1. Introduction
This paper deals with the interval-valued optimal control problem, which plays an important role in studying uncertainty in optimization problems. The errors due to data uncertainty or imprecision have created the need to investigate certain real-world problems. Various scientists have contributed to this research direction. Among the techniques used, interval-valued optimization is an emerging branch dealing with the uncertainty of optimization problems. In this regard, variational inequalities, first introduced by Hartman and Stampacchia [
1], have been observed to be useful mathematical objects for studying optimization problems. Giannessi [
2] stated remarkable results on variational inequalities and complementarity problems. Moore [
3,
4] suggested interval analysis to study optimization problems determined by interval-valued functions. Stefanini and Bede [
5] continued by defining the generalized Hukuhara differentiability associated with interval-valued functions. In addition, some sufficiency and duality results for interval-valued programming problems have been established by Jayswal et al. [
6]. Additionally, Liu [
7] studied variational inequalities and optimization problems, and Treanţă [
8] contributed to the study of vector variational inequalities and multiobjective optimization problems. Jayswal et al. [
9] formulated and proved some results for multiple objective optimization problems and vector variational inequalities. Connections between the solutions of some interval-valued multiple objective optimization problems and vector variational inequalities have been derived by Zhang et al. [
10]. Jha et al. [
11], via the associated modified problems and saddle point criteria, presented several results for interval-valued variational problems. Treanţă [
12,
13] provided important connections between the notions of optimal solution, KT-pseudoinvex point, and a saddle-point of an interval-valued functional of the Lagrange type. Recently, Treanţă [
14,
15,
16] formulated optimality conditions for some multi-dimensional interval-valued variational problems. Additionally, Guo et al. [
17] established optimality conditions and duality results for a class of generalized convex interval-valued optimization problems. In [
18], Guo et al. provided a complete study on the properties of symmetric gH-derivative. More precisely, a necessary and sufficient condition for the symmetric gH-differentiability of interval-valued functions has been presented. Further, the authors clarified the relationship between the symmetric gH-differentiability and gH-differentiability. For more information and connected results on this topic, we direct the reader to the following research papers: Antczak [
19], Hanson [
1], Lodwick [
20], Myskova [
21], Wu [
22], Zhang et al. [
23], Zhang et al. [
24], Jayswal and Baranwal [
25], and references therein.
In this paper, we continue and improve the research mentioned above. Concretely, we establish some equivalence relations between LU-optimal solutions of the considered interval-valued optimal control problem and solutions of the associated weak variational control inequality. The present paper has several merits, as follows: (i) defining, by using the -order relation, the notion of -optimal solution for functionals determined by path-independent curvilinear integrals, (ii) formulating original and innovative proofs associated with the main results, and (iii) providing a mathematical context determined by infinite-dimensional function spaces and curvilinear integral-type functionals. These elements are new in the area of interval-valued optimal control problems. The limitations of the study: (1) the concept of “LU-convexity” is strongly used in our arguments; consequently, in our next research works, we will try to improve this aspect and replace it with a general one; (2) also, the well-posedness study of the considered problem is still an open problem that should be investigated.
The paper continues as follows:
Section 2 presents notations, preliminary ingredients, and definitions on interval-valued functional of the curvilinear integral type, (strictly) convex real-valued curvilinear integral type functional, and (strictly)
LU-convex interval-valued curvilinear integral type functional; in
Section 3, we state some existence results of
LU-optimal solutions for the considered interval-valued control problem, and of solutions for the corresponding weak variational control inequalities;
Section 4 formulates the conclusions of the paper.
2. Preliminaries
In this paper, we consider and denote the standard Euclidean spaces, is a domain in and is a piecewise differentiable curve joining the following two multiple variables of evolution , included in , and , is the current point in . Denote by , the operator associated with the total derivative, and let M represent the space of all piecewise smooth state functions , with as the first-order partial derivative of with respect to . Additionally, let N be the space of all continuous control functions , and K denotes the set of all closed and bounded intervals in . For , the real numbers indicate the lower bounds, and indicate the upper bounds of A and B, respectively. The interval operations are performed as follows:
is equivalent with and
if , then
.
In addition, the following conventions for any two intervals will be used:
Next, on the line of Treanţă [
12,
13,
15], we introduce the interval-valued functionals determined by curvilinear integrals, (strictly) convexity associated with real-valued curvilinear integral type functionals, and (strictly)
LU-convexity for interval-valued curvilinear integral type functionals. Additionally, we define the path independence of the involved curvilinear integrals.
