲ ← [❲ ✇✐], P ← [P ❡℘✐], ℘ ← ℘ ℘✐

• ✶✵✿ ❡♥❞ ❢♦r

✽ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❉✐s❝r❡t❡ ❊♠♣✐r✐❝❛❧ ■♥t❡r♣♦❧❛t✐♦♥ ▼❡t❤♦❞ ✭❉❊■▼✮

❚❤❡ ♣r♦❞✉❝t P❚❋ ✐s ❛ s❡❧❡❝t✐♦♥ ♦❢ ❡♥tr✐❡s

▲❡t ♠ = ✸✳ ❙✉♣♣♦s❡ t❤❡ ❉❊■▼✲❛❧❣♦r✐t❤♠ ❤❛s ❝❤♦s❡♥ ✐♥❞✐❝❡s ℘✶, ..., ℘♠ s✉❝❤ t❤❛t✿ ❆ss✉♠✐♥❣ t❤❛t ❋(·) ❛❝ts ♣♦✐♥t✇✐s❡✱ ✇❡ ♦❜t❛✐♥✿ ◆ ≈ ❯❚

ℓ ❲ (P❚❲ )−✶P❚❋(t, ❯ℓ⑦

②(t)) = ❯❚

ℓ ❲ (P❚❲ )−✶

• ℓ×♠

❋(t, P❚❯ℓ⑦ ②(t))

• ♠×✶

✾ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

❚❤❡ ♥♦♥❧✐♥❡❛r ✶❉ ❇✉r❣❡rs✬ ♠♦❞❡❧ ②t + ✶ ✷②✷ − ν②①

• ① = ❢ ,

(①, t) ∈ (✵, ▲) × (✵, ❚), ②(t, ✵) = ②(t, ▲) = ✵, t ∈ (✵, ❚), ②(✵, ①) = ②✵(①), ① ∈ (✵, ▲).

✶ ❋❊▼✲❞✐s❝r❡t✐③❛t✐♦♥ ✐♥ s♣❛❝❡ ❧❡❛❞s t♦✿

▼ ˙ ②(t) = −✶ ✷❇②✷(t) − ν❈②(t) + ❢, t > ✵ ②(✵) = ②✵

✷ ❚✐♠❡ ✐♥t❡❣r❛t✐♦♥ ✈✐❛ ✐♠♣❧✐❝✐t ❊✉❧❡r ✰ ◆❡✇t♦♥✬s ♠❡t❤♦❞ ✶✵ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

P❖❉✲❉❊■▼ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

❙✉♣♣♦s❡✱ Φℓ ✐s ❛♥ ▼✲♦rt❤♦❣♦♥❛❧ P❖❉ ❜❛s✐s✳ ❚❤❡ P❖❉ r❡❞✉❝❡❞ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

=■ℓ

• Φ❚

ℓ ▼Φℓ ˙

⑦ ②(t) = − ✶ ✷Φ❚

ℓ ❇(Φℓ⑦

②(t))✷ − νΦ❚

ℓ ❈Φℓ⑦

②(t) ⇒ ˙ ⑦ ②(t) = − ✶ ✷❇ℓ(Φℓ⑦ ②(t))✷ − ν❈ℓ⑦ ②(t) ◆❡①t✱ ♦❜t❛✐♥ ❲ ✈✐❛ ❛ tr✉♥❝❛t❡❞ ❙❱❉ ♦❢ [②✷(t✶), ..., ②✷(t♥s )] ❛♥❞ ❛♣♣❧② ❉❊■▼ t♦ t❤❡ ❝♦❧✉♠♥s ♦❢ ❲ ✳ ❚❤❡ P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ˙ ⑦ ②(t) = − ✶ ✷ ˜ ❇(˜ ❋⑦ ②(t))✷ − ν ˜ ❈⑦ ②(t), ✇✐t❤ ˜ ❇ = Φ❚

ℓ ❇❲ (P❚ ❲ )−✶ ∈ Rℓ×♠✱ ˜

❋ = P❚ Φℓ ∈ R♠×ℓ✱ ❛♥❞ ˜ ❈ = ❈ℓ ∈ Rℓ×ℓ✳

✶✶ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

P❖❉✲❉❊■▼ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

❙✉♣♣♦s❡✱ Φℓ ✐s ❛♥ ▼✲♦rt❤♦❣♦♥❛❧ P❖❉ ❜❛s✐s✳ ❚❤❡ P❖❉ r❡❞✉❝❡❞ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

