Robust Efficiency Conditions in Multiple-Objective Fractional Variational Control Problems

Abstract

The aim of this study is to investigate multi-dimensional vector variational problems considering data uncertainty in each of the objective functional and constraints. We establish the robust necessary and sufficient efficiency conditions such that any robust feasible solution could be the robust weakly efficient solution for the problems under consideration. Emphatically, we present robust efficiency conditions for multi-dimensional scalar, vector, and vector fractional variational problems by using the notion of a convex functional.

1. Introduction

Many real-world problems, including those in engineering, production planning, and finance, are addressed through multi-objective programming problems. Research over the years has been progressively centered on multi-objective programming problem in various areas of mathematics such as optimal control theory, game theory, statistics, and finance. Numerous researchers have extensively examined the necessary and sufficient optimality conditions and important duality theorems for multi-objective control problems [1,2,3,4]. Mititelu and Treanţă [5] established necessary and sufficient efficiency conditions for multi-objective control problems that incorporate multiple integrals.

Multi-dimensional (multi-time) optimization problems have recently found applications in many fields of the mathematical, economic, and engineering sciences. Additionally, a number of application-oriented problems that arise in a wide range of scientific and engineering disciplines, notably structural optimization, shape-optimization in fluid mechanics and medicine, and optimal process control, which require constraints as partial differential equations/inequations (PDEs and PDIs). The optimality conditions of control problems with constraints as nonlinear equalities and inequalities are developed and investigated by a lot of authors. Treanţă [6] derived efficiency conditions for a class of multi-objective fractional-control problems together with a multi-time concept involving partial derivatives of higher order. In [7], multi-time variational problems of some functionals governed by second-order Lagrangians are considered and optimality conditions are developed. The latest developments in this field can be seen in [8,9,10].

Since empirical mechanisms are extremely complicated and frequently involve uncertainty in the original data, many researchers have focused on optimization problems incorporating uncertainty in control problems. Treanţă and Das [11] discussed robustness in optimization problems with curvilinear integrals having applications in mechanical work. A new modified robust control problem involving mixed constraints and second-order PDEs is discussed in [12]. Baranwal et al. [13] formulated two important duals in the literature, namely, Mond Weir and Wolfe type, for a given multi-time control problem amidst data uncertainty involving constraints as partial differential equations of the first order and further established robust duality theorems. Recently, for a robust variational control problem incorporating data uncertainty, the necessary and sufficient optimality conditions are established in [14].

Encouraged by the aforementioned research work, we study multi-dimensional a robust variational control problem involving partial derivatives of the first order in constraints. Considering uncertainty in both objective functional and constraints, we aim to investigate robust efficiency conditions for scalar, multi-objective, and multi-objective fractional-control problems using the notion of a convex functional. The limitations of the proposed approach could be the convexity assumptions. However, we can overcome these limitations by using a generalized convexity (for example, invexity). This can be the topic of a new future work.

The article is organized in the following manner. We start Section 2 by providing some important preliminaries. Further, we present the formulation of the robust multi-dimensional variational control problem (scalar, vector, and vector fractional) concerning uncertainty in both objective functional and constraints. The non-fractional-control problem associated with the robust fractional-control problem is also provided. In Section 3, we develop robust necessary efficiency conditions for robust multi-dimensional scalar, vector, and vector fractional-control problems defined in Section 2. Subsequently, Section 4 provides robust sufficient efficiency conditions for the problems under consideration using the convexity assumptions of involved integrals.

2. Problem Formulation and Preliminaries

Consider the euclidean spaces of the dimensions $p,q,r$, and n as ${\mathbb{R}}^p,{\mathbb{R}}^q,{\mathbb{R}}^r$, and ${\mathbb{R}}^n$, respectively, and a compact subset denoted by $▵$ in ${\mathbb{R}}^p$. Define the multi-time argument $t=\left(t^{\alpha }\right)$, $\alpha =\overline{1,p} (\mathrm{or},\mathrm{equivalently},\alpha \in \{1,\dots ,p\}$, such that t $\in ▵$. Define the state functions having continuous first-order partial derivatives as $\varrho =\left({\varrho }^i\right):▵\to {\mathbb{R}}^q$, where $\varrho \in \Lambda$, and denote the continuous control functions in the space M as $\mu =\left({\mu }^j\right):▵\to {\mathbb{R}}^r$. Next, the following rules are considered for any two vectors $\mathit{a},\mathit{b} \in {\mathbb{R}}^n$:

(i)

$\mathit{a}<\mathit{b}\iff {\mathit{a}}_{\mathit{k}}<{\mathit{b}}_{\mathit{k}},\forall \mathit{k}=\overline{\mathit{1},\mathit{n}},$

(ii)

$\mathit{a}=\mathit{b}\iff {\mathit{a}}_{\mathit{k}}={\mathit{b}}_{\mathit{k}},\forall \mathit{k}=\overline{\mathit{1},\mathit{n}},$

(iii)

$\mathit{a}\leqq \mathit{b}\iff {\mathit{a}}_{\mathit{k}}\le {\mathit{b}}_{\mathit{k}},\forall \mathit{k}=\overline{\mathit{1},\mathit{n}},$

(iv)

$\mathit{a}\le \mathit{b}\iff {\mathit{a}}_{\mathit{k}}\le {\mathit{b}}_{\mathit{k}},\forall \mathit{k}=\overline{\mathit{1},\mathit{n}}\mathrm{and}{\mathit{a}}_{\mathit{k}}<{\mathit{b}}_{\mathit{k}}\mathrm{for}\mathrm{some}\mathit{k}.$

Additionally, we denote $\mathrm{\Omega }$ as $\mathrm{\Omega }:=(t,\varrho (t),\mu (t\left)\right)$, $\overline{\mathrm{\Omega }}$ as $\overline{\mathrm{\Omega }}:=(t,\overline{\varrho }\left(t\right),\overline{\mu }\left(t\right)),dv=d{t}^1\dots d{t}^p$, ${\displaystyle {\varrho }_{\alpha }\left(t\right):=\frac{\partial \varrho }{\partial t^{\alpha }}\left(t\right)}$. Now, we consider the following multi-dimensional uncertain vector control problem$\left(\mathcal{UVCP}\right)$, considering data uncertainty in each one of the objective functional and constraints:

$${\displaystyle \left(\mathcal{UVCP}\right)\underset{(\varrho ,\mu )}{min}\left\{\mathcal{I}(\varrho ,\mu ):={\int }_{▵}f(\mathrm{\Omega },\varsigma )dv\right\}}$$

$$\mathrm{subject\; to}$$

$${\displaystyle \varphi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\tau )\leqq 0,}$$

$${\displaystyle \psi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\eta )=0,}$$

$${\displaystyle t\in ▵,\varrho \left(t_0\right)={\varrho }_0,\varrho \left(t_1\right)={\varrho }_1,}$$

where

$${\displaystyle {\int }_{▵}f(\mathrm{\Omega },\varsigma )dv:=\left({\displaystyle {\int }_{▵}{f}_1(\mathrm{\Omega },{\varsigma }_1)dv,\dots ,{\int }_{▵}{f}_p(\mathrm{\Omega },{\varsigma }_p)dv}\right),}$$

and

$$f_{\kappa }:\Lambda \times M\times V_{\kappa }\to \mathbb{R},\kappa =\overline{1,p};f=(f_1,\dots ,f_p),$$

$${\varphi }_l:J^1\left(▵,{\mathbb{R}}^q\right)\times M\times T_l\to {\mathbb{R}}^m,l=\overline{1,m};\varphi =({\varphi }_1,\dots ,{\varphi }_m),$$

$${\psi }_{\zeta }:J^1\left(▵,{\mathbb{R}}^q\right)\times M\times N_b\to {\mathbb{R}}^n,b=\overline{1,n},\psi =({\psi }_1,\dots ,{\psi }_n),$$

are functionals belonging to the class of continuous first-order partial derivatives (almost everywhere), the uncertainty parameters are represented as $\varsigma =\left({\varsigma }_{\kappa }\right),\tau =\left({\tau }_l\right)$ and $\eta =\left({\eta }_b\right)$, belonging to convex compact subsets $V=\left(V_{\kappa }\right)\subset {\mathbb{R}}^p,T=\left(T_l\right)\subset {\mathbb{R}}^m$ and $N=\left(N_b\right)\subset {\mathbb{R}}^n$, respectively. Additionally, define the first-order jet bundle for $▵$ and ${\mathbb{R}}^q$ as $J^1\left(▵,{\mathbb{R}}^q\right)$.

