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■♥tr♦❞✉❝t✐♦♥ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ◗❱■s ◆✉♠❡r✐❝❛❧ ✐ss✉❡s ❈♦♥❝❧✉s✐♦♥ ❇❛❝❦✇❛r❞ ❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❛♥❞ ◗✉❛s✐✲❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s✿ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♠♣✉❧s❡ ❝♦♥tr♦❧s ✐♥ ✜♥❛♥❝❡ ❍✉②ê♥ P❍❆▼ P▼❆ ❯♥✐✈❡rs✐té P❛r✐s ✼✱ ❛♥❞ ■♥st✐t✉t ❯♥✐✈❡rs✐t❛✐r❡ ❞❡ ❋r❛♥❝❡ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ✿ ■✳ ❑❤❛rr♦✉❜✐ ✭P▼❆✱ ❈❘❊❙❚✮✱ ❏✳ ▼❛ ❛♥❞ ❏✳ ❩❤❛♥❣ ✭❯❙❈✮ ❲♦r❦s❤♦♣ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ▲✐♥③✱ ❖❝t♦❜❡r ✷✷✱ ✷✵✵✽ ❍✉②ê♥ P❍❆▼ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❛♥❞ ◗❱■s

■♥tr♦❞✉❝t✐♦♥ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ◗❱■s ◆✉♠❡r✐❝❛❧ ✐ss✉❡s ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ❈♦♥s✐❞❡r t❤❡ ♣❛r❛❜♦❧✐❝ ◗✉❛s✐✲❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t② ✭◗❱■✮ ✿ ♠✐♥ [ − ∂ t ✈ − L ✈ − ❢ , ✈ − H ✈ ] = ✵ , ✈ ( ❚ , . ) = ❣ , ✭✶✮ ✇❤❡r❡ L ✐s t❤❡ s❡❝♦♥❞ ♦r❞❡r ❧♦❝❛❧ ♦♣❡r❛t♦r L ✈ ( t , ① ) = ❜ ( ① ) . ❉ ① ✈ ( t , ① ) + ✶ ✷ tr ( σσ ′ ( ① ) ❉ ✷ ① ✈ ( t , ① )) ❛♥❞ H ✐s t❤❡ ♥♦♥❧♦❝❛❧ ♦♣❡r❛t♦r H ❡ ✈ ( t , ① ) H ✈ ( t , ① ) = s✉♣ ❡ ∈ ❊ ✇✐t❤ H ❡ ✈ ( t , ① ) = ✈ ( t , ① + γ ( ① , ❡ )) + ❝ ( ① , ❡ ) . ❍✉②ê♥ P❍❆▼ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❛♥❞ ◗❱■s

■♥tr♦❞✉❝t✐♦♥ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ◗❱■s ◆✉♠❡r✐❝❛❧ ✐ss✉❡s ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✭■■✮ ❚❤❡ ◗❱■ ✭✶✮ ✐s t❤❡ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✐♠♣✉❧s❡ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ✭s❡❡ ❇❡♥s♦✉ss❛♥✲▲✐♦♥s ✽✷ ♦r Ø❦s❡♥❞❛❧✲❙✉❧❡♠ ✵✻✮ ✿ � ❚ � � � ❣ ( ❳ α ❢ ( ❳ α ❝ ( ❳ α ✈ ( t , ① ) = s✉♣ α ❊ ❚ ) + s ) ❞s + ✐ , ξ ✐ ) τ − t t <τ ✐ ≤ s ✇✐t❤ • ❝♦♥tr♦❧s ✿ α = ( τ ✐ , ξ ✐ ) ✐ ✇❤❡r❡ ( τ ✐ ) ✐ t✐♠❡ ❞❡❝✐s✐♦♥s ✿ ♥♦♥❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ st♦♣♣✐♥❣ t✐♠❡s ( ξ ✐ ) ✐ ❛❝t✐♦♥ ❞❡❝✐s✐♦♥s ✿ s❡q✉❡♥❝❡ ♦❢ r✳✈✳ s✳t✳ ξ ✐ ∈ F τ ✐ ✈❛❧✉❡❞ ✐♥ ❊ ✱ • ❝♦♥tr♦❧❧❡❞ ♣r♦❝❡ss ❳ α ❞❡✜♥❡❞ ❜② � s � s � ❳ α ❜ ( ❳ α σ ( ❳ α γ ( ❳ α s = ① + ✉ ) ❞✉ + ✉ ) ❞❲ ✉ + ✐ , ξ ✐ ) τ − t t t <τ ✐ ≤ s ❍✉②ê♥ P❍❆▼ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❛♥❞ ◗❱■s

