Page 134 - IJOCTA-15-1

R. Bagri, G. Sachdev, D. Agarwal / IJOCTA, Vol.15, No.1, pp.123-136 (2025)

T
τ β ζ κ (Π, ζ κ , µ) dν.
Z Z
T
T
λ θ(Λ, σ κ , b)dν − λ θ(Π, ζ κ , b)dν ≧ Further, applying integration by parts,
Ω Ω
Z
T
η(t, σ, ζ, σ κ , ζ κ , ω, ϱ)λ θ ζ (Π, ζ κ , b)dν Z T T T
Ω χ Ψ(Λ, ¯ a)+λ θ(Λ, σ κ , b)+τ β(Λ, σ κ , µ) dν
Ω
Z
Z

T
T
T
T
+ D κ η(t, σ, ζ, σ κ , ζ κ , ω, ϱ)λ θ ζ κ (Π, ζ κ , b)dν+ − χ Ψ(Π, ¯ a)+λ θ(Π, ζ κ , b)+τ β(Π, ζ κ , µ) dν
Ω Ω

Z
T
ξ(t, σ, ζ, σ κ , ζ κ , ω, ϱ)λ θ ϱ (Π, ζ κ , b)dν (16) T (Π, ζ κ , b)
≧η(t, σ, ζ, σ κ , ζ κ , ω, ϱ) λ θ ζ κ
Ω
and !
T t 1
+τ β ζ κ (Π, ζ κ , µ) dν
t 0
Z Z
T
T
τ β(Λ, σ κ , µ)− τ β(Π, ζ κ , µ)dν ≧ Z
T
Ω Ω − D κ η(t, σ, ζ, σ κ , ζ κ , ω, ϱ) λ θ ζ κ (Π, ζ κ , b)dν+
Z
Ω
T
η(t, σ, ζ, σ κ , ζ κ , ω, ϱ)τ β ζ (Π, ζ κ , µ)dν
T
Ω τ β ζ κ (Π, ζ κ , µ) dν+
Z
T
+ D κ η(t, σ, ζ, σ κ , ζ κ , ω, ϱ)τ β ζ κ (Π, ζ κ , µ)dν+ Z T
Ω + D κ η(t, σ, ζ, σ κ , ζ κ , ω, ϱ) λ θ ζ κ (Π, ζ κ , b)dν+
Z Ω
T
ξ(t, σ, ζ, σ κ , ζ κ , ω, ϱ)τ β ϱ (Π, ζ κ , µ)dν. (17) T (Π, ζ κ , µ) dν
τ β ζ κ
Ω
Adding inequalities (15), (16) and (17), we obtain which on using boundary conditions of Definition
2, give
Z
Z
T
T
T
χ Ψ(Λ, ¯ a)+λ θ(Λ, σ κ , b)+τ β(Λ, σ κ , µ)dν
T
T
T
χ Ψ(Λ, ¯ a)+λ θ(Λ, σ κ , b)+τ β(Λ, σ κ , µ) dν−
Ω
Ω Z
Z
T T T
T
T
T
χ Ψ(Π, ¯ a)+λ θ(Π, ζ κ , b)+τ β(Π, ζ κ , µ) dν, ≧ χ Ψ(Π, ¯ a)+λ θ(Π, ζ κ , b)+τ β(Π, ζ κ , µ)dν,
Ω Ω
Z
a contradiction to inequality (14). Hence, the re-
T
≧ η(t, σ, ζ, σ κ , ζ κ , ω, ϱ) χ Ψ ζ (Π, ¯ a) sult is derived.
Ω

T
T
+λ θ ζ (Π, ζ κ , b)+τ β ζ (Π, ζ κ , µ) dν Pertinent Case : Consider the following bi-
objective multi-dimensional variational control
Z

T
+ D κ η(t, σ, ζ, σ κ , ζ κ , ω, ϱ) λ θ ζ κ (Π, ζ κ , b)dν+ problem with data uncertainty
Ω !
Z Z

T
τ β ζ κ (Π, ζ κ , ¯µ) dν (UVP1) min Ψ 1 (Λ, a 1 ), Ψ 2 (Λ, a 2 ) =
(σ,ω)
Z Ω Ω

T
+ ξ(t, σ, ζ, σ κ , ζ κ , ω, ϱ) χ Ψ ϱ (Π, ¯ a)+ Z Z !
Ω 2 2
min (σ +a 1 ω)dν, (a 2 −ω )dν
T
T
λ θ ϱ (Π, ζ κ , b)+τ β ϱ (Π, ζ κ , µ) dν. (σ,ω) Ω Ω
Using (10) and (11), the above inequality implies subject to
the following θ(Λ, σ κ (t), b)=2b+σ−7≦0, b∈[1, 2] (1∗)
−5ω =0, µ∈[0, 1] (2∗)
β(Λ, σ κ (t), µ)=µσ t 1
Z
1 3
T
T
T
χ Ψ(Λ, ¯ a)+λ θ(Λ, σ κ , b)+τ β(Λ, σ κ , µ) dν σ(0, 0)= , σ(1, 1)= , (3∗)
Ω 2 2
1
Z h i

T
T
T
− χ Ψ(Π, ¯ a)+λ θ(Π, ζ κ , b)+τ β(Π, ζ κ , µ) dν, a 1 ∈ 0, 2 , a 2 ∈[0, 3], [t 1 , t 2 ]∈[0, 0]×[1, 1].
Ω
The robust counterpart to the variational control
Z
h
T
≧ η(t, σ, ζ, σ κ , ζ κ , ω, ϱ) D κ λ θ ζ κ (Π, ζ κ , b)+ problem (UVP1) is written as :
Ω
Z
! 2
(RUVP1) min max (σ +a 1 ω)dν,
i
T (σ,ω) 1
2
τ β ζ κ (Π, ζ κ , µ) dν+ Ω a 1 ∈[0, ]
!
Z
Z
2
T
+ D κ η(t, σ, ζ, σ κ , ζ κ , ω, ϱ) λ θ ζ κ (Π, ζ κ , b)dν+ max (a 2 −ω )dν
Ω a 2 ∈[0,3]
Ω
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