Search Results (13)

This research focuses on the theoretical asymptotic stability and long-time decay of the zero solution for a system of time-fractional nonlinear Schrödinger delay equations (NSDEs) in the context of the Caputo fractional derivative. Using the fractional Halanay inequality, we demonstrate theoretically when the considered system decays and behaves asymptotically, employing an energy function in the sense of the L₂ norm. Together with utilizing the finite difference method for the spatial variables, we investigate the long-time stability for the semi-discrete system. Furthermore, we operate the L1 scheme to approximate the Caputo fractional derivative and analyze the long-time stability of the fully discrete system through the discrete energy of the system. Moreover, we demonstrate that the proposed numerical technique energetically captures the long-time behavior of the original system of NSDEs. Finally, we provide numerical examples to validate the theoretical results. Full article

This paper is concerned with exponential synchronization for a class of coupled neural networks with hybrid delays and stochastic distributed delayed impulses. First of all, based on the average impulsive interval method, total probability formula and ergodic theory, several novel impulsive Halanay differential inequalities are established. Two types of stochastic impulses, i.e., stochastic distributed delayed impulses with dependent property and Markov property have been taken into account, respectively. Secondly, some criteria on exponential synchronization in the mean square of a class of coupled neural networks with stochastic distributed delayed impulses are acquired by combining the proposed lemmas and graph theory. The validity of the theoretical results is demonstrated by several numerical simulation examples. Full article

In this paper, we focus on the asymptotic stability of the trajectories governed by the differential inequalities with mixed delays using the fixed-point theorem. It is interesting that the Halanay inequality is a special case of the differential inequality studied in this paper. Our results generalize and improve the existing results on Halanay inequality. Finally, three numerical examples are utilized to illustrate the effectiveness of the obtained results. Full article

In this paper, we ponder a kind of discrete-time fractional-order complex-valued fuzzy BAM neural network. Firstly, in order to guarantee the quasi-projective synchronization of the considered networks, an original quantitative control strategy is designed. Next, by virtue of the relevant definitions and properties of the Mittag-Leffler function, we propose a novel discrete-time fractional-order Halanay inequality, which is more efficient for disposing of the discrete-time fractional-order models with time delays. Then, based on the new lemma, fractional-order h-difference theory, and comparison principle, we obtain some easy-to-verify synchronization criteria in terms of algebraic inequalities. Finally, numerical simulations are provided to check the accuracy of the proposed theoretical results. Full article

This paper focuses on the asymptotic behavior of nonautonomous neural networks with delays. We establish criteria for analyzing the asymptotic behavior of nonautonomous recurrent neural networks with delays by means of constructing some new generalized Halanay inequalities. We do not require to constructi any complicated Lyapunov function and our results improve some existing works. Lastly, we provide some illustrative examples to demonstrate the effectiveness of the obtained results. Full article

Very recently, a different generalization of real-valued neural networks (RVNNs) to multidimensional domains beside the complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and Clifford-valued neural networks (ClVNNs) has appeared, namely octonion-valued neural networks (OVNNs), which are not a subset of ClVNNs. They are defined on the octonion algebra, which is an 8D algebra over the reals, and is also the only other normed division algebra that can be defined over the reals beside the complex and quaternion algebras. On the other hand, fractional-order neural networks (FONNs) have also been very intensively researched in the recent past. Thus, the present work combines FONNs and OVNNs and puts forward a fractional-order octonion-valued neural network (FOOVNN) with neutral-type, time-varying, and distributed delays, a very general model not yet discussed in the literature, to our awareness. Sufficient criteria expressed as linear matrix inequalities (LMIs) and algebraic inequalities are deduced, which ensure the asymptotic and Mittag–Leffler synchronization properties of the proposed model by decomposing the OVNN system of equations into a real-valued one, in order to avoid the non-associativity problem of the octonion algebra. To accomplish synchronization, we use two different state feedback controllers, two different types of Lyapunov-like functionals in conjunction with two Halanay-type lemmas for FONNs, the free-weighting matrix method, a classical lemma, and Young’s inequality. The four theorems presented in the paper are each illustrated by a numerical example. Full article

