Differential Inclusion Theory Research Articles

Mono/multi-periodicity generated by impulses control in time-delayed memristor-based neural networks

This paper considers the mono/multi-periodicity problems in time-delayed memristor-based neural networks (MNNs) possessing impulsive effects, where the input–output activation functions are relaxed to be unbounded and nonmonotonic. Based on differential inclusions theory, several conditions are derived which ensure the existence and multiplicity of periodic positive solutions for impulsive time-delayed MNNs by exploiting Krasnoselskii’s fixed point theorem of multimap type. It should be noted that such periodic positive solutions in MNNs are generated by impulses control. The developed impulses control method generating mono/multi-periodicity of this paper is meaningful for the design and application of time-delayed MNNs circuits in a periodic environment.

An approximate solution of a differential inclusion with maximal monotone operator

The theory of differential inclusions has played a central role in many areas as biological systems, physical problems and population dynamics. The principle aim of our work is to compute explicitly the discrete approximate solution of a differential inclusion including a maximal monotone operator. Also we present a numerical application of our results for showing how to compute the discrete approximate solution of its corresponding differential inclusion.

Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy

This paper studies the solution behaviour of a general delayed predator-prey model with discontinuous prey control strategy. The positiveness and boundeness of the solution of the system is firstly investigated using the comparison theorem. Then the sufficient conditions are derived for the existence of positive periodic solutions using the differential inclusion theory and the topological degree theory. Furthermore, the positive periodic solution is proved to be globally exponentially stable by employing the generalized Lyapunov approach. The global finite-time convergence is also discussed for the system state. Finally, the numerical simulations of four examples are given to validate the correctness of the theoretical results.

Fixed-Time Synchronization of Delayed Fractional-Order Memristor-Based Fuzzy Cellular Neural Networks

In this article, we discussed the fixed-time synchronization of fractional-order memristor-based fuzzy cellular neural networks (FMFCNNs) with time-varying delays. Based on the differential inclusion theory and discontinuous state feedback control technique, some useful novel criteria of the fixed-time synchronization for FMFCNNs are derived by constructing appropriate Lyapunov function. The results in our paper improve and generalize current literature research results significantly. Finally, the effectiveness of our control schemes for the fixed-time synchronization is demonstrated through numerical simulations.

Global dissipativity and finite-time synchronization of mixed time-varying delayed memristor-based neural networks with discontinuous activations

In this paper, the matters of dissipativity and finite time synchronization for memristor-based neural networks (MNNs) with mixed time-varying discontinuities are investigated. Firstly, under the framework of extending Filippov differential inclusion theory, several effective new criteria are derived. Then, the global dissipativity of Filippov solution to neural networks is proved by using generalized Halanay inequality and matrix measure method. Secondly, some novel sufficient conditions are introduced to guarantee the finite-time synchronization of the drive-response MNNs based on a simple Lyapunov function and two different feedback controllers. Finally, several numerical examples are given to verify the validity of the theoretical results.

Mixed-mode oscillations and the bifurcation mechanism for a Filippov-type dynamical system

In this paper, mixed-mode oscillations and bifurcation mechanism for a Filippov-type system including two time-scales in the frequency domain are demonstrated. According to classic Chua’s system, we investigate a non-smooth dynamical system including two time-scales. As there exists an order gap between the exciting frequency and the natural one, the whole external excitation term can be considered as a slow-changing parameter, which results in two smooth subsystems divided by the non-smooth boundary. In addition, the critical condition about fold bifurcation (FB) is studied, and by applying the Hopf bifurcation (HB) theorem, specific formulas for determining the existence of HBs are presented. By introducing an auxiliary parameter via differential inclusions theory, the non-smooth bifurcations on the boundary are discussed. Then, the equilibrium branches and the bifurcations are derived, and two typical cases associated with different bifurcations are considered. In light of the superposition between the bifurcation curve and the transformed phase portrait, the dynamical behaviours of the mixed-mode oscillations as well as sliding movement along the non-smooth boundary are obtained, which reveal the corresponding dynamical mechanism.

Finite-time synchronization of switched neural networks with state-dependent switching via intermittent control

In this paper, the finite-time synchronization problem for switched neural networks with state-dependent switching under intermittent control is discussed. Different from previous literature, this paper uses aperiodically intermittent control to make switched neural networks achieve finite-time synchronization. A new differential inequality is established to get our main results. Meanwhile, based on graph theory and differential inclusions theory, a proper Lyapunov function is proposed to derive a criterion to ensure finite-time synchronization of switched neural networks. And the convergence time, closely related to the topological structure of networks, is obtained and does not rely on control widths or rest widths for aperiodically intermittent control. Finally, a numerical example is presented to illustrate the effectiveness and feasibility of our results.

