On Solution Set Associated with a Class of Multiple Objective Control Models

Abstract

In this paper, necessary and sufficient efficiency conditions in new multi-cost variational models are formulated and proved. To this end, we introduce a new notion of $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ functionals determined by multiple integrals. To better emphasize the significance of the suggested $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ functionals and how they add to previous studies, we mention that the $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ and generalized $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I-typeI$ assumptions associated with the involved multiple integral functionals cover broader and more general classes of problems, where the convexity of the functionals is not fulfilled or the functionals considered are not of simple integral type. In addition, innovative proofs are provided for the main results.

1. Introduction

Hanson [1] formulated and studied a link between the calculus of variations and mathematical programming problems. In this way, some importance was given to variational programming models. Thus, optimality criteria and duality results were stated by Mond and Hanson [2] for certain optimization problems under convexity hypotheses. Motivated and inspired by the technique of Bector and Husain [3], Nahak and Nanda [4] and Bhatia and Mehra [5] have generalized the outcomes of Mond et al. [6] for multi-objective optimization problems determined by invariant convex (invex) or generalized B-invex functions. Similar theorems have been formulated and proved in Zalmai [7] for classes of fractional optimization problems with arbitrary norms, and by Liu [8] in the generalized fractional case implying $(F,\rho )$-convexity of the involved functions.

Functions of type-I have first been studied in Hanson and Mond [9]. Later, pseudo- and quasi-functions of type-I have been introduced by Rueda and Hanson [10] as a natural generalization of functions of type-I. On the other hand, conditions of optimality and duality theorems in generalized B-invex multi-objective optimization problems were provided by Bhatia and Mehra [5]. In this paper, we use $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ and generalized $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ functionals to establish sufficient optimality conditions for a new class of multi-objective variational control problems. The presence of control variables and the innovative proofs associated with the principal results derived in this study give this research paper an essential and crucial role, compared with previous research works. Moreover, due to the significance of multiple integral functionals in the physical sciences, this paper offers a theoretical basis for various applications. Concretely, the study’s outcomes are as follows:

• We extended the notion of functions of type-I for controlled multiple integral type functionals. More precisely, the $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ and generalized $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ assumptions associated with the multiple integral functionals cover broader and more general classes of problems, where the convexity of the functionals is not fulfilled or the functionals considered are not of simple integral type.
• Under $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ and generalized $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ hypotheses of the involved functionals, we have derived sufficient efficiency conditions for the considered class of extremization models.
• Additionally, for the controlled model, sufficient criteria for efficient solutions and properly efficient solutions have been established.
• The limitations of the presented results could be, for instance, the appearance of (path-dependent or path-independent) curvilinear integrals as functionals instead of multiple integrals. This situation would add additional conditions to the study of this class of problems. In addition, as issues which remain unsolved, we can mention the analysis of well-posedness associated with this type of extremization models and also establishing new efficiency criteria based on saddle-points or modified cost functional approach. Such research directions are still unexplored and, due to their importance and efficiency, warrant immediate investigation. Consequently, we demonstrate the applicability of the proposed technique to larger or more complex optimization problems.

In Section 2 of this paper, we introduce the $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ and generalized $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ multiple integral functionals for continuous-time extremization models. By considering these new types of functionals given by multiple integrals, we state various sufficient conditions of optimality in Section 3. In Section 4, we highlight the main merits of this study and present some further research directions. The main results obtained in this article include the results formulated in [4,5,11].

2. Preliminaries

Consider the following multi-objective variational control problem:

$$\begin{array}{c}\hfill \left(P\right)\underset{(\kappa ,\omega )}{min}\left\{{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,\kappa ,\omega )d\pi :=\left({\int }_{{\mathcal{D}}_{x,z}}{\psi }_1(\pi ,\kappa ,\omega )d\pi ,\dots ,{\int }_{{\mathcal{D}}_{x,z}}{\psi }_p(\pi ,\kappa ,\omega )d\pi \right)\right\}\end{array}$$

subject to

$$\kappa \left(x\right)=\alpha ,\kappa \left(z\right)=\beta ,$$

$$\begin{array}{c}\hfill \varphi (\pi ,\kappa ,{\kappa }_{\mu },\omega ):=g(\pi ,\kappa ,\omega )-{\kappa }_{\mu }\leqq 0,\pi \in {\mathcal{D}}_{x,z},\end{array}$$

where ${\mathcal{D}}_{x,z}$ is a compact subset of ${\mathbb{R}}^q$ with $x,z$ some fixed arbitrary boundary points in ${\mathcal{D}}_{x,z}$, ${\displaystyle {\kappa }_{\mu }:=\frac{\partial \kappa }{\partial {\pi }^{\mu }},\mu =\overline{1,q}}$, and $\varphi :{\mathcal{D}}_{x,z}\times C^1{\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)}^2\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\to {\mathbb{R}}^{nq}$ is continuously differentiable functional in all variables.

