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▼♦s❝♦✇ ■♥st✐t✉t❡ ♦❢ ❊❧❡❝tr♦♥✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝s

❋✉❧❧❡r P❤❡♥♦♠❡♥♦♥ ✐♥ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s

▲❛r✐s❛ ▼❛♥✐t❛

❧♠❛♥✐t❛❅❤s❡✳r✉

✶✹ ◆♦✈❡♠❜❡r ✷✵✶✽ ▼♦s❝♦✇✱ ❘✉ss✐❛

❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

T

ˆ

✵

(ϕ✵(x) + uϕ✶(x))dt → ♠✐♥, ˙ x = f✵(x) + uf✶(x) x(✵) ∈ B✵ ⊂ Rn, x(T) ∈ BT ⊂ Rn |u| ≤ ✶ ❍❡r❡ x ✐s ❛ st❛t❡ ✈❛r✐❛❜❧❡✱ u ✐s ❛ s❝❛❧❛r ❝♦♥tr♦❧✱ ϕi : Rn → R✱ fi : Rn → Rn✱ i = ✵, ✶✱ t❤❡ ❢✉♥❝t✐♦♥s ϕi, fi ❛r❡ s♠♦♦t❤ ❡♥♦✉❣❤✱ B✵, BT ❛r❡ s♠♦♦t❤ ♠❛♥✐❢♦❧❞s✳ ❚❤❡ ❛❞♠✐ss✐❜❧❡ ❝♦♥tr♦❧s u(t) ♥❡❡❞ t♦ ❜❡ ♠❡❛s✉r❛❜❧❡✱ t❤❡ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r✐❡s x(t) ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s✳

P♦♥tr②❛❣✐♥✬s ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡

❉❡✜♥❡ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ H = H✵(x, ψ) + uH✶(x, ψ), ✇❤❡r❡ H✵(x, ψ) = f✵(x)ψ − ✶

✷ϕ✵(x)✱

H✶(x, ψ) = f✶(x)ψ − ✶

✷ϕ✶(x)✳

❲❡ ❤❛✈❡ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ˙ x = ∂H ∂ψ , ˙ ψ = −∂H ∂x ✭✶✮ ❛♥❞ H (x (t) , ψ (t) , uopt (t)) = ♠❛①

✵≤u≤✶ H (x (t) , ψ (t) , u)

✭✷✮

❙✐♥❣✉❧❛r ❡①tr❡♠❛❧

❙✐♥❝❡ t❤❡ ❍❛♠✐❧t♦♥✐❛♥ H ✐s ❧✐♥❡❛r ✐♥ ✉✱ ❤❡♥❝❡ t♦ ♠❛①✐♠✐③❡ ✐t ♦✈❡r t❤❡ ✐♥t❡r✈❛❧ u ∈ [−✶, ✶] ✇❡ ♥❡❡❞ t♦ ✉s❡ ❜♦✉♥❞❛r② ✈❛❧✉❡s ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ s✐❣♥ ♦❢ H✶ = ψ✳ ❚❤❡ ♠❛①✐♠✉♠ ❝♦♥❞✐t✐♦♥ ②✐❡❧❞s✿ u = +✶ ❢♦r H✶ > ✵✱ u = −✶ ❢♦r H✶ < ✵✳ ❆♥ ❡①tr❡♠❛❧ (x(t), ψ(t)), t ∈ (t✵, t✶)✱ ✐s ❝❛❧❧❡❞ s✐♥❣✉❧❛r ✐❢ H✶(x(t), ψ(t)) = ✵ ❢♦r t ∈ (t✵, t✶)✳ ❚♦ ✜♥❞ t❤❡ ❝♦♥tr♦❧ ♦♥ s✐♥❣✉❧❛r ❡①tr❡♠❛❧ (x(t), ψ(t)) ♦♥❡ ♥❡❡❞s t♦ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ✐❞❡♥t✐t② H✶(x(t), ψ(t)) = ✵✳

