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Systems of fractional differential equations with a general form of fractional derivative are considered. A unique continuous solution is derived using the Banach fixed point theorem. Additionally, the dependence of the solution on the fractional order and on the initial conditions are studied. Then the stability of autonomous linear fractional differential systems with order 0<α<1 of the ψ‐Caputo derivative is investigated. Finally, an application of the theoretical results to the problem of the leader‐follower consensus for fractional multi‐agent systems is presented. Sufficient conditions are given to ensure that the tracking errors asymptotically converge to zero. The results of the paper are illustrated by some examples. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the initial value problem of nonlinear fractional differential equations involving a Caputo-type fractional derivative with respect to another function. Existence and uniqueness results for the problem are established by means of the some standard fixed point theorems. Next, we develop the Picard iteration method for solving numerically the problem and obtain results on the long-term behavior of solutions. Finally, we analyze a population growth model and a gross domestic product model with governing equations being fractional differential equations that we have introduced in this work. [ABSTRACT FROM AUTHOR]

In this note we present a counterexample to Theorem 24 in G.S.F. Frederico and D.F.M. Torres (2010) [3] . [ABSTRACT FROM AUTHOR]

We extend the second Noether theorem to fractional variational problems which are invariant under infinitesimal transformations that depend upon r arbitrary functions and their fractional derivatives in the sense of Caputo. Our main result is illustrated using the fractional Lagrangian density of the electromagnetic field. [ABSTRACT FROM PUBLISHER]

Abstract: We study operators that are generalizations of the classical Riemann–Liouville fractional integral, and of the Riemann–Liouville and Caputo fractional derivatives. A useful formula relating the generalized fractional derivatives is proved, as well as three relations of fractional integration by parts that change the parameter set of the given operator into its dual. Such results are explored in the context of dynamic optimization, by considering problems of the calculus of variations with general fractional operators. Necessary optimality conditions of Euler–Lagrange type and natural boundary conditions for unconstrained and constrained problems are investigated. Interesting results are obtained even in the particular case when the generalized operators are reduced to be the standard fractional derivatives in the sense of Riemann–Liouville or Caputo. As an application we provide a class of variational problems with an arbitrary kernel that give answer to the important coherence embedding problem. Illustrative optimization problems are considered. [Copyright &y& Elsevier]

Abstract: This paper presents the Euler–Lagrange equations for fractional variational problems with multiple integrals. The fractional Noether-type theorem for conservative and nonconservative generalized physical systems is proved. Our approach uses well-known notion of the Riemann–Liouville fractional derivative. [Copyright &y& Elsevier]

Abstract: We give a proper fractional extension of the classical calculus of variations by considering variational functionals with a Lagrangian depending on a combined Caputo fractional derivative and the classical derivative. Euler–Lagrange equations to the basic and isoperimetric problems as well as transversality conditions are proved. [Copyright &y& Elsevier]

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed. [ABSTRACT FROM AUTHOR]

Abstract: The study of fractional variational problems in terms of a combined fractional Caputo derivative is introduced. Necessary optimality conditions of Euler–Lagrange type for the basic, isoperimetric, and Lagrange variational problems are proved, as well as transversality and sufficient optimality conditions. This allows to obtain necessary and sufficient Pareto optimality conditions for multiobjective fractional variational problems. [Copyright &y& Elsevier]

Abstract: This paper presents the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrangian depending on the free end-points. The fractional derivatives are defined in the sense of Caputo. [Copyright &y& Elsevier]

We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann–Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler–Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string. [ABSTRACT FROM AUTHOR]

Abstract: This article is devoted to the regular fractional Sturm–Liouville eigenvalue problem. By applying the methods of fractional variational analysis, we prove the existence of a countable set of orthogonal solutions and corresponding eigenvalues. Moreover, we formulate two results showing that the lowest eigenvalue is the minimum value for a certain variational functional. [Copyright &y& Elsevier]

This paper deals with a control strategy enforcing consensus in a fractional opinion formation model with leadership, where the interaction rates between followers and the influence rate of the leader are functions of deviations of opinions between agents. The fractional-order derivative determines the impact of the memory during the opinion evolution. The problem of leader-following consensus control is cast in the framework of nonlinear optimal control theory. We study a finite horizon optimal control problem, in which deviations of opinions between agents and with respect to the leader are penalized along with the control that is applied only to the leader. The existence conditions for optimal consensus control are proved and necessary optimality conditions for the considered problem are derived. The results of the paper are illustrated by some examples. [ABSTRACT FROM AUTHOR]