Let the set of feasible solutions associated with $\left(P\right)$ be denoted as

$$\mathcal{S}=\{(\kappa ,\omega )\in C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\mid \kappa \left(x\right)=\alpha ,\kappa \left(z\right)=\beta ,$$

$$\varphi (\pi ,\kappa ,{\kappa }_{\mu },\omega )\leqq 0,\pi \in {\mathcal{D}}_{x,z}\}.$$

In relation to $\left(P\right)$, we introduce the following minimization problem $\left(P_r^{\ast }\right)$, for each $r=1,\dots ,p$, as follows:

$$\begin{array}{cc}\hfill & \left(P_r^{*}\right)\underset{(\kappa ,\omega )}{min}{\int }_{{\mathcal{D}}_{x,z}}{\psi }_r(\pi ,\kappa ,\omega )d\pi \hfill \\ \hfill & \mathrm{subject}\mathrm{to}\kappa \left(x\right)=\alpha ,\kappa \left(z\right)=\beta ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}{\psi }_i(\pi ,\kappa ,\omega )d\pi \le {\int }_{{\mathcal{D}}_{x,z}}{\psi }_i(\pi ,{\kappa }^{*},{\omega }^{*})d\pi ,i=1,2,\dots ,p,i\ne r,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\varphi }_s(\pi ,\kappa ,{\kappa }_{\mu },\omega )\le 0,s=1,\dots ,n,\pi \in {\mathcal{D}}_{x,z}.\hfill \end{array}$$

Next, by considering Geoffrion [12] and Nahak and Nanda [4], we formulate the following definitions.

Definition 1. 

A point $({\kappa }^{*},{\omega }^{*})$ in $\mathcal{S}$ is named efficient solution of $\left(P\right)$ if we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}{\psi }_i(\pi ,{\kappa }^{*},{\omega }^{*})d\pi \ge {\int }_{{\mathcal{D}}_{x,z}}{\psi }_i(\pi ,\kappa ,\omega )d\pi ,\forall i\in \{1,\dots ,p\}\hfill \\ \hfill & ⟹{\int }_{{\mathcal{D}}_{x,z}}{\psi }_i\left(\pi ,{\kappa }^{*},{\omega }^{*}\right)d\pi ={\int }_{{\mathcal{D}}_{x,z}}{\psi }_i(\pi ,\kappa ,\omega )d\pi ,\forall i\in \{1,\dots ,p\},\hfill \end{array}$$

for all $(\kappa ,\omega )\in \mathcal{S}$.

Definition 2. 

A point $({\kappa }^{*},{\omega }^{*})$ in $\mathcal{S}$ is named properly efficient solution of $\left(P\right)$ if there exists a scalar $H>0$ satisfying

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}{\psi }_i\left(\pi ,{\kappa }^{*},{\omega }^{*}\right)d\pi -{\int }_{{\mathcal{D}}_{x,z}}{\psi }_i(\pi ,\kappa ,\omega )d\pi \hfill \\ \hfill & \le H\left({\int }_{{\mathcal{D}}_{x,z}}{\psi }_j(\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}{\psi }_j(\pi ,{\kappa }^{*},{\omega }^{*})d\pi \right),\forall i\in \{1,\dots ,p\}\hfill \end{array}$$

for j with

$${\int }_{{\mathcal{D}}_{x,z}}{\psi }_j(\pi ,\kappa ,\omega )d\pi >{\int }_{{\mathcal{D}}_{x,z}}{\psi }_j(\pi ,{\kappa }^{*},{\omega }^{*})d\pi$$

whenever $(\kappa ,\omega )$ is in $\mathcal{S}$ and

$${\int }_{{\mathcal{D}}_{x,z}}{\psi }_i(\pi ,\kappa ,\omega )d\pi <{\int }_{{\mathcal{D}}_{x,z}}{\psi }_i(\pi ,{\kappa }^{*},{\omega }^{*})d\pi ,$$

for all $(\kappa ,\omega )\in \mathcal{S}$.

Next, in accordance with Chankong and Haimes [13], we establish the following result.

Lemma 1. 

The pair $({\kappa }^{*},{\omega }^{*})$ is an efficient solution of $\left(P\right)$ if and only if $({\kappa }^{*},{\omega }^{*})$ is an optimal solution of $\left(P_r^{*}\right)$ for each $r=1,2,\dots ,p$.

According to Treanţă [14], following Mond and Hanson [2], we state the theorem.

Theorem 1. 

For each [normal] optimal solution $({\kappa }^{*},{\omega }^{*})$ of $\left(P_r^{*}\right),r=1,2,\dots ,p$, there exist ${\aleph }_{1r},\dots ,{\aleph }_{pr}$, with ${\aleph }_{rr}=1$, and the piecewise differentiable function $h_r$: ${\mathcal{D}}_{x,z}\to {\mathbb{R}}^n$ such that

$$\begin{array}{c}\hfill {\psi }_{r\kappa }(\pi ,{\kappa }^{*},{\omega }^{*})+\sum _{\begin{array}{c}i=1\\ i\ne r\end{array}}^p{\aleph }_{ir}{\psi }_{i\kappa }(\pi ,{\kappa }^{*},{\omega }^{*})+h_r{\left(\pi \right)}^T{\varphi }_{\kappa }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\end{array}$$

$$\begin{array}{c}={\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}({\psi }_{r{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\omega }^{*})+\sum _{\begin{array}{c}i=1\\ i\ne r\end{array}}^p{\aleph }_{ir}{\psi }_{i{\kappa }_{\mu }}\left(\pi ,{\kappa }^{*},{\omega }^{*}\right)+h_r{\left(\pi \right)}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})),\pi \in {\mathcal{D}}_{x,z}\hfill \end{array}$$

