Connections between Some Extremization Problems and Associated Inequalities

Abstract

In this paper, we investigate and characterize a family of optimization problems introduced by interval-valued functionals that are determined by curvilinear integrals. To this end, we first state the path independence and (strictly) $LU$ convexity properties of the considered functionals. Thereafter, we formulate the corresponding controlled variational inequalities. The main results of this paper provide some connections for the above-mentioned variational models. Since the objective functionals have a physical importance, an illustrative application is considered and studied by using the theoretical elements obtained in this study.

1. Introduction

The errors provided by imprecision involve the need to study various real-life problems by considering interval or fuzzy analysis. Many scientists have worked in the previous research area. Interval analysis represents a branch addressing the uncertainty in different optimization and variational problems. In this sense, the variational inequalities, defined and studied by Hartman and Stampacchia [1], are useful mathematical ingredients for investigating several variational or control problems. Very interesting results on variational inequalities and complementarity problems have been established by Giannessi [2]. Also, Moore [3,4] suggested analysis of intervals as an important tool for investigating optimization problems that have objective functions with interval values. Later, Stefanini and Bede [5] worked in this research direction by introducing the generalized Hukuhara-type differentiability for interval differential equations and interval-valued functions. Also, various duality and sufficiency results associated with some variational models with interval values were stated by Jayswal et al. [6]. Moreover, Liu [7] analyzed variational optimization problems and variational inequalities, and Treanţă [8] studied vector-type variational inequalities and multiple objective variational optimization problems. Moreover, Jayswal et al. [9] established various results for vector variational problems and inequalities. In addition, Zhang et al. [10] obtained the relationships between the solutions for some multiple objective variational models with interval values and the corresponding vector variational inequalities. Jha et al. [11] stated some results for interval-valued problems by introducing the associated modified models and optimality criteria of the saddle point type. Also, Treanţă [12] formulated some important links between the notions of an optimal solution, the KT point, and a saddle point associated with a Lagrange-type functional with interval values. In addition, Treanţă [13] presented some optimality conditions in interval-valued variational models. For more information on this subject, the reader can consult Lodwick [14], Hanson [1], and Antczak [15]. Alefeld and Mayer [16] gave an overview regarding the applications of interval arithmetic. Ishibuchi and Tanaka [17] formulated the ordering relation associated with compact intervals determined by the center and half-width of the considered intervals. Also, they established some solution concepts for the multi-objective extremization problems with interval values. Further, Wu [18], by using LU and CW partial ordering on the class of all real compact intervals, provided two solution concepts in the studied family of interval-valued problems.

In recent years, convex sets and convex functions have been converted to a variety of convexities, such as quasi-convexity [19], harmonic convexity [20], and strong convexity [21]. Moreover, we can mention the Schur-type convexity [22] and p-convexity [23]. The interest in convexity associated with various problems is remarkable. Thus, as applications of convex functions, several researchers have investigated multiple equalities and inequalities. Some important results in this regard include the Hardy-type inequality [24] and Gagliardo–Nirenberg inequality [25]. Also, we can include here the Olsen-type inequality [26], the Ostrowski-type inequality [27], and the Hermite–Hadamard inequality [28].

Motivated by the research works given above and having in mind the valuable papers of Myskova [29], Wu [30], and Zhang et al. [31], in this paper, a family of optimization problems introduced by interval-valued functionals determined by curvilinear integrals is investigated and characterized. To this end, the path independence and (strictly) LU convexity properties of the considered functionals are first stated. Thereafter, the corresponding controlled variational inequalities are formulated. The main results of this paper provide some connections for the above-mentioned variational models. Since the objective functionals have physical importance, an illustrative application is considered and studied using the theoretical elements obtained in this study. Work that is strongly connected to the current paper includes the study of Treanţă and Guo [32], where several relations among the solutions of some (weak) vector control variational inequalities and efficient solutions (weak, proper) associated with certain multiple-objective control variational problems are formulated and proved by using generalized convexity and differentiability (of Fréchet-type) hypotheses of the involved functionals. Moreover, an important ingredient for proving the main results derived in the paper is the notion of an invex set related to some given functions. In the present study, we relax the above-mentioned assumptions by imposing only (strictly) $LU$ convexity, without invex set hypotheses. A new element is the presence of curvilinear integral functionals instead of multiple integral functionals. Also, a strong motivation in support of this study, in the sense of potential impact on the level of knowledge, is given by the technique used here to transform an optimization problem into a problem with inequalities, which is much simpler to study, both involving curvilinear integral type functionals (an element of novelty in specialized literature and of great importance in applied sciences). In this regard, more precisely, the proper motivation for studying such issues is as follows.

Neumann boundary control: Find a control function ${\displaystyle x\in L^2(\Gamma )}$ that minimizes the cost functional

$${\displaystyle J\left(x(·)\right)=\frac{1}{2}{\int }_{\Omega }{(y\left(t\right)-y_d)}^{2}d{t}^1\cdots d{t}^m+\frac{\beta }{2}{\int }_{\Gamma }{x}^2\left(t\right)ds,}$$

where ${\displaystyle (y,x)}$ satisfies ${\displaystyle -\Delta y+y=f}$ in ${\displaystyle \Omega }$ and ${\displaystyle \frac{\partial y}{\partial n}=x}$ on ${\displaystyle \Gamma }$; the function f is a given source term; the function x is a control variable; and ${\displaystyle \Omega }$ is a bounded domain in ${\displaystyle R^m}$ with a boundary ${\displaystyle \Gamma }$ of class ${\displaystyle C^2}$. Since the term ${\displaystyle \frac{\beta }{2}{\int }_{\Gamma }{x}^2\left(t\right)ds}$ (see ${\displaystyle \beta >0}$) is proportional to the consumed energy, the minimizing of ${\displaystyle J}$ is a compromise between the energy consumption and finding x so that the distribution y is close to the desired profile $y_d$. We say that the control function is a boundary control because the control acts on ${\displaystyle \Gamma }$. The above-mentioned problem can be successfully studied by applying the techniques established in Treanţă and Guo [32] (for the first term given in $J\left(x\right(·\left)\right)$) and in this paper.

Next, the paper continues with a review of multidimensional variational control models. Some notations, preliminary concepts, and the problem formulation are presented in Section 3. In Section 4, we state the characterization theorems for the solutions associated with the mentioned variational models. Section 5 includes the conclusions and provides further research developments.

2. On Multidimensional Variational Control Models

Multidimensional variational control models have applications in different branches of mathematics, engineering, and economics, such as shape optimization in medicine and fluid mechanics, structural optimization, material inversion in geophysics, and optimal control of processes (see Jayswal et al. [33] for a detailed description). Thus, partial differential equation-constrained multidimensional variational control models have been given considerable importance in recent years.

