S S symmetry Article New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter Ricardo Almeida * and Natália Martins Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal; [email protected] * Correspondence: [email protected] Abstract: This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided. Keywords: fractional calculus; Euler–Lagrange equation; natural boundary conditions; time delay Citation: Almeida, R.; Martins, N. New Variational Problems with an MSC: 26A33; 49K05; 34A08 Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter. Symmetry 2021, 13, 592. 1. Introduction https://doi.org/10.3390/sym13040592 Fractional calculus refers to the integration and differentiation of a non-integer order and is as old as the classical (integer order) calculus [1]. It is a subject that has gained Academic Editors: Sergei D. Odintsov, much popularity and importance in the last few decades and has been applied in sev- Francisco Martínez González and eral fields of knowledge, such as mechanics [2,3], bioengineering [4], signal and image Mohammed KA Kaabar processing [5], physics [6,7], viscoelasticity [8], electrical engineering [9], economics [10], epidemiology [11,12], control theory [13,14], energy supply-demand systems [15], and Received: 15 March 2021 fuzzy problems [16]. Accepted: 1 April 2021 Published: 2 April 2021 One of the specificities of fractional calculus is that there are many definitions of fractional derivatives that allow the researcher to choose the one that best corresponds to a Publisher’s Note: MDPI stays neutral given problem. Some of the most commonly used fractional derivatives are the Riemann– with regard to jurisdictional claims in Liouville, the Erdélyi–Kober, the Caputo, the Hadamard, and the Grünwald–Letnikov published maps and institutional affil- derivatives. For a detailed study on this subject, see [1,17]. In this present work, we consider iations. fractional operators with respect to an arbitrary kernel (see [17] for the Riemann–Liouville sense and [18] for the Caputo sense). Fractional calculus of variations is a recent field that consists of minimizing or maxi- mizing functionals that depend on fractional operators. The first works in this scientific area are due to Riewe [3,19]. Since then, many papers were published on different topics of Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. the fractional calculus of variations for different types of fractional operators (see [2,20–29] This article is an open access article and the references therein). For more details, we recommend the works [30–32]. distributed under the terms and By considering a more general form of the fractional derivative, like the Caputo frac- conditions of the Creative Commons tional derivative with respect to an arbitrary kernel (see [18]), we can generalize different Attribution (CC BY) license (https:// fractional variational problems. In [23,33], necessary and sufficient optimality conditions creativecommons.org/licenses/by/ were proven for different variational problems depending on the Caputo fractional deriva- 4.0/). tive with respect to an arbitrary kernel. Symmetry 2021, 13, 592. https://doi.org/10.3390/sym13040592 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 592 2 of 17 In [33], the following problem was studied: determine x 2 C1([a, b], R) and z 2 R that extremize: Z b J ( ) = ( ) (C g,g )( ) (C d,g )( ) ( ) ( ) x, z : L t, x t , Da+ x t , Db− x t , x a , x b , z dt, (1) a 2 1([ ] × 6 ) C g,g C d,g where L C a, b R , R and Da+ x and Db− x denote, respectively, the left and right g-Caputo fractional derivatives of x of order g and d, with g, d 2]0, 1[ (see Definition3 ). The main results of [33] are optimality conditions for variational problems with or without isoperimetric and holonomic constraints. The aim of this paper is to generalize these previous results. It is important to mention that this type of generalized fractional varia- tional problem cannot be solved using the classical theory. Moreover, since the g-Caputo fractional derivatives are generalizations of several fractional derivatives