Fuzzy Optimal Control Problem of Several Variables

Mustafa, Altyeb Mohammed; Gong, Zengtai; Osman, Mawia

doi:https://doi.org/10.1155/2019/2182640

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 2182640 | https://doi.org/10.1155/2019/2182640

Fuzzy Optimal Control Problem of Several Variables

Altyeb Mohammed Mustafa,^1,2Zengtai Gong ,¹and Mawia Osman¹

Academic Editor: Remi Léandre

Received24 Sept 2019

Accepted26 Nov 2019

Published29 Dec 2019

Abstract

The purpose of this paper is to establish the necessary conditions for a fuzzy optimal control problem of several variables. Also, we define fuzzy optimal control problems involving isoperimetric constraints and higher order differential equations. Then, we convert these problems to fuzzy optimal control problems of several variables in order to solve these problems using the same solution method. The main results of this paper are illustrated throughout three examples, more specifically, a discussion on the strong solutions (fuzzy solutions) of our problems.

1. Introduction

Optimal control theory is considered as a modern extension of the classical calculus of variations; however, it differs from calculus of variations in that it uses control variables to optimize the function. The development of the mathematical theory for optimal control began in the early 1950’s, partially in response to problems in various branches of engineering and economics. The study of classical optimal control theory from different viewpoints greatly attracted the attention of many mathematicians, and the detailed arguments can be found in many textbooks, for instance, [1], and references therein. Moreover, optimal control strategy, i.e., solving necessary conditions for optimality, can be applied in several fields, such as economy, biology, and process engineering (for more details, see [1–5]).

On the other hand, uncertainty is inherent in most dynamical systems in its input, output, and manner, and fuzziness is a kind of uncertainty very common in real-world problems [6]. In 1965, Zadeh introduced the concepts of fuzzy sets and fuzzy numbers in [7], followed up in 1972 by Chang and Zadeh when they proposed the concept of the fuzzy derivative in [8]. A large number of researches have been studied in various aspects of the theory and applications of these notions; one of these research lines has been the fuzzy optimal control problem. In the past few decades, the fuzzy optimal control problem has received growing attention, and many results of researches have been reported in the literature ([9–19] and references therein).

Recently, a lot of works done in the field of the fuzzy optimal control problem have only examined problems with one control and one dependent state variable; however, many times, we will wish to examine fuzzy optimal control problems which arise in a wide variety of scientific and engineering applications such as physics, chemical engineering, and economy, with more variables (more controls and more states). It seems that it is a good idea to consider fuzzy optimal control problems of several variables and discuss how to handle such problems. Further, treating a special type of fuzzy optimal control problems such as problems having a type of constraint known as an isoperimetric constraint and problems involving higher order differential equations has been presented. In [11], the modified fuzzy Euler-Lagrange condition was established for the fuzzy Isoperimetric Variational Problem (IVP), which is considered as a fuzzy constrained variational problem, but, in this paper, we overcome the fuzzy optimal control problem involving the isoperimetric constraint, which is considered as a fuzzy constrained optimal control problem.

The main aim of this paper is to derive the necessary conditions of the fuzzy optimal control problem of several variables based on the concepts of differentiability and integrability of a fuzzy valued function parameterized by the left- and right-hand functions of its α-level set and variational approaches, in order to provide the solutions of this problem. However, the solutions of the fuzzy optimal control problem of several variables, optimal controls, and corresponding optimal states are not always fuzzy functions. Thus, to guarantee that the solutions of the fuzzy optimal control problem of several variables are always fuzzy functions, we will introduce the concepts of strong (fuzzy) and weak solutions of this problem.

The rest of this paper is organized as follows: In Section 2, we recall some basic terminologies and definitions used in the present paper. In Section 3, we establish our main results concerning the necessary conditions of the fuzzy optimal control problem of several variables and treating two special cases of the fuzzy optimal control problem. Additionally, we propose the definitions of strong (fuzzy) and weak solutions of our problem. In Section 4, we give three examples that can serve to illustrate our main results. In Section 5, we present some concluding remarks.

