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❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ ❛ s❡♠✐❧✐♥❡❛r ❤❡❛t ❡q✉❛t✐♦♥ s✉❜❥❡❝t t♦ st❛t❡ ❛♥❞ ❝♦♥tr♦❧ ❝♦♥str❛✐♥ts

❆①❡❧ ❑rö♥❡r

❍✉♠❜♦❧❞t✲❯♥✐✈❡rs✐tät ❇❡r❧✐♥

❘■❈❆▼✱ ❖❝t♦❜❡r ✶✼✱ ✷✵✶✾

❈♦❧❧❛❜♦r❛t♦rs✿ ▼✳❙✳ ❆r♦♥♥❛ ✭❊s❝♦❧❛ ❞✳ ▼❛t❡♠❛t✐❝❛✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✮✱ ❋✳ ❇♦♥♥❛♥s ✭■◆❘■❆ ❙❛❝❧❛②✮

✶

❚❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

✷

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s ❛♥❞ ❛❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s

✸

❖♥ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡r

✹

❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ r❛❞✐❛❧✐t②

✺

❚❤❡ ●♦❤ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠ ❛♥❞ ❝r✐t✐❝❛❧ ❝♦♥❡

✻

❙❡❝♦♥❞ ♦r❞❡r s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷

❈♦♥t❡♥t

✶

❚❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

✷

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s ❛♥❞ ❛❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s

✸

❖♥ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡r

✹

❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ r❛❞✐❛❧✐t②

✺

❚❤❡ ●♦❤ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠ ❛♥❞ ❝r✐t✐❝❛❧ ❝♦♥❡

✻

❙❡❝♦♥❞ ♦r❞❡r s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸

❙t❛t❡ ❡q✉❛t✐♦♥

❈♦♥tr♦❧✿ u✱ ❙t❛t❡✿ y Ω ⊂ Rn✱ ♦♣❡♥ ❛♥❞ ❜♦✉♥❞❡❞ ✇✐t❤ s♠♦♦t❤ ❜♦✉♥❞❛r②✱ Q = Ω × (0, T)✱ Σ = ∂Ω × (0, T)✳      ˙ y(x, t) − ∆y(x, t) + γy3(x, t) = f(x, t) + y(x, t)

m

• i=0

ui(t)bi(x) ✐♥ Q, y = 0 ♦♥ Σ, y(·, 0) = y0 ✐♥ Ω, ✇✐t❤ y0 ∈ W 1,∞ (Ω), f ∈ L∞(Q), b ∈ W 1,∞(Ω)m+1✱ γ ≥ 0✱ u0 ≡ 1 ✐s ❛ ❝♦♥st❛♥t✱ ❛♥❞ u := (u1, . . . , um) ∈ L2(0, T)m✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹

❙t❛t❡ ❡q✉❛t✐♦♥

▲❡♠♠❛

❋♦r i = 0, . . . , m✱ t❤❡ ♠❛♣♣✐♥❣ ❞❡✜♥❡❞ ♦♥ L2(0, T) × L∞(Ω) × L∞(0, T; L2(Ω))✱ ❣✐✈❡♥ ❜② (ui, bi, y) → uibiy, ❤❛s ✐♠❛❣❡ ✐♥ L2(Q)✱ ✐s ♦❢ ❝❧❛ss C∞✱ ❛♥❞ s❛t✐s✜❡s uibiy2 ≤ ui2bi∞yL∞(0,T ;L2(Ω)). ❚❤❡ st❛t❡ ❡q✉❛t✐♦♥ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ✐♥ Y := H2,1(Q)✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✺

❙❡tt✐♥❣

❈♦st ❢✉♥❝t✐♦♥ J(u, y) := 1

2

• Q

(y(x, t) − yd(x))2dxdt + 1

2

• Ω

(y(x, T) − ydT (x))2dx +

m

• i=1

αi T ui(t)dt. ✇✐t❤ yd ∈ L∞(Q)✱ ydT ∈ W 1,∞ (Ω)✱ α ∈ Rm✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✻

❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

❈♦♥tr♦❧ ❝♦♥str❛✐♥ts u ∈ U❛❞✱ ✇❤❡r❡ U❛❞ = {u ∈ L2(0, T)m; ˇ ui ≤ u(t) ≤ ˆ ui, i = 1, . . . , m}, ❢♦r s♦♠❡ ❝♦♥st❛♥ts ˇ ui < ˆ ui✱ ❢♦r i = 1, . . . , m. ❙t❛t❡ ❝♦♥str❛✐♥ts gj(y(·, t)) :=

• Ω

cj(x)y(x, t)dx + dj ≤ 0, ❢♦r t ∈ [0, T], j = 1, . . . , q, ✇❤❡r❡ cj ∈ H2(Ω) ∩ H1

0(Ω) ❢♦r j = 1, . . . , q✱ ❛♥❞ d ∈ Rq✳

❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

❛❞

s✉❜❥❡❝t t♦ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ✭P✮

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✼

❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

❈♦♥tr♦❧ ❝♦♥str❛✐♥ts u ∈ U❛❞✱ ✇❤❡r❡ U❛❞ = {u ∈ L2(0, T)m; ˇ ui ≤ u(t) ≤ ˆ ui, i = 1, . . . , m}, ❢♦r s♦♠❡ ❝♦♥st❛♥ts ˇ ui < ˆ ui✱ ❢♦r i = 1, . . . , m. ❙t❛t❡ ❝♦♥str❛✐♥ts gj(y(·, t)) :=

• Ω

cj(x)y(x, t)dx + dj ≤ 0, ❢♦r t ∈ [0, T], j = 1, . . . , q, ✇❤❡r❡ cj ∈ H2(Ω) ∩ H1

0(Ω) ❢♦r j = 1, . . . , q✱ ❛♥❞ d ∈ Rq✳

❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ Min

u∈U❛❞

J(u, y[u]); s✉❜❥❡❝t t♦ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts. ✭P✮

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✼

❆✐♠✿ ❙❡❝♦♥❞✲♦r❞❡r ❛♥❛❧②s✐s ❚♦♦❧s✿ ❛❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s ✭❇♦♥♥❛♥s ❛♥❞ ❏❛✐ss♦♥ ✷✵✶✵✮ r❛❞✐❛❧✐t② t♦ ❞❡r✐✈❡ s❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✭❆r♦♥♥❛✱ ❇♦♥♥❛♥s ❛♥❞ ●♦❤ ✷✵✶✻✮

• ♦❤ tr❛♥s❢♦r♠

✭●♦❤ ✶✾✻✻✮

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✽

❘❡s✉❧ts

❙✳ ❆r♦♥♥❛✱ ❋✳ ❇♦♥♥❛♥s✱ ❆✳❑✳ ❙t❛t❡✲❝♦♥str❛✐♥❡❞ ❝♦♥tr♦❧✲❛✣♥❡ ♣❛r❛❜♦❧✐❝ ♣r♦❜❧❡♠s ■✿ ✜rst ❛♥❞ s❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✷✵✶✾✱ ♣r❡♣r✐♥t ❙✳ ❆r♦♥♥❛✱ ❋✳ ❇♦♥♥❛♥s✱ ❆✳❑✳ ❙t❛t❡ ❝♦♥str❛✐♥❡❞ ❝♦♥tr♦❧✲❛✣♥❡ ♣❛r❛❜♦❧✐❝ ♣r♦❜❧❡♠s ■■✿ ❙❡❝♦♥❞ ♦r❞❡r s✉✣❝✐❡♥t ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ✷✵✶✾✱ ♣r❡♣r✐♥t

❏✳❋✳ ❇♦♥♥❛♥s✱ ❙✐♥❣✉❧❛r ❛r❝s ✐♥ t❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ ❛ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥✱ ✷✵✶✸✱ ♣♣✳ ✷✽✶✲✷✾✷✱ ♣r♦❝ ✶✶t❤ ■❋❆❈ ❲♦r❦s❤♦♣ ♦♥ ❆❞❛♣t❛t✐♦♥ ❛♥❞ ▲❡❛r♥✐♥❣ ✐♥ ❈♦♥tr♦❧ ❛♥❞ ❙✐❣♥❛❧ Pr♦❝❡ss✐♥❣ ✭❆▲❈❖❙P✮✱ ❈❛❡♥✱ ❋✳ ●✐r✐ ❡❞✳✱ ❏✉❧② ✸✲✺✱ ✷✵✶✸✳ ▼✳ ❙✳ ❆r♦♥♥❛✱ ❏✳❋✳ ❇♦♥♥❛♥s✱ ❆✳❑✳✱ ❖♣t✐♠❛❧ ❈♦♥tr♦❧ ♦❢ ■♥✜♥✐t❡ ❉✐♠❡♥s✐♦♥❛❧ ❇✐❧✐♥❡❛r ❙②st❡♠s✿ ❆♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ ❍❡❛t ❛♥❞ ❲❛✈❡ ❊q✉❛t✐♦♥s✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✶✻✽ ✭✶✮ ✭✷✵✶✽✮ ✼✶✼✲✼✺✼✱ ❡rr❛t✉♠✿ ▼❛t❤✳ Pr♦❣r❛♠♠✐♥❣ ❙❡r✳ ❆✱ ❱♦❧✳ ✶✼✵ ✭✷✵✶✽✮✳ ▼✳ ❙✳ ❆r♦♥♥❛✱ ❏✳ ❋✳ ❇♦♥♥❛♥s✱ ❆✳ ❑✳✱ ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ✐♥ ❛ ❝♦♠♣❧❡① s♣❛❝❡ s❡tt✐♥❣❀ ❛♣♣❧✐❝❛t✐♦♥ t♦ t❤❡ ❙❝❤rö❞✐♥❣❡r ❡q✉❛t✐♦♥✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✳ ✺✼ ✭✷✮ ✭✷✵✶✾✮ ✶✸✾✵✕✶✹✶✷✳ ▼✳ ❙✳ ❆r♦♥♥❛✱ ❏✳ ❋✳ ❇♦♥♥❛♥s✱ ❇✳ ❙✳ ●♦❤✱ ❙❡❝♦♥❞ ♦r❞❡r ❛♥❛❧②s✐s ♦❢ ❝♦♥tr♦❧✲❛✣♥❡ ♣r♦❜❧❡♠s ✇✐t❤ s❝❛❧❛r st❛t❡ ❝♦♥str❛✐♥t✱ ▼❛t❤✳ Pr♦❣r❛♠✳ ✶✻✵ ✭✶✲✷✱ ❙❡r✳ ❆✮ ✭✷✵✶✻✮ ✶✶✺✕✶✹✼✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✾

❋✉rt❤❡r r❡s✉❧ts

❊✳ ❈❛s❛s✱ ❉✳ ❲❛❝❤s♠✉t❤✱ ●✳ ❲❛❝❤s♠✉t❤✱ ❙❡❝♦♥❞✲♦r❞❡r ❛♥❛❧②s✐s ❛♥❞ ♥✉♠❡r✐❝❛❧

❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r ❜❛♥❣✲❜❛♥❣ ❜✐❧✐♥❡❛r ❝♦♥tr♦❧ ♣r♦❜❧❡♠s✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✳ ✺✻

✭✻✮ ✭✷✵✶✽✮ ✹✷✵✸✕✹✷✷✼✳ ❊✳ ❈❛s❛s✱ ❋✳ ❚rö❧t③s❝❤✱ ❆✳ ❯♥❣❡r✱ ❙❡❝♦♥❞ ♦r❞❡r s✉✣❝✐❡♥t ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ❛

