1 Introduction

In optimization theory, the fractional optimization problem plays a vital role in decision theory, economics, game theory, portfolio selection, etc. A fractional optimization problem aims to optimize the ratio of objective functions concerning certain constraints, such as minimizing investment upon return, production cost upon outcome, and risk upon profit. Dinkelbach [7] and Jagannathan [8] utilized the parametric approach for solving a fractional optimization problem in which the fractional objective function is transformed into an equivalent non-fractional objective function. Afterward, many researchers have used this method to solve various classes of fractional optimization problems. Antczak and Pitea [2] have given the parametric optimality and duality results for a multi-time fractional variational problem under the assumption of generalized convexity.

Treanta and Mititelu [23] have given significant duality results for a multi-dimensional fractional control problem using the parametric approach. Further, several authors have shared their insights on this topic, see for instance, [5, 6, 9, 17, 18, 20] and references therein.

On the other hand, uncertain optimization problems, i.e., optimization problems with uncertain data, are a fast-expanding major optimization topic. Inadequate information, old sources, a large volume of data, sample disparity, and other factors lead to data uncertainty, creating problems in finding a solution. The robust method is one of the most useful methods to study the optimization problem in the face of data uncertainty. This method aims to reduce the maximum possible uncertainty of the problem. So many researchers investigated optimization problems, including data uncertainty, to produce new results (see, for example, [3, 4, 10, 21, 22] and references therein). Further, Jeyakumar et al. [12] investigated the robust optimality and duality results for an uncertain convex linear optimization problem. Jayswal et al. [11] have formulated a multi-time controlled optimization problem with uncertainty and efficiently solved it using the penalty function method.

Moreover, the practical views of fractional optimization problems inspire many researchers to study this problem in the face of data uncertainty. Antczak [1] has given the parametric robust necessary and sufficient optimality conditions for an uncertain fractional programming problem. Kim and Kim [15] studied a fractional robust optimization problem and given the optimality and duality results utilizing the parametric approach. After that, many significant studies have been done in this area, for example, [13, 14, 16] and references therein.

Motivated by the research works mentioned above, we introduce the multi-dimensional fractional control optimization problem with first-order PDE constraints in the face of uncertainty in the objective functional. Then using the parametric approach, we present new robust optimality conditions and duality results. This paper is structured in the following manner: Sect. 2 contains basic concepts and lemmas that assist in presenting the main results of this article. Moreover, we formulated the multi-dimensional fractional control optimization problem with uncertainty in the objective functional (FP) and a non-fractional problem associated with it and their robust counterparts. Section 3 states and proves the robust optimality conditions for the considered problem (FP). Section 4 presents the parametric robust dual problem associated with the primal problem (FP) and proves the weak and strong robust duality theorems. Finally, Sect. 5 brings this paper to a conclusion.

2 Problem Formulation and Preliminaries

In this section we are considering some basic notations that will assist in framing the problem and presenting the significant results.

\(\Rightarrow \) Consider the finite dimensional Euclidean spaces \(R^m\), \(R^n\) and \(R^k\). Let \(t = (t^\alpha ),~ \alpha = \overline{1,m},~a=(a^i),~i=\overline{1,n}\) and \( c = (c^j),~ j = \overline{1,k}\) are the local coordinates of \(R^m\), \(R^n\) and \(R^k\) respectively.

\(\Rightarrow \) Let \(\mathcal {T} = \mathcal {T}_{t_0,t_1} \subset R^m\) is a hyperparallelepiped, fixed by the diagonally opposite points \(t_0 = (t_0^\alpha )\) and \(t_1 = (t_1^\alpha ),~\alpha = \overline{1,m}\) in \(R^m\), as well as \(d\omega = dt^1 \wedge \dots \wedge \textrm{d}t^m\) denote the volume element in \(R^m \supset \mathcal {T}\).

\(\Rightarrow \) Let \(X \subset R^n\) be the space of piecewise smooth state function \(a(t): \mathcal {T} \mapsto R^n\) endowed with norm

$$\begin{aligned}&\Vert a(t) \Vert = \Vert a(t) \Vert _\infty + \sum ^m_{\alpha =1} \Vert a_\alpha (t) \Vert _\infty , \end{aligned}$$

where \(\Vert a(t) \Vert _\infty = \max (|a^1(t) |,~ | a^2(t) |, \dots ,~ | a^n(t)|)\) is uniform norm and \(a_\alpha (t) = \frac{\partial a(t)}{\partial t^\alpha }\).

\(\Rightarrow \) Let \(C \subset R^k\) be the space of continuous control function \(c(t): \mathcal {T} \mapsto R^k\) endowed with uniform norm \(\Vert \cdot \Vert _\infty \).

\(\Rightarrow \) For any two points, \(x =(x^\ell ) \) and \(y= (y^\ell ),~ \ell = \overline{1,l}\) in \(R^l\), the following convention will be used in this paper

$$\begin{aligned}&x=y \Leftrightarrow x^\ell = y^\ell ,~ x \le y \Leftrightarrow x^\ell \le y^\ell , ~ x< y \Leftrightarrow x^\ell < y^\ell , ~~ \ell = \overline{1,l}. \end{aligned}$$

Considering the above mathematical tools, we formulate the following multi-dimensional first- order PDE constraint fractional control optimization problem with data uncertainty (FP) as:

$$\begin{aligned}&\text {(FP)} \quad \min _{(a(\cdot ),c(\cdot ))} \frac{\int _\mathcal {T} f(t,a(t),c(t),u) \textrm{d}\omega }{\int _\mathcal {T} g(t,a(t),c(t),v) \textrm{d}\omega }\\&\quad \text {subject to} \nonumber \\&\quad \mathcal {Y}_\beta (t,a(t),c(t)) \le 0,~~ \beta = \overline{1,q},\\&\quad \frac{\partial a^i}{\partial t^\alpha }=\mathcal {H}^i_\alpha (t,a(t),c(t)),~i=\overline{1,n}, ~\alpha =\overline{1,m},\\&\quad a(t_0)=a_0,~a(t_1)=a_1, \end{aligned}$$

where u and v are uncertain parameter for some convex compact subsets of \(U \subset R\) and \(V \subset R\), respectively. Further, \(t \in \mathcal {T},~ f: \mathcal {T} \times X \times C \times U \mapsto R,~ g: \mathcal {T} \times X \times C \times V \mapsto R \setminus \{0\},~\mathcal {Y}_\beta : \mathcal {T} \times X \times C \mapsto R,~ \beta = \overline{1,q},~ \mathcal {H}^i_\alpha : \mathcal {T} \times X \times C \mapsto R,~ i=\overline{1,n},~\alpha = \overline{1,m},\) are continuously differentiable functionals. The first-order PDE constraints \(\mathcal {H}^i_\alpha \) also satisfy the complete integrability conditions (closeness conditions) \(D_\gamma \mathcal {H}_\alpha = D_\alpha \mathcal {H}_\gamma ,~ \alpha ,\gamma =\overline{1,m},\alpha \ne \gamma , i=\overline{1,n}\), where \(D_\gamma \) is the total derivative.

The robust counterpart to (FP) which reduces all the possible uncertainties present in (FP), is constructed as follows:

$$\begin{aligned}&\text {(RFP)} \quad \min _{(a(\cdot ),c(\cdot ))} \frac{\int _\mathcal {T} \max _{u \in U} f(t,a(t),c(t),u) \textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V}g(t,a(t),c(t),v) \textrm{d}\omega } \nonumber \\&\quad \text {subject to} \nonumber \\&\quad \mathcal {Y}_\beta (t,a(t),c(t)) \le 0,~~ \beta = \overline{1,q},\\&\quad \frac{\partial a^i}{\partial t^\alpha }=\mathcal {H}^i_\alpha (t,a(t),c(t)),~i=\overline{1,n}, ~\alpha =\overline{1,m},\\&\quad a(t_0)=a_0,~a(t_1)=a_1, \end{aligned}$$

where \(f, g, \mathcal {Y} =(\mathcal {Y}_\beta )\) and \(\mathcal {H}=(\mathcal {H}^i_\alpha )\) are same as defined for (FP).

