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■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s ✐♥ ❈♦♥tr♦❧ Pr♦❜❧❡♠s ✇✐t❤ ❙t❛t❡ ❈♦♥str❛✐♥ts

❆✳❱✳ ❆r✉t②✉♥♦✈ P❛❞♦✈❛✱ ❙❡♣t❡♠❜❡r✱ ✷✵✶✼

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t

❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ✇✐t❤ ✐♥❡q✉❛❧✐t② st❛t❡ ❝♦♥str❛✐♥ts✳ ▼✐♥✐♠✐③❡ ϕ(p), s✉❜❥❡❝t t♦ ˙ x = f (x, u, t), p = (x✵, x✶) ∈ S, u(t) ∈ U ❛✳❛✳ t ∈ [t✵, t✶], g(x(t), t) ≤ ✵ ∀ t ∈ [t✵, t✶]. ✭✶✮ ❍❡r❡✱ ˙ x = dx dt ✱ t ∈ [✵, ✶] ❞❡s✐❣♥❛t❡s t✐♠❡✱ x ✐s t❤❡ st❛t❡ ✈❛r✐❛❜❧❡ ✇❤✐❝❤ t❛❦❡s ✈❛❧✉❡s ✐♥ Rn❀ x✵ = x(t✵)✱ x✶ = x(t✶)❀ ✈❡❝t♦r u ∈ U ⊂ Rm ✐s t❤❡ ❝♦♥tr♦❧ ♣❛r❛♠❡t❡r✳ ❚❤❡ s❡ts S✱ U ❛r❡ ❝❧♦s❡❞✳ ❱❡❝t♦r p = (x✵, x✶) ✐s t❡r♠❡❞ ❡♥❞♣♦✐♥t✳ ❆ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥ u : [t✵, t✶] → U ✐s t❡r♠❡❞ ❝♦♥tr♦❧ ✭♦r✱ ❝♦♥tr♦❧ ❢✉♥❝t✐♦♥✮✳

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡

❉❡✜♥✐t✐♦♥ ❚❤❡ ❝♦♥tr♦❧ ♣r♦❝❡ss (x∗(t), u∗(t)) ♦❢ ✭✶✮ s❛t✐s✜❡s t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ♣r♦✈✐❞❡❞ t❤❛t t❤❡r❡ ❡①✐st ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs✿ ❛ ♥✉♠❜❡r λ ∈ [✵, ✶]✱ ❛ ❢✉♥❝t✐♦♥ ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ψ : [t✵, t✶] → Rn✱ ❛♥❞ ❛ ❇♦r❡❧ ♠❡❛s✉r❡ η ∈ C ∗([t✵, t✶])✱ η ≥ ✵✱ s✉❝❤ t❤❛t λ + sup

t∈[t✵,t✶]

|ψ(t)| = ✶, dψ(t) = −H′

x(x∗(t), u∗(t), ψ(t), t)dt + g′ x(x∗(t), t)dη, t ∈ [t✵, t✶],

(ψ(t✵), −ψ(t✶)) ∈ λϕ′(p∗) + NS(p∗), max

u∈U H(x∗(t), u, ψ(t), t) = H(x∗(t), u∗(t), ψ(t), t) ❛✳❛✳ t ∈ [t✵, t✶],

• [t✵,t✶]
• g(x∗(t), t), dη
• = ✵.

❍❡r❡✱ H(x, u, ψ, t) := ψ, f (x, u, t)✱ p∗ = (x∗(t✵), x∗(t✶))✳

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡

❆✳❨❛ ❉✉❜♦✈✐ts❦✐✐ ❛♥❞ ❆✳❆✳ ▼✐❧②✉t✐♥ ♣r♦♣♦s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥tr♦❧ ♣r♦❝❡ss (x∗(t), u∗(t)) ✐s ♦♣t✐♠❛❧ t♦ ♣r♦❜❧❡♠ ✭✶✮✳ ❚❤❡♥✱ t❤❡ ♣r♦❝❡ss (x∗(t), u∗(t)) s❛t✐s✜❡s t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡✳

