Differential Inclusion Theory Research Articles

A Distributed k-Winners-Take-All Model With Binary Consensus Protocols.

This article concentrates on solving the k -winners-take-all (k WTA) problem with large-scale inputs in a distributed setting. We propose a multiagent system with a relatively simple structure, in which each agent is equipped with a 1-D system and interacts with others via binary consensus protocols. That is, only the signs of the relative state information between neighbors are required. By virtue of differential inclusion theory, we prove that the system converges from arbitrary initial states. In addition, we derive the convergence rate as O(1/t) . Furthermore, in comparison to the existing models, we introduce a novel comparison filter to eliminate the resolution ratio requirement on the input signal, that is, the difference between the k th and (k+1) th largest inputs must be larger than a positive threshold. As a result, the proposed distributed k WTA model is capable of solving the k WTA problem, even when more than two elements of the input signal share the same value. Finally, we validate the effectiveness of the theoretical results through two simulation examples.

New results on synchronization control of memristor-based quaternion-valued fuzzy neural networks with delayed impulses

A model of memristor-based quaternion-valued fuzzy neural networks (MQVFNNs) with delayed impulses and time-varying delays is established, and the synchronization issue is investigated in this paper. Firstly, to avoid non-commutativity, the MQVFNNs are decomposed into real-valued delayed impulsive neural networks. Secondly, a novel lemma for delayed impulsive system is introduced based on Halanay differential inequality. Then, by employing the theory of differential inclusion and the proposed lemma, some sufficient conditions are deduced to guarantee the global exponential synchronization of memristive quaternion-valued fuzzy neural networks under a simple delayed nonlinear control scheme. Finally, a numerical simulation is given to demonstrate the effectiveness of our theoretical results.

Fixed-time synchronization of discontinuous fuzzy competitive neural networks via quantized control

This article concentrates on fixed-time synchronization for discontinuous fuzzy competitive neural networks including time-varying delays. Firstly, to cut down the signal transmission cost, two kinds of nontrivial control strategies, namely quantized control protocols without the linear part, are developed to investigate fixed-time synchronization of discontinuous fuzzy competitive neural networks based on two different types of Lyapunov functions. Then, with the aid of inequality skills and differential inclusion theory, several uncomplicated synchronization criteria are attained and the synchronized time possessing less conservativeness is effectively evaluated. A numerical example is presented at last to testify the validity of theoretical analysis.

Finite-time synchronization of discontinuous network systems via two control schemes

This article considers network systems (NSs) which can realize random switch among different NSs based on Bernoulli random variables. As a result, the systems will be the networks with linear coupling, nonlinear coupling or partial communication failure. Discontinuous activation function is introduced, which is overcome by the theories of differential inclusions. Finite-time (FET) synchronization is established under the designed intermittent adaptive control and the reason why it cannot get the results of prespecified-time (PDT) synchronization is analysed, where PDT synchronization is a special case of FET synchronization. Moreover, PDT synchronization is realized via a feedback controller which switches based on the state information of systems. In addition, simulation experiments are provided to verify the control methods and synchronization results.

Synchronization robustness analysis of memristive-based neural networks with deviating arguments and stochastic perturbations

<abstract><p>In this article, we investigate the robustness of memristive-based neural networks (MNNs) with deviating arguments (DAs) and stochastic perturbations (SPs). Based on the set-valued mapping method, differential inclusion theory and Gronwall inequalities, we derive the upper bounds for the width of DAs and the intensity of SPs. When the DAs and SPs are smaller than these upper bounds, the MNNs maintains exponential synchronization. Finally, several specific simulation examples demonstrate the effectiveness of the results.</p></abstract>

Periodic dynamics of predator-prey system with Beddington–DeAngelis functional response and discontinuous harvesting

This paper investigates a delayed predator-prey model with discontinuous harvesting and Beddington–DeAngelis functional response. Using the theory of differential inclusion theory, the existence of positive solutions in the sense of Filippov is discussed. Under reasonable assumptions and periodic disturbances, the existence of positive periodic solutions of the model is studied based on the theory of Mawhin’s coincidence degree. Finally, through numerical simulation, the correctness and feasibility of the conclusions are verified.