A general formulation of such an extremization model is given below:

$$\begin{array}{cc}\hfill \left(\mathrm{Problem}\right)\underset{\left(p\right(·),q(·\left)\right)}{min}& {\int }_{\Omega }\left(f_1(t,p\left(t\right),q\left(t\right)),\cdots ,f_k(t,p\left(t\right),q\left(t\right))\right)dt\hfill \\ \hfill \mathrm{subject} \mathrm{to}& g_i(t,p\left(t\right),q\left(t\right))\leqq 0,\hfill \\ & {\displaystyle \frac{\partial p^j}{\partial t^{\alpha }}}=h_j^{\alpha }(t,p\left(t\right),q\left(t\right)),\hfill \\ & p\left(t_0\right)=p_0,p\left(t_1\right)=p_1,\hfill \end{array}$$

where $f=\left(f_l\right):\Omega \times P\times Q\to {\mathbb{R}}^k,l=\overline{1,k}$ (objective or cost functional) and where the inequality constraint $g=\left(g_i\right):\Omega \times P\times Q\to {\mathbb{R}}^u,i=\overline{1,u}$ and the equality constraint $h=\left(h_j^{\alpha }\right):\Omega \times P\times Q\to {\mathbb{R}}^{rm},j=\overline{1,m},\alpha =\overline{1,r}$ are considered to be of $C^{\infty }$-class. In addition, we consider $h^{\alpha }$ to satisfy $D_{\beta }{h}^{\alpha }=D_{\alpha }{h}^{\beta },\alpha ,\beta =\overline{1,r},\alpha \ne \beta ,$ where ${\displaystyle D_{\alpha }:=\frac{\partial }{\partial t^{\alpha }}}$.

A pair $(\overline{p}\left(t\right),\overline{q}\left(t\right))\in P\times Q$ is said to be a feasible point or feasible solution to (Problem) if all the considered constraint functionals are satisfied. The feasible set of solutions can be written as follows:

$$\begin{array}{cc}\hfill DOM=\{(p\left(t\right),q\left(t\right))\in P\times Q:g_i(t,p\left(t\right),q\left(t\right))\leqq 0,h_j^{\alpha }(t,p\left(t\right),& q\left(t\right))={\displaystyle \frac{\partial p^j}{\partial t^{\alpha }}},\hfill \\ & p\left(t_0\right)=p_0,p\left(t_1\right)=p_1\}.\hfill \end{array}$$

In multi-objective optimization, a unique feasible solution that optimizes all the objectives, in general, does not exist. Therefore, the concepts of a weak Pareto solution and a Pareto solution play a crucial role in solving such optimization problems.

Definition 1. 

A feasible solution $(\overline{p}\left(t\right),\overline{q}\left(t\right))\in DOM$ is called an efficient solution to (Problem) if there does not exist $\left(p\right(t),q(t\left)\right)\in DOM$ satisfying

$$\begin{array}{c}\hfill {\int }_{\Omega }f(t,p\left(t\right),q\left(t\right))dt\leqq {\int }_{\Omega }f(t,\overline{p}\left(t\right),\overline{q}\left(t\right))dt,\end{array}$$

with at least one strict inequality.

Definition 2. 

A feasible solution $(\overline{p}\left(t\right),\overline{q}\left(t\right))\in DOM$ is called a weak efficient solution to (Problem) if there does not exist $\left(p\right(t),q(t\left)\right)\in DOM$ satisfying

$$\begin{array}{c}\hfill {\int }_{\Omega }f(t,p\left(t\right),q\left(t\right))dt<{\int }_{\Omega }f(t,\overline{p}\left(t\right),\overline{q}\left(t\right))dt.\end{array}$$

In general, the multidimensional functional is classified as follows:

I. Curvilinear integral cost functional

$$\begin{array}{cc}\hfill \left(\mathbf{P}\mathbf{1}\right)& \underset{\left(p\right(·),q(·\left)\right)}{min}{\int }_{\Gamma }{f}_{\alpha }(t,p\left(t\right),q\left(t\right))d{t}^{\alpha }\hfill \\ & =\left({\int }_{\Gamma }{f}_{\alpha }^1(t,p\left(t\right),q\left(t\right))d{t}^{\alpha },\dots ,{\int }_{\Gamma }{f}_{\alpha }^k(t,p\left(t\right),q\left(t\right))d{t}^{\alpha }\right),\hfill \end{array}$$

where $f_{\alpha }^l:\Omega \times P\times Q\to \mathbb{R},\alpha =\overline{1,r},l=\overline{1,k}$, are $C^1$-class functionals, and $\Gamma$ is a curve (piecewise smooth) included in $\Omega$ that joins $t_0$ and $t_1$.

II. Multiple integral cost functional

$$\begin{array}{cc}\hfill \left(\mathbf{P}\mathbf{2}\right)\underset{\left(p\right(·),q(·\left)\right)}{min}& {\int }_{\Omega }f(t,p\left(t\right),q\left(t\right))dt\hfill \\ & =\left({\int }_{\Omega }{f}_1(t,p\left(t\right),q\left(t\right))dt,\dots ,{\int }_{\Omega }{f}_k(t,p\left(t\right),q\left(t\right))dt\right),\hfill \end{array}$$

where $f:\Omega \times P\times Q\to {\mathbb{R}}^k$ is a vector-valued $C^1$-class functional.

Necessary and Sufficient Conditions of Efficiency

Necessary and sufficient conditions of optimality are based on differential calculus, which plays a crucial role in generating the optimal solutions in the considered optimization problems.

Theorem 1 

(Fritz John-type necessary conditions of efficiency). If the pair $(\overline{p}\left(t\right),\overline{q}\left(t\right))\in DOM$ is an efficient solution to (Problem), then there exist ${\theta }_l\in \mathbb{R},l=\overline{1,k},{\mu }_i\left(t\right)\in {\mathbb{R}}_{+},i=\overline{1,u},$ and ${\gamma }_{\alpha }^j\left(t\right)\in \mathbb{R},\alpha =\overline{1,r},j=\overline{1,m}$, satisfying (for all $t\in \Omega ,$ except at discontinuities)

$$\begin{array}{cc}\hfill {\theta }_l{\displaystyle \frac{\partial f_l}{\partial p^{\varsigma }}}& (t,\overline{p}\left(t\right),\overline{q}\left(t\right))+{\mu }_i\left(t\right){\displaystyle \frac{\partial g_i}{\partial p^{\varsigma }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))\hfill \\ \hfill & +{\gamma }_{\alpha }^j\left(t\right){\displaystyle \frac{\partial h_j^{\alpha }}{\partial p^{\varsigma }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))+{\displaystyle \frac{\partial {\gamma }_{\alpha }^j}{\partial t^{\alpha }}}\left(t\right)=0,\varsigma =\overline{1,m},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\theta }_l{\displaystyle \frac{\partial f_l}{\partial q^{\tau }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))& +{\mu }_i\left(t\right){\displaystyle \frac{\partial g_i}{\partial q^{\tau }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))\hfill \\ & +{\gamma }_{\alpha }^j\left(t\right){\displaystyle \frac{\partial h_j^{\alpha }}{\partial q^{\tau }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))=0,\tau =\overline{1,n},\hfill \end{array}$$

$$\begin{array}{c}\hfill {\mu }_i\left(t\right)g_i(t,\overline{p}\left(t\right),\overline{q}\left(t\right))=0,\theta \ge 0,\mu \left(t\right)\ge 0.\end{array}$$

To ensure that $\theta >0$, some restrictions are imposed on the constraint functions, and these restrictions are known as constraint qualifications. In this regard, we formulate the following result.

