Differential Inclusion Theory Research Articles

Finite-/fixed-time synchronization for Cohen-Grossberg neural networks with discontinuous or continuous activations via periodically switching control.

This paper is concerned with finite-/fixed-time synchronization for a class of Cohen-Grossberg neural networks with discontinuous or continuous activations and mixed time delays. Based on the finite-time stability theory, Lyapunov stability theory, the concept of Filippov solution and the differential inclusion theory, some useful finite-/fixed-time synchronization sufficient conditions for the considered Cohen-Grossberg neural networks are established by designing two kinds of novel periodically switching controllers. Instead of using uninterrupted high control strength, the periodically switching controller in each period is used with high strength control in one stage and weak strength in the other. It can overcome the effects caused by the uncertainties of Filippov solution induced by discontinuous neuron activation functions and reduce the control cost. Besides, the period switching control rate is closely related to the settling time T. Finally, two numerical examples are given to demonstrate the effectiveness and feasibility of the obtained results.

Quantized intermittent control tactics for exponential synchronization of quaternion-valued memristive delayed neural networks

This article studies the global exponential synchronization (GES) of quaternion-valued memristive delayed neural networks (QVMDNNs) by quantized intermittent control (QIC). Without decomposing the original systems into usual real-valued or complex-valued ones, the discussed system is directly processed in both cases of the differential inclusion theory. Based on two QIC strategies and applying Lyapunov functional method and inequality techniques, several new delay-dependent criteria in the form of real-valued algebraic inequalities are directly derived to assure the GES of the concerned system. Compared to the traditional feedback control, the QIC strategy can reduce the control expenses and shorten time to achieve the GES. Ultimately, two examples are given to validate the effectiveness and advantages of the proposed outcomes.

Fixed-Time Synchronization of Delayed Memristive Neural Networks with Discontinuous Activations

In this paper, the fixed-time synchronization problem for a class of memristive neural networks with discontinuous neuron activation functions and mixed time-varying delays is investigated. With the help of the fixed-time stability theory, under the framework of Filippov solution and differential inclusion theory, several new and useful sufficient criteria for fixed-time synchronization are obtained by designing two types of energy-saving and simple controllers for the considered systems. Compared with the traditional fixed-time synchronization controller, the controllers used in this paper only have one power exponent term, which is a function of the system state error rather than a constant. Moreover, some previous relevant works are especially improved. Finally, two numerical examples are given to show the correctness and the effectiveness of the obtained theoretical results.

Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications

Abstract This paper is concerned with the existence and global exponential stability of the periodic solution of delayed Cohen–Grossberg neural networks (CGNNs) with discontinuous activation functions. The activations considered herein are non-decreasing but not required to be Lipschitz or continuous. Based on differential inclusion theory, Lyapunov functional theory and Leary–Schauder alternative theorem, some sufficient criteria are derived to ensure the existence and global exponential stability of the periodic solution. In order to show the superiority of the obtained results, an application and some detailed comparisons between some existing related results and our results are presented. Finally, some numerical examples are also illustrated.

Novel Criteria of Stability for Delayed Memristive Quaternionic Neural Networks: Directly Quaternionic Method

In this paper, we fixate on the stability of varying-time delayed memristive quaternionic neural networks (MQNNs). With the help of the closure of the convex hull of a set the theory of differential inclusion, MQNN are transformed into variable coefficient continuous quaternionic neural networks (QNNs). The existence and uniqueness of the equilibrium solution (ES) for MQNN are concluded by exploiting the fixed-point theorem. Then a derivative formula of the quaternionic function’s norm is received. By utilizing the formula, the M-matrix theory, and the inequality techniques, some algebraic standards are gained to affirm the global exponential stability (GES) of the ES for the MQNN. Notably, compared to the existing work on QNN, our direct quaternionic method operates QNN as a whole and markedly reduces computing complexity and the gained results are more apt to be verified. The two numerical simulation instances are provided to evidence the merits of the theoretical results.