The robust counterpart for the above-defined robust multi-dimensional vector variational control problem $\left(\mathcal{UVCP}\right)$ is presented as follows:

$${\displaystyle \left(RoUVCP\right)\underset{(\varrho ,\mu )}{min}{\int }_{▵}\underset{\varsigma \in V}{max}f(\mathrm{\Omega },\varsigma )dv}$$

$$\mathrm{subject\; to}$$

$${\displaystyle \varphi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\tau )\leqq 0,\tau \in T}$$

$${\displaystyle \psi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\eta )=0,\eta \in N}$$

$${\displaystyle t\in ▵,\varrho \left(t_0\right)={\varrho }_0,\varrho \left(t_1\right)={\varrho }_1.}$$

We define the feasible solutions set of $\left(RoUVCP\right)$, also known as the robust feasible solution set for the problem $\left(\mathcal{UVCP}\right)$, as follows:

$${\displaystyle Z=\{(\varrho ,\mu )\in \Lambda \times M:\varphi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\tau )\leqq 0,}$$

$${\displaystyle \psi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\eta )=0,\varrho \left(t_0\right)={\varrho }_0,\varrho \left(t_1\right)={\varrho }_1,t\in ▵,\tau \in T,\eta \in N\}.}$$

Now, for the robust multi-dimensional vector variational control problem $\left(\mathcal{UVCP}\right)$, we develop the concept of the robust weakly efficient solution. It is utilized to establish the robust efficiency conditions for the problem $\left(\mathcal{UVCP}\right)$.

Definition 1.

A point $(\overline{\varrho },\overline{\mu })\in Z$ is considered to be the robust weakly efficient solution for the given multi-dimensional vector variational problem $\left(\mathcal{UVCP}\right)$ if

$$\begin{array}{c}\hfill {\int }_{▵}\underset{\varsigma \in V}{max}f(\overline{\mathrm{\Omega }},\varsigma )dv<{\int }_{▵}\underset{\varsigma \in V}{max}f(\mathrm{\Omega },\varsigma )dv,\end{array}$$

for all feasible points $(\varrho ,\mu )$ in Z.

Definition 2.

A point $(\overline{\varrho },\overline{\mu })\in Z$ is considered to be the robust efficient solution for the given multi-dimensional vector variational problem $\left(\mathcal{UVCP}\right)$ if

$$\begin{array}{c}\hfill {\int }_{▵}\underset{\varsigma \in V}{max}f(\overline{\mathrm{\Omega }},\varsigma )dv\le {\int }_{▵}\underset{\varsigma \in V}{max}f(\mathrm{\Omega },\varsigma )dv,\end{array}$$

for all feasible points $(\varrho ,\mu )$ in Z.

Subsequently, the following robust multi-dimensional uncertain vector fractional-control problem $\left(UVFCP\right)$ is developed as:

$${\displaystyle \left(UVFCP\right)\underset{(\varrho ,\mu )}{min}\left\{\mathcal{J}(\varrho ,\mu ):=\left(\frac{{\displaystyle {\int }_{▵}{f}_1\left(\mathrm{\Omega },{\varsigma }_1\right)dv}}{{\displaystyle {\int }_{▵}{g}_1\left(\mathrm{\Omega },{\gamma }_1\right)dv}},\dots ,\frac{{\displaystyle {\int }_{▵}{f}_p\left(\mathrm{\Omega },{\varsigma }_p\right)dv}}{{\displaystyle {\int }_{▵}{g}_p\left(\mathrm{\Omega },{\gamma }_p\right)dv}}\right)\right\}}$$

$$\mathrm{subject\; to} (\varrho ,\mu )\in Z,$$

where it is assumed that ${\displaystyle {\int }_{▵}{g}_{\kappa }\left(\mathrm{\Omega },{\gamma }_{\kappa }\right)dv>0,\kappa =\overline{1,p}}$, $\varsigma =\left({\varsigma }_{\kappa }\right),\gamma =\left({\gamma }_{\kappa }\right),\tau =\left({\tau }_l\right)$ and $\eta =\left({\eta }_b\right)$ represent the uncertainty parameters of the convex compact subsets $\Lambda =\left({\Lambda }_{\kappa }\right)\subset {\mathbb{R}}^p, Y=\left(Y_{\kappa }\right)\subset {\mathbb{R}}^p, T=\left(T_l\right)\subset {\mathbb{R}}^m$, and $N=\left(N_b\right)\subset {\mathbb{R}}^n$, respectively.

The robust counterpart for the above-defined robust multi-dimensional vector fractional variational control problem $\left(\mathcal{UVCP}\right)$ is defined as follows:

$${\displaystyle \left(RoUVFCP\right)\underset{(\varrho ,\mu )}{min}\left(\frac{{\displaystyle {\int }_{▵}\underset{{\varsigma }_1\in V_1}{max}{f}_1\left(\mathrm{\Omega },{\varsigma }_1\right)dv}}{{\displaystyle {\int }_{▵}\underset{{\gamma }_1\in Y_1}{min}{g}_1\left(\mathrm{\Omega },{\gamma }_1\right)dv}},\dots ,\frac{{\displaystyle {\int }_{▵}\underset{{\varsigma }_p\in V_p}{max}{f}_p\left(\mathrm{\Omega },{\varsigma }_p\right)dv}}{{\displaystyle {\int }_{▵}\underset{{\gamma }_p\in Y_p}{min}{g}_p\left(\mathrm{\Omega },{\gamma }_p\right)dv}}\right)}$$

$$\mathrm{subject\; to}$$

$${\displaystyle \varphi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\tau )\leqq 0,t\in ▵,\tau \in T}$$

$${\displaystyle \psi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\eta )=0,t\in ▵,\eta \in N}$$

$${\displaystyle \varrho \left(t_0\right)={\varrho }_0,\varrho \left(t_1\right)={\varrho }_1.}$$

Definition 3.

A point $(\overline{\varrho },\overline{\mu })\in Z$ is considered to be the robust weakly efficient solution for the given multi-dimensional vector fractional variational problem $\left(UVFCP\right)$ if

$${\displaystyle \frac{{\displaystyle {\int }_{▵}\underset{\varsigma \in V}{max}f\left(\overline{\mathrm{\Omega }},\varsigma \right)dv}}{{\displaystyle {\int }_{▵}\underset{\gamma \in Y}{min}g\left(\overline{\mathrm{\Omega }},\gamma \right)dv}}<\frac{{\displaystyle {\int }_{▵}\underset{\varsigma \in V}{max}f\left(\mathrm{\Omega },\varsigma \right)dv}}{{\displaystyle {\int }_{▵}\underset{\gamma \in Y}{min}g\left(\mathrm{\Omega },\gamma \right)dv}},}$$

for all feasible points $(\varrho ,\mu )$ in Z.

Definition 4.

A point $(\overline{\varrho },\overline{\mu })\in Z$ is considered to be the robust efficient solution for the given multi-dimensional vector fractional variational problem $\left(UVFCP\right)$ if

$${\displaystyle \frac{{\displaystyle {\int }_{▵}\underset{\varsigma \in V}{max}f\left(\overline{\mathrm{\Omega }},\varsigma \right)dv}}{{\displaystyle {\int }_{▵}\underset{\gamma \in Y}{min}g\left(\overline{\mathrm{\Omega }},\gamma \right)dv}}\le \frac{{\displaystyle {\int }_{▵}\underset{\varsigma \in V}{max}f\left(\mathrm{\Omega },\varsigma \right)dv}}{{\displaystyle {\int }_{▵}\underset{\gamma \in Y}{min}g\left(\mathrm{\Omega },\gamma \right)dv}},}$$

for all feasible points $(\varrho ,\mu )$ in Z.

Now, on the line of Jagannathan [15], and following Mititelu and Treanţă [5], we present a non-fractional uncertain scalar control problem ${\left(NonFUSCP\right)}_w$, associated with $\left(UVFCP\right)$, as follows:

$${\displaystyle {\left(NonFUSCP\right)}_w\underset{(\varrho ,\mu )}{min}{\int }_{▵}\left[f_w\left(\mathrm{\Omega },{\varsigma }_w\right)-Y_w^0{g}_w\left(\mathrm{\Omega },{\gamma }_w\right)\right]dv}$$

$$\mathrm{subject\; to}$$

$${\displaystyle (\varrho ,\mu )\in Z}$$

$${\displaystyle {\int }_{▵}\left[f_{\kappa }\left(\mathrm{\Omega },{\varsigma }_{\kappa }\right)-Y_{\kappa }^0{g}_{\kappa }\left(\mathrm{\Omega },{\gamma }_{\kappa }\right)\right]dv\le 0,\kappa =\overline{1,p},\kappa \ne w,}$$

where ${\displaystyle Y_{\kappa }^0}$ is a real positive scalar (see Treanţă [5]).

