Characterization Results of Solution Sets Associated with Multiple-Objective Fractional Optimal Control Problems

: This paper investigates some duality results of a mixed type for a class of multiple objective fractional optimal control problems. More precisely, by considering the Wolfe-and Mond–Weir-type dualities, we formulate a robust mixed-type dual problem and, under suitable convexity assumptions of the involved functionals, we establish some equivalence results between the solution sets of the considered models. Essentially, we investigate robust weak, robust strong, and robust strict converse-type duality results. To the best of the authors’ knowledge, robust duality results for such problems are new in the specialized literature


Introduction
In the research paper [1], Hanson applied the duality theory from mathematical programming to a new class of functions named invex functions.In this regard, Craven and Glover [2] established that invex functions are characterized as functions where the stationary/critical points become global minima.As a generalization of the work of Mond and Hanson [3], Mond and Smart [4] formulated some sufficiency and duality results in scalar variational control problems.Also, duality theorems have been stated for linear fractional variational problems by Aggarwal et al. [5].Mukherjee and Rao [6] presented mixed dual problems associated with multiobjective variational problems and established dualities under ρ-invexity hypotheses.Historically, multiobjective variational problems governed by equality and inequality restrictions have been of great importance and interest (including conditions of optimality, dual problems, and various areas of applicability), and we have only failed to consider the following researchers: Zhian and Qingkai [7], Zalmai [8], Mititelu [9], Hachimi and Aghezzaf [10], Chen [11], Kim and Kim [12], and Nahak and Nanda [13].Gulati et al. [14] studied optimality conditions and the associated duality for a class of multiobjective control problems.Arana-Jiménez et al. [15] investigated a necessary and sufficient condition for duality in some multiobjective variational problems.Khazafi et al. [16] discussed sufficiency and duality for multiobjective control problems under generalized (B, ρ)-type I functions.Zhang et al. [17] analyzed the sufficiency and duality for multiobjective variational control problems under G-invexity assumptions.
Recently, Das et al. [18] provided sufficient KKT-type second-order optimality conditions for a class of set-valued fractional minimax problems.Under contingent epi-derivative and generalized second-order cone convexity hypotheses, the authors formulated some duals for the considered problem.Khan and Al-Solamy [19] discussed, for a non-differentiable minimax fractional programming problem, the optimality condition for an optimal solution and a dual model.Mititelu and Treanţȃ [20] formulated some efficiency conditions in vector control problems generated by multiple integrals.Sharma [21] presented a higher-order duality for variational control problems.Oliveira and Silva [22] studied sufficient optimality conditions for some multiobjective control problems.In the last decade, Treanţȃ [23] and his collaborators investigated some classes of multi-dimensional multiobjective variational control problems.In this direction, Treanţȃ and Mititelu [24] formulated duality results in multi-dimensional vector fractional control problems by considering (ρ, b)-quasiinvexity assumptions.
Most optimization problems that occur in practice have several objective functions that must be optimized simultaneously.This type of problem, of considerable interest, includes various branches of mathematical sciences, design engineering, and game theory.Because of the increasing complexity of the environment, the initial data often suffer from inaccuracy.For example, in the modeling of many processes in industry and economy in order to make decisions, it is not always possible to have complete information about the parameters and variables involved.Therefore, an adequate uncertainty framework is necessary to formulate the model, and new methods have to be adapted or developed to provide optimal or efficient solutions in a certain sense.In order to tackle the uncertainty in an optimization problem, robust and interval-valued optimization represents some growing branches of applied mathematics and may provide an alternative choice for considering the uncertainty.Over time, several researchers and mathematicians have been interested to obtain many solution procedures in interval analysis and robust control.In order to formulate necessary and sufficient optimality conditions and duality theorems for different types of robust and interval-valued variational problems, various approaches have been proposed.
In this paper, under the motivation of the above-mentioned research papers and by considering suitable convexity hypotheses for the involved integral-like functionals, a mixed-type dual model is developed for the multiple objective fractional optimal control problem determined by multiple integral functionals defined in Ritu et al. [25].More specifically, this paper is essentially a natural continuation of the studies stated in Mititelu and Treanţȃ [20] and Ritu et al. [25].In this regard, by using the robust necessary efficiency conditions established in Ritu et al. [25], we investigate robust weak, robust strong, and robust strict converse-type duality results.The limitations of the existing works and the main credits of this paper are the following: (i) the presence of mixed constraints involving partial derivatives, (ii) the presence of the uncertainty data both in the cost functionals but also in the constraint functionals, and (iii) the combination of parametric and robust approaches to study the considered class of problems.

Preliminaries
Let us start with the standard Euclidean spaces R p , R q , R r , and R n , and a compact set in R p , denoted by S. Define the multi-time variable t = (t α ), α = 1, p, such that t ∈ S. Also, consider the space (denoted by A) of state functions with continuous firstorder partial derivatives as λ = (λ i ) : S → R q and consider the continuous control functions in the space B as π = (π j ) : S → R r .Additionally, we use the abbreviations: . Next, we formulate the rules that are considered for any two points a, b ∈ R n : n and a s < b s for some s.
The robust multiple objective fractional optimal control problem is formulated (see, also, Mititelu and Treanţȃ [20], Treanţȃ and Mititelu [24], and Ritu et al. [25]) as: where are C 1 -class functionals (almost everywhere); the jet bundle of first-order associated with S and R q is stated as J 1 S, R q ; also, we assume S z (Υ, γ )dt > 0, = 1, p, and ς = (ς ), γ = (γ ), σ = (σ l ), and δ = (δ s ) represent the uncertainty parameters of the com- The robust counterpart for (P ) is introduced as follows: (RP ) min The feasible solution set of (RP ), known as the robust feasible solution set for (P ), is denoted as follows: Next, we consider the following parametric scalar optimal control problem corresponding to (P ) as follows: The robust counterpart associated to (P w ) is given by: Definition 1.A feasible pair ( λ, π) is named as a robust weak optimal solution for (P w ) if: for all feasible pairs (λ, π).