■♥tr♦❞✉❝t✐♦♥ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ◗❱■s ◆✉♠❡r✐❝❛❧ ✐ss✉❡s ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✭■■■✮ ❱❛r✐♦✉s ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ✐♠♣✉❧s❡ ❝♦♥tr♦❧s ✿ • ❋✐♥❛♥❝✐❛❧ ♠♦❞❡❧❧✐♥❣ ✇✐t❤ ❞✐s❝r❡t❡ tr❛♥s❛❝t✐♦♥ ❞❛t❡s✱ ❞✉❡ ❡✳❣✳ t♦ ✜①❡❞ tr❛♥s❛❝t✐♦♥ ❝♦sts ♦r ❧✐q✉✐❞✐t② ❝♦♥str❛✐♥ts • ❖♣t✐♠❛❧ ♠✉❧t✐♣❧❡ st♦♣♣✐♥❣ ✿ s✇✐♥❣ ♦♣t✐♦♥s • ❋✐r♠✬s ✐♥✈❡st♠❡♥t ❛♥❞ r❡❛❧ ♦♣t✐♦♥s ✿ ♠❛♥❛❣❡♠❡♥t ♦❢ ♣♦✇❡r ♣❧❛♥ts✱ ✈❛❧✉❛t✐♦♥ ♦❢ ❣❛s st♦r❛❣❡✱ ✳✳✳ . . . • ▼♦r❡ ❣❡♥❡r❛❧❧② t♦ ♠♦❞❡❧s ✇✐t❤ ❝♦♥tr♦❧ ♣♦❧✐❝✐❡s t❤❛t ❞♦ ♥♦t ❛❝❝✉♠✉❧❛t❡ ✐♥ t✐♠❡✳ → ▼❛♥② ♣❛♣❡rs ✦ ❍✉②ê♥ P❍❆▼ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❛♥❞ ◗❱■s

❈❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ✿ ❉❡❝♦✉♣❧❡ t❤❡ ◗❱■ ✭✶✮ ❜② ❞❡✜♥✐♥❣ ❜② ✐t❡r❛t✐♦♥ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✈ ♥ ♥ ✿ ♠✐♥ ✵ ✭✷✮ t ✈ ♥ ✈ ♥ ❢ ✈ ♥ ✈ ♥ ✈ ♥ ✶ ❚ ❣ ✶ ✶ ✶ ❛ss♦❝✐❛t❡❞ t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ t✐♠❡ ♣r♦❜❧❡♠s ✭r❡✢❡❝t❡❞ ❇❙❉❊s✮ ❋✉rt❤❡r♠♦r❡✱ t♦ ❝♦♠♣✉t❡ ✈ ♥ ✶ ✱ ✇❡ ♥❡❡❞ t♦ ❦♥♦✇ ✈ ♥ ♦♥ t❤❡ ✇❤♦❧❡ ❞♦♠❛✐♥ ❤❡❛✈② ❝♦♠♣✉t❛t✐♦♥s ✿ ♥✉♠❡r✐❝❛❧❧② ❝❤❛❧❧❡♥❣✐♥❣ ✦ ■♥tr♦❞✉❝t✐♦♥ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ◗❱■s ◆✉♠❡r✐❝❛❧ ✐ss✉❡s ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✭■❱✮ • ▼❛✐♥ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❞✐✣❝✉❧t② ✐♥ t❤❡ ◗❱■ ✭✶✮ ✿ ❚❤❡ ♦❜st❛❝❧❡ t❡r♠ ❝♦♥t❛✐♥s t❤❡ s♦❧✉t✐♦♥ ✐ts❡❧❢ ■t ✐s ♥♦♥❧♦❝❛❧ ❍✉②ê♥ P❍❆▼ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❛♥❞ ◗❱■s

