DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020027 DYNAMICAL SYSTEMS SERIES S Volume 13, Number 3, March 2020 pp. 485{501 GENERALIZED VARIATIONAL CALCULUS IN TERMS OF MULTI-PARAMETERS INVOLVING ATANGANA-BALEANU'S DERIVATIVES AND APPLICATION G. M. Bahaa Department of Mathematics, College of Science, Taibah University Al-Madinah Al-Munawarah, Saudi Arabia Abstract. In this paper, the generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's Derivatives are discussed. We consider the Hilfers generalized fractional derivative that in sense Atangana- Baleanu derivatives. We develop integration by parts formulas for the gener- alized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variation- al formulations can be obtained by setting different parameters to different values. The fractional Euler-Lagrange equations of fractional Lagrangians for constrained systems contains a fractional Hilfer-Atangana-Baleanu's deriva- tives with multi parameters are investigated. We also define fractional gener- alized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed. We present a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP. 1. Introduction. Fractional calculus represents a generalization of ordinary differ- entiation and integration to arbitrary order. During the last decades the fractional calculus started to be used in various fields, e.g. physics, engineering, biology and many important results were obtained [1]-[6],[25], [31]. There exists many definitions of a fractional derivative. A commonly known fractional derivatives are the classical Riemann-Liouville and Caputo derivatives. Fractional derivatives and integrals of these Riemann-Liouville and Caputo types have a vast number of applications across many fields of science and engineering. For example, they can be used to model controllability, viscoelastic flows, chaotic 2010 Mathematics Subject Classification. 26A33, 34A08, 49J20. Secondary: 35R11, 49J15, 49K20, 45D10. Key words and phrases. Fractional optimal control, fractional Lagrangian and Hamiltonian, fractional Euler-Lagrange equations, fractional Atangana-Baleanu's derivatives, Riemann-Liouville and Caputo's derivatives. ∗ Permanent address: Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt. Email: Bahaa−[email protected]. The author is supported by Taibah University, Dean of Scientific Research. 485 486 G. M. BAHAA systems, Stokes problems, thermo-elasticity, several vibration and diffusion process- es, bioengineering problems, and many other complex phenomena (see e.g. [3]-[6] [12]-[20], [25], [27], [31]-[36] and references therein). Fractional optimal control problems involving that classical derivatives have at- tracted several authors in the last two decades, and many techniques have been developed for solving such problems involving these classical fractional derivatives. Agrawal [3,4] presented a general formulation and proposed a numerical method to solve such problems. In that paper, the fractional derivative was defined in the Riemann-Liouville sense and the formulation was obtained by means of frac- tional variation principle and the Lagrange multiplier technique. Using the same technique, Frederico et al. [27] obtained a Noether-like theorem for the fraction- al optimal control problem in the sense of Caputo. In [39, 40], Mophu studied the fractional optimal control for the diffusion equation involving the classical Riemann- Liouville derivatives. In [41], Ozdemir investigated the fractional optimal control problem of a distributed system in cylindrical coordinates whose dynamics are de- fined in the classical Riemann-Liouville sense. The FC brings new features in describing complex behaviors of the real-world phenomena with memory effects. However, the description of systems with memory effect is still a big challenge for researchers, since the classic type of FDs with singular kernel cannot characterize always properly the nonlocal dynamics. Hence, it seems there is a need of new FDs with nonsingular kernel to better describe the non-locality of complex systems. One of the best candidates among existing kernels is the one based on Mittag-Leffler function ( see e.g. [1,2,9, 10, 11, 19, 21, 23, 24, 28, 29, 30, 37] and references therein). Recently, Caputo and Fabrizio [22] have proposed a new definition of fractional derivatives: Z x CF α B(α) d −α Dc+f(x) = exp 1−α (x − y) f(y) dy; 1 − α dt c valid for 0 < α < 1, with B(α) being a normalisation function satisfying B(0) = B(1) = 1. The basic challenge they were addressing was whether it is possible to construct another type of fractional operator which has nonsingular kernel and which can better describe in some cases the dynamics of non-local phenomena. The Caputo{Fabrizio definition has already found applications in areas such