The set of all feasible solutions to (RFP) (which is also the set of robust feasible solution to (FP)) is defined as:

$$\begin{aligned}&\displaystyle \mathcal {D} = \{(a(t),c(t)) \in (X,C) ~ | ~\mathcal {Y}(t,a(t),c(t)) \le 0, ~\frac{\partial a}{\partial t}=\mathcal {H}(t,a(t),c(t)),\\&\displaystyle a(t_0)=a_0,~a(t_1)=a_1,~t \in \mathcal {T}\}. \end{aligned}$$

Now, on the line of Dinkelbach [7], and Antczak and Pitea [2], we present a non-fractional problem (NFP) associated to (FP) as:

$$\begin{aligned}&\text {(NFP)} \quad \min _{(a(\cdot ),c(\cdot ))} \Big \{ \int _\mathcal {T} f(t,a(t),c(t),u) \textrm{d}\omega - \mathcal {P}(t,a(t),c(t),u,v) \int _\mathcal {T} g(t,a(t),c(t),v) \textrm{d}\omega \Big \} \nonumber \\&\qquad \text {subject to} \nonumber \\&\qquad \mathcal {Y}_\beta (t,a(t),c(t)) \le 0,~~ \beta = \overline{1,q},\\&\qquad \frac{\partial a^i}{\partial t^\alpha }=\mathcal {H}^i_\alpha (t,a(t),c(t)),~i=\overline{1,n}, ~\alpha =\overline{1,m},\\&\qquad a(t_0)=a_0,~a(t_1)=a_1, \end{aligned}$$

where, parameter \(\mathcal {P}(t,a(t),c(t),u,v) \in R_+.\)

The robust counterpart to (NFP) is given by

$$\begin{aligned}&\text {(RNFP)} \quad \min _{(a(\cdot ),c(\cdot ))}\Big \{ \int _\mathcal {T} \max _{u \in U} f(t,a(t),c(t),u) \textrm{d}\omega - \mathcal {P}(t,a(t),c(t),u,v) \int _\mathcal {T} \min _{v \in V}g(t,a(t),c(t),v) \textrm{d}\omega \Big \} \nonumber \\&\qquad \text {subject to} \nonumber \\&\qquad \mathcal {Y}_\beta (t,a(t),c(t)) \le 0,~~ \beta = \overline{1,q},\\&\qquad \frac{\partial a^i}{\partial t^\alpha }=\mathcal {H}^i_\alpha (t,a(t),c(t)),~i=\overline{1,n}, ~\alpha =\overline{1,m},\\&\qquad a(t_0)=a_0,~a(t_1)=a_1. \end{aligned}$$

Remark 2.1

We can clearly observe that \(\mathcal {D}\) is the set of robust feasible solutions to (NFP) (feasible solutions to (RNFP)).

For simplicity of presentation, we use some notions throughout the article given as: \(a=a(t),~b=b(t),~ c=c(t),~z=z(t),~\overline{a}=\overline{a}(t),~\overline{b}=\overline{b}(t),~\overline{c}=\overline{c}(t),~\overline{z}=\overline{z}(t), ~ \pi =(t,a(t),c(t)),~\overline{\pi }=(t,\overline{a}(t),\overline{c}(t)),~\hat{\pi }=(t,\hat{a}(t),\hat{c}(t)),~\varrho =(t,b(t),z(t)),~\overline{\varrho }=(t,\overline{b}(t),\overline{z}(t)),~\hat{\varrho }=(t,\hat{b}(t),\hat{z}(t))\).

Definition 2.1

A point \((\overline{a},\overline{c}) \in \mathcal {D}\) is said to be a robust optimal solution to (FP), if

$$\begin{aligned}&\frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u) \textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V}g(\overline{\pi },v) \textrm{d}\omega } \le \frac{\int _\mathcal {T} \max _{u \in U} f(\pi ,u) \textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V}g(\pi ,v) \textrm{d}\omega }, ~\forall (a,c) \in \mathcal {D}. \end{aligned}$$

Definition 2.2

A point \((\overline{a},\overline{c}) \in \mathcal {D}\) is said to be a robust optimal solution to (NFP), if

$$\begin{aligned}&\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u) \textrm{d}\omega - \mathcal {P}(\overline{\pi },u,v) \int _\mathcal {T} \min _{v \in V}g(\overline{\pi },v) \textrm{d}\omega \\&\qquad \le \int _\mathcal {T} \max _{u \in U} f(\pi ,u) \textrm{d}\omega - \mathcal {P}(\overline{\pi },u,v) \int _\mathcal {T} \min _{v \in V} g(\pi ,v) \textrm{d}\omega ,\quad \forall (a,c) \in \mathcal {D}. \end{aligned}$$

Remark 2.2

The robust optimal solutions to (FP) and (NFP) are also the optimal solution to (RFP) and (RNFP), respectively.

Following the footsteps of Manesh et al. [16], we present an equivalent relation between the robust optimal solutions to (FP) and (NFP), through given lemma.

Lemma 2.1

Let \((\overline{a},\overline{c}) \in \mathcal {D}\) is a robust optimal solution to (FP) then there exist \(\mathcal {P}(\overline{\pi },u,v) \in R_+\) such that \((\overline{a},\overline{c})\) is a robust optimal solution to (NFP). Conversely, if \((\overline{a},\overline{c}) \in \mathcal {D}\) is a robust optimal solution to (NFP), with \(\mathcal {P}(\overline{\pi },u,v)=\frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\overline{\pi },v)\textrm{d}\omega }\), then \((\overline{a},\overline{c})\) is a robust optimal solution to (FP).

Proof

Let \((\overline{a},\overline{c}) \in \mathcal {D}\) is a robust optimal solution to (FP), but not a robust optimal solution in (NFP), then there exist \((a,c) \in \mathcal {D}\), such that

$$\begin{aligned}&\int _\mathcal {T} \max _{u \in U} f(\pi ,u) d\omega - \mathcal {P}(\overline{\pi },u,v) \int _\mathcal {T} \min _{v \in V} g(\pi ,v) \textrm{d}\omega \\&\qquad < \int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u) \textrm{d}\omega - \mathcal {P}(\overline{\pi },u,v) \int _\mathcal {T} \min _{v \in V}g(\overline{\pi },v) \textrm{d}\omega . \end{aligned}$$

In particular, if we take \(\mathcal {P}(\overline{\pi },u,v)=\frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\overline{\pi },v)\textrm{d}\omega }\), then

$$\begin{aligned}&\int _{\mathcal {T}} \max _{u \in U} f(\pi ,u)\textrm{d}\omega - \frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\overline{\pi },v)\textrm{d}\omega } \int _{\mathcal {T}} \min _{v \in V} g(\pi ,v) \textrm{d}\omega \\&\quad < \int _{\mathcal {T}} \max _{u \in U} f(\overline{\pi },u) \textrm{d}\omega - \frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\overline{\pi },v)\textrm{d}\omega } \int _{\mathcal {T}} \min _{v \in V} g(\overline{\pi },v) \textrm{d}\omega . \end{aligned}$$

Further, the above inequality yields

$$\begin{aligned}&\frac{\int _\mathcal {T} \max _{u \in U} f(\pi ,u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\pi ,v) \textrm{d}\omega } < \frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u) \textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\overline{\pi },v) \textrm{d}\omega }, \end{aligned}$$

which contradicts \((\overline{a},\overline{c})\) is a robust optimal solution to (FP).

Conversely, let \((\overline{a},\overline{c}) \in \mathcal {D}\) is a robust optimal solution to (NFP) and \(\mathcal {P}(\overline{\pi },u,v)=\frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\overline{\pi },v)\textrm{d}\omega }\). Suppose, contrary to the result that \((\overline{a},\overline{c})\) is not a robust optimal solution to (FP), then there exist \((a,c) \in \mathcal {D}\), such that

$$\begin{aligned}&\frac{\int _\mathcal {T} \max _{u \in U} f(\pi ,u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\pi ,v) \textrm{d}\omega } < \frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u) \textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\overline{\pi },v) \textrm{d}\omega }. \end{aligned}$$

As \(\mathcal {P}(\overline{\pi },u,v)=\frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\overline{\pi },v)\textrm{d}\omega }\), the above inequality becomes

$$\begin{aligned}&\frac{\int _\mathcal {T} \max _{u \in U} f(\pi ,u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\pi ,v) \textrm{d}\omega }< \mathcal {P}(\overline{\pi },u,v), \\&\quad \text {or} ~~\int _\mathcal {T} \max _{u \in U} f(\pi ,u) \textrm{d}\omega - \mathcal {P}(\overline{\pi },u,v) \int _\mathcal {T} \min _{v \in V} g(\pi ,v) \textrm{d}\omega < 0. \end{aligned}$$

Again, since \(\mathcal {P}(\overline{\pi },u,v)=\frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\overline{\pi },v)\textrm{d}\omega }\), we obtain

$$\begin{aligned}&\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u) \textrm{d}\omega - \mathcal {P}(\overline{\pi },u,v) \int _\mathcal {T} \min _{v \in V}g(\overline{\pi },v) \textrm{d}\omega = 0. \end{aligned}$$

On combining the above two inequalities, we have

$$\begin{aligned} \int _\mathcal {T} \max _{u \in U} f(\pi ,u) \textrm{d}\omega&- \mathcal {P}(\overline{\pi },u,v) \int _\mathcal {T} \min _{v \in V} g(\pi ,v) \textrm{d}\omega \\&\qquad < \int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u) \textrm{d}\omega - \mathcal {P}(\overline{\pi },u,v) \int _\mathcal {T} \min _{v \in V}g(\overline{\pi },v) \textrm{d}\omega , \end{aligned}$$

which contradicts that \((\overline{a},\overline{c})\) is a robust optimal solution to (RNFP). Hence the proof is complete \(\square \)

This article aims to prove the sufficiency of the parametric necessary optimality condition and the duality results. So, we present here the required definition of convex and concave functionals as follows:

Definition 2.3

[11] A functional \(\int _\mathcal {T} f(\pi ,\overline{u}) \textrm{d}\omega \) is said to be convex at \((\overline{a},\overline{c})\), if the following inequality

$$\begin{aligned}&\int _\mathcal {T} f(\pi ,\overline{u}) \textrm{d}\omega -\int _\mathcal {T} f(\overline{\pi },\overline{u}) \textrm{d}\omega \\&\quad \ge \int _\mathcal {T} \Big \{ (a-\overline{a}) \frac{\partial f}{\partial x^i} (\overline{\pi },\overline{u}) + (c-\overline{c}) \frac{\partial f}{\partial c^j} (\overline{\pi },\overline{u}) \Big \} \textrm{d}\omega ,~ \forall (a,c) \in (X,C), \end{aligned}$$

holds.