Theorem 2 

(Kuhn–Tucker-type necessary conditions of efficiency). If the pair $(\overline{p}\left(t\right),\overline{q}\left(t\right))\in DOM$ is an efficient solution to (Problem) and the constraint conditions hold, then there exist ${\theta }_l\in \mathbb{R},l=\overline{1,k},{\mu }_i\left(t\right)\in \mathbb{R},i=\overline{1,u},{\gamma }_{\alpha }^j\left(t\right)\in \mathbb{R},\alpha =\overline{1,r},j=\overline{1,m}$ satisfying (for all $t\in \Omega ,$ except at discontinuities)

$${\theta }_l{\displaystyle \frac{\partial f_l}{\partial p^{\varsigma }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))+{\mu }_i\left(t\right){\displaystyle \frac{\partial g_i}{\partial p^{\varsigma }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))+{\gamma }_{\alpha }^j\left(t\right){\displaystyle \frac{\partial h_j^{\alpha }}{\partial p^{\varsigma }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))+{\displaystyle \frac{\partial {\gamma }_{\alpha }^j}{\partial t^{\alpha }}}\left(t\right)=0,$$

$$\varsigma =\overline{1,m},$$

$${\theta }_l{\displaystyle \frac{\partial f_l}{\partial q^{\tau }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))+{\mu }_i\left(t\right){\displaystyle \frac{\partial g_i}{\partial q^{\tau }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))+{\gamma }_{\alpha }^j\left(t\right){\displaystyle \frac{\partial h_j^{\alpha }}{\partial q^{\tau }}}(t,\overline{p}\left(t\right),\overline{q}\left(t\right))=0,$$

$$\tau =\overline{1,n},$$

$${\mu }_i\left(t\right)g_i(t,\overline{p}\left(t\right),\overline{q}\left(t\right))=0,{\theta }_l>0,\mu \left(t\right)\ge 0.$$

The above-mentioned Kuhn–Tucker conditions are not sufficient for a feasible solution to be considered the optimal solution to an optimization problem. The sufficiency of these conditions is formulated in the following theorem.

Theorem 3 

(Kuhn–Tucker sufficient conditions of efficiency). If the pair $(\overline{p}\left(t\right),\overline{q}\left(t\right))\in DOM$ satisfies the necessary conditions of efficiency given the above and if the involved functionals are convex at $(\overline{p}\left(t\right),\overline{q}\left(t\right))$, then the pair $(\overline{p}\left(t\right),\overline{q}\left(t\right))$ is an efficient solution to (Problem).

Definition 3. 

A functional $F(p,q)={\int }_{\Omega }f(t,p\left(t\right),p_{\sigma }\left(t\right),q\left(t\right))dt$ is named convex at $(\overline{p},\overline{q})$ if

$$\begin{array}{cc}\hfill F(p,q)-F(\overline{p},\overline{q})\geqq & {\int }_{\Omega }(p\left(t\right)-\overline{p}\left(t\right))f_p(t,\overline{p}\left(t\right),{\overline{p}}_{\sigma }\left(t\right),\overline{q}\left(t\right))dt\hfill \\ & +{\int }_{\Omega }(p_{\sigma }\left(t\right)-{\overline{p}}_{\sigma }\left(t\right))f_{p_{\sigma }}(t,\overline{p}\left(t\right),{\overline{p}}_{\sigma }\left(t\right),\overline{q}\left(t\right))dt\hfill \\ & +{\int }_{\Omega }(q\left(t\right)-\overline{q}\left(t\right))f_q(t,\overline{p}\left(t\right),{\overline{p}}_{\sigma }\left(t\right),\overline{q}\left(t\right))dt\hfill \end{array}$$

is fulfilled for $(p,q)$ on $P\times Q$.

3. Notations, Preliminary Concepts, and Problem Formulation

In the following, we consider ${\mathbb{R}}^q,{\mathbb{R}}^k$, and ${\mathbb{R}}^n$ to be Euclidean spaces. For a compact set in ${\mathbb{R}}^q$, denoted by $\mathcal{S}$, let us take $K\subset \mathcal{S}$ to be a curve that joins the fixed points $t_1=\left(t_1^{\xi }\right),t_2=$$\left(t_2^{\xi }\right),\xi =\overline{1,q}$, and $t=\left(t^{\xi }\right),\xi =\overline{1,q}$, to be an element of $\mathcal{S}$.

Consider $D_{\xi }:={\displaystyle \frac{d}{d{t}^{\xi }}}$, with $\xi =\overline{1,q}$, and denote by $\mathfrak{P}$ the family of all functions $\mathfrak{d}:\mathcal{S}\mapsto {\mathbb{R}}^n$ (piecewise differentiable functions, state functions), with ${\displaystyle \frac{\partial \mathfrak{d}}{\partial t^{\xi }}}\left(t\right):={\mathfrak{d}}_{\xi }\left(t\right)$ as the partial derivatives of the first order. Further, denote by $\mathfrak{B}$ the space of all functions $\mu :\mathcal{S}\mapsto {\mathbb{R}}^k$ (piecewise continuous functions, control functions) and by $\mathfrak{C}$ the class of all compact intervals in $\mathbb{R}$.

