Global Asymptotic Stability of Recurrent Neural Networks With Multiple Time-Varying Delays

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

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.

Mode-dependent stochastic stability criteria of fuzzy Markovian jumping neural networks with mixed delays

This paper investigates the stochastic stability of fuzzy Markovian jumping neural networks with mixed delays in mean square. The mixed delays include time-varying delay and continuously distributed delay. By using the Lyapunov functional method, Jensen integral inequality, the generalized Jensen integral inequality, linear convex combination technique and the free-weight matrix method, several novel sufficient conditions are derived to ensure the global asymptotic stability of the equilibrium point of the considered networks in mean square. The proposed results, which do not require the differentiability of the activation functions, can be easily checked via Matlab software. Finally, two numerical examples are given to demonstrate the effectiveness and less conservativeness of our theoretical results over existing literature.

Convergence analysis of the FOCUSS algorithm

Focal Underdetermined System Solver (FOCUSS) is a powerful and easy to implement tool for basis selection and inverse problems. One of the fundamental problems regarding this method is its convergence, which remains unsolved until now. We investigate the convergence of the FOCUSS algorithm in this paper. We first give a rigorous derivation for the FOCUSS algorithm by exploiting the auxiliary function. Following this, we further prove its convergence by stability analysis.

Further improvement on delay-dependent robust stability criteria for neutral-type recurrent neural networks with time-varying delays

This paper is concerned with the problem of improved delay-dependent robust stability criteria for neutral-type recurrent neural networks (NRNNs) with time-varying delays. Combining the Lyapunov-Krasovskii functional with linear matrix inequality (LMI) techniques and integral inequality approach (IIA), delay-dependent robust stability conditions for RNNs with time-varying delay, expressed in terms of quadratic forms of state and LMI, are derived. The proposed methods contain the least number of computed variables while maintaining the effectiveness of the robust stability conditions. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the present method. Numerical examples are included to show that the proposed method is effective and can provide less conservative results.

Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays

This paper deals with the problem of existence and uniform stability analysis of fractional-order complex-valued neural networks with constant time delays. Complex-valued recurrent neural networks is an extension of real-valued recurrent neural networks that includes complex-valued states, connection weights, or activation functions. This paper explains sufficient condition for the existence and uniform stability analysis of such networks. Three numerical simulations are delineated to substantiate the effectiveness of the theoretical results.

A mathematical model of cancer treatment by radiotherapy

A periodic mathematical model of cancer treatment by radiotherapy is presented and studied in this paper. Conditions on the coexistence of the healthy and cancer cells are obtained. Furthermore, sufficient conditions on the existence and globally asymptotic stability of the positive periodic solution, the cancer eradication periodic solution, and the cancer win periodic solution are established. Some numerical examples are shown to verify the validity of the results. A discussion is presented for further study.

Controllability of asynchronous Boolean multiplex control networks

In this article, the controllability of asynchronous Boolean multiplex control networks (ABMCNs) is studied. First, the model of Boolean multiplex control networks under Harvey' asynchronous update is presented. By means of semi-tensor product approach, the logical dynamics is converted into linear representation, and a generalized formula of control-depending network transition matrices is achieved. Second, a necessary and sufficient condition is proposed to verify that only control-depending fixed points of ABMCNs can be controlled with probability one. Third, using two types of controls, the controllability of system is studied and formulae are given to show: (a) when an initial state is given, the reachable set at time s under a group of specified controls; (b) the reachable set at time s under arbitrary controls; (c) the specific probability values from a given initial state to destination states. Based on the above formulae, an algorithm to calculate overall reachable states from a specified initial state is presented. Moreover, we also discuss an approach to find the particular control sequence which steers the system between two states with maximum probability. Examples are shown to illustrate the feasibility of the proposed scheme.