Definition 2.4

A functional \(\int _\mathcal {T} f(\pi ,\overline{u}) \textrm{d}\omega \) is said to be concave at \((\overline{a},\overline{c})\), if the following inequality

$$\begin{aligned}&\int _\mathcal {T} f(\pi ,\overline{u}) \textrm{d}\omega -\int _\mathcal {T} f(\overline{\pi },\overline{u}) \textrm{d}\omega \\&\quad \le \int _\mathcal {T} \Big \{ (a-\overline{a}) \frac{\partial f}{\partial x^i} (\overline{\pi },\overline{u}) + (c-\overline{c}) \frac{\partial f}{\partial c^j} (\overline{\pi },\overline{u}) \Big \} \textrm{d}\omega ,~ \forall (a,c) \in (X,C), \end{aligned}$$

holds.

3 Parametric Robust Optimality Conditions

In this section, we establish the parametric robust necessary optimality conditions for the problem (FP) and prove it’s sufficiency under the hypotheses of convexity and concavity over the involved functionals.

Now, on the line of Jayswal et al. [11] and Mititelu and Treanţă [19], we define the robust KT point for (FP) as follows:

Definition 3.1

The point \((\overline{a},\overline{c}) \in \mathcal {D}\) is said to be a robust KT point to (FP), if \(\max _{u \in U}f(\pi ,u)\) \( = f(\pi ,\overline{u})\), \(\min _{v \in V}g(\pi ,v) = g(\pi ,\overline{v})\), and there exist piecewise smooth Lagrange multipliers \(\overline{\mu } = (\overline{\mu }_\beta ) \in R^q_+,~\overline{\lambda } = (\overline{\lambda }^i_\alpha ) \in R^{nm}\), such that

$$\begin{aligned}&\frac{\partial f}{\partial a^i} (\overline{\pi },\overline{u}) -\mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \frac{\partial g}{\partial a^i} (\overline{\pi },\overline{v}) + \overline{\mu }_\beta \frac{\partial \mathcal {Y}_\beta }{\partial a^i} (\overline{\pi }) + \overline{\lambda }^i_\alpha \frac{\partial \mathcal {H}^i_\alpha }{\partial a^i} (\overline{\pi }) + \frac{\partial \overline{\lambda }^i_\alpha }{\partial t^\alpha }=0,~i=\overline{1,n}, \end{aligned}$$
(1)
$$\begin{aligned}&\quad \frac{\partial f}{\partial c^j} (\overline{\pi },\overline{u}) - \mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \frac{\partial g}{\partial c^j} (\overline{\pi },\overline{v}) + \overline{\mu }_\beta \frac{\partial \mathcal {Y}_\beta }{\partial c^j} (\overline{\pi }) + \overline{\lambda }^i_\alpha \frac{\partial \mathcal {H}^i_\alpha }{\partial c^j} (\overline{\pi })=0,~j=\overline{1,k}, \end{aligned}$$
(2)
$$\begin{aligned}&\quad \overline{\mu }_\beta \mathcal {Y}_\beta (\overline{\pi }) = 0, ~ \overline{\mu }_\beta \ge 0, \end{aligned}$$
(3)

hold for all \(t \in \mathcal {T}\), except at discontinuities.

Remark 3.1

The above conditions (1)–(3) are said to be the parametric robust necessary optimality conditions for (FP).

Theorem 3.1

Let a point \((\overline{a},\overline{c}) \in \mathcal {D}\) be a robust normal optimal solution to (FP). Then, \((\overline{a},\overline{c})\) is a robust KT point.

Proof

Let the constraint conditions (for the existence of multipliers) hold. Following the direction of Mititelu and Treanţă [19], if \((\overline{a},\overline{c}) \in \mathcal {D}\) be a robust optimal solution to (FP), then there exist the piecewise smooth Lagrange multipliers \(\theta \in R,~\overline{\mu } = (\overline{\mu }_\beta ) \in R^q_+,~\overline{\lambda } = (\overline{\lambda }^i_\alpha ) \in R^{nm}\), such that the following inequalities

$$\begin{aligned}&\theta \Big (\frac{\partial f}{\partial a^i} (\overline{\pi },\overline{u}) - \mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \frac{\partial g}{\partial a^i} (\overline{\pi },\overline{v}) \Big ) + \overline{\mu }_\beta \frac{\partial \mathcal {Y}_\beta }{\partial a^i} (\overline{\pi }) + \overline{\lambda }^i_\alpha \frac{\partial \mathcal {H}^i_\alpha }{\partial a^i} (\overline{\pi }) + \frac{\partial \overline{\lambda }^i_\alpha }{\partial t^\alpha }=0,~i=\overline{1,n}, \end{aligned}$$
(4)
$$\begin{aligned}&\quad \theta \Big (\frac{\partial f}{\partial c^j} (\overline{\pi },\overline{u}) - \mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \frac{\partial g}{\partial c^j} (\overline{\pi },\overline{v})\Big ) + \overline{\mu }_\beta \frac{\partial \mathcal {Y}_\beta }{\partial c^j} (\overline{\pi }) + \overline{\lambda }^i_\alpha \frac{\partial \mathcal {H}^i_\alpha }{\partial c^j} (\overline{\pi })=0,~j=\overline{1,k}, \end{aligned}$$
(5)
$$\begin{aligned}&\quad \overline{\mu }_\beta \mathcal {Y}_\beta (\overline{\pi }) = 0, ~ \overline{\mu }_\beta \ge 0, \end{aligned}$$
(6)

hold for all \(t \in \mathcal {T}\), except at discontinuities. Since the robust optimal solution \((\overline{a},\overline{c}) \in \mathcal {D}\) to (FP) is said to be a robust normal optimal solution, if \(\theta > 0\), then we can assume that \(\theta = 1\), without losing generality. This completes the proof. \(\square \)

Next we proof the sufficiency of the parametric robust necessary optimality conditions (1)–(3).

Theorem 3.2

Let \((\overline{a}, \overline{c}) \in \mathcal {D}\) and there exist the piecewise smooth Lagrange multipliers \(\overline{\mu } = (\overline{\mu }_\beta ) \in R^q_+,~\overline{\lambda } = (\overline{\lambda }^i_\alpha ) \in R^{nm}\), such that the parametric robust necessary optimality conditions (1)–(3) hold for all \(t\in \mathcal {T}\), except at discontinuities and \(\max _{u \in U} f(\pi , u) = f(\pi , \overline{u})\), \(\min _{v \in V} g(\pi ,v) = g(\pi , \overline{v})\), also the parameter \(\mathcal {P}(\overline{\pi },u,v)=\frac{\int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u)\textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V} g(\overline{\pi },v)d\omega }\). Further assume that the functionals \(\int _\mathcal {T} f(\pi , \overline{u})\textrm{d}\omega ,\) \(\int _\mathcal {T} \langle \overline{\mu }_\beta , \mathcal {Y}_\beta (\pi ) \rangle \textrm{d}\omega ,~\beta = \overline{1, q}\) and \(\int _\mathcal {T} \langle \overline{\lambda }^i_\alpha , (\mathcal {H}^i_\alpha (\pi ) -\frac{\partial a^i}{\partial t^\alpha } )\rangle \textrm{d}\omega ,~ i= \overline{1,n},~\alpha = \overline{1, m}\) are convex and \(\int _\mathcal {T} g(\pi , \overline{v})\textrm{d}\omega \) is concave on \(\mathcal {D}\). Then, \((\overline{a}, \overline{c})\) is a robust optimal solution to (FP).