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❉❡❣❡♥❡r❛❝② ♦❢ t❤❡ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡

❙✉♣♣♦s❡ t❤❛t ♦♥❡ ♦❢ t❤❡ ❡♥❞♣♦✐♥ts✱ ❡✐t❤❡r x✵✱ ♦r x✶✱ ✐s ✜①❡❞✳ ■♥ t❤✐s ❝❛s❡✱ ❛s ✐s ❡❛s② t♦ ✈❡r✐❢②✱ t❤❡ ❉✉❜♦✈✐ts❦✐✐✲▼✐❧②✉t✐♥ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❝❛♥ ❜❡ s❛t✐s✜❡❞ ❜② ❛♥② ❢❡❛s✐❜❧❡ ♣❛✐r ❝♦♥tr♦❧✴tr❛❥❡❝t♦r②✳ ■♥❞❡❡❞✱ ✐♥ ♦r❞❡r t♦ ❡♥s✉r❡ t❤✐s ✇❡ ♦♥❧② ♥❡❡❞ t♦ ❝♦♥s✐❞❡r t❤❡ s❡t ♦❢ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs ✭✐♥ ❝❛s❡ t❤❡ ❧❡❢t ❡♥❞♣♦✐♥t ✐s ✜①❡❞✮✿ λ = ✵, η = δ(t✵), ψ(·) : ψ(t) = ✵ ∀ t ∈ (t✵, t✶], ❛♥❞ t❤❡ ♠✉❧t✐♣❧✐❡rs λ = ✵, η = δ(t✶), ψ(·) : ψ(t) = ✵ ∀ t ∈ [t✵, t✶), ✐❢ t❤❡ r✐❣❤t ❡♥❞♣♦✐♥t ✐s ✜①❡❞✳

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❉❡❣❡♥❡r❛❝② ♦❢ t❤❡ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡

❆✳❨❛✳ ❉✉❜♦✈✐ts❦✐✐ ❛♥❞ ❱✳❆✳ ❉✉❜♦✈✐ts❦✐✐ ❢♦✉♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r❡st✐♥❣ ❡①❛♠♣❧❡ ✭✶✾✽✺✮✳ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r n = m = k = ✶✱ t✵ = ✵✱ t✶ = ✶✱                ✶

✵

u(t)dt → min, ˙ x = tu, x✵ = ✵, u(t) ∈ [−✶, ✶], x(t) ≥ ✵. ❚❤❡ ♦♣t✐♠❛❧ ♣r♦❝❡ss ✐s x = u = ✵✱ ❜✉t t❤❡r❡ ❛r❡ ♦♥❧② ❞❡❣❡♥❡r❛t❡ ♠✉❧t✐♣❧✐❡rs s❛t✐s❢②✐♥❣ t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ✐t✳

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❈♦♥❞✐t✐♦♥s ❢♦r ◆♦♥✲❞❡❣❡♥❡r❛❝②

❉❡✜♥✐t✐♦♥ ❚❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❛r❡ s❛✐❞ t♦ ❜❡ r❡❣✉❧❛r ♣r♦✈✐❞❡❞ t❤❛t ❢♦r ❛❧❧ x, t : g(x, t) ≤ ✵ t❤❡r❡ ❡①✐sts z = z(x, t) s✉❝❤ t❤❛t ∂gj ∂x (x, t), z

• > ✵ ∀ j ∈ J(x, t).

❍❡r❡✱ J(x, t) := {j : gj(x, t) = ✵}✳

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❈♦♥❞✐t✐♦♥s ❢♦r ◆♦♥✲❞❡❣❡♥❡r❛❝②

