❱■❊❚◆❆▼ ❆❈❆❉❊▼❨ ❖❋ ❙❈■❊◆❈❊ ❆◆❉ ❚❊❈❍◆❖▲❖●❨ ■◆❙❚■❚❯❚❊ ❖❋ ▼❆❚❍❊▼❆❚■❈❙ ◆●❯❨❊◆ ❚❍■ ◆●❖❈ ❖❆◆❍ ❉❆❚❆ ❆❙❙■▼■▲❆❚■❖◆ ■◆ ❍❊❆❚ ❈❖◆❉❯❈❚■❖◆ ❙♣❡❝✐❛❧✐t②✿ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ❙♣❡❝✐❛❧✐t② ❈♦❞❡✿ ✻✷ ✹✻ ✵✶ ✵✸ ❙❯▼▼❆❘❨ ❖❋ ❆ ❚❍❊❙■❙ ❋❖❘ ❚❍❊ ❉❊●❘❊❊ ❖❋ ❉❖❈❚❖❘ ❖❋ P❍■▲❖❙❖P❍❨ ■◆ ▼❆❚❍❊▼❆❚■❈❙ ❍❆◆❖■ ✕ ✷✵✶✼ ❚❤✐s ✇♦r❦ ❤❛s ❜❡❡♥ ❝♦♠♣❧❡t❡❞ ❛t ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❱✐❡t♥❛♠ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② Supervisor: Prof Dr Habil Dinh Nho Hào ❘❡❢❡r❡❡ ✶✿ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ❘❡❢❡r❡❡ ✷✿ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ❘❡❢❡r❡❡ ✸✿ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ❚♦ ❜❡ ❞❡❢❡♥❞❡❞ ❛t ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❱✐❡t♥❛♠ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✿ ❆t ✳✳✳✳✳✳✳✳✳✳♦✬❝❧♦❝❦✱✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ❚❤❡ t❤❡s✐s ✐s ♣✉❜❧✐❝❧② ❛✈❛✐❧❛❜❧❡ ❛t✿ ✲ ❚❤❡ ◆❛t✐♦♥❛❧ ▲✐❜r❛r②✳ ✲ ❚❤❡ ▲✐❜r❛r② ♦❢ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✳ ■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ♣r❡❞✐❝t✐♦♥ ♦❢ ❛♥ ❡✈♦❧✉t✐♦♥ ♣r♦❝❡ss r❡q✉✐r❡s ✐ts ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✇❤✐❝❤ ✐s ✉♥❢♦rt✉✲ ♥❛t❡❧② ♥♦t ❛❧✇❛②s ❛✈❛✐❧❛❜❧❡ ♦r ♣r❡❝✐s❡❧② ❣✐✈❡♥ ✐♥ ♣r❛❝t✐❝❡✳ ❉❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ✐s t❤❡ ♣r♦❝❡ss ♦❢ r❡❝♦♥str✉❝t✐♥❣ ♠♦❞❡❧ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❢r♦♠ ♠❡❛s✉r❡❞ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ t❤❡ ✜rst ❣✉❡ss ✜❡❧❞ ✐♥ ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ t❤❡ ❞②♥❛♠✐❝❛❧ s②st❡♠✳ ❉❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ✐s ❡①t❡♥s✐✈❡❧② ✉s❡❞ ✐♥ ♠❡t❡♦r♦❧♦❣②✱ ♦❝❡❛♥♦❣r❛♣❤②✱ ✇❡❛t❤❡r ❢♦r❡❝❛st✱ ❡♥✈✐r♦♥♠❡♥t❛❧ ♣♦❧❧✉t✐♦♥✱ ♣r♦❝❡ss✐♥❣✱ ✐♥❞✉s✲ tr✐❛❧ ♣r♦❞✉❝t✐♦♥✱ ✳ ✳ ✳ ✳ ❋♦r s✉r✈❡②s✱ ♠❡t❤♦❞s ✐♥ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥✱ ✇❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ♦✉r ♣❛♣❡rs ❬✶✱ ✷✱ ✸❪ ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ♣r♦❝❡ss ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥ ✐s ♠♦❞❡❧❧❡❞ ❜② ❛ s②st❡♠ ♦❢ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s dU + AU = F, dt ✭✵✳✶✮ ✇❤❡r❡ U ✐s t❤❡ ✈❡❝t♦r r❡♣r❡s❡♥t✐♥❣ t❤❡ st❛t❡ ✈❛r✐❛❜❧❡s t❤❛t ✇❡ ✇❛♥t t♦ ✧♣r❡❞✐❝t✧✱ A ✐s ❛♥ ❡❧❧✐♣t✐❝ ♦♣❡r❛t♦r ✐♥ t❤❡ s♣❛❝❡ ✈❛r✐❛❜❧❡s ❛♥❞ F ✐s t❤❡ ✈❡❝t♦r ♦❢ ❡①t❡r✐♦r ❢♦r❝❡ ❛❝t✐♥❣ ♦♥ t❤❡ s②st❡♠✳ ❚❤❡ ❣♦❛❧ ♦❢ ♣r❡❞✐❝t✐♦♥ ✐s t♦ ✜♥❞ ❛ ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ U ❞✉r✐♥❣ ❛ ♣❡r✐♦❞ ♦❢ t✐♠❡ ♦❢ ❧❡♥❣t❤ T ✳ ❚❤❡ ♣r♦❜❧❡♠ ✇❡ ❛r❡ ❢❛❝❡❞ ✇✐t❤ ✐s t❤❛t ✇❡ ❞♦ ♥♦t ❦♥♦✇ t❤❡ ✐♥✐t✐❛❧ ❞❛t❛ ❢♦r U ❜❡❢♦r❡ s♦♠❡ t✐♠❡ ♠♦♠❡♥t T0 ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧ ❢r♦♠ T0 ♦♥✳ ❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ♦❜s❡r✈❡ ✭♠❡❛s✉r❡✮ U s♦♠❡❤♦✇✱ s❛②✱ ❜② CU ✇✐t❤ C ❜❡✐♥❣ ❛ ❧✐♥❡❛r ♦♣❡r❛t♦r✳ ❚❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ♣r♦❜❧❡♠ ✐s t♦ ❞❡t❡r♠✐♥❡ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❛t ❛ t✐♠❡ ❜❡❢♦r❡ T0 ❢r♦♠ ♠❡❛s✉r❡♠❡♥ts ❛♥❞ t❤❡♥ ✉s❡ ✐t t♦ s♦❧✈❡ t❤❡ ❛❜♦✈❡ s②st❡♠ ❢♦r ♣r❡❞✐❝t✐♦♥✳ ❚❤✐s ♣r♦❜❧❡♠ ✐s ✉♥❢♦rt✉♥❛t❡❧② ✐❧❧✲♣♦s❡❞✳ ❆ ♣r♦❜❧❡♠ ✐s s❛✐❞ t♦ ❜❡ ✇❡❧❧✲♣♦s❡❞ ✐♥ s❡♥s❡ ♦❢ ❍❛❞❛♠❛r❞ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ s❛t✐s✜❡❞✿ ✐✮ ❊①✐st❡♥❝❡✿ ❚❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠✳ ✐✐✮ ❯♥✐q✉❡♥❡ss✿ ❚❤❡ s♦❧✉t✐♦♥ ✐s ✉♥✐q✉❡✳ ✐✐✐✮ ❙t❛❜✐❧✐t②✿ ❚❤❡ s♦❧✉✲ t✐♦♥ ❝♦♥t✐♥✉♦✉s❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❛t❛ ✭✐♥ s♦♠❡ ❛♣♣r♦♣r✐❛t❡ t♦♣♦❧♦❣✐❡s✮✳ ■❢ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥s ✐s ♥♦t ❢✉❧✜❧❧❡❞✱ t❤❡ ♣r♦❜❧❡♠ ✐s s❛✐❞ t♦ ❜❡ ✐❧❧✲♣♦s❡❞ ✭♦r ✐♠♣r♦♣❡r❧② ♣♦s❡❞✮✳ ❚❤❡ ✐❧❧✲♣♦s❡❞♥❡ss ♦❢ ❛ ♣r♦❜❧❡♠ ❝❛✉s❡s s❡r✐♦✉s tr♦✉❜❧❡s ✐♥ ✐ts ♥✉♠❡r✐❝❛❧ s♦❧✉t✐♦♥✳ ■❢ ❛ ♣r♦❜❧❡♠ ✐s ♥♦t st❛❜❧❡✱ ❛ s♠❛❧❧ ❡rr♦r ✐♥ t❤❡ ❞❛t❛ ♠❛② ❝❛✉s❡ ❛r❜✐tr❛r✐❧② ❧❛r❣❡ ❡rr♦rs ✐♥ ✐ts s♦❧✉t✐♦♥✱ ♠❡❛♥✇❤✐❧❡ ✐♥ ♣r❛❝t✐❝❡ ✇❡ ❛❧✇❛②s ❤❛✈❡ ♠❡❛s✉r❡♠❡♥t ♥♦✐s❡✱ ❞✐s❝r❡t✐③❛t✐♦♥ ❛♥❞ ❝♦♠✲ ♣✉t❡r r♦✉♥❞✲♦✛ ❡rr♦rs ✇❤✐❝❤ ♠❛❦❡ ❝❧❛ss✐❝❛❧ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ✉♥st❛❜❧❡✳ ❚✐❦❤♦♥♦✈ ❆✳◆✳ ✐♥ ✶✾✹✸ r❡❛❧✐③❡❞ t❤❛t ❛ ♣r♦❜❧❡♠ ✐s ✉♥st❛❜❧❡ s✐♥❝❡ ✐t ❧❛❝❦s ✐♥❢♦r♠❛t✐♦♥✱ t♦ r❡st♦r❡ ✐ts st❛❜✐❧✐t② ✇❡ s❤♦✉❧❞ ✐♠♣♦s❡ s♦♠❡ ❛ ♣r✐♦r✐ ✐♥❢♦r♠❛t✐♦♥✳ ❚✐❦❤♦♥♦✈ t❤❡♥ ♣♦✐♥t❡❞ ♦✉t t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ✜♥❞✐♥❣ st❛❜❧❡ s♦❧✉t✐♦♥s t♦ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s✳ ❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s ❤❛s ✶ ❜❡❡♥ t❤❡♥ r❡❛❧✐③❡❞ ❜② ▲❛✈r❡♥t✬❡✈ ▼✳▼✳✱ ❏♦❤♥ ❋✳✱ P✉❝❝✐ ❈✳✱ ■✈❛♥♦✈ ❱✳❑✳ ✇❤♦ ❝❛♥ ❜❡ ❝♦♥✲ s✐❞❡r❡❞ ❛s t❤❡ ❢♦✉♥❞❡rs ♦❢ t❤❡ t❤❡♦r② ♦❢ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s✳ ■♥ ✶✾✻✸✱ ❚✐❦❤♦♥♦✈ ♣✉❜❧✐s❤❡❞ ❤✐s ❝❡❧❡❜r❛t❡❞ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❛♥❞ s✐♥❝❡ t❤❡♥ ✐♥✈❡rs❡ ❛♥❞ ✐❧❧✲♣♦s❡❞ ♣r♦❜❧❡♠s ❜❡❝❛♠❡ ❛♥ ❛❝t✐✈❡ ❜r❛♥❝❤ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ s❝✐❡♥❝❡✳ ❚❤✐s t❤❡s✐s ✐s ❞❡✈♦t❡❞ t♦ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ✐♥ ❤❡❛t ❝♦♥❞✉❝t✐♦♥✱ ♥❛♠❡❧② ✐t ❛✐♠s ❛t ❞❡t❡r✲ ♠✐♥✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✭✵✳✶✮ ❞❡s❝r✐❜✐♥❣ ❤❡❛t ❝♦♥❞✉❝t✐♦♥ ✉s✐♥❣ t❤r❡❡ t②♣❡s ♦❢ ♦❜s❡r✈❛t✐♦♥✿ ✶✮ ♦❜s❡r✈❛t✐♦♥ ❛t t❤❡ ✜♥❛❧ t✐♠❡ ♠♦♠❡♥t✱ ✷✮ ✐♥t❡r✐♦r ✭✐♥t❡❣r❛❧✮ ♦❜s❡r✈❛t✐♦♥s✱ ✸✮ ❜♦✉♥❞❛r② ♦❜s❡r✈❛t✐♦♥s✳ ❲❡ ♥♦✇ ❢♦r♠✉❧❛t❡ ♦✉r ♣r♦❜❧❡♠s ♠♦r❡ ♣r❡❝✐s❡❧②✳ ▲❡t Ω ❜❡ ❛♥ ♦♣❡♥ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ Rn , n ≥ 1, ✇✐t❤ ❜♦✉♥❞❛r② ∂Ω✳ ❉❡♥♦t❡ Q = Ω × (0, T ] ✇✐t❤ T > ❜❡✐♥❣ ❣✐✈❡♥✱ S = ∂Ω × (0, T ]✳ ▲❡t aij , i, j ∈ {1, 2, , n}, b ∈ L∞ (Q), i, j ∈ {1, 2, , n}, aij = aji , n λ ξ Rn aij (x, t)ξi ξj ≤ λ−1 ξ ≤ Rn , ∀ξ ∈ Rn , i,j=1 ≤ b(x, t) ≤ µ ❛✳❡✳ ✐♥ Q, v ∈ L2 (Ω), g ∈ L2 (S), f ∈ L2 (Q), λ ✐s ♣♦s✐t✐✈❡ ❝♦♥st❛♥t ❛♥❞ µ ≥ ❈♦♥s✐❞❡r t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ n ∂ ∂u − ∂t i,j=1 ∂xi aij (x, t) ∂u ∂xj + b(x, t)u = f ✐♥ Q, ✭✵✳✷✮ u|t=0 = v ✐♥ Ω, ✭✵✳✸✮ ✇✐t❤ ❡✐t❤❡r ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ u = ♦♥ S ✭✵✳✹✮ ♦r ◆❡✉♠❛♥♥ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ n ∂u = (aij (x, t)uxj ) cos(ν, xi ) = g ∂N i,j=1 ♦♥ S, ✭✵✳✺✮ ✇❤❡r❡ ν ✐s t❤❡ ♦✉t❡r ♥♦r♠❛❧ ✈❡❝t♦r t♦ S ✳ ❲❡ s❡❡ t❤❛t t❤❡ s②st❡♠ ✭✵✳✷✮✱ ✭✵✳✹✮ ♦r ✭✵✳✷✮✱ ✭✵✳✺✮ ❛r❡ ❡①❛♠♣❧❡s ♦❢ s②st❡♠ ♦❢ t❤❡ ❢♦r♠ ✭✵✳✶✮✳ ❲❤❡♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ❡q✉❛t✐♦♥ ✭✵✳✷✮✱ ❞❛t❛ v, g ❛♥❞ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ✭s♦✉r❝❡✮ f ❛r❡ ❣✐✈❡♥✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❞❡t❡r♠✐♥✐♥❣ u(x, t) ❢r♦♠ t❤❡ s②st❡♠ ✭✵✳✷✮✕✭✵✳✹✮ ♦r ✭✵✳✷✮✱ ✭✵✳✸✮✱ ✭✵✳✺✮ ✐s ❝❛❧❧❡❞ t❤❡ ❞✐r❡❝t ♣r♦❜❧❡♠✳ ❚❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ✭❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥✮ ❝♦♥s✐❞❡r❡❞ ✐♥ t❤✐s t❤❡s✐s ✐s t❤❛t ♦❢ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v ❢r♦♠ ♦♥❡ ♦❢ t❤❡ ❛❜♦✈❡ t❤r❡❡ t②♣❡s ♦❢ ♦❜s❡r✈❛t✐♦♥✿ ❉❡♥♦t✐♥❣ t❤❡ s♦❧✉t✐♦♥ t♦ ✭✵✳✷✮✕✭✵✳✹✮ ♦r ✭✵✳✷✮✱ ✭✵✳✸✮✱ ✭✵✳✺✮ ❜② u(v) t♦ ❡♠♣❤❛s✐③❡ ✐ts ❞❡♣❡♥❞❡♥❝❡ ♦♥ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v ❛♥❞ s✉♣♣♦s✐♥❣ t❤❛t ✇❡ ♦❜s❡r✈❡ u ❜② Cu(v) ✇✐t❤ ✷ C ❜❡✐♥❣ ❛ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦r✳ ❚❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ✐s t♦ ❞❡t❡r♠✐♥❡ v ✇❤❡♥ Cu(v) ✐s ❣✐✈❡♥✱ s❛②✱ ❜② z ✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ ✭✵✳✻✮ Cu(v) = z ❚❤✐s ♣r♦❜❧❡♠ ✐s ✐❧❧✲♣♦s❡❞ ❛s ✇❡ ✇✐❧❧ s❡❡ t❤❛t ✐♥ ♦✉r ❝❤♦s❡♥ s♣❛❝❡s t❤❡ ♦♣❡r❛t♦r ❢r♦♠ v t♦ Cu(v) ✐s ❝♦♠♣❛❝t✳ ❍♦✇❡✈❡r✱ ❝❤❛r❛❝t❡r✐③✐♥❣ ✐ts ❞❡❣r❡❡ ♦❢ ✐❧❧✲♣♦s❡❞♥❡ss ✐s ♥♦t ❛♥ ❡❛s② t❛s❦✳ o ❉❡♥♦t✐♥❣ t❤❡ s♦❧✉t✐♦♥ t♦ ✭✵✳✷✮✕✭✵✳✹✮ ✭♦r ✭✵✳✷✮✱ ✭✵✳✸✮✱ ✭✵✳✺✮✮ ✇✐t❤ v ≡ ❜② u✱ ✇❡ s❡❡ t❤❛t o t❤❡ ♦♣❡r❛t♦r ❢r♦♠ v t♦ Cv := Cu(v) − C u ✐s ❜♦✉♥❞❡❞ ❛♥❞ ❧✐♥❡❛r✳ ❚❤✉s✱ ✐♥st❡❛❞ ♦❢ st✉❞②✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ✭✵✳✻✮✱ ✇❡ ❤❛✈❡ t♦ ❞❡❛❧ ✇✐t❤ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ❡q✉❛t✐♦♥ o Cv = z − C u ❚❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ s✐♥❣✉❧❛r ✈❛❧✉❡s ♦❢ C ✭♦r ❡✐❣❡♥✈❛❧✉❡s ♦❢ C ∗ C ✮ ❝❤❛r❛❝t❡r✐③❡s t❤❡ ✐❧❧✲♣♦s❡❞♥❡ss ♦❢ t❤❡ ♣r♦❜❧❡♠✳ ❍♦✇❡✈❡r✱ ✉♣ t♦ ♥♦✇ t❤❡r❡ ❛r❡ ✈❡r② ❢❡✇ r❡s✉❧ts ♦♥ t❤✐s q✉❡s✲ t✐♦♥ ✐♥ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ❢♦r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✳ ❙♦♠❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❤❛s ❜❡❡♥ ♦❜t❛✐♥❡❞✱ ❜✉t ♦♥❧② ✐♥ ✈❡r② s✐♠♣❧❡ ❝❛s❡s✳ ■♥ t❤✐s t❤❡s✐s✱ ❛s ❛ ❜②✲♣r♦❞✉❝t ♦❢ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞ ❢♦r ✜♥❞✐♥❣ v ✱ ✇❡ ♣r♦♣♦s❡ ❛ ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡ ❢♦r ❡st✐♠❛t✐♥❣ s✐♥❣✉❧❛r ✈❛❧✉❡s ♦❢ C ✇❤✐❝❤ ✇❡ ✇✐❧❧ ♣r❡s❡♥t ❜❡❧♦✇✳ ❚♦ ✜♥❞ v ✱ ✇❡ ♠✐♥✐♠✐③❡ t❤❡ ♠✐s✜t ❢✉♥❝t✐♦♥❛❧ J0 (v) = Cu(v) − z 2 H ✇✐t❤ r❡s♣❡❝t t♦ v ∈ L2 (Ω)✳ ❍♦✇❡✈❡r✱ t❤✐s ♣r♦❜❧❡♠ ✐s ✐❧❧✲♣♦s❡❞✱ ✇❡ ✇✐❧❧ ♠✐♥✐♠✐③❡ t❤❡ r❡❣✉❧❛r✐③❡❞ ❢✉♥❝t✐♦♥❛❧ Jγ (v) = Cu(v) − z 2 H + γ v − v∗ 2 L2 (Ω) ✇✐t❤ r❡s♣❡❝t t♦ v ∈ L2 (Ω)✱ ✐♥st❡❛❞✳ ❍❡r❡✱ v ∗ ✐s ❛♥ ❛ ♣r✐♦r✐ ❡st✐♠❛t❡ t♦ v ✱ · H ✐s t❤❡ ♥♦r♠ ♦❢ ❛♥ ❛♣♣r♦♣r✐❛t❡ ❍✐❧❜❡rt s♣❛❝❡ H ❛♥❞ γ > t❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r✳ ■t ✇✐❧❧ ❜❡ ♣r♦✈❡❞ t❤❛t t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ t❤✐s ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✱ t❤❡ ❢✉♥❝t✐♦♥❛❧ Jγ ✐s ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ✐ts ❣r❛❞✐❡♥t ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ✈✐❛ ❛♥ ❛❞❥♦✐♥t ♣r♦❜❧❡♠✳ ❚♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ♥✉♠❡r✐❝❛❧❧②✱ ✇❡ s❤❛❧❧ ❛♣♣❧② t❤❡ s♣❧✐tt✐♥❣ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞ t♦ ❞✐s❝r❡t✐③❡ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ❛♥❞ ♣r♦✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♠❡t❤♦❞✳ ❚❤❡ r❡❛s♦♥ ✇❡ ❝❤♦♦s❡ t❤❡ s♣❧✐tt✐♥❣ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞ ✐s t❤❛t t❤✐s ♠❡t❤♦❞ ✐s ❡❛s② ❢♦r ❝♦❞✐♥❣ ❛♥❞ ✐t s♣❧✐ts ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❜❧❡♠s ✐♥t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❜❧❡♠s ❛♥❞ ❤❡♥❝❡ ✈❡r② ❢❛st✳ ❲❡ ♥♦t❡ t❤❛t ♦♥❡ ❝❛♥ ❞✐s❝r❡t✐③❡ t❤❡ ♣r♦❜❧❡♠s ❜② t❤❡ ✜♥✐t❡ ❡❧❡♠❡♥t ♠❡t❤♦❞✱ ❤♦✇❡✈❡r✱ s✐♥❝❡ t❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ ♦✉r ♣r♦❜❧❡♠s ❞❡♣❡♥❞ ♦♥ t✐♠❡✱ ✐t ✐s ❡❛s✐❡r t♦ ✉s❡ t❤❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞✳ ◆♦✇ ✇❡ r❡t✉r♥ t♦ ❡st✐♠❛t✐♥❣ t❤❡ s✐♥❣✉❧❛r ✈❛❧✉❡s ♦❢ C ✳ ❲❡ ❤❛✈❡ ∇J0 (v) = C ∗ (Cv − (z − o o C u)) ✇✐t❤ C ∗ ❜❡✐♥❣ t❤❡ ❛❞❥♦✐♥t ♦♣❡r❛t♦r ♦❢ C ✳ ■❢ ✇❡ ❝❤♦♦s❡ z = C u✱ t❤❡♥ ∇J0 (v) = C ∗ Cv ✳ ❯♥❢♦rt✉♥❛t❡❧②✱ t❤❡ ❡①♣❧✐❝✐t ❢♦r♠ ♦❢ C ✐s ♥♦r♠❛❧❧② ♥♦t ❛✈❛✐❧❛❜❧❡✱ ❛♥❞ ❡✈❡♥ ✐t ✐s ❛✈❛✐❧❛❜❧❡✱ ✐t ✐s ♥♦t ❡❛s② t♦ ❛♥❛❧②③❡ t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦✉r ♦❢ ✐ts s✐♥❣✉❧❛r ✈❛❧✉❡s✳ ❍♦✇❡✈❡r✱ ❛s ✇❡ ❝❛♥ ✸ ❝❛❧❝✉❧❛t❡ ∇J0 (v) = C ∗ Cv ✈✐❛ t❤❡ s♦❧✉t✐♦♥ t♦ ❛♥ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ❢♦r ❛♥② v ✱ ✇❡ ❝❛♥ ❛♣♣❧② ▲❛♥❝③♦s✬ ❛❧❣♦r✐t❤♠ t♦ ❡st✐♠❛t❡ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ C ∗ C ✳ ❚❤❡ ❝♦♥t❡♥t ♦❢ t❤✐s t❤❡s✐s ✐s ❛s ❢♦❧❧♦✇s✳ ■♥ ❈❤❛♣t❡r ✶ ✇❡ s✉♠♠❛r✐③❡ s♦♠❡ ❜❛s✐❝ r❡s✉❧ts ❢♦r t❤❡ ❞✐r❡❝t ♣r♦❜❧❡♠s✱ t❤❡✐r ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ❛♣♣r♦①✐♠❛t✐♦♥s ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts✱ ❛❧s♦ s♦♠❡ st❛♥❞❛r❞ ❛❧❣♦r✐t❤♠s ❧✐❦❡ t❤❡ ❝♦♥❥✉❣❛t❡ ❣r❛❞✐❡♥t ♠❡t❤♦❞✱ ▲❛♥❝③♦s✬ ❛❧❣♦r✐t❤♠✱✳ ✳ ✳ ❛r❡ ♣r❡s❡♥t❡❞ t❤❡r❡✳ ❲❡ ♥♦t❡ t❤❛t s✐♥❝❡ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ ✭✵✳✷✮✕✭✵✳✹✮ ♦r t♦ t❤❡ ◆❡✉♠❛♥♥ ♣r♦❜❧❡♠ ✭✵✳✷✮✱ ✭✵✳✸✮✱ ✭✵✳✺✮ ✐s ✉♥❞❡rst♦♦❞ ✐♥ t❤❡ ✇❡❛❦ s❡♥s❡✱ t❤❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞ ❢♦r t❤❡♠ ✐s ❝♦♠♣❧✐❝❛t❡❞ ❛♥❞ t❤❡ ♣r♦♦❢ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♠❡t❤♦❞ ✐s ♥♦t tr✐✈✐❛❧✳ ❈❤❛♣t❡r ✷ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ r❡❝♦♥str✉❝t✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v ✐♥ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ ✭✵✳✷✮✕✭✵✳✹✮ ❢r♦♠ t❤❡ ♦❜s❡r✈❛t✐♦♥ ❛t t❤❡ ✜♥❛❧ t✐♠❡ ♠♦♠❡♥t✿ Cu := u(x, T ) = ξ(x)✳ ❲❡ s♦♠❡t✐♠❡s ✇r✐t❡ u(x, t; v) ♦r u(v) ✐♥st❡❛❞ ♦❢ u(x, t) t♦ ❡♠♣❤❛s✐③❡ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ u ♦♥ v ✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ❣❡♥❡r❛❧ ❛♣♣r♦❛❝❤ st❛t❡❞ ❛❜♦✈❡ t♦ r❡❝♦♥str✉❝t v ✱ ✇❡ ♠✐♥✐♠✐③❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ J0 (v) := u(·, T ; v) − ξ 2L2 (Ω) ✇✐t❤ r❡s♣❡❝t t♦ v ∈ L2 (Ω)✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t t❤❡ ♦♣❡r❛t♦r v → Cu(v) : L2 (Ω) → L2 (Ω) ✐s ❝♦♠♣❛❝t✱ ❤❡♥❝❡ t❤❡ ♣r♦❜❧❡♠ ♦❢ s♦❧✈✐♥❣ t❤❡ ❡q✉❛t✐♦♥ Cu(v) = ξ ✐s ✐❧❧✲♣♦s❡❞✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❛❜♦✈❡ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐s ❛❧s♦ ✐❧❧✲♣♦s❡❞✳ ❚♦ st❛❜✐❧✐③❡ ✐t ✇❡ ♠✐♥✐♠✐③❡ t❤❡ ❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❡❞ ❢✉♥❝t✐♦♥❛❧ γ u(·, T ; v) − ξ 2L2 (Ω) + v − v ∗ 2L2 (Ω) , 2 ✇✐t❤ γ > ❜❡✐♥❣ ❛ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r ❛♥❞ v ∗ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ v ✳ ❲❡ ♣r♦✈❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥❛❧ Jγ ✐s ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ❞❡r✐✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ❢♦r ✐ts ❣r❛❞✐❡♥t Jγ (v) := ∇Jγ (v) = p(x, 0) + γ(v − v ∗ ), ✇✐t❤ p(x, t) ❜❡✐♥❣ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ n ∂p ∂p ∂ − a (x, t) + b(x, t)p = ✐♥ Q, − ij ∂t i,j=1 ∂xj ∂xi p(x, T ) = u(x, T ; v) − ξ p(x, t) = ♦♥ S ✐♥ Ω, ❚❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐s t❤❡♥ ❞✐s❝r❡t✐③❡❞ ❜② t❤❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞ ✐♥ s♣❛❝❡ ✈❛r✐✲ ❛❜❧❡s ❛♥❞ ✐t ✐s ♣r♦✈❡❞ t❤❛t t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡t✐③❡❞ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❝♦♥✈❡r❣❡s ✭✇❡❛❦❧② ♦r str♦♥❣❧② ❞❡♣❡♥❞s ♦♥ t❤❡ s♠♦♦t❤♥❡ss ♦❢ t❤❡ ❞❛t❛✮ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❝♦♥t✐♥✲ ✉♦✉s ♦♥❡✳ ❚♦ ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡ t❤❡ ♦r✐❣✐♥❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✇❡ ❢✉rt❤❡r ❞✐s❝r❡t✐③❡ ✐t ✐♥ t✐♠❡✱ ❛♥❞ ❢♦r t❤✐s ♣✉r♣♦s❡ ✇❡ ✉s❡ t❤❡ s♣❧✐tt✐♥❣ ♠❡t❤♦❞ ✇❤✐❝❤ s♣❧✐ts ♠✉❧t✐✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❜❧❡♠s ✐♥t♦ ❛ s❡q✉❡♥❝❡ ♦❢ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ♦♥❡s✳ ❲❡ ❞❡r✐✈❡ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❣r❛❞✐✲ ❡♥t ♦❢ t❤❡ ❢✉❧❧② ❞✐s❝r❡t✐③❡❞ ❢✉♥❝t✐♦♥❛❧ ✈✐❛ ❛♥ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ❛♥❞ t❤❡♥ ❛♣♣❧② t❤❡ ❝♦♥❥✉❣❛t❡ ✹ ❣r❛❞✐❡♥t ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ✐t ♦♥ ❝♦♠♣✉t❡r✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✐s t❤❡♥ t❡st❡❞ ♦♥ s❡✈❡r❛❧ ❜❡♥❝❤✲ ♠❛r❦ ❡①❛♠♣❧❡s t♦ s❤♦✇ t❤❡ ❡✣❝✐❡♥❝② ♦❢ ♦✉r ❛♣♣r♦❛❝❤✳ ❲❡ ❛❧s♦ ❛♣♣❧② ▲❛♥❝③♦s✬ ❛❧❣♦r✐t❤♠ t♦ ❡st✐♠❛t❡ t❤❡ s✐♥❣✉❧❛r ✈❛❧✉❡s ♦❢ t❤❡ ❞✐s❝r❡t✐③❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♦♣❡r❛t♦r C ✳ ❍❡r❡✱ ❛s ❛❜♦✈❡✱ o o C ✐s t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r ❢r♦♠ v ✐♥t♦ Cv := Cu(v) − C u ✇✐t❤ u ❜❡✐♥❣ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ ✭✵✳✷✮✕✭✵✳✹✮ ✇✐t❤ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✳ ❈❤❛♣t❡r ✸ st✉❞✐❡s t❤❡ r❡❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v ✐♥ ✭✵✳✷✮✕✭✵✳✹✮ ❢r♦♠ N ✐♥t❡❣r❛❧ ♦❜s❡r✈❛t✐♦♥s li u = hi (t), t ∈ (τ, T ), τ ≥ 0, i = 1, 2, , N, ✇❤❡r❡ ωi ∈ L1 (Ω), i = 1, 2, , N ❛r❡ ♥♦♥✲♥❡❣❛t✐✈❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥s ✇✐t❤ Ω ωi (x)dx > ωi (x)u(x, t)dx = hi (t), t ∈ (τ, T ), i = 1, , N li u(x, t) = ✭✵✳✼✮ Ω ▲❡t ✉s ❞✐s❝✉ss ♦❜s❡r✈❛t✐♦♥s ✭✵✳✼✮✳ ❋✐rst✱ ❛♥② ♠❡❛s✉r❡♠❡♥t ✐s ❛♥ ❛✈❡r❛❣❡❞ ♣r♦❝❡ss✱ t❤❛t ✐s ♦❢ t❤❡ ❢♦r♠ ✭✵✳✼✮✳ ❙❡❝♦♥❞✱ s✉❝❤ ❛ ❦✐♥❞ ♦❢ ♦❜s❡r✈❛t✐♦♥s ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♣♦✐♥t ♦❜s❡r✈❛t✐♦♥s✳ ■♥❞❡❡❞✱ ❧❡t xi ∈ Ω, i = 1, , N ❜❡ ❣✐✈❡♥✱ Si ❜❡ ❛ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ xi ✱ ❛♥❞ ωi (x) ❜❡ ❝❤♦s❡♥ ❛s , ✐❢ x ∈ Si , ωi (x) = |Si | 0, ♦t❤❡r✇✐s❡ ✇❤❡r❡ |Si | ✐s t❤❡ ✈♦❧✉♠❡ ♦❢ Si ✳ ❲❡ s❡❡ t❤❛t |Si | ✐s s✐♠✐❧❛r t♦ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ ✐♥str✉♠❡♥t ❛♥❞ ✇❤❡♥ ✇❡ ❧❡t |Si | t❡♥❞ t♦ ③❡r♦ ✇❡ ❤❛✈❡ t❤❡ ♣♦✐♥t ♦❜s❡r✈❛t✐♦♥✳ ■t s❤♦✉❧❞ ❜❡ ♥♦t❡❞ ❛❧s♦ t❤❛t ❛s t❤❡ s♦❧✉t✐♦♥ t♦ ✭✵✳✷✮✕✭✵✳✹✮ ✐s ✉♥❞❡rst♦♦❞ ✐♥ t❤❡ ✇❡❛❦ s❡♥s❡✱ ✐ts ✈❛❧✉❡ ❛t ❝❡rt❛✐♥ ♣♦✐♥t ❞♦❡s ♥♦t ❛❧✇❛②s ❤❛✈❡ ❛ ♠❡❛♥✐♥❣✱ ❜✉t ✐t ❞♦❡s ✐♥ t❤❡ ❛❜♦✈❡ ❛✈❡r❛❣❡❞ s❡♥s❡✳ ❚❤✐r❞✱ ✇✐t❤ t❤✐s ❦✐♥❞ ♦❢ ♦❜s❡r✈❛t✐♦♥s✱ t❤❡ ❞❛t❛ ♥❡❡❞ ♥♦t ❜❡ ❛❧✇❛②s ❛✈❛✐❧❛❜❧❡ ❛t t❤❡ ✇❤♦❧❡ s♣❛❝❡ ❞♦♠❛✐♥ ❛♥❞ ❛t ❛♥② t✐♠❡✳ ❚❤✉s✱ ♦✉r ♣r♦❜❧❡♠ s❡tt✐♥❣ ✐s ♥❡✇ ❛♥❞ ♠♦r❡ ♣r❛❝t✐❝❛❧ t❤❛♥ t❤❡ r❡❧❛t❡❞ ♦♥❡s✱ ✇❤❡r❡ ♦♥❡ r❡q✉✐r❡s ❡✐t❤❡r ✶✮ t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ u(x, T ) ✐♥ t❤❡ ✇❤♦❧❡ s♣❛t✐❛❧ ❞♦♠❛✐♥ Ω t❤❛t ✐s ❤❛r❞❧② r❡❛❧✐③❡❞ ✐♥ ♣r❛❝t✐❝❡✱ ✷✮ ♦r t❤❡ ♠❡❛s✉r❡♠❡♥ts ♦❢ u ✐♥ ω × (τ, T )✱ ✇❤❡r❡ ω ✐s ❛ s✉❜❞♦♠❛✐♥ ♦❢ Ω ❛♥❞ τ > ✐s ❛ ❝♦♥st❛♥t✳ ❆s ✐♥ ❈❤❛♣t❡r ✷✱ t♦ r❡❝♦♥str✉❝t v ❢r♦♠ t❤❡ ♦❜s❡r✈❛t✐♦♥s ✭✵✳✼✮✱ ✇❡ ♠✐♥✐♠✐③❡ t❤❡ ❚✐❦❤♦♥♦✈ ❢✉♥❝t✐♦♥❛❧ Jγ (v) = N li u(v) − hi i=1 L2 (τ,T ) + γ v − v∗ 2 L2 (Ω) ✇✐t❤ r❡s♣❡❝t t♦ v ∈ L2 (Ω) ✇✐t❤ γ > ❜❡✐♥❣ t❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r ❛♥❞ v ∗ ❛♥ ❡st✐♠❛t✐♦♥ ♦❢ v ✳ ❲❡ ❛♣♣r♦❛❝❤ t♦ t❤✐s ♣r♦❜❧❡♠ ❛s ✐♥ ❈❤❛♣t❡r ✷ ❛♥❞ ♦❜t❛✐♥ s✐♠✐❧❛r r❡s✉❧ts✳ ❚❤❡ ❧❛st ❝❤❛♣t❡r ✐s ❞❡✈♦t❡❞ t♦ t❤❡ ❝❛s❡ ♦❢ ♦❜s❡r✈❛t✐♦♥s ❛t t❤❡ ❜♦✉♥❞❛r②✳ ❲❡ s✉♣♣♦s❡ t❤❛t ♦✉r ❡✈♦❧✉t✐♦♥ s②st❡♠ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ ◆❡✉♠❛♥♥ ♣r♦❜❧❡♠ ✭✵✳✷✮✱ ✭✵✳✸✮✱ ✭✵✳✺✮✳ ❚❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ✇❡ st✉❞② ✐s t♦ r❡❝♦♥str✉❝t t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v ✐♥ ✭✵✳✸✮✱ ✇❤❡♥ t❤❡ s♦❧✉t✐♦♥ u ✐s ❣✐✈❡♥ ♦♥ ❛ ♣❛rt ♦❢ t❤❡ ❜♦✉♥❞❛r② S ✳ ◆❛♠❡❧②✱ ❧❡t Γ ⊂ ∂Ω ❛♥❞ ❞❡♥♦t❡ Σ = Γ × (0, T )✳ ❖✉r ❛✐♠ ✐s t♦ r❡❝♦♥str✉❝t t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v ❢r♦♠ t❤❡ ✐♠♣r❡❝✐s❡ ♠❡❛s✉r❡♠❡♥t ϕ ♦❢ t❤❡ s♦❧✉t✐♦♥ u ♦♥ Σ✿ u|Σ − ϕ L2 (Σ) ≤ ✺ ❚❤❡ ✉♥✐q✉❡♥❡ss ♦❢ t❤❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ t❤❡♦r② ♦❢ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s✳ ❘❡❝❡♥t❧②✱ ❑❧✐❜❛♥♦✈ ♣r♦✈❡❞ s♦♠❡ st❛❜✐❧✐t② ❡st✐♠❛t❡s ❢♦r t❤✐s ✐♥✈❡rs❡ ♣r♦❜❧❡♠✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ✉♣ t♦ ♥♦✇ t❤❡r❡ ❛r❡ ✈❡r② ❢❡✇ st✉❞✐❡s ♦♥ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ❢♦r t❤✐s ♣r♦❜❧❡♠✳ ❚❤❡ ✇♦r❦ ❜② ❇✉❧②❝❤ë✈ ❡t ❛❧ ✐s t❤❡ ♦♥❧② r❡❢❡r❡♥❝❡ ♦♥ t❤✐s ❛s♣❡❝t ✇❤✐❝❤ ✇❡ ❤❛✈❡ ❢♦✉♥❞✳ ❚❤✉s✱ t❤❡ s♦❧✉t✐♦♥ ♠❡t❤♦❞ ❢♦r t❤✐s ♣r♦❜❧❡♠ ♣r♦♣♦s❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ✐s ❛ ♥❡✇ ❝♦♥tr✐❜✉t✐♦♥ t♦ ✜❡❧❞✳ ❲❡ ♥♦t❡ t❤❛t t❤✐s ♣r♦❜❧❡♠ ❤❛s t❤❡ r♦♦t ✐♥ t❤❡ ✐♥✈❡rs❡ ❤❡❛t ❝♦♥❞✉❝t✐♦♥ ♣r♦❜❧❡♠ ✭■❍❈P✮ ✇❤❡r❡ ♦♥❡ ❞❡t❡r♠✐♥❡s t❤❡ s✉r❢❛❝❡ t❡♠♣❡r❛t✉r❡ ❛♥❞ ❤❡❛t ✢✉① ♦♥ ❛♥ ❛❝❝❡ss✐❜❧❡ ♣❛rt ♦❢ t❤❡ ❜♦✉♥❞❛r② ❢r♦♠ t❤❡ s✉r❢❛❝❡ t❡♠♣❡r❛t✉r❡ ❛♥❞ t❤❡ s✉r❢❛❝❡ ❤❡❛t ✢✉① ♦♥ t❤❡ ❛❝❝❡ss✐❜❧❡ ♣❛rt ♦❢ ✐t✳ ❚❤❡ t②♣✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥ ❢♦r ■❍❈P r❡q✉✐r❡s t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❣✐✈❡♥✳ ❍♦✇❡✈❡r✱ ❛s ■❍❈P ❝❛♥ ❜❡ r❡❣❛r❞❡❞ ❛s ❛ ♥♦♥✲❝❤❛r❛❝t❡r✐st✐❝ ❈❛✉❝❤② ♣r♦❜❧❡♠ ❢♦r ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s✱ ♥♦ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐s ♥❡❡❞❡❞✳ ❯s✐♥❣ t❤❡ s❛♠❡ ♠❡t❤♦❞ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡rs ✷ ❛♥❞ ✸✱ ❢♦r r❡❝♦♥str✉❝t✐♥❣ v ❢r♦♠ u|Σ ✇❡ ♠✐♥✐♠✐③❡ t❤❡ ❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❡❞ ❢✉♥❝t✐♦♥❛❧ γ Jγ (v) = u(v) − ϕ 2L2 (Σ) + v − v ∗ 2L2 (Ω) 2 ∗ ✇✐t❤ γ > ❜❡✐♥❣ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r ❛♥❞ v ❛ ❝❡rt❛✐♥ ❡st✐♠❛t✐♦♥ ♦❢ v ✳ ❲❡ ♦❜t❛✐♥ s✐♠✐❧❛r r❡s✉❧ts t♦ t❤❛t ♦❢ ❈❤❛♣t❡r ✷✳ ❈❤❛♣t❡r ✶ ❆✉①✐❧✐❛r② r❡s✉❧ts ✶✳✶ ❉✐r❡❝t ❛♥❞ ❆❞❥♦✐♥t ♣r♦❜❧❡♠s ▲❡t Ω ❜❡ ❛♥ ♦♣❡♥ ❜♦✉♥❞❡❞ ❞♦♠❛✐♥ ✐♥ Rn , n ≥ ✇✐t❤ ❜♦✉♥❞❛r② ∂Ω✳ ❉❡♥♦t❡ Q = Ω × (0, T ], S = ∂Ω × (0, T ]✳ ▲❡t aij , i, j ∈ {1, 2, , n}, b ∈ L∞ (Q), aij = aji , i, j ∈ {1, 2, , n}, n λ ξ Rn aij (x, t)ξi ξj ≤ λ−1 ξ ≤ i,j=1 ≤ b(x, t) ≤ µ ❛✳❡✳ ✐♥ Q, v ∈ L2 (Ω), g ∈ L2 (S), f ∈ L2 (Q), λ ✐s ♣♦s✐t✐✈❡ ❝♦♥st❛♥t ❛♥❞ µ ≥ ✻ Rn , ∀ξ ∈ Rn , ❈♦♥s✐❞❡r t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ n ∂ ∂u − ∂t i,j=1 ∂xi aij (x, t) ∂u ∂xj + b(x, t)u = f ✐♥ Q, ✭✶✳✶✮ u|t=0 = v ✐♥ Ω, ✭✶✳✷✮ ✇✐t❤ ❡✐t❤❡r ❉✐r✐❝❤❧❡t ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ u = ♦♥ S, ✭✶✳✸✮ ♦r ◆❡✉♠❛♥♥ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ n ∂u = (aij (x, t)uxj ) cos(ν, xi )|S = g ∂N i,j=1 ♦♥ S, ✭✶✳✹✮ ✇❤❡r❡ ν ✐s t❤❡ ♦✉t❡r ♥♦r♠❛❧ ✈❡❝t♦r t♦ S ✳ ❲❤❡♥ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✭✶✳✶✮✱ ❞❛t❛ v, g ❛♥❞ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ f ❛r❡ ❣✐✈❡♥✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✉♥✐q✉❡❧② s♦❧✈✐♥❣ u(x, t) ❢r♦♠ t❤❡ s②st❡♠ ✭✶✳✶✮✕✭✶✳✸✮ ♦r ✭✶✳✶✮✱ ✭✶✳✷✮✱ ✭✶✳✹✮ ✐s ❝❛❧❧❡❞ ❞✐r❡❝t ♣r♦❜❧❡♠✳ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ ✭✶✳✶✮✲✭✶✳✸✮✱ ❛♥❞ t❤❡ ◆❡✉♠❛♥♥ ♣r♦❜❧❡♠ ✭✶✳✶✮✱ ✭✶✳✷✮✱ ✭✶✳✹✮ ❛r❡ ✉♥❞❡rst♦♦❞ ✐♥ t❤❡ ✇❡❛❦ s❡♥s❡ ❛s ❢♦❧❧♦✇s✿ ❉❡✜♥✐t✐♦♥ ✶✳✶✳✶✳ ❆ ✇❡❛❦ s♦❧✉t✐♦♥ ✐♥ W (0, T ; H01(Ω)) ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✳✶✮✲✭✶✳✸✮ ✐s ❛ ❢✉♥❝✲ t✐♦♥ u(x, t) ∈ W (0, T ; H01 (Ω)) s❛t✐s❢②✐♥❣ t❤❡ ✐❞❡♥t✐t② n T ut , η H −1 (Ω),H01 (Ω) dt aij (x, t) + Q i,j=1 ∂u ∂η + b(x, t)uη − f η dxdt = 0, ∂xj ∂xi ∀η ∈ L2 (0, T ; H01 (Ω)) ❛♥❞ u|t=0 = v ✐♥ Ω ❉❡✜♥✐t✐♦♥ ✶✳✶✳✷✳ ❆ ✇❡❛❦ s♦❧✉t✐♦♥ ✐♥ W (0, T ; H 1(Ω)) ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✶✳✶✮✱ ✭✶✳✷✮✱ ✭✶✳✹✮ ✐s ❛ ❢✉♥❝t✐♦♥ u(x, t) ∈ W (0, T ; H (Ω)) s❛t✐s❢②✐♥❣ t❤❡ ✐❞❡♥t✐t② n T ut , η (H (Ω)) ,H (Ω) dt + aij (x, t) Q = f ηdxdt + gηdζdt, Q i,j=1 ∂u ∂η + b(x, t)uη dxdt ∂xj ∂xi ∀η ∈ L2 (0, T ; H (Ω)) S ❛♥❞ u|t=0 = v ✐♥ Ω ❚❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ t♦ ✭✶✳✶✮✕✭✶✳✸✮ ✐s t❤❛t ♦❢ t❤❡ ❢♦r♠✿ n −pt − ∂ ∂xi i,j=1 aij (x, t) ∂p ∂xj + bp = aQ ✐♥ Q, p(ζ, t) = ♦♥ S, p(x, T ) = aΩ ✼ ✐♥ Ω, ✇❤❡r❡ aQ ∈ L2 (Q) ❛♥❞ aΩ ∈ L2 (Ω)✳ ❲❡ ❞❡✜♥❡ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤✐s ♣r♦❜❧❡♠ ✐s ❛ ❢✉♥❝t✐♦♥ p ∈ W (0, T ; H01 (Ω)) s❛t✐s❢②✐♥❣ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ n T −(pt , v)H −1 (Ω),H01 (Ω) dt + aij pv + bpv dxdt = Q aQ vdxdt, Q i,j=1 ∀v ∈ L2 (0, T ; H01 (Ω)), p(T ) = aΩ ❙✐♠✐❧❛r❧②✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ t♦ t❤❡ ◆❡✉♠❛♥♥ ♣r♦❜❧❡♠ ✭✶✳✶✮✱ ✭✶✳✷✮✱ ✭✶✳✹✮ ❛s ❢♦❧❧♦✇s n −pt − ∂ ∂xi i,j=1 aij (x, t) ∂p ∂xj + bp = aQ ✐♥ Q, ∂N p = aS ♦♥ S, p(x, T ) = aΩ ✐♥ Ω, ✇❤❡r❡ aQ ∈ L2 (Q), aS ∈ L2 (S), ❛♥❞ aΩ ∈ L2 (Ω)✳ ❲❡ ❞❡✜♥❡ t❤❡ s♦❧✉t✐♦♥ t♦ t❤✐s ♣r♦❜❧❡♠ ❜② ❛ ❢✉♥❝t✐♦♥ p ∈ W (0, T ; H (Ω)) s❛t✐s❢②✐♥❣ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ n T −(pt , v)(H (Ω))∗ ,H (Ω) dt + Q i,j=1 aS vdζdt, aQ vdxdt + aij pv + bpv dxdt = Q S ∀v ∈ L2 (0, T ; H (Ω)), p(T ) = aΩ ✶✳✷ ❋✐♥✐t❡ ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞ ❢♦r ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❞✐r❡❝t ♣r♦❜❧❡♠s ▲❡t Ω = (0, L) ❛♥❞ Q = (0, L) × (0, T ), S = {0, 1} × (0, T )✳ ❲❡ s✉❜❞✐✈✐❞❡ t❤❡ ✐♥t❡r✈❛❧ (0, L) ✐♥t♦ Nx s✉❜✐♥t❡r✈❛❧s ❜② t❤❡ ✉♥✐❢♦r♠ ❣r✐❞ = x0 < x1 < · · · < xNx = L ✇✐t❤ xi+1 − xi = h = L/Nx ✳ ❉❡♥♦t❡ ❜② ui (t) ✭♦r ui ✐❢ t❤❡r❡ ✐s ♥♦ ❝♦♥❢✉s✐♦♥✮ t❤❡ ✈❛❧✉❡ ♦❢ u ❛t x = xi ❲❡ ❛❧s♦ ✉s❡ t❤❡ s✐♠✐❧❛r ♥♦t❛t✐♦♥ ❢♦r η ✳ ❚❤❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ ✭✶✳✶✮✕✭✶✳✸✮ ❤❛s ♥♦✇ t❤❡ ❢♦r♠ ∂u ∂ ∂u − a(x, t) ∂t ∂x ∂x + b(x, t)u = f u|t=0 = v ✐♥ Ω, u(0, t) = u(L, t) = ✐♥ (0, T ] ✽ ✐♥ Q, ✭✶✳✺✮ ✶✳✸✳✶✳ ■♥t❡r♣♦❧❛t✐♦♥s ♦❢ ❣r✐❞ ❢✉♥❝t✐♦♥s ❲❡ s✉♣♣♦s❡ t❤❛t t❤❡ ❞♦♠❛✐♥ Ω ✐s t❤❡ ♦♣❡♥ ♣❛r❛❧❧❡❧❡♣✐♣❡❞ Ω = (0, L1 ) × (0, L2 ) × · · · × (0, Ln ) ❲❡ s✉❜❞✐✈✐❞❡ Ω ✐♥t♦ s♠❛❧❧ ❝❡❧❧s ❜② r❡❝t❛♥❣✉❧❛r ✉♥✐❢♦r♠ ❣r✐❞ s♣❡❝✐✜❡❞ ❜② i = x0i < x1i = hi < · · · < xN i = Li , i = 1, , n ❍❡r❡ • hi = Li /Ni ✐s t❤❡ ❣r✐❞ s✐③❡ ✐♥ t❤❡ xi ✲❞✐r❡❝t✐♦♥✱ i = 1, , n ❛♥❞ ∆h := h1 · · · hn ❀ • k := (k1 , , kn )✱ ≤ ki ≤ Ni ❛♥❞ xk := (xk11 , , xknn ) ✐s t❤❡ ❣r✐❞ ♣♦✐♥t❀ • h := (h1 , , hn ) ✐s t❤❡ ✈❡❝t♦r ♦❢ s♣❛t✐❛❧ ❣r✐❞ s✐③❡❀ • ei ✱ i = 1, , n ❜❡✐♥❣ t❤❡ ✉♥✐t ✈❡❝t♦r ✐♥ t❤❡ xi ✲❞✐r❡❝t✐♦♥✳ ❉❡♥♦t❡ ❜② ω(k) :={x ∈ Ω : ki hi < xi < (ki + 1)hi , ∀i = 1, , n − 1}, ωi+ (k) :={x ∈ Ω : ki hi ≤ xi ≤ (ki + 1)hi , (kj − 0.5)hj ≤ xj ≤ (kj + 0.5)hj , ∀i = j} ¯ ✐s ❞❡♥♦t❡❞ ❜② Ω ¯ h ✳ ❚❤❛t ✐s ❚❤❡ s❡t ♦❢ t❤❡ ✐♥❞✐❝❡s ♦❢ ❛❧❧ ❣r✐❞ ♣♦✐♥ts ❜❡❧♦♥❣✐♥❣ t♦ Ω ¯ h := {k = (k1 , , kn ) : ≤ ki ≤ Ni , ∀i = 1, , n} Ω ❚❤❡ s❡t ♦❢ t❤❡ ✐♥❞✐❝❡s ♦❢ ❛❧❧ ✐♥t❡r✐♦r ❣r✐❞ ♣♦✐♥ts ✐s ❞❡♥♦t❡❞ ❜② Ωh ✱ t❤❛t ✐s Ωh := {k = (k1 , , kn ) : ≤ ki ≤ Ni − 1, ∀i = 1, , n} ❚❤❡ s❡t ♦❢ t❤❡ ✐♥❞✐❝❡s ♦❢ ❛❧❧ ❜♦✉♥❞❛r② ❣r✐❞ ♣♦✐♥ts ✐s ❞❡♥♦t❡❞ ❜② Πh ✱ t❤❛t ✐s ¯ h \ Ωh Πh = Ω ▼♦r❡♦✈❡r✱ ✇❡ ✉s❡ s♦♠❡ ♥♦t❛t✐♦♥s ❢♦r t❤❡ s✉❜s❡t ♦❢ Πh ❛s ❢♦❧❧♦✇s Πilh = {k = (k1 , k2 , , kn ) : ki = 0}, i = 1, , n, Πir h = {k = (k1 , k2 , , kn ) : ki = Ni }, i = 1, , n ❚❤❡ ♣♦✐♥ts ❤❛✈✐♥❣ ✐♥❞❡① k ✇✐t❤ ki = ♦r Li ✱ i = 1, , n ❛r❡ ❝❛❧❧❡❞ t❤❡ ❝♦r♥❡r ♣♦✐♥ts ❛♥❞ t❤❡ s❡t ♦❢ t❤❡ ✐♥❞✐❝❡s ♦❢ ❛❧❧ ❝♦r♥❡r ♣♦✐♥ts ✐s ❞❡♥♦t❡❞ ❜② Π0 ✳ ❲❡ ❛❧s♦ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts Ωih := {k = (k1 , , kn ) : ≤ ki ≤ Ni − 1, ≤ kj ≤ Nj , ∀j = i}, ✇✐t❤ i = 1, , n ❋♦r t❤❡ ❣r✐❞ ❢✉♥❝t✐♦♥ u ¯ ✇❡ ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r♣♦❧❛t✐♦♥s ✐♥ Q✿ ˜¯(x, t) := u¯k (t), (x, t) ∈ ω(k) × (0, T ) ✶✮ P✐❡❝❡✇✐s❡ ❝♦♥st❛♥t✿ u ✶✶ ✷✮ ▼✉❧t✐✲❧✐♥❡❛r✿ n uˆ¯(x, t) := u¯k (t) + u¯kxi (t)(xi − ki hi ) + i=1 u¯kxi xj (t)(xi − ki hi )(xj − kj hj ) 1≤i ❜❡✐♥❣ t❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r ❛♥❞ v ∗ ❛♥ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v ✇❤✐❝❤ ❝❛♥ ❜❡ s❡t t♦ ③❡r♦✳ ❲❡ ❛❧s♦ ✐♥tr♦❞✉❝❡ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ n ∂ ∂p ∂p − − a (x, t) + b(x, t)p = ✐♥ Q, ij ∂t i=1 ∂xj ∂xi ✭✷✳✸✮ p(x, T ) = u(x, T ; v) − ξ(x) ✐♥ Ω, p = ♦♥ S Jγ (v) := J0 (v) + ✷✳✷ ▼❛✐♥ r❡s✉❧ts ❚❤❡♦r❡♠ ✷✳✷✳✶✳ ❚❤❡ ❢✉♥❝t✐♦♥❛❧ Jγ ✐s ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ✐ts ❣r❛❞✐❡♥t ∇Jγ (v) ❛t v ❤❛s t❤❡ ❢♦r♠ ∇Jγ (v) = p(x, 0) + γ(v − v ∗ ), ✇❤❡r❡ p(x, t) ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ✭✷✳✸✮✳ ❲❡ s✉♣♣♦s❡ t❤❛t t❤❡ ✜rst ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ ✭✷✳✶✮ ❤❛s ♥♦ ♠✐①❡❞ ❞❡r✐✈❛✲ t✐✈❡✱ t❤❛t ✐s aij = ✐❢ i = j ✱ ❛♥❞ ✇❡ ❞❡♥♦t❡ aii ❜② ❛♥❞ ❣❡t t❤❡ s②st❡♠ ✭✶✳✶✶✮✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❛ss✉♠❡ Ω ✐s t❤❡ ♦♣❡♥ ♣❛r❛❧❧❡❧❡♣✐♣❡❞✳ ❲❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ Jγ (v) ❜② Jγh (¯ v) = h k∈Ωh γ |¯ u(T ; v¯) − ξ¯k |2 + h k |¯ v k − v¯∗ |2 ✭✷✳✹✮ kΩh ˆ¯(x, T ; v¯)✱ ✇❤❡r❡ uˆ¯ ✐s t❤❡ ❛♥❞ ♠✐♥✐♠✐③❡ t❤✐s ❢✉♥❝t✐♦♥❛❧ ♦✈❡r ❛❧❧ ❣r✐❞ ❢✉♥❝t✐♦♥s v¯✳ ❙❡t Ch v¯ = u ♣✐❡❝❡✇✐s❡ ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ ♦❢ u ¯✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ♦♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❞✐s❝r❡t✐③❡❞ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✭✷✳✹✮✳ ❚❤❡♦r❡♠ ✷✳✷✳✷✳ ▲❡t ❝♦♥❞✐t✐♦♥ ✷✮ ♦❢ ❚❤❡♦r❡♠ ✶✳✸✳✷ ✐s s❛t✐s✜❡❞✳ ❚❤❡♥ t❤❡ ❧✐♥❡❛r ✐♥t❡r♣♦✲ ❧❛t✐♦♥ uˆ¯ ♦❢ t❤❡ s♦❧✉t✐♦♥ uγ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✷✳✹✮ ❝♦♥✈❡r❣❡s ✐♥ L2 (Ω) t♦ t❤❡ s♦❧✉t✐♦♥ vγ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✷✳✷✮ ❛s h t❡♥❞s t♦ ③❡r♦✳ ❲❡ ❞✐s❝r❡t✐③❡ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥❛❧ J0 (v) ❛s ❢♦❧❧♦✇s J0h (¯ v ) := [uk,M (¯ v ) − ξ k ]2 k∈Ωh ✶✻ ❍❡r❡ ✇❡ ✉s❡ t❤❡ ♥♦t❛t✐♦♥ uk,M (¯ v ) t♦ ✐♥❞✐❝❛t❡ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v¯ ❛♥❞ M r❡♣r❡s❡♥ts t❤❡ ✐♥❞❡① ♦❢ t❤❡ ✜♥❛❧ t✐♠❡✳ ❚❤❡♦r❡♠ ✷✳✷✳✸✳ ❚❤❡ ❣r❛❞✐❡♥t ∇J0h (¯v) ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥❛❧ J0h ❛t v¯ ✐s ❣✐✈❡♥ ❜② ∇J0h (¯ v ) = A0 ∗ η0, ✇❤❡r❡ η = (η0 , , η M ) s❛t✐s✜❡s t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ η M −1 = uM (¯ v) − ξ η m = Am+1 ∗ η m+1 , m = M − 2, M − 3, , ❍❡r❡✱ t❤❡ ♠❛tr✐① (Am )∗ ❤❛s ❢♦r♠ ∆t m ∆t m −1 ∆t m ∆t m −1 Λ1 )(E1 + Λ1 ) (En − Λn )(En + Λ ) 4 4 n ∆t m ∆t m −1 ∆t m ∆t m −1 × (En − Λn )(En + Λn ) (E1 − Λ1 )(E1 + Λ ) 4 4 (Am )∗ = (E1 − ❙♦♠❡ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s ❛r❡ t❡st❡❞ ♦♥ ❝♦♠♣✉t❡r t♦ ♣r♦✈❡ t❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞✳ ❚❤✐s ❝❤❛♣t❡r ✐s ✇r✐tt❡♥ ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ ♣❛♣❡r ❬✶❪ ❖❛♥❤ ◆✳❚✳◆✳✱ ❆ s♣❧✐tt✐♥❣ ♠❡t❤♦❞ ❢♦r ❛ ❜❛❝❦✇❛r❞ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥ ✇✐t❤ t✐♠❡✲ ❞❡♣❡♥❞❡♥t ❝♦❡✣❝✐❡♥ts✳ ❈♦♠♣✉t❡rs ✫ ▼❛t❤❡♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s ✻✺✭✷✵✶✸✮✱ ✶✼✕✷✽✳ ❈❤❛♣t❡r ✸ ❉❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❜② t❤❡ ✐♥t❡❣r❛❧ ♦❜s❡r✈❛t✐♦♥s ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ st✉❞② t❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ♣r♦❜❧❡♠ ♦❢ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v(x) ✐♥ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ ✭✶✳✶✮✕✭✶✳✸✮ ❢r♦♠ N ✐♥t❡❣r❛❧ ♦❜s❡r✈❛t✐♦♥s li u = hi (t), t ∈ (τ, T ), τ ≥ 0, i = 1, 2, , N r❡❣❛r❞❡❞ ❛s ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♣♦✐♥t✇✐s❡ s♣❛t✐❛❧ ♦❜s❡r✈❛t✐♦♥s✿ ❙✉♣♣♦s❡ t❤❛t ✇❡ ❛r❡ ❣✐✈❡♥ N ♥♦♥♥❡❣❛t✐✈❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥s ωi ∈ L1 (Ω) ✇✐t❤ ✶✼ Ω ωi (x)dx > ❛♥❞ ✇❡ ❝❛♥ ♦❜s❡r✈❡ u ❜② ωi (x)u(x, t)dx = hi (t), t ∈ [τ, T ], i = 1, , N li u(x, t) = Ω ■t ✐s r❡q✉✐r❡❞ t♦ r❡❝♦♥str✉❝t t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v ❢r♦♠ t❤❡s❡ ♦❜s❡r✈❛t✐♦♥s✳ ❆s ✐♥ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✱ ✇❡ r❡❢♦r♠✉❧❛t❡ t❤✐s ♣r♦❜❧❡♠ ❛s ❛ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ❛✐♠s ❛t ♠✐♥✐♠✐③✐♥❣ ❛ ♠✐s✜t ❢✉♥❝t✐♦♥❛❧ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❧❡❛st sq✉❛r❡s✳ ❆s t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✐s ✐❧❧✲♣♦s❡❞✱ ✇❡ st❛❜✐❧✐③❡ ✐t ❜② ✉s✐♥❣ t❤❡ ❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞✳ ■t ✐s ♣r♦✈❡❞ t❤❛t t❤❡ r❡❣✉❧❛r✐③❡❞ ❢✉♥❝t✐♦♥❛❧ ✐s ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r ✐ts ❣r❛❞✐❡♥t ✐s ❞❡r✐✈❡❞ ✈✐❛ ❛♥ ❛❞❥♦✐♥t ♣r♦❜❧❡♠✳ ❚❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✐s ✜rst ❞✐s❝r❡t✐③❡❞ ✐♥ s♣❛❝❡ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts ♦❢ t❤❡ ♠❡t❤♦❞ ❛r❡ ♣r♦✈❡❞✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s t❤❡♥ ❢✉❧❧② ❞✐s❝r❡t✐③❡❞ ❛♥❞ ✐t ✐s ♣r♦✈❡❞ t❤❛t t❤❡ ❞✐s❝r❡t✐③❡❞ ❢✉♥❝t✐♦♥❛❧ ✐s ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❆ ❢♦r♠✉❧❛ ❢♦r ✐ts ❣r❛❞✐❡♥t ✐s ❞❡r✐✈❡❞ ✈✐❛ ❛ ❞✐s❝r❡t❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ❛♥❞ t❤❡ ❝♦♥❥✉❣❛t❡ ❣r❛❞✐❡♥t ♠❡t❤♦❞ ✐s ❛♣♣❧✐❡❞ t♦ ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡ t❤❡ ❞✐s❝r❡t✐③❡❞ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠✳ ❙♦♠❡ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s ❛r❡ ♣r♦✈✐❞❡❞ t♦ s❤♦✇ t❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠✳ ✸✳✶ Pr♦❜❧❡♠ s❡tt✐♥❣ ❛♥❞ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞ ❲❡ r❡✇r✐t❡ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ ✭✶✳✶✮✕✭✶✳✸✮ ✇✐t❤ t❤❡ ♥❡✇ ✐♥❞❡①✐♥❣✿ n ∂u ∂ ∂u − a (x, t) + b(x, t)u = f ✐♥ Q, ij ∂t i,j=1 ∂xi ∂xj ✭✸✳✶✮ u|t=0 = v ✐♥ Ω, u = ♦♥ S ❉❛t❛ ❛ss✐♠✐❧❛t✐♦♥✿ ❙✉♣♣♦s❡ t❤❛t v ✐s ♥♦t ❣✐✈❡♥ ❛♥❞ ✇❡ ❤❛✈❡ t♦ r❡❝♦♥str✉❝t ✐t ❢r♦♠ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ u t♦ ✭✸✳✶✮ ❢r♦♠ N ✐♥t❡❣r❛❧ ♦❜s❡r✈❛t✐♦♥s li u = hi (t), t ∈ (τ, T ), τ ≥ 0, i = 1, 2, , N ✱ ✇✐t❤ ωi (x)u(x, t)dx = hi (t), t ∈ [τ, T ], i = 1, , N li u(x, t) = Ω ❘❡♠❛r❦ ✸✳✶✳✶✳ ■❢ v ∈ H01 (Ω)✱ aij , b ∈ C ([0, T ]; L∞ (Ω)), i, j = 1, , n ❛♥❞ t❤❡r❡ ❡①✐sts ❛ ❝♦♥st❛♥t µ1 ✇❤✐❝❤ s❛t✐s✜❡s |∂aij /∂t|, |∂b/∂t| ≤ µ1 ✱ t❤❡♥ t❤❡ ♦♣❡r❛t♦r C ♠❛♣s v ∈ L2 (Ω) t♦ (l1 u(x, t; v), · · · , lN u(x, t; v)) ✐s ❝♦♠♣❛❝t ❢r♦♠ L2 (Ω) t♦ (L2 (τ, T ))N ✳ ❍❡♥❝❡ t❤❡ ♣r♦❜❧❡♠ ♦❢ r❡❝♦♥str✉❝t✐♥❣ v ❢r♦♠ Cu(v) = (h1 , · · · , hN ) ∈ (L2 (τ, T ))N ✐s ✐❧❧✲♣♦s❡❞✳ ❲❡ ♥♦✇ ❝♦♠❜✐♥❡ t❤❡ ❧❡❛st sq✉❛r❡s ♣r♦❜❧❡♠ ✇✐t❤ ❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❛t✐♦♥ ❛s ❢♦❧❧♦✇s✿ ♠✐♥✐♠✐③❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ γ Jγ (v) = J0 (v) + v − v ∗ 2 L2 (Ω) = N li u(v) − hi i=1 L2 (τ,T ) + γ v − v∗ 2 L2 (Ω) ✇✐t❤ r❡s♣❡❝t t♦ v ∈ L2 (Ω) ✇✐t❤ γ ❜❡✐♥❣ ♣♦s✐t✐✈❡ ❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r✳ ✶✽ ✭✸✳✷✮ ❈♦♥s✐❞❡r t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠✿ n ∂p ∂ ∂p − − a (x, t) + b(x, t)p = ij ∂t i,j=1 ∂xi ∂xj N ωi (li u − hi )χ(τ,T ) (t) ✐♥ Q, i=1 ✭✸✳✸✮ p = ♦♥ S, p(x, T ) = ✐♥ Ω, ✇❤❡r❡ χ(τ,T ) (t) = 1✱ ✐❢ t ∈ (τ, T ) ❛♥❞ χ(τ,T ) (t) = 0✱ ♦t❤❡r✇✐s❡✳ ✸✳✷ ▼❛✐♥ r❡s✉❧ts ❚❤❡♦r❡♠ ✸✳✷✳✶✳ ❚❤❡ ❣r❛❞✐❡♥t ∇J0(v) ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥❛❧ J0(v) ❛t v ✐s ❣✐✈❡♥ ❜② ∇J0 (v) = p(x, 0), ✇❤❡r❡ p(x, t) ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ✭✸✳✸✮✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ✜rst ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠ ✭✸✳✶✮ ❤❛s ♥♦ ♠✐①❡❞ ❞❡r✐✈❛t✐✈❡✱ t❤❛t ✐s aij = ✐❢ i = j ✱ ❛♥❞ ✇❡ ❞❡♥♦t❡ aii ❜② ❛♥❞ ❣❡t t❤❡ s②st❡♠ ✭✶✳✶✶✮✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❛ss✉♠❡ Ω ✐s t❤❡ ♦♣❡♥ ♣❛r❛❧❧❡❧❡♣✐♣❡❞✳ ❲❡ ❞✐s❝r❡t✐③❡ t❤✐s ♣r♦❜❧❡♠ ❜② t❤❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ ♠❡t❤♦❞ ✐♥ s♣❛❝❡ ✈❛r✐❛❜❧❡s ❛♥❞ ❣❡t t❤❡ s②st❡♠ ♦❢ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭✶✳✶✺✮✳ ❲❡ ❤❛✈❡ t❤❡ ✜rst✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ❢♦r t❤✐s ♣r♦❜❧❡♠ ❛s ❢♦❧❧♦✇s Jγ (v) = C ∗ (Cv − ) + γ(v − v ∗ ) = p(x, 0) + γ(v − v ∗ ) = ❲❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ Jγ ❛s ❢♦❧❧♦✇s✳ ❋✐rst✱ ✇❡ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❢✉♥❝t✐♦♥❛❧ li (v) ❜② lih u¯(¯ v ) = ∆h ω ¯ ik u¯k (¯ v ), i = 1, , N k∈Ωh ¯o ❲❡ ❞❡✜♥❡ u ❜② t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ u d¯ + (Λ1 + · · · + Λn )¯ u − F¯ = 0, dt u¯(0) = 0, ✇❤❡r❡ F¯ ✐s ❞❡✜♥❡❞ ❛s ✐♥ ✭✶✳✶✺✮✱ ❛♥❞ u ¯[¯ v ] t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ u d¯ + (Λ1 + · · · + Λn )¯ u = 0, dt u¯(0) = v¯ ❚❤❡♥✱ ¯o lih u¯(¯ v ) = lih u¯[¯ v ] + lih u = ∆h ¯o k ω ¯ ik u , ω ¯ ik u¯k [¯ v ] + ∆h k∈Ωh k∈Ωh ✶✾ i = 1, , N ✭✸✳✹✮ v ] ✐s ❧✐♥❡❛r ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ L2 (Ω) ✐♥t♦ L2 (τ, T )✱ ❢♦r ❍❡♥❝❡ t❤❡ ♦♣❡r❛t♦r Cih v˜ ¯ = lih u¯[¯ i = 1, , N ✳ ❋✉rt❤❡r♠♦r❡✱ ✐❢ p¯† ✐s ❛ s♦❧✉t✐♦♥ t♦ t❤❡ ❈❛✉❝❤② ♣r♦❜❧❡♠ ¯ dp† ¯ = 0, + (Λ1 + · · · + Λn )p¯† − G dt p¯† (T ) = 0, ˜ ∗ ¯ = {¯ g = p¯† (x, 0) ¯ ik ✐s ❞❡✜♥❡❞ ✐♥ ❢♦r♠✉❧❛ ✭✶✳✶✸✮✱ t❤❡♥ Cih ✇✐t❤ G ωik g(t), k ∈ Ωh } ✇❤❡r❡ ω ❚❤✉s✱ t❤❡ ❞✐s❝r❡t✐③❡❞ ✈❡rs✐♦♥ ♦❢ C ❤❛s t❤❡ ❢♦r♠ Ch := (C1h , , CN h ) ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥❛❧ Jγ ✐s ♥♦✇ ❛♣♣r♦①✐♠❛t❡❞ ❛s ❢♦❧❧♦✇s✿ γ Jγh (vh ) : = Chv˜¯ − 2(L2 (τ,T ))N + v¯ − v¯∗ 2 N γ ✭✸✳✺✮ = Cih v˜¯ − i 2L2 (τ,T ) + v¯ − v¯∗ 2 i=1 o ❍❡r❡✱ ❢♦r s✐♠♣❧✐❝✐t② ♦❢ ♥♦t❛t✐♦♥✱ ✇❡ ❛❣❛✐♥ s❡t i = hi −lih u ¯✳ ❋♦r t❤✐s ❞✐s❝r❡t✐③❡❞ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✇❡ ❤❛✈❡ t❤❡ ✜rst✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ v − v¯∗ ) = Ch∗ (Ch v¯ − ) + γ(¯ ■❢ ✇❡ s✉♣♣♦s❡ t❤❛t ❝♦♥❞✐t✐♦♥ ✷✮ ♦❢ ❚❤❡♦r❡♠ ✶✳✸✳✸ ✐s s❛t✐s✜❡❞✱ t❤❡♥ Ch v¯ − Cv (L2 (τ,T ))N ❛♥❞ Ch∗ g − Cg L2 (τ,T ) t❡♥❞ t♦ ③❡r♦ ❛s h t❡♥❞s t♦ ③❡r♦✳ ❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t✳ ❚❤❡♦r❡♠ ✸✳✷✳✷✳ ▲❡t ❝♦♥❞✐t✐♦♥ ✷✮ ♦❢ ❚❤❡♦r❡♠ ✶✳✸✳✸ ❜❡ s❛t✐s✜❡❞✳ ❚❤❡♥ t❤❡ s♦❧✉t✐♦♥ vˆ¯hγ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✸✳✺✮ ❝♦♥✈❡r❣❡s t♦ t❤❡ s♦❧✉t✐♦♥ vγ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✸✳✷✮ ✐♥ L2 (Ω) ❛s h t❡♥❞s t♦ ③❡r♦✳ ❚♦ s♦❧✈❡ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ♥✉♠❡r✐❝❛❧❧② ✇❡ ♥❡❡❞ t♦ ❞✐s❝r❡t✐③❡ t❤❡ s②st❡♠ ✭✶✳✶✶✮ ✐♥ t✐♠❡✳ ❲❡ ❞✐s❝r❡t✐③❡ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥❛❧ J0 (v) ❛s ❢♦❧❧♦✇s J0h,∆t (¯ v) := N M v ) − hm ωik uk,m (¯ i ] , [ i=1 m= k∈Ωh ✇❤❡r❡ ✐s t❤❡ ✜rst ✐♥❞❡① ❢♦r ✇❤✐❝❤ ∆t > τ ✱ ❛♥❞ uk,m (¯ v ) s❤♦✇s ✐ts ❞❡♣❡♥❞❡♥❝❡ ♦♥ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v¯ ❛♥❞ m ✐s t❤❡ ✐♥❞❡① ♦❢ ❣r✐❞ ♣♦✐♥ts ♦♥ t✐♠❡ ❛①✐s✳ ◆♦t❛t✐♦♥ ωik = ωi (xk ) ✐♥❞✐❝❡s t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ωi (x) ✐♥ Ωh ❛t ♣♦✐♥ts xk ✳ ◆♦r♠❛❧❧②✱ ✇❡ t❛❦❡ ❛s t❤❡ ✐ts ❛✈❡r❛❣❡ ♦✈❡r t❤❡ ❝❡❧❧ ✇❤❡r❡ xk ✐s ❧♦❝❛t❡❞ ❛s ❞❡✜♥❡❞ ❜② ✭✶✳✶✸✮✳ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ J0h,∆t (¯ v ) ✐s s❤♦✇♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✸✳✷✳✸✳ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ J0h,∆t ❛t v¯ ✐s ❣✐✈❡♥ ❜② ∇J0h,∆t (¯ v ) = A0 ∗ · · · A −1 ∗ η −1 , ✇❤❡r❡ η s❛t✐s✜❡s t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ η m = (Am+1 )∗ η m+1 + N i=1 ωik um+1 (¯ v ) − hm+1 ], i ωik [ k∈Ωh η M = ✷✵ m = M − 1, M − 2, , − 1, ❍❡r❡ t❤❡ ♠❛tr✐① (Am )∗ ✐s ❣✐✈❡♥ ❜② ∆t m −1 ∆t m ∆t m −1 ∆t m Λ1 )(E1 + Λ1 ) (En − Λn )(En + Λ ) 4 4 n ∆t m ∆t m −1 ∆t m ∆t m −1 × (En − Λn )(En + Λn ) (E1 − Λ1 )(E1 + Λ ) 4 4 (Am )∗ = (E1 − ■♥ t❤❡ ❧❛st ♣❛rt ♦❢ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ♣r❡s❡♥t s♦♠❡ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s t♦ ♣r♦✈❡ t❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞✳ ❚❤✐s ❝❤❛♣t❡r ✐s ✇r✐tt❡♥ ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ ♣❛♣❡r ❬✷❪ ❍à♦ ❉✳◆✳ ❛♥❞ ❖❛♥❤ ◆✳❚✳◆✳✱ ❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐♥ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ❢r♦♠ ✐♥t❡❣r❛❧ ♦❜s❡r✈❛t✐♦♥s✳ ■♥✈❡rs❡ Pr♦❜❧❡♠s ✐♥ ❙❝✐❡♥❝❡ ❛♥❞ ❊♥❣✐♥❡❡r✐♥❣ ✭t♦ ❛♣♣❡❛r✮ ❞♦✐✿ ✶✵✳✶✵✽✵✴✶✼✹✶✺✾✼✼✳✷✵✶✻✳✶✷✷✾✼✼✽✳ ❈❤❛♣t❡r ✹ ❉❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❜② t❤❡ ❜♦✉♥❞❛r② ♦❜s❡r✈❛t✐♦♥s ■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ st✉❞② t❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ♦❢ ❡st✐♠❛t✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v(x) ✐♥ t❤❡ ◆❡✉♠❛♥♥ ♣r♦❜❧❡♠ ✭✶✳✶✮✱ ✭✶✳✷✮✱ ✭✶✳✹✮ ❢r♦♠ t❤❡ ❜♦✉♥❞❛r② ♦❜s❡r✈❛t✐♦♥ u|Σ = ϕ(ζ, t), (ζ, t) ∈ Σ = Γ × (0, T ), Γ ⊂ ∂Ω✳ ❲❡ r❡❢♦r♠✉❧❛t❡ ✐t ❛s ❛ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ♦❢ ♠✐♥✐♠✐③✐♥❣ ❛ ♠✐s✜t ❢✉♥❝t✐♦♥❛❧✳ ❆s t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✐s ✐❧❧✲♣♦s❡❞✱ ✇❡ st❛❜✐❧✐③❡ ✐t ❜② ✉s✐♥❣ t❤❡ ❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞✳ ■t ✐s ♣r♦✈❡❞ t❤❛t t❤❡ r❡❣✉❧❛r✐③❡❞ ❢✉♥❝t✐♦♥❛❧ ✐s ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r ✐ts ❣r❛❞✐❡♥t ✐s ❞❡r✐✈❡❞ ✈✐❛ ❛♥ ❛❞❥♦✐♥t ♣r♦❜❧❡♠✳ ❚❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✐s ✜rst ❞✐s❝r❡t✐③❡❞ ✐♥ s♣❛❝❡ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ r❡s✉❧ts ♦❢ t❤❡ ♠❡t❤♦❞ ❛r❡ ♣r♦✈❡❞✳ ❚❤❡ ♣r♦❜❧❡♠ ✐s t❤❡♥ ❢✉❧❧② ❞✐s❝r❡t✐③❡❞ ❛♥❞ ✐t ✐s ♣r♦✈❡❞ t❤❛t t❤❡ ❞✐s❝r❡t✐③❡❞ ❢✉♥❝t✐♦♥❛❧ ✐s ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❆ ❢♦r♠✉❧❛ ❢♦r ✐ts ❣r❛❞✐❡♥t ✐s ❞❡r✐✈❡❞ ✈✐❛ ❛ ❞✐s❝r❡t❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ❛♥❞ t❤❡ ❝♦♥❥✉❣❛t❡ ❣r❛❞✐❡♥t ♠❡t❤♦❞ ✐s ❛♣♣❧✐❡❞ t♦ ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡ t❤❡ ❞✐s❝r❡t✐③❡❞ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠✳ ❙♦♠❡ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s ❛r❡ ♣r♦✈✐❞❡❞ t♦ s❤♦✇ t❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡ ♣r♦♣♦s❡❞ ❛❧❣♦r✐t❤♠✳ ✷✶ ✹✳✶ Pr♦❜❧❡♠ s❡tt✐♥❣ ❛♥❞ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ n ∂u ∂ − ∂t i,j=1 ∂xi aij (x, t) ∂u ∂xj + b(x, t)u = f u|t=0 = v ∂u = g, ∂N ✇❤❡r❡ ∂u ∂N |S := n i,j=1 (aij (x, t)uxj ) cos(ν, xi )|S ✐♥ Q, ✭✹✳✶✮ ✐♥ Ω, ✭✹✳✷✮ ♦♥ S, ✭✹✳✸✮ ❛♥❞ ν ✐s t❤❡ ♦✉t❡r ♥♦r♠❛❧ t♦ S ✳ ❉❛t❛ ❛ss✐♠✐❧❛t✐♦♥✿ ❘❡❝♦♥str✉❝t t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v(x) ✐♥ ✭✹✳✶✮✕✭✹✳✸✮ ❢r♦♠ t❤❡ ♦❜s❡r✈❛t✐♦♥s ♦❢ t❤❡ s♦❧✉t✐♦♥ u ♦♥ ❛ ♣❛rt ♦❢ t❤❡ ❜♦✉♥❞❛r② S ✳ ◆❛♠❡❧②✱ ❧❡t Γ ⊂ ∂Ω ❛♥❞ ❞❡♥♦t❡ Σ = Γ × (0, T )✳ ❖✉r ❛✐♠ ✐s t♦ r❡❝♦♥str✉❝t t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ v ❢r♦♠ t❤❡ ✐♠♣r❡❝✐s❡ ♠❡❛s✉r❡♠❡♥t ϕ ∈ L2 (Σ) ♦❢ t❤❡ s♦❧✉t✐♦♥ u ♦♥ Σ✿ u|Σ − ϕ L2 (Σ) ≤ ❘❡♠❛r❦ ✹✳✶✳✶✳ ❚♦ s❡❡ t❤❡ ✐❧❧✲♣♦s❡❞♥❡ss ♦❢ t❤❡ ♣r♦❜❧❡♠✱ ❧❡t ♥♦t❡ t❤❛t ✐❢ u ∈ H 1,1 (Q)✳ ❍❡♥❝❡ t❤❡ ♦♣❡r❛t♦r ♠❛♣♣✐♥❣ v t♦ u|Σ ✐s ❝♦♠♣❛❝t ❢r♦♠ L2 (Ω) t♦ L2 (Σ)✳ ❚❤✉s✱ t❤❡ ♣r♦❜❧❡♠ ♦❢ r❡❝♦♥str✉❝t✐♥❣ v ❢r♦♠ u|Σ ✐s ✐❧❧✲♣♦s❡❞✳ ◆❡①t✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ n ∂p ∂ ∂p aij (x, t) + b(x, t)p = ✐♥ Q, − − ∂xi ∂t i,j=1 ∂xj ∂p = u(v) − ϕ χΣ ∂N p(x, T ) = ✐♥ Ω, ♦♥ S, ✭✹✳✹✮ ✇❤❡r❡ χΣ (ξ, t) = ✐❢ (ξ, t) ∈ Σ ❛♥❞ ③❡r♦ ♦t❤❡r✇✐s❡✳ ❚❤✐s r❡❝♦♥str✉❝t✐♥❣ ♣r♦❜❧❡♠ ✐s r❡❢♦r✲ ♠✉❧❛t❡❞ ❛s t❤❡ ❧❡❛st sq✉❛r❡s ♣r♦❜❧❡♠ ♦❢ ♠✐♥✐♠✐③✐♥❣ ✐ts ❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❡❞ ❢✉♥❝t✐♦♥❛❧ γ u(v) − ϕ 2L2 (Σ) + v − v ∗ 2L2 (Ω) ✭✹✳✺✮ 2 ♦✈❡r L2 (Ω) ✇✐t❤ γ > t❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r✱ v ∗ ❛♥ ❡st✐♠❛t✐♦♥ ♦❢ v ✇❤✐❝❤ ❝❛♥ ❜❡ s❡t ❜② ③❡r♦✳ Jγ (v) = ✹✳✷ ▼❛✐♥ r❡s✉❧ts ❚❤❡♦r❡♠ ✹✳✷✳✶✳ ❚❤❡ ❢✉♥❝t✐♦♥❛❧ Jγ ✐s ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛♥❞ ∇Jγ (v) = p(x, 0) + γ(v(x) − v ∗ (x)), ✇❤❡r❡ p(x, t) ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ✭✹✳✹✮✳ ✷✷ o ❉❡♥♦t❡ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠ ✭✹✳✶✮✕✭✹✳✸✮ ✇✐t❤ v ≡ ❜② u ❛♥❞ ❞❡♥♦t❡ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠ ✭✹✳✶✮✕✭✹✳✸✮ ✇✐t❤ f ≡ 0, g ≡ ❜② u0 ✳ ❲❡ ♥♦✇ t✉r♥ t♦ ❛♣♣r♦①✐♠❛t✐♥❣ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✭✹✳✺✮✱ ✇❡ ❤❛✈❡ Jγ (v) = o Cv − (ϕ − u|Σ ) 2 L2 (Σ) + γ v − v∗ 2 L2 (Ω) ❛♥❞ o Jγ (v) = C ∗ Cv − (ϕ − u|Σ ) + γ(v − v ∗ ) = p(x, 0) + γ(v − v ∗ ), ✇❤❡r❡ Cv = u0 (v)|Σ ❛♥❞ p ✐s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠✳ ❚❤✉s✱ t❤❡ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ ✐s o C ∗ Cv − (ϕ − u|Σ ) + γ(v − v ∗ ) = p(x, 0) + γ(v − v ∗ ) = ❉❡♥♦t❡ ❜② Ch v = u ˆ0h |Σ ✇❡ ❤❛✈❡ t❤❛t Cv − Ch