Contact-force distribution optimization and control for quadruped robots using both gradient and adaptive neural networks

This paper investigates optimal feet forces' distribution and control of quadruped robots under external disturbance forces. First, we formulate a constrained dynamics of quadruped robots and derive a reduced-order dynamical model of motion/force. Consider an external wrench on quadruped robots; the distribution of required forces and moments on the supporting legs of a quadruped robot is handled as a tip-point force distribution and used to equilibrate the external wrench. Then, a gradient neural network is adopted to deal with the optimized objective function formulated as to minimize this quadratic objective function subjected to linear equality and inequality constraints. For the obtained optimized tip-point force and the motion of legs, we propose the hybrid motion/force control based on an adaptive neural network to compensate for the perturbations in the environment and approximate feedforward force and impedance of the leg joints. The proposed control can confront the uncertainties including approximation error and external perturbation. The verification of the proposed control is conducted using a simulation.

Novel neural control for a class of uncertain pure-feedback systems

This paper is concerned with the problem of adaptive neural tracking control for a class of uncertain pure-feedback nonlinear systems. Using the implicit function theorem and backstepping technique, a practical robust adaptive neural control scheme is proposed to guarantee that the tracking error converges to an adjusted neighborhood of the origin by choosing appropriate design parameters. In contrast to conventional Lyapunov-based design techniques, an alternative Lyapunov function is constructed for the development of control law and learning algorithms. Differing from the existing results in the literature, the control scheme does not need to compute the derivatives of virtual control signals at each step in backstepping design procedures. Furthermore, the scheme requires the desired trajectory and its first derivative rather than its first n derivatives. In addition, the useful property of the basis function of the radial basis function, which will be used in control design, is explored. Simulation results illustrate the effectiveness of the proposed techniques.

Exponential input-to-state stability of recurrent neural networks with multiple time-varying delays

In this paper, input-to-state stability problems for a class of recurrent neural networks model with multiple time-varying delays are concerned with. By utilizing the Lyapunov-Krasovskii functional method and linear matrix inequalities techniques, some sufficient conditions ensuring the exponential input-to-state stability of delayed network systems are firstly obtained. Two numerical examples and its simulations are given to illustrate the efficiency of the derived results.

Multistability of two kinds of recurrent neural networks with activation functions symmetrical about the origin on the phase plane

In this paper, we investigate multistability of two kinds of recurrent neural networks with time-varying delays and activation functions symmetrical about the origin on the phase plane. One kind of activation function is with zero slope at the origin on the phase plane, while the other is with nonzero slope at the origin on the phase plane. We derive sufficient conditions under which these two kinds of n-dimensional recurrent neural networks are guaranteed to have (2m+1)(n) equilibrium points, with (m+1)(n) of them being locally exponentially stable. These new conditions improve and extend the existing multistability results for recurrent neural networks. Finally, the validity and performance of the theoretical results are demonstrated through two numerical examples.

Boundedness and complete stability of complex-valued neural networks with time delay

In this paper, the boundedness and complete stability of complex-valued neural networks (CVNNs) with time delay are studied. Some conditions to guarantee the boundedness of the CVNNs are derived using local inhibition. Moreover, under the boundedness conditions, a compact set that globally attracts all the trajectories of the network is also given. Additionally, several conditions in terms of real-valued linear matrix inequalities (LMIs) for complete stability of the CVNNs are established via the energy minimization method and the approach that converts the complex-valued LMIs to real-valued ones. Examples with simulation results are given to show the effectiveness of the theoretical analysis.

On stabilization of stochastic Cohen-Grossberg neural networks with mode-dependent mixed time-delays and Markovian switching

The globally exponential stabilization problem is investigated for a general class of stochastic Cohen-Grossberg neural networks with both Markovian jumping parameters and mixed mode-dependent time-delays. The mixed time-delays consist of both discrete and distributed delays. This paper aims to design a memoryless state feedback controller such that the closed-loop system is stochastically exponentially stable in the mean square sense. By introducing a new Lyapunov-Krasovskii functional that accounts for the mode-dependent mixed delays, stochastic analysis is conducted in order to derive delay-dependent criteria for the exponential stabilization problem. Three numerical examples are carried out to demonstrate the feasibility of our delay-dependent stabilization criteria.