❉❡✜♥✐t✐♦♥ ❚❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❛r❡ s❛✐❞ t♦ ❜❡ ❝♦♠♣❛t✐❜❧❡ ❛t p∗ ✇✐t❤ t❤❡ ❡♥❞♣♦✐♥t ❝♦♥str❛✐♥ts ♣r♦✈✐❞❡❞ t❤❛t ∃ ε > ✵ : {p ∈ R✷n : |p∗ − p| ≤ ε, p ∈ S} ⊆ {p ∈ R✷n : g(x✵, t✵) ≤ ✵, g(x✶, t✶) ≤ ✵}. ❚❤❡ ❝♦♠♣❛t✐❜✐❧✐t② ♦❢ ❝♦♥str❛✐♥ts ✐s ♥♦t ❛♥ ❡①tr❛ r❡q✉✐r❡♠❡♥t✳ ■t ❝❛♥ ❛❧✇❛②s ❜❡ ❛❝❤✐❡✈❡❞ ❜② r❡♣❧❛❝✐♥❣ t❤❡ s❡t S ✇✐t❤ t❤❡ s❡t S ∩ {p ∈ R✷n : g(x✵, t✵) ≤ ✵, g(x✶, t✶) ≤ ✵}.

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❈♦♥❞✐t✐♦♥s ❢♦r ◆♦♥✲❞❡❣❡♥❡r❛❝②

❈♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥ Γ(x, u, t) = g′

x(x, t)f (x, u, t) + g′ t(x, t).

❉❡✜♥✐t✐♦♥ ❆ ❢❡❛s✐❜❧❡ ❛r❝ x∗(t) ✐s s❛✐❞ t♦ ❜❡ ❝♦♥tr♦❧❧❛❜❧❡ ❛t t❤❡ ❡♥❞♣♦✐♥ts ✇✳r✳t✳ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts✱ ♣r♦✈✐❞❡❞ t❤❛t t❤❡r❡ ❡①✐st ✈❡❝t♦rs γr ∈ ❝♦♥✈ Γ(xr, U, r) s✉❝❤ t❤❛t✿ (−✶)rγj

r < ✵ ∀ j ∈ J(xr, r), r = ✵, ✶.

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

◆♦♥✲❞❡❣❡♥❡r❛t❡ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡

❚❤❡♦r❡♠ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥tr♦❧ ♣r♦❝❡ss (x∗(t), u∗(t)) ✐s ♦♣t✐♠❛❧ ✐♥ ♣r♦❜❧❡♠ ✭✶✮✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❛r❡ r❡❣✉❧❛r ❛♥❞ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❡♥❞♣♦✐♥ts ❝♦♥str❛✐♥ts ❛t p∗✱ ❛♥❞ t❤❡ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞✳ ❚❤❡♥✱ t❤❡ ♣r♦❝❡ss (x∗(t), u∗(t)) s❛t✐s✜❡s t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡✱ ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ str❡♥❣t❤❡♥❡❞ ♥♦♥✲tr✐✈✐❛❧✐t② ❝♦♥❞✐t✐♦♥ ✐s ✈❛❧✐❞ λ + ℓ(t ∈ [t✵, t✶] : |ψ(t)| = ✵) > ✵, ✇❤❡r❡ ℓ st❛♥❞s ❢♦r t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

P♦✐♥t✇✐s❡ ❈♦♥tr♦❧❧❛❜✐❧✐t②

❉❡✜♥✐t✐♦♥ ❆ ❢❡❛s✐❜❧❡ ❛r❝ x∗(t) ✐s s❛✐❞ t♦ ❜❡ ❝♦♥tr♦❧❧❛❜❧❡ ✇✳r✳t✳ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts✱ ♣r♦✈✐❞❡❞ t❤❛t ✐t ✐s ❝♦♥tr♦❧❧❛❜❧❡ ❛t t❤❡ ❡♥❞♣♦✐♥ts ❛♥❞ ❢♦r ❡✈❡r② t ∈ (t✵, t✶) t❤❡r❡ ❡①✐st ✈❡❝t♦rs γr,t ∈ Rk✱ r = ✵, ✶✱ s✉❝❤ t❤❛t γr,t ∈ ❝♦♥✈ Γ(x∗(t), U, t), (−✶)rγj

r,t < ✵ ∀ j ∈ J(x∗(t), t), r = ✵, ✶.