=■ℓ

• Φ❚

ℓ ▼Φℓ ˙

⑦ ②(t) = − ✶ ✷Φ❚

ℓ ❇(Φℓ⑦

②(t))✷ − νΦ❚

ℓ ❈Φℓ⑦

②(t) ⇒ ˙ ⑦ ②(t) = − ✶ ✷❇ℓ(Φℓ⑦ ②(t))✷ − ν❈ℓ⑦ ②(t) ◆❡①t✱ ♦❜t❛✐♥ ❲ ✈✐❛ ❛ tr✉♥❝❛t❡❞ ❙❱❉ ♦❢ [②✷(t✶), ..., ②✷(t♥s )] ❛♥❞ ❛♣♣❧② ❉❊■▼ t♦ t❤❡ ❝♦❧✉♠♥s ♦❢ ❲ ✳ ❚❤❡ P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ˙ ⑦ ②(t) = − ✶ ✷ ˜ ❇(˜ ❋⑦ ②(t))✷ − ν ˜ ❈⑦ ②(t), ✇✐t❤ ˜ ❇ = Φ❚

ℓ ❇❲ (P❚ ❲ )−✶ ∈ Rℓ×♠✱ ˜

❋ = P❚ Φℓ ∈ R♠×ℓ✱ ❛♥❞ ˜ ❈ = ❈ℓ ∈ Rℓ×ℓ✳

✶✶ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

0.5 1 0.5 1 1 t ℓ = 3, m = 13 x yℓ(t, x)

✶✷ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

0.5 1 0.5 1 1 t ℓ = 5, m = 13 x yℓ(t, x)

✶✸ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

0.5 1 0.5 1 1 t ℓ = 7, m = 13 x yℓ(t, x)

✶✹ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

0.5 1 0.5 1 1 t ℓ = 9, m = 13 x yℓ(t, x)

✶✺ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

0.5 1 0.5 1 1 t ℓ = 11, m = 13 x yℓ(t, x)

✶✻ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

❈♦♠♣✉t❛t✐♦♥❛❧ ❙♣❡❡❞✉♣ ❬✶❪

❈♦♥❝❧✉s✐♦♥✿ ❍✐❣❤ ❛❝❝✉r❛❝② ♦❢ t❤❡ P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ♠♦❞❡❧✳ ❇✉t ✐s ✐t ❛❧s♦ ❢❛st❡r❄

POD POD−DEIM 0.5 1 1.5 2 Speedup ν = 0.01, N = 80 POD POD−DEIM 1 2 3 4 5 Speedup ν = 0.001, N = 200 POD POD−DEIM 10 20 30 40 50 60 Speedup ν = 0.0001, N = 800

❙♣❛t✐❛❧ ❞✐s❝r❡t✐③❛t✐♦♥ ♦❢ t❤❡ ❢✉❧❧ ♠♦❞❡❧ ❞❡♣❡♥❞s ♦♥ ✈✐s❝♦s✐t② ♣❛r❛♠❡t❡r ν ❝❤♦♦s❡ ℓ, ♠ s✉❝❤ t❤❛t r❡❧❛t✐✈❡ ▲✷✲❡rr♦r ✐♥ O(✶✵−✹)

✶✼ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❆♣♣❧✐❝❛t✐♦♥✿ ▼❖❘ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥

❈♦♠♣✉t❛t✐♦♥❛❧ ❙♣❡❡❞✉♣ ❬✷❪

❋♦r ❛ ✜①❡❞ ν = ✵.✵✶✱ ✇❡ ❝♦✉❧❞ s❤♦✇ t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ♠♦❞❡❧ ♦❢ t❤❡ ❢✉❧❧✲♦r❞❡r ❞✐♠❡♥s✐♦♥ ◆✳

80 500 1.000 1.500 2.000 2.500 3.000 0.02 0.04 0.06 0.08 0.1 0.12 0.14 N tPDEsol [s] ν = 0.01 POD POD−DEIM 80 500 1.000 1.500 2.000 2.500 3.000 200 400 600 800 1000 N SP

(2)

