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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 ⊂ R n ✱ ♦♣❡♥ ❛♥❞ ❜♦✉♥❞❡❞ ✇✐t❤ s♠♦♦t❤ ❜♦✉♥❞❛r②✱ Q = Ω × (0 , T ) ✱ Σ = ∂ Ω × (0 , T ) ✳  m �   y ( x, t ) − ∆ y ( x, t ) + γy 3 ( x, t ) = f ( x, t ) + y ( x, t ) ˙ u i ( t ) b i ( x ) ✐♥ Q, i =0   y = 0 ♦♥ Σ , y ( · , 0) = y 0 ✐♥ Ω , ✇✐t❤ y 0 ∈ W 1 , ∞ f ∈ L ∞ ( Q ) , b ∈ W 1 , ∞ (Ω) m +1 ✱ γ ≥ 0 ✱ u 0 ≡ 1 ✐s ❛ ❝♦♥st❛♥t✱ (Ω) , 0 ❛♥❞ u := ( u 1 , . . . , u m ) ∈ L 2 (0 , T ) m ✳ ❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✹

❙t❛t❡ ❡q✉❛t✐♦♥ ▲❡♠♠❛ ❋♦r i = 0 , . . . , m ✱ t❤❡ ♠❛♣♣✐♥❣ ❞❡✜♥❡❞ ♦♥ L 2 (0 , T ) × L ∞ (Ω) × L ∞ (0 , T ; L 2 (Ω)) ✱ ❣✐✈❡♥ ❜② ( u i , b i , y ) �→ u i b i y, ❤❛s ✐♠❛❣❡ ✐♥ L 2 ( Q ) ✱ ✐s ♦❢ ❝❧❛ss C ∞ ✱ ❛♥❞ s❛t✐s✜❡s � u i b i y � 2 ≤ � u i � 2 � b i � ∞ � y � L ∞ (0 ,T ; L 2 (Ω)) . ❚❤❡ st❛t❡ ❡q✉❛t✐♦♥ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ✐♥ Y := H 2 , 1 ( Q ) ✳ ❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✺

❙❡tt✐♥❣ ❈♦st ❢✉♥❝t✐♦♥ � � J ( u, y ) := 1 ( y ( x, t ) − y d ( x )) 2 d x d t + 1 ( y ( x, T ) − y dT ( x )) 2 d x 2 2 Ω Q � T m � + α i u i ( t )d t. 0 i =1 ✇✐t❤ y d ∈ L ∞ ( Q ) ✱ y dT ∈ W 1 , ∞ (Ω) ✱ α ∈ R m ✳ 0 ❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✻

❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ s✉❜❥❡❝t t♦ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts ✭P✮ ❛❞ ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥tr♦❧ ❝♦♥str❛✐♥ts u ∈ U ❛❞ ✱ ✇❤❡r❡ U ❛❞ = { u ∈ L 2 (0 , T ) m ; ˇ u i ≤ u ( t ) ≤ ˆ u i , i = 1 , . . . , m } , ❢♦r s♦♠❡ ❝♦♥st❛♥ts ˇ u i < ˆ u i ✱ ❢♦r i = 1 , . . . , m. ❙t❛t❡ ❝♦♥str❛✐♥ts � g j ( y ( · , t )) := c j ( x ) y ( x, t )d x + d j ≤ 0 , ❢♦r t ∈ [0 , T ] , j = 1 , . . . , q, Ω ✇❤❡r❡ c j ∈ H 2 (Ω) ∩ H 1 0 (Ω) ❢♦r j = 1 , . . . , q ✱ ❛♥❞ d ∈ R q ✳ ❆①❡❧ ❑rö♥❡r ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♦❢ P❉❊s ❖❝t♦❜❡r✱ ✷✵✶✾ ✼

❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❈♦♥tr♦❧ ❝♦♥str❛✐♥ts u ∈ U ❛❞ ✱ ✇❤❡r❡ U ❛❞ = { u ∈ L 2 (0 , T ) m ; ˇ u i ≤ u ( t ) ≤ ˆ u i , i = 1 , . . . , m } , ❢♦r s♦♠❡ ❝♦♥st❛♥ts ˇ u i < ˆ u i ✱ ❢♦r i = 1 , . . . , m. ❙t❛t❡ ❝♦♥str❛✐♥ts � g j ( y ( · , t )) := c j ( x ) y ( x, t )d x + d j ≤ 0 , ❢♦r t ∈ [0 , T ] , j = 1 , . . . , q, Ω ✇❤❡r❡ c j ∈ H 2 (Ω) ∩ H 1 0 (Ω) ❢♦r j = 1 , . . . , q ✱ ❛♥❞ d ∈ R q ✳ ❖♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ Min J ( u, y [ u ]); s✉❜❥❡❝t t♦ t❤❡ st❛t❡ ❝♦♥str❛✐♥ts . ✭P✮ u ∈U ❛❞ ❆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✱ ✷✵✶✾ ✾