♥♦♥❧✐♥❡❛r ❡❧❧✐♣t✐❝ ❝♦♥tr♦❧ ♣r♦❜❧❡♠✱ ❏✳ ❢♦r ❆♥❛❧②s✐s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s ✭❩❆❆✮ ✶✺

✭✶✾✾✻✮ ✻✽✼✕✼✵✼✳ ❏✳ ❋✳ ❇♦♥♥❛♥s✱ ❙❡❝♦♥❞✲♦r❞❡r ❛♥❛❧②s✐s ❢♦r ❝♦♥tr♦❧ ❝♦♥str❛✐♥❡❞ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ♦❢

s❡♠✐❧✐♥❡❛r ❡❧❧✐♣t✐❝ s②st❡♠s✱ ❆♣♣❧✳ ▼❛t❤✳ ❖♣t✐♠✳ ✸✽ ✭✸✮ ✭✶✾✾✽✮ ✸✵✸✕✸✷✺✳

❊✳ ❈❛s❛s✱ ▼✳ ▼❛t❡♦s✱ ❆✳ ❘ös❝❤ ❊rr♦r ❡st✐♠❛t❡s ❢♦r s❡♠✐❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ❝♦♥tr♦❧ ♣r♦❜❧❡♠s ✐♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❚✐❦❤♦♥♦✈ t❡r♠✱ ❙■❆▼ ❏✳ ❈♦♥tr♦❧ ❖♣t✐♠✳✱ ✺✼✭✹✮✱ ✷✺✶✺✕✷✺✹✵✱ ✷✵✶✾✳ ❊✳ ❈❛s❛s✱ ▼✳ ▼❛t❡♦s✱ ❋✳ ❚rö❧t③s❝❤✱ ◆❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r

♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✐♥ ❢✉♥❝t✐♦♥ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❝♦♥tr♦❧ t❤❡♦r②✱ ✐♥✿

Pr♦❝❡❡❞✐♥❣s ♦❢ ✷✵✵✸ ▼❖❉❊✲❙▼❆■ ❈♦♥❢❡r❡♥❝❡✱ ❱♦❧✳ ✶✸ ♦❢ ❊❙❆■▼ Pr♦❝❡❡❞✐♥❣s✱ ❊❉P ❙❝✐❡♥❝❡s✱ ✷✵✵✸✱ ♣♣✳ ✶✽✕✸✵✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✵

❊①✐st❡♥❝❡

❈♦♠♣❛❝t♥❡ss ❬▲✐♦♥s ✶✾✽✸❪ ❛♥❞ ❬❊❞✇❛r❞s ✶✾✻✺❪✿ ❋♦r ❛♥② p ∈ [1, 10)✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❥❡❝t✐♦♥ ✐s ❝♦♠♣❛❝t✿ Y ֒ → Lp(0, T; L10(Ω)), ✇❤❡♥ n ≤ 3✳ ❚❤❡ ♠❛♣♣✐♥❣ u → y[u] ✐s s❡q✉❡♥t✐❛❧❧② ✇❡❛❦❧② ❝♦♥t✐♥✉♦✉s ❢r♦♠ L2(0, T)m ✐♥t♦ Y ✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✶

❈♦♥t❡♥t

✶

❚❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

✷

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s ❛♥❞ ❛❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s

✸

❖♥ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡r

✹

❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ r❛❞✐❛❧✐t②

✺

❚❤❡ ●♦❤ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠ ❛♥❞ ❝r✐t✐❝❛❧ ❝♦♥❡

✻

❙❡❝♦♥❞ ♦r❞❡r s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✷

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s

■♠♣❧✐❝✐t ❢✉♥❝t✐♦♥ t❤❡♦r❡♠✿ u → y[u] ✐s ♦❢ ❝❧❛ss C∞ ❢r♦♠ L2(0, T)m t♦ Y ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ▲❛❣r❛♥❣✐❛♥ ♦❢ ♣r♦❜❧❡♠ (P) ✐s✱ ❝❤♦♦s✐♥❣ t❤❡ ♠✉❧t✐♣❧✐❡r ♦❢ t❤❡ st❛t❡ ❡q✉❛t✐♦♥ t♦ ❜❡ (p, p0) ∈ L2(Q) × H−1(Ω) ❛♥❞ t❛❦✐♥❣ β ∈ R+✱ dµ ∈ M+(0, T), L[β, p, p0, dµ](u, y) := βJ(u, y) − p0, y(·, 0) − y0H1

0 (Ω)

+

• Q

p

• ∆y(x, t) − γy3(x, t) + f(x, t) +

m

• i=0

ui(t)bi(x)y(x, t) − ˙ y(x, t)

• dxdt

+

q

• j=1

T gj(y(·, t))dµj(t).

❍❡r❡✿ M+(0, T) ♣♦s✐t✐✈❡ ✜♥✐t❡ ❘❛❞♦♥ ♠❡❛s✉r❡s❀ ✇❡ ✐❞❡♥t✐❢② ✐t ✇✐t❤ t❤❡ s❡t BV (0, T)q

0,+ := {µ ∈ BV (0, T)q; µ(T) = 0, dµ ≥ 0}.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✸

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s

❋♦r ❡❛❝❤ z ∈ L2(0, T; H2(Ω)) ❛♥❞ (x, t) ∈ Q, (Az)(x, t) := −∆z(x, t) + 3γ¯ y(x, t)2z(x, t) −

m

• i=0

¯ ui(t)bi(x)z(x, t). ❈♦st❛t❡ ❡q✉❛t✐♦♥✿ ❢♦r ❛♥② t❤❡r❡ ❡①✐st ✇✐t❤ ❆❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s ✭❇♦♥♥❛♥s ✫ ❏❛✐ss♦♥ ✷✵✶✵✮ ✭❈❙✮ ✇❤❡r❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✹

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s

❋♦r ❡❛❝❤ z ∈ L2(0, T; H2(Ω)) ❛♥❞ (x, t) ∈ Q, (Az)(x, t) := −∆z(x, t) + 3γ¯ y(x, t)2z(x, t) −

m

• i=0

¯ ui(t)bi(x)z(x, t). ❈♦st❛t❡ ❡q✉❛t✐♦♥✿ ❢♦r ❛♥② z ∈ Y t❤❡r❡ ❡①✐st p ∈ L2(Q) ✇✐t❤

• Q

p( ˙ z + Az)dxdt + p0, z(·, 0)H1

0 (Ω) =

q

• j=1

T

• Ω

cjzdxdµj(t) + β

• Q

(¯ y − yd)zdxdt + β

• Ω

(¯ y(x, T) − ydT (x))z(x, T)dx. ❆❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s ✭❇♦♥♥❛♥s ✫ ❏❛✐ss♦♥ ✷✵✶✵✮ ✭❈❙✮ ✇❤❡r❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✹

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s

❋♦r ❡❛❝❤ z ∈ L2(0, T; H2(Ω)) ❛♥❞ (x, t) ∈ Q, (Az)(x, t) := −∆z(x, t) + 3γ¯ y(x, t)2z(x, t) −

m

• i=0

¯ ui(t)bi(x)z(x, t). ❈♦st❛t❡ ❡q✉❛t✐♦♥✿ ❢♦r ❛♥② z ∈ Y t❤❡r❡ ❡①✐st p ∈ L2(Q) ✇✐t❤

• Q

p( ˙ z + Az)dxdt + p0, z(·, 0)H1

0 (Ω) =

q

• j=1

T

• Ω

cjzdxdµj(t) + β

• Q

(¯ y − yd)zdxdt + β

• Ω

(¯ y(x, T) − ydT (x))z(x, T)dx. ❆❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s ✭❇♦♥♥❛♥s ✫ ❏❛✐ss♦♥ ✷✵✶✵✮ p1 := p +

q

• j=1

cjµj; p1

0 := p0 + q

• j=1

cjµj(0), ✭❈❙✮ ✇❤❡r❡ µ ∈ BV (0, T)q

0,+ ❛ss♦❝✐❛t❡❞ ✇✐t❤ dµ✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✹

▲❡♠♠❛

▲❡t (p, p0, µ) ∈ L2(Q) × H−1(Ω) × BV (0, T)q

0,+ s❛t✐s❢② t❤❡ ✇❡❛❦ ❢♦r♠✉❧❛t✐♦♥✱ ❛♥❞ ❧❡t

(p1, p1

0) ❜❡ ❛ss♦❝✐❛t❡❞ ❝♦st❛t❡s✳ ❚❤❡♥

p1 ∈ Y, p1(0) = p1

0,

− ˙ p1 + Ap1 = β(¯ y − yd) +

q

• j=1

µjAcj, p1(·, T) = β(¯ y(·, T) − ydT ). ▼♦r❡♦✈❡r✱ p(x, 0) ❛♥❞ p(x, T) ❛r❡ ✇❡❧❧✲❞❡✜♥❡❞ ✐♥ H1

0(Ω) ✐♥ ✈✐❡✇ ♦❢ ✭❈❙✮✱ ❛♥❞ ✇❡ ❤❛✈❡

p(·, 0) = p0, p(·, T) = β(¯ y(·, T) − ydT ). Pr♦♦❢✿ ■♥t❡❣r❛t✐♦♥ ❜② ♣❛rts

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✺

Pr♦♦❢ ❜② ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts

❘❡♠❡♠❜❡r p1 := p +

q

• j=1

cjµj; p1

0 := p0 + q

• j=1

cjµj(0). ✭❈❙✮

❲✐t❤ ψ = z(·, 0) ✇❡ ❤❛✈❡

q

• j=1
• Q

cjµj ˙ zdxdt +

q

• j=1

µj(0)cj, ψL2(Ω) = −

q

• j=1

T

• Ω

cjzdxdµj(t). ❚❤❡ ❧❛tt❡r ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ r❡✇r✐tt❡♥ ❛s

• Q

(p1 − p) ˙ zdxdt + p1

0 − p0, ψH1

0 (Ω) = −

q

• j=1

T

• Ω

cjzdxdµj(t). ✭✶✳✶✮

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✻

Pr♦♦❢

❚❤❛t ♠❡❛♥s✱ ✇❡ ❤❛✈❡

• Q

(p1 − p) ˙ zdxdt + p1

0 − p0, ψH1

0 (Ω) = −

q

• j=1

T

• Ω

cjzdxdµj(t). ❛♥❞ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❝♦st❛t❡ ❡q✉❛t✐♦♥ ❛♥❞ ✇❡ ♦❜t❛✐♥✱ ✇✐t❤ ✱ t❤❛t

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✼

Pr♦♦❢

❚❤❛t ♠❡❛♥s✱ ✇❡ ❤❛✈❡

• Q

(p1 − p) ˙ zdxdt + p1

0 − p0, ψH1

0 (Ω) = −

q

• j=1

T

• Ω

cjzdxdµj(t). ❛♥❞ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❝♦st❛t❡ ❡q✉❛t✐♦♥

