On Weak Variational Control Inequalities via Interval Analysis

: This paper deals with the connections between the interval-valued optimal control problem and the associated weak variational control inequality. More precisely, by considering the (strictly) LU-convexity and path independence properties of the involved curvilinear integral functionals, we establish a result on the existence of LU-optimal solutions for the interval-valued optimal control problem under study, and a result on the existence of solutions for the associated weak variational control inequality


Introduction
This paper deals with the interval-valued optimal control problem, which plays an important role in studying uncertainty in optimization problems.The errors due to data uncertainty or imprecision have created the need to investigate certain real-world problems.Various scientists have contributed to this research direction.Among the techniques used, interval-valued optimization is an emerging branch dealing with the uncertainty of optimization problems.In this regard, variational inequalities, first introduced by Hartman and Stampacchia [1], have been observed to be useful mathematical objects for studying optimization problems.Giannessi [2] stated remarkable results on variational inequalities and complementarity problems.Moore [3,4] suggested interval analysis to study optimization problems determined by interval-valued functions.Stefanini and Bede [5] continued by defining the generalized Hukuhara differentiability associated with intervalvalued functions.In addition, some sufficiency and duality results for interval-valued programming problems have been established by Jayswal et al. [6].Additionally, Liu [7] studied variational inequalities and optimization problems, and Treanţȃ [8] contributed to the study of vector variational inequalities and multiobjective optimization problems.Jayswal et al. [9] formulated and proved some results for multiple objective optimization problems and vector variational inequalities.Connections between the solutions of some interval-valued multiple objective optimization problems and vector variational inequalities have been derived by Zhang et al. [10].Jha et al. [11], via the associated modified problems and saddle point criteria, presented several results for interval-valued variational problems.Treanţȃ [12,13] provided important connections between the notions of optimal solution, KT-pseudoinvex point, and a saddle-point of an interval-valued functional of the Lagrange type.Recently, Treanţȃ [14][15][16] formulated optimality conditions for some multi-dimensional interval-valued variational problems.Additionally, Guo et al. [17] established optimality conditions and duality results for a class of generalized convex interval-valued optimization problems.In [18], Guo et al. provided a complete study on the properties of symmetric gH-derivative.More precisely, a necessary and sufficient condition for the symmetric gH-differentiability of interval-valued functions has been presented.Further, the authors clarified the relationship between the symmetric gH-differentiability and gH-differentiability.For more information and connected results on this topic, we direct the reader to the following research papers: Antczak [19], Hanson [1], Lodwick [20], Myskova [21], Wu [22], Zhang et al. [23], Zhang et al. [24], Jayswal and Baranwal [25], and references therein.
In this paper, we continue and improve the research mentioned above.Concretely, we establish some equivalence relations between LU-optimal solutions of the considered interval-valued optimal control problem and solutions of the associated weak variational control inequality.The present paper has several merits, as follows: (i) defining, by using the LU-order relation, the notion of LU-optimal solution for functionals determined by pathindependent curvilinear integrals, (ii) formulating original and innovative proofs associated with the main results, and (iii) providing a mathematical context determined by infinitedimensional function spaces and curvilinear integral-type functionals.These elements are new in the area of interval-valued optimal control problems.The limitations of the study: (1) the concept of "LU-convexity" is strongly used in our arguments; consequently, in our next research works, we will try to improve this aspect and replace it with a general one; (2) also, the well-posedness study of the considered problem is still an open problem that should be investigated.
The paper continues as follows: Section 2 presents notations, preliminary ingredients, and definitions on interval-valued functional of the curvilinear integral type, (strictly) convex real-valued curvilinear integral type functional, and (strictly) LU-convex intervalvalued curvilinear integral type functional; in Section 3, we state some existence results of LU-optimal solutions for the considered interval-valued control problem, and of solutions for the corresponding weak variational control inequalities; Section 4 formulates the conclusions of the paper.