Proof

We assume on contrary that \((\overline{a}, \overline{c})\) is not a robust optimal solution to (FP), so from Lemma 2.1, \((\overline{a}, \overline{c})\) be not a robust optimal solution to (NFP). Then, there exists \((a,c)\in \mathcal {D}\) such that

$$\begin{aligned}&\int _\mathcal {T} \max _{u \in U} f(\pi ,u) \textrm{d}\omega - \mathcal {P}(\overline{\pi },u,v) \int _\mathcal {T} \min _{v \in V} g(\pi ,v) \textrm{d}\omega \\&\qquad < \int _\mathcal {T} \max _{u \in U} f(\overline{\pi },u) \textrm{d}\omega - \mathcal {P}(\overline{\pi },u,v) \int _\mathcal {T} \min _{v \in V}g(\overline{\pi },v) \textrm{d}\omega . \end{aligned}$$

Since, \(\max _{u \in U} f(\pi , u) = f(\pi , \overline{u})\) and \(\min _{v \in V} g(\pi ,v) = g(\pi , \overline{v})\), we get

$$\begin{aligned}&\int _\mathcal {T} f(\pi ,\overline{u}) \textrm{d}\omega -\mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \int _\mathcal {T} g(\pi ,\overline{v}) \textrm{d}\omega < \int _\mathcal {T} f(\overline{\pi },\overline{u}) \textrm{d}\omega -\mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \int _\mathcal {T} g(\overline{\pi },\overline{v}) \textrm{d}\omega . \end{aligned}$$
(7)

Also, since \(\int _\mathcal {T} f(\pi ,\overline{u})\textrm{d}\omega \) is convex and \(\int _\mathcal {T} g(\pi ,\overline{v})\textrm{d}\omega \) is concave on \(\mathcal {D}\), we have

$$\begin{aligned}&\int _\mathcal {T} f(\pi ,\overline{u}) \textrm{d}\omega -\int _\mathcal {T} f(\overline{\pi },\overline{u}) \textrm{d}\omega \ge \int _\mathcal {T} \Big \{ (a-\overline{a}) \frac{\partial f}{\partial a^i} (\overline{\pi },\overline{u}) +(c-\overline{c}) \frac{\partial f}{\partial c^j} (\overline{\pi },\overline{u}) \Big \} \textrm{d}\omega , \end{aligned}$$
(8)

and

$$\begin{aligned}&\int _\mathcal {T} g(\pi ,\overline{v}) \textrm{d}\omega -\int _\mathcal {T} g(\overline{\pi },\overline{v}) \textrm{d}\omega \le \int _\mathcal {T} \Big \{ (a-\overline{a}) \frac{\partial g}{\partial a^i} (\overline{\pi },\overline{v}) + (c-\overline{c}) \frac{\partial g}{\partial c^j} (\overline{\pi },\overline{v}) \Big \} \textrm{d}\omega . \end{aligned}$$
(9)

On multiplying the inequality (9) with \(\mathcal {P}(\overline{\pi },\overline{u},\overline{v})\) and then subtracting it from the inequality (8), we obtain

$$\begin{aligned}&\int _\mathcal {T} f(\pi ,\overline{u}) \textrm{d}\omega -\mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \int _\mathcal {T} g(\pi ,\overline{v}) \textrm{d}\omega - \int _\mathcal {T} f(\overline{\pi },\overline{u})\textrm{d}\omega -\mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \int _\mathcal {T} g(\overline{\pi },\overline{v}) \}\textrm{d}\omega \\&\qquad \ge \int _\mathcal {T} (a-\overline{a})\frac{\partial f}{\partial a^i} (\overline{\pi },\overline{u}) \textrm{d}\omega -\mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \int _\mathcal {T}(a-\overline{a}) \frac{\partial g}{\partial a^i} (\overline{\pi },\overline{v}) \textrm{d}\omega \\&\qquad \qquad + \int _\mathcal {T} (c-\overline{c})\frac{\partial f}{\partial c^j} (\overline{\pi },\overline{u}) \textrm{d}\omega -\mathcal {P}(\overline{\pi },\overline{u},\overline{v})\int _\mathcal {T}(c-\overline{c}) \frac{\partial g}{\partial c^j} (\overline{\pi },\overline{v}) \textrm{d}\omega , \end{aligned}$$

which, together with the inequality (7), yields

$$\begin{aligned}&\int _\mathcal {T} (a-\overline{a})\frac{\partial f}{\partial a^i} (\overline{\pi },\overline{u}) \textrm{d}\omega -\mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \int _\mathcal {T} (a-\overline{a})\frac{\partial g}{\partial a^i} (\overline{\pi },\overline{v}) \textrm{d}\omega \nonumber \\&\qquad + \int _\mathcal {T} (c-\overline{c})\frac{\partial f}{\partial c^j} (\overline{\pi },\overline{u}) \textrm{d}\omega -\mathcal {P}(\overline{\pi },\overline{u},\overline{v})\int _\mathcal {T} (c-\overline{c}) \frac{\partial g}{\partial c^j} (\overline{\pi },\overline{v}) \textrm{d}\omega < 0. \end{aligned}$$
(10)

From the hypotheses that the functionals, \(\int _\mathcal {T} \langle \overline{\mu }_\beta , \mathcal {Y}_\beta (\pi ) \rangle \textrm{d}\omega \) and \(\int _\mathcal {T} \langle \overline{\lambda }^i_\alpha , (\mathcal {H}^i_\alpha (\pi ) - \frac{\partial a^i}{\partial t^\alpha }) \rangle \textrm{d}\omega \) are convex on \(\mathcal {D}\), we obtain

$$\begin{aligned}&\int _\mathcal {T} \Big \{\langle \overline{\mu }_\beta , \mathcal {Y}_\beta (\pi ) \rangle - \langle \overline{\mu }_\beta , \mathcal {Y}_\beta (\overline{\pi }) \rangle \Big \}\textrm{d}\omega \ge \int _\mathcal {T} (a-\overline{a}) \langle \overline{\mu }_\beta , \frac{\partial \mathcal {Y}_\beta }{\partial a^i}(\overline{\pi }) \rangle \textrm{d}\omega + \int _\mathcal {T} (c-\overline{c}) \langle \overline{\mu }_\beta , \\&\quad \frac{\partial \mathcal {Y}_\beta }{\partial c^j} (\overline{\pi }) \textrm{d}\omega ,\\&\quad \text {and}\\&\quad \int _\mathcal {T} \Big \{\langle \overline{\lambda }^i_\alpha , (\mathcal {H}^i_\alpha (\pi ) - \frac{\partial a^i}{\partial t^\alpha }) \rangle - \langle \overline{\lambda }^i_\alpha ,(\mathcal {H}^i_\alpha (\overline{\pi }) - \frac{\partial a^i}{\partial t^\alpha }) \rangle \Big \}\textrm{d}\omega \ge \int _\mathcal {T} (a-\overline{a}) \Big \{\langle \overline{\lambda }^i_\alpha , \frac{\partial \mathcal {H}^i_\alpha }{\partial a^i}(\overline{\pi }) \rangle \\&\quad + \frac{\partial \overline{\lambda }^i_\alpha }{\partial t^\alpha }\Big \} \textrm{d}\omega + \int _\mathcal {T} (c-\overline{c}) \langle \overline{\lambda }^i_\alpha , \frac{\partial \mathcal {H}^i_\alpha }{\partial c^j}(\overline{\pi }) \rangle \textrm{d}\omega . \end{aligned}$$

From the robust feasibility of (a, c) in (FP) and the necessary optimality condition (3), it follows for the above two inequalities

$$\begin{aligned}&\int _\mathcal {T} (a-\overline{a}) \langle \overline{\mu }_\beta , \frac{\partial \mathcal {Y}_\beta }{\partial a^i}(\overline{\pi }) \rangle \textrm{d}\omega + \int _\mathcal {T} (c-\overline{c}) \langle \overline{\mu }_\beta , \frac{\partial \mathcal {Y}_\beta }{\partial c^j} (\overline{\pi }) \textrm{d}\omega \le 0, \end{aligned}$$
(11)
$$\begin{aligned}&\int _\mathcal {T} (a-\overline{a}) \Big \{\langle \overline{\lambda }^i_\alpha , \frac{\partial \mathcal {H}^i_\alpha }{\partial a^i}(\overline{\pi }) \rangle + \frac{\partial \overline{\lambda }^i_\alpha }{\partial t^\alpha }\Big \} \textrm{d}\omega + \int _\mathcal {T} (c-\overline{c}) \langle \overline{\lambda }^i_\alpha , \frac{\partial \mathcal {H}^i_\alpha }{\partial c^j}(\overline{\pi }) \rangle \textrm{d}\omega \le 0. \end{aligned}$$
(12)