• Q

p( ˙ z + Az)dxdt + p0, z(·, 0)H1

0 (Ω) =

q

• j=1

T

• Ω

cjzdxdµj(t) + β

• Q

(¯ y − yd)zdxdt + β

• Ω

(¯ y(x, T) − ydT (x))z(x, T)dx. ❛♥❞ ✇❡ ♦❜t❛✐♥✱ ✇✐t❤ ✱ t❤❛t

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✼

Pr♦♦❢

❚❤❛t ♠❡❛♥s✱ ✇❡ ❤❛✈❡

• Q

(p1 − p) ˙ zdxdt + p1

0 − p0, ψH1

0 (Ω) = −

q

• j=1

T

• Ω

cjzdxdµj(t). ❛♥❞ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❝♦st❛t❡ ❡q✉❛t✐♦♥

• Q

p( ˙ z + Az)dxdt + p0, z(·, 0)H1

0 (Ω) =

q

• j=1

T

• Ω

cjzdxdµj(t) + β

• Q

(¯ y − yd)zdxdt + β

• Ω

(¯ y(x, T) − ydT (x))z(x, T)dx. ❛♥❞

• Q

(p1 − p)Az =

• Q

q

• j=1

cjµjAz ✇❡ ♦❜t❛✐♥✱ ✇✐t❤ ✱ t❤❛t

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✼

Pr♦♦❢

❚❤❛t ♠❡❛♥s✱ ✇❡ ❤❛✈❡

• Q

(p1 − p) ˙ zdxdt + p1

0 − p0, ψH1

0 (Ω) = −

q

• j=1

T

• Ω

cjzdxdµj(t). ❛♥❞ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❝♦st❛t❡ ❡q✉❛t✐♦♥

• Q

p( ˙ z + Az)dxdt + p0, z(·, 0)H1

0 (Ω) =

q

• j=1

T

• Ω

cjzdxdµj(t) + β

• Q

(¯ y − yd)zdxdt + β

• Ω

(¯ y(x, T) − ydT (x))z(x, T)dx. ❛♥❞

• Q

(p1 − p)Az =

• Q

q

• j=1

cjµjAz ✇❡ ♦❜t❛✐♥✱ ✇✐t❤ ϕ = ˙ z + Az✱ t❤❛t

• Q

p1ϕdxdt + p1

0, ψH1

0 (Ω)

= β

• Q

(¯ y − yd)zdxdt + β

• Ω

(¯ y(x, T) − ydT (x))z(x, T)dx +

• Q

q

• j=1

cjµjAz.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✼

❙♦✱ ✇❡ ❤❛✈❡

• Q

p1ϕdxdt + p1

0, ψH1

0 (Ω)

= β

• Q

(¯ y − yd)zdxdt + β

• Ω

(¯ y(x, T) − ydT (x))z(x, T)dx +

• Q

q

• j=1

cjµjAz. ❙✐♥❝❡ A ✐s s②♠♠❡tr✐❝✱ ✇❡ s❡❡ t❤❛t p1 ✐s s♦❧✉t✐♦♥ ✐♥ Y ✳

• ❚❤✐s s❤♦✇s t❤❡ st❛t❡♠❡♥t ♦❢ t❤❡ ❧❡♠♠❛✿

▲❡t (p, p0, µ) ∈ L2(Q) × H−1(Ω) × BV (0, T)q

0,+ s❛t✐s❢② t❤❡ ✇❡❛❦ ❢♦r♠✉❧❛t✐♦♥✱ ❛♥❞ ❧❡t

(p1, p1

0) ❜❡ ❛ss♦❝✐❛t❡❞ ❝♦st❛t❡s✳ ❚❤❡♥

p1 ∈ Y, p1(0) = p1

0,

− ˙ p1 + Ap1 = β(¯ y − yd) +

q

• j=1

µjAcj, p1(·, T) = β(¯ y(·, T) − ydT ). ▼♦r❡♦✈❡r✱ p(x, 0) ❛♥❞ p(x, T) ❛r❡ ✇❡❧❧✲❞❡✜♥❡❞ ✐♥ H1

0(Ω) ✐♥ ✈✐❡✇ ♦❢ ✭❈❙✮✱ ❛♥❞ ✇❡ ❤❛✈❡

p(·, 0) = p0, p(·, T) = β(¯ y(·, T) − ydT ).

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✽

❲❡ ❦♥♦✇ p1 := p +

q

• j=1

cjµj; p1

0 := p0 + q

• j=1

cjµj(0), ✭❈❙✮ ❛♥❞ s✐♥❝❡ p1 ❛♥❞ cjµj ❜❡❧♦♥❣ t♦ L∞(0, T; H1

0(Ω)) ✇❡ ❤❛✈❡

p ∈ L∞(0, T; H1

0(Ω)).

❈♦r♦❧❧❛r②

■❢ t❤❡♥ ❛♥❞ Pr♦♦❢✿ ❚❤✐s ❢♦❧❧♦✇s ❞✐r❡❝t❧② ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ❢♦r ❛♥❞ ✭❈❙✮✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✾

❲❡ ❦♥♦✇ p1 := p +

q

• j=1

cjµj; p1

0 := p0 + q

• j=1

cjµj(0), ✭❈❙✮ ❛♥❞ s✐♥❝❡ p1 ❛♥❞ cjµj ❜❡❧♦♥❣ t♦ L∞(0, T; H1

0(Ω)) ✇❡ ❤❛✈❡

p ∈ L∞(0, T; H1

0(Ω)).

❈♦r♦❧❧❛r②

■❢ µ ∈ H1(0, T)q t❤❡♥ p ∈ Y ❛♥❞ − ˙ p + Ap = β(¯ y − yd) +

q

• j=1

cj ˙ µj. Pr♦♦❢✿ ❚❤✐s ❢♦❧❧♦✇s ❞✐r❡❝t❧② ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ❢♦r p1 ❛♥❞ ✭❈❙✮✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✶✾

❘❡❞✉❝❡❞ ♣r♦❜❧❡♠

❙❡t F(u) := J(u, y[u]), G: L2(0, T)m → C([0, T])q, G(u) := g(y[u]). ❘❡❞✉❝❡❞ ♣r♦❜❧❡♠✿ min

u∈U❛❞

F(u); G(u) ∈ K, ✭❘P✮ ✇✐t❤ K := C([0, T])q

− ❝❧♦s❡❞ ❝♦♥✈❡① ❝♦♥❡✳

■ts ✐♥t❡r✐♦r ✐s t❤❡ s❡t ♦❢ ❢✉♥❝t✐♦♥s ✐♥ C([0, T])q ✇✐t❤ ♥❡❣❛t✐✈❡ ✈❛❧✉❡s✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ r❡❞✉❝❡❞ ♣r♦❜❧❡♠ ✭❘P✮ ✐s q✉❛❧✐✜❡❞ ❛t ✐❢✿ t❤❡r❡ ❡①✐sts

❛❞ s✉❝❤ t❤❛t

s❛t✐s✜❡s

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✵

❘❡❞✉❝❡❞ ♣r♦❜❧❡♠

❙❡t F(u) := J(u, y[u]), G: L2(0, T)m → C([0, T])q, G(u) := g(y[u]). ❘❡❞✉❝❡❞ ♣r♦❜❧❡♠✿ min

u∈U❛❞

F(u); G(u) ∈ K, ✭❘P✮ ✇✐t❤ K := C([0, T])q

− ❝❧♦s❡❞ ❝♦♥✈❡① ❝♦♥❡✳

■ts ✐♥t❡r✐♦r ✐s t❤❡ s❡t ♦❢ ❢✉♥❝t✐♦♥s ✐♥ C([0, T])q ✇✐t❤ ♥❡❣❛t✐✈❡ ✈❛❧✉❡s✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ r❡❞✉❝❡❞ ♣r♦❜❧❡♠ ✭❘P✮ ✐s q✉❛❧✐✜❡❞ ❛t ¯ u ✐❢✿ t❤❡r❡ ❡①✐sts u ∈ U❛❞ s✉❝❤ t❤❛t v := u − ¯ u s❛t✐s✜❡s G(¯ u) + DG(¯ u)v ∈ int(K).

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✵

▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r

❲❡ s❛② t❤❛t (β, p, dµ) ✐s ❛ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r ✐❢ ✐t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ✜rst✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✿ dµ ✐s ❝♦♠♣❧❡♠❡♥t❛r② t♦ t❤❡ st❛t❡ ❝♦♥str❛✐♥t✱ p ✐s t❤❡ ❝♦st❛t❡✱ (β, dµ) = 0✳ ❙❡tt✐♥❣ Ψ(t) := βα(t) +

• Ω

b(x)¯ y(x, t)p(x, t)dx ♦♥❡ ❤❛s✿ T Ψ(t)(u(t) − ¯ u(t))dt ≥ 0, ❢♦r ❡✈❡r② u ∈ U❛❞. ❉❡♥♦t❡ t❤❡ s❡t ♦❢ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs (β, p, dµ) ❜② Λ(¯ u, ¯ y)✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✶

▲❡♠♠❛

▲❡t (¯ u, y[¯ u]) ❜❡ ❛♥ L2✲❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ (P)✳ ❚❤❡♥✿ t❤❡ ❛ss♦❝✐❛t❡❞ s❡t Λ ♦❢ ♠✉❧t✐♣❧✐❡rs ✐s ♥♦♥❡♠♣t②✱ ✐❢ ✐♥ ❛❞❞✐t✐♦♥ t❤❡ q✉❛❧✐✜❝❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s ❛t ¯ u✱ t❤❡♥ t❤❡r❡ ✐s ♥♦ s✐♥❣✉❧❛r ♠✉❧t✐♣❧✐❡r✱ ❛♥❞ ✇❡ ❝❛❧❧ Λ1 := {(p, dµ) ✇✐t❤ (1, p, dµ) ∈ Λ(¯ u, ¯ y).} Pr♦♦❢✿ ✭✐✮ ❙❡t ▲❡t ❜❡ ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭❘P✮✳ ❇②✱ ❡✳❣✳✱ ❬❇♦♥♥❛♥s ✫ ❙❤❛♣✐r♦✱ Pr♦♣✳ ✸✳✶✽❪✱ s✐♥❝❡ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r✱ t❤❡r❡ ❡①✐sts ❛ ❣❡♥❡r❛❧✐③❡❞ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r s✉❝❤ t❤❛t ❛♥❞

❛❞

❉✉❡ t♦ t❤❡ ❝♦st❛t❡ ❡q✉❛t✐♦♥✱ t❤❡ ❧❛tt❡r ❝♦♥❞✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ❛❜♦✈❡✳ ✭✐✐✮ ❋♦❧❧♦✇s ❜② ❬❇♦♥♥❛♥s ✫ ❙❤❛♣✐r♦✱ Pr♦♣✳ ✸✳✶✻❪✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✷

