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This paper deals with the problems of quasi‐projective synchronization (QPS) and finite‐time synchronization (FTS) for a kind of delayed fractional‐order BAM neural networks (DFOBAMNNs). In order to reach the goals of synchronization and more accurately gauge of settling time and error level, several fresh quantized controllers are structured to make the utmost of confined communication resources. Then, based on the finite‐time theorem, quantized control strategy, Lyapunov function theory and properties of Mittag–Leffler function as well as inequality analysis techniques, some plentiful criteria are formed to set up a relation between control gains and quantization parameters. In addition, the corresponding error bound of QPS and guages of the settling time on FTS are also given. Finally, a few numerical examples are introduced to validate the effectiveness of the presented control protocols. [ABSTRACT FROM AUTHOR]

In this paper, without transforming the quaternion-valued neural networks into two equivalent complex-valued systems or four equivalent real-valued systems, a novel class of fractional-order quaternion-valued neural networks with leakage and discrete delays is investigated. To this end, two novel inequalities are established with the aid of properties of quaternion and Caputo fractional derivative. By exploiting Lyapunov method, our established inequalities, fractional-order Razumikhin theorem and some analysis techniques, some criteria ensuring the global asymptotical synchronization and the global Mittag-Leffler synchronization of the considered networks are obtained through designing suitable controllers. Finally, two numerical examples are utilized to show the derived theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, spatial diffusions are introduced to fractional-order coupled networks and the problem of synchronization is investigated for fractional-order coupled neural networks with reaction-diffusion terms. First, a new fractional-order inequality is established based on the Caputo partial fractional derivative. To realize asymptotical synchronization, two types of adaptive coupling weights are considered, namely: 1) coupling weights only related to time and 2) coupling weights dependent on both time and space. For each type of coupling weights, based on local information of the node’s dynamics, an edge-based fractional-order adaptive law and an edge-based fractional-order pinning adaptive scheme are proposed. Furthermore, some new analytical tools, including the method of contradiction, L’Hopital rule, and Barbalat lemma are developed to establish adaptive synchronization criteria of the addressed networks. Finally, an example with numerical simulations is provided to illustrate the validity and effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the leader-following consensus problem of fractional-order multi-agent systems is considered via adaptive pinning control. The dynamics of leader and all followers with linear and nonlinear functions are investigated, respectively. We assume that the node should be pinned if its in-degree is less than its out-degree in the paper. Under this assumption and based on the stability theory of fractional-order differential systems, some leader-following consensus criteria are derived, which are easily obtained by matrix inequalities. The control of each agent using local information is designed and detailed analysis of the leader-following consensus is presented. The design technique is based on algebraic graph theory and the Riccati inequality. Several simulation examples are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM PUBLISHER]

The leader-following consensus problem of fractional-order multi-agent systems is considered. In the system, the dynamics of each agent and leader are nonlinear systems. The control of each agent using local information is designed and detailed analysis of the leader-following consensus is presented. The design technique is based on algebraic graph theory and Lyapunov method. Several simulation examples are presented as a proof of concept. [ABSTRACT FROM AUTHOR]

Abstract This paper addresses the problem with respect to finite-time (FT) synchronization for fractional-order (FO) complex networks. Based on hybrid feedback control technique, Lyapunov approach, and some novel analysis techniques of fractional calculation, some sufficient conditions are obtained to guarantee FT synchronization, and the estimation bound of the setting time for synchronization is given. It is meaningful to find that the setting time is dependent on fractional order, controller parameters and initial value of networks. At last, numerical simulations are presented to illustrate the effectiveness and applicability of our proposed control method and verify our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating discrete-time fractional-order delayed quaternion-valued neural networks (DFDQNNs) by utilizing direct quaternion approach. Firstly, a novel lemma and its two corresponding corollaries have been proposed for estimating nabla fractional difference of the quaternion-valued Lyapunov function. Then, the existence and uniqueness of equilibrium point for DFDQNNs is proved by constructing a new quaternion-valued contraction mapping. In addition, by means of our designed Lyapunov functions and the effective feedback controller as well as neoteric nabla difference inequalities, some sufficient criteria have been obtained to ensure the global Mittag–Leffler stability and Mittag–Leffler synchronization of DFDQNNs, respectively. Finally, some numerical examples are provided to verify the yielded results. [ABSTRACT FROM AUTHOR]

The issues of complete synchronization (CS) and finite-time synchronization (F-TS) for a class of fractional-order fuzzy neural networks are addressed based on nonlinear feedback control in this paper. First, a fractional-order finite-time convergence principle is established by virtue of fractional calculus basic theory and reduction to absurdity. Next, two novel nonlinear controllers, namely the adaptive nonlinear controller and discontinuous nonlinear controller, are designed. Then some easily validated criteria to guarantee CS and F-TS are derived with the help of some useful analysis techniques and our newly established convergence principle. Moreover, the settling time of F-TS is effectively estimated. Finally, some numerical results are presented to show the validity of derived theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the global projective synchronization of fractional-order neural networks is investigated. First, a sufficient condition in the sense of Caputo’s fractional derivation to ensure the monotonicity of the continuous and differential functions and a new fractional-order differential inequality are derived, which play central roles in the investigation of the fractional adaptive control. Based on the preparation and some analysis techniques, some novel criteria are obtained to realize projective synchronization of fractional-order neural networks via combining open loop control and adaptive control. As some special cases, several control strategies are given to ensure the realization of complete synchronization, anti-synchronization and the stabilization of the addressed neural networks. Finally, an example with numerical simulations is given to show the effectiveness of the obtained results. [Copyright &y& Elsevier]

