Page 138 - IJOCTA-15-1

R. Bagri, G. Sachdev, D. Agarwal / IJOCTA, Vol.15, No.1, pp.123-136 (2025)
h i
T T ¯
Z Z (Π,ζ κ (t), µ) =0,
−D κ λ θ ζ κ (Π, ζ κ (t), b)+τ β ζ κ
T
¯
T
¯

¯ χ Ψ Λ, ¯ a dν − ¯ χ Ψ Π, ¯ a dν (23)
Ω Ω
χ Ψ ϱ (Π, ¯ a)+λ θ ϱ (Π, ζ κ (t), b)+
Z Z T T
λ θ(Λ, ¯σ κ , b)dν −
λ θ(Π, ζ κ , b)dν+
+ ¯ T ¯ ¯ ¯ T ¯ ¯ ¯ ¯ τ β ϱ (Π, ζ κ (t), µ)=0, (24)
T
Ω Ω
Z Z
T
T
¯ ¯
¯
¯ τ β(Λ, ¯σ κ , ¯µ)dν − ¯ τ β(Π, ζ κ , ¯µ)dν T
Ω Ω λ θ(Π, ζ κ , b)≧0, (25)
Z
T
τ β(Π, ζ κ , µ)=0, (26)
¯
T
¯
¯
> η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) ¯χ Ψ ζ Π, ¯ a +
Ω ζ(t 0 )=δ 0 , ζ(t 1 )=δ 1 , (27)
¯ T ¯ ¯ ¯ T ¯ ¯ λ≧0, χ>0, τ >0, (28)
λ θ ζ (Π, ζ κ , b)+ ¯τ β ζ (Π, ζ κ , ¯µ)

¯ ¯ ¯ T ¯ ¯ ¯ for ¯a ∈A, b∈B, µ∈M .
+D κ η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) λ θ ζ κ (Π, ζ κ , b)

T ¯ ¯
+¯τ β ζ κ (Π, ζ κ , ¯µ) dν Denote T MW = {(ζ, ϱ, χ, λ, τ, ¯ a, b, µ) satisfying
Z
conditions (23)−(28)} as the set of feasible so-
¯
¯
T
¯
+ ξ(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) ¯χ Ψ ϱ Π, ¯ a + lutions to (MWD).
Ω

¯ ¯ ¯
¯ ¯
T
¯ T
λ θ ϱ (Π, ζ κ , b)+ ¯τ β ϱ (Π, ζ κ , ¯µ) dν, Theorem 6. [Weak Duality Theorem] Assume
that (σ, ω)∈T and (ζ, ϱ, χ, λ, τ, ¯ a, b, µ)∈ T MW .
P p Further, if
Using (3), (18) and ¯χ≧0, i=1 ¯ χ i =1, the above Z
inequality gives T T
(i) λ θ(., ., b)dν +τ β(., ., µ) dν is
Ω
quasi-invex w.r.t. η and ξ at (ζ, ϱ);
Z
Z
¯
¯
¯
T
0> η(t, ¯σ, ζ, ¯σ κ , ζ ϱ , ¯ω, ¯ϱ) ¯χ Ψ ζ Π, ¯ a (ii) Ψ(., ¯ a)dν is strictly pseudo-invex w.r.t.
Ω
Ω
T
¯ ¯
¯ T
¯ ¯ ¯
+λ θ ζ (Π, ζ κ , b)+ ¯τ β ζ (Π, ζ κ , ¯µ) η and ξ at (ζ, ϱ);

¯ ¯ ¯ T ¯ ¯ ¯ then, the following inequality cannot hold
+D κ η(t, ¯σ, ζ, ¯σ κ , ζ ϱ , ¯ω, ¯ϱ) λ θ ζ κ (Π, ζ κ , b)
Z Z
T ¯ ¯
Ψ(Λ, ¯ a)dν < Ψ(Π, ¯ a)dν.
+¯τ β ζ κ (Π, ζ κ , ¯µ) dν
Ω Ω
Z

¯
¯
¯
T
+ ξ(t, ¯σ, ζ, ¯σ κ , ζ ϱ , ¯ω, ¯ϱ) ¯χ Ψ ϱ Π, ¯ a +
Ω Proof. Let (σ, ω) and (ζ, ϱ, χ, λ, τ, ¯ a, b, µ) be ro-

¯
T
¯ T
¯ ¯ ¯
λ θ ϱ (Π, ζ κ , b)+ ¯τ β ϱ (Π, ζ κ , ¯µ) dν, bust feasible solutions of (RUVP) and (MWD), re-
spectively. As λ≧0,
which conflicts with equation (19). Thus, (¯σ, ¯ω)=
¯
(ζ, ¯ϱ). □
Z Z
T
T
λ θ(Λ, σ κ , b)dν ≦ 0≦ λ θ(Π, ζ κ , b)dν
Ω Ω
4. Mond-Weir type dual and duality and
theorems
Z Z
T
T
In this section, the Mond Weir type dual is pre- τ β(Λ, σ κ , µ)dν = 0= τ β(Π, ζ κ , µ)dν,
sented. For this dual, the duality results under Ω Ω
weaker conditions are derived which extend its Hence,
application to vast area. The robust Mond-Weir
Z
type dual associated with the multi-time vector λ θ(Λ, σ κ , b)+τ β(Λ, σ κ , µ) dν ≦

T
T
variational control problem (RUVP) is defined as Ω
below : Z T T
λ θ(Π, ζ κ , b)+τ β(Π, ζ κ , µ) dν.
Ω

Z
Thus, hypothesis (i) gives
(MWD) max Ψ(Π, ¯ a)dν =
(ζ(.),ϱ(.)) Ω
Z

!
Z Z T
η(t, σ, ζ, σ κ , ζ κ , ω, ϱ) λ θ ζ (Π, ζ κ , b)+
Ψ 1 (Π, ¯ a 1 )dν, ..., Ψ p (Π, ¯ a p )dν Ω
Ω Ω
T
τ β ζ (Π, ζ κ , µ) dν
subject to Z
T
T
T
T
χ Ψ ζ (Π,¯ a)+λ θ ζ (Π, ζ κ (t), b)+τ β ζ (Π,ζ κ (t), µ) + D κ η(t, σ, ζ, σ κ , ζ κ , ω, ϱ) λ θ ζ κ (Π, ζ κ , b)+
Ω
132