For $\mathfrak{Z}=\left[{\mathfrak{z}}^L,{\mathfrak{z}}^U\right],\mathfrak{Y}=\left[{\mathfrak{y}}^L,{\mathfrak{y}}^U\right]\in \mathfrak{C}$, the interval operations are considered as follows:

• $\mathfrak{Z}=\mathfrak{Y}⟺{\mathfrak{z}}^L={\mathfrak{y}}^L,{\mathfrak{z}}^U={\mathfrak{y}}^U$,
• if ${\mathfrak{z}}^L={\mathfrak{z}}^U=\mathfrak{z}$, then $\mathfrak{Z}=[\mathfrak{z},\mathfrak{z}]=\mathfrak{z}$,
• $\mathfrak{Z}+\mathfrak{Y}=\left[{\mathfrak{z}}^L+{\mathfrak{y}}^L,{\mathfrak{z}}^U+{\mathfrak{y}}^U\right]$,
• $-\mathfrak{Z}=-\left[{\mathfrak{z}}^L,{\mathfrak{z}}^U\right]=\left[-{\mathfrak{z}}^U,-{\mathfrak{z}}^L\right]$,
• $\mathfrak{Z}-\mathfrak{Y}=\left[{\mathfrak{z}}^L-{\mathfrak{y}}^U,{\mathfrak{z}}^U-{\mathfrak{y}}^L\right]$,
• $\kappa +\mathfrak{Z}=\left[\kappa +{\mathfrak{z}}^L,\kappa +{\mathfrak{z}}^U\right],\kappa \in \mathbb{R}$,
• $\kappa \mathfrak{Z}=\left[\kappa {\mathfrak{z}}^L,\kappa {\mathfrak{z}}^U\right],\kappa \in \mathbb{R},\kappa \ge 0$,
• $\kappa \mathfrak{Z}=\left[\kappa {\mathfrak{z}}^U,\kappa {\mathfrak{z}}^L\right],\kappa \in \mathbb{R},\kappa <0$.

In addition, for $\mathfrak{Z},\mathfrak{Y}\in \mathfrak{C}$, we consider the $LU$ order:

• $\mathfrak{Z}{⪯}_{LU}\mathfrak{Y}⟺{\mathfrak{z}}^L\le {\mathfrak{y}}^L,{\mathfrak{z}}^U\le {\mathfrak{y}}^U$,
• $\mathfrak{Z}{\prec }_{LU}\mathfrak{Y}⟺\mathfrak{Z}{⪯}_{LU}\mathfrak{Y},\mathfrak{Z}\ne \mathfrak{Y}$.

In the following, along the same lines as Ciontescu and Treanţă [34] and Saeed and Treanţă [35], we state some definitions that are useful for establishing the main theorems of this study.

Definition 4 

(Treanţă and Saeed [36]). We say that the following functional determined by a curvilinear integral

$${\displaystyle Q:\mathfrak{P}\times \mathfrak{B}\mapsto \mathfrak{C},Q(\mathfrak{d},\mu )={\int }_K{\theta }_{\nu }\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }}$$

is called a functional with interval values or an interval-valued functional if it can be written as

$${\displaystyle Q(\mathfrak{d},\mu )=\left[{\int }_K{\theta }_{\nu }^L\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu },{\int }_K{\theta }_{\nu }^U\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }\right],}$$

where

$$\begin{gathered} {\displaystyle Q^L(\mathfrak{d},\mu ),Q^U(\mathfrak{d},\mu ):\mathfrak{P}\times \mathfrak{B}\mapsto \mathbb{R},} \\ {\displaystyle Q^L(\mathfrak{d},\mu ):={\int }_K{\theta }_{\nu }^L\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu },} \\ {\displaystyle Q^U(\mathfrak{d},\mu ):={\int }_K{\theta }_{\nu }^U\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu },} \end{gathered}$$

are real-valued functionals (or functionals with real values), and where ${\theta }_{\nu }:\mathcal{S}\times \mathfrak{P}\times \mathfrak{P}\times \mathfrak{B}\to \mathfrak{C},{\theta }_{\nu }=[{\theta }_{\nu }^L,{\theta }_{\nu }^U],\nu =\overline{1,q}$, satisfies the following inequality:

$${\displaystyle {\int }_K{\theta }_{\nu }^L\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }\le {\int }_K{\theta }_{\nu }^U\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }.}$$

Definition 5 

(Treanţă and Saeed [36]). An interval-valued functional Q is said to be independent of the path if the functionals with real values $Q^L$ and $Q^U$ are independent of the path or, equivalently, the conditions

$$D_{\xi }{\theta }_{\nu }^L=D_{\nu }{\theta }_{\xi }^L$$

and

$$D_{\xi }{\theta }_{\nu }^U=D_{\nu }{\theta }_{\xi }^U$$

are fulfilled for $\nu \ne \xi$.

Definition 6 

(Treanţă and Saeed [36]). We say that a real-valued functional determined by a curvilinear integral

$${\displaystyle J:\mathfrak{P}\times \mathfrak{B}\mapsto \mathbb{R},J(\mathfrak{d},\mu )={\int }_K{h}_{\nu }\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }}$$

is called (strictly) convex at $({\mathfrak{d}}^0,{\mu }^0)\in \mathfrak{P}\times \mathfrak{B}$ if the following inequality

$$\begin{gathered} {\displaystyle J(\mathfrak{d},\mu )-J({\mathfrak{d}}^0,{\mu }^0)\ge (>){\int }_K\frac{\partial h_{\nu }}{\partial \mathfrak{d}}\left(t,{\mathfrak{d}}^0\left(t\right),{\mathfrak{d}}_{\xi }^0\left(t\right),{\mu }^0\left(t\right)\right)\left(\mathfrak{d}\left(t\right)-{\mathfrak{d}}^0\left(t\right)\right)d{t}^{\nu }} \\ {\displaystyle +{\int }_K\frac{\partial h_{\nu }}{\partial {\mathfrak{d}}_{\xi }}\left(t,{\mathfrak{d}}^0\left(t\right),{\mathfrak{d}}_{\xi }^0\left(t\right),{\mu }^0\left(t\right)\right)D_{\xi }\left(\mathfrak{d}\left(t\right)-{\mathfrak{d}}^0\left(t\right)\right)d{t}^{\nu }} \\ {\displaystyle +{\int }_K\frac{\partial h_{\nu }}{\partial \mu }\left(t,{\mathfrak{d}}^0\left(t\right),{\mathfrak{d}}_{\xi }^0\left(t\right),{\mu }^0\left(t\right)\right)\left(\mu \left(t\right)-{\mu }^0\left(t\right)\right)d{t}^{\nu }} \end{gathered}$$

is satisfied for all $(\mathfrak{d},\mu )\in \mathfrak{P}\times \mathfrak{B}$.

We say that a real-valued functional determined by a curvilinear integral $J:\mathfrak{P}\times \mathfrak{B}\mapsto \mathbb{R}$ is (strictly) convex on $\mathfrak{P}\times \mathfrak{B}$ if the above definition is valid for every $({\mathfrak{d}}^0,{\mu }^0)$ in $\mathfrak{P}\times \mathfrak{B}$.

Definition 7 

(Treanţă and Saeed [36]). We say that an interval-valued functional determined by a curvilinear integral

$$Q:\mathfrak{P}\times \mathfrak{B}\mapsto \mathfrak{C},Q(\mathfrak{d},\mu )={\int }_K{\theta }_{\nu }\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }=[Q^L(\mathfrak{d},\mu ),Q^U(\mathfrak{d},\mu )]$$

is called $LU$-convex at $({\mathfrak{d}}^0,{\mu }^0)\in \mathfrak{P}\times \mathfrak{B}$ if the associated real-valued functionals determined by a curvilinear integral

$${\displaystyle Q^L(\mathfrak{d},\mu )={\int }_K{\theta }_{\nu }^L\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }}$$

and

$${\displaystyle Q^U(\mathfrak{d},\mu )={\int }_K{\theta }_{\nu }^U\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }}$$

are convex at $({\mathfrak{d}}^0,{\mu }^0)\in \mathfrak{P}\times \mathfrak{B}$.