Finite-time and fixed-time synchronization control of fuzzy Cohen-Grossberg neural networks

This paper aims to investigate the finite-time and fixed-time robust synchronization of fuzzy Cohen-Grossberg neural networks with discontinuous activations. To deal with the discontinuous property, the framework of Filippov solution is invoked to solve the inexistence of the classical solutions. For the purpose of achieving fixed-time synchronization, we turn to investigating the fixed-time stability problem of the error system between the drive-response systems. By the functional differential inclusions theory, inequality technique and the non-smooth Lyapunov-Krasovskii functional and designing a simple discontinuous state-feedback control law for the response neural system, some sufficient algebraic criteria are derived to achieve synchronization within a fixed time, and the settling time is given. Moreover, based on the fixed-time robust synchronization and under the same basic assumptions, we further complete the finite-time synchronization of the addressed drive-response systems by designing a simple switching adaptive controller, and the upper bound of the settling time is also estimated. It should be regarded as the first time to study the finite-time synchronization of fuzzy Cohen-Grossberg neural networks with discontinuous activations based on the adaptive control. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed synchronization strategies.

Distributed Nash Equilibrium Seeking for Quadratic Games with Security

Considering a game with quadratic cost functions, this paper presents a distributed algorithm with security, whereby each player updates its strategy variable without using its private data and still achieves the Nash equilibrium. By using the theory of differential inclusions, Lyapunov function and invariance principle, the algorithm is proved to be convergent. Our algorithm can be used when it is required to seek the Nash equilibrium without disclosure of private data.

Global Asymptotic Stability and Adaptive Ultimate Mittag–Leffler Synchronization for a Fractional-Order Complex-Valued Memristive Neural Networks With Delays

This paper investigates some dynamic behaviors for a fractional-order complex-valued memristive neural networks (FCVMNNs) with delays. A new mathematical expression of the complex-value memductance (memristance) is proposed according to the feature of the complex-valued memristor-based neural networks and a new class of FCVMNNs with delays is designed. Based on the framework of Filippov solution and differential inclusion theory, the sufficient conditions are given first to guarantee the global uniform asymptotic stability for the new FCVMNNs with delays, by using fractional-order Leibniz rule and a Razumikhin-type method. In addition, a complex-value adaptive controller is designed to achieve ultimate Mittag–Leffler synchronization between two FCVMNNs with delays. Numerical simulations are given to show the effectiveness of the theoretical results.

Antiperiodic dynamical behaviors of discontinuous neutral‐type Cohen‐Grossberg neural networks with mixed time delays

AbstractThis article presents a class of neutral‐type Cohen‐Grossberg neural networks with discontinuous activations and mixed time delays. Based on the functional differential inclusions theory, inequality technique, and nonsmooth analysis theory with Lyapunovlike approach, some new sufficient criteria are given to ascertain the existence, uniqueness, and globally exponential stability of the antiperiodic solution. Finally, two topical numerical examples and the corresponding computer simulations are delineated to substantiate the correctness of our theoretical predictions. The obtained results of this article are new and complement some related earlier works.

Stability analysis of almost periodic solutions of discontinuous BAM neural networks with hybrid time-varying delays and D operator

In the paper, we are concerned with a class of discontinuous BAM neural networks with hybrid time-varying delays and D operator. Based on the concept of Filippov solution, by means of the differential inclusions theory and the non-smooth analysis theory with Lyapunov-like approach, some new and novel sufficient conditions are derived to guarantee the existence, uniqueness and global exponential stability of almost-periodic solution of our proposed neural network model. To the authors’ knowledge, the results established in the paper are the only available results on the BAM neural networks, connecting the three main characteristics, i.e., discontinuous activation functions, hybrid time-varying delays and D operator. Some previous works in the literature are significantly extend and complement. Finally, two topical simulation examples are given to show the effectiveness of the established main results.

Bordering in Europe: Differential Inclusion

Borders exist and operate not only at formal frontiers but also outside and within European countries. Furthermore, borders both exclude and include migrants. This article advances the theory of differential inclusion. It provides a more nuanced understanding of processes of bordering that comprehends irregular migration as a phenomenon that is not only officially denounced and combatted but also unofficially tolerated and facilitated by European states. The analysis reveals that differential inclusion is transforming citizenship in Europe away from officially desired equality toward unofficially tolerated stratification. Employing Foucault’s notion of “assemblage”, the article also seeks to shed light on not only whose interests are undermined but also on whose interests are served by differential inclusion.

Finite Time Stability Analysis of Fractional-Order Complex-Valued Memristive Neural Networks with Proportional Delays

In this paper, finite time stability analysis of fractional-order complex-valued memristive neural networks with proportional delays is investigated. Under the framework of Filippov solution and differential inclusion theory, by using H$$\ddot{o}$$lder inequality, Gronwall inequality and inequality scaling skills, some sufficient conditions are derived to ensure the finite-time stability of concerned fractional-order complex-valued memristive neural networks with fractional order $$\alpha $$: $$0<\alpha <1/2$$ and $$1/2\le \alpha <1$$. In the end, two numerical examples are provided to illustrate the availability of the obtained results.

