On a Dual Pair of Multiobjective Interval-Valued Variational Control Problems

: In this paper, by using the new concept of ( (cid:36) , ψ , ω ) -quasiinvexity associated with interval-valued path-independent curvilinear integral functionals, we establish some duality results for a new class of multiobjective variational control problems with interval-valued components. More concretely, we formulate and prove weak, strong, and converse duality theorems under ( (cid:36) , ψ , ω ) quasiinvexity hypotheses for the considered class of optimization problems.

The present paper, motivated by the aforementioned research works and practical reasons, establishes weak, strong, and converse Mond-Weir duality results for a new class of multiobjective optimization problems with interval-valued components governed by path-independent curvilinear integral functionals. The main novelty elements of this work are represented by the necessary LU-efficiency conditions derived using some recent research papers of the author; the notion of ( , ψ, ω)-quasiinvexity associated with intervalvalued path-independent curvilinear integral functionals; and the presence of a partition associated with a set of indices used for the inequality-type constraints.
In the following, we organize the paper as follows: in Section 2, we present notations, preliminary mathematical tools, and the problem formulation we are going to study; in Section 3, we establish the main results of this paper-namely, weak, strong, and con-verse Mond-Weir dualities are formulated and proved for the new class of multiobjective optimization problems; finally, in Section 4, we conclude the paper.

Preliminaries
Throughout this paper, we consider Ω as a compact domain in the Euclidean real space R m and denote by t = (t α ), α = 1, m, u = (u j ), j ∈ 1, k, and a = (a i ), i = 1, n, the points in Ω, R k , and R n , respectively. Further, consider that Ω ⊃ γ is a piecewise smooth curve joining the different points t 1 = t 1 1 , . . . , t m 1 , t 2 = t 1 2 , . . . , t m 2 in Ω. Now, we define the following continuously differentiable functions and we accept that the following Lagrange densities satisfy the closeness conditions (complete integrability conditions) where D η is the total derivative operator. Denote by A the space of all piecewise smooth state functions a : Ω → R n , and by U the space of all piecewise continuous control functions u : Ω → R k , endowed with the induced norm. Additionally, in this paper, for any two p-tuples l = l 1 , . . . , l p , c = c 1 , . . . , c p in R p , we use the following partial ordering l = c ⇔ l r = c r , l ≤ c ⇔ l r ≤ c r , l < c ⇔ l r < c r , l c ⇔ l ≤ c, l = c, r = 1, p.
Let K be the set of all closed and bounded real intervals. Denote by I = [i L , i U ] a closed and bounded real interval, where i L and i U indicate the lower and upper bounds of I, respectively. Throughout this paper, the interval operations are performed as follows: Definition 1 (Treanţȃ [9]). Let I, J ∈ K. We write I ≤ J if and only if i L ≤ j L and i U ≤ j U . Further, we write I < J if and only if i L < j L and i U < j U .
Definition 2 (Treanţȃ [9]). A function f : Ω × R n × R k → K, defined by where f L (t, a(t), u(t)) and f U (t, a(t), u(t)) are real-valued functions and satisfy the condition f L (t, a(t), u(t)) ≤ f U (t, a(t), u(t)); t ∈ Ω is said to be an interval-valued function.
For α = 1, m, we consider the following vector continuously differentiable functions with interval-valued components (closed 1-forms) which, for r = 1, p, generate the interval-valued path-independent curvilinear integral functionals (see Einstein summation): Further, in accordance with Treanţȃ and Mititelu [17,26], following Treanţȃ [9], in order to formulate and prove the main results included in this paper, we introduce the concept of ( , ψ, ω)-quasiinvexity associated with an interval-valued path-independent curvilinear integral functional. For α = 1, m, we consider an interval-valued continuously differentiable function: where a ς (t) := ∂a ∂t ς (t) and for a ∈ A and u ∈ U , we introduce the following interval-valued path-independent curvilinear integral functional: Furthermore, let be a real number, ψ : A × U × A × U → [0, ∞) be a positive functional, and ω (a, u), (a 0 , u 0 ) be a real-valued function on (A × U ) 2 .
Next, for α = 1, m, we consider the vector continuously differentiable function with interval-valued components

Definition 4. The vector path-independent curvilinear integral functional with interval-valued components
is said to be ( , ψ, ω)-quasiinvex (strictly ( , ψ, ω)-quasiinvex) at a 0 , u 0 ∈ A × U with respect to ν and τ, if each interval-valued component of the vector is ( , ψ, ω)-quasiinvex (strictly ( , ψ, ω)quasiinvex) at a 0 , u 0 ∈ A × U with respect to ν and τ. Now, we are in a position to formulate the following new class of multiobjective fractional variational control problems with interval-valued components, called the Primal Problem (in short, PP): where, for r = 1, p, we have denoted and it is assumed that G r (a, u) > [0, 0], ∀(a, u) ∈ A × U .

Definition 5.
A feasible solution (a 0 , u 0 ) ∈ D in (PP) is called an LU-efficient solution if there is no other (a, u) ∈ D such that K(a, u) K(a 0 , u 0 ).
Theorem 1 (Weak Duality). Let (a, u) ∈ D be a feasible solution of the multiobjective variational control problem with interval-valued components (PP) and (b, w, θ, φ, ϕ) ∈ be a feasible solution of the multiobjective variational control problem with interval-valued components (DP). Further, assume that the following conditions are fulfilled: (a) Each functional with respect to ν and τ, or, equivalently, each interval-valued path-independent curvilinear integral functional is ( ϑ,2 , ψ, ω)-quasiinvex at (b, w) with respect to ν and τ.
Then, the infimum of (PP) is greater than or equal to the supremum of (DP).
By direct computation, we get but, applying the condition ν(t 1 ) = ν(t 2 ) = 0 and the result "A total divergence is equal to a total derivative.", we get Consequently, and applying the hypothesis (d) and ω 2 ((a, u), (b, w)) ≥ 0, we get a contradiction. Therefore, the infimum of (PP) is greater than or equal to the supremum of (DP).
The next, according to Treanţȃ and Mititelu [17], establishes a strong duality between the two considered multiobjective optimization problems with interval-valued components.
The following theorem formulates a converse duality result associated with the considered multiobjective optimization problems with interval-valued components.

Remark 1.
If, for r = 1, p and (a, u) ∈ A × U , each interval-valued path-independent curvilinear integral functional γ g r α (t, a(t), u(t))dt α is equal to 1, then we obtain primal and dual multiobjective nonfractional variational control problems with interval-valued components and the corresponding Mond-Weir duality results.

Conclusions
In this paper, we have studied a dual pair of multiobjective variational control problems with interval-valued components. More precisely, based on the new notion of ( , ψ, ω)quasiinvexity associated with interval-valued path-independent curvilinear integral functionals, we have established weak, strong, and converse duality results for the considered class of optimization problems. Moreover, by considering the physical meaning of the curvilinear integrals (mechanical work) and the importance of Interval Analysis in the applied sciences and engineering, this research work can be seen as a starting point for further investigations.