(1)

$$\begin{array}{c}\hfill {\psi }_{r\omega }(\pi ,{\kappa }^{*},{\omega }^{*})+\sum _{\begin{array}{c}i=1\\ i\ne r\end{array}}^p{\aleph }_{ir}{\psi }_{i\omega }(\pi ,{\kappa }^{*},{\omega }^{*})+h_r{\left(\pi \right)}^T{\varphi }_{\omega }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})=0,\pi \in {\mathcal{D}}_{x,z}\end{array}$$

(2)

$$\begin{array}{c}\hfill h_r{\left(\pi \right)}^T\varphi \left(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*}\right)=0,\pi \in {\mathcal{D}}_{x,z}\end{array}$$

(3)

$$\begin{array}{c}\hfill h_r\left(\pi \right)\geqq 0,\pi \in {\mathcal{D}}_{x,z}\end{array}$$

(4)

$$\begin{array}{c}\hfill {\aleph }_{ir}\ge 0,i=1,2,\dots ,p,i\ne r.\end{array}$$

(5)

are satisfied, for all $\pi \in {\mathcal{D}}_{x,z}$, except at discontinuities.

Definition 3. 

Let ψ and ϕ be two real-valued functionals. The pair $(\psi ,\varphi )$ is said to be $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $(w,\zeta )\in \mathcal{S}\subset C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)$ with respect to $ϵ,\xi$, and θ if there exist ${\sigma }_0,{\sigma }_1:{\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}_{+}$, $ϵ,\theta :{\mathcal{D}}_{x,z}\times {\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}^n$, $\xi :{\mathcal{D}}_{x,z}\times {\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}^l$, with

$${ϵ(\pi ,\kappa ,\omega ,\kappa ,\omega )=ϵ(\pi ,\kappa ,\omega ,w,\zeta )|}_{\pi =x,z}=\xi (\pi ,\kappa ,\omega ,\kappa ,\omega )=0,{\vartheta }_0,{\vartheta }_1\in \mathbb{R},$$

such that, for all $(\kappa ,\omega )\in \mathcal{S}$, we have

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,w,\zeta )\left\{{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,w,\zeta )d\pi \right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \ge {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\kappa }(\pi ,w,\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{{\kappa }_{\mu }}(\pi ,w,\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\omega }(\pi ,w,\zeta )+{\vartheta }_0{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \hfill \end{array}$$

(6)

and

$$\begin{array}{cc}\hfill & -{\sigma }_1(\kappa ,\omega ,w,\zeta ){\int }_{{\mathcal{D}}_{x,z}}\varphi (\pi ,w,w_{\nu },\zeta )d\pi \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \ge {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\kappa }(\pi ,w,w_{\nu },\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,w,w_{\nu },\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\omega }(\pi ,w,w_{\nu },\zeta )+{\vartheta }_1{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi .\hfill \end{array}$$

If in the previous definition, the relation given in $\left(6\right)$ is valid as strict inequality, the pair $(\psi ,\varphi )$ is semi-strictly-$({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $(w,\zeta )\in \mathcal{S}\subset C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)$ with respect to $ϵ,\xi$, and θ.

Definition 4. 

The pair $(\psi ,\varphi )$ is said to be quasi-$({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $(w,\zeta )\in \mathcal{S}\subset C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)$ with respect to $ϵ,\xi$, and θ, if there exist ${\sigma }_0,{\sigma }_1:{\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}_{+}$, $ϵ,\theta :{\mathcal{D}}_{x,z}\times {\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}^n$, $\xi :{\mathcal{D}}_{x,z}\times {\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}^l$, with

$${ϵ(\pi ,\kappa ,\omega ,\kappa ,\omega )=ϵ(\pi ,\kappa ,\omega ,w,\zeta )|}_{\pi =x,z}=\xi (\pi ,\kappa ,\omega ,\kappa ,\omega )=0,{\vartheta }_0,{\vartheta }_1\in \mathbb{R},$$

such that, for all $(\kappa ,\omega )\in \mathcal{S}$, the inequality

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,w,\zeta )\left\{{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,w,\zeta )d\pi \right\}\le 0\hfill \end{array}$$

implies

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\kappa }(\pi ,w,\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{{\kappa }_{\mu }}(\pi ,w,\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\omega }(\pi ,w,\zeta )+{\vartheta }_0{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \le 0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & -{\sigma }_1(\kappa ,\omega ,w,\zeta ){\int }_{{\mathcal{D}}_{x,z}}\varphi (\pi ,w,w_{\nu },\zeta )d\pi \le 0\hfill \end{array}$$

involves

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\kappa }(\pi ,w,w_{\nu },\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,w,w_{\nu },\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\omega }(\pi ,w,w_{\nu },\zeta )+{\vartheta }_1{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \le 0.\hfill \end{array}$$

Definition 5. 