ν = 0.01 POD POD−DEIM

❈♦♠♣✉t❛t✐♦♥ t✐♠❡ ❢♦r s♦❧✈✐♥❣ t❤❡ P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ✐s ❛❧♠♦st ❝♦♥st❛♥t ✭r✐❣❤t✮ P❖❉✲❉❊■▼ ❛❧♠♦st ✹ t✐♠❡s ❢❛st❡r t❤❛♥ ♣✉r❡ P❖❉ ✭❧❡❢t✮

✶✽ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

P❉❊✲❝♦♥str❛✐♥❡❞ ♦♣t✐♠✐③❛t✐♦♥ ▼✐♥✐♠✐③❡ ♠✐♥

✉ J (②(✉), ✉),

✇❤❡r❡ ② ✐s t❤❡ s♦❧✉t✐♦♥ t♦ ❛ ♥♦♥❧✐♥❡❛r✱ ♣♦ss✐❜❧② t✐♠❡✲❞❡♣❡♥❞❡♥t ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✱ ❝(②, ✉) = ✵. J ✐s ❝❛❧❧❡❞ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✱ ✐♥ ♦r❞❡r t♦ ❡✈❛❧✉❛t❡ J ✱ ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ ❝(②, ✉) = ✵ ❢♦r ②(✉)✱ s♦❧✈❡ ✇✐t❤ ❛❧❣♦r✐t❤♠s ❢♦r ✉♥❝♦♥str❛✐♥❡❞ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✳

✶✾ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠

❆ ◆❡✇t♦♥✲t②♣❡ ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠

▼✐♥✐♠✐③❡ J (②(✉), ✉) ✐♥ ✉ ✉s✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡✳

✷✵ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠

• r❛❞✐❡♥t ❝♦♠♣✉t❛t✐♦♥ ✈✐❛ ❛❞❥♦✐♥ts

❈♦♥s✐❞❡r t❤❡ ▲❛❣r❛♥❣✐❛♥ ❢✉♥❝t✐♦♥ L(②, ✉, λ) = J (②, ✉) + λ❚❝(②, ✉) ❛♥❞ ✐♠♣♦s❡ t❤❡ ③❡r♦✲❣r❛❞✐❡♥t ❝♦♥❞✐t✐♦♥ ∇②L(②, ✉, λ) = ✵✳ ❲❡ ❞❡r✐✈❡ t❤❡ ❛❞❥♦✐♥t ❡q✉❛t✐♦♥✿ ❝②(②(✉), ✉)❚λ = −∇②J (②(✉), ✉) ❆❧❣♦r✐t❤♠ ✷ ❈♦♠♣✉t✐♥❣ ∇ ˆ

J (✉) ✈✐❛ ❛❞❥♦✐♥ts ❬❍❡✐♥❦❡♥s❝❤❧♦ss✱ ✷✵✵✽❪

✶✿ ❋♦r ❛ ❣✐✈❡♥ ❝♦♥tr♦❧ ✉✱ s♦❧✈❡ ❝(②, ✉) = ✵ ❢♦r t❤❡ st❛t❡ ②(✉) ✷✿ ❙♦❧✈❡ t❤❡ ❛❞❥♦✐♥t ❡q✉❛t✐♦♥ ❝②(②(✉), ✉)❚λ = −∇②J (②(✉), ✉) ❢♦r λ(✉) ✸✿ ❈♦♠♣✉t❡ ∇ ˆ

J (✉) = ∇✉J (②(✉), ✉) + ❝✉(②(✉), ✉)❚λ(✉)

✷✶ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❙❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠

• r❛❞✐❡♥t ❝♦♠♣✉t❛t✐♦♥ ✈✐❛ ❛❞❥♦✐♥ts

❈♦♥s✐❞❡r t❤❡ ▲❛❣r❛♥❣✐❛♥ ❢✉♥❝t✐♦♥ L(②, ✉, λ) = J (②, ✉) + λ❚❝(②, ✉) ❛♥❞ ✐♠♣♦s❡ t❤❡ ③❡r♦✲❣r❛❞✐❡♥t ❝♦♥❞✐t✐♦♥ ∇②L(②, ✉, λ) = ✵✳ ❲❡ ❞❡r✐✈❡ t❤❡ ❛❞❥♦✐♥t ❡q✉❛t✐♦♥✿ ❝②(②(✉), ✉)❚λ = −∇②J (②(✉), ✉) ❆❧❣♦r✐t❤♠ ✸ ❈♦♠♣✉t✐♥❣ ∇ ˆ

