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This paper addresses the problem of designing secure control for networked multi-agent systems (MASs) under Denial-of-Service (DoS) attacks. We propose a constructive design method based on the interaction topology. The MAS with a non-attack communication topology, modeled by quasi-Abelian Cayley graphs subject to DoS attacks, can be represented as a switched system. Using switching theory, we provide easily implementable sufficient conditions for the networked MAS to remain asymptotically stable despite DoS attacks. Our results are applicable to both continuous-time and discrete-time systems, as well as to discrete-time systems with variable steps or systems that combine discrete and continuous times. [ABSTRACT FROM AUTHOR]

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 work offers a real-world application for the study of fuzzy dynamic equations. First, we propose the novel concept of granular delta differentiability for fuzzy-valued functions defined on time scales with the help of the relative distance measure fuzzy arithmetic and horizontal membership functions. Then, fundamental foundations of fuzzy calculus on time scales are provided. Discussion on the Malthusian model defined on particular time scales to illustrate the proposed approach is presented. [ABSTRACT FROM AUTHOR]

In this paper, the study of multiobjective variational problems with time delay is introduced. We provide methods for identifying Pareto optimal solutions. We also prove necessary and sufficient optimality conditions for the isoperimetric problem with multiple constraints and delayed arguments. [ABSTRACT FROM AUTHOR]

This paper deals with an opinion formation model, that obeys a nonlinear system of fractional-order differential equations. We introduce a virtual leader in order to attain a consensus. Sufficient conditions are established to ensure that the opinions of all agents globally asymptotically approach the opinion of the leader. We also address the problem of designing optimal control strategies for the leader so that the followers tend to consensus in the most efficient way. A variational integrator scheme is applied to solve the leader–follower optimal control problem. Finally, in order to verify the theoretical analysis, several particular examples are presented. [ABSTRACT FROM AUTHOR]

This paper deals with control strategies for discrete-time fractional multi-agent systems. By using the discrete fractional order operator we introduce memory effects to the considered problem. Necessary optimality conditions for discrete-time fractional optimal control problems with single- and double-summator dynamics are proved, and simulation results are presented. [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 paper, a leader-following consensus of discrete-time multi-agent systems with nonlinear intrinsic dynamics is investigated. We propose and prove conditions ensuring a leader-following consensus. Numerical examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider variational problems with delayed argument. Simple proofs of the first and second Noether theorems are given. Applying our results to some concrete problems we obtain conservation laws for systems with delay. [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]

We prove optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points. Problems of calculus of variations of this type cannot be solved using the classical theory. [ABSTRACT FROM AUTHOR]

We extend the second Noether theoremto variational problems on time scales. As corollarieswe obtain the classical second Noether theorem, the second Noether theorem for the h-calculus and the second Noether theorem for the q-calculus. [ABSTRACT FROM AUTHOR]

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 introduce differential, integral, and variational delta embeddings. We prove that the integral delta embedding of the Euler–Lagrange equations and the variational delta embedding coincide on an arbitrary time scale. In particular, a new coherent embedding for the discrete calculus of variations that is compatible with the least-action principle is obtained. [Copyright &y& Elsevier]

The calculus of variations on time scales is considered. We propose a new approach to the subject that consists of applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to the calculus of variations on time scales is very useful: it allows to unify the delta and nabla approaches previously considered in the literature. Generalized versions of the Euler–Lagrange necessary optimality conditions are obtained, both for the basic problem of the calculus of variations and isoperimetric problems. As particular cases one gets the recent delta and nabla results. [ABSTRACT FROM PUBLISHER]

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]

We present a unified treatment to control problems on an arbitrary time scale by introducing the study of forward-backward optimal control problems. Necessary optimality conditions for delta-nabla isoperimetric problems are proved, and previous results in the literature are obtained as particular cases. As an application of the results of the paper we give necessary and sufficient Pareto optimality conditions for delta-nabla bi-objective optimal control problems. [ABSTRACT FROM PUBLISHER]

In this work we study a quasilinear system of equations on time scales. Using the Krasnoselskii fixed-point theorem, sufficient conditions are given for the existence of positive solutions. [ABSTRACT FROM AUTHOR]

Abstract: We prove the Euler–Lagrange delta-differential equations for problems of the calculus of variations on arbitrary time scales with delta-integral functionals depending on higher-order delta derivatives. [ABSTRACT FROM AUTHOR]

Abstract: The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical calculus of variations as particular cases. In this note we follow Leitmann’s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale. [ABSTRACT FROM AUTHOR]

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]

We introduce the notion of strong local minimizer for the problems of the calculus of variations on time scales. Simple examples show that on a time scale a weak minimum is not necessarily a strong minimum. A time scale form of the Weierstrass necessary optimality condition is proved, which enables to include and generalize in the same result both continuous-time and discrete-time conditions [ABSTRACT FROM AUTHOR]

We address the problem of obtaining well-defined criteria for multiple criteria optimal control problems. Necessary and sufficient conditions for an objective functional to be nonessential are proved. The results provide effective tools for determining nonessential objectives in multiobjective optimal control problems. [ABSTRACT FROM AUTHOR]

We analyze a bounded confidence model, introduced by Krause, on isolated time scales. In this model, each agent takes into account only the assessments of the agents whose opinions are not too far away from its own opinion. We show that the behavior of the model depends strongly on the graininess function μ: If μ takes values in the interval ]0,1], then our discrete time scale model behaves similarly to the classical one, but if μ takes values in ]1,+ ∞[, then the model has different properties. Simulations are performed to validate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd. [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]

The aim of this paper is to analyze the behavior of Hegselmann‐Krause–type models. General sufficient conditions for achieving consensus are provided. Model predictive control is applied to the Hegselmann‐Krause system in the context of time scales. Numerical simulations show that the predictive control mechanism has the ability to steer the system to attain a consensus. Moreover, faster consensus speed is observed when compared with the classical model. Finally, in the case of the discrete time scale and all agents being neighbors at the initial time, conditions guaranteeing an average consensus are given. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze the behavior of Hegselmann‐Krause–type models. General sufficient conditions for achieving consensus are provided. Model predictive control is applied to the Hegselmann‐Krause system in the context of time scales. Numerical simulations show that the predictive control mechanism has the ability to steer the system to attain a consensus. Moreover, faster consensus speed is observed when compared with the classical model. Finally, in the case of the discrete time scale and all agents being neighbors at the initial time, conditions guaranteeing an average consensus are given. [ABSTRACT FROM AUTHOR]

A leader-following consensus for Caputo fractional multi-agent systems with nonlinear intrinsic dynamics is investigated. The second Lyapunov method is used to design a control protocol ensuring a consensus for two types of multi-agent systems. Contrary to the previous studies on leader-following consensus, the investigation covers systems with bounded and unbounded time-dependent Lipschitz coefficients in the intrinsic dynamics. Moreover, coupling strength describing the interactions between agents is considered to be a function of time. [ABSTRACT FROM AUTHOR]