2. Preliminaries

Throughout this paper, denotes the class of fuzzy subsets of the real axis. A fuzzy set on is a mapping . For each fuzzy set , we denote its α-level set by and defined by for any . The support of we denote by supp, where . The closure of supp defines the 0-level set of ; thus, where denotes the closure of set . Fuzzy set is called a fuzzy number if (1) is a normal fuzzy set, i.e., there exists an such that (2) is a convex fuzzy set, i.e., for any and (3) is upper semicontinuous on (4) is compact

In the rest of this paper, we use to denote the fuzzy number space.

It is clear that the level set is bounded closed interval in for all , where and denote the left-hand and right-hand endpoints of , respectively. Obviously, any can be regarded as a fuzzy number defined by

In particular, fuzzy zero is defined as if and otherwise.

Let and . For any , we can define the addition and scalar multiplication , respectively, as

Using level set, we can also define the addition and scalar multiplication , respectively, as

Let be a fuzzy number, the opposite of is denoted by and characterized by [20]. In the case that , we have for all .

The binary operation “.” in can be extended to the binary operation “” of two fuzzy numbers by using the extension principle. Let and be fuzzy numbers, then

Using -level set, the product is defined by in the case that and .

Lemma 1 (see [21]). If and satisfy the following conditions:(1) is a bounded increasing function(2) is a bounded decreasing function(3)(4) and , for all (5) and then defined by is a fuzzy number with . Conversely, if is a fuzzy number with , then the functions and satisfy conditions (1)-(5).

Define by where and . is called the distance between fuzzy numbers and .

It should be noted that satisfies the following properties: (1) is a complete metric space(2)(3), where and

A special class of fuzzy numbers is the class of triangular fuzzy numbers. We say that the fuzzy number is triangular if , , and . The triangular fuzzy number is generally denoted by .

Definition 2 (partial ordering [9]). Let , we write , if and for all . We also write , if and there exists such that or . Furthermore, , if and . In other words, , if for all .
In the sequel, we say that are comparable if either or and noncomparable otherwise.

Definition 3 (gH-difference [22]). Suppose that , where and for all , the generalized Hukuhara difference of two fuzzy numbers and (gH-difference for short) is defined by If exists as a fuzzy number, then its -level set is for all .

Definition 4 (fuzzy valued function [9]). The function is called a fuzzy valued function if is assigned a fuzzy number for any . We also denote , where and . Therefore, any fuzzy valued function may be understood by and being, respectively, a bounded increasing function of and a bounded decreasing function of for . Also, it holds for any .

Definition 5 (continuity of a fuzzy valued function [23]). We say that is continuous at , if both and are continuous functions at for all .

Definition 6 (gH-differentiability of a fuzzy valued function [24]). Let and be such that , then the gH-derivative of a fuzzy valued function at is defined as If , we say that is generalized Hukuhara differentiable (gH-differentiable for short) at . Also, we say that is -gH-differentiable at if and is -gH-differentiable at if

Definition 7 (th order gH-differentiability of a fuzzy valued function [25]). Let . We say that is th order gH-differentiable at whenever the function is gH-differentiable of the order , , at and if there exist such that

Definition 8 (switching point [26]). We say that a point is a switching point for the differentiability of if in any neighborhood of there exist points such that (i)type(I): at (11) holds while (12) does not hold and at (12) holds while (12) does not hold or(ii)type(II): at (12) holds while (12) does not hold and at (11) holds while (12) does not hold

Definition 9 (see [21]). Let . We say that is fuzzy-Riemann integrable to if for any , there exists such that for any division of with the norms , we have where denotes the fuzzy summation. We choose to write . Furthermore, for any ,

Theorem 10 (see [24]). If is gH-differentiable with no switching point in the interval , then we have

Theorem 11 (see [24]). Let be a continuous fuzzy valued function. Then, is gH-differentiable and .