J (✉) ✈✐❛ ❛❞❥♦✐♥ts ❬❍❡✐♥❦❡♥s❝❤❧♦ss✱ ✷✵✵✽❪

✶✿ ❋♦r ❛ ❣✐✈❡♥ ❝♦♥tr♦❧ ✉✱ s♦❧✈❡ ❝(②, ✉) = ✵ ❢♦r t❤❡ st❛t❡ ②(✉) ✷✿ ❙♦❧✈❡ t❤❡ ❛❞❥♦✐♥t ❡q✉❛t✐♦♥ ❝②(②(✉), ✉)❚λ = −∇②J (②(✉), ✉) ❢♦r λ(✉) ✸✿ ❈♦♠♣✉t❡ ∇ ˆ

J (✉) = ∇✉J (②(✉), ✉) + ❝✉(②(✉), ✉)❚λ(✉)

✷✶ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡ ❋✐rst✲♦r❞❡r ♠❡t❤♦❞s✿ ❇❋●❙ ❛♥❞ ❙P●

❋✐rst✲♦r❞❡r ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠s

■♥st❡❛❞ ♦❢ s♦❧✈✐♥❣ ∇✷J (②❦, ✉❦)s❦ = −∇J (②❦, ✉❦), ✜rst✲♦r❞❡r ♠❡t❤♦❞s ❛♣♣r♦①✐♠❛t❡ t❤❡ ❍❡ss✐❛♥ ✈✐❛ ❍❦ ❛♥❞ s♦❧✈❡ ❍❦s❦ = −∇J (②❦, ✉❦). ❲❡ ❤❛✈❡ ✉s❡❞ ▼❛t❧❛❜ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ♦❢ t❤❡ ❇❋●❙ ❛♥❞ t❤❡ ❙P● ♠❡t❤♦❞✱ ❊✈❛❧✉❛t✐♦♥ ♦❢ J ❛♥❞ ❣r❛❞✐❡♥t ❝♦♠♣✉t❛t✐♦♥ ❛s s❡❡♥ ❜❡❢♦r❡✱ ❙P● ❡❛s✐❧② ❛❧❧♦✇s t♦ ✐♥❝❧✉❞❡ ❜♦✉♥❞s ♦♥ t❤❡ ❝♦♥tr♦❧✱ ✐✳❡✳ ✉❧♦✇❡r ≤ ✉(t, ①) ≤ ✉✉♣♣❡r ✇❤✐❝❤ ✐s ✉s❡❞ ✐♥ ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s

✷✷ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❖♣t✐♠❛❧ ❈♦♥tr♦❧ ♣r♦❜❧❡♠ ❢♦r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ▼✐♥✐♠✐③❡ ♠✐♥

✉

✶ ✷ ❚

✵

▲

✵

[②(t, ①) − ③(t, ①)]✷ + ω✉✷(t, ①) ❞① ❞t, ✇❤❡r❡ ② ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ♥♦♥❧✐♥❡❛r ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ②t + ✶ ✷②✷ − ν②①

• ①

= ❢ + ✉, (①, t) ∈ (✵, ▲) × (✵, ❚), ②(t, ✵) = ②(t, ▲) = ✵, t ∈ (✵, ❚), ②(✵, ①) = ②✵(①), ① ∈ (✵, ▲). ✉ ✐s t❤❡ ❝♦♥tr♦❧ t❤❛t ❞❡t❡r♠✐♥❡s ② ③ ✐s t❤❡ ❞❡s✐r❡❞ st❛t❡

✷✸ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❈♦♥tr♦❧ ❣♦❛❧

❲❡ ✇❛♥t t♦ ❝♦♥tr♦❧ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t ✐t st❛②s ✐♥ t❤❡ ❞❡s✐r❡❞ st❛t❡ ③(t, ·) = ②✵, ∀t✿

0.5 1 0.5 1 0.5 1 1.5 t uncontrolled state x y(t,x) 0.5 1 0.5 1 0.5 1 1.5 t desired state x z(t,x)