The pair $(\psi ,\varphi )$ is said to be strongly pseudo-$({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $(w,\zeta )\in \mathcal{S}\subset C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)$ with respect to $ϵ,\xi$, and θ, if there exist ${\sigma }_0,{\sigma }_1:{\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}_{+}$, $ϵ,\theta :{\mathcal{D}}_{x,z}\times {\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}^n$, $\xi :{\mathcal{D}}_{x,z}\times {\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}^l$, with ${ϵ(\pi ,\kappa ,\omega ,\kappa ,\omega )=ϵ(\pi ,\kappa ,\omega ,w,\zeta )|}_{\pi =x,z}=$$\xi (\pi ,\kappa ,\omega ,\kappa ,\omega )=0,{\vartheta }_0,{\vartheta }_1\in \mathbb{R}$, such that, for all $(\kappa ,\omega )\in \mathcal{S}$, the following relation

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\kappa }(\pi ,w,\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{{\kappa }_{\mu }}(\pi ,w,\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\omega }(\pi ,w,\zeta )+{\vartheta }_0{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \ge 0\hfill \end{array}$$

involves

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,w,\zeta )\left\{{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,w,\zeta )d\pi \right\}\ge 0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\kappa }(\pi ,w,w_{\nu },\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,w,w_{\nu },\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\omega }(\pi ,w,w_{\nu },\zeta )+{\vartheta }_1{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \ge 0\hfill \end{array}$$

implies

$$\begin{array}{cc}\hfill & -{\sigma }_1(\kappa ,\omega ,w,\zeta ){\int }_{{\mathcal{D}}_{x,z}}\varphi (\pi ,w,w_{\nu },\zeta )d\pi \ge 0.\hfill \end{array}$$

Definition 6. 

The pair $(\psi ,\varphi )$ is said to be pseudo- quasi- $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $(w,\zeta )\in \mathcal{S}\subset C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)$ with respect to $ϵ,\xi$, and θ, if there exist ${\sigma }_0,{\sigma }_1:{\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}_{+}$, $ϵ,\theta :{\mathcal{D}}_{x,z}\times {\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}^n$, $\xi :{\mathcal{D}}_{x,z}\times {\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}^l$, with ${ϵ(\pi ,\kappa ,\omega ,\kappa ,\omega )=ϵ(\pi ,\kappa ,\omega ,w,\zeta )|}_{\pi =x,z}=$$\xi (\pi ,\kappa ,\omega ,\kappa ,\omega )=0,{\vartheta }_0,{\vartheta }_1\in \mathbb{R}$, such that, for all $(\kappa ,\omega )\in \mathcal{S}$, are verified the conditions: if

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,w,\zeta )\left\{{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,w,\zeta )d\pi \right\}\le 0\hfill \end{array}$$

we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\kappa }(\pi ,w,\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{{\kappa }_{\mu }}(\pi ,w,\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\omega }(\pi ,w,\zeta )+{\vartheta }_0{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \le 0,\hfill \end{array}$$

and from

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\kappa }(\pi ,w,w_{\nu },\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,w,w_{\nu },\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\omega }(\pi ,w,w_{\nu },\zeta )+{\vartheta }_1{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \ge 0\hfill \end{array}$$

there result that

$$\begin{array}{cc}\hfill & -{\sigma }_1(\kappa ,\omega ,w,\zeta ){\int }_{{\mathcal{D}}_{x,z}}\varphi (\pi ,w,w_{\nu },\zeta )d\pi \ge 0.\hfill \end{array}$$

If, in the previous definition, we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\kappa }(\pi ,w,w_{\nu },\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,w,w_{\nu },\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\omega }(\pi ,w,w_{\nu },\zeta )+{\vartheta }_1{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \ge 0\hfill \end{array}$$

there result that

$$\begin{array}{cc}\hfill & -{\sigma }_1(\kappa ,\omega ,w,\zeta ){\int }_{{\mathcal{D}}_{x,z}}\varphi (\pi ,w,w_{\nu },\zeta )d\pi >0,\hfill \end{array}$$

then we say that the pair $(\psi ,\varphi )$ is strictly pseudo-quasi-$({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $(w,\zeta )\in \mathcal{S}\subset C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)$ with respect to $ϵ,\xi$, and θ.

Definition 7. 

The pair $(\psi ,\varphi )$ is said to be strongly pseudo-quasi-$({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $(w,\zeta )\in \mathcal{S}\subset C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)$ with respect to $ϵ,\xi$, and θ, if there exist ${\sigma }_0,{\sigma }_1:{\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}_{+}$, $ϵ,\theta :{\mathcal{D}}_{x,z}\times {\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}^n$, $\xi :{\mathcal{D}}_{x,z}\times {\left(C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)\right)}^2\to {\mathbb{R}}^l$, with ${ϵ(\pi ,\kappa ,\omega ,\kappa ,\omega )=ϵ(\pi ,\kappa ,\omega ,w,\zeta )|}_{\pi =x,z}=$$\xi (\pi ,\kappa ,\omega ,\kappa ,\omega )=0,{\vartheta }_0,{\vartheta }_1\in \mathbb{R}$, such that, for all $(\kappa ,\omega )\in \mathcal{S}$, are verified the conditions:

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\kappa }(\pi ,w,\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{{\kappa }_{\mu }}(\pi ,w,\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\omega }(\pi ,w,\zeta )+{\vartheta }_0{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \ge 0\hfill \end{array}$$

implies

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,w,\zeta )\left\{{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,w,\zeta )d\pi \right\}\ge 0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & -{\sigma }_1(\kappa ,\omega ,w,\zeta ){\int }_{{\mathcal{D}}_{x,z}}\varphi (\pi ,w,w_{\nu },\zeta )d\pi \le 0\hfill \end{array}$$

gives

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\kappa }(\pi ,w,w_{\nu },\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,w,w_{\nu },\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\varphi }_{\omega }(\pi ,w,w_{\nu },\zeta )+{\vartheta }_1{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \le 0.\hfill \end{array}$$

If, in the previous definition, we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\kappa }(\pi ,w,\zeta )+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{{\kappa }_{\mu }}(\pi ,w,\zeta )\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,w,\zeta )}^T{\psi }_{\omega }(\pi ,w,\zeta )+{\vartheta }_0{\parallel \theta (\pi ,\kappa ,\omega ,w,\zeta )\parallel }^2]d\pi \ge 0\hfill \end{array}$$

implies

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,w,\zeta )\left\{{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}\psi (\pi ,w,\zeta )d\pi \right\}>0,\hfill \end{array}$$

then we say that the pair $(\psi ,\varphi )$ is strictly, strongly pseudo-quasi-$({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $(w,\zeta )\in \mathcal{S}\subset C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)$ with respect to $ϵ,\xi$, and θ.

Lemma 2 

(Nahak and Behera [15]). Every $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ pair of functions is strongly pseudo-$({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$. The converse is false.

Remark 1. 

Taking into account all the above-mentioned definitions, their dependencies on each other (e.g., implications, subset relations) are the classical ones, namely, as well as connections between (strictly, strongly) invex, quasi-invex, and pseudo-invex functions (see Mititelu and Stancu-Minasian [16]). In addition, as we all know, the notion of invexity is the first version of generalized convexity; thus, the geometric insights or intuitive explanations for the proposed functional types follow from the properties of classical convexity.

3. Necessary and Sufficient Efficiency Conditions in (P)

Xiuhong [17] and Zhian and Qingkai [18] presented a duality theory for multiobjective control problems. However, the functions involved in their study were not of multiple integral type and the hypotheses used there are different from those used in this section. In addition, in the following, we establish some sufficient conditions of efficiency for the problem considered.

Theorem 2 

(Necessary efficiency conditions). If $({\kappa }^{*},{\omega }^{*})\in C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)$ is a properly efficient solution of $\left(P\right)$, which is assumed to be a normal optimal solution for $\left(P_r^{*}\right),$$r=1,2,\dots ,p$, then there exist ${\aleph }^{*}\in {\mathbb{R}}^p$ and $h^{*}:{\mathcal{D}}_{x,z}\to {\mathbb{R}}^n$ a piecewise differentiable function such that

$$\begin{array}{c}\hfill {{\aleph }^{*}}^T{\psi }_{\kappa }(\pi ,{\kappa }^{*},{\omega }^{*})+h^{*}{\left(\pi \right)}^T{\varphi }_{\kappa }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\end{array}$$

$$\begin{array}{c}\hfill ={\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}\left({{\aleph }^{*}}^T{\psi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\omega }^{*})+h^{*}{\left(\pi \right)}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\right),\pi \in {\mathcal{D}}_{x,z}\end{array}$$

(7)

$$\begin{array}{c}\hfill {{\aleph }^{*}}^T{\psi }_{\omega }(\pi ,{\kappa }^{*},{\omega }^{*})+h^{*}{\left(\pi \right)}^T{\varphi }_{\omega }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})=0,\pi \in {\mathcal{D}}_{x,z}\end{array}$$

(8)

$$\begin{array}{c}\hfill h^{*}{\left(\pi \right)}^T\varphi \left(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*}\right)=0,\pi \in {\mathcal{D}}_{x,z}\end{array}$$

(9)

$$\begin{array}{c}\hfill h^{*}\left(\pi \right)\geqq 0,{\aleph }^{*}\ge 0,\pi \in {\mathcal{D}}_{x,z}.\end{array}$$

(10)

Proof. 

Taking into account Lemma 1 and Theorem 1, the proof is apparent. □

Theorem 3 

(Sufficient efficiency conditions). Suppose $({\kappa }^{*},{\omega }^{*})\in \mathcal{S}$ and there exist ${\aleph }^{*}\in {\mathbb{R}}^p,{\aleph }^{*}>0$ and $h^{*}:{\mathcal{D}}_{x,z}\to {\mathbb{R}}^n$ a piecewise differentiable function satisfying relations (7) - (10), for all $\pi \in {\mathcal{D}}_{x,z}$. If $({{\aleph }^{*}}^T\psi ,h^{*}{\left(\pi \right)}^T\varphi )$ is $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $({\kappa }^{*},{\omega }^{*})\in \mathcal{S}\subset C^1\left({\mathcal{D}}_{x,z},{\mathbb{R}}^n\right)\times C\left({\mathcal{D}}_{x,z},{\mathbb{R}}^l\right)$, with respect to $ϵ,\xi$, and θ, and ${\vartheta }_0,{\vartheta }_1\in \mathbb{R}$, ${\vartheta }_0+{\vartheta }_1\ge 0$, with ${\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})>0$, for all $(\kappa ,\omega )\in \mathcal{S}$, then $({\kappa }^{*},{\omega }^{*})$ is a properly efficient solution in $\left(P\right)$.