❖r❞❡r ♦❢ ❛ s✐♥❣✉❧❛r ❡①tr❡♠❛❧

❲❡ s❛② t❤❛t ❛ ♥✉♠❜❡r q ✐s ❛♥ ♦r❞❡r ♦❢ ❛ s✐♥❣✉❧❛r tr❛❥❡❝t♦r② ✐✛ ∂ ∂u dk dtk

• (✶)

H✶(x, ψ) = ✵, k = ✵, . . . , ✷q − ✶, ∂ ∂u d✷q dt✷q

• (✶)

H✶(x, ψ) = ✵ ✐♥ s♦♠❡ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ t❤❡ s✐♥❣✉❧❛r tr❛❥❡❝t♦r② (x(t), ψ(t))✳ ■t ✐s ❦♥♦✇♥ t❤❛t ❢♦r ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s ❛ s✐♥❣✉❧❛r ❛r❝ ♦❢ ❡✈❡♥ ♦r❞❡r ✐s ❥♦✐♥❡❞ ✇✐t❤ ❛ ❝❤❛tt❡r✐♥❣ tr❛❥❡❝t♦r②✳ ❆ ❝❤❛tt❡r✐♥❣ tr❛❥❡❝t♦r② ✐s ❛ tr❛❥❡❝t♦r② ✇✐t❤ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦♥tr♦❧ s✇✐t❝❤✐♥❣s ✐♥ ❛ ✜♥✐t❡ t✐♠❡ ✐♥t❡r✈❛❧✳

❋✉❧❧❡r ♣r♦❜❧❡♠

▼✐♥✐♠✐③❡ ˆ ∞

✵

s✷(t)dt ✭✸✮ s✉❜❥❡❝t t♦ ¨ s(t) = u(t), −✶ ≤ u(t) ≤ ✶ ✇✐t❤ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s s(✵) = a, ˙ s(✵) = b ✭✹✮

❖♣t✐♠❛❧ ❢❡❡❞❜❛❝❦ ❝♦♥tr♦❧

❖♣t✐♠❛❧ ❙♦❧✉t✐♦♥s ✐♥ ❋✉❧❧❡r Pr♦❜❧❡♠

❉❡♥♦t❡ ˙ s = v

◮ ❚❤❡ ❝✉r✈❡

s = −Cv ✷sgnv ✐s t❤❡ ♦♣t✐♠❛❧ s✇✐t❝❤✐♥❣ s❡t ♦❢ t❤❡ ❋✉❧❧❡r Pr♦❜❧❡♠✳ ❍❡r❡ C ≈ ✵, ✹✹✹✻✷✸ . . .✳

◮ ❚✇✐st✐♥❣ ❛r♦✉♥❞ t❤❡ ♦r✐❣✐♥ t❤❡ ♦♣t✐♠❛❧ tr❛❥❡❝t♦r✐❡s ❛tt❛✐♥

t❤❡ ♦r✐❣✐♥ ✐♥ ❛ ✜♥✐t❡ t✐♠❡ ❛♥❞ ✐♥t❡rs❡❝t t❤❡ s✇✐t❝❤✐♥❣ ❝✉r✈❡ ❛t ❛ ❝♦✉♥t❛❜❧❡ s❡t ♦❢ ♣♦✐♥ts✳

◮ ❚❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ❡q✉❛❧s ✶ ❢r♦♠ t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡ ♦❢

t❤❡ s✇✐t❝❤✐♥❣ ❝✉r✈❡ ❛♥❞ ❡q✉❛❧s −✶ ❢r♦♠ t❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ♦❢ ✐t✳

❖♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❋✉❧❧❡r ♣r♦❜❧❡♠

H(s, v, φ, ψ) = −✶ ✷s✷ + vφ + uψ = H✵ + uH✶ ▲❡t ( s (t) , v (t) , u (t)) ❜❡ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ✐♥ t❤❡ ♣r♦❜❧❡♠✳ ❚❤❡♥ t❤❡r❡ ❡①✐st ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s φ (t) , ψ (t) s✉❝❤ t❤❛t ˙ φ = − ∂H