as diffusion modelling [34] and mass-spring-damper systems [8]. However, some issues were pointed out against both derivatives, including the one in Caputo sense and the one in Riemann-Liouville sense. As Sheikh [44] pointed out, the CF fractional derivative as the kernel in integral was non-singular but was still nonlocal. Some researchers also concluded that the operator was not a derivative with fractional order but a filter with fractional parameter. The fractional parameter can then be viewed as a filter regulator. Atangana and Baleanu introduced a new operator with fractional order based upon the generalized Mittag-Leffler function [10]. Their operators have all the benefits of that of the CF derivative in addition to the kernel being nonlocal and non-singular. The non-locality of the kernel gives better description of the memory within the structure with different scale. Here we shall mostly be considering a more recently developed definition for fractional differintegrals, due to [10]. This new type of calculus addresses the same underlying challenge as that of Caputo and Fabrizio, but it uses a kernel which is GENERALIZED VARIATIONAL CALCULUS IN TERMS OF MULTI-PARAMETERS 487 non-local as well as non-singular, namely the Mittag-Leffler function: Z t ABR α B(α) d −α α Da+f(t) = 1−α f(x)Eα 1−α (t − x) dx; dt a Z t ABC α B(α) 0 −α α Da+f(t) = 1−α f (x)Eα 1−α (t − x) dx: a In this way we are able to describe a different type of dynamics of non-local com- plex systems. In fact the classical fractional calculus and the one corresponding to the Mittag-Leffler nonsingular kernel complement each other in the attempt to better describe the hidden aspects of non-local dynamical systems. Fractional cal- culus based on the non-singular Mittag-Leffler kernel is more easily used from the numerical viewpoint, and this has been studied for example in [24]. We note that the Mittag-Leffler function is already known to be highly useful in fractional calculus [38]. Applications of the new AB formula have been explored in fields as diverse as chaos theory [7], heat transfer [10], and variational problems [2]. Furthermore, it is natural to address the same questions about the fractional integrator and applications of these new operators in the theory of control and related fractional variational Euler-Lagrange and Hamilton equations (see [2, 24, 28]). Besides, we expect to obtain some new terms in all generalized formulae from the classical fractional calculus, and this aspect will be important for the related applications. Some basic properties of the new AB differintegrals have already been proven in several recent papers: for example, the original paper [10] established the formulae for Laplace transforms of AB differintegrals and some Lipschitz-type inequalities; the paper [2] considered integration by parts identities and Euler-Lagrange equa- tions; and the paper [24] established, using Laplace transforms, analogues of the Newton{Leibniz formula for the integral of a derivative. However, much of the ground-level theory of this new model of fractional calculus has not yet been fully developed, and this paper aims to add to this basic theory by establishing new fundamental results in the field. Our main results is that to compare the results of fractional Euler-Lagrange equations for the classical Hilfer Riemann{Liouville and Caputo fractional deriva- tives which stated in [32, 33] corresponding to Hilfer- Atangana-Baleanu's fractional derivatives which defined in [10]. In this study, by using the Hilfer-Atangana- Baleanu's derivative we propose to generalize the notion of equivalent Lagrangian for the fractional case. For a given classical Lagrangian there are several proposed methods to obtain the fractional Euler-Lagrange equations and the corresponding Hamiltonians. However, the fractional dynamics depends on the fractional deriva- tives used to construct the Lagrangian to start with, therefore the existence of several options can be used to treat a specific physical system. In this respect, application of the Hilfer -Atangana-Baleanu's derivative to the fractional dynamics will bring new opportunities in studying the constrained systems mainly because the Atangana-Baleanu's derivative contains both the left and the right derivatives. In addition, the fractional derivative of a function is given by a definite integral, thus depends on the values of the function over the entire interval. Therefore, the fractional derivatives are suitable to model systems with long range interactions in space and/or time (memory) and process with many scales of space and/or time involved. 488 G. M. BAHAA The plan of this paper is as follows: In section 2 we collect notations, definitions involving the Atangana-Baleanu fractional time derivative.