❋✉rt❤❡r♠♦r❡✱ t♦ ❝♦♠♣✉t❡ ✈ ♥ ✶ ✱ ✇❡ ♥❡❡❞ t♦ ❦♥♦✇ ✈ ♥ ♦♥ t❤❡ ✇❤♦❧❡ ❞♦♠❛✐♥ ❤❡❛✈② ❝♦♠♣✉t❛t✐♦♥s ✿ ♥✉♠❡r✐❝❛❧❧② ❝❤❛❧❧❡♥❣✐♥❣ ✦ ■♥tr♦❞✉❝t✐♦♥ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ◗❱■s ◆✉♠❡r✐❝❛❧ ✐ss✉❡s ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✭■❱✮ • ▼❛✐♥ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❞✐✣❝✉❧t② ✐♥ t❤❡ ◗❱■ ✭✶✮ ✿ ❚❤❡ ♦❜st❛❝❧❡ t❡r♠ ❝♦♥t❛✐♥s t❤❡ s♦❧✉t✐♦♥ ✐ts❡❧❢ ■t ✐s ♥♦♥❧♦❝❛❧ ◮ ❈❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ✿ ❉❡❝♦✉♣❧❡ t❤❡ ◗❱■ ✭✶✮ ❜② ❞❡✜♥✐♥❣ ❜② ✐t❡r❛t✐♦♥ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ( ✈ ♥ ) ♥ ✿ ♠✐♥ [ − ∂ t ✈ ♥ + ✶ − L ✈ ♥ + ✶ − ❢ , ✈ ♥ + ✶ − H ✈ ♥ ] = ✵ , ✈ ♥ + ✶ ( ❚ , . ) = ❣ ✭✷✮ → ❛ss♦❝✐❛t❡❞ t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ t✐♠❡ ♣r♦❜❧❡♠s ✭r❡✢❡❝t❡❞ ❇❙❉❊s✮ ❍✉②ê♥ P❍❆▼ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❛♥❞ ◗❱■s

■♥tr♦❞✉❝t✐♦♥ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ◗❱■s ◆✉♠❡r✐❝❛❧ ✐ss✉❡s ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✭■❱✮ • ▼❛✐♥ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❞✐✣❝✉❧t② ✐♥ t❤❡ ◗❱■ ✭✶✮ ✿ ❚❤❡ ♦❜st❛❝❧❡ t❡r♠ ❝♦♥t❛✐♥s t❤❡ s♦❧✉t✐♦♥ ✐ts❡❧❢ ■t ✐s ♥♦♥❧♦❝❛❧ ◮ ❈❧❛ss✐❝❛❧ ❛♣♣r♦❛❝❤ ✿ ❉❡❝♦✉♣❧❡ t❤❡ ◗❱■ ✭✶✮ ❜② ❞❡✜♥✐♥❣ ❜② ✐t❡r❛t✐♦♥ t❤❡ s❡q✉❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ( ✈ ♥ ) ♥ ✿ ♠✐♥ [ − ∂ t ✈ ♥ + ✶ − L ✈ ♥ + ✶ − ❢ , ✈ ♥ + ✶ − H ✈ ♥ ] = ✵ , ✈ ♥ + ✶ ( ❚ , . ) = ❣ ✭✷✮ → ❛ss♦❝✐❛t❡❞ t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ♦♣t✐♠❛❧ st♦♣♣✐♥❣ t✐♠❡ ♣r♦❜❧❡♠s ✭r❡✢❡❝t❡❞ ❇❙❉❊s✮ → ❋✉rt❤❡r♠♦r❡✱ t♦ ❝♦♠♣✉t❡ ✈ ♥ + ✶ ✱ ✇❡ ♥❡❡❞ t♦ ❦♥♦✇ ✈ ♥ ♦♥ t❤❡ ✇❤♦❧❡ ❞♦♠❛✐♥ → ❤❡❛✈② ❝♦♠♣✉t❛t✐♦♥s ✿ ♥✉♠❡r✐❝❛❧❧② ❝❤❛❧❧❡♥❣✐♥❣ ✦ ❍✉②ê♥ P❍❆▼ ❇❙❉❊s ✇✐t❤ ❝♦♥str❛✐♥❡❞ ❥✉♠♣s ❛♥❞ ◗❱■s