▲❡♠♠❛

▲❡t (¯ u, y[¯ u]) ❜❡ ❛♥ L2✲❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ (P)✳ ❚❤❡♥✿ t❤❡ ❛ss♦❝✐❛t❡❞ s❡t Λ ♦❢ ♠✉❧t✐♣❧✐❡rs ✐s ♥♦♥❡♠♣t②✱ ✐❢ ✐♥ ❛❞❞✐t✐♦♥ t❤❡ q✉❛❧✐✜❝❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s ❛t ¯ u✱ t❤❡♥ t❤❡r❡ ✐s ♥♦ s✐♥❣✉❧❛r ♠✉❧t✐♣❧✐❡r✱ ❛♥❞ ✇❡ ❝❛❧❧ Λ1 := {(p, dµ) ✇✐t❤ (1, p, dµ) ∈ Λ(¯ u, ¯ y).} Pr♦♦❢✿ ✭✐✮ ❙❡t L[β, dµ](u) := βF(u) +

q

• j=1

T Gj(u)(t)dµj(t). ▲❡t ¯ u ❜❡ ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭❘P✮✳ ❇②✱ ❡✳❣✳✱ ❬❇♦♥♥❛♥s ✫ ❙❤❛♣✐r♦✱ Pr♦♣✳ ✸✳✶✽❪✱ s✐♥❝❡ K ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r✱ t❤❡r❡ ❡①✐sts ❛ ❣❡♥❡r❛❧✐③❡❞ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r (β, dµ) ∈ R+ × NK(G(¯ u)) s✉❝❤ t❤❛t (β, dµ) = 0 ❛♥❞ − DuL[β, dµ](¯ u) ∈ NU❛❞(¯ u). ❉✉❡ t♦ t❤❡ ❝♦st❛t❡ ❡q✉❛t✐♦♥✱ t❤❡ ❧❛tt❡r ❝♦♥❞✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ❛❜♦✈❡✳ ✭✐✐✮ ❋♦❧❧♦✇s ❜② ❬❇♦♥♥❛♥s ✫ ❙❤❛♣✐r♦✱ Pr♦♣✳ ✸✳✶✻❪✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✷

▲❡♠♠❛

▲❡t (¯ u, y[¯ u]) ❜❡ ❛♥ L2✲❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ (P)✳ ❚❤❡♥✿ t❤❡ ❛ss♦❝✐❛t❡❞ s❡t Λ ♦❢ ♠✉❧t✐♣❧✐❡rs ✐s ♥♦♥❡♠♣t②✱ ✐❢ ✐♥ ❛❞❞✐t✐♦♥ t❤❡ q✉❛❧✐✜❝❛t✐♦♥ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s ❛t ¯ u✱ t❤❡♥ t❤❡r❡ ✐s ♥♦ s✐♥❣✉❧❛r ♠✉❧t✐♣❧✐❡r✱ ❛♥❞ ✇❡ ❝❛❧❧ Λ1 := {(p, dµ) ✇✐t❤ (1, p, dµ) ∈ Λ(¯ u, ¯ y).} Pr♦♦❢✿ ✭✐✮ ❙❡t L[β, dµ](u) := βF(u) +

q

• j=1

T Gj(u)(t)dµj(t). ▲❡t ¯ u ❜❡ ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭❘P✮✳ ❇②✱ ❡✳❣✳✱ ❬❇♦♥♥❛♥s ✫ ❙❤❛♣✐r♦✱ Pr♦♣✳ ✸✳✶✽❪✱ s✐♥❝❡ K ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r✱ t❤❡r❡ ❡①✐sts ❛ ❣❡♥❡r❛❧✐③❡❞ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r (β, dµ) ∈ R+ × NK(G(¯ u)) s✉❝❤ t❤❛t (β, dµ) = 0 ❛♥❞ − DuL[β, dµ](¯ u) ∈ NU❛❞(¯ u). ❉✉❡ t♦ t❤❡ ❝♦st❛t❡ ❡q✉❛t✐♦♥✱ t❤❡ ❧❛tt❡r ❝♦♥❞✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ❛❜♦✈❡✳ ✭✐✐✮ ❋♦❧❧♦✇s ❜② ❬❇♦♥♥❛♥s ✫ ❙❤❛♣✐r♦✱ Pr♦♣✳ ✸✳✶✻❪✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✷

❈♦♥t❛❝t s❡ts

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡t (¯ u, ¯ y) ❜❡ ❛♥ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r②✳ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❝♦♥tr♦❧ ❝♦♥str❛✐♥ts✿ ˇ Ii := {t ∈ [0, T]; ¯ ui(t) = ˇ ui}, ˆ Ii := {t ∈ [0, T]; ¯ ui(t) = ˆ ui}, Ii := ˇ Ii ∪ ˆ Ii. t❤ st❛t❡ ❝♦♥str❛✐♥t✱ ✱ ✐s

• ✐✈❡♥

✱ ✇❡ ❝❛❧❧ ❛ ♠❛①✐♠❛❧ st❛t❡ ❝♦♥str❛✐♥❡❞ ❛r❝ ❢♦r t❤❡ t❤ st❛t❡ ❝♦♥str❛✐♥ts✱ ✐❢ ❝♦♥t❛✐♥s ❜✉t ✐t ❝♦♥t❛✐♥s ♥♦ ♦♣❡♥ ✐♥t❡r✈❛❧ str✐❝t❧② ❝♦♥t❛✐♥✐♥❣ ✳ ❲❡ ❞❡✜♥❡ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛ ♠❛①✐♠❛❧ ✭❧♦✇❡r ♦r ✉♣♣❡r✮ ❝♦♥tr♦❧ ❜♦✉♥❞ ❝♦♥str❛✐♥ts ❛r❝✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✸

❈♦♥t❛❝t s❡ts

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡t (¯ u, ¯ y) ❜❡ ❛♥ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r②✳ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❝♦♥tr♦❧ ❝♦♥str❛✐♥ts✿ ˇ Ii := {t ∈ [0, T]; ¯ ui(t) = ˇ ui}, ˆ Ii := {t ∈ [0, T]; ¯ ui(t) = ˆ ui}, Ii := ˇ Ii ∪ ˆ Ii. jt❤ st❛t❡ ❝♦♥str❛✐♥t✱ j = 1, . . . , q✱ ✐s IC

j := {t ∈ [0, T]; gj(¯

y(·, t)) = 0}.

• ✐✈❡♥

✱ ✇❡ ❝❛❧❧ ❛ ♠❛①✐♠❛❧ st❛t❡ ❝♦♥str❛✐♥❡❞ ❛r❝ ❢♦r t❤❡ t❤ st❛t❡ ❝♦♥str❛✐♥ts✱ ✐❢ ❝♦♥t❛✐♥s ❜✉t ✐t ❝♦♥t❛✐♥s ♥♦ ♦♣❡♥ ✐♥t❡r✈❛❧ str✐❝t❧② ❝♦♥t❛✐♥✐♥❣ ✳ ❲❡ ❞❡✜♥❡ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛ ♠❛①✐♠❛❧ ✭❧♦✇❡r ♦r ✉♣♣❡r✮ ❝♦♥tr♦❧ ❜♦✉♥❞ ❝♦♥str❛✐♥ts ❛r❝✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✸

❈♦♥t❛❝t s❡ts

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡t (¯ u, ¯ y) ❜❡ ❛♥ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r②✳ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❝♦♥tr♦❧ ❝♦♥str❛✐♥ts✿ ˇ Ii := {t ∈ [0, T]; ¯ ui(t) = ˇ ui}, ˆ Ii := {t ∈ [0, T]; ¯ ui(t) = ˆ ui}, Ii := ˇ Ii ∪ ˆ Ii. jt❤ st❛t❡ ❝♦♥str❛✐♥t✱ j = 1, . . . , q✱ ✐s IC

j := {t ∈ [0, T]; gj(¯

y(·, t)) = 0}.

• ✐✈❡♥ 0 ≤ a < b ≤ T✱ ✇❡ ❝❛❧❧ (a, b) ❛ ♠❛①✐♠❛❧ st❛t❡ ❝♦♥str❛✐♥❡❞ ❛r❝ ❢♦r t❤❡ jt❤

st❛t❡ ❝♦♥str❛✐♥ts✱ ✐❢ IC

j ❝♦♥t❛✐♥s (a, b) ❜✉t ✐t ❝♦♥t❛✐♥s ♥♦ ♦♣❡♥ ✐♥t❡r✈❛❧ str✐❝t❧②

❝♦♥t❛✐♥✐♥❣ (a, b)✳ ❲❡ ❞❡✜♥❡ ✐♥ t❤❡ s❛♠❡ ✇❛② ❛ ♠❛①✐♠❛❧ ✭❧♦✇❡r ♦r ✉♣♣❡r✮ ❝♦♥tr♦❧ ❜♦✉♥❞ ❝♦♥str❛✐♥ts ❛r❝✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✸

❋✐rst ♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥

Ψp

i (t) = αi +

• Ω

bi(x)¯ y(x, t)p(x, t)dx, ❢♦r i = 1, . . . , m, ♦♥❡ ❤❛s Ψp ∈ L∞(0, T)m ❛♥❞

m

• i=1

T Ψp

i (t)(ui(t) − ¯

ui(t))dt ≥ 0, ❢♦r ❡✈❡r② u ∈ U❛❞. ✭✶✳✷✮

❈♦r♦❧❧❛r②

❚❤❡ ✜rst ♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ✐s ❡q✉✐✈❛❧❡♥t t♦ {t ∈ [0, T]; Ψp

i (t) > 0} ⊆ ˇ

Ii, {t ∈ [0, T]; Ψp

i (t) < 0} ⊆ ˆ

Ii, ❢♦r ❡✈❡r② (p, dµ) ∈ Λ1.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✹

❈♦♥t❡♥t

✶

❚❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

✷

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s ❛♥❞ ❛❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s

✸

❖♥ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡r

✹

❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ r❛❞✐❛❧✐t②

✺

❚❤❡ ●♦❤ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠ ❛♥❞ ❝r✐t✐❝❛❧ ❝♦♥❡

✻

❙❡❝♦♥❞ ♦r❞❡r s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✺

❍②♣♦t❤❡s✐s

❋✐♥✐t❡ ❛r❝ ♣r♦♣❡rt②✿

• t❤❡ ❝♦♥t❛❝t s❡ts ❢♦r t❤❡ st❛t❡ ❛♥❞ ❜♦✉♥❞ ❝♦♥str❛✐♥ts ❛r❡✱

✉♣ t♦ ❛ ✜♥✐t❡ s❡t✱ t❤❡ ✉♥✐♦♥ ♦❢ ✜♥✐t❡❧② ♠❛♥② ♠❛①✐♠❛❧ ❛r❝s✳ ❚❤❡r❡ ❡①✐st ❥✉♥❝t✐♦♥ ♣♦✐♥ts 0 =: τ0 < · · · < τr := T, s✉❝❤ t❤❛t t❤❡ ✐♥t❡r✈❛❧s (τk, τk+1) ❛r❡ ♠❛①✐♠❛❧ ❛r❝s ✇✐t❤ ❝♦♥st❛♥t ❛❝t✐✈❡ ❝♦♥str❛✐♥ts✳

❉❡✜♥✐t✐♦♥

❋♦r ❧❡t ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ✐♥❞❡①❡s ♦❢ ❛❝t✐✈❡ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞ ❝♦♥str❛✐♥ts✱ ❛♥❞ st❛t❡ ❝♦♥str❛✐♥ts✱ ♦♥ t❤❡ ♠❛①✐♠❛❧ ❛r❝ ✱ ❛♥❞ s❡t ✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✻