Stability analysis for neural networks with time-varying delay based on quadratic convex combination

In this paper, a novel method is developed for the stability problem of a class of neural networks with time-varying delay. New delay-dependent stability criteria in terms of linear matrix inequalities for recurrent neural networks with time-varying delay are derived by the newly proposed augmented simple Lyapunov-Krasovski functional. Different from previous results by using the first-order convex combination property, our derivation applies the idea of second-order convex combination and the property of quadratic convex function which is given in the form of a lemma without resorting to Jensen's inequality. A numerical example is provided to verify the effectiveness and superiority of the presented results.

Stability for neural networks with time-varying delays via some new approaches

This paper considers the problem of delay-dependent stability criteria for neural networks with time-varying delays. First, by constructing a newly augmented Lyapunov-Krasovskii functional, a less conservative stability criterion is established in terms of linear matrix inequalities. Second, by proposing novel activation function conditions which have not been proposed so far, further improved stability criteria are proposed. Finally, three numerical examples used in the literature are given to show the improvements over the existing criteria and the effectiveness of the proposed idea.

Improved delay-dependent robust stability criteria for recurrent neural networks with time-varying delays

In this paper, the problem of improved delay-dependent robust stability criteria for recurrent neural networks (RNNs) with time-varying delays is investigated. Combining the Lyapunov-Krasovskii functional with linear matrix inequality (LMI) techniques and integral inequality approach (IIA), delay-dependent robust stability conditions for RNNs with time-varying delay, expressed in terms of quadratic forms of state and LMI, are derived. The proposed methods contain the least numbers of computed variables while maintaining the effectiveness of the stability conditions. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the present method. Numerical examples are included to show that the proposed method is effective and can provide less conservative results.

Robust exponential stability of uncertain stochastic neural networks with distributed delays and reaction-diffusions

This paper considers the problem of stability analysis for uncertain stochastic neural networks with distributed delays and reaction-diffusions. Two sufficient conditions for the robust exponential stability in the mean square of the given network are developed by using a Lyapunov-Krasovskii functional, an integral inequality, and some analysis techniques. The conditions, which are expressed by linear matrix inequalities, can be easily checked. Two simulation examples are given to demonstrate the reduced conservatism of the proposed conditions.

Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects

This paper focuses on the hybrid effects of parameter uncertainty, stochastic perturbation, and impulses on global stability of delayed neural networks. By using the Ito formula, Lyapunov function, and Halanay inequality, we established several mean-square stability criteria from which we can estimate the feasible bounds of impulses, provided that parameter uncertainty and stochastic perturbations are well-constrained. Moreover, the present method can also be applied to general differential systems with stochastic perturbation and impulses.

Global stability of complex-valued recurrent neural networks with time-delays

Since the last decade, several complex-valued neural networks have been developed and applied in various research areas. As an extension of real-valued recurrent neural networks, complex-valued recurrent neural networks use complex-valued states, connection weights, or activation functions with much more complicated properties than real-valued ones. This paper presents several sufficient conditions derived to ascertain the existence of unique equilibrium, global asymptotic stability, and global exponential stability of delayed complex-valued recurrent neural networks with two classes of complex-valued activation functions. Simulation results of three numerical examples are also delineated to substantiate the effectiveness of the theoretical results.

Solving the assignment problem using continuous-time and discrete-time improved dual networks

The assignment problem is an archetypal combinatorial optimization problem. In this brief, we present a continuous-time version and a discrete-time version of the improved dual neural network (IDNN) for solving the assignment problem. Compared with most assignment networks in the literature, the two versions of IDNNs are advantageous in circuit implementation due to their simple structures. Both of them are theoretically guaranteed to be globally convergent to a solution of the assignment problem if only the solution is unique.