Definition 8 

(Treanţă and Saeed [36]). If the real-valued functionals determined by a curvilinear integral ${\displaystyle Q^L(\mathfrak{d},\mu )}$ and ${\displaystyle Q^U(\mathfrak{d},\mu )}$ are convex at $({\mathfrak{d}}^0,{\mu }^0)\in \mathfrak{P}\times \mathfrak{B}$ and either

$${\displaystyle Q^L(\mathfrak{d},\mu )={\int }_K{\theta }_{\nu }^L\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }}$$

or

$${\displaystyle Q^U(\mathfrak{d},\mu )={\int }_K{\theta }_{\nu }^U\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }}$$

is strictly convex at $({\mathfrak{d}}^0,{\mu }^0)\in \mathfrak{P}\times \mathfrak{B}$, then the interval-valued functional given by the curvilinear integral

$$Q:\mathfrak{P}\times \mathfrak{B}\mapsto \mathfrak{C},Q(\mathfrak{d},\mu )={\int }_K{\theta }_{\nu }\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }$$

is strictly $LU$-convex at $({\mathfrak{d}}^0,{\mu }^0)\in \mathfrak{P}\times \mathfrak{B}$.

In the following, we introduce a family of extremization problems driven by functionals with interval values determined by curvilinear integrals:

$$\begin{array}{c}\left(Problem\right){\displaystyle \underset{\left(\mathfrak{d}\right(·),\mu (·\left)\right)}{min}}{\int }_K{\theta }_{\nu }\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)d{t}^{\nu }\\ \mathrm{subject}\mathrm{to}\\ M_{\lambda }\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)\le 0,\lambda =\overline{1,u}\\ N_{\pi }\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right):={\displaystyle \frac{\partial \mathfrak{d}}{\partial t^{\xi }}}\left(t\right)-Q_{\pi }\left(t,\mathfrak{d}\left(t\right),\mu \left(t\right)\right)=0,\pi =\overline{1,r}\\ {\mathfrak{d}|}_{t=t_1,t_2}=\mathrm{fixed},\end{array}$$

where $t\in \mathcal{S},{\theta }_{\nu }:\mathcal{S}\times \mathfrak{P}\times \mathfrak{P}\times \mathfrak{B}\mapsto \mathfrak{C},M_{\lambda }:\mathcal{S}\times \mathfrak{P}\times \mathfrak{P}\times \mathfrak{B}\mapsto \mathbb{R}$, and $N_{\pi }:\mathcal{S}\times \mathfrak{P}\times \mathfrak{P}\times \mathfrak{B}\mapsto \mathbb{R}$ are $C^1$-class functions.

The feasible solution set associated with $\left(Problem\right)$ is defined as follows:

$$\begin{gathered} {\displaystyle \mathcal{L}=\{(\mathfrak{d},\mu )\in \mathfrak{P}\times \mathfrak{B}\mid M_{\lambda }\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)\le 0,} \\ {\displaystyle N_{\pi }\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)=0,\mathfrak{d}{|}_{t=t_1,t_2}=\mathrm{fixed}\},} \end{gathered}$$

assuming it is a convex subset of $\mathfrak{P}\times \mathfrak{B}$.

Further, for simplicity, we consider the following abbreviations: $\mathfrak{d}=\mathfrak{d}\left(t\right),\mu =\mu \left(t\right),\gamma =\left(t,\mathfrak{d}\left(t\right),{\mathfrak{d}}_{\xi }\left(t\right),\mu \left(t\right)\right)$, ${\gamma }^0=\left(t,{\mathfrak{d}}^0\left(t\right),{\mathfrak{d}}_{\xi }^0\left(t\right),{\mu }^0\left(t\right)\right),{\theta }_{\nu ,\mathfrak{d}}={\displaystyle \frac{\partial {\theta }_{\nu }}{\partial \mathfrak{d}}},{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}={\displaystyle \frac{\partial {\theta }_{\nu }}{\partial {\mathfrak{d}}_{\xi }}}$, and ${\theta }_{\nu ,\mu }={\displaystyle \frac{\partial {\theta }_{\nu }}{\partial \mu }}$.

Definition 9 

(Treanţă and Saeed [36]). A pair $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ is called a (strong) $LU$-optimal solution in $\left(Problem\right)$ if the inequality

$${\int }_K{\theta }_{\nu }\left({\gamma }^0\right)d{t}^{\nu }{⪯}_{LU}\left({\prec }_{LU}\right){\int }_K{\theta }_{\nu }\left(\gamma \right)d{t}^{\nu }$$

is valid for all $(\mathfrak{d},\mu )\in \mathcal{L}$.

Now, we formulate the following controlled variational inequalities associated with the above-mentioned optimization problem. $\left(Problem\right)$: Let us find a pair $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ for which there exists no other feasible pair $(\mathfrak{d},\mu )\in \mathcal{L}$ satisfying:

$$\begin{gathered} {\displaystyle \left(Inequality\right){\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^L\left({\gamma }^0\right)+{\theta }_{\nu ,\mathfrak{d}}^U\left({\gamma }^0\right)\right\}\left(\mathfrak{d}-{\mathfrak{d}}^0\right)d{t}^{\nu }} \\ {\displaystyle +{\int }_K\left\{{\theta }_{\nu ,\mu }^L\left({\gamma }^0\right)+{\theta }_{\nu ,\mu }^U\left({\gamma }^0\right)\right\}\left(\mu -{\mu }^0\right)d{t}^{\nu }} \\ {\displaystyle +{\int }_K\left\{{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^L\left({\gamma }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^U\left({\gamma }^0\right)\right\}{D}_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)d{t}^{\nu }\le 0.} \end{gathered}$$

4. Main Results

The main results of this paper present some connections for the above-mentioned variational models, $\left(Problem\right)$ and $\left(Inequality\right)$. In the next theorems, we relax the assumptions formulated in Treanţă and Guo [32] by imposing only (strictly) $LU$ convexity without invex set hypotheses for curvilinear integral functionals (instead of multiple integral functionals). Finally, since the functionals defined as objectives have a physical importance, an illustrative application is considered and studied, taking into account the theoretical elements obtained in this paper.

The next theorem states under which conditions a solution in $\left(Inequality\right)$ also becomes an $LU$-optimal solution to $\left(Problem\right)$.

Theorem 4. 

