Exponential stability for delayed complex-valued neural networks with reaction-diffusion terms

In this study, we investigate reaction-diffusion complex-valued neural networks with mixed delays. The mixed delays include both time-varying and infinite distributed delays. Criteria are derived to ensure the existence, uniqueness, and exponential stability of the equilibrium state of the addressed system on the basis of the M-matrix properties and homeomorphism mapping theories as well as the vector Lyapunov function method. The results demonstrate the positive effect of reaction-diffusion on the stability, which further improves the existing conditions. Finally, the analysis of several examples is compared to the present results to verify the correctness and reduced conservatism of the primary results.

Synchronization of Cohen-Grossberg fuzzy cellular neural networks with time-varying delays

In this paper, a class of Cohen-Grossberg fuzzy cellular neural networks (CGFCNNs) with time-varying delays are considered. Initially, the sufficient conditions are proposed to ascertain the existence and uniqueness of the solutions for the considered dynamical system via homeomorphism mapping principle. Then synchronization of the considered delayed neural networks is analyzed by utilizing the drive-response (master-slave) concept, in terms of a linear matrix inequality (LMI), the Lyapunov-Krasovskii (LK) functional, and also using some free weighting matrices. Next, this result is extended so as to establish the robust synchronization of a class of delayed CGFCNNs with polytopic uncertainty. Sufficient conditions are proposed to ascertain that the considered delayed networks are robustly synchronized by using a parameter-dependent LK functional and LMI technique. The restriction on the bounds of derivative of the time delays to be less than one is relaxed. In particular, the concept of fuzzy theory is greatly extended to study the synchronization with polytopic uncertainty which differs from previous efforts in the literature. Finally, numerical examples and simulations are provided to illustrate the effectiveness of the obtained theoretical results.

Homeomorphism Mapping Based Neural Networks for Finite Time Constraint Control of a Class of Nonaffine Pure-Feedback Nonlinear Systems

This paper proposes a new scheme for solving finite time neural networks adaptive tracking control issue for the nonaffine pure-feedback nonlinear system. The procedure, based on homeomorphism mapping and backstepping, effectively deals with constraint control and design difficulty induced by pure-feedback structure. The most outstanding novelty is that finite time adaptive law is proposed for training weights of neural networks. Furthermore, by combining finite time adaptive law and Lyapunov-based arguments, a valid finite time adaptive neural networks controller design algorithm is presented to ensure that system is practical finite stable (PFS) rather than uniformly ultimately bounded (UUB). Because of using the finite time adaptive law to training weights of neural networks, the closed-loop error system signals are in assurance of bounded in finite time. Benchmark simulations have well demonstrated effectiveness and efficiency of the proposed approach.

Impulsive Disturbances on the Dynamical Behavior of Complex-Valued Cohen-Grossberg Neural Networks with Both Time-Varying Delays and Continuously Distributed Delays

This paper studies the global exponential stability for a class of impulsive disturbance complex-valued Cohen-Grossberg neural networks with both time-varying delays and continuously distributed delays. Firstly, the existence and uniqueness of the equilibrium point of the system are analyzed by using the corresponding property of M-matrix and the theorem of homeomorphism mapping. Secondly, the global exponential stability of the equilibrium point of the system is studied by applying the vector Lyapunov function method and the mathematical induction method. The established sufficient conditions show the effects of both delays and impulsive strength on the exponential convergence rate. The obtained results in this paper are with a lower level of conservatism in comparison with some existing ones. Finally, three numerical examples with simulation results are given to illustrate the correctness of the proposed results.

Globalμ-Stability of Impulsive Complex-Valued Neural Networks with Leakage Delay and Mixed Delays

The impulsive complex-valued neural networks with three kinds of time delays including leakage delay, discrete delay, and distributed delay are considered. Based on the homeomorphism mapping principle of complex domain, a sufficient condition for the existence and uniqueness of the equilibrium point of the addressed complex-valued neural networks is proposed in terms of linear matrix inequality (LMI). By constructing appropriate Lyapunov-Krasovskii functionals, and employing the free weighting matrix method, several delay-dependent criteria for checking the globalμ-stability of the complex-valued neural networks are established in LMIs. As direct applications of these results, several criteria on the exponential stability, power-stability, and log-stability are obtained. Two examples with simulations are provided to demonstrate the effectiveness of the proposed criteria.

An Improved Genetic Algorithm Based on K2(m) Triangulation of Continuous Self-Mapping

In this paper an improved genetic algorithm based on the simplex self-mapping fixed point algorithm is proposed. With this algorithm, the optimal problem of n-dimensional closure function will be transformed as the solution of approximate fixed point problem of n-dimensional standard simplexes by homeomorphism mapping. The genetic operators relying on the integer labels are designed. In this case, whether every individual loading simplex of the population is a completely labeled simplex can be used as an objective convergence criterion. The simulation results demonstrate that the proposed algorithm is valid and effective.

GLOBAL EXPONENTIAL STABILITY OF CHUA'S REACTION–DIFFUSION CNN

In this paper, the global exponential stability of Chua's reaction–diffusion CNN system is investigated. For this system, some sufficient conditions ensuring the existence and global exponential stability of the equilibrium point is derived by using homeomorphism mapping, the property of coefficient matrix and analytical techniques. Finally, three illustrative examples are given to show the effectiveness of our results.

On weak concepts of stability

The manifold in this paper is assumed to be connected differentiable of class C∞. Let Dr(M) and Ӿr(M) be the set of all diffeomorphisms and vector fields of class Cr on a manifold M with Whitney Cr topology, respectively. In [2], the concept of weak stability is defined. The definition is equivalent to the following ((2.1) of this paper); f∈Dr(M) or X ∈ Ӿr(M) is weakly (allowably) stable if and only if there is a neighborhood U of f or X in Dr(M) or Ӿr(M) such that for any (a suitable) g or Y ∈ U the set of all elements topologically equivalent to g or Y is dense in U, respectively. Here, f, g ∈ Dr(M) are said to be topologically equivalent if they are topologically conjugate and X, Y ∈ Ӿr(M) are said to be topologically equivalent if there is a homeomorphism mapping any trajectory of X onto a trajectory of Y preserving the orientations of the trajectories. Similarly, weak Ω-stability is defined for f and X.