Definition 1. A curvilinear integral type functionalis named an interval-valued functional if it is formulated aswhereare real-valued curvilinear integral type functionals, with , satisfying the condition Definition 2. An interval-valued curvilinear integral type functional Ψ is called path-independent if the real-valued curvilinear integral type functionals and are path-independent, that is, the following equalities and are satisfied, for .
Definition 3. A real-valued curvilinear integral type functionalis named (strictly) convex atiffor all . If the above inequality is valid for each , then the real-valued curvilinear integral type functional is named (strictly) convex on .
Definition 4. An interval-valued curvilinear integral type functionalis called -convex at if both the real-valued curvilinear integral type functionals and are convex at . Definition 5. If the real-valued curvilinear integral type functionals and are convex at and or/and is strictly convex at , then the interval-valued curvilinear integral type functionalis named strictly -convex at . Taking into account the above-mentioned mathematical framework, we state the following interval-valued optimization problem:where and are -class functions. Let us denote the feasible solution set to asand consider that Ω is a convex subset of . For simplicity, we use the following notations: , and .
Definition 6. A point is named a (strong) -optimal solution to iffor all . Now, on the line of Treanţă [8], in order to provide some characterizations of the solution set associated with the interval-valued optimal control problem , we formulate the following variational control inequalities: * find in such a way there exists no , fulfilling the following variational control inequality: * find in such a way that there exists no , fulfilling the following weak variational control inequality: * find in such a way that, for all , the following split variational control inequalitiesare satisfied. 3. Main Results
This section, via solutions of the weak variational control inequality , formulates and proves an existence result for the -optimal solutions of the interval-valued optimal control problem .
For a better understanding of the mathematical context associated with the control problems under study, we state some recent auxiliary results provided by Tareq [
26] (see Theorems 1–4).
The next result, by considering the solution associated with the variational control inequality , provides a sufficient condition for a pair to become an -optimal solution to .
Theorem 1. Consider is a solution to and is -convex at . Then is an -optimal solution to .
The following theorem represents the reciprocal of the previous result.
Theorem 2. Consider is an -optimal solution to and is strictly -convex at . Then, is a solution to .
In the following, by using the solution associated with the split variational control inequality , the next theorem provides a sufficient condition for a pair to become a strong -optimal solution to .
Theorem 3. Consider fulfills and is -convex at . Then, is a strong -optimal solution to .
The following result provides the conditions such that the reciprocal of the previous result is satisfied.
Theorem 4. Consider is a strong -optimal solution to and is -convex at . Then, fulfills .
In the following, we state and prove the main results of the present paper. The next result, by considering the solution associated with the variational control inequality , provides a sufficient condition for a pair to become an -optimal solution to .
Theorem 5. Consider is an -optimal solution to and is -convex at . Then, fulfills the weak variational control inequality .
Proof. By hypothesis,
is an
-optimal solution to
. Thus, there exists no
satisfying
equivalent with
Thus, there exists no
fulfilling
Now, we proceed by contradiction and assume that
is not a solution to the weak variational control inequality
. In consequence, there exists
so that
Since the functional
is
-convex at
, we obtain
and
for all
. From the above inequalities, we obtain
for all
, which, together with the inequality
, yields the following inequality:
for a point
, which contradicts the inequality
.
□
The following theorem represents the converse result of the previous one.
Theorem 6. Consider fulfills the weak variational control inequality and the interval-valued functional is strictly -convex at . Then, is an -optimal solution to .
Proof. By hypothesis,
is a solution of
. Therefore, there exists no
, fulfilling
Now, we proceed by contradiction and assume that
is not an
-optimal solution to
. Thus, there exists
, fulfilling
equivalent with
or
or
From strict
-convexity property of
, it results
and
or
and
or
and
for all
. On combining the above three inequalities with the inequalities (4)–(6), respectively, we obtain
From the above inequalities, it follows that
holds, for
, which contradicts the inequality
. □
Application. Let us extremize the interval-valued curvilinear integral cost functional given by
subject to
The feasible solution set of
is given by
and, taking into account the notations used in this paper, we have
. Let us assume that we are only interested in the affine state and control real-valued functions and
. It can be easily seen that the real-valued functionals
and
are (strictly) convex at
. Thus, the interval-valued functional
is strictly
-convex at
. By imposing the condition
it results
involving
Since the following inequality
holds at
, for all
, then it is a solution to variational control inequality
. By using Theorem 6, it follows that
is an
-optimal solution of
. Indeed, it can be verified that the inequality
is satisfied.