Page 130 - IJOCTA-15-1

R. Bagri, G. Sachdev, D. Agarwal / IJOCTA, Vol.15, No.1, pp.123-136 (2025)
problems with nonlinear constraints and estab- 2. Preliminaries
lished the optimality conditions. Recent develop-
ments in control theory can be seen in. 11,12 With Consider the following notations in developing the
the aid of a multi-time notion using partial deriva- results :
s
q
r
p
n
tives of higher order, Mititelu and Trean t¸˘a 13 de- • R , R , R , R and R to be finite dimen-
veloped efficiency conditions for multi-objective sional standard Euclidean spaces;
s
fractional control problem. In, 14 multi-time con- • Ω is a compact subset in R ;
trol issues of various functionals regulated by • t=(t κ ), κ=1, s, where t ∈Ω is the multi-
higher order Lagrangians are investigated. Re- ple time argument;
s
q
cently, Jayswal et al. 15 established robust dual- • σ =(σ i ) :Ω⊆R →R are the state func-
q
ity for uncertain multi-dimensional vector control tions in the space K ⊆R , having contin-
problem involving convex integrals. Further, the uous first order partial derivatives;
s
efficiency conditions for multi-objective fractional • ω =(ω j ) :Ω⊆R →R r are control func-
r
control problem in the face of data uncertainty tions which are continuous in Υ⊆R ;
¯
ˆ
has been developed by Ritu et al. 16 The second or- • Λ=(t, σ(t), ω(t)), Λ=(t, ¯σ(t), ¯ω(t)), Λ=
der subdifferentials have been introduced in 17 and (t, ˆσ(t), ˆω(t));
the significant robust duality results for multi- • dν =dt 1 dt 2 ...dt s , t 0 =(t 01 , t 02 , ..., t 0s ), t 1 =
objective control problem are demonstrated. (t 11 , t 12 , ..., t 1s );
The conceptual framework of invexity introduced • σ κ (t)= ∂σ(t) , D κ ≡ ∂ .
by Hanson 18 holds immense importance in mathe- ∂t κ ∂t κ
matical programming problems. Its development Take the following convention of vectors α, γ ∈ R n
is crucial as it eases the theory of convexity per- :
taining to duality results since each stationary
(i) α<γ ⇔α k <γ k , ∀k =1, n,
point for an invex function is a global minimum. (ii) α≦γ ⇔α k ≤γ k , ∀k =1, n,
Mond et al. 19 established duality results for vari-
(iii) α≤γ ⇔α k ≤γ k , ∀k =1, n and α k <γ k
ational problem using invexity, which were then
for some k.
extended for multi-objective problem by Nahak
and Nanda. 20 Thereafter, Mititelu 21 formulated Defining continuously differentiable functions

p
Wolfe type dual for multi-time control problem Ψ:Ω×K ×Υ×A→R , θ :J 1 Ω, R q ×Υ×
and discussed duality results based on invexity m 1 q n
hypotheses. B →R , β :J Ω, R ×Υ×M →R , where

The articles cited above served as motivation J 1 Ω, R q is the jet bundle of first order re-
and source of inspiration to present a multi- q p m
lated with Ω and R ; a ∈A⊆R , b∈B ⊆R and
dimensional vector variational control problem in- µ∈M ⊆R are parameters of uncertainty. The
n
corporating data uncertainty. To the author’s uncertain vector variational control problem in-
best knowledge, this work is new and extends the corporating multiple integrals is formulated as
earlier studied model. Table 1 highlights the re- :
search gap and novelty of this study. Sectionwise,
the paper is arranged as follows : Section 2 in- Z
cludes preliminaries, fundamental notations and (UVP) min Ψ(Λ, a)dν =
(σ,ω)
the model to be discussed in the article is in- Ω
!
troduced. In addition, the sufficient conditions Z Z
are demonstrated. Section 3 presents Wolfe type Ψ 1 (Λ, a 1 )dν, ..., Ψ p (Λ, a p )dν
Ω Ω
dual of the variational problem under considera-
tion. In order to relate the robust solutions of the subject to
primal and dual, important robust duality results θ(Λ, σ κ (t), b)≦0, b∈B
viz. weak, strong and strict converse duality the- β(Λ, σ κ (t), µ)=0, µ∈M
orems are established. These duality results have
t∈Ω, σ(t 0 )=δ 0 , σ(t 1 )=δ 1 ,
been developed under invexity assumptions of in-
R
volved functionals. Moreover, a pertinent case is where Ψ(Λ, a)dν is a functional on the domain
Ω
also procured to validate the weak duality theo- of set of functions Ψ.
rem. In Section 4, the Mond Weir type dual is The robust counterpart to the variational control
provided and the corresponding significant dual- problem (UVP) is framed as :
ity results are derived under generalized invex-
Z
ity assumptions. Eventually, Section 5 draws the (RUVP) min maxΨ(Λ, a)dν =
work to a conclusion. (σ,ω) Ω a∈A
124