The long-term behavior of the weak solution of a fractional delayed reaction–diffusion equation with a generalized Caputo derivative is investigated. By using the classic Galerkin approximation method and comparison principal, the existence and uniqueness of the solution is proved in the sense of weak solution. In addition, the global attracting set of the considered system is obtained, with the help of the Sobolev embedding theorem and Halanay inequality. Full article

A class of generalized Halanay inequalities is studied via the Banach fixed point method and comparison principle. The conditions to ensure the boundedness and stability of the zero solution are obtained in this study. This research provides a new approach to the study of the boundedness and stability of Halanay inequality. Numerical examples and simulation results verify the validity and superiority of the conclusions obtained in this study. Full article

The lack of a conventional Lyapunov theory for fractional-order (FO) systems makes it difficult to study the dynamics of fractional-order neural networks (FONNs). Instead, the existing literature derives necessary conditions for various dynamic properties of FONNs using Halanay-type lemmas. However, when these lemmas are used, the results are frequently more conservative than those produced for integer-order neural networks (NNs). In order to provide sufficient criteria that are less conservative than those found in other research, a novel application of the Halanay-type lemma is made within this study. Thus, for extremely general FONNs containing neutral-type, time-varying, and distributed delays, sufficient conditions presented by way of linear matrix inequalities (LMIs) and algebraic inequalities are achieved. For the FO scenario, a model this broad and including so many different kinds of delays is developed for the first time. Additionally, a novel form of Lyapunov-like function is built, which results in less stringent algebraic inequalities. One of the first times in the setting of FONNs, the free-weighting matrix method is also used to further lower the conservativeness of the obtained conditions. Based on different Lyapunov-type functions, three theorems are developed regarding the asymptotic stability of the proposed networks. Three numerical simulations are used to demonstrate the theoretical developments. Full article

This paper mainly investigates the exponential stability of switched neural networks (SNNs) with partial state reset and time-varying delays, in which partial state reset means that only a fraction of the states can be reset at each switching instant. Moreover, both stable and unstable subsystems are also taken into account and therefore, switched systems under consideration can take several switched systems as special cases. The comparison principle, the Halanay-like inequality, and the time-dependent switched Lyapunov function approach are used to obtain sufficient conditions to ensure that the considered SNNs with delays and partial state reset are exponentially stable. Numerical examples are provided to demonstrate the reliability of the developed results. Full article

This study on the local stability of quaternion-valued neural networks is of great significance to the application of associative memory and pattern recognition. In the research, we study local Lagrange exponential stability of quaternion-valued neural networks with time delays. By separating the quaternion-valued neural networks into a real part and three imaginary parts, separating the quaternion field into $3^{4n}$ subregions, and using the intermediate value theorem, sufficient conditions are proposed to ensure quaternion-valued neural networks have $3^{4n}$ equilibrium points. According to the Halanay inequality, the conditions for the existence of $2^{4n}$ local Lagrange exponentially stable equilibria of quaternion-valued neural networks are established. The obtained stability results improve and extend the existing ones. Under the same conditions, quaternion-valued neural networks have more stable equilibrium points than complex-valued neural networks and real-valued neural networks. The validity of the theoretical results were verified by an example. Full article

We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity of the corresponding system, namely, we obtain the existence of global absorbing set. Besides, some available results are improved in this work. The existence of a global attracting set is still an open problem. Full article

In this paper, the consensus problem of discrete multiagent systems with time varying sampling periods is studied. Firstly, with thorough analysis of various delays among agents, the control input of each agent is designed with consideration of sending delay and receiving delay. With construction of discrete dynamics of state error vector, it is proved by applying Halanay inequality that consensus of the system can be reached. Further, the definition of bounded consensus is proposed in the situation where environmental disturbances exist. In order to handle this problem, the Halanay inequality is extended into a more general one with boundedness property. Based on the new Halanay inequality obtained, the boundedness of consensus error is guaranteed. At last, simulation examples are presented to demonstrate the theoretical conclusions. Full article