v ❞✐s❝r❡t❡ ✈❡rs✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧ ✭✹✳✺✮ ✐s L2 (Σ) ✭✹✳✻✮ t❡♥❞s t♦ ③❡r♦ ❛s h t❡♥❞s t♦ ③❡r♦✳ ❚❤❡ γ oˆ Ch v − (ϕˆh − uh |Σ ) 2L2 (Σ) + vˆh − vˆh∗ 2 ❢♦r ✇❤✐❝❤ ✇❡ ❤❛✈❡ t❤❡ ✜rst✲♦r❞❡r ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥ oˆ vh − vˆh∗ ) = Ch∗ Ch v − (ϕˆh − uh |Σ ) + γ(ˆ Jγh (v) = L2 (Ω) ❲❡ ♥♦t❡ t❤❛t t♦ ❡✈❛❧✉❛t❡ Ch∗ ✇❡ ❤❛✈❡ t♦ s♦❧✈❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞✐s❝r❡t✐③❡❞ ❛❞❥♦✐♥t ♣r♦❜❧❡♠✱ ❜✉t t❤❡ ◆❡✉♠❛♥♥ ❝♦♥❞✐t✐♦♥ ✐♥ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ H 1,1 (S)✱ t❤❡r❡❢♦r❡ p ✐s ♥♦t ✐♥ H 1,1 (Q)✱ ❤❡♥❝❡ ✇❡ ❞♦ ♥♦t ❤❛✈❡ t❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ Ch∗ z t♦ C ∗ z ✐♥ L2 (Ω)✳ ❍♦✇❡✈❡r✱ ✇❤❡♥ ✇❡ ❞✐s❝r❡t✐③❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ✇❡ ❤❛✈❡ t♦ ♠♦❧❧✐❢② t❤❡ ◆❡✉♠❛♥♥ ❞❛t❛ ❜② t❤❡ ❝♦♥✈♦❧✉t✐♦♥ ✇✐t❤ ❙t❡❦❧♦✈✬s ❦❡r♥❡❧✱ t❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ ❛ ♥❡✇ ❛♣♣r♦①✐♠❛t❡ ❞❛t❛ ✐♥ H 1,1 (S)✳ ❙✐♥❝❡ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ✐s st❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❞❛t❛✱ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ✇✐t❤ ♠♦❧❧✐✜❡❞ ❞❛t❛ p¯ ❛♣♣r♦①✐♠❛t❡s t❤❡ s♦❧✉t✐♦♥ p ♦❢ ❛❞❥♦✐♥t ♣r♦❜❧❡♠✳ ◆♦✇ ✇❡ ❛♣♣❧② t❤❡ ❛❜♦✈❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡ s❝❤❡♠❡ t♦ t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ ✇✐t❤ ♠♦❧❧✐✜❡❞ ❞❛t❛ t♦ ❣❡t ✐ts ♠✉❧t✐✲❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ pˆ ¯h s✉❝❤ t❤❛t pˆ¯h → p¯ ✐♥ L2 ([0, T ], L2 (Ω)) ❛♥❞ pˆ¯h (t) → p¯(t) ✇❡❛❦❧② ✐♥ H (Ω) ❢♦r ❛❧❧ t ∈ [0, T ]✳ ❚❤✉s✱ ✐♥ t❤✐s ✇❛②✱ ✐♥st❡❛❞ ♦❢ t❤❡ ❛❞❥♦✐♥t ♦♣❡r❛t♦r S∗h ✱ ✇❡ ❞❡✜♥❡❞ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ Cˆh∗ ♦❢ C ❢♦r ✇❤✐❝❤ Ch∗ z − Cˆh∗ z L2 (Ω) t❡♥❞s t♦ ③❡r♦ ❢♦r ❛❧❧ z ❜❡✐♥❣ ♠✉❧t✐✲❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥s ♦♥ Ωh ✳ ▲❡t vˆhγ ❜❡ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ oˆ Cˆh∗ Ch v − (ϕˆh − uh |Σ ) + γ(ˆ vh − vˆh∗ ) = Pr♦♣♦s✐t✐♦♥ ✹✳✷✳✷✳ ▲❡t vγ ❜❡ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ❚❤❡♥ vˆhγ ❝♦♥✈❡r❣❡s t♦ vγ ✐♥ L2 (Ω) ❛s h t❡♥❞s t♦ ③❡r♦✳ ✭✹✳✻✮ ❛♥❞ γ > 0✳ ❚❤❡ ❢✉❧❧② ❞✐s❝r❡t✐③❡❞ ✈❡rs✐♦♥ ♦❢ Jγ ❤❛s t❤❡ ❢♦r♠ J0h,∆t (¯ v) := M [uk,m (¯ v ) − ϕk,m ]2 k∈Γh m=1 ✷✸ ✭✹✳✼✮ ❋♦r ♠✐♥✐♠✐③✐♥❣ t❤❡ ♣r♦❜❧❡♠ ✭✹✳✼✮ ❜② t❤❡ ❝♦♥❥✉❣❛t❡ ❣r❛❞✐❡♥t ♠❡t❤♦❞✱ ✇❡ ✜rst ❝❛❧❝✉❧❛t❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥❛❧ J0h,∆t (¯ v ) ❛♥❞ ✐t ✐s s❤♦✇♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✹✳✷✳✸✳ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ J0h,∆t ❛t v¯ ✐s ❣✐✈❡♥ ❜② ∗ v ) = A0 η , ∇J0h,∆t (¯ ✇❤❡r❡ η = (η0 , , η M ) s❛t✐s✜❡s t❤❡ ❛❞❥♦✐♥t ♣r♦❜❧❡♠ m m+1 ∗ m+1 )η + ψ m+1 , η = (A η M −1 = ψ M , η M = 0, m = M − 2, M − , 0, ✇✐t❤ ψm = {ψk,m := uk,m (¯v) − ϕk,m , k ∈ Γh }, m = 0, 1, , M ❛♥❞ t❤❡ ♠❛tr✐❝❡s (Am )∗ ❛♥❞ (B m )∗ ❜❡✐♥❣ ❣✐✈❡♥ ❜② ∆t m ∆t m −1 ∆t m ∆t m −1 Λ1 )(E1 + Λ1 ) (En − Λn )(En + Λ ) 4 4 n ∆t m −1 ∆t m −1 ∆t m ∆t m × (En − Λn )(En + Λn ) (E1 − Λ1 )(E1 + Λ ) , 4 4 ∆t m ∆t m −1 ∆t m ∆t m −1 (B m )∗ = (En − Λn )(En + Λn ) (E1 − Λ1 )(E1 + Λ ) 4 4 (Am )∗ = (E1 − ❙✐♠✐❧❛r t♦ t❤❡ ♣r❡✈✐♦✉s ❝❤❛♣t❡r✱ ✇❡ ❛❧s♦ ♣r❡s❡♥t s♦♠❡ ♥✉♠❡r✐❝❛❧ ✐❧❧✉str❛t✐♦♥s t♦ ♣r♦✈❡ t❤❡ ❡✣❝✐❡♥❝② ♦❢ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞✳ ❚❤✐s ❝❤❛♣t❡r ✐s ✇r✐tt❡♥ ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ ♣❛♣❡r ❬✸❪ ❍à♦ ❉✳◆✳ ❛♥❞ ❖❛♥❤ ◆✳❚✳◆✳✱ ❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐♥ ♣❛r❛❜♦❧✐❝ ❡q✉❛✲ t✐♦♥s ❢r♦♠ ❜♦✉♥❞❛r② ♦❜s❡r✈❛t✐♦♥s✳ ❏♦✉r♥❛❧ ♦❢ ■♥✈❡rs❡ ❛♥❞ ■❧❧✲P♦s❡❞ Pr♦❜❧❡♠s ✷✹✭✷✵✶✻✮✱ ♥♦✳ ✷✱ ✶✾✺✕✷✷✵✳ ✷✹ ❚❤❡ ❛✉t❤♦r✬s ♣✉❜❧✐❝❛t✐♦♥s r❡❧❛t❡❞ t♦ t❤❡ t❤❡s✐s ✶✳ ◆❣✉②❡♥ ❚❤✐ ◆❣♦❝ ❖❛♥❤✱ ❆ s♣❧✐tt✐♥❣ ♠❡t❤♦❞ ❢♦r ❛ ❜❛❝❦✇❛r❞ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥ ✇✐t❤ t✐♠❡✲❞❡♣❡♥❞❡♥t ❝♦❡✣❝✐❡♥ts✱ ❈♦♠♣✉t❡rs ✫ ▼❛t❤❡♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s ✻✺✭✷✵✶✸✮✱ ✶✼✕✷✽✳ ✭❈❤❛♣t❡r ✷✮ ✷✳ ❉✐♥❤ ◆❤♦ ❍à♦ ❛♥❞ ◆❣✉②❡♥ ❚❤✐ ◆❣♦❝ ❖❛♥❤✱ ❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐♥ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ❢r♦♠ ✐♥t❡❣r❛❧ ♦❜s❡r✈❛t✐♦♥s✱ ■♥✈❡rs❡ Pr♦❜❧❡♠s ✐♥ ❙❝✐❡♥❝❡ ❛♥❞ ❊♥❣✐♥❡❡r✐♥❣ ✭t♦ ❛♣♣❡❛r✮ ❞♦✐✿ ✶✵✳✶✵✽✵✴✶✼✹✶✺✾✼✼✳✷✵✶✻✳✶✷✷✾✼✼✽✳ ✭❈❤❛♣t❡r ✸✮ ✸✳ ❉✐♥❤ ◆❤♦ ❍à♦ ❛♥❞ ◆❣✉②❡♥ ❚❤✐ ◆❣♦❝ ❖❛♥❤✱ ❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ✐♥ ♣❛r❛❜♦❧✐❝ ❡q✉❛t✐♦♥s ❢r♦♠ ❜♦✉♥❞❛r② ♦❜s❡r✈❛t✐♦♥s✱ ❏♦✉r♥❛❧ ♦❢ ■♥✈❡rs❡ ❛♥❞ ■❧❧✲P♦s❡❞ Pr♦❜❧❡♠s ✷✹✭✷✵✶✻✮✱ ♥♦✳ ✷✱ ✶✾✺✕✷✷✵✳ ✭❈❤❛♣t❡rs ✶ ❛♥❞ ✹✮ ❚❤❡ r❡s✉❧ts ♦❢ t❤❡ t❤❡s✐s ❤❛✈❡ ❜❡❡♥ ♣r❡s❡♥t❡❞ ❛t ✶✳ ✽t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ✧■♥✈❡rs❡ Pr♦❜❧❡♠s✿ ▼♦❞❡❧✐♥❣ ✫ ❙✐♠✉❧❛t✐♦♥✧✱ ✷✸✕✷✽ ▼❛②✱ ✷✵✶✻✱ Ö❧ü❞❡♥✐③✱ ❋❡t❤✐②❡✱ ❚✉r❦❡②❀ ✷✳ ▼✐♥✐✲✇♦r❦s❤♦♣ ♦♥ ✧❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ P❉❊s✧✱ ✷✾ ❏✉♥❡ ✷✵✶✻✱ ❱✐❡t♥❛♠ ■♥✲ st✐t✉t❡ ❢♦r ❆❞✈❛♥❝❡❞ ❙t✉❞② ✐♥ ▼❛t❤❡♠❛t✐❝s❀ ✸✳ ❱✐❡t♥❛♠✲❑♦r❡❛ ✇♦r❦s❤♦♣ ♦♥ s❡❧❡❝t❡❞ t♦♣✐❝s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✷✵✕✷✹ ❋❡❜r✉❛r②✱ ✷✵✶✼✱ ❉❛ ◆❛♥❣✱ ❱✐❡t♥❛♠❀ ✹✳ ✶✷t❤ ❲♦r❦s❤♦♣ ♦♥ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ❙❝✐❡♥t✐✜❝ ❈♦♠♣✉t✐♥❣✱ ✷✸✕✷✺ ❆♣r✐❧✱ ✷✵✶✹✱ ❇❛ ❱✐✱ ❱✐❡t♥❛♠❀ ✺✳ ✶✸t❤ ❲♦r❦s❤♦♣ ♦♥ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ❙❝✐❡♥t✐✜❝ ❈♦♠♣✉t✐♥❣✱ ✷✸✕✷✺ ❆♣r✐❧✱ ✷✵✶✺✱ ❇❛ ❱✐✱ ❱✐❡t♥❛♠❀ ✻✳ P❤❉ ❙t✉❞❡♥t ❈♦♥❢❡r❡♥❝❡✱ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❱✐❡t♥❛♠ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✱ ✷✵✶✶✱ ✷✵✶✷✱ ✷✵✶✸✱ ✷✵✶✹❀ ✼✳ ❙❡♠✐♥❛r ❛t ❉❡♣❛rt♠❡♥t ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❱✐❡t♥❛♠ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣②✳ ... ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❱✐❡t♥❛♠ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② Supervisor: Prof Dr Habil Dinh Nho Hào ❘❡❢❡r❡❡ ✶✿ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ❘❡❢❡r❡❡ ✷✿ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