❋✐❣✉r❡✿ ❯♥❝♦♥tr♦❧❧❡❞ ✭✉ ≡ ✵✮ ❛♥❞ ❞❡s✐r❡❞ st❛t❡ ❢♦r ν = ✵.✵✶✳

✷✹ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t

✶ ❉✐s❝r❡t✐③❡ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ✐♥ t✐♠❡

❛♥❞ s♣❛❝❡

✷ ❆♣♣❧② ❛❞❥♦✐♥ts ✐♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ ❣r❛❞✐❡♥t ❛♥❞ ❍❡ss✐❛♥ ✸ ❆♣♣❧② ✜rst✲♦r❞❡r ♦r s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✹ ❊①♣❧♦r❡ t❤❡ ✉s❛❣❡ ♦❢ ❛ P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ♠♦❞❡❧ ✷✺ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t

✶ ❉✐s❝r❡t✐③❡ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ✐♥ t✐♠❡

❛♥❞ s♣❛❝❡

✷ ❆♣♣❧② ❛❞❥♦✐♥ts ✐♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ ❣r❛❞✐❡♥t ❛♥❞ ❍❡ss✐❛♥ ✸ ❆♣♣❧② ✜rst✲♦r❞❡r ♦r s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✹ ❊①♣❧♦r❡ t❤❡ ✉s❛❣❡ ♦❢ ❛ P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ♠♦❞❡❧ ✷✺ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t

✶ ❉✐s❝r❡t✐③❡ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ✐♥ t✐♠❡

❛♥❞ s♣❛❝❡

✷ ❆♣♣❧② ❛❞❥♦✐♥ts ✐♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ ❣r❛❞✐❡♥t ❛♥❞ ❍❡ss✐❛♥ ✸ ❆♣♣❧② ✜rst✲♦r❞❡r ♦r s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✹ ❊①♣❧♦r❡ t❤❡ ✉s❛❣❡ ♦❢ ❛ P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ♠♦❞❡❧ ✷✺ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t

✶ ❉✐s❝r❡t✐③❡ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ✐♥ t✐♠❡

❛♥❞ s♣❛❝❡

✷ ❆♣♣❧② ❛❞❥♦✐♥ts ✐♥ ♦r❞❡r t♦ ❝♦♠♣✉t❡ ❣r❛❞✐❡♥t ❛♥❞ ❍❡ss✐❛♥ ✸ ❆♣♣❧② ✜rst✲♦r❞❡r ♦r s❡❝♦♥❞✲♦r❞❡r ♦♣t✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✹ ❊①♣❧♦r❡ t❤❡ ✉s❛❣❡ ♦❢ ❛ P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ♠♦❞❡❧ ✷✺ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

◆✉♠❡r✐❝❛❧ t❡sts

◆❡✇t♦♥✲t②♣❡ ♠❡t❤♦❞ ❢♦r t❤❡ ❢✉❧❧✲♦r❞❡r ❇✉r❣❡rs✬ ♠♦❞❡❧✿

1 1 −0,5 1 1.5 t x y(t,x)

❦ = ✵ ✭✉♥❝♦♥tr♦❧❧❡❞✮

1 1 −0,5 1 1,5 t x y(t,x)

❦ = ✶

1 1 −0,5 1 1,5 t x y(t,x)

❦ = ✷

1 1 −0,5 1 1,5 t x y(t,x)

❦ = ✸

1 1 −0.5 1 1.5 t x y(t,x)

❦ = ✹

1 1 −0.5 1 1.5 t x y(t,x)

❦ = ✺

✷✻ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ❛t ❡❛❝❤ ✐t❡r❛t✐♦♥✿

1 1 −6 7 u(t,x) x t

❦ = ✵ ✭✐♥✐t✐❛❧✮

1 1 −6 7 t x u(t,x)

❦ = ✶

1 1 −6 7 t x u(t,x)

❦ = ✷

1 1 −6 7 t x u(t,x)

❦ = ✸

1 1 −6 7 t x u(t,x)

❦ = ✹

1 1 −6 7 t x u(t,x)

❦ = ✺

✷✼ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❲❡ ♣r♦♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣♦r✐t❤♠ ❢♦r P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ✿

✷✽ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❋✐♥❛❧ st❛t❡ ❛♥❞ ❝♦♥tr♦❧ ♦❢ t❤❡ P❖❉✲❉❊■▼ r❡❞✉❝❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✿