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

P♦✐♥t✇✐s❡ ❈♦♥tr♦❧❧❛❜✐❧✐t②

❚❤❡♦r❡♠ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥tr♦❧ ♣r♦❝❡ss (x∗(t), u∗(t)) ✐s ♦♣t✐♠❛❧ ✐♥ ♣r♦❜❧❡♠ ✭✶✮✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❛r❡ r❡❣✉❧❛r ❛♥❞ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❡♥❞♣♦✐♥ts ❝♦♥str❛✐♥ts ❛t p∗✱ ❛♥❞ t❤❡ ♣♦✐♥t✇✐s❡ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞✳ ❚❤❡♥✱ t❤❡ ♣r♦❝❡ss (x∗(t), u∗(t)) s❛t✐s✜❡s t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡✱ ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ str❡♥❣t❤❡♥❡❞ ♥♦♥✲tr✐✈✐❛❧✐t② ❝♦♥❞✐t✐♦♥ ✐s ✈❛❧✐❞ λ + |ψ(t)| = ✵ ∀ t ∈ (t✵, t✶).

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ✐♥ t❤❡ ●❛♠❦r❡❧✐❞③❡ ❢♦r♠

❈♦♥s✐❞❡r t❤❡ ❡①t❡♥❞❡❞ ❍❛♠✐❧t♦♥✲P♦♥tr②❛❣✐♥ ❢✉♥❝t✐♦♥✿ ¯ H(x, u, ψ, µ, t) = ψ, f (x, u, t) − µ, Γ(x, u, t) . ❉❡✜♥✐t✐♦♥ ❚❤❡ ❝♦♥tr♦❧ ♣r♦❝❡ss (x∗(t), u∗(t)) ♦❢ ✭✶✮ s❛t✐s✜❡s t❤❡ ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ♣r♦✈✐❞❡❞ t❤❛t t❤❡r❡ ❡①✐st ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs λ ∈ [✵, ✶]✱ ψ ∈ W✶,∞([t✵, t✶])✱ ❛♥❞ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s µj✱ j = ✶, ..., k✱ s✉❝❤ t❤❛t ˙ ψ(t) = − ¯ H′

x(x∗(t), u∗(t), ψ(t), µ(t), t) ❛✳❛✳ t ∈ [t✵, t✶],

(ψ(t✵), −ψ(t✶)) ∈ λϕ′(p∗) + (µ(t✵)g′

x(x∗ ✵, t✵), −µ(t✶)g′ x(x∗ ✶, t✶)) + NS(p∗),

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ✐♥ t❤❡ ●❛♠❦r❡❧✐❞③❡ ❢♦r♠

❉❡✜♥✐t✐♦♥ max

u∈U

¯ H(x∗(t), u, ψ(t), µ(t), t) = ¯ H(x∗(t), u∗(t), ψ(t), µ(t), t) ❛✳❛✳ t ∈ [t✵, t✶], t✶

t✵

• g(x∗(t), t), dµ(t)
• = ✵,

λ + ℓ

• t ∈ [t✵, t✶] : ψ(t) − µ(t)g′

x(x∗(t), t) = ✵

• > ✵.

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ✐♥ t❤❡ ●❛♠❦r❡❧✐❞③❡ ❢♦r♠

❚❤❡♦r❡♠ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥tr♦❧ ♣r♦❝❡ss (x∗(t), u∗(t)) ✐s ♦♣t✐♠❛❧ t♦ ♣r♦❜❧❡♠ ✭✶✮✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❛r❡ r❡❣✉❧❛r ❛♥❞ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❡♥❞♣♦✐♥ts ❝♦♥str❛✐♥ts ❛t t❤❡ ♣♦✐♥t p∗✱ ❛♥❞ t❤❡ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞✳ ❚❤❡♥✱ t❤❡ ♣r♦❝❡ss (x∗(t), u∗(t)) s❛t✐s✜❡s ♥♦♥✲❞❡❣❡♥❡r❛t❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ✐♥ t❤❡ ●❛♠❦r❡❧✐❞③❡ ❢♦r♠✳