On adding the inequalities (10), (11) and (12), we obtain

$$\begin{aligned}&\int _\mathcal {T} (a-\overline{a})\frac{\partial f}{\partial a^i} (\overline{\pi },\overline{u})\textrm{d}\omega -\mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \int _\mathcal {T} (a-\overline{a}) \frac{\partial g}{\partial a^i} (\overline{\pi },\overline{v}) \textrm{d}\omega + \int _\mathcal {T} (a-\overline{a}) \langle \overline{\mu }_\beta , \frac{\partial \mathcal {Y}_\beta }{\partial a^i}(\overline{\pi }) \rangle \textrm{d}\omega \nonumber \\&\qquad + \int _\mathcal {T} (a-\overline{a}) \langle \overline{\lambda }^i_\alpha , \frac{\partial \mathcal {H}^i_\alpha }{\partial a^i}(\overline{\pi }) \rangle \textrm{d}\omega + \int _\mathcal {T} (a-\overline{a}) \frac{\partial \overline{\lambda }^i_\alpha }{\partial t^\alpha } \textrm{d}\omega + \int _\mathcal {T} (c-\overline{c}) \frac{\partial f}{\partial c^j} (\overline{\pi },\overline{u}) \textrm{d}\omega \nonumber \\&\quad - \mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \int _\mathcal {T} (c-\overline{c}) \frac{\partial g}{\partial c^j} (\overline{\pi },\overline{v})\textrm{d}\omega + \int _\mathcal {T} (c-\overline{c}) \langle \overline{\mu }_\beta , \frac{\partial \mathcal {Y}_\beta }{\partial c^j} (\overline{\pi }) \rangle \textrm{d}\omega \nonumber \\&\qquad + \int _\mathcal {T} (c-\overline{c}) \langle \overline{\lambda }^i_\alpha , \frac{\partial \mathcal {H}^i_\alpha }{\partial c^j}(\overline{\pi }) \rangle \textrm{d}\omega < 0, \end{aligned}$$
(13)

On the other hand, if we integrate the necessary optimality conditions (1) and (2) after multiplying them with the terms \((a-\overline{a})\) and \((c-\overline{c})\), respectively, and then adding the resulting equations, we get

$$\begin{aligned}&\int _\mathcal {T} (a-\overline{a})\frac{\partial f}{\partial a^i} (\overline{\pi },\overline{u})\textrm{d}\omega -\mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \int _\mathcal {T} (a-\overline{a}) \frac{\partial g}{\partial a^i} (\overline{\pi },\overline{v}) \textrm{d}\omega \\&\qquad + \int _\mathcal {T} (a-\overline{a}) \langle \overline{\mu }_\beta , \frac{\partial \mathcal {Y}_\beta }{\partial a^i}(\overline{\pi }) \rangle \textrm{d}\omega + \int _\mathcal {T} (a-\overline{a}) \langle \overline{\lambda }^i_\alpha , \frac{\partial \mathcal {H}^i_\alpha }{\partial a^i}(\overline{\pi }) \rangle \textrm{d}\omega \nonumber \\&\qquad + \int _\mathcal {T} (a-\overline{a}) \frac{\partial \overline{\lambda }^i_\alpha }{\partial t^\alpha } \textrm{d}\omega + \int _\mathcal {T} (c-\overline{c}) \frac{\partial f}{\partial c^j} (\overline{\pi },\overline{u}) \textrm{d}\omega \nonumber \\&\qquad - \mathcal {P}(\overline{\pi },\overline{u},\overline{v}) \int _\mathcal {T} (c-\overline{c}) \frac{\partial g}{\partial c^j} (\overline{\pi },\overline{v})\textrm{d}\omega + \int _\mathcal {T} (c-\overline{c}) \langle \overline{\mu }_\beta , \frac{\partial \mathcal {Y}_\beta }{\partial c^j} (\overline{\pi }) \rangle \textrm{d}\omega \nonumber \\&\qquad + \int _\mathcal {T} (c-\overline{c}) \langle \overline{\lambda }^i_\alpha , \frac{\partial \mathcal {H}^i_\alpha }{\partial c^j}(\overline{\pi }) \rangle \textrm{d}\omega = 0, \end{aligned}$$

which opposes the inequality (13). Hence the theorem. \(\square \)

Following example numerically illustrate the Theorem 3.2.

Example 3.1

Let \(X = C = R\), \(U=[-1,1],~V=[-3,3]\) and \(t \in \mathcal {T} \subset R^2\) fixed by the diagonally opposite points \(t_0:= (t^1_0, t^2_ 0)=(0,0),~t_1:= (t^1_ 1, t^2_1)=(1,1) \in R^2\). We consider the following multi-dimensional fractional control optimization problem with data uncertainty in objective functional as:

$$\begin{aligned}&\text {(FP1)} ~~\min _{(a(\cdot ),c(\cdot ))} \Bigg \{\frac{\int _\mathcal {T} f(\pi ,u) \textrm{d}t^1\textrm{d}t^2}{\int _\mathcal {T} g(\pi ,v) \textrm{d}t^1\textrm{d}t^2} = \frac{\int _\mathcal {T} [a(\textit{e}^a-1)+2c^2+u] \textrm{d}t^1\textrm{d}t^2}{\int _\mathcal {T} [-\frac{a^2}{4} -c^2-2v] \textrm{d}t^1\textrm{d}t^2}\Bigg \}, \\&\quad \text {subject to} \nonumber \\&\quad \mathcal {Y}(\pi )=4a^2-9 \le 0,\\&\quad \frac{\partial a}{\partial t^\alpha }= \mathcal {H}_\alpha (\pi ) =3-c,~~\alpha =\{1,2\}\\&\quad a(0,0)=0,~a(1,1)=4. \end{aligned}$$

The parametric form associated with (FP1) is defined as follows:

$$\begin{aligned}&\text {(NFP1)} ~~\min _{(a(\cdot ),c(\cdot ))} \int _\mathcal {T} \{ a(\textit{e}^a-1)+2c^2+u \} \textrm{d}t^1\textrm{d}t^2 - \mathcal {P} \int _\mathcal {T} \Big (-\frac{a^2}{4} -c^2-2v\Big ) \textrm{d}t^1\textrm{d}t^2, \nonumber \\&\quad \text {subject to} \nonumber \\&\quad 4a^2-9 \le 0,\\&\quad \frac{\partial a}{\partial t^\alpha }=3-c,~~\alpha =\{1,2\}\\&\quad a(0,0)=0,~a(1,1)=4. \end{aligned}$$

where \(\mathcal {P} = \mathcal {P}(\pi ,u,v) \in R_+\). Then, the robust counterpart for (NFP1) is given by

$$\begin{aligned}&\text {(RNFP1)} ~~\min _{(a(\cdot ),c(\cdot ))}\int _\mathcal {T} \max _{u \in U} \{a(\textit{e}^a-1)+2c^2+u \}\textrm{d}t^1\textrm{d}t^2 \nonumber \\&\quad - \mathcal {P} \int _\mathcal {T} \min _{v \in V}\Big (-\frac{a^2}{4} -c^2-2v\Big ) \textrm{d}t^1\textrm{d}t^2, \nonumber \\&\quad \text {subject to} \nonumber \\&\quad \quad 4a^2-9 \le 0, \end{aligned}$$
(14)
$$\begin{aligned}&\quad \quad \frac{\partial a}{\partial t^\alpha }=3-c,~~\alpha =\{1,2\} \end{aligned}$$
(15)
$$\begin{aligned}&\quad \quad a(0,0)=0,~a(1,1)=4. \end{aligned}$$
(16)

The set of robust feasible solutions to (NFP1) is

$$\begin{aligned} \mathcal {D}=\Big \{(a,c)\in (X,C): - \frac{3}{2} \le a \le \frac{3}{2},~ \frac{\partial a}{\partial t^1}=\frac{\partial a}{\partial t^2},~ a(0,0)=0,~ a(1,1)=4\Big \}. \end{aligned}$$

By direct computation, we observe that \((\overline{a},\overline{c})=(2(t^1+t^2),1) \in \mathcal {D}\) at \(t^1=t^2=0\) satisfies the necessary optimality conditions (1)–(3) with parameter \(\mathcal {P} = \frac{3}{5}\), uncertainty parameters \(\overline{u}=1,\overline{v}=-3\) and Lagrange multipliers \(\overline{\mu }=0,\overline{\lambda }_1=\overline{\lambda }_2=\frac{13}{5}\). Furthermore, one can easily verify that the functionals \(\int _{ \mathcal {T}} f(\overline{\pi },\overline{u})\textrm{d}t^1\textrm{d}t^2,~ \int _{ \mathcal {T}} \langle \overline{\mu }, \mathcal {Y}(\overline{\pi }) \rangle \textrm{d}t^1\textrm{d}t^2, \int _{ \mathcal {T}} \langle \overline{\lambda }_\alpha , (\mathcal {H}_\alpha (\overline{\pi })-\frac{\partial a}{\partial t^\alpha }) \rangle \textrm{d}t^1\textrm{d}t^2,~\alpha =\{1,2\}\) are convex on \(\mathcal {D}\) and the functional \(\int _{ \mathcal {T}} g(\overline{\pi },\overline{v})\textrm{d}t^1\textrm{d}t^2\) is concave on \(\mathcal {D}\). Thus, all the hypotheses of Theorem 3.2 are fulfilled. Now it is remaining to verify that \((\overline{a},\overline{c})\) is robust optimal solution to (FP1).