Intermittent Discrete Observation Control for Synchronization of Stochastic Neural Networks.

In this paper, to investigate the exponential synchronization of stochastic neural networks, a new periodically intermittent discrete observation control (PIDOC) is first proposed. Different from the existing periodically intermittent control, our control in control time is feedback control based on discrete-time state observations (FCDSOs) instead of a continuous-time one. By employing the Lyapunov method, graph theory, and theory of differential inclusions, the exponential synchronization of stochastic neural networks with a discontinuous right-hand side is realized by PIDOC and some sufficient conditions are presented. Especially, when control width tends to control period, PIDOC will be reduced to a general FCDSO and we give some detailed discussions. Then, we provide some corollaries about synchronization in mean square, asymptotical synchronization in mean square, and exponential synchronization of stochastic neural networks under FCDSO. Finally, some numerical simulations are provided to demonstrate our analytical results.

Finite-time projective synchronization of fractional-order complex-valued memristor-based neural networks with delay

This paper studies the finite-time projective synchronization of fractional-order complex-valued memristor-based neural networks (FCVMNNs) with delay. By applying the set-valued map, the differential inclusion theory and Gronwall–Bellman integral inequalities, some sufficient criteria are established to achieve the finite time projective synchronization of the FCVMNNs. The upper bound of the settling time for synchronization is also estimated. Moreover, two numerical examples are designed to verify the correctness and effectiveness of the obtained theoretical results.

Complex Projection Synchronization of Fractional-Order Complex-Valued Memristive Neural Networks with Multiple Delays

In this paper, the complex projection synchronization problem of fractional-order complex-valued memristive neural networks is investigated, in which the projection factor is set to complex value and multiple time delays are considered. Under the framework of set-valued mapping and differential inclusion theory, a hybrid control strategy is designed to analyze the complex projection synchronization problem of the system. Moreover, some criterion to ensure the synchronization of drive response network is obtained by applying the stability theorem and comparison principle of the fractional order systems with multiple time delays. Finally, numerical simulation example is provided to verify the correctness and effectiveness of the complex projection synchronization strategy.

Global quasi-synchronization and global anti-synchronization of delayed neural networks with discontinuous activations via non-fragile control strategy

This paper deals with the global quasi-synchronization and global anti-synchronization issue of delayed neural networks with discontinuous activations (DNNDA) by means of the non-fragile control strategy. In this paper, a class of DNNDA model is presented. Furthermore, applying the Lyapunov-Krasovskii function with matrix measure, the sufficient conditions are put forward to ensure that such DNNDA drive system and DNNDA response system are global quasi-synchronized. In addition, based on a novel Lur′e-Postnikov Lyapunov function, differential inclusion theory and inequality analysis technique, the sufficient conditions to achieve the global anti-synchronization of the DNNDA drive system and DNNDA response system are proposed in term of the Linear matrix inequalities (LMIs). Finally, two typical numerical simulations are provided to show the effectiveness of the obtained results.

Robust fixed-time synchronization of discontinuous Cohen–Grossberg neural networks with mixed time delays

This paper aims to investigate the fixed-time synchronization (i.e., synchronization in fixed-time sense) of Cohen–Grossberg drive-response neural networks with discontinuous neuron activations and mixed time delays (both time-varying discrete delay and distributed delay). To accomplish the target of fixed-time synchronization, a novel discontinuous feedback control procedure is firstly designed for the response neural networks. Then, under the framework of Filippov solutions, by means of functional differential inclusions theory, inequality technique and the nonsmooth analysis theory with Lyapunov-like approach, some sufficient criteria are derived to design the control parameters for achieving fixed-time synchronization of the proposed drive-response systems. Finally, two numerical examples are presented to illustrate the proposed methodologies.

Stability analysis of fractional order memristive discontinuous neural networks with partial state control

This paper investigates exponential stability of fractional order memristive discontinuous neural networks (FMDNNs). Under the framework of the fractional order Filippov solution and differential inclusion theory, the global existence of the solution for the FMDNNs is studied by a given growth conditions. Based on fractional stability theoryand the properties of the Mittag Leffler function, some new criteria for the stability of FMDNNs are obtained by using effective partial state impulsive control, which only needs to control a small fraction of the states. At every impulsive moment, these states of the trajectory far away from the desired trajectory will be firstly controlled. The relations between stable region and fractional order α, control parameters and control rate are discussed. Finally, a numerical simulation is given to verify the effectiveness of the theoretical analysis.