❍②♣♦t❤❡s✐s

❋✐♥✐t❡ ❛r❝ ♣r♦♣❡rt②✿

• t❤❡ ❝♦♥t❛❝t s❡ts ❢♦r t❤❡ st❛t❡ ❛♥❞ ❜♦✉♥❞ ❝♦♥str❛✐♥ts ❛r❡✱

✉♣ t♦ ❛ ✜♥✐t❡ s❡t✱ t❤❡ ✉♥✐♦♥ ♦❢ ✜♥✐t❡❧② ♠❛♥② ♠❛①✐♠❛❧ ❛r❝s✳ ❚❤❡r❡ ❡①✐st ❥✉♥❝t✐♦♥ ♣♦✐♥ts 0 =: τ0 < · · · < τr := T, s✉❝❤ t❤❛t t❤❡ ✐♥t❡r✈❛❧s (τk, τk+1) ❛r❡ ♠❛①✐♠❛❧ ❛r❝s ✇✐t❤ ❝♦♥st❛♥t ❛❝t✐✈❡ ❝♦♥str❛✐♥ts✳

❉❡✜♥✐t✐♦♥

❋♦r k = 0, . . . , r − 1, ❧❡t ˇ Bk, ˆ Bk, Ck ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ✐♥❞❡①❡s ♦❢ ❛❝t✐✈❡ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞ ❝♦♥str❛✐♥ts✱ ❛♥❞ st❛t❡ ❝♦♥str❛✐♥ts✱ ♦♥ t❤❡ ♠❛①✐♠❛❧ ❛r❝ (τk, τk+1)✱ ❛♥❞ s❡t Bk := ˇ Bk ∪ ˆ Bk✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✻

❋♦r v : [0, T] → X✱ X ❇❛♥❛❝❤ s♣❛❝❡✱ ✇❡ ❞❡♥♦t❡ ✭✐❢ t❤❡② ❡①✐st✮ ✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❧✐♠✐ts ❛t τ ∈ [0, T] ❜② v(τ±)✱ ✇✐t❤ v(0−) := v(0), v(T+) := v(T) ❛♥❞ t❤❡ ❥✉♠♣ ❜② [v(τ)] := v(τ+) − v(τ−). ❲❡ ❞❡♥♦t❡ t❤❡ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❜② ◆♦t❡ t❤❛t ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ ❢♦r ❡❛❝❤

▲❡♠♠❛

▲❡t ❤❛✈❡ ❧❡❢t ❛♥❞ r✐❣❤t ❧✐♠✐ts ❛t ✳ ❚❤❡♥

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✼

❋♦r v : [0, T] → X✱ X ❇❛♥❛❝❤ s♣❛❝❡✱ ✇❡ ❞❡♥♦t❡ ✭✐❢ t❤❡② ❡①✐st✮ ✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❧✐♠✐ts ❛t τ ∈ [0, T] ❜② v(τ±)✱ ✇✐t❤ v(0−) := v(0), v(T+) := v(T) ❛♥❞ t❤❡ ❥✉♠♣ ❜② [v(τ)] := v(τ+) − v(τ−). ❲❡ ❞❡♥♦t❡ t❤❡ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❜② g(1)

j

(¯ y(·, t)) := d dtgj(¯ y(·, t)) =

• Ω

cj(x) ˙ ¯ y(x, t)dx, j = 1, . . . , q. ◆♦t❡ t❤❛t g(1)

j

(¯ y(·, t)) ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ L1(0, T), ❢♦r ❡❛❝❤ j = 1, . . . , q.

▲❡♠♠❛

▲❡t ❤❛✈❡ ❧❡❢t ❛♥❞ r✐❣❤t ❧✐♠✐ts ❛t ✳ ❚❤❡♥

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✼

❋♦r v : [0, T] → X✱ X ❇❛♥❛❝❤ s♣❛❝❡✱ ✇❡ ❞❡♥♦t❡ ✭✐❢ t❤❡② ❡①✐st✮ ✐ts ❧❡❢t ❛♥❞ r✐❣❤t ❧✐♠✐ts ❛t τ ∈ [0, T] ❜② v(τ±)✱ ✇✐t❤ v(0−) := v(0), v(T+) := v(T) ❛♥❞ t❤❡ ❥✉♠♣ ❜② [v(τ)] := v(τ+) − v(τ−). ❲❡ ❞❡♥♦t❡ t❤❡ t✐♠❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❜② g(1)

j

(¯ y(·, t)) := d dtgj(¯ y(·, t)) =

• Ω

cj(x) ˙ ¯ y(x, t)dx, j = 1, . . . , q. ◆♦t❡ t❤❛t g(1)

j

(¯ y(·, t)) ✐s ❛♥ ❡❧❡♠❡♥t ♦❢ L1(0, T), ❢♦r ❡❛❝❤ j = 1, . . . , q.

▲❡♠♠❛

▲❡t ¯ u ❤❛✈❡ ❧❡❢t ❛♥❞ r✐❣❤t ❧✐♠✐ts ❛t τ ∈ (0, T)✳ ❚❤❡♥ [Ψp

i (τ)][¯

ui(τ)] = [g(1)

j

(¯ y(·, τ))][µj(τ)] = 0, i = 1, . . . , m, j = 1, . . . , q.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✼

▲♦❝❛❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥

❋♦r ✜①❡❞ k ✐♥ {0, . . . , r − 1} ❛♥❞ ♠❛①✐♠❛❧ ❛r❝ (τk, τk+1), s❡tt✐♥❣ Mij(t) :=

• Ω

bi(x)cj(x)¯ y(x, t)dx, 1 ≤ i ≤ m, 1 ≤ j ≤ q. ▲❡t ¯ Mk(t) ✭♦❢ s✐③❡ | ¯ Bk| × |Ck|✮ ❞❡♥♦t❡ t❤❡ s✉❜♠❛tr✐① ♦❢ M(t) ❤❛✈✐♥❣ r♦✇s ✇✐t❤ ✐♥❞❡① ✐♥ ¯ Bk ❛♥❞ ❝♦❧✉♠♥s ✇✐t❤ ✐♥❞❡① ✐♥ Ck✳

❍②♣♦t❤❡s✐s

❆ss✉♠❡ ❢♦r ❛♥❞ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t ❢♦r ❛❧❧ ❛✳❡✳ ♦♥ ❢♦r ✭✶✳✸✮ ❚❤✐s ❤②♣♦t❤❡s✐s ✇❛s ❛❧r❡❛❞② ✉s❡❞ ✐♥ ❛ ❞✐✛❡r❡♥t s❡tt✐♥❣ ✭✐✳❡✳ ❤✐❣❤❡r✲♦r❞❡r st❛t❡ ❝♦♥str❛✐♥ts ✐♥ t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✮ ✐♥ ❡✳❣✳ ❬❇♦♥♥❛♥s✱ ❍❡r♠❛♥t ✷✵✵✾❀ ▼❛✉r❡r ✶✾✼✾❪✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✽

▲♦❝❛❧ ❝♦♥tr♦❧❧❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥

❋♦r ✜①❡❞ k ✐♥ {0, . . . , r − 1} ❛♥❞ ♠❛①✐♠❛❧ ❛r❝ (τk, τk+1), s❡tt✐♥❣ Mij(t) :=

• Ω

bi(x)cj(x)¯ y(x, t)dx, 1 ≤ i ≤ m, 1 ≤ j ≤ q. ▲❡t ¯ Mk(t) ✭♦❢ s✐③❡ | ¯ Bk| × |Ck|✮ ❞❡♥♦t❡ t❤❡ s✉❜♠❛tr✐① ♦❢ M(t) ❤❛✈✐♥❣ r♦✇s ✇✐t❤ ✐♥❞❡① ✐♥ ¯ Bk ❛♥❞ ❝♦❧✉♠♥s ✇✐t❤ ✐♥❞❡① ✐♥ Ck✳

❍②♣♦t❤❡s✐s

❆ss✉♠❡ |Ck| ≤ | ¯ Bk|, ❢♦r k = 0, . . . , r − 1, ❛♥❞

• t❤❡r❡ ❡①✐sts α > 0, s✉❝❤ t❤❛t | ¯

Mk(t)λ| ≥ α|λ|, ❢♦r ❛❧❧ λ ∈ R|Ck|, ❛✳❡✳ ♦♥ (τk, τk+1), ❢♦r k = 0, . . . , r − 1. ✭✶✳✸✮ ❚❤✐s ❤②♣♦t❤❡s✐s ✇❛s ❛❧r❡❛❞② ✉s❡❞ ✐♥ ❛ ❞✐✛❡r❡♥t s❡tt✐♥❣ ✭✐✳❡✳ ❤✐❣❤❡r✲♦r❞❡r st❛t❡ ❝♦♥str❛✐♥ts ✐♥ t❤❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✮ ✐♥ ❡✳❣✳ ❬❇♦♥♥❛♥s✱ ❍❡r♠❛♥t ✷✵✵✾❀ ▼❛✉r❡r ✶✾✼✾❪✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✽

❍②♣♦t❤❡s✐s

❲❡ ❛ss✉♠❡ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❛t ❝♦rr❡s♣♦♥❞✐♥❣ ❥✉♥❝t✐♦♥ ♣♦✐♥ts✱ t❤❡ ❝♦♥tr♦❧ ¯ u ❤❛s ❧❡❢t ❛♥❞ r✐❣❤t ❧✐♠✐ts ❛t t❤❡ ❥✉♥❝t✐♦♥ ♣♦✐♥ts τk ∈ (0, T)✳ ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s❡s ❛♥❞ t❤❡ ❧❡♠♠❛ ✭♦♥ t❤❡ ❥✉♠♣s✮ ✇❡ ♦❜t❛✐♥

❚❤❡♦r❡♠

✭✐✮ ❋♦r t❤❡ ❛ss♦❝✐❛t❡❞ st❛t❡ ❜❡❧♦♥❣s t♦ ✳ ✭✐✐✮ ❋♦r ❡✈❡r② ♦♥❡ ❤❛s t❤❛t ❛♥❞ ✐s ❡ss❡♥t✐❛❧❧② ❜♦✉♥❞❡❞ ✐♥ ✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✾

❍②♣♦t❤❡s✐s

❲❡ ❛ss✉♠❡ ❞✐s❝♦♥t✐♥✉✐t② ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ❛t ❝♦rr❡s♣♦♥❞✐♥❣ ❥✉♥❝t✐♦♥ ♣♦✐♥ts✱ t❤❡ ❝♦♥tr♦❧ ¯ u ❤❛s ❧❡❢t ❛♥❞ r✐❣❤t ❧✐♠✐ts ❛t t❤❡ ❥✉♥❝t✐♦♥ ♣♦✐♥ts τk ∈ (0, T)✳ ❯♥❞❡r t❤❡ ❤②♣♦t❤❡s❡s ❛♥❞ t❤❡ ❧❡♠♠❛ ✭♦♥ t❤❡ ❥✉♠♣s✮ ✇❡ ♦❜t❛✐♥

❚❤❡♦r❡♠

✭✐✮ ❋♦r u ∈ L∞(0, T)m, t❤❡ ❛ss♦❝✐❛t❡❞ st❛t❡ y[u] ❜❡❧♦♥❣s t♦ C( ¯ Q)✳ ✭✐✐✮ ❋♦r ❡✈❡r② (p, dµ) ∈ Λ1, ♦♥❡ ❤❛s t❤❛t µ ∈ W 1,∞(0, T)q ❛♥❞ p ✐s ❡ss❡♥t✐❛❧❧② ❜♦✉♥❞❡❞ ✐♥ Q✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✷✾

❈♦♥t❡♥t

✶

❚❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

✷

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s ❛♥❞ ❛❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s

✸

❖♥ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡r

✹

❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ r❛❞✐❛❧✐t②

✺

❚❤❡ ●♦❤ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠ ❛♥❞ ❝r✐t✐❝❛❧ ❝♦♥❡

✻

❙❡❝♦♥❞ ♦r❞❡r s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✵

❙❡❝♦♥❞ ✈❛r✐❛t✐♦♥

❋♦r (p, dµ) ∈ Λ1, s❡t κ(x, t) := 1 − 6γ¯ y(x, t)p(x, t), ❛♥❞ ❝♦♥s✐❞❡r t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠ Q[p, dµ](z, v) :=

• Q
• κz2 + 2p

m

• i=1

vibiz

• dxdt +
• Ω

z(x, T)2dx.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✶

▲❡t (u, y) ❜❡ ❛ tr❛❥❡❝t♦r②✱ ❛♥❞ s❡t (δy, v) := (y − ¯ y, u − ¯ u). ❲❡ ❤❛✈❡      d dtδy + Aδy =

m

• i=1

vibiy − 3γ¯ y(δy)2 − γ(δy)3 ✐♥ Q, δy = 0 ♦♥ Σ, δy(·, 0) = 0 ✐♥ Ω.