The robust counterpart to ${\left(NonFUSCP\right)}_w$ is given by

$${\displaystyle \left(RoNonFUSCP\right)\underset{(\varrho ,\mu )}{min}\left\{{\int }_{▵}\underset{{\varsigma }_w\in V_w}{max}{f}_w\left(\mathrm{\Omega },{\varsigma }_w\right)dv-Y_w^0{\int }_{▵}\underset{{\gamma }_w\in Y_w}{min}{g}_w\left(\mathrm{\Omega },{\gamma }_w\right)dv\right\}}$$

$$\mathrm{subject\; to}$$

$${\displaystyle (\varrho ,\mu )\in Z}$$

$${\displaystyle {\int }_{▵}\left[f_{\kappa }\left(\mathrm{\Omega },{\varsigma }_{\kappa }\right)-Y_{\kappa }^0{g}_{\kappa }\left(\mathrm{\Omega },{\gamma }_{\kappa }\right)\right]dv\le 0,\kappa =\overline{1,p},\kappa \ne w,}$$

Definition 5.

A point $(\overline{\varrho },\overline{\mu })\in Z$ is considered to be the robust weak optimal solution for the given multi-dimensional non-fractional-control problem ${\left(NonFUSCP\right)}_w$ if

$${\displaystyle {\int }_{▵}\underset{{\varsigma }_w\in V_w}{max}{f}_w\left(\overline{\mathrm{\Omega }},{\varsigma }_w\right)dv-Y_w^0{\int }_{▵}\underset{{\gamma }_w\in Y_w}{min}{g}_w\left(\overline{\mathrm{\Omega }},{\gamma }_w\right)dv}$$

$${\displaystyle <{\int }_{▵}\underset{{\varsigma }_w\in V_w}{max}{f}_w\left(\mathrm{\Omega },{\gamma }_w\right)dv-Y_w^0{\int }_{▵}\underset{{\gamma }_w\in Y_w}{min}{g}_w\left(\mathrm{\Omega },{\gamma }_w\right)dv,}$$

for all feasible points $(\varrho ,\mu )$ in ${\left(NonFUSCP\right)}_w$.

Definition 6.

A point $(\overline{\varrho },\overline{\mu })\in Z$ is considered to be the robust optimal solution in the robust multi-dimensional non-fractional-control problem ${\left(NonFUSCP\right)}_w$ if

$${\displaystyle {\int }_{▵}\underset{{\varsigma }_w\in V_w}{max}{f}_w\left(\overline{\mathrm{\Omega }},{\varsigma }_w\right)dv-Y_w^0{\int }_{▵}\underset{{\gamma }_w\in Y_w}{min}{g}_w\left(\overline{\mathrm{\Omega }},{\gamma }_w\right)dv}$$

$${\displaystyle \le {\int }_{▵}\underset{{\varsigma }_w\in V_w}{max}{f}_w\left(\mathrm{\Omega },{\gamma }_w\right)dv-Y_w^0{\int }_{▵}\underset{{\gamma }_w\in Y_w}{min}{g}_w\left(\mathrm{\Omega },{\gamma }_w\right)dv,}$$

for all feasible points $(\varrho ,\mu )$ in ${\left(NonFUSCP\right)}_w$.

Remark 1.

The robust (weak) efficient/optimal solutions to $\left(UVFCP\right)$ and ${\left(NonFUSCP\right)}_w$ are also (weak) efficient/optimal solutions to $\left(RoUVFCP\right)$ and $\left(RoNonFUSCP\right)$, respectively.

3. Robust Necessary Efficiency Conditions in Scalar, Vector, and Vector Fractional Variational Problems

We start with the formulation of a multi-dimensional robust uncertain scalar control problem $\left(USCP\right)$ as follows:

$${\displaystyle \left(USCP\right)\underset{(\varrho ,\mu )}{min}{\int }_{▵}P(\mathrm{\Omega },\varsigma )dv}$$

$$\mathrm{subject\; to} (\varrho ,\mu )\in Z.$$

The robust counterpart for the above robust multi-dimensional scalar variational problem $\left(USCP\right)$ is developed as follows:

$${\displaystyle \left(RoUSCP\right)\underset{(\varrho ,\mu )}{min}{\int }_{▵}\underset{\varsigma \in V\subset \mathbb{R}}{max}P(\mathrm{\Omega },\varsigma )dv}$$

$$\mathrm{subject\; to} (\varrho ,\mu )\in Z.$$

Definition 7.

A point $(\overline{\varrho },\overline{\mu })\in Z$ is considered to be the robust weak optimal solution for the given robust multi-dimensional scalar variational problem $\left(USCP\right)$ if

$$\begin{array}{c}\hfill {\int }_{▵}\underset{\varsigma \in V}{max}P(\overline{\mathrm{\Omega }},\varsigma )dv<{\int }_{▵}\underset{\varsigma \in V}{max}P(\mathrm{\Omega },\varsigma )dv,\end{array}$$

for all feasible points $(\varrho ,\mu )$ in $\left(USCP\right)$.

Definition 8.

A point $(\overline{\varrho },\overline{\mu })\in Z$ is considered to be the robust optimal solution for the given robust multi-dimensional scalar variational problem $\left(USCP\right)$ if

$$\begin{array}{c}\hfill {\int }_{▵}\underset{\varsigma \in V}{max}P(\overline{\mathrm{\Omega }},\varsigma )dv\le {\int }_{▵}\underset{\varsigma \in V}{max}P(\mathrm{\Omega },\varsigma )dv,\end{array}$$

for all feasible points $(\varrho ,\mu )$ in $\left(USCP\right)$.

For a given feasible solution ${\displaystyle (\overline{\varrho },\overline{\mu })}$, the following result develops the robust necessary optimality conditions for the given robust scalar variational problem $\left(USCP\right)$ (see, Additionally, Treanţă [16]):

Theorem 1

(Robust necessary efficiency conditions for $\left(USCP\right)$). Consider a robust weak optimal solution $(\overline{\varrho },\overline{\mu })\in Z$ for the given robust scalar variational problem $\left(USCP\right)$ and ${max}_{\varsigma \in V}P(\mathrm{\Omega },\varsigma )$$=P(\mathrm{\Omega },\overline{\varsigma })$. In this case, $\overline{\zeta }\in \mathbb{R}$ exists as the scalar, $\overline{\rho }=\left({\overline{\rho }}_l\left(t\right)\right)\in {\mathbb{R}}_{+}^m,\overline{\theta }=\left({\overline{\theta }}_{\xi }\left(t\right)\right)\in {\mathbb{R}}^n$ as the piecewise smooth functions, and $\overline{\tau }\in T$, $\overline{\eta }\in N$ as the parameters of uncertainty satisfying

$${\displaystyle \overline{\zeta }{P}_{\varrho }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })}$$

$${\displaystyle -D_{\alpha }\left[{\overline{\rho }}^T{\varphi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\right]=0,}$$

(1)

$${\displaystyle \overline{\zeta }{P}_{\mu }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })=0,}$$

(2)

$${\displaystyle {\overline{\rho }}^T\varphi (\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })=0,\overline{\rho }\geqq 0,}$$

(3)

$$\overline{\zeta }\ge 0,$$

(4)

for all $t\in ▵$, except at discontinuities.

Proof. 

We start the proof by defining the variations in parameters ${\displaystyle \overline{\varrho }\left(t\right)}$ and ${\displaystyle \overline{\mu }\left(t\right)}$, as ${\displaystyle \overline{\varrho }\left(t\right)+{\epsilon }_{1}h\left(t\right)}$ and ${\displaystyle \overline{\mu }\left(t\right)+{\epsilon }_2\mathsf{m}\left(t\right)}$, respectively, (considering the parameters ${\epsilon }_1,{\epsilon }_2$ as “small” as required in the variational arguments). Next, the cost functional and the constraints, which are dependent on ${\displaystyle ({\epsilon }_1,{\epsilon }_2)}$, are transformed as below

$${\displaystyle \mathcal{J}({\epsilon }_1,{\epsilon }_2)={\int }_{▵}P\left(t,\overline{\varrho }\left(t\right)+{\epsilon }_{1}h\left(t\right),\overline{\mu }\left(t\right)+{\epsilon }_2\mathsf{m}\left(t\right),\overline{\varsigma }\right)dv,}$$

$${\displaystyle \mathcal{K}({\epsilon }_1,{\epsilon }_2)={\int }_{▵}\varphi \left(t,\overline{\varrho }\left(t\right)+{\epsilon }_{1}h\left(t\right),{\overline{\varrho }}_{\alpha }\left(t\right)+{\epsilon }_1{h}_{\alpha }\left(t\right),\overline{\mu }\left(t\right)+{\epsilon }_2\mathsf{m}\left(t\right),\overline{\tau }\right)dv}$$

and

$${\displaystyle \mathcal{L}({\epsilon }_1,{\epsilon }_2)={\int }_{▵}\psi \left(t,\overline{\varrho }\left(t\right)+{\epsilon }_{1}h\left(t\right),{\overline{\varrho }}_{\alpha }\left(t\right)+{\epsilon }_1{h}_{\alpha }\left(t\right),\overline{\mu }\left(t\right)+{\epsilon }_2\mathsf{m}\left(t\right),\overline{\eta }\right)dv.}$$

