**4. Positives and negatives of formal influence of cyberspace**

The usage of information technologies and especially communication in cyberspace has its positives but also negatives. Modern communication technologies, similarly to other tools, can act as a good servant but a bad master. It is very difficult today to find the borderline between these two polarities because the bound between us and them is so strong that we are more or less unable to reliably distinguish and realize how much they influence us. Middle-aged and older generation, having lived without the influence of modern media, is more likely to debate this than younger, or the so-called digital generation, as they were growing up surrounded by new technologies that became an inseparable part of their life. Therefore it is extremely important to learn to see the perspective, build up a mental a psychological protective barrier when approaching media, and distinguish positives from negatives in communication in cyberspace.

We can now speak more on positives and negatives of the four formal influenced areas (time, dimension, structure of thinking, and identification) in cyberspace:

1.Time. When we communicate in cyberspace, we experience enormous speeding up of information transfer, which nowadays reaches almost the speed of light. Then there is a huge increase of amount of information, which still grows exponentially. This means we can access almost any information quickly, but selecting and processing are more demanding and time-consuming, which lead to sketchiness. The lack of time further causes another effect—deprivation of time that should be dedicated to holiday, family, bringing up children, and so on. Paradoxically, one may be killing the time by surfing the Internet, chatting to friends, or sending emails simply to maintain the feeling of being engaged or belonging to a group. Besides this, information in cyberspace is not stored and communicated linearly, but hypertextually, in a fanlike pattern, which leads to favoritism of simultaneous time over linear time. With linear time being broken comes also lack of interest in the past, history, culture, and traditions but also indifference regarding the future. This is typical of the modern digital

**21**

*Cyberspace as a New Existential Dimension of Man DOI: http://dx.doi.org/10.5772/intechopen.88156*

> youth. Bauerlein [16] carried out research at high schools and universities in the USA and found out that year after year students are less and less aware of history and civic education, and generally their knowledge in subjects that have something to do with history is less and less adequate. Volko [17] carried a knowledge research at one of the Slovak universities and acquired a similar result, which he commented: "Quality of general knowledge is, mildly speaking, inadequate. Students that will in the future work in media, struggle when asked to say for example when Slovak National Uprising started, they cannot

2. Space. A positive aspect of communication in cyberspace is in its ability to defeat geographical locations. We can now communicate with someone who lives in Australia or the USA not only orally but also visually. We even can watch events happening in various places on the globe. This may bring its negative aspect we can lose the sense of value of the real surrounding, our homeland, traditions, and culture in a given place. With communication on the Internet, importance of such a place declines, and people lose their roots. The Internet and also globalization tear the bound between geographical location and social role. With no geographical and social roots, one can easily become homeless in cyberspace.

Cyberspace of electronic media does not only consist of online telecommunication or online tele-seeing of the world; it is a world of new opportunities in virtualization of reality. Virtualization of reality may take various modes of reality or creation of brand new, fantastic worlds. Communication, or contact with virtual worlds, brings some pros and some contrast. In playing games young people can learn manual and visual skills and learn about the world but also become completely

3.Structure of thinking. In communication within cyberspace, some structural changes in thinking and consequently in learning occur. Each media has its own semiotics, and the most fundamental media, for example, spoken word, found their new cultural epochs. Therefore, media are tools for our thinking and learning. Bystřický [18], for instance, says that "we also use different ways of thinking with increased use of technologies, not in terms of changing the actual availability of such abilities; we rather fundamentally alter strategy of their use." Thinking in cyberspace is influenced by discontinuity of images, short texts, and similar, which does not help us to train concentration and continual refinement of ideas. On the contrary, a text in a newspaper or book requires us to concentrate and pursue the logical chain of ideas that are expressed. Book and newspaper thus develop abstract and logically continual thinking, while television and especially the Internet nourish visual and discontinuous thinking. According to G. Sartori, image-based media, such as television and the Internet, alter the way we think, imagine, and learn. He is convinced that a new type of human is rising—*homo videns*—whose perception and knowledge are greatly modified by media images. In his idea, the turn from conceptual language of texts to media images also brings deprivation of abstract thinking, and, as Solík [19] adds, also emotional changes. We do not need to think; seeing a picture is enough. Sartori [13] explains: "Television brings metamorphosis that affects the very core of *Homo sapiens*. It is not a mere tool for communication, but also an anthropological instrument that constructs a new kind of human existence." *Homo sapiens* then changes to *homo videns*, which introduces decline in erudition and cultural decadence. It is similar in the case of the Internet that, unlike television, is interactive. If used by

immersed and become addicted or virtualize the real world.

define holocaust or think of two Slovak classical music composers."

*Cyberspace*

group, identify oneself with the group, or change identity. Our identity can then be

We can call the changes that we studied in this part of subchapter (changes in time, space, structures of thinking, and identification with technologies) formal, because they are results of using mental or physical connection to information technologies. Of course we could mention other formal influences, and we may, for example, study changes in the attention, memory, social contacts, and more. In the background of this approach is the idea of technological determinism, such as M. McLuhan, L. A. White, J. Lohisse, and other authors. The starting point for this approach is the idea that new communication technologies have a profound impact on human cognitive changes and consequently changes in culture and society. Along with formal influences, there are also changes based on content influence. We see content as particular communicated information that may take various forms—perhaps as symbols (images, sounds, and so on) or meanings (scientific, social, entertaining, and similar). Formal influence of cyberspace, though harder to be recognized as it is not a direct product of communication, has a stronger influence on shaping a man than communicated content, because it structurally changes his ideas and thinking. S. Gáliková Tolnaiová [15] calls the formal type of influence stronger version and the second, content type, weaker version of media influence. It is chiefly the first—formal type—that contributes to the new anthropogenesis, influencing man mentally, psychically, and also physically to certain degree.

constructed in accord with our participation in various groups.