We observe that the inequality

$$\begin{aligned}&\int _\mathcal {T} f(\pi ,\overline{u}) \textrm{d}t^1\textrm{d}t^2 -\mathcal {P} \int _\mathcal {T} g(\pi ,\overline{v}) \textrm{d}t^1\textrm{d}t^2 - \int _\mathcal {T} f(\overline{\pi },\overline{u}) \textrm{d}t^1\textrm{d}t^2 -\mathcal {P} \int _\mathcal {T}g(\overline{\pi },\overline{v}) \textrm{d}t^1\textrm{d}t^2\\&\quad =\int _\mathcal {T} \Big \{ a(\textit{e}^a-1)+2c^2+\overline{u} \Big \} \textrm{d}t^1\textrm{d}t^2 - \mathcal {P}\int _\mathcal {T} \Big (-\frac{a^2}{4} -c^2-2\overline{v}\Big ) \textrm{d}t^1\textrm{d}t^2\\&\qquad - \int _\mathcal {T} \Big \{\overline{a}(\textit{e}^{\overline{a}}-1)+2\overline{c}^2+\overline{u} \Big \} \textrm{d}t^1\textrm{d}t^2 - \mathcal {P}\int _\mathcal {T}\Big (-\frac{\overline{a}^2}{4} -\overline{c}^2-2\overline{v}\Big ) \textrm{d}t^1\textrm{d}t^2\\&\quad \ge 0, \end{aligned}$$

holds for all \((a,c) \in \mathcal {D}\) at \((\overline{a},\overline{c})=(0,1)\), therefore \((\overline{a},\overline{c})\) is a robust optimal solution to (NFP1). Since, parameter \(\mathcal {P}= \frac{\int _\mathcal {T} f(\overline{\pi },\overline{u}) \textrm{d}t^1\textrm{d}t^2}{\int _\mathcal {T} g(\overline{\pi },\overline{v}) \textrm{d}t^1\textrm{d}t^2} = \frac{3}{5}\), thus from Lemma 2.1\((\overline{a},\overline{c})\) is also robust optimal solution to (FP1).

4 Parametric Robust Dual Problem

In this section, we formulate parametric dual problem for (FP) as follows:

$$\begin{aligned}&\text {(DFP)}~~ \max _{(b(\cdot ),z(\cdot ))} \mathcal {P}(\varrho ,u,v) = \frac{\int _\mathcal {T} f(\varrho ,u) \textrm{d}\omega }{\int _\mathcal {T} g(\varrho ,v)\textrm{d}\omega }\\&\quad \text {subject to}\\&\quad \frac{\partial f}{\partial b^i} (\varrho ,u) - \mathcal {P}(\varrho ,u,v) \frac{\partial g}{\partial b^i} (\varrho ,v) +\mu _\beta \frac{\partial \mathcal {Y}_\beta }{\partial b^i} (\varrho ) + \lambda ^i_\alpha \frac{\partial \mathcal {H}^i_\alpha }{\partial b^i} (\varrho ) + \frac{\partial \lambda ^i_\alpha }{\partial t^\alpha }=0,~i=\overline{1,n}, \\&\quad \frac{\partial f}{\partial z^j} (\varrho ,u) - \mathcal {P}(\varrho ,u,v) \frac{\partial g}{\partial z^j} (\varrho ,v) + \mu _\beta \frac{\partial \mathcal {Y}_\beta }{\partial z^j} (\varrho ) + \lambda ^i_\alpha \frac{\partial \mathcal {H}^i_\alpha }{\partial z^j} (\varrho )=0,~j=\overline{1,k}, \\&\quad \mu _\beta \mathcal {Y}_\beta (\varrho ) \ge 0,~\mu \in R^q_+, \\&\quad \frac{\partial b^i}{\partial t^\alpha }=\mathcal {H}^i_\alpha (\varrho ),~\lambda \in R^{nm},\\&\quad b(t_0)=a_0,~b(t_1)=a_1, \end{aligned}$$

where \(u \in U,~v \in V\).

The robust counterpart for (DFP) is given by

$$\begin{aligned} \text {(RDFP)}~~ \max _{(b(\cdot ),z(\cdot ))} \mathcal {P}(\varrho ,\overline{u},\overline{v}) =\frac{\int _\mathcal {T} \max _{u \in U} f(\varrho ,u) \textrm{d}\omega }{\int _\mathcal {T} \min _{v \in V}g(\varrho ,v)\textrm{d}\omega } \end{aligned}$$

subject to

$$\begin{aligned}&\displaystyle \frac{\partial f}{\partial b^i} (\varrho ,\overline{u}) - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \frac{\partial g}{\partial b^i} (\varrho ,\overline{v}) +\mu _\beta \frac{\partial \mathcal {Y}_\beta }{\partial b^i} (\varrho ) + \lambda ^i_\alpha \frac{\partial \mathcal {H}^i_\alpha }{\partial b^i} (\varrho ) + \frac{\partial \lambda ^i_\alpha }{\partial t^\alpha }=0,~i=\overline{1,n},\nonumber \\ \end{aligned}$$
(17)
$$\begin{aligned}&\displaystyle \frac{\partial f}{\partial z^j} (\varrho ,\overline{u}) - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \frac{\partial g}{\partial z^j} (\varrho ,\overline{v}) + \mu _\beta \frac{\partial \mathcal {Y}_\beta }{\partial z^j} (\varrho ) + \lambda ^i_\alpha \frac{\partial \mathcal {H}^i_\alpha }{\partial z^j} (\varrho )=0,~j=\overline{1,k},\qquad \end{aligned}$$
(18)
$$\begin{aligned}&\displaystyle \mu _\beta \mathcal {Y}_\beta (\varrho ) \ge 0,~\mu \in R^q_+, \end{aligned}$$
(19)
$$\begin{aligned}&\displaystyle \frac{\partial b^i}{\partial t^\alpha }=\mathcal {H}^i_\alpha (\varrho ),~\lambda \in R^{nm}, \end{aligned}$$
(20)
$$\begin{aligned}&\displaystyle b(t_0)=a_0,~b(t_1)=a_1, \end{aligned}$$
(21)

where \(\max _{u \in U} f (\varrho , u)=f (\varrho , \overline{u}),~ \min _{v \in V} g (\varrho , v)=g (\varrho , \overline{v})\). The set of all feasible solutions to (RDFP) (which is also the robust feasible solution set to the problem (DFP)) is denoted by

$$\begin{aligned} \mathcal {D}_p = \{(b,z,\mu ,\lambda ,u,v) \in X \times C \times R^q_+ \times R^{nm} \times U\times V: \text {satisfying conditions} (17)-(21)\}. \end{aligned}$$

Next, we prove the weak robust duality for (FP), which asserts that the value of the objective functional of the dual problem over its feasible set is not greater than the value of the objective functional of the primal problem.

Theorem 4.1

(Weak robust duality) Let \((a,c) \in \mathcal {D}\) and \((b,z,\mu ,\lambda ,\overline{u},\overline{v}) \in \mathcal {D}_p\), also \(\max _{u \in U} f(\cdot ,u) = f(\cdot ,\overline{u})\) and \(\min _{v \in V} g(\cdot ,v) = g(\cdot ,\overline{v})\). If the functionals \(\int _\mathcal {T}f(\pi , \overline{u})\textrm{d}\omega -\mathcal {P}(\varrho ,\overline{u},\overline{v})\int _\mathcal {T}g(\pi , \overline{v})\textrm{d}\omega \), \(\int _\mathcal {T} \langle \mu _\beta , Y_\beta (\pi ) \rangle \textrm{d}\omega \), \(\beta = \overline{1,q},~\int _\mathcal {T} \langle \lambda ^i_\alpha , H^i_\alpha (\pi ) -\frac{\partial a^i}{\partial t^\alpha } \rangle \textrm{d}\omega ,~\alpha =\overline{1,m},~i=\overline{1,n}\) are convex at (b, z). Then, the following inequality

$$\begin{aligned} \frac{\int _{\mathcal {T}}\max _{u \in U} f(\pi ,u)\textrm{d}\omega }{\int _{\mathcal {T}}\min _{v \in V} g(\pi ,v)\textrm{d}\omega }\ge \mathcal {P}(\varrho ,\overline{u},\overline{v}), \end{aligned}$$

holds.