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ✐♥ t❤❡ ●❛♠❦r❡❧✐❞③❡ ❢♦r♠

❘❊▼❆❘❑✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ s❡t NS(p∗) ✐s ❝♦♥✈❡①✳ ❚❤❡♥✱ ✐♥ t❤❡ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡✱ ✇❡ ❝❛♥ ❛❞❞✐t✐♦♥❛❧❧② r❡q✉✐r❡ t❤❛t µ ✐s ❝♦♥t✐♥✉♦✉s ❛t t✵, t✶✱ ❛♥❞ µ(t✶) = ✵✱ ✰ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ tr❛♥s✈❡rs❛❧✐t② ❝♦♥❞✐t✐♦♥ ✐s ✈❛❧✐❞ (ψ(t✵), −ψ(t✶)) ∈ λϕ′(p∗) + NS(p∗), ✰ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ♥♦♥✲tr✐✈✐❛❧✐t② ❝♦♥❞✐t✐♦♥ ✐s ✈❛❧✐❞ λ + max

t∈[t✵,t✶] |ψ(t)| + ❱❛r |t✶ t✵µ = ✵.

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❘❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦ s❡ts ♦❢ ◆❖❈

❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t g ✐s t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡✱ t❤❡ t✇♦ ❛❜♦✈❡ ❢♦r♠s ♦❢ t❤❡ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ❚❤✐s ✐s s♦ ❞✉❡ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs ❝❤❛♥❣❡✿ µ(t) :=

• [t,t✶]

dη ∀ t < t✶, µ(t✶) = ✵, φ(t) := ψ(t) + µ(t)g′

x(x∗(t), t), t ∈ [t✵, t✶].

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❙♦♠❡ ❋✉rt❤❡r ❘❡✜♥❡♠❡♥ts

❚❤❡s❡ r❡s✉❧ts ✇❡r❡ ❧❛t❡r r❡✜♥❡❞ r❡❣❛r❞✐♥❣ t❤❡ ❝♦♥t✐♥✉✐t② ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♠❡❛s✉r❡✳ ◆❛♠❡❧②✱ ✐t ✇❛s ♣r♦✈❡❞ t❤❛t✱ ✉♥❞❡r t❤❡ r❡❣✉❧❛r✐t② ❛ss✉♠♣t✐♦♥s ♣r♦♣♦s❡❞ ❜② ●❛♠❦r❡❧✐❞③❡✱ t❤❡ ♠❡❛s✉r❡✲♠✉❧t✐♣❧✐❡r ✐s ❝♦♥t✐♥✉♦✉s✳ ▼♦r❡♦✈❡r✱ ✉♥❞❡r t❤❡ ❛❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ❞❛t❛ ❛r❡ t✇✐❝❡ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ✇✳r✳t✳ u✱ ✐t ✇❛s ♣r♦✈❡❞ t❤❛t t❤❡ ♠❡❛s✉r❡ ❡♥❥♦②s ❡✈❡♥ t❤❡ ❍☎ ♦❧❞❡r ♣r♦♣❡rt②✱ t❤❛t ✐s✱ |µ(t) − µ(s)| ≤ ❝♦♥st

• |t − s|

∀ t, s ∈ [t✵, t✶]. ❚❤❡ ❛❜♦✈❡ r❡s✉❧ts ✇❡r❡ ❛❧s♦ ❣❡♥❡r❛❧✐③❡❞ ♦♥t♦ t❤❡ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✇✐t❤ ❡q✉❛❧✐t② st❛t❡ ❝♦♥str❛✐♥ts✳