Finite-Time and Fixed-Time Synchronization of Delayed Memristive Neural Networks via Adaptive Aperiodically Intermittent Adjustment Strategy.

This article investigates the finite-time and fixed-time synchronization for memristive neural networks (MNNs) with mixed time-varying delays under the adaptive aperiodically intermittent adjustment strategy. Different from previous works, this article first employs the aperiodically intermittent adjustment feedback control and adaptive control to drive the MNNs to achieve synchronization in finite time and fixed time. First of all, according to the theories of set-valued mappings and differential inclusions, the error MNNs is derived, and its finite-time and fixed-time stability problems are discussed by applying the Lyapunov function method and some LMI techniques. Moreover, by meticulously designing an effective aperiodically intermittent adjustment with adaptive updating law, sufficient conditions that guarantee the finite-time and fixed-time synchronization of the drive-response MNNs are obtained, and the settling time is explicitly estimated. Finally, three numerical examples are provided to illustrate the validity of the obtained theoretical results.

Output-Feedback Adaptive Neural Network Control for Uncertain Nonsmooth Nonlinear Systems With Input Deadzone and Saturation.

Nonsmooth nonlinear systems can model many practical processes with discontinuous property and are difficult to be stabilized by classical control methods like smooth nonlinear systems. This article considers the output-feedback adaptive neural network (NN) control problem for nonsmooth nonlinear systems with input deadzone and saturation. First, the nonsmooth input deadzone and saturation is converted to a smooth function of affine form with bounded estimation error by means of the mean-value theorem. Second, with the help of approximation theorem and Filippov's differential inclusion theory, the given nonsmooth system is converted to an equivalent smooth system model. Then, by introducing a proper logarithmic barrier Lyapunov function (BLF), an output-feedback adaptive NN strategy is set up by constructing an appropriate observer and adopting the adaptive backstepping technique. A new stability criterion is established to guarantee that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Finally, comparative simulations through Chua's oscillator are offered to verify the effectiveness of the proposed control algorithm.

Finite/fixed-time synchronization of inertial memristive neural networks by interval matrix method for secure communication

This paper investigates the finite/fixed-time synchronization problem of delayed inertial memristive neural networks (DIMNNs) using interval matrix-based methods within a unified control framework. By employing set-valued mapping and differential inclusion theory, two distinct methods are applied to handle the switching behavior of memristor parameters: the maximum absolute value method and the interval matrix method. Based on these different approaches, two control strategies are proposed to select appropriate control parameters, enabling the system to achieve finite and fixed-time synchronization, respectively. Additionally, the resulting theoretical criteria differ based on the chosen control strategy, with one expressed in algebraic form and the other in the form of linear matrix inequalities (LMIs). Numerical simulations demonstrate that the interval matrix method outperforms the maximum absolute value method in terms of handling memristor parameter switching, achieving faster finite/fixed-time synchronization. Furthermore, the theoretical results are extended to the field of image encryption, where the response system is utilized for decryption and expanding the keyspace.

Global Dissipativity and Quasi-Mittag-Leffler Synchronization of Fractional-Order Discontinuous Complex-Valued Neural Networks.

This article is concerned with fractional-order discontinuous complex-valued neural networks (FODCNNs). Based on a new fractional-order inequality, such system is analyzed as a compact entirety without any decomposition in the complex domain which is different from a common method in almost all literature. First, the existence of global Filippov solution is given in the complex domain on the basis of the theories of vector norm and fractional calculus. Successively, by virtue of the nonsmooth analysis and differential inclusion theory, some sufficient conditions are developed to guarantee the global dissipativity and quasi-Mittag-Leffler synchronization of FODCNNs. Furthermore, the error bounds of quasi-Mittag-Leffler synchronization are estimated without reference to the initial values. Especially, our results include some existing integer-order and fractional-order ones as special cases. Finally, numerical examples are given to show the effectiveness of the obtained theories.