The following optimal control problem has a solution $(0,0)$, as a consequence of ${\displaystyle (\overline{\varrho },\overline{\mu })}$ being a robust weak optimal solution for $\left(USCP\right)$

$${\displaystyle \underset{{\epsilon }_1,{\epsilon }_2}{min}\mathcal{J}({\epsilon }_1,{\epsilon }_2)}$$

subject to

$${\displaystyle \mathcal{K}({\epsilon }_1,{\epsilon }_2)\leqq 0,\mathcal{L}({\epsilon }_1,{\epsilon }_2)=0,}$$

$${\displaystyle h\left(t_0\right)=h\left(t_1\right)=\mathsf{m}\left(t_0\right)=\mathsf{m}\left(t_1\right)=0.}$$

As a result, the following Fritz John conditions are satisfied with the existence of $\overline{\zeta }\in \mathbb{R}$ as scalar, $\overline{\rho }=\left({\overline{\rho }}_l\left(t\right)\right)\in {\mathbb{R}}_{+}^m, \overline{\theta }=\left({\overline{\theta }}_{\xi }\left(t\right)\right)\in {\mathbb{R}}^n$ as the piecewise smooth functions, and $\overline{\tau }\in T$ and $\overline{\eta }\in N$ as the uncertain parameters such that

$${\displaystyle \overline{\zeta }\nabla \mathcal{J}(0,0)+{\overline{\rho }}^T\nabla \mathcal{K}(0,0)+{\overline{\theta }}^T\nabla \mathcal{L}(0,0)=0,}$$

(5)

$${\displaystyle {\overline{\rho }}^T\mathcal{K}(0,0)=0,\overline{\rho }\geqq 0,}$$

$$\overline{\zeta }\ge 0,$$

where the gradient of $\varphi$ at $(\varrho ,y)$ is represented as ${\displaystyle \nabla \varphi (\varrho ,y)}$. We can reinterpret the first equation in $\left(5\right)$ as below:

$${\displaystyle {\int }_{▵}(\overline{\zeta }\frac{\partial P}{\partial {\overline{\varrho }}^i}{h}^i+{\overline{\rho }}^T\frac{\partial \varphi }{\partial {\overline{\varrho }}^i}{h}^i+{\overline{\rho }}^T\frac{\partial \varphi }{\partial {\overline{\varrho }}_{\alpha }^i}{h}_{\alpha }^i}$$

$$+{\overline{\theta }}^T\frac{\psi }{\partial {\overline{\varrho }}^i}{h}^i+{\overline{\theta }}^T\frac{\partial \psi }{\partial {\overline{\varrho }}_{\alpha }^i}{h}_{\alpha }^i)dv=0,$$

$${\displaystyle {\int }_{▵}\left(\overline{\zeta }\frac{\partial P}{\partial {\overline{\mu }}^j}{\mathsf{m}}^j+{\overline{\rho }}^T\frac{\partial \varphi }{\partial {\overline{\mu }}^j}{\mathsf{m}}^j+{\overline{\theta }}^T\frac{\partial \psi }{\partial {\overline{\mu }}^j}{\mathsf{m}}^j\right)dv=0,}$$

or, equivalently, it can be written as

$${\displaystyle {\int }_{▵}(\overline{\zeta }\frac{\partial P}{\partial {\overline{\varrho }}^i}+{\overline{\rho }}^T\frac{\partial \varphi }{\partial {\overline{\varrho }}^i}-D_{\alpha }{\overline{\rho }}^T\frac{\partial \varphi }{\partial {\overline{\varrho }}_{\alpha }^i}}$$

$$+{\overline{\theta }}^T\frac{\partial \psi }{\partial {\overline{\varrho }}^i}-D_{\alpha }{\overline{\theta }}^T\frac{\partial \psi }{\partial {\overline{\varrho }}_{\alpha }^i})h^{i}dv=0,$$

$${\displaystyle {\int }_{▵}\left(\overline{\zeta }\frac{\partial P}{\partial {\overline{\mu }}^j}+{\overline{\rho }}^T\frac{\partial \varphi }{\partial {\overline{\mu }}^j}+{\overline{\theta }}^T\frac{\partial \psi }{\partial {\overline{\mu }}^j}\right){\mathsf{m}}^{j}dv=0,}$$

by using the divergence formula, integration by parts, and the boundary conditions.

Now, employing the fundamental result of the calculus of variations, it follows that

$${\displaystyle \overline{\zeta }\frac{\partial P}{\partial {\overline{\varrho }}^i}+{\overline{\rho }}^T\frac{\partial \varphi }{\partial {\overline{\varrho }}^i}-D_{\alpha }{\overline{\rho }}^T\frac{\partial \varphi }{\partial {\overline{\varrho }}_{\alpha }^i}}$$

$$+{\overline{\theta }}^T\frac{\partial \psi }{\partial {\overline{\varrho }}^i}-D_{\alpha }{\overline{\theta }}^T\frac{\partial \psi }{\partial {\overline{\varrho }}_{\alpha }^i}=0,i=\overline{1,q},$$

$${\displaystyle \overline{\zeta }\frac{\partial P}{\partial {\overline{\mu }}^j}+{\overline{\rho }}^T\frac{\partial \varphi }{\partial {\overline{\mu }}^j}+{\overline{\theta }}^T\frac{\partial \psi }{\partial {\overline{\mu }}^j}=0,j=\overline{1,r},}$$

or, equivalently,

$${\displaystyle \overline{\zeta }\frac{\partial P}{\partial \varrho }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })}$$

$${\displaystyle -D_{\alpha }\left[{\overline{\rho }}^T{\varphi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\right]=0,}$$

$${\displaystyle \overline{\zeta }\frac{\partial P}{\partial \mu }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })=0.}$$

The last two conditions in relation $(\ast )$,

$${\displaystyle {\overline{\rho }}^T\mathcal{K}(0,0)=0,\overline{\rho }\geqq 0,}$$

$$\overline{\zeta }\ge 0$$

gives

$${\displaystyle {\overline{\rho }}^T\varphi (\overline{\mathrm{\Omega }},{\overline{\rho }}_{\alpha }\left(t\right),\overline{\tau })=0,\overline{\rho }\geqq 0,}$$

$$\overline{\zeta }\ge 0$$

This completes the proof. □

We next move on to the following multi-dimensional vector variational problem

$${\displaystyle \left(\mathcal{UVCP}\right)\underset{(\varrho ,\mu )}{min}\left\{\mathcal{I}(\varrho ,\mu ):={\int }_{▵}f(\mathrm{\Omega },\varsigma )dv\right\}}$$

$${\displaystyle \mathrm{subject\; to} (\varrho ,\mu )\in Z,}$$

where

$${\displaystyle {\int }_{▵}f(\mathrm{\Omega },\varsigma )dv:=\left({\displaystyle {\int }_{▵}{f}_1(\mathrm{\Omega },{\varsigma }_1)dv,\dots ,{\int }_{▵}{f}_p(\mathrm{\Omega },{\varsigma }_p)dv}\right).}$$

Theorem 2

(Robust necessary efficiency conditions for $\left(\mathcal{UVCP}\right)$). Consider a robust weakly efficient solution $(\overline{\varrho },\overline{\mu })\in Z$ for the given robust multi-dimensional vector variational problem $\left(\mathcal{UVCP}\right)$ and ${max}_{\varsigma \in V}f(\mathrm{\Omega },\varsigma )=f(\mathrm{\Omega },\overline{\varsigma })$. Then, $\overline{\zeta }=\left({\overline{\zeta }}^k\right)\in {\mathbb{R}}^p$ exist as the scalars, $\overline{\rho }=\left({\overline{\rho }}_l\left(t\right)\right)\in {\mathbb{R}}_{+}^m,\overline{\theta }=\left({\overline{\theta }}_{\xi }\left(t\right)\right)\in {\mathbb{R}}^n$ as the piecewise smooth functions, and $\overline{\tau }\in T$, $\overline{\eta }\in N$ as the parameters of uncertainty satisfying the following conditions:

$${\displaystyle {\overline{\zeta }}^{\kappa }\frac{\partial f_{\kappa }}{\partial \varrho }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })}$$

$${\displaystyle -D_{\alpha }\left[{\overline{\rho }}^T{\varphi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\right]=0,}$$

$${\displaystyle {\overline{\zeta }}^{\kappa }\frac{\partial f_{\kappa }}{\partial \mu }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })=0,}$$

$${\displaystyle {\overline{\rho }}^T\varphi (\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })=0,\overline{\rho }\geqq 0,}$$

$${\displaystyle {\overline{\zeta }}^{\kappa }\ge 0,}$$

for all $t\in ▵$, except at discontinuities.