**4. Positives and negatives of formal influence of cyberspace**

The usage of information technologies and especially communication in cyberspace has its positives but also negatives. Modern communication technologies, similarly to other tools, can act as a good servant but a bad master. It is very difficult today to find the borderline between these two polarities because the bound between us and them is so strong that we are more or less unable to reliably distinguish and realize how much they influence us. Middle-aged and older generation, having lived without the influence of modern media, is more likely to debate this than younger, or the so-called digital generation, as they were growing up surrounded by new technologies that became an inseparable part of their life. Therefore it is extremely important to learn to see the perspective, build up a mental a psychological protective barrier when approaching media, and distinguish positives from negatives in communication in cyberspace.

We can now speak more on positives and negatives of the four formal influenced areas (time, dimension, structure of thinking, and identification) in cyberspace:

1.Time. When we communicate in cyberspace, we experience enormous speeding up of information transfer, which nowadays reaches almost the speed of light. Then there is a huge increase of amount of information, which still grows exponentially. This means we can access almost any information quickly, but selecting and processing are more demanding and time-consuming, which lead to sketchiness. The lack of time further causes another effect—deprivation of time that should be dedicated to holiday, family, bringing up children, and so on. Paradoxically, one may be killing the time by surfing the Internet, chatting to friends, or sending emails simply to maintain the feeling of being engaged or belonging to a group. Besides this, information in cyberspace is not stored and communicated linearly, but hypertextually, in a fanlike pattern, which leads to favoritism of simultaneous time over linear time. With linear time being broken comes also lack of interest in the past, history, culture, and traditions but also indifference regarding the future. This is typical of the modern digital

**20**

youth. Bauerlein [16] carried out research at high schools and universities in the USA and found out that year after year students are less and less aware of history and civic education, and generally their knowledge in subjects that have something to do with history is less and less adequate. Volko [17] carried a knowledge research at one of the Slovak universities and acquired a similar result, which he commented: "Quality of general knowledge is, mildly speaking, inadequate. Students that will in the future work in media, struggle when asked to say for example when Slovak National Uprising started, they cannot define holocaust or think of two Slovak classical music composers."

2. Space. A positive aspect of communication in cyberspace is in its ability to defeat geographical locations. We can now communicate with someone who lives in Australia or the USA not only orally but also visually. We even can watch events happening in various places on the globe. This may bring its negative aspect we can lose the sense of value of the real surrounding, our homeland, traditions, and culture in a given place. With communication on the Internet, importance of such a place declines, and people lose their roots. The Internet and also globalization tear the bound between geographical location and social role. With no geographical and social roots, one can easily become homeless in cyberspace.

Cyberspace of electronic media does not only consist of online telecommunication or online tele-seeing of the world; it is a world of new opportunities in virtualization of reality. Virtualization of reality may take various modes of reality or creation of brand new, fantastic worlds. Communication, or contact with virtual worlds, brings some pros and some contrast. In playing games young people can learn manual and visual skills and learn about the world but also become completely immersed and become addicted or virtualize the real world.

3.Structure of thinking. In communication within cyberspace, some structural changes in thinking and consequently in learning occur. Each media has its own semiotics, and the most fundamental media, for example, spoken word, found their new cultural epochs. Therefore, media are tools for our thinking and learning. Bystřický [18], for instance, says that "we also use different ways of thinking with increased use of technologies, not in terms of changing the actual availability of such abilities; we rather fundamentally alter strategy of their use." Thinking in cyberspace is influenced by discontinuity of images, short texts, and similar, which does not help us to train concentration and continual refinement of ideas. On the contrary, a text in a newspaper or book requires us to concentrate and pursue the logical chain of ideas that are expressed. Book and newspaper thus develop abstract and logically continual thinking, while television and especially the Internet nourish visual and discontinuous thinking. According to G. Sartori, image-based media, such as television and the Internet, alter the way we think, imagine, and learn. He is convinced that a new type of human is rising—*homo videns*—whose perception and knowledge are greatly modified by media images. In his idea, the turn from conceptual language of texts to media images also brings deprivation of abstract thinking, and, as Solík [19] adds, also emotional changes. We do not need to think; seeing a picture is enough. Sartori [13] explains: "Television brings metamorphosis that affects the very core of *Homo sapiens*. It is not a mere tool for communication, but also an anthropological instrument that constructs a new kind of human existence." *Homo sapiens* then changes to *homo videns*, which introduces decline in erudition and cultural decadence. It is similar in the case of the Internet that, unlike television, is interactive. If used by

someone culturally illiterate, only what seems to be interesting will be picked, namely, entertainment. We could see this aspect of structuring in thinking as negative. Positive aspect could be seen in rapid access to information and, under certain condition, also access to information offered by "collective intelligence," collective source of information, for example Wikipedia.

Another structural change in thinking in communication in cyberspace occurs in net-based or hypertext-based source of information. We could describe this type of communication or information as rhizomorph. Eco [20] used this term to distinguish it from the previous, treelike (*arborescent*) thinking. A picture of tree, for example, in the Middle Ages thinking (*arbor porphyriana*), represented a neat structure of hierarchy-based and logical thinking, from the essence of being, all the way down to its peripheral symptoms. However, rhizomatic thinking is non-systematic, incomplete, and netlike and has no beginning and no end. Thus the Internet, based on its own technological and netlike (rhizomatic) structure, promotes "loop connections" and consequent disintegration of the so-called linear code. U. Eco explains that rhizome excuses and supports disharmony, because rhizome creates loop-like processes. Eco even says that "To think means, in rhizome, to advance blindly and rely only on assumptions." The Internet, characteristic for its hypertext, or perhaps rhizomatic connection of information, will not support abstract, linear, and logical thinking, which may constitute a threat for modern society. Spitzer [21] states that digital natives instead of thinking in hermeneutic circle (from fragments to the whole picture and the other way round) get only superficial information surfing on the Internet: "Digital natives do not go through this hermeneutic circle: they haphazardly click here and there and never return to a good source; they look only horizontally (do not dig deeper)." Višňovský [22] notices that there is a difference between printed information and online information. When we read online, we do not read horizontally, line by line, but slide vertically along the text.