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❇✐❜❧✐♦❣r❛♣❤②

❆r✉t②✉♥♦✈✱ ❆✳❱✳✱ ❚②♥②❛♥s❦✐②✱ ◆✳❚✳✿ ❚❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ✐♥ ❛ ♣r♦❜❧❡♠ ✇✐t❤ ♣❤❛s❡ ❝♦♥str❛✐♥ts✳ ❙♦✈✐❡t ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❡r ❛♥❞ ❙②st❡♠ ❙❝✐❡♥❝❡s✱ ✷✸ ✭✶✾✽✺✮ ❆r✉t②✉♥♦✈✱ ❆✳❱✳✿ ❖♥ ◆❡❝❡ss❛r② ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s ✐♥ ❛ Pr♦❜❧❡♠ ✇✐t❤ P❤❛s❡ ❈♦♥str❛✐♥ts✳ ❙♦✈✐❡t ▼❛t❤✳ ❉♦❦❧✳✱ ✸✶✱ ✶ ✭✶✾✽✺✮ ❉✉❜♦✈✐ts❦✐✐✱ ❆✳❨❛✳✱ ❉✉❜♦✈✐ts❦✐✐✱ ❱✳❆✳✿ ◆❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r str♦♥❣ ♠✐♥✐♠✉♠ ✐♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✇✐t❤ ❞❡❣❡♥❡r❛t✐♦♥ ♦❢ ❡♥❞♣♦✐♥t ❛♥❞ ♣❤❛s❡ ❝♦♥str❛✐♥ts✳ ❯s♣✳ ▼❛t✳ ◆❛✉❦✱ ✹✵✱ ✷ ✭✶✾✽✺✮ ❆r✉t②✉♥♦✈✱ ❆✳❱✳✿ ❖♥ t❤❡ t❤❡♦r② ♦❢ t❤❡ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r st❛t❡ ❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✇✐t❤ st❛t❡ ❝♦♥str❛✐♥ts✳ ❉♦❦❧❛❞② ❆◆ ❙❙❙❘✱ ✸✵✹✱ ✶ ✭✶✾✽✾✮ ❆r✉t②✉♥♦✈✱ ❆✳❱✳✿ ❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✿ ❆❜♥♦r♠❛❧ ❛♥❞ ❉❡❣❡♥❡r❛t❡ Pr♦❜❧❡♠s✳ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥✳ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡r ✭✷✵✵✵✮

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳

❇✐❜❧✐♦❣r❛♣❤②

❉✳ ❑❛r❛♠③✐♥✳ ◆❡❝❡ss❛r② ❡①tr❡♠✉♠ ❝♦♥❞✐t✐♦♥s ✐♥ t❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ✇✐t❤ st❛t❡ ❝♦♥str❛✐♥ts✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ❱♦❧ ✹✼✱ ◆♦✳ ✼✳ ❆✳❱✳ ❆r✉t②✉♥♦✈✱ ❉✳❨✉✳ ❑❛r❛♠③✐♥✱ ❋✳▲✳ P❡r❡✐r❛✳ ❚❤❡ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ❢♦r ❖♣t✐♠❛❧ ❈♦♥tr♦❧ Pr♦❜❧❡♠s ✇✐t❤ ❙t❛t❡ ❈♦♥str❛✐♥ts ❜② ❘✳❱✳ ●❛♠❦r❡❧✐❞③❡✿ ❘❡✈✐s✐t❡❞✳ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✭✷✵✶✶✮ ❱♦❧✳ ✶✹✾✳ ❆✳❱✳ ❆r✉t②✉♥♦✈✱ ❉✳❨✉✳ ❑❛r❛♠③✐♥✳ ❖♥ ❙♦♠❡ ❈♦♥t✐♥✉✐t② Pr♦♣❡rt✐❡s ♦❢ t❤❡ ▼❡❛s✉r❡ ▲❛❣r❛♥❣❡ ▼✉❧t✐♣❧✐❡r ❢r♦♠ t❤❡ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡ ❢♦r ❙t❛t❡ ❈♦♥str❛✐♥❡❞ Pr♦❜❧❡♠s✳ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❈♦♥tr♦❧ ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥ ✭✷✵✶✺✮✱ ❱♦❧✳ ✺✸✱ ◆♦✳ ✹✳ ❆✳❱✳ ❆r✉t②✉♥♦✈✱ ❉✳❨✉✳ ❑❛r❛♠③✐♥✳ ◆♦♥✲❞❡❣❡♥❡r❛t❡ ♥❡❝❡ss❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ✇✐t❤ ❡q✉❛❧✐t②✲t②♣❡ st❛t❡ ❝♦♥str❛✐♥ts ✭✷✵✶✻✮ ❏♦✉r♥❛❧ ♦❢ ●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥✱ ❱♦❧✳ ✻✹✱ ◆♦✳ ✹✳

❆✳❱✳ ❆r✉t②✉♥♦✈ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s✳✳✳