Proof. 

Due to $({{\aleph }^{*}}^T\psi ,h^{*}{\left(\pi \right)}^T\varphi )$ is $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $({\kappa }^{*},{\omega }^{*})\in \mathcal{S}$, with respect to $ϵ,\xi$, and $\theta$, and ${\vartheta }_0,{\vartheta }_1\in \mathbb{R}$, ${\vartheta }_0+{\vartheta }_1\ge 0$, with ${\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})>0$, for all $(\kappa ,\omega )\in \mathcal{S}$, it follows

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\left\{{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,{\kappa }^{*},{\omega }^{*})d\pi \right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \ge {\int }_{{\mathcal{D}}_{x,z}}\left[ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left({\aleph }^{*}\right)}^T{\psi }_{\kappa }(\pi ,{\kappa }^{*},{\omega }^{*})+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left({\aleph }^{*}\right)}^T{\psi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\omega }^{*})\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left.+\xi {(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left({\aleph }^{*}\right)}^T{\psi }_{\omega }(\pi ,{\kappa }^{*},{\omega }^{*})+{\vartheta }_0{\parallel \theta (\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\parallel }^2\right]d\pi \hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & -{\sigma }_1(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*}){\int }_{{\mathcal{D}}_{x,z}}{\left(h^{*}\right)}^T\varphi (\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})d\pi \hfill \\ \hfill & \ge {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left(h^{*}\right)}^T{\varphi }_{\kappa }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\hfill \\ \hfill & +{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left(h^{*}\right)}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\hfill \\ \hfill & +\xi {(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left(h^{*}\right)}^T{\varphi }_{\omega }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})+{\vartheta }_1{\parallel \theta (\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\parallel }^2]d\pi .\hfill \end{array}$$

Since ${\left(h^{*}\right)}^T\varphi (\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})=0$, we obtain

$$\begin{array}{cc}\hfill 0& \ge {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left(h^{*}\right)}^T{\varphi }_{\kappa }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\hfill \\ \hfill & +{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left(h^{*}\right)}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\hfill \\ \hfill & +\xi {(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left(h^{*}\right)}^T{\varphi }_{\omega }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})+{\vartheta }_1{\parallel \theta (\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\parallel }^2]d\pi .\hfill \end{array}$$

By adding the above relations, we obtain

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\left\{{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,{\kappa }^{*},{\omega }^{*})d\pi \right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \ge {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T\left[{\left({\aleph }^{*}\right)}^T{\psi }_{\kappa }(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_{\kappa }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\right]\hfill \end{array}$$

$$\begin{array}{c}\hfill +{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T[{\left({\aleph }^{*}\right)}^T{\psi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})]\end{array}$$

$$\begin{array}{c}\hfill +\xi {(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T[{\left({\aleph }^{*}\right)}^T{\psi }_{\omega }(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_{\omega }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})]\end{array}$$

$$\begin{array}{cc}\hfill & \left.+({\rho }_0+{\rho }_1){\parallel \theta (\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\parallel }^2\right]d\pi .\hfill \end{array}$$

(11)

The relation in $\left(11\right)$ becomes

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\left\{{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,{\kappa }^{*},{\omega }^{*})d\pi \right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \ge {\int }_{{\mathcal{D}}_{x,z}}\left[ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T\right({\left({\aleph }^{*}\right)}^T{\psi }_{\kappa }(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_{\kappa }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\hfill \end{array}$$

$$\begin{array}{c}\hfill -{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}({\left({\aleph }^{*}\right)}^T{\psi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*}))\end{array}$$

$$\begin{array}{c}\hfill +\xi {(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T({\left({\aleph }^{*}\right)}^T{\psi }_{\omega }(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_{\omega }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*}))\end{array}$$

$$\begin{array}{cc}\hfill & \left.+({\rho }_0+{\rho }_1){\parallel \theta (\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\parallel }^2\right]d\pi .\hfill \end{array}$$

Considering the relations in $\left(7\right),\left(8\right)$, and ${\rho }_0+{\rho }_1\ge 0$, we get

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\left\{{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,{\kappa }^{*},{\omega }^{*})d\pi \right\}\ge 0.\hfill \end{array}$$

We know, by hypothesis, that ${\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})>0$, so, it follows

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,{\kappa }^{*},{\omega }^{*})d\pi \ge 0,\hfill \end{array}$$

or, equivalent,

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,\kappa ,\omega )d\pi \ge {\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,{\kappa }^{*},{\omega }^{*})d\pi ,\hfill \end{array}$$

which means that $({\kappa }^{*},{\omega }^{*})$ minimizes ${\displaystyle {\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,\kappa ,\omega )d\pi }$ over $\mathcal{S}$, with ${\aleph }^{*}>0$. Therefore, by using Theorem 1 in Bector and Husain [3], $({\kappa }^{*},{\omega }^{*})$ is a properly efficient solution in $\left(P\right)$. □

Theorem 4. 