4.Identification. In communication on the Internet, there are two sorts of identifications: mental and physical connection with media that helps us get inside cyberspace or mental identification with content in cyberspace. The first type of identification constitutes mental or mentally physical cyborgism. It is currently possible to connect technology (artificial arm) to the nervous system and control it by thought. We can expect similar applications also in the field of information technology—for example, Google's Google Glass and later possibly electronic contact lenses. Some technologies may, in the future, be implemented also in the human body. This could bring its positives for some people who suffer from injuries after accidents and also provide immediate access to information. On the other hand, it could bring a fundamental dependence on technology and potential danger of abusing this technology to spy on people or control them.

At the present time, self-identification with content in cyberspace through social groups, or avatars, is still more and more common. The effort to find one's place among a social group and be able to share one's knowledge and experiences may be taken as desirable. One can sometimes feel the need to live a better life in cyberspace, for example, in a videogame called Second life. This can induce therapeutic, liberating effects. People can feel a need to become somebody else in life and demonstrate this also in social life, as we can see, for example, in a videogame called cosplay (a portmanteau of the words costume play). In Japan, but now also in other counties in the Western world, cartoons and cartoon characters are idols for teenagers. Young people identify themselves with these characters, which manifest the most in their costumes. Sometimes this new identity is so strong that young people will not want

**23**

**Author details**

Trnava, Slovakia

Slavomír Gálik\* and Sabína Gáliková Tolnaiová

provided the original work is properly cited.

\*Address all correspondence to<EMAIL_ADDRESS>
Faculty of Mass Media Communication, University of Ss. Cyril and Methodius in

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Cyberspace as a New Existential Dimension of Man DOI: http://dx.doi.org/10.5772/intechopen.88156*

**5. Conclusion**

to abandon the idea [9]. Everything depends on the extent and manifestation of such identification. If it causes alienation or addiction, it becomes a negative situation. Analysis of positive and negative changes in communication in cyberspace reveals that we need media education. D. Petranová [23] explains it is critical thinking that is the most important objective, and this can help us treat media with reserve, analyze information correctly, think independently, free ourselves from

We have known communication in cyberspace, especially in the cyberspace Internet, for slightly over a generation span, and we can already say that it has significantly influenced our cultural and social life; it even initiated a new existential dimension. The Internet cyberspace is a medium through which we create our ideas, communicate, and learn. Basing on analysis of older types of media, for example, written word and printed word, we know that these managed to restructure human thinking and acting completely. This leads us to believe that something similar is happening, and will be happening, also in connection with the Internet cyberspace. Media, including the Internet, influence us simply because we use them. The mere fact that we are connected to the Internet and use it in our communication in cyberspace is all what it takes; how we use it is not so important. We call the first type of influence, which is the result of being connected to technology, formal influence. The second kind of influence, triggered by communicating certain content, is defined as content influence. In this article we tried to point out that formal type changes are more crucial and paradigmatical and even constitute a new anthropogenesis. We specifically studied changes in our ideas of time and space, structure of thinking, and identity in cyberspace. These changes do not manifest merely in communication in cyberspace but affect also our everyday life. This is the reason why it is necessary to know their scope, positives, and negatives. New communication technologies influence our mentality but also our physical body. The question is how much is just enough to refine our personality, knowledge, and freedom and how much is simply too much, so they will start dictating and conducting us. We therefore need to learn to trust media with reserve, be critical, and spend at least part of the time we have without the influence of media, especially away from the Internet cyberspace.

stereotypes, and so on. This all should improve our personal freedom.

#### *Cyberspace as a New Existential Dimension of Man DOI: http://dx.doi.org/10.5772/intechopen.88156*

to abandon the idea [9]. Everything depends on the extent and manifestation of such identification. If it causes alienation or addiction, it becomes a negative situation.

Analysis of positive and negative changes in communication in cyberspace reveals that we need media education. D. Petranová [23] explains it is critical thinking that is the most important objective, and this can help us treat media with reserve, analyze information correctly, think independently, free ourselves from stereotypes, and so on. This all should improve our personal freedom.


**Author details**

Jonathan Montomoli\*, Raffaele Mitri and Emiliano Gamberini Infermi Hospital, AUSL Romagna, Rimini, Italy

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**RECENT INTERFEROMETRY APPLICATIONS IN TOPOGRAPHY AND ASTRONOMY**

Edited by **Ivan Padron**

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*Article* **Numerical Method for Solving Fractional Order Optimal Control Problems with Free and Non-Free Terminal Time**

**Oday I. Al-Shaher 1,2, M. Mahmoudi 1 and Mohammed S. Mechee 3,\***

**Abstract:** The optimal control theory in mathematics aims to study the finding of control for a dynamic system over time, where an objective function is optimized. It has a broad range of applications in engineering, operations research, and science. The main purpose of this study is to provide numerical algorithms for two cases of optimal control problems of fractional order that involve fractional order derivatives with free and non-free terminal time. In addition to comparing the numerical results for three test problems with exact solutions of these problems, various computer simulations are also introduced.