∂s = s,

˙ ψ = − ∂H

∂v = −φ,

• u = +✶ ❢♦r ψ > ✵

❛♥❞

• u = −✶ ❢♦r ψ < ✵✳

■❢ ψ = ✵ ❢♦r t ∈ (t✵, t✶) t❤❡♥ ❛♥ ❡①tr❡♠❛❧ (s (t) , v (t) , φ (t) , ψ (t)) , t ∈ (t✵, t✶) , ✐s ❛ s✐♥❣✉❧❛r ♦♥❡✳

❙✐♥❣✉❧❛r ❈♦♥tr♦❧

❉❡♥♦t❡ z = (s, v, φ, ψ)✳ ❲❡ ❤❛✈❡✿ H✶ (z (t)) = ψ (t) ≡ ✵, d dt H✶ (z (t)) = ✵ ⇒ −φ (t) = ✵ d✷ dt✷H✶ (z (t)) = ✵ ⇒ −s (t) = ✵, d✸ dt✸H✶ (z (t)) = ✵ ⇒ −v (t) = ✵, d✹ dt✹H✶ (z (t)) = ✵ ⇒ −u (t) = ✵. ✭✺✮ ❚❤❡ s✐♥❣✉❧❛r ❡①tr❡♠❛❧ ✐♥ t❤❡ ❋✉❧❧❡r ♣r♦❜❧❡♠ s = ✵, v = ✵✳

n ✲❧✐♥❦ ✐♥✈❡rt❡❞ ♣❡♥❞✉❧✉♠

n ✲❧✐♥❦ ✐♥✈❡rt❡❞ ♣❡♥❞✉❧✉♠

M ✐s t❤❡ ❝❛rt ♠❛ss✱ s ✐s t❤❡ ❝❛rt ♣♦s✐t✐♦♥✱ g ✐s t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ ❣r❛✈✐t②✱ u ✐s t❤❡ ❢♦r❝❡ ❛♣♣❧✐❡❞ t♦ t❤❡ ❝❛rt✱ γi ✐s t❤❡ ❛♥❣❧❡ ♦❢ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ it❤ ❧✐♥❦ ❢r♦♠ t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡✱ mi ✐s t❤❡ ♠❛ss ♦❢ t❤❡ it❤ ❧✐♥❦✱ ri ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❧♦✇❡r ❡♥❞ ♦❢ t❤❡ it❤ ❧✐♥❦ t♦ ✐ts ❝❡♥t❡r ♦❢ ♠❛ss✱ Ii ✐s t❤❡ ♠♦♠❡♥t ♦❢ ✐♥❡rt✐❛ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❡♥t❡r ♦❢ ♠❛ss ♦❢ t❤❡ it❤ ❧✐♥❦✱ ❛♥❞ li ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ it❤ ❧✐♥❦ (i = ✶, . . . , n)✳

n ✲❧✐♥❦ ✐♥✈❡rt❡❞ ♣❡♥❞✉❧✉♠✳ ▼♦t✐♦♥ ❡q✉❛t✐♦♥s

❚❤❡ ❡q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥ ❛r❡ a✶✶¨ s +

n

• i=✶

a✶,i+✶ ¨ γi ❝♦s γi −

n

• i=✶

a✶,i+✶ ˙ γi

✷ s✐♥ γi = u

a✶,i+✶¨ s ❝♦s γi + ai+✶,i+✶¨ γi +

n

• j=✶

ai+✶,j+✶¨ γj ❝♦s(γi + γj) − ✭✻✮ −

n

• j=✶

ai+✶,j+✶ ˙ γ✷

j s✐♥(γi + γj) − bi s✐♥ γi = ✵,

i = ✶, . . . , n.