Let us assume the pair $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ is a solution in $\left(Inequality\right)$ and that the functional with interval values defined by a curvilinear integral ${\displaystyle {\int }_K{\theta }_{\nu }\left(\gamma \right)d{t}^{\nu }}$ is $LU$-convex at $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$. In this case, the pair $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ becomes an $LU$-optimal solution in $\left(Problem\right)$.

Proof. 

By assumption, the point $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ is a solution in $\left(Inequality\right)$. Now, let us suppose, on the contrary, that the pair $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ is not an $LU$-optimal solution in $\left(Problem\right)$. In this situation, there exists $(\mathfrak{d},\mu )\in \mathcal{L}$ fulfilling

$${\displaystyle {\int }_K{\theta }_{\nu }\left(\gamma \right)d{t}^{\nu }{\prec }_{LU}{\int }_K{\theta }_{\nu }\left({\gamma }^0\right)d{t}^{\nu },}$$

which is equivalent to

$${\int }_K{\theta }_{\nu }^L\left(\gamma \right)d{t}^{\nu }<{\int }_K{\theta }_{\nu }^L\left({\gamma }^0\right)d{t}^{\nu }\mathrm{and}{\int }_K{\theta }_{\nu }^U\left(\gamma \right)d{t}^{\nu }\le {\int }_K{\theta }_{\nu }^U\left({\gamma }^0\right)d{t}^{\nu }$$

(1)

or

$${\displaystyle {\int }_K{\theta }_{\nu }^L\left(\gamma \right)d{t}^{\nu }\le {\int }_K{\theta }_{\nu }^L\left({\gamma }^0\right)d{t}^{\nu }\mathrm{and}{\int }_K{\theta }_{\nu }^U\left(\gamma \right)d{t}^{\nu }<{\int }_K{\theta }_{\nu }^U\left({\gamma }^0\right)d{t}^{\nu }}$$

(2)

or

$${\displaystyle {\int }_K{\theta }_{\nu }^L\left(\gamma \right)d{t}^{\nu }<{\int }_K{\theta }_{\nu }^L\left({\gamma }^0\right)d{t}^{\nu }\mathrm{and}{\int }_K{\theta }_{\nu }^U\left(\gamma \right)d{t}^{\nu }<{\int }_K{\theta }_{\nu }^U\left({\gamma }^0\right)d{t}^{\nu }.}$$

(3)

By considering that the functional ${\displaystyle {\int }_K{\theta }_{\nu }\left(\gamma \right)d{t}^{\nu }}$ is $LU$-convex at $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$, it follows that

$$\begin{gathered} {\displaystyle {\int }_K{\theta }_{\nu }^L\left(\gamma \right)d{t}^{\nu }-{\int }_K{\theta }_{\nu }^L\left({\gamma }^0\right)d{t}^{\nu }} \\ {\displaystyle \ge {\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^L\left({\gamma }^0\right)\left(\mathfrak{d}-{\mathfrak{d}}^0\right)+{\theta }_{\nu ,\mu }^L\left({\gamma }^0\right)\left(\mu -{\mu }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^L\left({\gamma }^0\right)D_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)\right\}d{t}^{\nu }} \end{gathered}$$

and

$$\begin{gathered} {\displaystyle {\int }_K{\theta }_{\nu }^U\left(\gamma \right)d{t}^{\nu }-{\int }_K{\theta }_{\nu }^U\left({\gamma }^0\right)d{t}^{\nu }} \\ {\displaystyle \ge {\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^U\left({\gamma }^0\right)\left(\mathfrak{d}-{\mathfrak{d}}^0\right)+{\theta }_{\nu ,\mu }^U\left({\gamma }^0\right)\left(\mu -{\mu }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^U\left({\gamma }^0\right)D_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)\right\}d{t}^{\nu }} \end{gathered}$$

for every $(\mathfrak{d},\mu )\in \mathcal{L}$. By considering the inequalities $\left(1\right)-\left(3\right)$ and the above-mentioned ones, it results in:

$$\begin{gathered} {\displaystyle {\int }_K[\left\{{\theta }_{\nu ,\mathfrak{d}}^L\left({\gamma }^0\right)+{\theta }_{\nu ,\mathfrak{d}}^U\left({\gamma }^0\right)\right\}\left(\mathfrak{d}-{\mathfrak{d}}^0\right)+\left\{{\theta }_{\nu ,\mu }^L\left({\gamma }^0\right)+{\theta }_{\nu ,\mu }^U\left({\gamma }^0\right)\right\}\left(\mu -{\mu }^0\right)} \\ {\displaystyle +\left\{{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^L\left({\gamma }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^U\left({\gamma }^0\right)\right\}{D}_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)]d{t}^{\nu }\le 0,} \end{gathered}$$

which contradicts the idea that $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ is a solution to $\left(Inequality\right)$. □

The next theorem states under which conditions an $LU$-optimal solution to $\left(Problem\right)$ also becomes a solution for $\left(Inequality\right)$. In other words, this is the reciprocal result of the above theorem. It gives sufficient criteria such that an $LU$-optimal solution to $\left(Problem\right)$ also becomes a solution for $\left(Inequality\right)$.

Theorem 5. 

If the pair $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ is an $LU$-optimal solution to $\left(Problem\right)$ and the functional with interval values ${\displaystyle {\int }_K-{\theta }_{\nu }\left(\gamma \right)d{t}^{\nu }}$ is strictly $LU$-convex at $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$, then the pair $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ also becomes a solution for $\left(Inequality\right)$.

Proof. 

Since, by hypothesis, the pair $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ is an $LU$-optimal solution to $\left(Problem\right)$, therefore, by definition, there exists no other pair $(\mathfrak{d},\mu )\in \mathcal{L}$ fulfilling

$${\int }_K{\theta }_{\nu }\left(\gamma \right)d{t}^{\nu }{\prec }_{LU}{\int }_K{\theta }_{\nu }\left({\gamma }^0\right)d{t}^{\nu },$$

or, equivalently,

$$\begin{array}{cc}& {\int }_K{\theta }_{\nu }^L\left(\gamma \right)d{t}^{\nu }<{\int }_K{\theta }_{\nu }^L\left({\gamma }^0\right)d{t}^{\nu }\mathrm{and}{\int }_K{\theta }_{\nu }^U\left(\gamma \right)d{t}^{\nu }\le {\int }_K{\theta }_{\nu }^U\left({\gamma }^0\right)d{t}^{\nu },\hfill \\ & \mathrm{or}{\int }_K{\theta }_{\nu }^L\left(\gamma \right)d{t}^{\nu }\le {\int }_K{\theta }_{\nu }^L\left({\gamma }^0\right)d{t}^{\nu }\mathrm{and}{\int }_K{\theta }_{\nu }^U\left(\gamma \right)d{t}^{\nu }<{\int }_K{\theta }_{\nu }^U\left({\gamma }^0\right)d{t}^{\nu },\hfill \\ & \mathrm{or}{\int }_K{\theta }_{\nu }^L\left(\gamma \right)d{t}^{\nu }<{\int }_K{\theta }_{\nu }^L\left({\gamma }^0\right)d{t}^{\nu }\mathrm{and}{\int }_K{\theta }_{\nu }^U\left(\gamma \right)d{t}^{\nu }<{\int }_K{\theta }_{\nu }^U\left({\gamma }^0\right)d{t}^{\nu }.\hfill \end{array}$$