Finite-Time Multiparty Synchronization of T–S Fuzzy Coupled Memristive Neural Networks With Optimal Event-Triggered Control

This paper considers multiparty synchronization (MS) for coupled memristive neural networks (CMNNs) with a time delay. Some T-S type If-Then fuzzy rules are also introduced into the CMNNs. An event-triggered controller (ETC) is designed to achieve the MS in finite time, which avoids continuous control signals. Along with Lyapunov theory, differential inclusion theory, and inequalities, some criteria can be obtained to achieve the finite-time MS (FTMS), and the setting time (ST) of the FTMS can be calculated. By jointly considering ST, control inputs, error networks, and synchronization conditions, an optimization model is provided to get an optimal ETC (OETC). Further, the particle swarm optimization (PSO) algorithm is utilized to solve the optimization model. Thus, it gives a method to choose control gains and parameters of an event-triggered function (ETF). Finally, two examples are given to verify the theoretical results. Especially, two comparative experiments are proposed to demonstrate that the OETC can save more control energy and reduce the number of triggered times.

Asymptotic and Pinning Synchronization of Fractional-Order Nonidentical Complex Dynamical Networks with Uncertain Parameters

This paper is concerned with the asymptotic and pinning synchronization of fractional-order nonidentical complex dynamical networks with uncertain parameters (FONCDNUP). First of all, some synchronization criteria of FONCDNUP are proposed by using the stability of fractional-order dynamical systems and inequality theory. Moreover, a novel controller is derived by using the Lyapunov direct method and the differential inclusion theory. Next, based on the Lyapunov stability theory and pinning control techniques, a new group of sufficient conditions to assure the synchronization for FONCDNUP are obtained by adding controllers to the sub-nodes of networks. At last, two numerical simulations are utilized to illustrate the validity and rationality of the acquired results.

The pth Moment Exponential Synchronization of Drive-Response Memristor Neural Networks Subject to Stochastic Perturbations

In this paper, the p t h moment exponential synchronization problems of drive-response stochastic memristor neural networks are studied via a state feedback controller. The dynamics of the memristor neural network are nonidentical, consisting of both asymmetrically nondelayed and delayed coupled, state-dependent, and subject to exogenous stochastic perturbations. The pth moment exponential synchronization of these drive-response stochastic memristor neural networks is guaranteed under some testable and computable sufficient conditions utilizing differential inclusion theory and Filippov regularization. Finally, the correctness and effectiveness of our theoretical results are demonstrated through a numerical example.

Fixed-time periodic stabilization of discontinuous reaction–diffusion Cohen–Grossberg neural networks

This paper aims to study the fixed-time stabilization of a class of delayed discontinuous reaction–diffusion Cohen–Grossberg neural networks. Firstly, by providing some relaxed conditions containing indefinite functions and based on inequality techniques, a new fixed-time stability lemma is given, which can improve the traditional ones. Secondly, based on state-dependent switching laws, the periodic wave solution of the formulated networks is transformed into the periodic solution of ordinary differential system. By utilizing differential inclusions theory and coincidence theorem, the existence of periodic solutions is obtained. Thirdly, based on the new fixed-time stability lemma, the periodic solutions are stabilized at zero in a fixed-time, which is a new topic on reaction–diffusion networks. Moreover, the established criteria are all delay-dependent, which are less conservative than the previous delay-independent ones for ensuring the stabilization of delayed reaction–diffusion networks. Finally, two examples give numerical explanations of the proposed results and highlight the influence of delays.

Multistability of switched complex-valued neural networks with state-dependent switching rules

The paper focuses on the multistability problems of the switched complex-valued neural networks with state-dependent switching rules. Based on the differential inclusions theory and fixed point theorem, several sufficient conditions are derived to ascertain that there exist 25n equilibria, 9n of which are locally ex for n-neuron switched complex-valued neural networks. The number of stable equilibria of an n-neuron switched complex-valued neural network increases significantly from 4n to 9n compared with the conventional complex-valued neural networks. Finally, four numerical examples are presented to substantiate the theoretical results.