Abstract: In this paper, a class of fractional-order neural networks is investigated. First, -exponential stability is introduced as a new type of stability and some effective criteria are derived for such kind of stability of the addressed networks by handling a new fractional-order differential inequality. Based on the results, the existence and -exponential stability of the equilibrium point are considered. Besides, the synchronization of fractional chaotic networks is also proposed. Finally, several examples with numerical simulations are given to show the effectiveness of the obtained results. [Copyright &y& Elsevier]

In this paper, the problem of synchronization for a class of fractional-order delayed spatio-temporal neural networks with Dirichlet boundary conditions is investigated. Firstly, based on the L'Hopital law, a Caputo fractional partial differential inequality with p -norm is established to improve the previous results. Besides, a generalized fractional-order Halanay inequality is introduced by rigorous theoretical deduction. Additionally, to realize synchronization, two different control schemes are designed for the networks, respectively, and several criteria are derived based on the developed fractional inequalities. Finally, numerical examples and simulations are provided to verify the correctness of the results. [ABSTRACT FROM AUTHOR]

In this letter, we indicate that the conditions in Lemma 2 in Yu et al. (2014) are not sufficient. To correct the mistake, a new version is given, which provides a sufficient and necessary condition in the sense of Caputo’s fractional derivation to verify the monotonicity of the addressed function. Furthermore, a new fractional-order differential inequality is derived, which shows that the main results in Yu et al. (2014) are still true although Lemma 2 has to be modified. [ABSTRACT FROM AUTHOR]

This paper is committed to explore synchronization issues for a class of nabla fractional-order fuzzy neural networks with time delays. First, we establish two useful nabla fractional inequalities by feat of the theory of nabla fractional calculus and N -transform together with nabla convolution. Next, we design two distinctive controllers including nonlinear delay feedback controller and hybrid nonlinear delay controller, some easily verified quasi-synchronization (QS) criteria and complete synchronization (CS) criteria are derived by virtue of newly proposed inequalities therein and some inequality techniques. Finally, the correctness of theoretical results is indicated finally through numerical simulations. What is more important, theoretical results therein can offer some new avenues about QS and CS analysis for nabla fractional dynamical networks. [ABSTRACT FROM AUTHOR]

In this paper, under the framework of Filippov solutions, quasi-synchronization issue of fractional-order complex-valued neural networks (FCNNs) with discontinuous activations and uncertainties is investigated. Firstly, using Laplace transform and the property of Mittag-Leffler function, a novel fractional differential inequality is derived. Then combining the newly constructed inequality with delay feedback control scheme, several quasi-synchronization conditions are obtained for the considered FCNNs by means of non-decomposable method. Eventually, a numerical example is provided to substantiate validation of the proposed results. [ABSTRACT FROM AUTHOR]

This paper focuses on finite-time synchronization and parameter identification (FTSPI) problem of uncertain fractional-order complex dynamical networks. By utilizing the properties of fractional calculus and some analysis techniques, several sufficient conditions are given to guarantee FTSPI via designing suitable controller and parameter update law. In addition, the estimation bound of the setting time is also given. Finally, numerical simulations are provided to illustrate the effectiveness of our obtained theoretical results. • The novel controller and parameter identification law are designed. • Several simple FTSPI criteria are obtained. • The estimation bound of the setting time is explicitly evaluated. [ABSTRACT FROM AUTHOR]

In this paper, without separating complex-valued neural networks into two real-valued systems, the finite-time synchronization is addressed for a class of fully complex-valued neural networks with fractional-order. Firstly, a new fractional-order differential inequality is established to improve some existing results in the real domain. Besides, to avoid the traditional separation method, the sign function of complex numbers is proposed and some properties about it are derived. Under the proposed sign function framework, by designing some novel and effective control schemes, constructing nontrivial Lyapunov functions and developing some new inequality methods in complex domain, several criteria of finite-time synchronization are derived and the settling-time of synchronization is effectively estimated. Finally, the effectiveness of the theoretical results is demonstrated by some numerical examples. [ABSTRACT FROM AUTHOR]

This paper studies quasi-projective synchronization (QPS) and complete synchronization (CS) for a class of fractional-order complex-valued neural networks with time delays by designing suitable controllers. To realize QPS and CS, linear feedback controller and adaptive controller are designed, and a novel fractional-order differential inequality is built by means of Laplace transform and properties of Mittag-Leffler function. By utilizing Lyapunov method, our proposed inequality, fractional-order Razumikhin theorem and some complex analysis techniques, some effective criteria are derived to ensure QPS and CS of the considered networks. Furthermore, the error bound of QPS is obtained. Finally, some numerical results are given to demonstrate the effectiveness of the presented theoretical results. [ABSTRACT FROM AUTHOR]

Inspired by Lemma 1 in [36], a new fractional differential inequality is derived by means of inequality techniques and reduction to absurdity. • Without the participation of sign function, two kinds of finite-time control strategies are designed. • Non-separation method is first employed to explore the robust F-TS for the uncertain F-OQ-VNNs. • Easily-verified robust F-TS criteria are derived, and settling time is explicitly reckoned. In this paper, robust finite-time synchronization (F-TS) issue is addressed for a class of uncertain fractional-order quaternion-valued neural networks by employing non-separation method instead of separation method. First, a general fractional differential inequality is developed to provide new insight into the research about finite-time stability and synchronization of fractional-order systems. Next, quaternion-valued feedback controller and quaternion-valued adaptive controller are designed. On the basis of the newly developed inequality, quaternion inequality techniques, together with the properties of fractional calculus and reduction to absurdity, some easily-verified algebraic criteria for robust F-TS are established, and the settling time for robust F-TS is explicitly reckoned, which depends on not only the controller parameters but also the initial values and order of the considered systems. Eventually, numerical results are provided to substantiate our robust F-TS criteria. [ABSTRACT FROM AUTHOR]