Preliminaries
In this paper, we consider R m , R n and R k denote the standard Euclidean spaces, Θ is a domain in R m and C ⊂ Θ is a piecewise differentiable curve joining the following two multiple variables of evolution t 0 = t γ 0 , t 1 = t γ 1 , γ = 1, m, included in Θ, and t = (t γ ), γ = 1, m, is the current point in Θ. Denote by D γ , γ = 1, m, the operator associated with the total derivative, and let M represent the space of all piecewise smooth state functions σ : Θ → R n , with ∂σ ∂t γ (t) := σ γ (t) as the first-order partial derivative of σ with respect to t γ , γ = 1, m.Additionally, let N be the space of all continuous control functions τ : Θ → R k , and K denotes the set of all closed and bounded intervals in R.
the lower bounds, and a U , b U indicate the upper bounds of A and B, respectively.The interval operations are performed as follows: In addition, the following conventions for any two intervals A, B ∈ K will be used: Next, on the line of Treanţȃ [12,13,15], we introduce the interval-valued functionals determined by curvilinear integrals, (strictly) convexity associated with real-valued curvi-linear integral type functionals, and (strictly) LU-convexity for interval-valued curvilinear integral type functionals.Additionally, we define the path independence of the involved curvilinear integrals.

Definition 1. A curvilinear integral type functional
where are real-valued curvilinear integral type functionals, with Definition 2. An interval-valued curvilinear integral type functional Ψ is called path-independent if the real-valued curvilinear integral type functionals Ψ L and Ψ U are path-independent, that is, the following equalities Definition 3. A real-valued curvilinear integral type functional If the above inequality is valid for each (σ 0 , τ 0 ) ∈ M × N, then the real-valued curvilinear integral type functional H : M × N → R is named (strictly) convex on M × N.

Main Results
This section, via solutions of the weak variational control inequality (WVC), formulates and proves an existence result for the LU-optimal solutions of the interval-valued optimal control problem (IVP).
For a better understanding of the mathematical context associated with the control problems under study, we state some recent auxiliary results provided by Tareq [26] (see Theorems 1-4).
The next result, by considering the solution associated with the variational control inequality (VC), provides a sufficient condition for a pair (σ 0 , τ 0 ) ∈ Ω to become an LU-optimal solution to (IVP).
The following theorem represents the reciprocal of the previous result.
Theorem 2. Consider (σ 0 , τ 0 ) ∈ Ω is an LU-optimal solution to (IVP) and In the following, by using the solution associated with the split variational control inequality (SVC), the next theorem provides a sufficient condition for a pair (σ 0 , τ 0 ) ∈ Ω to become a strong LU-optimal solution to (IVP).
In the following, we state and prove the main results of the present paper.The next result, by considering the solution associated with the variational control inequality (VC), provides a sufficient condition for a pair (σ 0 , τ 0 ) ∈ Ω to become an LU-optimal solution to (IVP).
The following theorem represents the converse result of the previous one.
Application.Let us extremize the interval-valued curvilinear integral cost functional given by (IVP1) min The feasible solution set of (IVP1) is given by and, taking into account the notations used in this paper, we have Let us assume that we are only interested in the affine state and control real-valued functions and t ∈ are (strictly) convex at (σ 0 , τ 0 ) = 1  3 t 1 + t 2 + 6, 8 3 .Thus, the interval-valued functional is strictly LU-convex at (σ 0 , τ 0 ) = , for all (σ, τ) = (σ 0 , τ 0 ) ∈ Ω, then it is a solution to variational control inequality (WVC).By using Theorem 6, it follows that (σ 0 , τ 0 ) is an LU-optimal solution of (IVP1).Indeed, it can be verified that the inequality is satisfied.

Conclusions
By considering the solutions of a new class of weak variational control inequalities, denoted by (WVC) , in this paper we have established new results on the existence of LU-optimal solutions for the associated interval-valued optimal control problem, denoted by (IVP).The derived results have been generated by the LU-convexity property and path independence of the involved functionals determined by curvilinear integrals.As further developments of the results stated in this paper, we mention the formulation of the associated duality theory, well-posedness results, and saddle-point optimality criteria.