Proof. 

Consider ${\displaystyle (\overline{\varrho },\overline{\mu })\in Z}$ to be the robust weakly efficient solution for the uncertain multi-dimensional vector variational problem $\left(\mathcal{UVCP}\right)$. Consequently, the inequality ${\displaystyle \mathcal{I}(\varrho ,\mu )\le \mathcal{I}(\overline{\varrho },\overline{\mu }),\forall (\varrho ,\mu )\in Z}$ does not hold true. Moreover, for the point ${\displaystyle (\overline{\varrho },\overline{\mu })}$, define the neighborhood ${\displaystyle N_{\kappa }}$ in Z where ${\displaystyle \kappa \in \left\{1,2,\dots p\right\}}$ such that ${\displaystyle {\mathcal{I}}_{\kappa }(\varrho ,\mu )>{\mathcal{I}}_{\kappa }(\overline{\varrho },\overline{\mu }),\forall (\varrho ,\mu )\in N_{\kappa }}$. In view of this, ${\displaystyle (\overline{\varrho },\overline{\mu })}$ is a robust weak optimal solution for the following robust multi-dimensional scalar variational problem:

$${\displaystyle {\left(USCP\right)}_{\kappa }\underset{(\varrho ,\mu )}{min}\left\{{\displaystyle {\mathcal{I}}_{\kappa }(\varrho ,\mu )={\int }_{▵}{f}_{\kappa }\left(\mathrm{\Omega },{\varsigma }_k\right)dv}\right\}}$$

$${\displaystyle \mathrm{subject\; to} (\varrho ,\mu )\in Z.}$$

Now, applying Theorem 1, the following conditions are satisfied with the existence of ${\overline{\zeta }}_{\kappa }\in \mathbb{R}$, the piecewise smooth functions ${\overline{\rho }}_{\kappa }=\left({\overline{\rho }}_{\kappa ,l}\left(t\right)\right)\in {\mathbb{R}}_{+}^m, \overline{{\theta }_{\kappa }}=\left({\overline{\theta }}_{\kappa ,\xi }\left(t\right)\right)\in {\mathbb{R}}^n$, and the uncertainty parameters $\overline{\tau }\in T$, $\overline{\eta }\in N$ satisfying the following conditions (taking no summation over $\kappa$):

$${\displaystyle {\overline{\zeta }}_k\frac{\partial f_k}{\partial \varrho }(\overline{\mathrm{\Omega }},{\overline{\varsigma }}_k)+{\overline{\rho }}_k^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}_k^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })}$$

$${\displaystyle -D_{\alpha }\left[{\overline{\rho }}_k^T{\varphi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}_k^T{\psi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\right]=0,}$$

$${\displaystyle {\overline{\zeta }}_k\frac{\partial f_k}{\partial \mu }(\overline{\mathrm{\Omega }},{\overline{\varsigma }}_k)+{\overline{\rho }}_k^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}_k^T{\psi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })=0,}$$

$${\displaystyle {\overline{\rho }}_k^T\varphi (\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })=0,{\overline{\rho }}_k\geqq 0,}$$

$${\overline{\zeta }}_k\ge 0,$$

for all $t\in ▵$, except at discontinuities.

Implementing the required notations as follows: ${\displaystyle S=\sum _{\kappa =1}^p{\overline{\zeta }}_{\kappa }}$, with ${\displaystyle {\overline{\zeta }}^{\kappa }=\frac{{\overline{\zeta }}_{\kappa }}{S}}$ when ${\displaystyle {\mathcal{I}}_{\kappa }(\varrho ,\mu )>{\mathcal{I}}_{\kappa }(\overline{\varrho },\overline{\mu })}$, and ${\displaystyle {\overline{\zeta }}^{\kappa }=0}$ when ${\displaystyle {\mathcal{I}}_{\kappa }(\varrho ,\mu )\le {\mathcal{I}}_{\kappa }(\overline{\varrho },\overline{\mu })}$, ${\displaystyle \overline{\rho }=\frac{{\overline{\rho }}_{\kappa }}{S}}$, ${\displaystyle \overline{\theta }=\frac{{\overline{\theta }}_{\kappa }}{S}}$, the proof is complete. □

The robust multi-dimensional vector fractional variational problem $\left(UVFCP\right)$ is now considered as follows:

$${\displaystyle \left(UVFCP\right)\underset{(\varrho ,\mu )}{min}\left\{\mathcal{J}(\varrho ,\mu ):=\left(\frac{{\displaystyle {\int }_{▵}{f}_1\left(\mathrm{\Omega },{\varsigma }_1\right)dv}}{{\displaystyle {\int }_{▵}{g}_1\left(\mathrm{\Omega },{\gamma }_1\right)dv}},\dots ,\frac{{\displaystyle {\int }_{▵}{f}_p\left(\mathrm{\Omega },{\varsigma }_p\right)dv}}{{\displaystyle {\int }_{▵}{g}_p\left(\mathrm{\Omega },{\gamma }_p\right)dv}}\right)\right\}}$$

$${\displaystyle \mathrm{subject\; to} (\varrho ,\mu )\in Z,}$$

where it is assumed that ${\displaystyle {\int }_{▵}{g}_{\kappa }\left(\mathrm{\Omega },{\gamma }_{\kappa }\right)dv>0,\kappa =\overline{1,p}}$.

Equivalently, we have ${\left(NonFUSCP\right)}_w$ (on the line of Jagannathan [15])

$${\displaystyle {\left(NUSFCP\right)}_w\underset{(\varrho ,\mu )}{min}{\int }_{▵}\left[f_w\left(\mathrm{\Omega },{\varsigma }_w\right)-Y_w^0{g}_w\left(\mathrm{\Omega },{\gamma }_w\right)\right]dv}$$

$$\mathrm{subject\; to}$$

$${\displaystyle (\varrho ,\mu )\in Z}$$

$${\displaystyle {\int }_{▵}\left[f_{\kappa }\left(\mathrm{\Omega },{\varsigma }_{\kappa }\right)-Y_{\kappa }^0{g}_{\kappa }\left(\mathrm{\Omega },{\gamma }_{\kappa }\right)\right]dv\le 0,\kappa =\overline{1,p},\kappa \ne w,}$$

where ${\displaystyle Y_{\kappa }^0}$ is a real positive scalar.

Theorem 3

(Robust necessary efficiency conditions for $\left(UVFCP\right)$). Consider a robust weakly efficient solution $(\overline{\varrho },\overline{\mu })\in Z$ for the given robust vector fractional variational problem $\left(UVFCP\right)$. Then, $\overline{\zeta }=\left({\overline{\zeta }}^w\right)\in {\mathbb{R}}^p$ exist as the scalars, $\overline{\rho }=\left({\overline{\rho }}_l\left(t\right)\right)\in {\mathbb{R}}_{+}^m, \overline{\theta }=\left({\overline{\theta }}_{\xi }\left(t\right)\right)\in {\mathbb{R}}^n$ as the piecewise smooth functions, and $\overline{\tau }\in T$, $\overline{\eta }\in N$ as the parameters of uncertainty satisfying the following conditions:

$${\displaystyle {\overline{\zeta }}^w\left[\frac{\partial f_w}{\partial \varrho }\left(\overline{\mathrm{\Omega }},{\overline{\varsigma }}_w\right)-Y_w^0\frac{\partial g_w}{\partial \varrho }\left(\overline{\mathrm{\Omega }},{\overline{\gamma }}_w\right)\right]+{\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })}$$

$${\displaystyle -D_{\alpha }\left[{\overline{\rho }}^T{\varphi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\right]=0,}$$

$${\displaystyle {\overline{\zeta }}^w\left[\frac{\partial f_w}{\partial \mu }\left(\overline{\mathrm{\Omega }},{\overline{\varsigma }}_w\right)-Y_w^0\frac{\partial g_w}{\partial \mu }\left(\overline{\mathrm{\Omega }},{\overline{\gamma }}_w\right)\right]+{\overline{\rho }}^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })=0,}$$

$${\displaystyle {\overline{\rho }}^T\varphi (\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })=0,\overline{\rho }\geqq 0,}$$

$$\overline{\zeta }\ge 0,$$

for all $t\in ▵$, except at discontinuities.

Proof. 