Thus, there exists no pair $(\mathfrak{d},\mu )\in \mathcal{L}$ satisfying

$${\displaystyle {\int }_K\left\{{\theta }_{\nu }^L\left(\gamma \right)+{\theta }_{\nu }^U\left(\gamma \right)\right\}d{t}^{\nu }<{\int }_K\left\{{\theta }_{\nu }^L\left({\gamma }^0\right)+{\theta }_{\nu }^U\left({\gamma }^0\right)\right\}d{t}^{\nu }.}$$

(4)

Now, to the contrary, let us assume that the pair $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$ is not a solution to $\left(Inequality\right)$. In consequence, there exists $(\mathfrak{d},\mu )\in \mathcal{L}$ fulfilling

$$\begin{gathered} {\displaystyle {\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^L\left({\gamma }^0\right)+{\theta }_{\nu ,\mathfrak{d}}^U\left({\gamma }^0\right)\right\}\left(\mathfrak{d}-{\mathfrak{d}}^0\right)d{t}^{\nu }+{\int }_K\left\{{\theta }_{\nu ,\mu }^L\left({\gamma }^0\right)+{\theta }_{\nu ,\mu }^U\left({\gamma }^0\right)\right\}\left(\mu -{\mu }^0\right)d{t}^{\nu }} \\ {\displaystyle +{\int }_K\left\{{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^L\left({\gamma }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^U\left({\gamma }^0\right)\right\}{D}_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)d{t}^{\nu }\le 0.} \end{gathered}$$

(5)

Since the functional ${\displaystyle {\int }_K-{\theta }_{\nu }\left(\gamma \right)d{t}^{\nu }}$ is strictly $LU$-convex at $({\mathfrak{d}}^0,{\mu }^0)\in \mathcal{L}$, for all $(\mathfrak{d},\mu )\in \mathcal{L}$, we get

$$\begin{gathered} {\displaystyle {\int }_K{\theta }_{\nu }^L\left(\gamma \right)d{t}^{\nu }-{\int }_K{\theta }_{\nu }^L\left({\gamma }^0\right)d{t}^{\nu }} \\ {\displaystyle <{\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^L\left({\gamma }^0\right)\left(\mathfrak{d}-{\mathfrak{d}}^0\right)+{\theta }_{\nu ,\mu }^L\left({\gamma }^0\right)\left(\mu -{\mu }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^L\left({\gamma }^0\right)D_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)\right\}d{t}^{\nu }} \end{gathered}$$

and

$$\begin{gathered} {\displaystyle {\int }_K{\theta }_{\nu }^U\left(\gamma \right)d{t}^{\nu }-{\int }_K{\theta }_{\nu }^U\left({\gamma }^0\right)d{t}^{\nu }} \\ {\displaystyle \le {\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^U\left({\gamma }^0\right)\left(\mathfrak{d}-{\mathfrak{d}}^0\right)+{\theta }_{\nu ,\mu }^U\left({\gamma }^0\right)\left(\mu -{\mu }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^U\left({\gamma }^0\right)D_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)\right\}d{t}^{\nu },} \end{gathered}$$

or

$$\begin{gathered} {\int }_K{\theta }_{\nu }^L\left(\gamma \right)d{t}^{\nu }-{\int }_K{\theta }_{\nu }^L\left({\gamma }^0\right)d{t}^{\nu } \\ {\displaystyle \le {\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^L\left({\gamma }^0\right)\left(\mathfrak{d}-{\mathfrak{d}}^0\right)+{\theta }_{\nu ,\mu }^L\left({\gamma }^0\right)\left(\mu -{\mu }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^L\left({\gamma }^0\right)D_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)\right\}d{t}^{\nu }} \end{gathered}$$

and

$$\begin{gathered} {\displaystyle {\int }_K{\theta }_{\nu }^U\left(\gamma \right)d{t}^{\nu }-{\int }_K{\theta }_{\nu }^U\left({\gamma }^0\right)d{t}^{\nu }} \\ {\displaystyle <{\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^U\left({\gamma }^0\right)\left(\mathfrak{d}-{\mathfrak{d}}^0\right)+{\theta }_{\nu ,\mu }^U\left({\gamma }^0\right)\left(\mu -{\mu }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^U\left({\gamma }^0\right)D_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)\right\}d{t}^{\nu },} \end{gathered}$$

or

$$\begin{gathered} {\displaystyle {\int }_K{\theta }_{\nu }^L\left(\gamma \right)d{t}^{\nu }-{\int }_K{\theta }_{\nu }^L\left({\gamma }^0\right)d{t}^{\nu }} \\ {\displaystyle <{\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^L\left({\gamma }^0\right)\left(\mathfrak{d}-{\mathfrak{d}}^0\right)+{\theta }_{\nu ,\mu }^L\left({\gamma }^0\right)\left(\mu -{\mu }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^L\left({\gamma }^0\right)D_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)\right\}d{t}^{\nu }} \end{gathered}$$

and

$$\begin{gathered} {\displaystyle {\int }_K{\theta }_{\nu }^U\left(\gamma \right)d{t}^{\nu }-{\int }_K{\theta }_{\nu }^U\left({\gamma }^0\right)d{t}^{\nu }} \\ {\displaystyle <{\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^U\left({\gamma }^0\right)\left(\mathfrak{d}-{\mathfrak{d}}^0\right)+{\theta }_{\nu ,\mu }^U\left({\gamma }^0\right)\left(\mu -{\mu }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^U\left({\gamma }^0\right)D_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)\right\}d{t}^{\nu }.} \end{gathered}$$