From now, we use to denote the space of all fuzzy valued functions that have continuous gH-derivatives on and to denote the space of all fuzzy valued functions that have th continuous gH-derivatives on .

3. Fuzzy Optimal Control of Several Variables

This section is aimed at deriving the necessary conditions for the fuzzy optimal control problem of several variables. For this purpose, the fuzzy optimal control problem of several variables is introduced at first, then using fuzzy variational approaches, the problem is solved.

Consider the following fuzzy optimal control problem of several variables: where are assumed to be functions of class with respect to all their arguments. The fuzzy states for and the fuzzy controls for are functions of . Furthermore, are assumed to be -gH-differentiable functions for all . In problem (18), we make no requirements on and . In other words, , , or are all acceptable.

We say that an admissible fuzzy curve is the solution of problem (18), if for all admissible curve of problem (18)

It is well know that, from Definition 2, the above inequality holds if and only if for all , where the level set of fuzzy curves , , , and are characterized, respectively, by for and .

Definition 12 (fuzzy Hamiltonian function). We define fuzzy Hamiltonian function as

Theorem 13 (necessary conditions for problem (18)). Assume that is a vector of admissible fuzzy states and is a vector of admissible fuzzy controls. Then, the necessary conditions for and to be optimal solutions for (18) arefor all , , , and .

Proof. Let us first consider the variation of , , , and defined, respectively, by for and . Using Theorems 10 and 11 and because the problem is a minimization problem, we can rewrite as The increment of , denoted by is where Using the gH-difference, and without sake of generality, we consider Since if and only if for all , then are optimal solutions for the crisp functions and . Suppose that and denote the first variation of and , respectively. From the classical theory of optimal control, we know that if are optimal, then it is necessary that and are zero. In order to find the first variation of and , we need to evaluate the derivatives in the integrand of and along the optimal trajectory; then, we obtain Subsequently, on optimal trajectories, the first variation of and is zero, i.e., for all variations. Now, we are ready to introduce the fuzzy Lagrange multiplier functions by considering the integral for . Using the gH-difference and without sake of generality, we consider the level set of , respectively, as for . In the remainder of the proof, we will ignore similar arguments. We start by computing the variation of (38): It is clear that from the definition of gH-difference, the first term of Equation (40) is zero. Integrating the last term on the RHS of (41) by parts and because is specified, i.e., (and ) for all , then, we arrive at By considering the following summations, we can rewrite as Similarly, when we consider (39) with for all , and the same summation with small change (the right-hand functions of instead of the left-hand functions), we arrive at Since and for all , then and . Further, we can replace the conditions and by and , respectively [10]. Then, we have Summing Equations (46) and (47) and using the definition of the fuzzy Hamiltonian function, we have Integrating the first term of (48) considering and for all and removing the terms involving , and , therefore, the necessary conditions follow.

Note 1. It should be noted that in problem (18), if is specified, then has no boundary conditions for all .

3.1. Isoperimetric Constraints

In this part, we turn our attention to a special type of fuzzy optimal control problem, defined as

This type of constraint is known as an isoperimetric constraint, where , and are assumed to be functions of class with respect to all their arguments. To establish the solution method for this type of problem, we convert this problem to a more familiar form, the fuzzy optimal control problem of several variables, by introducing a second state variable , and let

Therefore, if we use Theorems 10 and 11, then we have

Subsequently, problem (49) is transformed into

This problem can now be solved using Theorem 13, i.e., the developed method for solving fuzzy optimal control problems of several variables.

3.2. Higher Order Differential Equations

Here, we deal with problems involving higher order differential equations. Consider the following problem: for . Theorem 13 does not directly deal with this type of problem. But it is easy to convert this problem to the fuzzy optimal control problem of several variables by introducing state variables defined, respectively, by

Then, problem (53) is transformed into which can be solved by using Theorem 13, i.e., the developed method for solving fuzzy optimal control problems of several variables.