Pr♦♣♦s✐t✐♦♥

▲❡t (p, dµ) ∈ Λ1✱ ❛♥❞ ❧❡t (u, y) ❜❡ ❛ tr❛❥❡❝t♦r②✳ ❚❤❡♥ L[p, dµ](u, y, p) − L[p, dµ](¯ u, ¯ y, p) = T Ψp(t) · v(t)dt + 1

2Q[p, dµ](δy, v) − γ

• Q

p(δy)3dxdt.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✷

❈r✐t✐❝❛❧ ❝♦♥❡

❋♦r ¯ u ∈ L2 ✇❡ ❞❡✜♥❡ C :=                (z[v], v) ∈ Y × L2(0, T)m; vi(t)Ψp

i (t) = 0 ❛✳❡✳ ♦♥ [0, T],

❢♦r ❛❧❧ (p, dµ) ∈ Λ1 vi(t) ≥ 0 ❛✳❡✳ ♦♥ ˇ Ii, vi(t) ≤ 0 ❛✳❡✳ ♦♥ ˆ Ii, ❢♦r i = 1, . . . , m,

• Ω

cj(x)z[v](x, t)dx ≤ 0 ♦♥ IC

j , ❢♦r j = 1, . . . , q

               .

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✸

❙tr✐❝t ❝r✐t✐❝❛❧ ❝♦♥❡

■♠♣♦s✐♥❣ t❤❛t t❤❡ ❧✐♥❡❛r✐③❛t✐♦♥ ♦❢ ❛❝t✐✈❡ ❝♦♥str❛✐♥ts ✐s ③❡r♦ Cs :=      (z[v], v) ∈ Y × L2(0, T)m; vi(t) = 0 ❛✳❡✳ ♦♥ Ii, ❢♦r i = 1, . . . , m,

• Ω

cj(x)z[v](x, t)dx = 0 ♦♥ IC

j , ❢♦r j = 1, . . . , q

     . ❍❡♥❝❡✱ ❝❧❡❛r❧② Cs ⊆ C, ❛♥❞ Cs ✐s ❛ ❝❧♦s❡❞ s✉❜s♣❛❝❡ ♦❢ Y × L2(0, T)m.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✹

❘❛❞✐❛❧✐t② ♦❢ ❝r✐t✐❝❛❧ ❞✐r❡❝t✐♦♥s

❍②♣♦t❤❡s✐s✿ ✉♥✐❢♦r♠ ❞✐st❛♥❝❡ t♦ ❝♦♥tr♦❧ ❜♦✉♥❞s ✇❤❡♥❡✈❡r t❤❡② ❛r❡ ♥♦t ❛❝t✐✈❡✱ ❆r♦♥♥❛ ❡t ❛❧✳ ✷✵✶✻✿ ❛ ❝r✐t✐❝❛❧ ❞✐r❡❝t✐♦♥ (z, v) ✐s q✉❛s✐ r❛❞✐❛❧ ✐❢ t❤❡r❡ ❡①✐sts τ0 > 0 s✉❝❤ t❤❛t✱ ❢♦r τ ∈ [0, τ0], t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ s❛t✐s✜❡❞✿ max

t∈[0,T ]

• gj(¯

y(·, t)) + τg′

j(¯

y(·, t))z(t)

• = o(τ 2),

❢♦r j = 1, . . . , q, ˇ ui ≤ ¯ ui(t) + τvi(t) ≤ ˆ ui, ❛✳❡✳ ♦♥ [0, T], ❢♦r i = 1, . . . , m.

❈♦r♦❧❧❛r②

❚❤❡ s❡t ♦❢ q✉❛s✐ r❛❞✐❛❧ ❝r✐t✐❝❛❧ ❞✐r❡❝t✐♦♥s ♦❢ ✐s ❞❡♥s❡ ✐♥

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✺

❘❛❞✐❛❧✐t② ♦❢ ❝r✐t✐❝❛❧ ❞✐r❡❝t✐♦♥s

❍②♣♦t❤❡s✐s✿ ✉♥✐❢♦r♠ ❞✐st❛♥❝❡ t♦ ❝♦♥tr♦❧ ❜♦✉♥❞s ✇❤❡♥❡✈❡r t❤❡② ❛r❡ ♥♦t ❛❝t✐✈❡✱ ❆r♦♥♥❛ ❡t ❛❧✳ ✷✵✶✻✿ ❛ ❝r✐t✐❝❛❧ ❞✐r❡❝t✐♦♥ (z, v) ✐s q✉❛s✐ r❛❞✐❛❧ ✐❢ t❤❡r❡ ❡①✐sts τ0 > 0 s✉❝❤ t❤❛t✱ ❢♦r τ ∈ [0, τ0], t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ s❛t✐s✜❡❞✿ max

t∈[0,T ]

• gj(¯

y(·, t)) + τg′

j(¯

y(·, t))z(t)

• = o(τ 2),

❢♦r j = 1, . . . , q, ˇ ui ≤ ¯ ui(t) + τvi(t) ≤ ˆ ui, ❛✳❡✳ ♦♥ [0, T], ❢♦r i = 1, . . . , m.

❈♦r♦❧❧❛r②

❚❤❡ s❡t ♦❢ q✉❛s✐ r❛❞✐❛❧ ❝r✐t✐❝❛❧ ❞✐r❡❝t✐♦♥s ♦❢ Cs ✐s ❞❡♥s❡ ✐♥ Cs.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✺

❚❤❡♦r❡♠ ✭❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥✮

▲❡t t❤❡ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r② (¯ u, ¯ y) ❜❡ ❛♥ L∞✲❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ (P)✳ ❚❤❡♥ max

(p,dµ)∈Λ1 Q[p, dµ](z, v) ≥ 0,

❢♦r ❛❧❧ (z, v) ∈ Cs.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✻

❈♦♥t❡♥t

✶

❚❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

✷

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s ❛♥❞ ❛❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s

✸

❖♥ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡r

✹

❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ r❛❞✐❛❧✐t②

✺

❚❤❡ ●♦❤ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠ ❛♥❞ ❝r✐t✐❝❛❧ ❝♦♥❡

✻

❙❡❝♦♥❞ ♦r❞❡r s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✼

• ♦❤ tr❛♥s❢♦r♠
• ✐✈❡♥ ❛ ❝r✐t✐❝❛❧ ❞✐r❡❝t✐♦♥ (z, v)✱ s❡t

w(t) := t v(s)ds; B(x, t) := ¯ y(x, t)b(x); ζ(x, t) := z(x, t) − B(x, t) · w(t), ❜❛s❡❞ ♦♥ ❬●♦❤ ✶✾✻✻❪✳ ❲❡ ❤❛✈❡ ❙✐♥❝❡ ✐t ❢♦❧❧♦✇s t❤❛t ✭✶✳✹✮ ✇❤❡r❡ ❢♦r

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✽

• ♦❤ tr❛♥s❢♦r♠
• ✐✈❡♥ ❛ ❝r✐t✐❝❛❧ ❞✐r❡❝t✐♦♥ (z, v)✱ s❡t

w(t) := t v(s)ds; B(x, t) := ¯ y(x, t)b(x); ζ(x, t) := z(x, t) − B(x, t) · w(t), ❜❛s❡❞ ♦♥ ❬●♦❤ ✶✾✻✻❪✳ ❲❡ ❤❛✈❡ ˙ ζ + Aζ =

• ˙

z + Az −

m

• i=1

viBi

• =0

−

m

• i=1

wi(ABi + ˙ Bi), ζ(·, 0) = 0. ❙✐♥❝❡ ˙ Bi = bi ˙ ¯ y ✐t ❢♦❧❧♦✇s t❤❛t ˙ ζ(x, t) + (Aζ)(x, t) = B1(x, t) · w(t), ζ(·, 0) = 0, ✭✶✳✹✮ ✇❤❡r❡ B1

i := −fbi + 2∇¯

y · ∇bi + ¯ y∆bi − 2γ¯ y3bi, ❢♦r i = 1, . . . , m.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✽

▲❡♠♠❛ ✭❚r❛♥s❢♦r♠❡❞ s❡❝♦♥❞ ✈❛r✐❛t✐♦♥✮

❲❡ ❝❛♥ ❞❡✜♥❡ ❛ q✉❛❞r❛t✐❝ ❢♦r♠ Q s✉❝❤ t❤❛t ❢♦r v ∈ L2(0, T)m, ❛♥❞ w ∈ AC([0, T])m ❣✐✈❡♥ ❜② t❤❡ ●♦❤ tr❛♥s❢♦r♠✱ ❛♥❞ ❢♦r ❛❧❧ (p, dµ) ∈ Λ1✱ ✇❡ ❤❛✈❡ Q[p, dµ](z[v], v) = Q[p, dµ](ζ[w], w, w(T)).