 Introduction**

Optimal control is the study of finding a dynamic control system over time in order to optimize an objective function. It has many uses in operations research, engineering, and science. For instance, the dynamic system could be a spacecraft with controls that correspond to rocket thrusters, and the goal could be to reach the moon using the least amount of fuel. In terms of result, the dynamic system could be a country's economy with the goal to minimize unemployment; in this scenario, the controls could be fiscal and monetary policy. It is also possible to integrate operations research issues into the framework of optimal control theory by using a dynamic system. Additionally, a branch of mathematics and physics known as the fractional dynamics examines how objects and systems behave by differentiating fractional orders. Research on fractional dynamical systems has produced novel findings that have attracted the interest of a significant audience of professionals, including mathematicians, physicists, applied researchers, and practitioners. This is due to the topic's wide applications in science and technology. In contrast to integer-order models, however, fractional-order models offer the potential to express non-local relations in time and space using power law memory kernels [1]. Consequently, this indicates that they offer more accurate and more realistic results. Moreover, the standard integral and differential calculus are generalized to any order in fractional calculus. If the order of the fractional derivative operator is an integer m, we obtain an m-fold integral when m is negative and the traditional derivative of order m when m is positive. Furthermore, for the review of the literature on numerical studies of fractional optimal control problems (FOCPs), Agrawal [2] preformed a formulation and numerical scheme for FOCPs, the work in [3] introduced the numerical solution of some types of FOCPs, while Bhrawy et al. [4] introduced an accurate numerical technique for solving FOCPs. Furthermore, a new method for the numerical solution of FOCPs was introduced in [5]. Furthermore, to solve multidimensional FOCPs with a quadratic performance index, the authors of [6] developed a practical numerical method for the purpose of solving FOCPs, and Doha et al. [7] investigated an effective numerical method based on the shifted orthonormal Jacobi polynomials. However, the

**Citation:** Al-Shaher, O.I.; Mahmoudi, M.; Mechee, M.S. Numerical Method for Solving Fractional Order Optimal Control Problems with Free and Non-Free Terminal Time. *Symmetry* **2023**, *15*, 624. https://doi.org/10.3390/ sym15030624

Academic Editor: Francisco Martínez González

Received: 24 January 2023 Revised: 14 February 2023 Accepted: 23 February 2023 Published: 2 March 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

generalized differential transform approach was used in [8] to introduce the numerical solutions of the coupled space-and-time Burgers equations. Lotfi et al. [9] introduced a numerical technique for solving FOCPs, Pooseh et al. [10] introduced a numerical scheme to solve FOCPs, Zhao and Li [11] solved the time–space fractional telegraph equation using the fractional difference-finite element, and Mechee and Senu [12] studied the numerical solution of fractional differential equations of Lane–Emden type by the method of collocation. For the space fractional diffusion equations, Zhou et al. [13] studied the quasi-compact finite difference schemes, and Bhrawy et al. [14] investigated a new Jacobi spectral collocation approach for fractional coupled Schrödinger systems and 1 + 1 fractional Schrödinger equations. At the same time, for the review of the literature on Legendre polynomials, using a Chebyshev–Legendre operational technique, Bhrawy et al. [15] solved the fractional optimal control for dynamical systems problems (FOCDSs). In fact, Yousefi et al. [16] employed a Legendre multiwavelet collocation approach in order to solve the FOCPs. In contrast, Bhrawy and Ezz-Eldien [17] used a new Legendre operational technique for solving delay FOCPs, in similar to Dirichlet boundary conditions, Heydari et al. [18] solved fractional partial differential equations (FPDEs) using the Legendre wavelets method. On the other hand, for the solution of fractional sub-diffusion and reaction sub-diffusion equations, Doha et al. [7] utilized an effective Legendre spectral tau matrix formulation, Khan and Khalil [19] provided a new approach that is based on Legendre polynomials. In parallel to these researchers, Sweilam and Al-Ajami [20] solved some types of FOCPs using the Legendre spectral-collocation method; additionally, some authors studied different cases of fractional differential equations. To solve the space fractional order diffusion equation, Sweilam et al. [21] utilized the second sort of shifted Chebyshev polynomials, but a discrete method for solving FOCPs was introduced in [22], while ref. [23] established a fractional adaptation strategy for lateral control of an AGV; whereas, Pinto and Tenreiro Machado [24] introduced the fractional dynamics of computer virus propagation, Pooseh et al. [25] studied the FOCPs with free terminal time by using operational matrices of Bernstein polynomials, Jafari and Tajadodi [26] solved the FOCPs, and Jesus and Tenreiro Machado [27] investigated the fractional control of heat diffusion systems. Thereafter, for a review of more literature on the applications, Ahmad and El-Khazali [28] introduced the fractional-order dynamical models of love and David et al. [29] studied fractional-order calculus, meanwhile, analog fractional-order controllers for temperature and motor control applications were studied in [30,31] introduced a 2D dynamic analysis of the model of disturbances in the calcium neuronal model and its implications in neurodegenerative disease; the work in [32] introduced the fractional sub-equation method and its applications to nonlinear fractional PDEs, whereas Kreyszig [33] studied historical apologia, fundamental ideas, as well as certain applications. Lastly, a fractional-order iterative learning control (FOILC) design challenge for linear time-varying systems with nonuniform trial durations was addressed in [34]. Additionally, a closed-loop FOILC updating legislation has been provided for activities with variable trial lengths. A central idea that unifies the coordination, prioritization, and execution of digital transformations within a firm was investigated in [35] in organizations that needed to build management procedures to oversee initiatives to investigate new digital technologies. For the purpose of tracking control of fractional-order linear systems, Zhao et al. [36] developed a revolutionary FOILC approach. In the meantime, the same beginning condition assumption is relaxed with the introduction of an initial state learning mechanism. For the FOCPs exposed to fractional systems with equality and inequality constraints, Sabermahani and Ordokhani [37] investigated fractional optimal control issues using the Fibonacci wavelets and Galerkin approach.

The free and non-free terminal time optimal control for dynamical systems (OCDSs) is introduced in this study. Additionally, the direct search approach to the unconstrained optimization problem is investigated. The proposed numerical methods for solving the optimal control problems of fractional orders with free and non-free terminal time are then constructed. The algorithm of the known procedure as Hooke and Jeeves's method is used in the computation.

#### **2. Main Problem**

A dynamic system's optimal control problem is the task of determining the control law that minimizes a performance index in terms of the state and control variables. Many authors have recently studied a wide range of optimization issues related to the integer optimal control of differential systems. In this research, we propose a novel numerical method for approximating the solutions of the fractional optimal control systems in both cases with free- and non-free terminal time.