n ✲❧✐♥❦ ✐♥✈❡rt❡❞ ♣❡♥❞✉❧✉♠✳

❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ✐♥✐t✐❛❧ st❛t❡ ♦❢ t❤❡ s②st❡♠ ✐s ✐♥ ❛ s✉❝✐❡♥t❧② s♠❛❧❧ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ t❤❡ ✉♣♣❡r ✉♥st❛❜❧❡ ❡q✉✐❧✐❜r✐✉♠ ♣♦s✐t✐♦♥ γ✶ = ˙ γ✶ = · · · = γn = ˙ γn ≡ ✵. ✭✼✮ ❲❡ st✉❞② t❤❡ ♣r♦❜❧❡♠ ♦❢ st❛❜✐❧✐③❛t✐♦♥ ♦❢ t❤❡ ♣❡♥❞✉❧✉♠ ✐♥ t❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ♣♦s✐t✐♦♥ ✭✼✮ ✐♥ t❤❡ s❡♥s❡ ♦❢ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥❛❧

∞

ˆ

✵

γ, γ dt → ♠✐♥, ✭✽✮

▲✐♥❡❛r✐③❡❞ ♠♦❞❡❧✳ ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

∞

ˆ

✵

Kx (t) , x (t) dt → ♠✐♥ ✭✾✮ ♦♥ t❤❡ tr❛❥❡❝t♦r✐❡s ♦❢ t❤❡ s②st❡♠ ¨ x (t) − Λx (t) = Iu (t) ✭✶✵✮ ✇✐t❤ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s x(✵) = x✵, ˙ x(✵) = y✵. ✭✶✶✮ ❍❡r❡✱ t❤❡ ❝♦♥tr♦❧ u (t) ✐s ❛ ❜♦✉♥❞❡❞ s❝❛❧❛r ❢✉♥❝t✐♦♥✿ |u (t) | ≤ ✶, ✭✶✷✮

❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✳ ◆♦t❛t✐♦♥

x ∈ Rn ❛r❡ ♣❤❛s❡ ✈❛r✐❛❜❧❡s✱ I ✐s t❤❡ ✈❡❝t♦r ❝♦♥s✐st✐♥❣ ♦❢ ✶✬s✱ K ✐s ❛ ❝♦♥st❛♥t s②♠♠❡tr✐❝ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ n × n ♠❛tr✐①✱ Λ ✐s ❛ ❝♦♥st❛♥t ❞✐❛❣♦♥❛❧ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ n × n ♠❛tr✐①✱ Λ = ❞✐❛❣ {λ✶, λ✷, . . . , λn}✱ λ✶, . . . , λn > ✵✳

P♦♥tr②❛❣✐♥ ▼❛①✐♠✉♠ Pr✐♥❝✐♣❧❡

H(x, y, φ, ψ) = −✶ ✷Kx, x + y, φ + Λx, ψ + I, ψu ˙ x = y ˙ y = Λx + Iu ˙ φ = Kx − Λψ ✭✶✸✮ ˙ ψ = −φ u (t) = sgn H✶(t) = sgn I, ψ (t) ✭✶✹✮

❙✐♥❣✉❧❛r s♦❧✉t✐♦♥

H✶(t) = I, ψ (t) ≡ ✵, dH✶ dt = −I, φ, d✷H✶ dt✷ = −I, Kx − Λψ, d✸H✶ dt✸ = −I, Ky + Λφ, d✹H✶ dt✹ = −I, K (Λx + Iu) + Λ (Kx − Λψ) = ✭✶✺✮ = −I, (KΛ + ΛK) x + I, Λ✷ψ − uKI, I

❙✐♥❣✉❧❛r ❝♦♥tr♦❧

S = { I, φ = ✵, I, Kx − Λψ = ✵, I, Ky + Λφ = ✵, −I, (KΛ + ΛK) x + I, Λ✷ψ = ✵

• uoc (t) = −I, (KΛ + ΛK) x (t) + I, Λ✷ψ (t)

σ ❍❡r❡ σ = KI, I.❙✐♥❝❡ |u (t) | ✶ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❞♦♠❛✐♥