Improved fixed‐time stability results and application to synchronization of discontinuous neural networks with state‐dependent switching

AbstractIn this article, we considered the fixed‐time stability (FXT) of a type of discontinuous dynamical system. First, we introduced some new FXT stability results via variable substitutions and using inequality techniques, which are more accurately estimate the upper bound of settling time (ST). Then, based on the established results and using the differential inclusion theory, we investigated the FXT synchronization of a type of general neural networks (NNs) with state‐dependent switching coefficients and discontinuous neuron activation functions via introducing a type of discontinuous controller which is more simple compared to existing results. In addition, the estimated upper bound of ST shown to be independent to the initial values of considered drive‐response networks, and it is much smaller and nearer to the real synchronization time than those given in the previously published works. Lastly, two numerical examples are given to demonstrate the feasibility of our derived theoretical results. We believe that the results of this article can give some new insights for the FXT stabilization and FXT synchronization analysis of state‐dependent switching dynamics with or without discontinuous right‐hand sides.

Global exponential stability analysis of anti-periodic of discontinuous BAM neural networks with time-varying delays

Discontinuous system is playing an increasingly important role in terms of both theory and applications. In this paper, we are concerned with discontinuous BAM (bidirectional associative memory) neural networks with time-varying delays. Under the basic framework of Filippov solution, by means of differential inclusions theory, inequality technique, fundamental solution matrix of coefficients and the non-smooth analysis theory with Lyapunov-like approach, some new sufficient criteria are given to ascertain the existence and globally exponential stability of the anti-periodic solutions for the considered BAM neural networks. Simulation results of two topical numerical examples are exploited to illustrate the improvement and advantages of the established theoretical results in comparison with some existing results. Some previous known results are extended and complemented.

Stability analysis for quaternion-valued inertial memristor-based neural networks with time delays

Quaternion-valued inertial memristor-based neural networks with time-varying delays are found to be effective in demonstrating complex dynamic behaviors. Considering the stability of the periodic solution for a novel neural network, we construct a class quaternion-valued inertial memristor-based neural network model by combining with the cellular neural networks and the inertial item with time-varying delay. Based on the continuation theorem of Mawhin’s coincidence degree theory, differential inclusion theory, and inequality techniques, some sufficient criteria for the existence of periodic solutions are obtained. Meanwhile, through constructing a novel Lyapunov functional method, several sufficient conditions have been derived to ensure the global exponential stability of periodic solutions. At last, an example is provided to illustrate the obtained theoretical results of the quaternion-valued inertial memristor-based neural networks model.

Synchronization in finite time for variable-order fractional complex dynamic networks with multi-weights and discontinuous nodes based on sliding mode control strategy

This paper is concerned with the global synchronization in finite time for variable-order fractional complex dynamic networks with multi-weights, where the dynamic nodes are modeled to be discontinuous, and subject to the local Hölder nonlinear growth in a neighborhood of continuous points. Firstly, an inequality with respect to variable-order fractional derivative for convex functions is proposed. On the basis of the proposed inequality, a global convergence principle in finite time for absolutely continuous functions is developed. Secondly, based on proposed convergence principle in finite time, a new sliding mode surface is presented, and an appropriate sliding mode control law is designed to drive the trajectory of the error system to the prescribed sliding mode surface in finite time and remain on it forever. In addition, on the basis of differential inclusions theory and Lur’e Postnikov-type convex Lyapunov function approach, the sufficient conditions with respect to the global stability in finite time are established in terms of linear matrix inequalities for the error system on designed sliding mode surface. Moreover, the upper bound of the settling time is explicitly evaluated. Finally, the effectiveness and correction of synchronization strategies are illustrated through two simulation experiments.

Finite-time synchronization of reaction-diffusion neural networks with time-varying parameters and discontinuous activations

The issue of finite-time synchronization (FTS) of discontinuous reaction-diffusion neural networks (DRDNNs) with time-varying coefficients is investigated here. First, the effects caused by discontinuous activations are handled by differential inclusion theory. Secondly, a relaxed Lyapunov function (RLF) method is introduced to design novel control algorithms, including state-feedback control and adaptive control, to achieve FTS of DRDNNs, and the upper-bound of the settling time is explicitly estimated. In this RLF method, the LF is allowed to be nonsmooth and possess an indefinite derivative.Moreover, the FTS results via generalized pinning state-feedback and generalized pinning adaptive control are presented as two corollaries. Finally, two numerical simulations are presented to substantiate the merits of the obtained results.