1 1 −0.5 1 t ν = 0.01 x Φℓ y(t)

ℓ = ♠ = ✼

1 1 −0.5 1 t ν = 0.01 x Φℓ y(t)

ℓ = ♠ = ✶✺

1 1 −9 5 t ν = 0.01 x u(t,x)

ℓ = ♠ = ✼

1 1 −9 5 t ν = 0.01 x u(t,x)

ℓ = ♠ = ✶✺

✷✾ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❈♦♠♣✉t❛t✐♦♥❛❧ ❙♣❡❡❞✉♣ ❬✸❪

❘❡❞✉❝❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ✉s✐♥❣ t❤❡ ◆❡✇t♦♥✲t②♣❡ ♠❡t❤♦❞✿

20 40 60 80 100 Speedup POD POD−DEIM ν = 0.01 ν = 0.001 ν = 0.0001

101.3 79.0 3.7 4.4 1.35 1.9

❛t ✜♥❛❧ st❛t❡✿ r❡❧❛t✐✈❡ ▲✷✲❡rr♦r ✐♥ O(✶✵−✷) ❝♦♠♣❛r❛❜❧❡ ✈❛❧✉❡ ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❛t ❝♦♥✈❡r❣❡♥❝❡ ✉s❡ s❛♠❡ st♦♣♣✐♥❣ ❝r✐t❡r✐❛ ❢♦r ❢✉❧❧✲♦r❞❡r ❛♥❞ r❡❞✉❝❡❞ ♠♦❞❡❧

✸✵ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❈♦♠♣✉t❛t✐♦♥❛❧ ❙♣❡❡❞✉♣ ❬✹❪

❙♦♠❡ ♦t❤❡r r❡s✉❧ts✳ ❋♦r ν = ✵.✵✵✵✶✱ ❧♦✇✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥tr♦❧ ❧❡❛❞s t♦ ❛ s♣❡❡❞✉♣ ♦❢ ∼ ✷✵ ❢♦r ❛❧❧ t❤r❡❡ ♦♣t✐♠✐③❛t✐♦♥ ♠❡t❤♦❞s✳

1 1 −30 15 t discrete control u(t,x) x

❙P● ❛❧❧♦✇s ❛ ❜♦✉♥❞❡❞ ❝♦♥tr♦❧ −✷ ≤ ✉(t, ①) ≤ ✷✳ ❋♦r ν = ✵.✵✵✵✶✱ ✇❡ ♦❜t❛✐♥❡❞ ❛ s♣❡❡❞✉♣ ♦❢ ✸.✻ ❢♦r P❖❉ ❛♥❞ ✽.✽ ❢♦r P❖❉✲❉❊■▼✳

1 1 −2 2

t bounded control u(t,x) x

✸✶ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❈♦♥❝❧✉❞✐♥❣ ❘❡♠❛r❦s

❲❤❛t ■ ❧❡❛r♥t✿ ❚❤❡ ❛❝❝✉r❛❝② ♦❢ t❤❡ r❡❞✉❝❡❞ ❇✉r❣❡rs✬ ♠♦❞❡❧ ✐s ♦❢ t❤❡ s❛♠❡ ♦r❞❡r ✇❤❡♥ P❖❉ ✐s ❡①t❡♥❞❡❞ ❜② ❉❊■▼✳ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ♦❢ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ✉s✐♥❣ P❖❉✲❉❊■▼ ❧❡❛❞s t♦ ❛ s♣❡❡❞✉♣ ♦❢ ∼ ✶✵✵ ❢♦r s♠❛❧❧ ν✳ ❋♦r t❤❡ r❡❞✉❝❡❞ ♠♦❞❡❧✱ ❛❧❧ ❞❡r✐✈❛t✐✈❡s ♥❡❡❞ t♦ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ r❡❞✉❝❡❞ ✈❛r✐❛❜❧❡✳ ❚❤✐s ❝❛♥ ❜❡ q✉✐t❡ ❤❛r❞ ✐♥ ♣r❛❝t✐❝❡✳