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✸✾

• ♦❤ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❝r✐t✐❝❛❧ ❝♦♥❡

❙❡t ♦❢ ♣r✐♠✐t✐✈❡s ♦❢ str✐❝t ❝r✐t✐❝❛❧ ❞✐r❡❝t✐♦♥ PC :=

• (ζ, w, w(T)) ∈ Y × H1(0, T)m × Rm;

(ζ, w) ✐s ❣✐✈❡♥ ❜② t❤❡ ●♦❤ tr❛♥s❢♦r♠ ❢♦r s♦♠❡ (z, v) ∈ Cs

• ,

❛♥❞ ❧❡t PC2 := ❝❧♦s✉r❡ ♦❢ PC ✐♥ Y × L2(0, T)m × Rm. ❲❡ ❝❛♥ ❣✐✈❡ ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❛ s✉♣❡rs❡t ✇❤✐❝❤ ❝♦✐♥❝✐❞❡s ✇✐t❤ ❢♦r s❝❛❧❛r ❝♦♥tr♦❧s ✭✐✳❡✳ ✮✳ ❲❡ ✇✐❧❧ ❢♦r♠✉❧❛t❡ t❤❡ s❡❝♦♥❞✲♦r❞❡r s✉✣❝✐❡♥t ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ♦♥ ❛ s✉♣❡rs❡t ✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✵

• ♦❤ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❝r✐t✐❝❛❧ ❝♦♥❡

❙❡t ♦❢ ♣r✐♠✐t✐✈❡s ♦❢ str✐❝t ❝r✐t✐❝❛❧ ❞✐r❡❝t✐♦♥ PC :=

• (ζ, w, w(T)) ∈ Y × H1(0, T)m × Rm;

(ζ, w) ✐s ❣✐✈❡♥ ❜② t❤❡ ●♦❤ tr❛♥s❢♦r♠ ❢♦r s♦♠❡ (z, v) ∈ Cs

• ,

❛♥❞ ❧❡t PC2 := ❝❧♦s✉r❡ ♦❢ PC ✐♥ Y × L2(0, T)m × Rm. − → ❲❡ ❝❛♥ ❣✐✈❡ ❛ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❛ s✉♣❡rs❡t PC′

2 ✇❤✐❝❤ ❝♦✐♥❝✐❞❡s ✇✐t❤ PC2 ❢♦r

s❝❛❧❛r ❝♦♥tr♦❧s ✭✐✳❡✳ m = 1✮✳ − → ❲❡ ✇✐❧❧ ❢♦r♠✉❧❛t❡ t❤❡ s❡❝♦♥❞✲♦r❞❡r s✉✣❝✐❡♥t ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ♦♥ ❛ s✉♣❡rs❡t PC2 ⊂ PC∗

2✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✵

❲❡ t❛❦❡ ❛ ❝❧♦s❡r ❧♦♦❦✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✶

❲❡ r❡❝❛❧❧

❋♦r ✜①❡❞ k ✐♥ {0, . . . , r − 1} ❛♥❞ ♠❛①✐♠❛❧ ❛r❝ (τk, τk+1), s❡tt✐♥❣ Mij(t) :=

• Ω

bi(x)cj(x)¯ y(x, t)dx, 1 ≤ i ≤ m, 1 ≤ j ≤ q.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✷

❋♦r ❛♥② (ζ, w, h) ∈ PC✱ ✐t ❤♦❧❞s wBk(t) = 1 τk+1 − τk τk+1

τk

wBk(s)ds, ❢♦r k = 0, . . . , r − 1. ✭✶✳✺✮ ❚❛❦❡ ❛♥❞ ❣✐✈❡♥ ❜② t❤❡ ●♦❤ tr❛♥s❢♦r♠✳ ▲❡t ❛♥❞ ❚❤❡♥ ♦♥ ✳ ❚❤❡r❡❢♦r❡✱ ❧❡tt✐♥❣ ❞❡♥♦t❡ t❤❡ t❤ ❝♦❧✉♠♥ ♦❢ t❤❡ ♠❛tr✐① ✱ ♦♥❡ ❤❛s ♦♥ ❢♦r ✭✶✳✻✮ ❲❡ ❝❛♥ r❡✇r✐t❡ ✭✶✳✺✮✲✭✶✳✻✮ ✐♥ t❤❡ ❢♦r♠ ♦♥ ✭✶✳✼✮ ✇❤❡r❡ ✐s ❛♥ ♠❛tr✐① ✇✐t❤ ❛♥❞

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✸

❋♦r ❛♥② (ζ, w, h) ∈ PC✱ ✐t ❤♦❧❞s wBk(t) = 1 τk+1 − τk τk+1

τk

wBk(s)ds, ❢♦r k = 0, . . . , r − 1. ✭✶✳✺✮ ❚❛❦❡ (z, v) ∈ Cs, ❛♥❞ (w, ζ[w]) ❣✐✈❡♥ ❜② t❤❡ ●♦❤ tr❛♥s❢♦r♠✳ ▲❡t k ∈ {0, . . . , r − 1} ❛♥❞ j ∈ Ck. ❚❤❡♥ 0 =

• Ω

cj(x)z(x, t)dx ♦♥ (τk, τk+1)✳ ❚❤❡r❡❢♦r❡✱ ❧❡tt✐♥❣ Mj(t) ❞❡♥♦t❡ t❤❡ jt❤ ❝♦❧✉♠♥ ♦❢ t❤❡ ♠❛tr✐① M(t)✱ ♦♥❡ ❤❛s Mj(t) · w(t) = −

• Ω

cj(x)ζ[w](x, t)dt, ♦♥ (τk, τk+1), ❢♦r j ∈ Ck. ✭✶✳✻✮ ❲❡ ❝❛♥ r❡✇r✐t❡ ✭✶✳✺✮✲✭✶✳✻✮ ✐♥ t❤❡ ❢♦r♠ ♦♥ ✭✶✳✼✮ ✇❤❡r❡ ✐s ❛♥ ♠❛tr✐① ✇✐t❤ ❛♥❞

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✸

❋♦r ❛♥② (ζ, w, h) ∈ PC✱ ✐t ❤♦❧❞s wBk(t) = 1 τk+1 − τk τk+1

τk

wBk(s)ds, ❢♦r k = 0, . . . , r − 1. ✭✶✳✺✮ ❚❛❦❡ (z, v) ∈ Cs, ❛♥❞ (w, ζ[w]) ❣✐✈❡♥ ❜② t❤❡ ●♦❤ tr❛♥s❢♦r♠✳ ▲❡t k ∈ {0, . . . , r − 1} ❛♥❞ j ∈ Ck. ❚❤❡♥ 0 =

• Ω

cj(x)z(x, t)dx ♦♥ (τk, τk+1)✳ ❚❤❡r❡❢♦r❡✱ ❧❡tt✐♥❣ Mj(t) ❞❡♥♦t❡ t❤❡ jt❤ ❝♦❧✉♠♥ ♦❢ t❤❡ ♠❛tr✐① M(t)✱ ♦♥❡ ❤❛s Mj(t) · w(t) = −

• Ω

cj(x)ζ[w](x, t)dt, ♦♥ (τk, τk+1), ❢♦r j ∈ Ck. ✭✶✳✻✮ ❲❡ ❝❛♥ r❡✇r✐t❡ ✭✶✳✺✮✲✭✶✳✻✮ ✐♥ t❤❡ ❢♦r♠ Ak(t)w(t) =

• Bkw
• (t),

♦♥ (τk, τk+1), ✭✶✳✼✮ ✇❤❡r❡ Ak(t) ✐s ❛♥ mk × m ♠❛tr✐① ✇✐t❤ mk := |Bk| + |Ck|, ❛♥❞ Bk : L2(0, T)m → H1(τk, τk+1)mk.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✸

▲❡t ck+1 ∈ Rm ❜❡ s✉❝❤ t❤❛t✱ ❢♦r s♦♠❡ νk+i✱ ck+1 = Ak+i(τk+1)⊤νk+i, ❢♦r i = 0, 1, ✭✶✳✽✮ ♠❡❛♥✐♥❣ t❤❛t ck+1 ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ r♦✇s ♦❢ Ak+i(τk+1) ❢♦r ❜♦t❤ i = 0, 1.

▲❡♠♠❛

▲❡t k = 0, . . . , r − 1, ❛♥❞ ❧❡t ck+1 s❛t✐s❢② ✭✶✳✽✮✳ ❚❤❡♥✱ t❤❡ ❥✉♥❝t✐♦♥ ❝♦♥❞✐t✐♦♥ ck+1 ·

• w(τ +

k+1) − w(τ − k+1)

• = 0,

✭✶✳✾✮ ❤♦❧❞s ❢♦r ❛❧❧ (ζ, w, h) ∈ PC2✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✹

❙❡t PC′

2 := {(ζ[w], w, h); w ∈ Ker(A − B), ✭✶✳✾✮ ❤♦❧❞s✱ ❢♦r ❛❧❧ c s❛t✐s❢②✐♥❣ ✭✶✳✽✮}.

❲❡ ❤❛✈❡ ♣r♦✈❡❞ t❤❛t PC2 ⊆ PC′

2.

■♥ t❤❡ ❝❛s❡ ♦❢ ❛ s❝❛❧❛r ❝♦♥tr♦❧ ✭m = 1✮ ✇❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡s❡ t✇♦ s❡ts ❝♦✐♥❝✐❞❡✳

Pr♦♣♦s✐t✐♦♥

■❢ t❤❡ ❝♦♥tr♦❧ ✐s s❝❛❧❛r✱ t❤❡♥ PC2 =          (ζ[w], w, h) ∈ Y × L2(0, T) × R; w ∈ Ker(A − B); w ✐s ❝♦♥t✐♥✉♦✉s ❛t ❇❇✱ ❇❈✱ ❈❇ ❥✉♥❝t✐♦♥s limt↓0 w(t) = 0 ✐❢ t❤❡ ✜rst ❛r❝ ✐s ♥♦t s✐♥❣✉❧❛r limt↑T w(t) = h ✐❢ t❤❡ ❧❛st ❛r❝ ✐s ♥♦t s✐♥❣✉❧❛r          .

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✺

❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥ ✐♥ tr❛♥s❢♦r♠❡❞ ✈❛r✐❛❜❧❡s

❚❤❡♦r❡♠

■❢ (¯ u, ¯ y) ✐s ❛♥ L∞✲❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ♣r♦❜❧❡♠ ✭P✮✱ t❤❡♥ max

(p,dµ)∈Λ1

• Q[p, dµ](ζ, w, h) ≥ 0,

♦♥ PC2.

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✻

❈♦♥t❡♥t

✶

❚❤❡ ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠

✷

❋✐rst ♦r❞❡r ❛♥❛❧②s✐s ❛♥❞ ❛❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s

✸

❖♥ t❤❡ r❡❣✉❧❛r✐t② ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡r

✹

❙❡❝♦♥❞ ♦r❞❡r ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ✉s✐♥❣ r❛❞✐❛❧✐t②

✺

❚❤❡ ●♦❤ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠ ❛♥❞ ❝r✐t✐❝❛❧ ❝♦♥❡

✻

❙❡❝♦♥❞ ♦r❞❡r s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✼

P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠

✭✐✮ ❆♥ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r② (¯ u, ¯ y) ✐s s❛✐❞ t♦ ❜❡ ❛ P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠ ✐❢ ❢♦r ❛❧❧ N > 0, t❤❡r❡ ❡①✐sts εN > 0 s✉❝❤ t❤❛t✱ (¯ u, ¯ y) ✐s ♦♣t✐♠❛❧ ❛♠♦♥❣ ❛❧❧ t❤❡ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r✐❡s (u, y) ✈❡r✐❢②✐♥❣ u − ˆ u∞ < N ❛♥❞ u − ˆ u1 < εN. ✭✐✐✮ ❆ s❡q✉❡♥❝❡ ✐s s❛✐❞ t♦ ❝♦♥✈❡r❣❡ t♦ ✐♥ t❤❡ P♦♥tr②❛❣✐♥ s❡♥s❡ ✐❢ ✐t ✐s ❜♦✉♥❞❡❞ ✐♥ ❛♥❞ ✳ ✭✐✐✐✮ ❲❡ s❛② t❤❛t ✐s ❛ P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠ s❛t✐s❢②✐♥❣ t❤❡ ✇❡❛❦ q✉❛❞r❛t✐❝ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥ ✐❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ❛❞♠✐ss✐❜❧❡ ✈❛r✐❛t✐♦♥s ❤❛✈✐♥❣ ❝♦♥✈❡r❣❡♥t t♦ ✐♥ t❤❡ P♦♥tr②❛❣✐♥ s❡♥s❡✱ ♦♥❡ ❤❛s ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ ❛♥❞ ✇❤❡r❡ ✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✽