#### Case I: Non-Free Terminal Time

Consider

$$\min\_{\mathbf{x}(\tau),\boldsymbol{\mu}(\tau)} \quad J(\tau,\mathbf{x}(\tau),\boldsymbol{\mu}(\tau)) \quad = \min\_{\mathbf{x}(\tau),\boldsymbol{\mu}(\tau)} \quad \int\_{\tau\_0}^{\tau\_1} P(\tau,\mathbf{x}(\tau),\boldsymbol{\mu}(\tau))d\tau,\tag{1}$$

subjected to the constricted dynamical system

$$a\dot{\mathbf{x}}(\tau) + \beta D^{\gamma}\mathbf{x}(\tau) = \gamma(\tau)\mathbf{x}(\tau) + f(\tau)u(\tau) + \mathbf{g}(\tau), \quad \tau\_0 \le \tau \le \tau\_1, \quad 0 \le \gamma \le 1. \tag{2}$$

The constricted boundary conditions are as follows:

$$\mathfrak{x}(\pi\_0) = \zeta, \quad \mathfrak{x}(\pi\_1) = \eta,\tag{3}$$

where *α*, *β* = 0,

#### Case II: Free Terminal Time

Consider the FOCP in the equations

$$\min\_{\mathbf{x}(\tau),\boldsymbol{\mu}(\tau),\boldsymbol{T}} \quad J(\tau,T,\mathbf{x}(\tau),\boldsymbol{\mu}(\tau)) \quad = \min\_{\mathbf{x}(\tau),\boldsymbol{\mu}(\tau),\boldsymbol{T}} \quad \int\_{\tau\_{0}}^{T} P(\tau,\mathbf{x}(\tau),\boldsymbol{\mu}(\tau))d\tau,\tag{4}$$

subjected to the constricted dynamical system in Equation (2) with the free terminal time:

$$\mathbf{x}(t\_0) = \mathbf{c}, \quad \mathbf{x}(T) = d,\tag{5}$$

where *T* is a free parameter.

Firstly, for using the proposed numerical approach, we use the basic polynomials to approximate the state variable *x*(*τ*) with the control variable *u*(*τ*), and the known functions *e*(*τ*), *f*(*τ*), and *g*(*τ*) are given. The second stage of the numerical method involves using a search method such as the Hooke and Jeeves method to optimize the parameters of the approximation in case of the problem of fractional order of optimal control systems with free terminal time together to the parameter of T in case of non-free terminal time. The manuscript is organized as follows. Section 3 introduces the basic definitions and background related to the problem of this study, while Section 4 presents the numerical methods and studies the proposed numerical method for solving FOCPs with free and non-free terminal time. Furthermore, Section 5 introduces the implementations of test examples for solving two types of fractional optimal control dynamical systems with free and non-free terminal time. Lastly, this paper ends with a discussion and conclusions in Section 6.

#### **3. Preliminary**

We have introduced the basic definitions and background related to the problem of this study.

*3.1. Basic Definitions of the Fractional Derivatives and (FOCDS) with Free and Non-Free Terminal Time*

The fundamental definitions of fractional derivatives as well as the free and non-free terminal times of fractional-order optimal control problems are introduced in this subsection.

**Definition 1.** *Non-Free Terminal Time (FOC) Problem*

*The (FOC) problem in Equations (1)–(3) is said to be a non-free terminal time if we have a constraint in t*1 *that means it is fixed else free terminal time (FOC) if t*1 = *T is not a fixed parameter. The following are two famous fractional derivatives since a large number of scholars have worked to establish a fractional derivative. In the literature, the fractional derivative often was presented in integral form. Two famous fractional derivatives are known as follows:*

*(i) Let f* : [*a*, ∞) → . *and a* > 0*. The fractional definition of f using the Riemann–Liouville derivative for α* ∈ [*n* − 1, *n*) *is defined by:*

**Definition 2.** *Riemann–Liouville Fractional Integral The left and right Riemann–Liouville (RL) fractional integral operators of order α* > 0

$$\prescript{a}{}{D}\_{\tau}^{a}y(\tau) \quad = \quad \frac{1}{\Gamma(n-a)} \frac{d^n}{d\tau^n} \int\_a^{\tau} \frac{y(\tau)}{(\tau-x)^{n-a-1}} d\tau,\tag{6}$$

*and*

$$\prescript{a}{}{D}\_{\tau}^{a}y(\tau) \quad = \quad \frac{1}{\Gamma(n-1)} \frac{d^n}{d\tau^n} \int\_a^{\tau} (\tau - s)^{n-a-1} y(s) ds,\tag{7}$$

*respectively, such that n is an integer and n* − 1 < *α* < *n*, *n* ∈ *N*. *Additionally, (ii) the Caputo derivative definition of f* , *for α* ∈ [*n* − 1, *n*)*, is defined as follows:*

**Definition 3.** *The Fractional Caputo Derivative*

$$^aD\_\tau^a f(\tau) \quad = \frac{1}{\Gamma(n-a)} \int\_a^\tau \frac{f^{(n)}(\tau)}{(\tau-x)^{a-n-1}} d\mathbf{x},\tag{8}$$

*where n is an integer and n* − 1 < *α* < *n, n* ∈ *N The fractional integral and derivative in the Definitions 2 and 3 satisfy the linearity properties for the fractional integrals and derivatives for α* > 0, *n* − 1 ≤ *α* < *n*.

#### *3.2. Hooke and Jeeves Direct Search Method Analysis*

In this subsection, the direct search method for solving the unconstrained optimization problem

$$\min\_{\mathbf{X}} \quad f(\mathbf{X})\_{\prime} \tag{9}$$

where the objective function f maps *n* into {+∞} and *X* = (*x*1, *x*2, *x*3, ... , *xn*), is introduced.