Proof

We proceed by the contradiction and assume that

$$\begin{aligned} \frac{\int _{\mathcal {T}}\max _{u \in U} f(\pi ,u)\textrm{d}\omega }{\int _{\mathcal {T}}\min _{v \in V} g(\pi ,v)\textrm{d}\omega } < \mathcal {P}(\varrho ,\overline{u},\overline{v}), \end{aligned}$$

Since \(\max _{u \in U} f(\cdot ,u) = f(\cdot ,\overline{u})\) and \(\min _{v \in V} g(\cdot ,v) = g(\cdot ,\overline{v})\), we have

$$\begin{aligned}&\frac{\int _{\mathcal {T}} f(\pi ,\overline{u})\textrm{d}\omega }{\int _{\mathcal {T}} g(\pi ,\overline{v})\textrm{d}\omega }< \mathcal {P}(\varrho ,\overline{u},\overline{v}). \nonumber \\&\quad \text {Or,}\quad \quad \quad \int _{\mathcal {T}} f(\pi ,\overline{u}) \textrm{d}\omega - \mathcal {P}(\varrho ,\overline{u},\overline{v})\int _{\mathcal {T}}g(\pi ,\overline{v})\textrm{d}\omega < 0, \end{aligned}$$
(22)

By assumption \(\int _\mathcal {T} f(\pi , \overline{u})\textrm{d}\omega -\mathcal {P}(\varrho ,\overline{u},\overline{v})\int _\mathcal {T} g(\pi , \overline{v})\textrm{d}\omega \) is convex at (b, z), we get

$$\begin{aligned}&\int _\mathcal {T} f(\pi , \overline{u})\textrm{d}\omega -\mathcal {P}(\varrho ,\overline{u},\overline{v})\int _\mathcal {T}g(\pi , \overline{v})\textrm{d}\omega - \int _\mathcal {T} f(\varrho , \overline{u}) \textrm{d}\omega - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \int _\mathcal {T} g(\varrho , \overline{v}) \textrm{d}\omega \nonumber \\&\quad \qquad ~ \ge \int _{\mathcal {T}} (a-b) \frac{\partial f}{\partial b} (\varrho ,\overline{u}) \textrm{d}\omega - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \int _{\mathcal {T}}(a-b)\frac{\partial g}{\partial b}(\varrho ,\overline{v})\textrm{d}\omega \nonumber \\&\qquad \qquad ~~~ + \int _{\mathcal {T}}(c-z) \frac{\partial f}{\partial z} (\varrho ,\overline{u})\textrm{d}\omega - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \int _{\mathcal {T}}(c-z)\frac{\partial g}{\partial z}(\varrho ,\overline{v}) \textrm{d}\omega . \end{aligned}$$
(23)

Since \(\mathcal {P}(\varrho ,\overline{u},\overline{v}) = \frac{\int _\mathcal {T} f(\varrho ,\overline{u}) \textrm{d}\omega }{\int _\mathcal {T} g(\varrho ,\overline{v})\textrm{d}\omega }\), on combining the inequalities (22) and (23), we obtain

$$\begin{aligned}&\int _{\mathcal {T}} (a-b) \frac{\partial f}{\partial b} (\varrho ,\overline{u}) \textrm{d}\omega - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \int _{\mathcal {T}}(a-b)\frac{\partial g}{\partial b}(\varrho ,\overline{v})\textrm{d}\omega \nonumber \\&\quad + \int _{\mathcal {T}}(c-z) \frac{\partial f}{\partial z} (\varrho ,\overline{u})\textrm{d}\omega - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \int _{\mathcal {T}}(c-z)\frac{\partial g}{\partial z}(\varrho ,\overline{v}) \textrm{d}\omega <0. \end{aligned}$$
(24)

From the assumption that the functionals \(~\int _\mathcal {T} \langle \mu _\beta , Y_\beta (\pi ) \rangle \textrm{d}\omega ,\beta = \overline{1,q}\) and \(\int _\mathcal {T} \langle \lambda ^i_\alpha , H^i_\alpha (\pi ) -\frac{\partial a^i}{\partial t^\alpha } \rangle \textrm{d}\omega ,\) \(~\alpha =\overline{1,m},~i=\overline{1,n}\) are convex at (b, z), we have

$$\begin{aligned}&\int _\mathcal {T} \langle \mu _\beta , Y_\beta (\pi ) \rangle \textrm{d}\omega -\int _\mathcal {T} \langle \mu _\beta , Y_\beta (\varrho ) \rangle \textrm{d}\omega \ge \int _{\mathcal {T}} \Big [ (a-b)\Big \langle \mu _\beta , \frac{\partial Y_\beta }{\partial b} (\varrho ) \Big \rangle + (c-z) \Big \langle \mu _\beta ,\nonumber \\&\qquad \qquad \qquad \frac{\partial Y_\beta }{\partial z} (\varrho ) \Big \rangle \Big ] \textrm{d}\omega , \end{aligned}$$
(25)
$$\begin{aligned}&\quad \text {and} \nonumber \\&\quad \int _\mathcal {T} \langle \lambda ^i_\alpha , H^i_\alpha (\pi ) -\frac{\partial a^i}{\partial t^\alpha } \rangle \textrm{d}\omega -\int _\mathcal {T} \langle \lambda ^i_\alpha , H^i_\alpha (\varrho ) - \frac{\partial b^i}{\partial t^\alpha } \rangle \textrm{d}\omega \ge \int _{\mathcal {T}} \Big [ (a-b)\Big \{\Big \langle \lambda ^i_\alpha , \frac{\partial H^i_\alpha }{\partial b} (\varrho ) \Big \rangle \nonumber \\&\qquad \qquad \quad + \frac{\partial \lambda ^i_\alpha }{\partial t^\alpha }\Big \} + (c-z)\Big \{\Big \langle \lambda ^i_\alpha , \frac{\partial H^i_\alpha }{\partial z} (\varrho ) \Big \rangle \Big \}\textrm{d}\omega . \end{aligned}$$
(26)

Since \((a,c) \in \mathcal {D}\) and \((b,z,\mu ,\lambda ,\overline{u},\overline{v}) \in \mathcal {D}_p\), it follows that

$$\begin{aligned}&\int _\mathcal {T} \langle \mu _\beta , Y_\beta (\pi ) \rangle \textrm{d}\omega -\int _\mathcal {T} \langle \mu _\beta , Y_\beta (\varrho ) \rangle \textrm{d}\omega \le 0,\\&\quad \int _\mathcal {T} \langle \lambda ^i_\alpha , H^i_\alpha (\pi ) -\frac{\partial a^i}{\partial t^\alpha } \rangle \textrm{d}\omega -\int _\mathcal {T} \langle \lambda ^i_\alpha , H^i_\alpha (\varrho ) - \frac{\partial b^i}{\partial t^\alpha } \rangle \textrm{d}\omega =0 \end{aligned}$$

In the virtue of the above relations, the inequalities (25) and (26) yield

$$\begin{aligned} \int _{\mathcal {T}} \Big [ (a-b)\Big \langle \mu _\beta , \frac{\partial Y_\beta }{\partial b} (\varrho ) \Big \rangle + (c-z) \Big \langle \mu _\beta , \frac{\partial Y_\beta }{\partial z} (\varrho ) \Big \rangle \Big ] \textrm{d}\omega \le 0, \end{aligned}$$
(27)

and

$$\begin{aligned}&\int _{\mathcal {T}} \Big [ (a-b)\Big \{\Big \langle \lambda ^i_\alpha , \frac{\partial H^i_\alpha }{\partial b} (\varrho ) \Big \rangle + \frac{\partial \lambda ^i_\alpha }{\partial t^\alpha }\Big \} + (c-z)\Big \langle \lambda ^i_\alpha , \frac{\partial H^i_\alpha }{\partial z} (\varrho ) \Big \rangle \textrm{d}\omega \le 0. \end{aligned}$$
(28)

On adding the inequalities (27), (28) and (24), we obtain

$$\begin{aligned}&\int _{\mathcal {T}} (a-b) \frac{\partial f}{\partial b} (\varrho ,\overline{u})\textrm{d}\omega - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \int _{\mathcal {T}} (a-b) \frac{\partial g}{\partial b}(\varrho ,\overline{v}) \textrm{d}\omega + \int _{\mathcal {T}} (a-b)\Big \langle \mu _\beta , \frac{\partial Y_\beta }{\partial b} (\varrho ) \Big \rangle \textrm{d}\omega \nonumber \\&\quad + \int _{\mathcal {T}} (a-b) \Big \langle \lambda ^i_\alpha , \frac{\partial H^i_\alpha }{\partial b} (\varrho ) \Big \rangle \textrm{d}\omega + \int _{\mathcal {T}} (a-b) \frac{\partial \lambda ^i_\alpha }{\partial t^\alpha } \textrm{d}\omega + \int _{\mathcal {T}} (c-z) \frac{\partial f}{\partial z} (\varrho ,\overline{u})\textrm{d}\omega \nonumber \\&\qquad - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \int _{\mathcal {T}} (c-z) \frac{\partial g}{\partial z}(\varrho ,\overline{v}) \textrm{d}\omega + \int _{\mathcal {T}} (c-z) \Big \langle \mu _\beta , \frac{\partial Y_\beta }{\partial z} (\varrho ) \Big \rangle \textrm{d}\omega \nonumber \\&\qquad +\int _{\mathcal {T}} (c-z) \Big \langle \lambda ^i_\alpha , \frac{\partial H^i_\alpha }{\partial z} (\varrho ) \Big \rangle \textrm{d}\omega <0. \end{aligned}$$
(29)