3.3. The Strong (Fuzzy) and Weak Solutions

The developed method for solving the fuzzy optimal control problem of several variables provides the solutions of this problem, optimal fuzzy controls and corresponding fuzzy states, by solving the necessary conditions introduced in Theorem 13. Meanwhile, to guarantee that the solutions of the fuzzy optimal control problem of several variables are always fuzzy functions, we propose the concepts of strong (fuzzy) and weak solutions of this problem. In the following definition, based on the conditions (1) and (2) of Lemma 1, we introduce the definition of strong (fuzzy) and weak solutions of the fuzzy optimal control problem of several variables (18).

Definition 14 (strong (fuzzy) and weak solutions). (1)Strong (Fuzzy) Solution. We say that and are strong(fuzzy) solutions of problem (18) if , and , for and , obtained from (23), (24), (25), (26), (27), and (28) satisfy their related properties defined in the conditions (1) and (2) of Lemma 1, for all and .(2)Weak Solution. We say that and are weak solutions of problem (18) if , , and , for and , obtained from (23), (24), (25), (26), (27), and (28) do not satisfy their related properties in the conditions (1) and (2) of Lemma 1; thus, we define and asfor all , , , and .

Note 2. In the next section, we will give three examples that can serve to illustrate our main results, more specifically, a discussion on the strong solutions (fuzzy solutions) of our problems.

4. Illustrative Examples

Example 1. Find the fuzzy control that

Solution 1. First, without loss of generality, we formulate the fuzzy Hamiltonian function as In fact, , then The necessary conditions for optimality of Theorem 13 with the initial conditions and the dynamical system of (57) give We solve Equations (60), (61), (62), (63), (64), (65), (66), (67), (68), and (69) analytically; then, we obtain We can easily show that , and are continuous increasing functions of , and , and are continuous decreasing functions of . Furthermore, for all . Therefore, the -level set of optimal fuzzy control and optimal fuzzy states and is characterized, respectively, by for all . Therefore, the above solutions are strong (fuzzy) solutions of problem (57).

Example 2. Find the fuzzy control that

Solution 2. First, we introduce a second state variable by then, problem (73) converts to

Second, without loss of generality, we formulate the fuzzy Hamiltonian function as

It is clear that

Now, applying Theorem 13, we find the following necessary conditions:

In the rest of the solution, we will ignore the similar cases; we only consider the left-hand functions of level set.

We begin by substituting (84) into differential Equation (82), and then, by solving the differential equation, we obtain

Substituting into (86) gives

After that, we substitute into Equation (78), then solve it with the appropriate condition, we obtain

By the same manner, we substitute into Equation (80) and solve it with the appropriate condition; then, we have

To find the value of constant of integration , we use the condition ; thus,

Therefore, for all , we arrive at

Similarly, if we consider the right-hand functions of level set and for all , we arrive at

We can easily show that , and are continuous increasing functions of and , and are continuous decreasing functions of . Moreover, for all . Therefore, the -level set of optimal fuzzy control and optimal fuzzy states and is characterized, respectively, by for all . Therefore, the above solutions are strong (fuzzy) solutions of problem (73).

Example 3. Find the fuzzy control that

Solution 3. We let and , then problem (97) converted to problem (57) in Example 1, i.e., problem (97) has the same solutions of problem (57).

5. Conclusion

In summary, we proved the necessary conditions for optimality of the fuzzy optimal control problem of several variables. Also, fuzzy optimal control problems involving isoperimetric constraint and higher order differential equation have been considered. By introducing new variables, we transformed these problems into fuzzy optimal control problems of several variables in order to use the developed method to solve these problems. The definitions of strong (fuzzy) and weak solutions of our problems have been introduced. By three examples, we discussed and summarized the applicability of our main results of this paper.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (61763044).

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Copyright © 2019 Altyeb Mohammed Mustafa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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