3.2.1. Algorithm of Hooke and Jeeves Method

3.2.2. The Convergence of Hooke and Jeeves Method

The set of points produced by the direct algorithm is consistently dense in the search region for all box selection methods. When *Nmax* = 1 and *Hmin* = 0, the proposed algorithm's properties of convergence are examined. The sequence of the solutions of the problem in Equation (9) is {*X*(0), *X*(1), ... , *X*(*k*), *X*(*k* + 1), ... }, which is obtained using the Hooke and Jeeves method. This sequence satisfies the convergence conditions according to the condition in step three.

Consider *ξ* ∈ *E* to be arbitrary, where

$$E = z\_m + h\_{meso}[-1;1]^n.$$

For each valid box selection approach, let {*δr*}∞ *r*=1 represent the points produced by strategy Γ. Let

$$\Delta(r) = \max\_{\Gamma} \max\_{\substack{\mathfrak{J} \in E \ \mathfrak{i} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{r}}} \min\_{\substack{\|\mathfrak{I}\| = \mathbf{1}, \mathbf{2}, \dots, \mathbf{r}}} ||\mathfrak{J}\_{\vec{\imath}} - \delta\_{\vec{\imath}}||.$$

Then, Δ(*r*) −→ 0 as *r* −→ ∞.

#### **4. The Numerical Method**

In this section, the proposed numerical method for solving (FOC) problems with freeand non-free terminal time is introduced.

#### *4.1. Proposed Algorithm*

In this subsection, we write the algorithm of the proposed numerical approach for approximating the solutions of (FOC) in two cases: (FOC) problems with free- and non-free terminal time. The steps of this algorithm are written as follows:

	- 1. Choose a suitable approximated base.

$$\Omega = \{ \Omega\_0(t), \Omega\_1(t), \Omega\_2(t), \dots, \Omega\_n(t) \},$$

2. Construct an approximated solution of (FOC),

$$\mathbf{x}(t) \quad = \sum\_{i=0}^{n} c\_i \Omega\_i(t) = c\_0 \Omega\_0(t) + c\_1 \Omega\_1(t) + \dots + c\_n \Omega\_n(t). \tag{10}$$

In Equations (1) and (2) which satisfy the boundary conditions in Equation (3) using the approximated base.

	- 1. Perform steps 1–4 in the previous algorithm;
	- 2. Use suitable minimizing search methods such as the Hooke and Jeeves method to find the best parameters (minimal) including the parameter T in Equation (1).

where T is the free parameter.

#### *4.2. Dual Discreet Problem*

• Algorithm of non-free terminal time (FOC) problem:

1. From Equation (10), consider

$$\mathbf{x}(t) = \sum\_{i=0}^{n} c\_i \Omega\_i(t),\tag{11}$$

with the boundary conditions leading

$$c\_1 = \gamma\_0 \frac{\eta - \frac{\zeta \Omega\_0(t\_1)}{\Omega\_0(t\_0)} - \sum\_{i=2}^{\eta} c\_i \left(\frac{\Omega\_i(t\_0)\Omega\_0(t\_1)}{\Omega\_0(t\_0)} - \Omega\_i(t\_1)\right)}{\Omega\_1(t\_1)}\,\tag{12}$$

where

$$\gamma\_0 = \frac{\Omega\_0(t\_0)\Omega\_1(t\_1) - \Omega\_1(t\_0)\Omega\_0(t\_1)}{\Omega\_0(t\_0)\Omega\_1(t\_1)}, \quad \Omega\_0$$

and

$$\omega\_0 = \frac{\zeta - c\_1 \Omega\_1(t\_0) - \sum\_{i=2}^n c\_i \Omega\_i(t\_0)}{\Omega\_0(t\_0)},\tag{13}$$

hence,

$$\begin{split} \mathbf{x}(t) &= \frac{\underline{\zeta} - \sum\_{i=2}^{n} c\_{i} \Omega\_{i}(t\_{0})}{\Omega\_{0}(t\_{0})} \Omega\_{0}(t) + \gamma\_{0} \frac{\eta - \frac{\zeta \Omega\_{0}(t\_{1})}{\Omega\_{0}(t\_{0})} - \sum\_{i=2}^{n} c\_{i} \left( \frac{\Omega\_{i}(t\_{0}) \Omega\_{0}(t\_{1})}{\Omega\_{0}(t\_{0})} - \Omega\_{i}(t\_{1}) \right)}{\Omega\_{1}(t\_{1})} \\ &+ \quad \left( \Omega\_{1}(t) - \frac{\Omega\_{1}(t\_{0})}{\Omega\_{0}(t\_{0})} \Omega\_{0}(t) \right) + \sum\_{i=2}^{n} c\_{i} \Omega\_{i}(t). \end{split} \tag{14}$$

where T is free parameter.

#### **5. Implementations (Numerical Examples)**

In this section, we introduce two types of dynamical problems. The numerical method introduced in Section 4 has been used for solving the optimal control problems of integer and fractional order with free and non-free terminal time.

**Example 1.** *Let us take into consideration the following (FOC) problem with non-free terminal time introduced by* [10,25]*.*

$$\min\_{\mathbf{x}(\tau),\boldsymbol{\mu}(\tau)} \quad f(\tau,\mathbf{x}(\tau),\boldsymbol{\mu}(\tau)) = \min\_{\mathbf{x}(\tau),\boldsymbol{\mu}(\tau)} \quad \int\_0^1 (\tau u(\tau) - (\gamma + 2)\mathbf{x}(\tau))^2 d\tau,\tag{15}$$

*subjected to the dynamic system*

$$D^{\gamma}\mathfrak{x}(\mathfrak{r}) + \mathfrak{x}(\mathfrak{r}) = \mathfrak{r}^2 + \mathfrak{u}(\mathfrak{r}),\tag{16}$$

*with the boundary conditions (BCs)*

$$\varkappa(0) = 0, \quad \varkappa(1) = \frac{2}{3 + \gamma}, \tag{17}$$

*where the exact solution is given by*

$$(\mathfrak{x}(\tau), \mathfrak{u}(\tau)) = (\frac{2\tau^2 + \gamma}{\Gamma(3+\gamma)}, \frac{2\tau^{1+\gamma}}{\Gamma(2+\gamma)}).$$