P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠

✭✐✮ ❆♥ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r② (¯ u, ¯ y) ✐s s❛✐❞ t♦ ❜❡ ❛ P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠ ✐❢ ❢♦r ❛❧❧ N > 0, t❤❡r❡ ❡①✐sts εN > 0 s✉❝❤ t❤❛t✱ (¯ u, ¯ y) ✐s ♦♣t✐♠❛❧ ❛♠♦♥❣ ❛❧❧ t❤❡ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r✐❡s (u, y) ✈❡r✐❢②✐♥❣ u − ˆ u∞ < N ❛♥❞ u − ˆ u1 < εN. ✭✐✐✮ ❆ s❡q✉❡♥❝❡ (vℓ) ⊂ L∞(0, T)m ✐s s❛✐❞ t♦ ❝♦♥✈❡r❣❡ t♦ 0 ✐♥ t❤❡ P♦♥tr②❛❣✐♥ s❡♥s❡ ✐❢ ✐t ✐s ❜♦✉♥❞❡❞ ✐♥ L∞(0, T)m ❛♥❞ vℓ1 → 0✳ ✭✐✐✐✮ ❲❡ s❛② t❤❛t ✐s ❛ P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠ s❛t✐s❢②✐♥❣ t❤❡ ✇❡❛❦ q✉❛❞r❛t✐❝ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥ ✐❢ t❤❡r❡ ❡①✐sts s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ❛❞♠✐ss✐❜❧❡ ✈❛r✐❛t✐♦♥s ❤❛✈✐♥❣ ❝♦♥✈❡r❣❡♥t t♦ ✐♥ t❤❡ P♦♥tr②❛❣✐♥ s❡♥s❡✱ ♦♥❡ ❤❛s ❢♦r s✉✣❝✐❡♥t❧② ❧❛r❣❡ ❛♥❞ ✇❤❡r❡ ✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✽

P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠

✭✐✮ ❆♥ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r② (¯ u, ¯ y) ✐s s❛✐❞ t♦ ❜❡ ❛ P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠ ✐❢ ❢♦r ❛❧❧ N > 0, t❤❡r❡ ❡①✐sts εN > 0 s✉❝❤ t❤❛t✱ (¯ u, ¯ y) ✐s ♦♣t✐♠❛❧ ❛♠♦♥❣ ❛❧❧ t❤❡ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r✐❡s (u, y) ✈❡r✐❢②✐♥❣ u − ˆ u∞ < N ❛♥❞ u − ˆ u1 < εN. ✭✐✐✮ ❆ s❡q✉❡♥❝❡ (vℓ) ⊂ L∞(0, T)m ✐s s❛✐❞ t♦ ❝♦♥✈❡r❣❡ t♦ 0 ✐♥ t❤❡ P♦♥tr②❛❣✐♥ s❡♥s❡ ✐❢ ✐t ✐s ❜♦✉♥❞❡❞ ✐♥ L∞(0, T)m ❛♥❞ vℓ1 → 0✳ ✭✐✐✐✮ ❲❡ s❛② t❤❛t (¯ u, ¯ y) ✐s ❛ P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠ s❛t✐s❢②✐♥❣ t❤❡ ✇❡❛❦ q✉❛❞r❛t✐❝ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥ ✐❢ t❤❡r❡ ❡①✐sts ρ > 0 s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ❛❞♠✐ss✐❜❧❡ ✈❛r✐❛t✐♦♥s (vℓ, δyℓ) ❤❛✈✐♥❣ (vℓ) ❝♦♥✈❡r❣❡♥t t♦ 0 ✐♥ t❤❡ P♦♥tr②❛❣✐♥ s❡♥s❡✱ ♦♥❡ ❤❛s F(uℓ) − F(¯ u) ≥ ρ(wℓ2

2 + |wℓ(T)|2),

❢♦r ℓ s✉✣❝✐❡♥t❧② ❧❛r❣❡ ❛♥❞ ✇❤❡r❡ wℓ(t) = t

0 vℓ(s)ds✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✽

❈♦♥s✐❞❡r t❤❡ ❝♦♥❞✐t✐♦♥ g′

j(¯

y(·, T))(ζ(·, T) + B(·, T)h) = 0, ✐❢ T ∈ IC

j ❛♥❞ [µj(T)] > 0, ❢♦r j = 1, . . . , q.

✭✶✳✶✵✮ ❲❡ ❞❡✜♥❡ PC∗

2 :=

• (ζ[w], w, h) ∈ Y × L2(0, T)m × Rm; wBk ✐s ❝♦♥st❛♥t ♦♥ ❡❛❝❤ ❛r❝❀

✭✶✳✹✮, ✭✶✳✻✮, ✭✶✳✶✶✮✭✐✮✲✭✐✐✮, ✭✶✳✶✵✮ ❤♦❧❞✳

• .

PC∗

2 ✐s ❛ s✉♣❡rs❡t ♦❢ PC2✳

❲❡ r❡❝❛❧❧ t❤❛t (ζ[w], w, h) ✐♥ PC s❛t✐s❢②      (i) wi = 0 ❛✳❡✳ ♦♥ (0, τ1), ❢♦r ❡❛❝❤ i ∈ B0, (ii) wi = hi ❛✳❡✳ ♦♥ (τr−1, T), ❢♦r ❡❛❝❤ i ∈ Br−1, (iii) g′

j(¯

y(·, T))[ζ(·, T) + B(·, T) · h] = 0 ✐❢ j ∈ Cr−1✳ ✭✶✳✶✶✮ ❛♥❞ ˙ ζ(x, t) + (Aζ)(x, t) = B1(x, t) · w(t), ζ(·, 0) = 0, ✭✶✳✹✮ Mj(t) · w(t) = −

• Ω

cj(x)ζ[w](x, t)dt, ♦♥ (τk, τk+1), ❢♦r j ∈ Ck. ✭✶✳✻✮

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹✾

❚❤❡♦r❡♠ ✭❙✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s✮

❛✮ ❆ss✉♠❡ ❛❞❞✐t✐♦♥❛❧ t❤❛t

✭✐✮ (¯ u, ¯ y) ✐s ❛ ❢❡❛s✐❜❧❡ tr❛❥❡❝t♦r② ✇✐t❤ ♥♦♥❡♠♣t② ❛ss♦❝✐❛t❡❞ s❡t ♦❢ ♠✉❧t✐♣❧✐❡rs Λ1❀ ✭✐✐✮ str✐❝t ❝♦♠♣❧❡♠❡♥t❛r✐t② ❢♦r ❝♦♥tr♦❧ ❛♥❞ st❛t❡ ❝♦♥str❛✐♥ts❀ ✭✐✐✐✮ ❢♦r ❡❛❝❤ (p, dµ) ∈ Λ1, Q[p, dµ](·) ✐s ❛ ▲❡❣❡♥❞r❡ ❢♦r♠ ♦♥ {(ζ[w], w, h) ∈ Y × L2(0, T)m × Rm}; ✭✐✈✮ t❤❡ ✉♥✐❢♦r♠ ♣♦s✐t✐✈✐t②✿ t❤❡r❡ ❡①✐sts ρ > 0 ✇✐t❤ max

(p,dµ)∈Λ1

• Q[p, dµ](ζ[w], w, h) ≥ ρ(w2

2 + |h|2), ❢♦r ❛❧❧ (w, h) ∈ PC∗ 2 ✳

❚❤❡♥ (¯ u, ¯ y) ✐s ❛ P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠ s❛t✐s❢②✐♥❣ t❤❡ ✇❡❛❦ q✉❛❞r❛t✐❝ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥✳ ❜✮ ❈♦♥✈❡rs❡❧②✱ ❢♦r ❛♥ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r② s❛t✐s❢②✐♥❣ ❛ ✭❝❡rt❛✐♥✮ q✉❛❞r❛t✐❝ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥✱ ✐t ❤♦❧❞s ❢♦r ❛❧❧ ✳

❆r♦♥♥❛✱ ❇♦♥♥❛♥s✱ ❑✳✱ ♣r❡♣r✐♥t✱ ✷✵✶✾✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✺✵

❚❤❡♦r❡♠ ✭❙✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s✮

❛✮ ❆ss✉♠❡ ❛❞❞✐t✐♦♥❛❧ t❤❛t

✭✐✮ (¯ u, ¯ y) ✐s ❛ ❢❡❛s✐❜❧❡ tr❛❥❡❝t♦r② ✇✐t❤ ♥♦♥❡♠♣t② ❛ss♦❝✐❛t❡❞ s❡t ♦❢ ♠✉❧t✐♣❧✐❡rs Λ1❀ ✭✐✐✮ str✐❝t ❝♦♠♣❧❡♠❡♥t❛r✐t② ❢♦r ❝♦♥tr♦❧ ❛♥❞ st❛t❡ ❝♦♥str❛✐♥ts❀ ✭✐✐✐✮ ❢♦r ❡❛❝❤ (p, dµ) ∈ Λ1, Q[p, dµ](·) ✐s ❛ ▲❡❣❡♥❞r❡ ❢♦r♠ ♦♥ {(ζ[w], w, h) ∈ Y × L2(0, T)m × Rm}; ✭✐✈✮ t❤❡ ✉♥✐❢♦r♠ ♣♦s✐t✐✈✐t②✿ t❤❡r❡ ❡①✐sts ρ > 0 ✇✐t❤ max

(p,dµ)∈Λ1

• Q[p, dµ](ζ[w], w, h) ≥ ρ(w2

2 + |h|2), ❢♦r ❛❧❧ (w, h) ∈ PC∗ 2 ✳

❚❤❡♥ (¯ u, ¯ y) ✐s ❛ P♦♥tr②❛❣✐♥ ♠✐♥✐♠✉♠ s❛t✐s❢②✐♥❣ t❤❡ ✇❡❛❦ q✉❛❞r❛t✐❝ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥✳ ❜✮ ❈♦♥✈❡rs❡❧②✱ ❢♦r ❛♥ ❛❞♠✐ss✐❜❧❡ tr❛❥❡❝t♦r② (¯ u, y[¯ u]) s❛t✐s❢②✐♥❣ ❛ ✭❝❡rt❛✐♥✮ q✉❛❞r❛t✐❝ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥✱ ✐t ❤♦❧❞s max

(p,dµ)∈Λ1

• Q[p, dµ](ζ[w], w, h) ≥ ρ(w2

2 + |h|2),

❢♦r ❛❧❧ (w, h) ∈ PC2✳

❆r♦♥♥❛✱ ❇♦♥♥❛♥s✱ ❑✳✱ ♣r❡♣r✐♥t✱ ✷✵✶✾✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✺✵

❙✉♠♠❛r②

❙❡❝♦♥❞✲♦r❞❡r ❛♥❛❧②s✐s ❢♦r s❡♠✐❧✐♥❡❛r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ✇✐t❤

◮ st❛t❡ ❝♦♥str❛✐♥ts✱ ◮ s❡✈❡r❛❧ ❝♦♥tr♦❧s✳

❚❡❝❤♥✐q✉❡s✿

◮ ❛❧t❡r♥❛t✐✈❡ ❝♦st❛t❡s✱ ◮ r❛❞✐❛❧✐t②✱ ◮ ●♦❤ tr❛♥s❢♦r♠❛t✐♦♥✳

❘❡s✉❧t✿

◮ ❙❡❝♦♥❞✲♦r❞❡r s✉✣❝✐❡♥t ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ✇✐t❤ ❣❛♣✳ ❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✺✶

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳

❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✺✷