This paper is concerned with global asymptotic stability of delayed neural networks. Notice that a Bessel–Legendre inequality plays a key role in deriving less conservative stability criteria for delayed neural networks. However, this inequality is in the form of Legendre polynomials and the integral interval is fixed on ${[{-}h,0]}$ . As a result, the application scope of the Bessel–Legendre inequality is limited. This paper aims to develop the Bessel–Legendre inequality method so that less conservative stability criteria are expected. First, by introducing a canonical orthogonal polynomial sequel, a canonical Bessel–Legendre inequality and its affine version are established, which are not explicitly in the form of Legendre polynomials. Moreover, the integral interval is shifted to a general one $ {[a,b]}$ . Second, by introducing a proper augmented Lyapunov–Krasovskii functional, which is tailored for the canonical Bessel–Legendre inequality, some sufficient conditions on global asymptotic stability are formulated for neural networks with constant delays and neural networks with time-varying delays, respectively. These conditions are proven to have a hierarchical feature: the higher level of hierarchy, the less conservatism of the stability criterion. Finally, three numerical examples are given to illustrate the efficiency of the proposed stability criteria.

Hierarchical Type Stability Criteria for Delayed Neural Networks via Canonical Bessel–Legendre Inequalities

In this brief, we establish a novel complex-valued memristive recurrent neural network (CVMRNN) to study its stability. As a generalization of real-valued memristive neural networks, CVMRNN can be separated into real and imaginary parts. By means of $M$ -matrix and Lyapunov function, the existence, uniqueness, and exponential stability of the equilibrium point for CVMRNNs are investigated, and sufficient conditions are presented. Finally, the effectiveness of obtained results is illustrated by two numerical examples.

Exponential Stability of Complex-Valued Memristive Recurrent Neural Networks

In this paper, we present a complex-valued projection neural network for solving constrained convex optimization problems of real functions with complex variables, as an extension of real-valued projection neural networks. Theoretically, by developing results on complex-valued optimization techniques, we prove that the complex-valued projection neural network is globally stable and convergent to the optimal solution. Obtained results are completely established in the complex domain and thus significantly generalize existing results of the real-valued projection neural networks. Numerical simulations are presented to confirm the obtained results and effectiveness of the proposed complex-valued projection neural network.

A Complex-Valued Projection Neural Network for Constrained Optimization of Real Functions in Complex Variables

There are two ultimate goals in multiobjective optimization. The primary goal is to obtain a set of Pareto-optimal solutions while the secondary goal is to obtain evenly distributed solutions to characterize the efficient frontier. In this paper, a collaborative neurodynamic approach to multiobjective optimization is presented to attain both goals of Pareto optimality and solution diversity. The multiple objectives are first scalarized using a weighted Chebyshev function. Multiple projection neural networks are employed to search for Pareto-optimal solutions with the help of a particle swarm optimization (PSO) algorithm in reintialization. To diversify the Pareto-optimal solutions, a holistic approach is proposed by maximizing the hypervolume (HV) using again a PSO algorithm. The experimental results show that the proposed approach outperforms three other state-of-the-art multiobjective algorithms (i.e., HMOEA/D, MOEA/DD, and NSGAIII) most of times on 37 benchmark datasets in terms of HV and inverted generational distance.

A Collaborative Neurodynamic Approach to Multiobjective Optimization

Redundancy resolution is a critical problem in the control of parallel Stewart platform. The redundancy endows us with extra design degree to improve system performance. In this paper, the kinematic control problem of Stewart platforms is formulated to a constrained quadratic programming. The Karush–Kuhn–Tucker conditions of the problem is obtained by considering the problem in its dual space, and then a dynamic neural network is designed to solve the optimization problem recurrently. Theoretical analysis reveals the global convergence of the employed dynamic neural network to the optimal solution in terms of the defined criteria. Simulation results verify the effectiveness in the tracking control of the Stewart platform for dynamic motions.

Songchuan Zhang

Papers

A complex-valued neural dynamical optimization approach and its stability analysis