*Using the approximation base* Ω(*t*) = {*τ*2, *τ*, 1}*, we have the approximation of x*(*τ*) *as*

$$\mathbf{x}(\pi) = c\_0 + c\_1 \pi + c\_2 \pi^2. \tag{18}$$

*If we use the BCs in Equation (17), we obtain c*0 = 0 *and c*1 = 2 3+*γ* − *c*2*. Then, the following approximation is obtained*

$$\mathbf{x}(t) = t(c\_2 + \frac{2}{3+\gamma} - c\_2t). \tag{19}$$

*Then, we have*

$$\mu(\tau) = \mathfrak{x}(\tau) = (1 + \tau^{1-\gamma})(c\_2 - \frac{2}{3+\gamma} + 2c\_2\tau) - \tau^2. \tag{20}$$

*Substitute Equations (19) and (20) in the problem of minimizing in the Equation (15) to obtain the optimal values of c*2, *and the non-free terminal parameter T*. *Hence, using the Hooke and Jeeves method for the problem in parameter c*2*, the approximation of the problem is plotted in Figure 1a.*

**Example 2.** *Consider the following integer-order optimal control problem with non-free terminal time:*

$$\min\_{\mathbf{x}(\tau),\boldsymbol{\mu}(\tau),T} J(\tau,\mathbf{x}(\tau),\boldsymbol{u}(\tau),T) = \min\_{\mathbf{x}(\tau),\boldsymbol{\mu}(\tau),T} \int\_{0}^{T} (\tau\boldsymbol{u}(\tau) - 2\mathbf{x}(\tau))^{2} d\tau,\tag{21}$$

*subjected to the dynamic system*

$$
\dot{\mathfrak{x}}(\pi) + \dot{\mathfrak{x}}(\pi) = \pi^2 + \mathfrak{u}(\pi),
\tag{22}
$$

*with the boundary conditions*

$$\mathfrak{x}(0) = 0, \quad \mathfrak{x}(1) = 1,\tag{23}$$

*where the exact solution is given by*

$$(\mathfrak{x}(\tau), \mathfrak{u}(\tau)) = (\tau(2-\tau), -\tau^2 + 2\tau + 2).$$

*Consider the solution of the optimization problem in Equations (21)–(23) written as follows*

$$\mathbf{x}(\tau) = \tau(1 + c\_2 - c\_2\tau). \tag{24}$$

*This satisfies the boundary conditions in Equation (23). Then, we have*

$$
\mu(\tau) = -\tau^2 + 2c\_2t + 3c\_2 - 1. \tag{25}
$$

*Substitute Equations (24) and (25) in minimizing the problem in Equation (21) to obtain the optimal values of c*2 = 0.989 *and T* = 0.997*. Hence, the approximation of the problem is written as x*(*τ*) = *τ*(2 − *τ*) *and then, it is plotted in Figure 1b.*

**Example 3.** *Let us consider the following optimal control problem of fractional order with non-free terminal time which was introduced by* [25]

$$\min\_{\mathbf{x}(\tau),\mu(\tau),T} \quad f(\tau,\mathbf{x}(\tau),\mu(\tau),T) = \min\_{\mathbf{x}(\tau),\mu(\tau),T} \int\_0^T (\tau u(\tau) - (\gamma+2)\mathbf{x}(\tau))^2 d\tau,\tag{26}$$

*subject to the control system*

$$D\_{\tau}^{\gamma}\chi(\tau) + \dot{\mathfrak{x}}(\tau) = \tau^2 + \mathfrak{u}(\tau),\tag{27}$$

*and the boundary conditions*

$$
\varkappa(0) = 0, \quad \varkappa(T) = 1. \tag{28}
$$

*Consider the solution of the optimization problem in Equations (26)–(28) written as follows:*

$$\mathbf{x}(\tau) = c\_2 \tau^2 + (\frac{1}{T} - c\_2 T)\tau,\tag{29}$$

*which satisfied the boundary conditions in Equation (28). Then, we have*

$$
\mu(\tau) = -\tau^2 + 2c\_2\tau + 3c\_2 - 1.\tag{30}
$$

*Substitute Equations (29) and (30) in minimizing the problem in Equation (26) to obtain the optimal solutions using the Hooke and Jeeves method. Hence, the approximation of the problem is plotted in Figure 1c.*

**Figure 1.** A Comparison of Numerical Solutions of (FrOCDS) Evaluated by the Hooke and Jeeves Direct Search Method for the Implementations in (**a**) Example 1, (**b**) Example 2, and (**c**) Example 3.

#### **6. Discussion and Conclusions**

The main purpose of this study is to introduce numerical methods for solving two cases of fractional-order optimal dynamical control systems with free and non-free terminal time. The study also offers a comparison of the numerical results obtained by using the proposed method with the exact solutions for three test problems. From the numerical results of the proposed method, we observe that the method is applicable to a class of (FOC) problems with free or non-free terminal time. Moreover, the proposed method achieves good agreement with exact solutions. As a result, the new method is efficient and provides encouraging results. This direction of this research can be extended in the future to new directions such as improving the numerical studies of stochastic optimal control problems to the continuity of the research in this domain.