An Improved Finite-Time and Fixed-Time Stable Synchronization of Coupled Discontinuous Neural Networks.

This article focuses on the finite-time and fixed-time synchronization of a class of coupled discontinuous neural networks, which can be viewed as a combination of the Hindmarsh-Rose model and the Kuramoto model. To this end, under the framework of Filippov solution, a new finite-time and fixed-time stable theorem is established for nonlinear systems whose right-hand sides may be discontinuous. Moreover, the high-precise settling time is given. Furthermore, by designing a discontinuous control law and using the theory of differential inclusions, some new sufficient conditions are derived to guarantee the synchronization of the addressed coupled networks achieved within a finite-time or fixed-time. These interesting results can be seemed as the supplement and expansion of the previous references. Finally, the derived theoretical results are supported by examples with numerical simulations.

Finite-time synchronization of discontinuous fuzzy neural networks with mixed time-varying delays and impulsive disturbances

In this paper, the finite-time synchronization issue is addressed for discontinuous fuzzy neural networks with mixed time-varying delays and impulsive disturbances. The dynamic impulsive gain about the impulsive moment is proposed as an alternative to the previous static impulsive gain. The time-varying impulsive gain can combine the synchronizing and desynchronizing impulses to directly observe their effects on the system, avoiding the classification discussion. In order to study the finite-time synchronization of the drive–response system, we obtain an equivalent system from the original fuzzy neural networks based on the concepts of Filippov solutions and differential inclusion theory. Furthermore, by constructing new Lyapunov functions that combine the 1-norm and the comparative system method, some algebraic sufficient conditions for the synchronization of the drive system and the response system in finite time are established, which are based on two types of novel non-chattering controllers. In addition, the parameters of the system itself can simply demonstrate the sufficient conditions that have been obtained her e, avoiding the time-consuming calculation of matrix inequalities. Moreover, the settling time estimation scheme have been established. Eventually, numerical examples are provided to demonstrate the effectiveness of the control scheme.

Finite-Time Synchronization of Neural Networks With Infinite Discrete Time-Varying Delays and Discontinuous Activations.

This article investigates finite-time synchronization of neural networks (NNs) with infinite discrete time-varying delays and discontinuous activations (DDNNs). By virtue of theory of differential inclusions, comparison strategies, and inequality techniques, finite-time synchronization of the underlying DDNNs can be developed via a discontinuous state feedback control law, and the synchronous settling time can be estimated. The delayed state feedback controller and finite-time stability theorem are not employed during the analysis. As a special case, finite-time synchronization of NNs with bounded delays and discontinuous activations is given. Finally, two examples are provided to illustrate the validity of the theories.

Finite-time synchronization of fractional-order delayed memristive fuzzy neural networks

This paper deals with the finite-time synchronization (FTS) for a class of fractional-order memristive fuzzy neural networks (FOMFNNs) with leakage and transmission delays. Two types of discontinuous control schemes with leakage and transmission delays are designed: one is the state feedback control and the other is the fractional-order adaptive control. Based on differential inclusion theory and fractional-order differential inequalities, several new delay-independent algebraic conditions are derived to guarantee the FTS of drive-response FOMFNNs with leakage and transmission delays. Moreover, the estimation of the upper bound of the settling time for the FTS is given as well. Ultimately, two numerical examples with simulations are furnished to demonstrate the feasibility and availability of the theoretical results.

Global asymptotic synchronization of inertial memristive Cohen–Grossberg neural networks with proportional delays

In this article, the global asymptotic synchronization (GAS) using inertial memristive Cohen-Grossberg neural networks (IMCGNNs) with proportional delays (PDs) as drive–response systems is studied. By utilizing differential inclusion theory (DIT) and appropriate variable transformation, the studied systems can be converted to first-order differential systems. Both the feedback controller and the adaptive controller are designed, which are easy to implement in hardware. By constructing Lyapunov functionals and using mean value inequality analysis skills, two GAS criteria of the studied systems are gained, which are embodied in the form of algebraic inequalities and are easy to verify. Ultimately, the numerical examples with simulations are applied to sustain the obtained consequences.