• −I, (KΛ + ΛK) x + I, Λ✷ψ
• ≤ σ

❍❛♠✐❧t♦♥✐❛♥ s②st❡♠ ✐♥ t❤❡ s✐♥❣✉❧❛r s✉r❢❛❝❡ S

                                       ˙ xk = yk, ˙ yk = λkxk + ✶

σ

• −

n

• i=✶

n

• j=✶

kij(λi + λj − (λ✶ + λ✷))xi+ +

n

• i=✸

(λ✶ − λi)(λ✷ − λi)ψi

• k = ✶, . . . , n;

˙ ψk = −φk, ˙ φk =

n

• j=✶

kkjxj − λkψk, k = ✸, . . . , n;

s✐♥❣✉❧❛r s✉r❢❛❝❡

❖♣t✐♠❛❧ s♦❧✉t✐♦♥ r❡❛❝❤❡s S ✐♥ ❛ ✜♥✐t❡ t✐♠❡ ✇✐t❤ ❛♥ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦♥tr♦❧ s✇✐t❝❤✐♥❣s ✭❝❤❛tt❡r✐♥❣ r❡❣✐♠❡✮✳ ❚❤❡♥ t❤❡ ♠♦t✐♦♥ ♣r♦❝❡❡❞s ❛❧♦♥❣ t❤❡ s✐♥❣✉❧❛r s✉r❢❛❝❡ ❛♥❞ ❛s②♠♣t♦t✐❝❛❧❧② ❛♣♣r♦❛❝❤✐♥❣ t❤❡ ♦r✐❣✐♥✳

▼♦❞❡❧ ♣r♦❜❧❡♠ ✇✐t❤ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥tr♦❧

∞

ˆ

✵

(x✷

✶(t) + x✷ ✷(t)) dt → ♠✐♥

˙ x✶ = y✶, ˙ x✷ = y✷ ˙ y✶ = u✶, ˙ y✷ = u✷ xi(✵) = s✵

i , yi(✵) = ri ✵,

i = ✶, ✷ u✷

✶ + u✷ ✷ ≤ ✶

❖♣t✐♠❛❧ ❙♦❧✉t✐♦♥s ♦❢ t❤❡ ▼♦❞❡❧ Pr♦❜❧❡♠

◮ ❖♣t✐♠❛❧ s♦❧✉t✐♦♥s✱ st❛rt✐♥❣ ❢r♦♠ ❛ s♠❛❧❧ ❡♥♦✉❣❤

♥❡✐❣❜♦✉r❤♦♦❞ ♦❢ t❤❡ ♦r✐❣✐♥✱ r❡❛❝❤ ③❡r♦ ✐♥ ✜♥✐t❡ t✐♠❡ T∗ ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ (x✵, y ✵)✳ ▼♦r❡♦✈❡r t❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ˆ u (t) ❞♦❡s ♥♦t ❤❛✈❡ ❛ ❧✐♠✐t ❛t t → T − ✵✳

◮ ❚❤❡r❡ ❡①✐st ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s ♦❢ t❤❡ ♠♦❞❡❧ ♣r♦❜❧❡♠ t❤❛t

r❡♣r❡s❡♥t ❧♦❣❛r✐t❤♠✐❝ s♣✐r❛❧s✿ x∗(t) = B✶t✷eiα ❧♥ |T ∗−t|, y ∗(t) = B✷teiα ❧♥ |T ∗−t|, u∗(t) = −eiα ❧♥ |T ∗−t|, α = ± √ ✺, B✶ = ✶ ✶✷✻ (✹ + iα) (✸ + iα) , B✷ = ✶ ✶✷✻ (✹ + iα) (✸ + iα) (✷ + iα) ❆s t → T ∗ t❤❡ ❝♦♥tr♦❧ u∗(t) ♠❛❦❡s ❝♦✉♥t❛❜❧② ♠❛♥② r♦t❛t✐♦♥s ❛❧♦♥❣ t❤❡ ❝✐r❝❧❡ S✶ ✐♥ ✜♥✐t❡ t✐♠❡✱ x∗ (t) , y ∗ (t) → ✵ ❛♥❞ s✇✐t❝❤❡s t♦ ❛ s✐♥❣✉❧❛r ♠♦❞❡ x = y = ✵✳