Assuming ${\displaystyle (\overline{\varrho },\overline{\mu })\in Z}$ to be the robust weakly efficient solution for the robust multi-dimensional vector fractional variational problem $\left(UVFCP\right)$. In addition, by taking into account ${\displaystyle f_w\left(\mathrm{\Omega },{\varsigma }_w\right)-Y_w^0{g}_w\left(\mathrm{\Omega },{\gamma }_w\right)}$ instead of ${\displaystyle f_{\kappa }\left(\mathrm{\Omega },{\varsigma }_{\kappa }\right)}$, the proof follows on the same lines of Theorem 1. □

Corollary 1

(Robust necessary efficiency conditions for ${\left(NonFUSCP\right)}_w$). Consider a robust weakly efficient solution $(\overline{\varrho },\overline{\mu })\in Z$ for the given robust vector fractional variational problem ${\left(NonFUSCP\right)}_w$. Then, $\overline{\zeta }=\left({\overline{\zeta }}^w\right)\in {\mathbb{R}}^p$ exist as the scalars, $\overline{\rho }=\left({\overline{\rho }}_l\left(t\right)\right)\in {\mathbb{R}}_{+}^m$ and $\overline{\theta }=\left({\overline{\theta }}_{\xi }\left(t\right)\right)\in {\mathbb{R}}^n$ as the piecewise smooth functions, and $\overline{\tau }\in T$ and $\overline{\eta }\in N$ as the parameters of uncertainty such that

$${\displaystyle {\overline{\zeta }}^w\left[G_w(\overline{\varrho },\overline{\mu })\frac{\partial f_w}{\partial \varrho }\left(\overline{\mathrm{\Omega }},{\overline{\varsigma }}_w\right)-F_w(\overline{\varrho },\overline{\mu })\frac{\partial g_w}{\partial \varrho }\left(\overline{\mathrm{\Omega }},{\overline{\gamma }}_w\right)\right]+{\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })}$$

$${\displaystyle +{\overline{\theta }}^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })-D_{\alpha }\left[{\overline{\rho }}^T{\varphi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\right]=0,}$$

$${\displaystyle {\overline{\zeta }}^w\left[G_w(\overline{\varrho },\overline{\mu })\frac{\partial f_w}{\partial \mu }\left(\overline{\mathrm{\Omega }},{\overline{\varsigma }}_w\right)-F_w(\overline{\varrho },\overline{\mu })\frac{\partial g_w}{\partial \mu }\left(\overline{\mathrm{\Omega }},{\overline{\gamma }}_w\right)\right]}$$

$${\displaystyle +{\overline{\rho }}^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })=0,}$$

$${\displaystyle {\overline{\rho }}^T\varphi (\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })=0,\overline{\rho }\geqq 0,}$$

$${\overline{\zeta }}^w\ge 0,$$

for all $t\in ▵$, except at discontinuities.

Proof. 

By denoting

$${\displaystyle Y_w^0=\frac{{\displaystyle {\int }_{▵}{f}_w\left(\overline{\mathrm{\Omega }},{\overline{\varsigma }}_w\right)dv}}{{\displaystyle {\int }_{▵}{g}_w\left(\overline{\mathrm{\Omega }},{\overline{\gamma }}_w\right)dv}}:=\frac{F_w(\overline{\varrho },\overline{\mu })}{G_w(\overline{\varrho },\overline{\mu })},w=\overline{1,p}}$$

and redefining the functions

$${\displaystyle {\overline{\rho }}^T:=G_w(\overline{\varrho },\overline{\mu }){\overline{\rho }}^T,{\overline{\theta }}^T:=G_w(\overline{\varrho },\overline{\mu }){\overline{\theta }}^T,}$$

which completes the proof. □

4. Robust Sufficient Efficiency Conditions in Scalar, Vector and Vector Fractional Variational Problems

For the considered robust multi-dimensional scalar, vector and vector fractional variational problems, the robust sufficient efficiency conditions are derived in this section. We begin by stating the sufficient conditions of efficiency for the aforementioned robust scalar control problem $\left(USCP\right)$. More specifically, we prove that if the involved functionals are convex, any robust feasible solution in $\left(USCP\right)$ that satisfies the conditions in Theorem 3, is a robust weak optimal solution to the problem under consideration.

Definition 9.

A functional ${\displaystyle {\int }_{▵}f(\mathrm{\Omega },\varsigma )dv}$ is said to be convex at $(\overline{\varrho },\overline{\mu })$, if

$${\displaystyle {\int }_{▵}f(\mathrm{\Omega },\varsigma )dv-{\int }_{▵}f(\overline{\mathrm{\Omega }},\varsigma )dv\ge {\int }_{▵}(\widehat{\varrho }-\overline{\varrho })f_{\varrho }(\overline{\mathrm{\Omega }},\varsigma )dv+{\int }_{▵}(\widehat{\mu }-\overline{\mu })f_{\mu }(\overline{\mathrm{\Omega }},\varsigma )dv}$$

holds for all $(\varrho ,\mu )$$\in \Lambda \times M.$

Theorem 4

(Robust sufficient efficiency conditions for $\left(USCP\right)$). Consider a robust feasible solution $(\overline{\varrho },\overline{\mu })\in Z$ for the robust scalar variational problem $\left(USCP\right)$ ensuring that the robust necessary optimality conditions stated in Theorem 1 are satisfied. Further, assume that ${\displaystyle {\int }_{▵}\overline{\zeta }P(\mathrm{\Omega },\overline{\varsigma })dv, {\int }_{▵}{\rho }^T\varphi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\overline{\tau })dv}$ and ${\displaystyle {\int }_{▵}{\overline{\theta }}^T\psi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\overline{\eta })dv}$ are convex at $(\overline{\varrho },\overline{\mu })$. Then, $(\overline{\varrho },\overline{\mu })$ is a robust weak optimal solution for the robust scalar variational problem $\left(USCP\right)$.

Proof. 

Suppose on the contradiction that $(\overline{\varrho },\overline{\mu })$ is not a robust weak optimal solution for the robust scalar variational control problem $\left(USCP\right)$. Consequently, $(\widehat{\varrho },\widehat{\mu })\in Z$ exists satisfying

$${\displaystyle {\int }_{▵}\underset{\varsigma \in V}{max}P(\widehat{\mathrm{\Omega }},\varsigma )dv<{\int }_{▵}\underset{\varsigma \in V}{max}P(\overline{\mathrm{\Omega }},\varsigma )dv.}$$

By assumption, ${max}_{\varsigma \in V}P(\mathrm{\Omega },\varsigma )=P(\mathrm{\Omega },\overline{\varsigma }),$ we obtain

$${\displaystyle {\int }_{▵}P(\widehat{\mathrm{\Omega }},\overline{\varsigma })dv<{\int }_{▵}P(\overline{\mathrm{\Omega }},\overline{\varsigma })dv.}$$

(6)

Since the necessary efficiency conditions (1)–(4) are satisfied at $(\overline{\varrho },\overline{\mu })$. Additionally, multiplying the Equation (1) by $(\widehat{\varrho }-\overline{\varrho })$ and Equation (2) by $(\widehat{\mu }-\overline{\mu })$ and integrating, we obtain,

$${\displaystyle {\int }_{▵}(\widehat{\varrho }-\overline{\varrho })\{\overline{\zeta }{P}_{\varrho }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })}$$

(7)

$${\displaystyle -D_{\alpha }\left[{\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\right]\}dv}$$

$${\displaystyle +{\int }_{▵}(\widehat{\mu }-\overline{\mu })\{\overline{\zeta }{P}_{\mu }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\}dv}$$

$${\displaystyle ={\int }_{▵}[(\widehat{\varrho }-\overline{\varrho })\{\overline{\zeta }{P}_{\varrho }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\}}$$

$${\displaystyle +({\widehat{\varrho }}_{\alpha }-{\overline{\varrho }}_{\alpha })\{{\overline{\rho }}^T{\varphi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\}]dv}$$

$${\displaystyle +{\int }_{▵}(\widehat{\mu }-\overline{\mu })\{\overline{\zeta }{P}_{\mu }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\}dv=0,}$$

by utilizing the divergence formula, using integration by parts, and using the boundary conditions in the problem under consideration.