**Author Contributions:** Literature Review, O.I.A.-S.; Investigation, O.I.A.-S.; Methodology, O.I.A.-S., and M. M.; Project administration, M.M.S.; Resources, O.I.A.-S.; Software, M.S.M.; Supervision, M.S.M. and M.M.; Validation, M.S.M., and M.M.; Writing—original draft, O.I.A.-S.; Writing—review and editing, M.S.M. and M.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

**Disclaimer/Publisher's Note:** The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.


**Abbreviations**

DM: diabetes mellitus; SGLT1: Na+/glucose cotransporter 1; OGTT: oral glucose tolerance test; AnF: fucoidan from *Ascophyllum nodosum*; KcF: fucoidan from *Kjellmaniella crassifolia*; LjF: fucoidan from *Laminaria japonica*; 2-NBDG: 2-Deoxy-2-[(7-nitro-2,1,3-benzoxadiazol-4-yl) amino]-D-glucose; SPR: surface plasmon resonance; ALP: alkaline phosphatase; GLP-1: glucagon-like peptide 1; DPP-4: dipeptidyl peptidase 4; aGLP-1: active GLP-1; tGLP-1: total GLP-1; HOMA-IR: homeostasis model assessment-insulin resistance; HbA1c: hemoglobin A1c; ddH2O: double-distilled water; DMEM: Dulbecco's Modified Eagle Medium; TEER: transepithelial electrical resistance; MEM: minimum Eagle's medium; BSA: bovine serum albumin; FvF: fucoidan from *Fucus vesiculosus*, MW: molecular weight; UAs: uronic acids; DMEM: Dulbecco's Modified Eagle Medium; Metf: metformin; db/db: leptin receptor-deficient; APTT: activated partial thromboplastin time; ROS: reactive oxygen species; JNK: c-Jun N-terminal kinase; Akt: protein kinase B.


*5.1. Diagnosis Versus Classification Criteria*

Faced with one or more compatible manifestations, the diagnosis of pSS must be evoked and investigated. Making a diagnosis is the basis of medical care. For the patient, it represents the end of questioning and diagnostic wandering. For the physician, the diagnosis makes it possible to clarify the management. Finally, for the researcher, the diagnosis makes it possible to create homogeneous groups around a consensus definition. Unfortunately, there is no single diagnostic test to confirm the diagnosis of pSS. Due to its protean and willingly insidious presentation, pSS is sometimes difficult to recognize and may delay diagnosis by more than 10 years. Sicca syndrome, fatigue and unspecific musculoskeletal pain can be wrongly taken for manifestations of age, anxio-depression or perimenopause in people with pSS. Systemic manifestations can sometimes precede sicca syndrome, resulting in an "occult pSS" [249]. For these various reasons, the gold standard for individual diagnosis of pSS remains the opinion of an expert clinician. To allow the study of the disease in groups of pSS patients, several consensuses have defined classification criteria allowing a common definition of what pSS is. The 3 most recent sets of classification criteria are presented in Table 2. By definition, classification criteria are specific but may lack sensitivity and should not be used blindly as diagnostic criteria but as a guide in clinical practice.

AECG = American European Consensus Group, SICCA = Sjögren's International Collaborative Clinical Alliance, ACR-EULAR = American College of Rheumatology—European League Against Rheumatism, UWSF = unstimulated whole saliva flow, RF = rheumatoid factor, ANA = antinuclear antibodies, ESSDAI = EULAR Sjögren's syndrome disease activity index.


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Synbiotics are preparations containing a mixture of a probiotic and a prebiotic, or a combination of probiotics and prebiotics (Davis & Milner, 2009; Holzapfel & Schillinger, 2002; Touhy et al., 2003). These preparations aim to improve the viability of the proven probiotic *in vivo* as well as stimulate the indigenous gut microflora. They provide both the beneficial microbial culture and a specific substrate that can be readily available for fermentation by this culture (Gallaher & Khil, 1999). The presence of the readily fermentable substrate could enhance the survival of the probiotic. The prebiotic ingredient could also offer protection of probiotic against gastric acidity and proteolysis, probably through steric

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**3**

*Turkey*

**The Impact of Probiotics**

Erdal Matur and Evren Eraslan

*University of Istanbul, Avcilar, Istanbul*

**on the Gastrointestinal Physiology**

Researches concerning probiotics were initiated by a Russian scientist named Elie Metchnikoff. He emphasized the importance of *Lactobacillus* species and fermented milk products present in the gastrointestinal tract for a healthy and long life. The term "probiotics" was first introduced in 1953 by Werner Kollath, and he defined probiotics as microbially derived factors that stimulate the growth of other microorganisms. Afterwards, the term "probiotics" was defined by Roy Fuller in 1989 as a live microbial feed supplement which beneficially affects the host by improving its intestinal microbial balance. Fuller's definition emphasizes the requirement of viability for probiotics and introduces the aspect of a beneficial effect on the host. Although, this definition has been widely used by the entire scientific world, according to the currently adopted definition by FAO/WHO, probiotics are: "Live microorganisms confer a health benefit on the host when administered

The most frequently used probiotic microorganisms are Lactobacillus and Bifidobacterium species. However, there are also much more bacteria and some yeast species used as probiotic. Bacteria and yeast species, which are used commonly as probiotic, are listed by Heyman & Menard (2002) as below. *Lactobacillus* species: *L. acidophilus, L. rhamnosus, L. gasseri, L. reuteri, L. bulgaricus, L plantarum, L. johnsonii, L. paracasei, L. casei, L. salivarius, L. lactis. Bifidobacterium* species: *B. bifidum, B. longum, B. breve, B. infantis, B. lactis, B. adolescentis.*  Other species: *Streptococcus thermophilus, Escherichia coli, Bacillus cereus, Clostridium butyricum, Enterococcus faecalis, Enterococcus faecium.* Yeast: *Saccharomyces boulardii, Saccharomyces cerevisiae*, VSL#3 (four strains of lactobacilli, three strains of bifidobacteria,

There is a mutual interaction between intestinal cells and microorganisms present in the gastrointestinal tract. Commensal microorganisms in the gastrointestinal tract have multidirectional effects on digestion, absorption and barrier function, secretory functions, or postnatal maturation of intestinal mucosa. Furthermore, it has been known that they change gene expression in intestinal cells (Hooper et al., 2001). The changes in the function of intestinal tract (e.g. the increase of motility, or disruption of carbohydrate digestion) also

**1. Introduction**

in adequate amounts*"* (FAO/WHO, 2001).

one strain of *Streptococcus salivarius* sp. *thermophilus*).

affect bacteria population and colonization.

*Department of Physiology, Faculty of Veterinary Medicine,*