From the above inequalities, we obtain

$$\begin{gathered} {\displaystyle {\int }_K\left\{{\theta }_{\nu }^L\left(\gamma \right)+{\theta }_{\nu }^U\left(\gamma \right)\right\}d{t}^{\nu }-{\int }_K\left\{{\theta }_{\nu }^L\left({\gamma }^0\right)+{\theta }_{\nu }^U\left({\gamma }^0\right)\right\}d{t}^{\nu }} \\ {\displaystyle <{\int }_K\left\{{\theta }_{\nu ,\mathfrak{d}}^L\left({\gamma }^0\right)+{\theta }_{\nu ,\mathfrak{d}}^U\left({\gamma }^0\right)\right\}\left(\mathfrak{d}-{\mathfrak{d}}^0\right)d{t}^{\nu }+{\int }_K\left\{{\theta }_{\nu ,\mu }^L\left({\gamma }^0\right)+{\theta }_{\nu ,\mu }^U\left({\gamma }^0\right)\right\}\left(\mu -{\mu }^0\right)d{t}^{\nu }} \\ {\displaystyle +{\int }_K\left\{{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^L\left({\gamma }^0\right)+{\theta }_{\nu ,{\mathfrak{d}}_{\xi }}^U\left({\gamma }^0\right)\right\}{D}_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)d{t}^{\nu },} \end{gathered}$$

which, together with inequality $\left(5\right)$, yields that

$${\int }_K\left\{{\theta }_{\nu }^L\left(\gamma \right)+{\theta }_{\nu }^U\left(\gamma \right)\right\}d{t}^{\nu }<{\int }_K\left\{{\theta }_{\nu }^L\left({\gamma }^0\right)+{\theta }_{\nu }^U\left({\gamma }^0\right)\right\}d{t}^{\nu }$$

holds for $(\mathfrak{d},\mu )\in \mathcal{L}$, which is a contradiction with $\left(4\right)$. □

As the functionals defined as objectives have a physical importance, an illustrative application is considered and studied below by using the theoretical elements obtained in this study.

Illustrative application. In the following illustrative example, we consider a variable force defined as ${\displaystyle \overline{F}({(\mu -4)}^2,{\mu }^2)}$. We aim to extremize the mechanical work determined by the above-mentioned variable force ${\displaystyle \overline{F}}$ such that, to move its current point along a given curve (piecewise smooth) ${\displaystyle K\subset \mathcal{S}={[0,3]}^2=[0,3]\times [0,3]}$, which links ${\displaystyle t_1=\left(0,0\right)}$ and ${\displaystyle t_2=\left(3,3\right)}$, such that the dynamic generated by

$${\displaystyle \begin{array}{cc}& \frac{\partial \mathfrak{d}}{\partial t^1}=\frac{\partial \mathfrak{d}}{\partial t^2}=-\mu +3,\hfill \\ & 81-{\mathfrak{d}}^2\le 0,\hfill \\ & \mathfrak{d}(3,3)=8,\mathfrak{d}(0,0)=6,\hfill \end{array}}$$

is satisfied.

According to the notations used in this paper, the feasible solution set associated with the above control problem is defined as

$$\begin{gathered} {\displaystyle \mathcal{L}=\{(\mathfrak{d},\mu )\in \mathfrak{P}\times \mathfrak{B}:81-{\mathfrak{d}}^2\le 0,\frac{\partial \mathfrak{d}}{\partial t^1}=\frac{\partial \mathfrak{d}}{\partial t^2}=3-\mu ,} \\ {\displaystyle \mathfrak{d}(0,0)=6,\mathfrak{d}(3,3)=8\},} \end{gathered}$$

and

$$\begin{gathered} {\displaystyle M\left(\gamma \right):=81-{\mathfrak{d}}^2} \\ {Q}_{\pi }\left(\gamma \right):=3-\mu ,\pi =1,2 \end{gathered}$$

for $t\in K\subset \mathcal{S}={[0,3]}^2$. Further, by considering only state functions that are affine and control real-valued functions in our study, and since the functionals with real values ${\displaystyle {\int }_K{\theta }^L\left(\gamma \right)d{t}^1+{\theta }^L\left(\gamma \right)d{t}^2}$ and ${\displaystyle {\int }_K{\theta }^U\left(\gamma \right)d{t}^1+{\theta }^U\left(\gamma \right)d{t}^2}$ are convex at $({\mathfrak{d}}^0,{\mu }^0)=\left({\displaystyle \frac{1}{3}}\left(t^1+t^2\right)+6,{\displaystyle \frac{8}{3}}\right)$, it results in a functional with interval values

$${\displaystyle Q(\mathfrak{d},\mu )=\left[{\int }_K{\theta }^L\left(\gamma \right)d{t}^1+{\theta }^L\left(\gamma \right)d{t}^2,{\int }_K{\theta }^U\left(\gamma \right)d{t}^1+{\theta }^U\left(\gamma \right)d{t}^2\right]}$$

that is $LU$-convex at $({\mathfrak{d}}^0,{\mu }^0)=\left({\displaystyle \frac{1}{3}}\left(t^1+t^2\right)+6,{\displaystyle \frac{8}{3}}\right)$.

By using the condition

$${\theta }^L\left(\gamma \right)\le {\theta }^U\left(\gamma \right),$$

we obtain

$${(\mu -4)}^2\le {\mu }^2,$$

which involves

$$\mu \ge 2.$$

Moreover, the inequality

$$\begin{gathered} {\displaystyle {\int }_K[\{{\theta }_{\mathfrak{d}}^L\left({\gamma }^0\right)+{\theta }_{\mathfrak{d}}^U\left({\gamma }^0\right)\}\left(\mathfrak{d}-{\mathfrak{d}}^0\right)+\{{\theta }_{{\mathfrak{d}}_{\xi }}^L\left({\gamma }^0\right)+{\theta }_{{\mathfrak{d}}_{\xi }}^U\left({\gamma }^0\right)\}{D}_{\xi }\left(\mathfrak{d}-{\mathfrak{d}}^0\right)} \\ {\displaystyle +\{{\theta }_{\mu }^L\left({\gamma }^0\right)+{\theta }_{\mu }^U\left({\gamma }^0\right)\}\left(\mu -{\mu }^0\right)](d{t}^1+d{t}^2)} \\ {\displaystyle ={\int }_K\left(2{\mu }^0-8+2{\mu }^0\right)\left(\mu -{\mu }^0\right)(d{t}^1+d{t}^2)} \\ {\displaystyle ={\int }_K\left(4{\mu }^0-8\right)\left(\mu -{\mu }^0\right)(d{t}^1+d{t}^2)\neg \le 0} \end{gathered}$$

is satisfied for all $(\mathfrak{d},\mu )\ne ({\mathfrak{d}}^0,{\mu }^0)\in S$ at ${\displaystyle \left({\mathfrak{d}}^0\left(t\right)=\frac{1}{3}\left(t^1+t^2\right)+6,{\mu }^0\left(t\right)=\frac{8}{3}\right)}$. As a result, it is a solution to the corresponding control inequality. Taking into account Theorem 4, we find that $({\mathfrak{d}}^0,{\mu }^0)$ is an $LU$-optimal solution for our extremization problem.

5. Conclusions

In this paper, we have introduced and studied a family of optimization problems driven by functionals with interval values defined by curvilinear integrals. In this regard, we have built a class of controlled variational inequalities. The main results of this paper provide some connections for the above-mentioned variational models. More precisely, the technique was used here to transform an optimization problem into a problem with inequalities, which is much simpler to study, with both involving curvilinear integral-type functionals. An immediate development of the results obtained in this study could be the consideration of the functional derivative associated with functionals of curvilinear integral type. In this way, new aspects can be generated based on the results presented in this paper.