Consider $({\kappa }^{*},{\omega }^{*})\in \mathcal{S}$ and there exist ${\aleph }^{*}\in {\mathbb{R}}^p,{\aleph }^{*}>0$ and $h^{*}:{\mathcal{D}}_{x,z}\to {\mathbb{R}}^n$ a piecewise differentiable function such that for all $\pi \in {\mathcal{D}}_{x,z}$ the relations (7) - (10) are satisfied. If $({{\aleph }^{*}}^T\psi ,h^{*}{\left(\pi \right)}^T\varphi )$ is $semi-strictly-({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $({\kappa }^{*},{\omega }^{*})$, with respect to $ϵ,\xi$, and θ, and ${\vartheta }_0,{\vartheta }_1\in \mathbb{R}$, ${\vartheta }_0+{\vartheta }_1\ge 0$, with ${\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})>0$, for all $(\kappa ,\omega )\in \mathcal{S}$, then $({\kappa }^{*},{\omega }^{*})$ is an efficient solution of $\left(P\right)$.

Proof. 

Let us suppose that $({\kappa }^{*},{\omega }^{*})$ is not an efficient solution of $\left(P\right)$. Thus, there exist $(\kappa ,\omega )\in \mathcal{S}$ and, for at least an index $s=1,2,\dots ,p$, we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}{f}_j(\pi ,\kappa ,\omega )d\pi \le {\int }_{{\mathcal{D}}_{x,z}}{f}_j(\pi ,{\kappa }^{*},{\omega }^{*})d\pi \hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_{{\mathcal{D}}_{x,z}}{f}_s(\pi ,\kappa ,\omega )d\pi <{\int }_{{\mathcal{D}}_{x,z}}{f}_s(\pi ,{\kappa }^{*},{\omega }^{*})d\pi .\hfill \end{array}$$

But, since ${\aleph }^{*}>0$ and ${\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})>0$, considering $\left(26\right)$, we get

$$\begin{array}{cc}\hfill & {\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\left\{{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,{\kappa }^{*},{\omega }^{*})d\pi \right\}\le 0.\hfill \end{array}$$

We take in consideration that $h^{*}{\left(\pi \right)}^T\varphi \left(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*}\right)=0,h^{*}\left(\pi \right)\ge 0,\pi \in {\mathcal{D}}_{x,z}$, and obtain

$$\begin{array}{cc}\hfill & -{\sigma }_1(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*}){\int }_{{\mathcal{D}}_{x,z}}{\left(h^{*}\right)}^T\varphi (\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})d\pi =0.\hfill \end{array}$$

As $({{\aleph }^{*}}^T\psi ,h^{*}{\left(\pi \right)}^T\varphi )$ is semi-strictly-$({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ at $({\kappa }^{*},{\omega }^{*})$, with respect to $ϵ,\xi$, and $\theta$, we have

$$\begin{array}{c}\hfill 0\ge {\sigma }_0(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})\left\{{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,\kappa ,\omega )d\pi -{\int }_{{\mathcal{D}}_{x,z}}{\left({\aleph }^{*}\right)}^T\psi (\pi ,{\kappa }^{*},{\omega }^{*})d\pi \right\}\end{array}$$

$$\begin{array}{c}\hfill >{\int }_{{\mathcal{D}}_{x,z}}\left[ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T\right({\left({\aleph }^{*}\right)}^T{\psi }_{\kappa }(\pi ,{\kappa }^{*},{\omega }^{*})+{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left({\aleph }^{*}\right)}^T{\psi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\omega }^{*})\end{array}$$

$$\begin{array}{c}\hfill +\xi {(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left({\aleph }^{*}\right)}^T{\psi }_{\omega }(\pi ,{\kappa }^{*},{\omega }^{*})+{\rho }_0\parallel \theta (\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*}){\parallel }^2]d\pi \end{array}$$

and

$$\begin{array}{c}\hfill 0=-{\sigma }_1(\kappa ,\omega ,{\kappa }^{*},{\omega }^{*}){\int }_{{\mathcal{D}}_{x,z}}{\left(h^{*}\right)}^T\varphi (\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})d\pi \end{array}$$

$$\begin{array}{cc}\hfill & >{\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left(h^{*}\right)}^T{\varphi }_{\kappa }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\hfill \\ \hfill & +{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left(h^{*}\right)}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\xi {(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T{\left(h^{*}\right)}^T{\varphi }_o(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})+{\rho }_1\parallel \theta (\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*}){\parallel }^2]d\pi .\hfill \end{array}$$

By joining the relations given above, we obtain

$$\begin{array}{c}\hfill {\int }_{{\mathcal{D}}_{x,z}}[ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T\left({\left({\aleph }^{*}\right)}^T{\psi }_{\kappa }(\pi ,{\kappa }^{*},{\omega }^{*})\right)+{\left(h^{*}\right)}^T{\varphi }_{\kappa }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*}))\end{array}$$

$$\begin{array}{c}\hfill +{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T({\left({\aleph }^{*}\right)}^T{\psi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*}))\end{array}$$

$$\begin{array}{c}\hfill +\xi {(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T({\left({\aleph }^{*}\right)}^T{\psi }_{\omega }(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_o(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*}))\end{array}$$

$$\begin{array}{c}\hfill ({\rho }_0+{\rho }_1)\parallel \theta (\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*}){\parallel }^2]d\pi <0,\end{array}$$