❙♣❤❡r✐❝❛❧ ■♥✈❡rt❡❞ P❡♥❞✉❧✉♠

B ✿ ♣♦✐♥t ♠❛ss m✱ S ✿ ♠♦✈❛❜❧❡ ❜❛s❡ ♦❢ ♠❛ss M B ✐s ❛tt❛❝❤❡❞ t♦ ❛ r✐❣✐❞ ♠❛ss❧❡ss r♦❞ ♦❢ ❧❡♥❣t❤ ℓ x✶✿ ❛♥❣❧❡ ❜❡t✇❡❡♥ SB ❛♥❞ Oηζ✱ x✷ ✿ ❛♥❣❧❡ ❜❡t✇❡❡♥ SB ❛♥❞ Oξζ (ξ, η) ✿ ♣♦s✐t✐♦♥ ♦❢ S✱ (u✶, u✷)✿ t❤❡ ❝♦♥tr♦❧ ❢♦r❝❡s

❈♦♥tr♦❧ ♣r♦❜❧❡♠ ❢♦r ❧✐♥❡❛r✐③❡❞ ♠♦❞❡❧

¨ x✶ = M + m ml gx✶ − ✶ Ml u✶, ¨ x✷ = M + m ml gx✷ − ✶ Ml u✷

∞

ˆ

✵

(x✷

✶(t) + x✷ ✷(t)) dt → ♠✐♥

˙ x = y, ˙ y = Kx + u, x(✵) = x✵, y(✵) = y ✵. ❍❡r❡ x, y, u ∈ R✷ ✱ K ✐s ❛ ✷ × ✷ ❞✐❛❣♦♥❛❧ ♠❛tr✐①✱ K = diag {k✶, k✷}✳ ❚❤❡ ❝♦♥tr♦❧ ❢♦r❝❡ ✐s ❜♦✉♥❞❡❞✿ u✷

✶ + u✷ ✷ ≤ ✶

▼❛✐♥ r❡s✉❧ts ❢♦r s♣❡r✐❝❛❧ ♣❡♥❞✉❧✉♠

■♥ ❛ s✉✣❝✐❡♥t❧② s♠❛❧❧ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ t❤❡ ♦r✐❣✐♥ t❤❡r❡ ❡①✐st s♦❧✉t✐♦♥s ♦❢ ✭✶✵✮✕✭✷✵✮ t❤❛t ❛tt❛✐♥ t❤❡ ✉♣♣❡r ❡q✉✐❧✐❜r✐✉♠ ♣♦s✐t✐♦♥ ❛♥❞ ❤❛✈❡ t❤❡ ❢♦r♠ ♦❢ ❧♦❣❛r✐t❤♠✐❝ s♣✐r❛❧s x(t) = Cx(T − t)✷ei̺ln|T−t|(✶ + gx (T − t)), y(t) = Cy(T − t)ei̺ln|T−t|(✶ + gy (T − t)), u(t) = −Cuei̺ln|T−t|(✶ + gu (T − t)), ❍❡r❡ ✵ < T < ∞ ✐s ❛ t✐♠❡ ❛t ✇❤✐❝❤ s♦❧✉t✐♦♥ ❤✐ts t❤❡ ♦r✐❣✐♥ ✭t❤❡ ❤✐tt✐♥❣ t✐♠❡✮✱ gx,y,u (T − t) → ✵ ❛s t → T✱ ̺ > ✵✱ Cx,y,u ∈ C✳

❘❡❢❡r❡♥❝❡s

❚❍❆◆❑ ❨❖❯ ✦