In contrast, as ${\displaystyle {\int }_{▵}\overline{\zeta }P(\mathrm{\Omega },\overline{\varsigma })dv}$ is convex at $(\overline{\varrho },\overline{\mu })$, we have

$${\displaystyle {\int }_{▵}\left\{\overline{\zeta }P(\widehat{\mathrm{\Omega }},\overline{\varsigma })-\overline{\zeta }P(\overline{\mathrm{\Omega }},\overline{\varsigma })\right\}dv\ge {\int }_{▵}(\widehat{\varrho }-\overline{\varrho })\overline{\zeta }{P}_{\varrho }(\overline{\mathrm{\Omega }},\overline{\varsigma })dv}$$

$${\displaystyle +{\int }_{▵}(\widehat{\mu }-\overline{\mu })\overline{\zeta }{P}_{\mu }(\overline{\mathrm{\Omega }},\overline{\varsigma })dv,}$$

which, along with the inequality (6), gives

$${\displaystyle {\int }_{▵}(\widehat{\varrho }-\overline{\varrho })\overline{\zeta }{P}_{\varrho }(\overline{\mathrm{\Omega }},\overline{\varsigma })dv+{\int }_{▵}(\widehat{\mu }-\overline{\mu })\overline{\zeta }{P}_{\mu }(\overline{\mathrm{\Omega }},\overline{\varsigma })dv<0.}$$

(8)

Once again, applying the fact that ${\displaystyle {\int }_{▵}{\overline{\rho }}^T\varphi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\overline{\tau })dv}$ is convex at $(\overline{\varrho },\overline{\mu })$, the following results:

$${\displaystyle {\int }_{▵}\left\{{\overline{\rho }}^T\varphi (\widehat{\mathrm{\Omega }},{\widehat{\varrho }}_{\alpha }\left(t\right),\overline{\tau })-{\overline{\rho }}^T\varphi (\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })\right\}dv\ge {\int }_{▵}(\widehat{\varrho }-\overline{\varrho }){\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })dv}$$

$${\displaystyle +{\int }_{▵}({\widehat{\varrho }}_{\alpha }-{\overline{\varrho }}_{\alpha }){\overline{\rho }}^T{\varphi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })dv+{\int }_{▵}(\widehat{\mu }-\overline{\mu }){\overline{\rho }}^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })dv.}$$

Now, by the robust feasibility of $(\widehat{\varrho },\widehat{\mu })$ in $\left(USCP\right)$ in the above inequality and condition (3), provide

$${\displaystyle {\int }_{▵}(\widehat{\varrho }-\overline{\varrho }){\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })dv+{\int }_{▵}({\widehat{\varrho }}_{\alpha }-{\overline{\varrho }}_{\alpha }){\overline{\rho }}^T{\varphi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })dv}$$

$${\displaystyle {\int }_{▵}(\widehat{\mu }-\overline{\mu }){\overline{\rho }}^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })dv\le 0.}$$

(9)

Further, by considering that ${\displaystyle {\int }_{▵}{\overline{\theta }}^T\psi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\overline{\eta })dv}$ is convex at $(\overline{\varrho },\overline{\mu })$ and the robust feasibility of $(\widehat{\varrho },\widehat{\mu })$ in $\left(USCP\right)$, we obtain

$${\displaystyle {\int }_{▵}(\widehat{\varrho }-\overline{\varrho }){\overline{\theta }}^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })dv+{\int }_{▵}({\widehat{\varrho }}_{\alpha }-{\overline{\varrho }}_{\alpha }){\overline{\theta }}^T{\psi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })dv}$$

$${\displaystyle +{\int }_{▵}(\widehat{\mu }-\overline{\mu }){\overline{\theta }}^T{\psi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })dv\le 0.}$$

(10)

On adding the inequalities (8)–(10), we have

$${\displaystyle {\int }_{▵}[(\widehat{\varrho }-\overline{\varrho })\{\overline{\zeta }{P}_{\varrho }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\varrho }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\}}$$

$${\displaystyle +({\widehat{\varrho }}_{\alpha }-{\overline{\varrho }}_{\alpha })\{{\overline{\rho }}^T{\varphi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{{\varrho }_{\alpha }}(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\}]dv}$$

$${\displaystyle +{\int }_{▵}(\widehat{\mu }-\overline{\mu })\{\overline{\zeta }{P}_{\mu }(\overline{\mathrm{\Omega }},\overline{\varsigma })+{\overline{\rho }}^T{\varphi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\tau })+{\overline{\theta }}^T{\psi }_{\mu }(\overline{\mathrm{\Omega }},{\overline{\varrho }}_{\alpha }\left(t\right),\overline{\eta })\}dv<0,}$$

which is a contradiction to the Equation (7) and, therefore, the proof is completed. □

Next, we demonstrate sufficient efficiency conditions for the robust multi-dimensional vector variational problem $\left(\mathcal{UVCP}\right)$.

Theorem 5

(Robust sufficient efficiency conditions for $\left(\mathcal{UVCP}\right)$). Consider a robust feasible solution ${\displaystyle (\overline{\varrho },\overline{\mu })\in Z}$ for the robust vector variational problem $\left(\mathcal{UVCP}\right)$, and $\overline{\zeta }=\left({\overline{\zeta }}^k\right)$ exists as the scalars, $\overline{\rho }=\left({\overline{\rho }}_l\left(t\right)\right)\in {\mathbb{R}}_{+}^m, \overline{\theta }=\left({\overline{\theta }}_{\xi }\left(t\right)\right)\in {\mathbb{R}}^n$ as the piecewise smooth functions, and $\overline{\tau }\in T$, $\overline{\eta }\in N$ as the parameters of uncertainty, fulfilling the conditions of Theorem 2. Further, assume that each functional ${\displaystyle {\int }_{▵}{\overline{\zeta }}^{\kappa }{f}_{\kappa }(\mathrm{\Omega },{\overline{\varsigma }}_{\kappa })dv}$ for ${\displaystyle \kappa =\overline{1,p}, {\int }_{▵}{\overline{\rho }}^T\varphi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\overline{\tau })dv}$ and ${\displaystyle {\int }_{▵}{\overline{\theta }}^T\psi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\overline{\eta })dv}$ are convex at $(\overline{\varrho },\overline{\mu })$. Then, $(\overline{\varrho },\overline{\mu })$ is a robust weakly efficient solution for the robust vector variational problem $\left(\mathcal{UVCP}\right)$.

Proof. 

The proof can be obtained on the same lines of Theorem 4 by replacing ${\displaystyle \overline{\zeta }}$ to $\overline{\zeta }\kappa =\frac{{\overline{\zeta }}_{\kappa }}{S}$, where ${\displaystyle S=\sum _{\kappa =1}^p{\overline{\zeta }}_{\kappa }}$ as in Theorem 2, and ${\displaystyle \overline{\rho }=\frac{{\overline{\rho }}_{\kappa }}{S}}$, ${\displaystyle \overline{\theta }=\frac{{\overline{\theta }}_{\kappa }}{S}}$. □

Next, we shall provide robust sufficient conditions of efficiency in the multi-dimensional vector fractional variational problem $\left(UVFCP\right)$ in the following result.

Theorem 6

(Robust sufficient efficiency conditions for $\left(UVFCP\right)$). Consider a robust feasible solution ${\displaystyle (\overline{\varrho },\overline{\mu })\in Z}$ for the vector fractional variational problem $\left(UVFCP\right)$. Then, $\overline{\zeta }=\left({\overline{\zeta }}^w\right)$ exist as the scalars, $\overline{\rho }=\left({\overline{\rho }}_l\left(t\right)\right)\in {\mathbb{R}}_{+}^m,\overline{\theta }=\left({\overline{\theta }}_{\xi }\left(t\right)\right)\in {\mathbb{R}}^n$ as the piecewise smooth functions, and $\overline{\tau }\in T$, $\overline{\eta }\in N$ as the parameters of uncertainty, satisfying the conditions formulated in Theorem 3. Further, suppose that each involved functionals ${\displaystyle {\int }_{▵}{\overline{\zeta }}^w\left[f_w(\mathrm{\Omega },{\overline{\varsigma }}_w)-Y_w^0{g}_w(\mathrm{\Omega },{\overline{\gamma }}_w)\right]dv}$ for ${\displaystyle w=\overline{1,p},{\int }_{▵}{\overline{\rho }}^T\varphi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\overline{\tau })dv,}$ and ${\displaystyle {\int }_{▵}{\overline{\theta }}^T\psi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\overline{\eta })dv}$ are convex at $(\overline{\varrho },\overline{\mu })$. Then, $(\overline{\varrho },\overline{\mu })$ is a robust weakly efficient solution for the robust multi-dimensional vector fractional-control problem $\left(UVFCP\right)$.

Proof. 