Moreover, from the robust feasibility of (a, c) and \((b,z,\mu ,\lambda ,\overline{u},\overline{v})\) for (FP) and (DFP), respectively the Eqs. (17)–(18) are satisfied by \((b,z,\mu ,\lambda ,\overline{u},\overline{v})\), therefore on multiplying \((a-b)\) and \((c-z)\) to Eqs. (17) and (18) respectively, and then adding after integrating the resulting expressions, we obtain

$$\begin{aligned}&\int _{\mathcal {T}} (a-b) \frac{\partial f}{\partial b} (\varrho ,\overline{u})\textrm{d}\omega - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \int _{\mathcal {T}} (a-b) \frac{\partial g}{\partial b}(\varrho ,\overline{v}) \textrm{d}\omega \\&\quad + \int _{\mathcal {T}} (a-b)\Big \langle \mu _\beta , \frac{\partial Y_\beta }{\partial b} (\varrho ) \Big \rangle \textrm{d}\omega \nonumber \\&\quad + \int _{\mathcal {T}} (a-b) \Big \langle \lambda ^i_\alpha , \frac{\partial H^i_\alpha }{\partial b} (\varrho ) \Big \rangle \textrm{d}\omega + \int _{\mathcal {T}} (a-b) \frac{\partial \lambda ^i_\alpha }{\partial t^\alpha } \textrm{d}\omega + \int _{\mathcal {T}} (c-z) \frac{\partial f}{\partial z} (\varrho ,\overline{u})\textrm{d}\omega \nonumber \\&\qquad - \mathcal {P}(\varrho ,\overline{u},\overline{v}) \int _{\mathcal {T}} (c-z) \frac{\partial g}{\partial z}(\varrho ,\overline{v}) \textrm{d}\omega + \int _{\mathcal {T}} (c-z) \Big \langle \mu _\beta , \frac{\partial Y_\beta }{\partial z} (\varrho ) \Big \rangle \textrm{d}\omega \nonumber \\&\qquad +\int _{\mathcal {T}} (c-z) \Big \langle \lambda ^i_\alpha , \frac{\partial H^i_\alpha }{\partial z} (\varrho ) \Big \rangle \textrm{d}\omega = 0, \end{aligned}$$

which contradicts to inequality (29). Hence the proof. \(\square \)

Example 4.1

From the Example 3.1, we have \( (\overline{a},\overline{c})=(0,1)\) be a robust optimal solution to (FP1). Now, we construct the Parametric robust dual problem associated with (FP1) as follows:

$$\begin{aligned}&\text {(DFP1)}~~ \max _{(b(\cdot ),z(\cdot ))} \mathcal {P} (\varrho ,u,v) = \frac{\int _\mathcal {T} [b(\textit{e}^b-1)+2z^2+u] \textrm{d}t^1\textrm{d}t^2}{\int _\mathcal {T} [-\frac{b^2}{4} -z^2-2v] \textrm{d}t^1\textrm{d}t^2}, \nonumber \\&\quad \text {subject to} \nonumber \\&\quad b\textit{e}^b+(\textit{e}^b-1)+\frac{b}{2}\mathcal {P}(\varrho ,u,v)+8b\mu +\frac{\partial \lambda _1}{\partial t^1}+\frac{\partial \lambda _2}{\partial t^2}=0, \end{aligned}$$
(30)
$$\begin{aligned}&\quad 4z+2z\mathcal {P}(\varrho ,u,v)-(\lambda _1+\lambda _2)=0, \end{aligned}$$
(31)
$$\begin{aligned}&\quad \mu (4b^2-9) \ge 0, ~\mu \ge 0, \end{aligned}$$
(32)
$$\begin{aligned}&\quad \frac{\partial b}{\partial t^\alpha } =3-z,~~\alpha =\{1,2\}, \end{aligned}$$
(33)
$$\begin{aligned}&\quad b(0,0)=0,~b(1,1)=4. \end{aligned}$$
(34)

Clearly \(\overline{b}=2(t^1+t^2)\) at \(t^1=t^2=0\), \(\overline{z}=1\), \(\overline{\mu }=0\), \(\overline{\lambda }_1=\overline{\lambda }_2=\frac{13}{5}\), \(\overline{u}=1,~\overline{v}=-3,~\mathcal {P}(\overline{\varrho },\overline{u},\overline{v})=\frac{3}{5}\) and \((\overline{b},\overline{z},\overline{\mu },\overline{\lambda },\overline{u},\overline{v})\) be a robust feasible solution to (DFP1). Further, one can observe that all involving functionals are convex at \((\overline{b},\overline{z},\overline{\mu },\overline{\lambda },\overline{u},\overline{v})\). Furthermore, the following inequality

$$\begin{aligned}&\frac{\int _{\mathcal {T}}\max _{u \in U} f(\overline{\pi },u)\textrm{d}t^1\textrm{d}t^2}{\int _{\mathcal {T}}\min _{v \in V} g(\overline{\pi },v)\textrm{d}t^1\textrm{d}t^2} - \mathcal {P}(\overline{\varrho },\overline{u},\overline{v}) \ge 0, \end{aligned}$$

holds, which validates the result of weak robust duality theorem.

Theorem 4.2

(Strong robust duality) Let (a, c) be a robust optimal solution to (FP) and \(\max _{u \in U}f(\cdot ,u)=f(\cdot ,\overline{u}),~\min _{v \in V}g(\cdot ,v)=g(\cdot ,\overline{v})\). Then there exist the piecewise smooth Lagrange functions \(\mu _\beta ,~\beta =\overline{1,q},~\lambda ^i_\alpha ,~i=\overline{1,n},~\alpha =\overline{1,m}\) such that \((a,c,\mu ,\lambda ,\overline{u},\overline{v})\) is robust feasible solution to (DFP) and that the values of the objective functions of (FP) and (DFP) are equal at these points. Further, if the assumptions of weak robust duality theorem hold then \((a,c,\mu ,\lambda ,\overline{u},\overline{v})\) is robust optimal solution to the (DFP).

Proof

Let (a, c) be a robust optimal solution to the (FP) and \(\max _{u \in U}f(\cdot ,u)=f(\cdot ,\overline{u})\),\(~\min _{v \in V}g(\cdot ,v)=g(\cdot ,\overline{v})\). Then there exist the piecewise smooth Lagrange functions \(\mu _\beta ,~\beta =\overline{1,q},~\lambda ^i_\alpha ,~i=\overline{1,n},~\alpha =\overline{1,m}\) such that conditions (1)-(2) hold, therefore \((a,c,\mu ,\lambda ,\overline{u},\overline{v})\) is robust feasible solution to (DFP). From the assumptions of weak duality theorem the following inequality

$$\begin{aligned} \frac{\int _{\mathcal {T}}\max _{u \in U} f(\pi ,u)\textrm{d}\omega }{\int _{\mathcal {T}}\min _{v \in V} g(\pi ,v)\textrm{d}\omega }\ge \mathcal {P}(\varrho ,\overline{u},\overline{v}), \end{aligned}$$

holds for any \((b,z,\mu ,\lambda ,\overline{u},\overline{v}) \in \mathcal {D}_p\). Since, \(\max _{u \in U}f(\cdot ,u)=f(\cdot ,\overline{u}), \min _{v \in V}g(\cdot ,v)=g(\cdot ,\overline{v})\), then we can write

$$\begin{aligned}&~~~~\frac{\int _{\mathcal {T}} f(\pi ,\overline{u})\textrm{d}\omega }{\int _{\mathcal {T}} g(\pi ,\overline{v})\textrm{d}\omega }\ge \mathcal {P}(\varrho ,\overline{u},\overline{v}),\\&\Rightarrow \frac{\int _{\mathcal {T}} f(\pi ,\overline{u})\textrm{d}\omega }{\int _{\mathcal {T}} g(\pi ,\overline{v})\textrm{d}\omega }\ge \frac{\int _{\mathcal {T}} f(\varrho ,\overline{u})\textrm{d}\omega }{\int _{\mathcal {T}} g(\varrho ,\overline{v})\textrm{d}\omega }. \end{aligned}$$

Hence, the above inequality concludes that \((a,c,\mu ,\lambda ,\overline{u},\overline{v})\) is robust optimal solution to (DFP). Also the corresponding values of objective functionals are equal at this point. \(\square \)

5 Conclusions

This research investigated a multi-dimensional fractional control optimization problem with data uncertainty in the objective functional (FP). We developed robust necessary and sufficient optimality conditions using the parametric approach. We also constructed a parametric robust dual problem for (FP) to prove robust duality theorems under convexity assumptions. The paper’s outcomes are additionally validated with appropriate numerical examples. To the best of our knowledge, in the field of multi-dimensional fractional control optimization problems together with uncertainty, the robust duality results provided in this research are new.

Moreover, studying the multi-dimensional fractional control optimization problem with data uncertainty in constraints and proving similar conclusions imposing the generalized convexity notion would be fascinating.