✸✷ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❋✉t✉r❡ ❘❡s❡❛r❝❤

❲❤❛t ■ st✐❧❧ ✇❛♥t t♦ ❧❡❛r♥✿ ❯s❡ t❤❡ P❖❉ ❜❛s✐s Φℓ ❛❧s♦ ❢♦r ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ♦❢ t❤❡ ❝♦♥tr♦❧✱ ✐✳❡✳ ✉(t) ≈ Φℓ⑦ ✉(t) =

ℓ

• ✐=✶

ϕ✐˜ ✉✐(t) ❊①t❡♥❞ ❇✉r❣❡rs✬ ♠♦❞❡❧ t♦ ✷❉✴✸❉ ▼♦r❡ s♦♣❤✐st✐❝❛t❡❞ ❝❤♦✐❝❡ ♦❢ r❡❞✉❝❡❞ ❞✐♠❡♥s✐♦♥s ℓ ❛♥❞ ♠

✸✸ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❚❤✐s ▼❛st❡r ♣r♦❥❡❝t ✇❛s s✉♣❡r✈✐s❡❞ ❜② ▼❛r✐❡❧❜❛ ❘♦❥❛s ❛♥❞ ▼❛rt✐♥ ✈❛♥ ●✐❥③❡♥✳ ❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦ ❆r❡ t❤❡r❡ ❛♥② q✉❡st✐♦♥s ♦r r❡♠❛r❦s❄ ❤tt♣s✿✴✴❣✐t❤✉❜✳❝♦♠✴▼❛♥✉❡❧▼❇❛✉♠❛♥♥✴▼❛st❡r❚❤❡s✐s

✸✹ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❚❤✐s ▼❛st❡r ♣r♦❥❡❝t ✇❛s s✉♣❡r✈✐s❡❞ ❜② ▼❛r✐❡❧❜❛ ❘♦❥❛s ❛♥❞ ▼❛rt✐♥ ✈❛♥ ●✐❥③❡♥✳ ❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦ ❆r❡ t❤❡r❡ ❛♥② q✉❡st✐♦♥s ♦r r❡♠❛r❦s❄ ❤tt♣s✿✴✴❣✐t❤✉❜✳❝♦♠✴▼❛♥✉❡❧▼❇❛✉♠❛♥♥✴▼❛st❡r❚❤❡s✐s

✸✹ ✴ ✸✹

■♥tr♦❞✉❝t✐♦♥ P❖❉✲❉❊■▼ ❛❧❣♦r✐t❤♠ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ❇✉r❣❡rs✬ ❡q✉❛t✐♦♥ ❖✉t❧♦♦❦ ▲✐t❡r❛t✉r❡

❋✉rt❤❡r ✐♥❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥✳✳✳

❙✳ ❈❤❛t✉r❛♥t❛❜✉t ❛♥❞ ❉✳ ❙♦r❡♥s❡♥ ◆♦♥❧✐♥❡❛r ▼♦❞❡❧ ❘❡❞✉❝t✐♦♥ ✈✐❛ ❉✐s❝r❡t❡ ❊♠♣✐r✐❝❛❧ ■♥t❡r♣♦❧❛t✐♦♥✳ ❙■❆▼ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥t✐✜❝ ❈♦♠♣✉t✐♥❣✱ ✷✵✶✵✳ ▼✳ ❍❡✐♥❦❡♥s❝❤❧♦ss ◆✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✐♠♣❧✐❝✐t❧② ❝♦♥str❛✐♥❡❞ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✳ ❚❡❝❤♥✐❝❛❧ r❡♣♦rt✱ ❉❡♣❛rt♠❡♥t ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ❘✐❝❡ ❯♥✐✈❡rs✐t②✱ ✷✵✵✽✳ ❑✳ ❑✉♥✐s❝❤ ❛♥❞ ❙✳ ❱♦❧❦✇❡✐♥ ❈♦♥tr♦❧ ♦❢ t❤❡ ❇✉r❣❡rs ❊q✉❛t✐♦♥ ❜② ❛ ❘❡❞✉❝❡❞✲❖r❞❡r ❆♣♣r♦❛❝❤ ❯s✐♥❣ Pr♦♣❡r ❖rt❤♦❣♦♥❛❧ ❉❡❝♦♠♣♦s✐t✐♦♥✳ ❏♦✉r♥❛❧ ♦❢ ❖♣t✐♠✐③❛t✐♦♥ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✶✾✾✾✳