By defining the functions

$${\displaystyle {\overline{\zeta }}^w\left[f_w(\mathrm{\Omega },{\overline{\varsigma }}_w)-Y_w^0{g}_w(\mathrm{\Omega },{\overline{\gamma }}_w)\right],w=\overline{1,p}}$$

instead of ${\overline{\zeta }}^{\kappa }{f}_{\kappa }(\mathrm{\Omega },{\overline{\varsigma }}_{\kappa })$. The proof follows in the similar way as in Theorem 5. □

Corollary 2

(Robust sufficient efficiency conditions for ${\left(NonFUSCP\right)}_w$). Consider a robust feasible solution ${\displaystyle (\overline{\varrho },\overline{\mu })\in Z}$ for the variational problem ${\left(NonFUSCP\right)}_w$. Then, $\overline{\zeta }=\left({\overline{\zeta }}^w\right)$ exists as the scalar and $\overline{\rho }=\left({\overline{\rho }}_l\left(t\right)\right)\in {\mathbb{R}}_{+}^m, \overline{\theta }=\left({\overline{\theta }}_{\xi }\left(t\right)\right)\in {\mathbb{R}}^n$ as the piecewise smooth functions, and $\overline{\tau }\in T$, $\overline{\eta }\in N$ exist as the parameters of uncertainty, satisfying the conditions formulated in Corollary 1. Further, suppose that each functional

$${\displaystyle {\int }_{▵}{\overline{\zeta }}^w\left[G_w(\overline{\varrho },\overline{\mu })f_w(\mathrm{\Omega },{\overline{\varsigma }}_w)-F_w(\overline{\varrho },\overline{\mu })g_w(\mathrm{\Omega },{\overline{\gamma }}_w)\right]dv,}$$

for ${\displaystyle w=\overline{1,p},{\int }_{▵}{\overline{\rho }}^T\varphi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\overline{\tau })dv,}$ and ${\displaystyle {\int }_{▵}{\overline{\theta }}^T\psi (\mathrm{\Omega },{\varrho }_{\alpha }\left(t\right),\overline{\eta })dv}$ are convex at $(\overline{\varrho },\overline{\mu })$. Then, $(\overline{\varrho },\overline{\mu })$ is a robust weakly efficient solution for the robust variational problem ${\left(NonFUSCP\right)}_w$.

Proof. 

The proof follows from Theorem 5 by replacing the functions

$${\displaystyle f_{\kappa }(\mathrm{\Omega },{\overline{\varsigma }}_{\kappa }),\kappa =\overline{1,p}}$$

with

$${\displaystyle G_w(\overline{\varrho },\overline{\mu })f_w(\mathrm{\Omega },{\overline{\varsigma }}_w)-F_w(\overline{\varrho },\overline{\mu })g_w(\mathrm{\Omega },{\overline{\gamma }}_w),w=\overline{1,p}.}$$

□

5. Illustrative Application

In this section, we present an application of the derived theoretical results. In this direction, we consider the following multi-dimensional multiple-objective optimization problem in the face of data uncertainty:

$$\left(\mathrm{P}\right)\underset{\left(\varrho \right(·),\mu (·\left)\right)}{min}{\int }_{▵}(e^{\varrho }+{\varsigma }_1,{\mu }^2+{\left({\varsigma }_2\varrho \right)}^2)d{t}^{1}d{t}^2$$

$$\mathrm{subject\; to}\hspace{1em}\tau (\varrho -2)(\varrho +1)\leqq 0,$$

(11)

$${\varrho }_{t^1}={\eta }_1^1-\mu ,$$

(12)

$${\varrho }_{t^2}={\eta }_2^1-\mu ,$$

(13)

$$\varrho (0,0)=0,\varrho \left(\frac{1}{3},\frac{1}{3}\right)=\frac{1}{2},$$

(14)

where $t=(t^1,t^2)\in ▵$ and ${\displaystyle {\int }_{▵}{g}_{\kappa }\left(\mathrm{\Omega },{\gamma }_{\kappa }\right)d{t}^{1}d{t}^2=1,\kappa =1,2}$.

The associated robust counterpart for the multi-dimensional multiple-objective optimization problem (P) is defined as:

$$\begin{array}{cc}\hfill \left(\mathrm{RP}\right)\underset{\left(\varrho \right(·),\mu (·\left)\right)}{min}& {\int }_{▵}\left(\underset{{\varsigma }_1\in V_1}{max}(e^{\varrho }+{\varsigma }_1),\underset{{\varsigma }_2\in V_2}{max}({\mu }^2+{\left({\varsigma }_2\varrho \right)}^2)\right)d{t}^{1}d{t}^2,\hfill \\ \hfill \mathrm{subject\; to}& \tau (\varrho -2)(\varrho +1)\leqq 0,\forall \tau \in T\hfill \\ & {\varrho }_{t^1}={\eta }_1^1-\mu ,\forall {\eta }_1^1\in N_1^1,\hfill \\ & {\varrho }_{t^2}={\eta }_2^1-\mu ,\forall {\eta }_2^1\in N_2^1,\hfill \\ & \varrho (0,0)=0,\varrho \left(\frac{1}{3},\frac{1}{3}\right)=\frac{1}{2},\hfill \end{array}$$

where $t=(t^1,t^2)\in ▵.$ Clearly $Z=\{(\varrho ,\mu )\in \Lambda \times M:-1\le \varrho \le 2\wedge {\varrho }_{t^1}={\eta }_1^1-\mu \wedge {\varrho }_{t^2}={\eta }_2^1-\mu ,\varrho (0,0)=0,\varrho \left(\frac{1}{3},\frac{1}{3}\right)=\frac{1}{2}\}$ is a robust feasible solution set to (P). Additionally, we assume that the state function is an affine one.

Let $(\overline{\varrho },\overline{\mu })$ be a robust feasible solution to the problem (P). The robust necessary efficiency conditions at $(\overline{\varrho },\overline{\mu })$ with the multipliers ${\overline{\zeta }}_1, {\overline{\zeta }}_2, \overline{\rho }, {\overline{\theta }}_1^1, {\overline{\theta }}_2^1$, ${max}_{\varsigma \in V}f(\mathrm{\Omega },\varsigma )$=$f(\mathrm{\Omega },\overline{\varsigma })$, and the uncertain parameters $\overline{\tau }, {\overline{\eta }}_1^1, {\overline{\eta }}_2^1$ are as follows:

$${\overline{\zeta }}_1{e}^{\overline{\varrho }}+2{\overline{\zeta }}_2{\overline{\varsigma }}_2^2\varrho +\overline{\rho }(2\overline{\varrho }-1)+{\left({\overline{\theta }}_1^1\right)}_{t^1}+{\left({\overline{\theta }}_2^1\right)}_{t^2}=0,$$

(15)

$${\overline{\zeta }}_{2}2\overline{\mu }-({\overline{\theta }}_1^1+{\overline{\theta }}_2^1)=0,$$

(16)

$$\overline{\rho }\overline{\tau }(\overline{\varrho }-2)(\overline{\varrho }+1)=0,$$

(17)

where relation (16) is satisfied if either $\overline{\rho }=0$ or $\overline{\tau }(\overline{\varrho }-2)(\overline{\varrho }+1)=0$, for all $t=(t^1,t^2)\in int▵$. Let $\overline{\rho }\ne 0$, then $\overline{\tau }(\overline{\varrho }-2)(\overline{\varrho }+1)=0$ implies that $\overline{\varrho }=-1,2$. Clearly, the boundary conditions (14) are failing at $\overline{\varrho }=-1,2$. Consequently, $\overline{\rho }=0$.

One can easily verifies that the robust necessary efficiency conditions (15)–(17) are satisfied at $(\overline{\varrho },\overline{\mu })=(\frac{3}{4}(t^1+t^2),\frac{3}{4})$ at $t^1=t^2=0$, i.e., $(0,\frac{1}{3})$ with the Lagrange multipliers ${\overline{\zeta }}_1=1, {\overline{\zeta }}_2=0, \overline{\rho }=0, {\overline{\theta }}_1^1+{\overline{\theta }}_2^1=\frac{3}{4}$ and the uncertain parameters ${\overline{\varsigma }}_1=2, {\overline{\varsigma }}_2=3, \overline{\tau }=1, {\overline{\eta }}_1^1=\frac{3}{2}, {\overline{\eta }}_2^1=\frac{3}{2}$. Further, it can also be easily verified that the involved functionals are convex at $(\overline{\varrho },\overline{\mu })=(0,\frac{3}{4})$. Hence, all of the conditions of the above theoretical results are satisfied, which ensure that $(0,\frac{3}{4})$ is also a robust weakly efficient solution to the problem (P).

6. Conclusions

In this study, we have introduced a new class of multi-dimensional variational control problems by considering data uncertainty in both objective functional and constraints. We have developed and established the robust necessary efficiency conditions for the scalar $\left(USCP\right)$, vector $\left(\mathcal{UVCP}\right)$, and vector fractional $\left(UVFCP\right)$ control problems based on the concepts of robust weakly efficient solution. Further, for a robust feasible solution to be a robust weakly efficient solution, the associated robust sufficient conditions need to have been obtained by using the assumption of convexity for the involved integrals.