or, equivalently,

$$\begin{array}{c}\hfill {\int }_{{\mathcal{D}}_{x,z}}\left[ϵ{(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T\right({\left({\aleph }^{*}\right)}^T{\psi }_{\kappa }(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_{\kappa }(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})\end{array}$$

$$\begin{array}{c}\hfill -{\displaystyle \frac{\partial }{\partial {\pi }^{\mu }}}({\left({\aleph }^{*}\right)}^T{\psi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_{{\kappa }_{\mu }}(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*})))\end{array}$$

$$\begin{array}{c}\hfill +\xi {(\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*})}^T({\left({\aleph }^{*}\right)}^T{\psi }_{\omega }(\pi ,{\kappa }^{*},{\omega }^{*})+{\left(h^{*}\right)}^T{\varphi }_o(\pi ,{\kappa }^{*},{\kappa }_{\mu }^{*},{\omega }^{*}))\end{array}$$

$$\begin{array}{c}\hfill ({\rho }_0+{\rho }_1)\parallel \theta (\pi ,\kappa ,\omega ,{\kappa }^{*},{\omega }^{*}){\parallel }^2]d\pi <0,\end{array}$$

which is contrary to Theorem 2 and ${\rho }_0+{\rho }_1\ge 0$. This means that our supposition that $({\kappa }^{*},{\omega }^{*})$ is not an efficient solution of $\left(P\right)$ is false. So, we obtain that $({\kappa }^{*},{\omega }^{*})$ is an efficient solution of $\left(P\right)$. □

Example 1. 

We formulate the constrained double-cost variational control problem:

$$\begin{array}{cc}\hfill \left(P\right)\underset{\left(\kappa \right(·),\omega (·\left)\right)}{min}& \left({\int }_{{\mathcal{D}}_{x,z}}({\kappa }^2+{\displaystyle \frac{5}{4}}\omega )d{\pi }^{1}d{\pi }^2,{\int }_{{\mathcal{D}}_{x,z}}{\omega }^{2}d{\pi }^{1}d{\pi }^2\right)\hfill \\ \hfill \mathrm{subject\; to}& \kappa (\kappa -3)\le 0,\hfill \\ & {\displaystyle \frac{\partial \kappa }{\partial {\pi }^1}}=2-\omega ,\hfill \\ & {\displaystyle \frac{\partial \kappa }{\partial {\pi }^2}}=2-\omega ,\hfill \\ & \kappa (0,0)=0,\kappa \left({\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{4}}\right)={\displaystyle \frac{1}{2}},\hfill \end{array}$$

where $\pi =({\pi }^1,{\pi }^2)\in {\mathcal{D}}_{x,z}={\left[0,{\displaystyle \frac{1}{4}}\right]}^2.$

For

$$\overline{\aleph }=({\overline{\aleph }}_1,{\overline{\aleph }}_2)=({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}),\overline{h}=0,\overline{\gamma }=({\overline{\gamma }}_1,{\overline{\gamma }}_2)=({\pi }^1{\pi }^2,{\pi }^1{\pi }^2),$$

the pair

$$(\overline{\kappa },\overline{\omega })=({\pi }^1+{\pi }^2,1)$$

is a feasible solution to the problem(P), satisfying relations given in (7)–(10) of Theorem 2. Further, for

$$\psi =({\psi }_1,{\psi }_2)=({\kappa }^2+{\displaystyle \frac{5}{4}}\omega ,{\omega }^2),$$

$$\varphi =\kappa (\kappa -3),$$

and

$$f=(f_1,f_2)=({\displaystyle \frac{\partial \kappa }{\partial {\pi }^1}}-2+\omega ,{\displaystyle \frac{\partial \kappa }{\partial {\pi }^2}}-2+\omega ),$$

it can be easily verified that $({\overline{\aleph }}^T\psi ,\overline{h}{\left(\pi \right)}^T\varphi )$ and $({\overline{\aleph }}^T\psi ,\overline{\gamma }{\left(\pi \right)}^{T}f)$ are $semi-strictly-(1,1)-(1,1)-type-I$ at $(\overline{\kappa },\overline{\omega })$, with respect to $ϵ=\kappa -\overline{\kappa },\xi =\omega -\overline{\omega }$, and $\theta \in {\mathbb{R}}^n$, for all $(\kappa ,\omega )\in \mathcal{S}$. In consequence, Theorem 4 is verified, that is, $(\overline{\kappa },\overline{\omega })$ is an efficient solution of $\left(P\right)$.

4. Conclusions and Further Research Directions

In this paper, we have established necessary and sufficient efficiency conditions in new classes of multi-cost variational models. In this regard, we have introduced the new concept of $({\vartheta }_0,{\vartheta }_1)-({\sigma }_0,{\sigma }_1)-type-I$ functionals determined by multiple integrals. Also, to this end, we used innovative proofs for the main results. As further research directions, we can mention the studies done by Treanţă [19,20,21] in its works by considering the new family of generalized functionals introduced in the current article. In terms of concreteness, the objectives are as follows:

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The appearance of (path-dependent or path-independent) curvilinear integrals as functionals instead of multiple integrals;

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The analysis of well-posedness associated with this type of extremization models;

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Establishing new efficiency criteria based on saddle-points or modified cost functional approach.