Page 132 - IJOCTA-15-1

R. Bagri, G. Sachdev, D. Agarwal / IJOCTA, Vol.15, No.1, pp.123-136 (2025)
Z !
ˆ
⇒ η(t, σ, ˆσ, σ κ , ˆσ κ , ω, ˆω)ψ σ (Λ, ˆσ κ , r)dν h ¯ T ¯ ¯ T ¯ i
−D κ λ θ σ κ (Λ,¯σ κ (t), b)+ ¯τ β σ κ (Λ, ¯σ κ (t), ¯µ) dν+
Ω
Z
ˆ
+ D κ η(t, σ, ˆσ, σ κ , ˆσ κ , ω, ˆω)ψ σ κ (Λ, ˆσ κ r)dν Z
¯
T
Ω ξ(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω) ¯χ Ψ ω (Λ, ¯ a)+
Z
ˆ
+ ξ(t, σ, ˆσ, σ κ , ˆσ κ , ω, ˆω)ψ ω (Λ, ˆσ κ , r)dν ≦0, Ω
!
Ω
λ θ ω (Λ, ¯σ κ (t), b)+ ¯τ β ω (Λ, ¯σ κ (t), ¯µ) dν =
∀(σ, ω)∈K ×Υ. ¯ T ¯ ¯ T ¯
Theorem 1. 16 [Necessary conditions for Z
T
¯
(RUVP)] Consider (¯σ, ¯ω)∈T as a robust weak ef- η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω) ¯χ Ψ σ Λ, ¯ a
Ω
ficient solution for the problem (RUVP). Then,
T
¯
¯
¯
¯ T
¯
p
m
there exist ¯χ∈R as the scalars, λ(t)∈R , ¯τ(t)∈ +λ θ σ (Λ, ¯σ κ , b)+ ¯τ β σ (Λ, ¯σ κ , ¯µ) dν
+
n p ¯ m n Z
R , ¯a ∈A⊆R , b∈B ⊆R and ¯µ∈M ⊆R as un-
¯
¯
+ ¯ T (Λ, ¯σ κ , b)
certain parameters satisfying D κ η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω) λ θ σ κ
Ω
¯
¯
T
¯ T
¯
¯
T
¯ χ Ψ σ (Λ, ¯ a)+λ θ σ (Λ, ¯σ κ (t), b)+ ¯τ β σ (Λ, ¯σ κ (t), ¯µ) T ¯
+¯τ β σ κ (Λ, ¯σ κ , ¯µ) dν
h i
¯
¯ T ¯ ¯ T (Λ, ¯σ κ (t), ¯µ) =0, Z
−D κ λ θ σ κ (Λ,¯σ κ (t), b)+ ¯τ β σ κ T ¯
+ ξ(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω) ¯χ Ψ ω Λ, ¯ a +
(1) Ω
¯
¯
¯
T
¯ T

¯ χ Ψ ω (Λ, ¯ a)+λ θ ω (Λ, ¯σ κ (t), b)+ λ θ ω (Λ, ¯σ κ , b)+ ¯τ β ω (Λ, ¯σ κ , ¯µ) dν =0. (6)
¯
T
¯ T
¯
¯
¯
T
¯ τ β ω (Λ, ¯σ κ (t), ¯µ)=0, (2)
Z
¯
¯
¯ T
λ θ(Λ, ¯σ κ (t),b)=0, (3) As Ψ(., ¯ a)dν is invex at (¯σ, ¯ω), there-
p Ω
X
¯
¯ χ≧0, ¯ χ i =1, λ≧0, ∀t∈Ω. (4) fore there exist η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω) and
ξ(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω) such that
i=1
Theorem 2. [Sufficient efficiency conditions Z Z

ˆ
¯
T
T
for (RUVP)] Consider a robust feasible so- ¯ χ Ψ Λ, ¯ a dν − ¯ χ Ψ Λ, ¯ a dν ≧
lution (¯σ, ¯ω)∈T for the robust vector varia- Ω Ω
Z
¯
tional control problem (RUVP) and there ex- η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)¯χ Ψ σ Λ, ¯ a dν
T

p ¯ m n
ists ¯χ∈R as the scalars, λ(t)∈R , ¯τ(t)∈R , Ω
+
p ¯ m n
¯ a ∈A⊂R , b∈B ⊆R and ¯µ∈M ⊆R as un-
Z
certain parameters, fulfilling the conditions T ¯
Z + ξ(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)¯χ Ψ ω Λ, ¯ a dν,
(1)−(4). Further, assume that if Ψ(., ¯ a)dν, Ω
Ω which along with (5) gives
Z Z
¯
T
¯ T
λ θ(., ., b)dν and ¯ τ β(., ., ¯µ)dν are in- Z T ¯
Ω Ω 0> η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)¯χ Ψ σ Λ, ¯ a dν
vex w.r.t. η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)∈R q and Ω
r
ξ(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)∈R at (¯σ, ¯ω). Then, (¯σ, ¯ω)
Z
is a robust weak efficient solution for the problem + ξ(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)¯χ Ψ ω Λ, ¯ a dν. (7)
T
¯

(RUVP). Ω
Z
Proof. Suppose, on the contrary that (¯σ, ¯ω) Also, invexity of ¯ T ¯
λ θ(., ., b)dν implies
is not a robust weak efficient solution for the prob- Ω
Z Z
lem (RUVP). Consequently, ∃(ˆσ, ˆω)∈T such that ¯ T ˆ ¯ ¯ T ¯ ¯
λ θ(Λ, ˆσ κ , b)dν −
λ θ(Λ, ¯σ κ , b)dν ≧
Z Z
Ω Ω
ˆ
¯
maxΨ Λ, a dν < maxΨ Λ, a dν Z
¯
¯
¯ T
Ω a∈A Ω a∈A η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)λ θ σ (Λ, ¯σ κ , b)dν
Since max a∈A Ψ(Λ, a)=Ψ(Λ, ¯a), we obtain Ω
Z Z
Z
ˆ
¯
Ψ Λ, ¯a dν < Ψ Λ, ¯a dν. (5) ¯ T ¯ ¯
Ω Ω + D κ η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)λ θ σ κ (Λ, ¯σ κ , b)dν+
Ω
Additionally, multiplying equations (1) and (2) by Z
¯
¯ T
¯
η and ξ, respectively and integrating, ξ(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)λ θ ω (Λ, ¯σ κ , b)dν
Ω

Z Now by robust feasibility of (ˆσ, ˆω)∈T and (3),
T
¯
η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω) ¯χ Ψ σ (Λ, ¯ a)+
Ω the above inequality yields
Z
¯
¯ T
¯
¯
¯
¯ T
T
¯
λ θ σ (Λ, ¯σ κ (t), b)+ ¯τ β σ (Λ, ¯σ κ (t), ¯µ) 0≧ η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)λ θ σ (Λ, ¯σ κ , b)dν
Ω
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