Global exponential synchronization via nonlinear feedback control for delayed inertial memristor-based quaternion-valued neural networks with impulses

The issue of global exponential synchronization is addressed for delayed impulsive and time-varying delayed inertial memristor-based quaternion-valued neural networks (IMQVNNs). First, the concept of the Filippov solution for IMQVNNs is introduced by applying differential inclusion theory. Then, a nonlinear feedback controller is designed in the skew field of quaternions to ensure the realization of global exponential synchronization for IMQVNNs. According to this controller, the global exponential synchronization criteria are established based on the Lyapunov stability theory. Moreover, a corollary for the real-valued system is given to compare with the existing conclusions. Finally, two examples are presented to show the validity of the results.

Extended Robust Exponential Stability of Fuzzy Switched Memristive Inertial Neural Networks With Time-Varying Delays on Mode-Dependent Destabilizing Impulsive Control Protocol.

This article investigates the problem of robust exponential stability of fuzzy switched memristive inertial neural networks (FSMINNs) with time-varying delays on mode-dependent destabilizing impulsive control protocol. The memristive model presented here is treated as a switched system rather than employing the theory of differential inclusion and set-value map. To optimize the robust exponentially stable process and reduce the cost of time, hybrid mode-dependent destabilizing impulsive and adaptive feedback controllers are simultaneously applied to stabilize FSMINNs. In the new model, the multiple impulsive effects exist between two switched modes, and the multiple switched effects may also occur between two impulsive instants. Based on switched analysis techniques, the Takagi-Sugeno (T-S) fuzzy method, and the average dwell time, extended robust exponential stability conditions are derived. Finally, simulation is provided to illustrate the effectiveness of the results.

Synchronization of coupled memristive competitive BAM neural networks with different time scales

In this paper, synchronization of coupled memristive competitive bidirectional associative memory (BAM) neural networks with different time scales is discussed. Two kinds of feedback controllers are designed such that the response system and the drive system can reach synchronization. By using the differential inclusions theory, and constructing a proper Lyapunov–Krasovskii functional, novel sufficient conditions are obtained to achieve asymptotical synchronization of competitive BAM neural networks. The proposed synchronization can be easily realized. An illustrative example is given to show the feasibility of our theoretical results.

Stability analysis of almost periodic solutions for discontinuous bidirectional associative memory (BAM) neural networks with discrete and distributed delays

Abstract This paper aims to discuss a class of discontinuous bidirectional associative memory (BAM) neural networks with discrete and distributed delays. By using the set-valued map, differential inclusions theory and fundamental solution matrix, the existence of almost-periodic solutions for the addressed neural network model is firstly discussed under some new conditions. Subsequently, based on the non-smooth analysis theory with Lyapunov-like strategy, the global exponential stability result of the almost-periodic solution for the proposed neural network system is also established without using any additional conditions. The results achieved in the paper extend some previous works on BAM neural networks to the discontinuous case and it is worth mentioning that it is the first time to investigate the almost-periodic dynamic behavior for the BAM neural networks like the form in this paper. Finally, in order to demonstrate the effectiveness of the theoretical schemes, simulation results of two topical numerical examples are delineated.

Periodicity and global exponential periodic synchronization of delayed neural networks with discontinuous activations and impulsive perturbations

In this paper, periodic dynamics and periodic synchronization of a class of delayed neural networks with discontinuous neuron activations and impulsive perturbations are investigated. Firstly, based on Filippov’s functional differential inclusion and impulsive differential inclusions theory, the existence of solutions of the considered neural network system is studied. Secondly, according to Krasnoselskii’s fixed point theorem on cones of set-valued analysis, sufficient conditions are provided for the existence of at least one periodic solution and at least two periodic solutions for the neural network systems. Furthermore, we also use Kakutani’s fixed point theorem to give sufficient criteria for the existence of at least one periodic solution of the neural network systems. These results shown that there are still one or more periodic solutions for the neural networks with discontinuous activations under the impulsive effects. Moreover, sufficient criteria are given to the exponential periodic synchronization under the switching state-feedback control strategy with or without delay. Finally, two numerical simulation examples are presented to verify the correctness and validity of our main conclusions.