and the varia- tional problem (1) is a generalization of several kinds of the calculus of variation problems, the results obtained in [33] not only generalize some known results, but also give new contributions to the theory of the fractional calculus of variations. In this paper, we prove optimality conditions for different fractional variational prob- lems that are generalizations of the one introduced in [33]. Namely, we prove the general- ized fractional variational principle for problems with optimal orders, with time delay and with arbitrary real order fractional derivatives. In addition, we prove sufficient optimality conditions for all of the problems considered in the paper. This main structure of this paper is as follows. In Section2, we present some prelimi- naries on fractional calculus. In Section3, we exhibit the main results. We finish the paper with two examples and some conclusions. 2. Preliminaries We begin with a brief review of some important concepts and results that will be used in this paper. In what follows, G represents the well-known Gamma function, and the integer part of g 2 R is denoted by [g]. Definition 1. [17] Let g be a positive real, g : [a, b] ! R a C1 function with positive derivative, and x 2 L1([a, b], R). The left Riemann–Liouville fractional integral of x of order g, with respect to the kernel g, is defined as: 1 Z t ( g,g )( ) = 0( )( ( ) − ( ))g−1 ( ) > Ia+ x t : g t g t g t x t dt, t a, G(g) a and the right derivative is given by: 1 Z b ( g,g )( ) = 0( )( ( ) − ( ))g−1 ( ) < Ib− x t : g t g t g t x t dt, t b. G(g) t Next, we present the definitions of the g-Riemann–Liouville fractional derivatives of a function x of order g. Definition 2. [17] Let g be a positive real, g : [a, b] ! R a Cn function with positive derivative, and x 2 L1([a, b], R). The left Riemann–Liouville fractional derivative of x of order g, with respect to the kernel g, is given by: g,g 1 d n n−g,g (D x)(t) := (I x)(t), t > a, a+ g0(t) dt a+ and the right derivative by: g,g 1 d n n−g,g (D x)(t) := − (I x)(t), t < b, b− g0(t) dt b− Symmetry 2021, 13, 592 3 of 17 where n = [g] + 1. Remark 1. It is easily seen that: 1. for certain choices of the kernel g, we recover well-known fractional derivatives, such as Riemann–Liouville (g(t) = t), Hadamard (g(t) = ln(t), a > 0), and Erdélyi–Kober fractional derivatives (g(t) = ts, s > 0); 2. if g = m 2 N, then: g,g 1 d m g,g 1 d m (D x)(t) = x(t) and (D x)(t) = − x(t). a+ g0(t) dt b− g0(t) dt Next, the concept of g-Caputo fractional derivatives of x of order g is presented, which is fundamental for the formulation of our problem. Definition 3. [18] Let g be a positive real and: [g] + 1 if g 2/ n = N g if g 2 N Let x, g be two real Cn functions defined on [a, b], where g satisfies g0(t) > 0. The left Caputo fractional derivative of x of order g, with respect to the kernel g, is defined as: g,g n−g,g 1 d n (C D x)(t) := I x (t), t > a, a+ a+ g0(t) dt and the right derivative as: g,g n−g,g 1 d n (C D x)(t) := I − x (t), t < b. b− b− g0(t) dt C g,g C g,g Remark 2. It is clear that if g is the identity, then Da+ x and Db− x are the usual Caputo fractional derivatives of x. Notice that if g = m 2 N, then: m g,g 1 d g,g 1 d m (C D x)(t) = x(t) and (C D x)(t) = − x(t). a+ g0(t) dt b− g0(t) dt Otherwise, 1 Z t 1 d n (C g,g )( ) = 0( )( ( ) − ( ))n−g−1 ( ) Da+ x t g t g t g t 0 x t dt G(n − g) a g (t) dt and: 1 Z b 1 d n (C g,g )( ) = 0( )( ( ) − ( ))n−g−1 − ( ) Db− x t g t g t g t 0 x t dt. G(n − g) t g (t) dt Since the integration by parts formula is of great importance in the calculus of varia- tions, we state here this basic result. Theorem 1. [18] Let x be a continuous function and y, g two Cn functions, with domain [a, b]. Then, Z b Z b x ( ) · (C g,g )( ) = ( ) · g,g ( ) 0( ) x t Da+ y t dt y t Db− 0 t g t dt a a g t=b "n−1 # 1 d k − x 1 d n−k−1 + − In g,g (t) · y(t) ∑ g0(t) dt b− g0 g0(t) dt k=0 t=a Symmetry 2021, 13, 592 4 of 17 and: Z b Z b x ( ) · (C g,g )( ) = ( ) · g,g ( ) 0( ) x t Db− y t dt y t Da+ 0 t g t dt a a g t=b "n−1 − − # 1 d k n−g,g x 1 d n k 1 + (−1)n−k I (t) · y(t) .