To the Theory of Differential Inclusions with Caputo Fractional Derivatives

We consider the Cauchy problem for a differential inclusion with a Caputo fractional derivative of order $$ \alpha \in (0, 1)$$ . It is assumed that the set-valued mapping defining the right-hand side of the inclusion has nonempty, convex, and compact values, is upper semicontinuous, and satisfies the sublinear growth condition. We prove that the solution set of the Cauchy problem is nonempty and compact and establish the nature of the dependence of the solution set on the initial data, the right-hand side of the inclusion, and the order of differentiation. The semigroup property of the solution set is established.

New fixed‐time synchronization control of discontinuous inertial neural networks via indefinite Lyapunov‐Krasovskii functional method

SummaryThe purpose of this article is to investigate the existence of periodic solutions and fixed‐time synchronization (FTS) of a class of discontinuous inertial neural networks (DINNs) with time‐varying delays. Due to the existence of the discontinuities, first, by using a generalized variable transformation, the original DINNs are transformed into a first‐order differential system. By using the differential inclusions theory and the set‐valued version of the Mawhin coincidence theorem, a new delay‐dependent criterion is derived to ensure the existence of periodic solutions. Furthermore, by designing some effective discontinuous control strategies and by constructing indefinite Lyapunov‐Krasovskii functional (LKF), algebraic criteria are derived to guarantee the FTS for the drive‐response system. The settling time is explicitly estimated. In comparison with the related results on INNs, it is the first time to study the existence result of the DINNs and the Lyapunov approaches applied to investigate the FTS are entirely different. Finally, the correctness of the main results is verified via numerical examples.

Periodic and homoclinic solutions of discontinuous Cohen–Grossberg neural networks with time-varying delays

In this paper, we investigate a class of discontinuous Cohen–Grossberg neural networks (DCGNNs) with time-varying delays. Under the framework of Filippov solution, by means of differential inclusions theory and the set-valued version of the Mawhin coincidence theorem, we firstly establish the existence results of 2kT-periodic solutions for the proposed DCGNNs. Secondly, by applying an approximation technique, we prove that the limit of the 2kT-periodic solutions exists and the limit is exactly the homoclinic solution of the proposed DCGNNs. Thirdly, by using the non-smooth analysis theory with Lyapunov-like approach, some new testable algebraic criteria are derived for ensuring the global exponential stability of the solutions. It should be regarded as the first time to study the homoclinic solutions of Cohen–Grossberg neural networks with discontinuous activations based on Filippov solution. Finally, convincing numerical examples and remarks are provided to substantiate the superiority and efficiency of obtained results.

A New Settling-time Estimation Protocol to Finite-time Synchronization of Impulsive Memristor-Based Neural Networks.

In this article, the issues of finite-time synchronization and finite-time adaptive synchronization for the impulsive memristive neural networks (IMNNs) with discontinuous activation functions (DAFs) and hybrid impulsive effects are probed into and elaborated on, where the stabilizing impulses (SIs), inactive impulses (IIs), and destabilizing impulses (DIs) are taken into account, respectively. Not resembling several earlier works, a more extensive range of impulses in the context of impulsive effects has been analyzed without using the known average impulsive interval strategy (AIIS). In light of the theories of differential inclusions and set-valued map, as well as impulsive control, new sufficient criteria with respect to the estimated settling time for synchronization of the related IMNNs are established using two types of switching control approaches, which sufficiently utilize information from not only the SIs, DIs, and DAFs but also the impulse sequences. Two simulation experiments are presented to the efficiency of the proposed results.

New criteria for finite-time stability of fractional order memristor-based neural networks with time delays

In this paper, the finite-time stability (FTS) problems for a class of fractional order memristor-based neural network (FOMNN) with time delays are investigated. Based on the method of steps, set-valued mapping, the theory of differential inclusion and fractional order Gronwall inequality, a new delay dependent criterion for the FTS of FOMNN with time delays and the fractional order 0<μ<1 is derived, which is less conservative than the existing criteria. Furthermore, a sufficient condition to ensure the FTS of FOMNN with the fractional-order 1<μ<2 is proposed. Finally, two examples are given to illustrate the validity of the proposed results.