Global convergence of serial Boolean networks based on algebraic representation

This paper analyzes the global convergence of serial Boolean networks (SBNs) via the semi-tensor product of matrices, and presents some new results. Firstly, an algebraic representation is obtained for SBNs, and an algorithm is established for the conversion between the algebraic representations of SBNs and the corresponding Boolean networks. Secondly, the non-equivalence of global convergence between SBNs and the corresponding Boolean networks is revealed, although they have the same fixed points. Thirdly, a necessary and sufficient condition is presented for the global convergence of SBNs. Finally, the obtained results are applied to the evolutionary behaviour analysis of evolutionary networked games with cascading myopic best response adjustment.     

Reliable stability and stabilizability for complex-valued memristive neural networks with actuator failures and aperiodic event-triggered sampled-data control

This study addresses the stability and stabilizability problems for complex-valued memristive neural networks (CVMNNs) with actuator failures via reliable aperiodic event-triggered sampled-data control. Different from the traditional control methods with time-triggered mechanism, an aperiodic event-triggered sampled-data control scheme is first proposed for CVMNNs. Taking the influence of actuator failures into account, a reliable controller is designed. In comparison with the existing control approaches, the one here is not only more applicable but effective to save the communication resources for CVMNNs. Then, a new Lyapunov–Krasovskii functional (LKF) is introduced, which can fully capture the information of sampling and complex-valued activation functions. Based on the LKF and some new estimation techniques, novel stability and stabilizability criteria are established, and the desired reliable aperiodic event-triggered sampled-data controller gains are obtained simultaneously. Finally, numerical simulations are provided to verify the effectiveness of the obtained theoretical results.     

Set stabilizability of switched Boolean control networks via Ledley antecedence solution

This paper studies two kinds of set stabilizability issues of switched Boolean control networks (SBCNs) by Ledley antecedence solution, that is, pointwise set stabilizability and set stabilizability under arbitrary switching signals. Firstly, based on the state transition matrix of SBCNs, the mode-dependent truth matrix is defined. Secondly, using the mode-dependent truth matrix in every step, a switching signal and the corresponding Ledley antecedence solutions are determined. Furthermore, a state feedback switching signal and a state feedback control are obtained for the pointwise set stabilizability. Thirdly, with the help of all mode-dependent truth matrices, the Ledley antecedence solutions are derived for a set of Boolean inclusions, which admits a state feedback control for the set stabilizability under arbitrary switching signals. Finally, an example is given to show the effectiveness of the proposed results.     

Output tracking of switched Boolean networks under open-loop/closed-loop switching signals

This paper addresses the output tracking problem of switched Boolean networks (SBNs) via the semi-tensor product method, and presents a number of new results. Firstly, the concept of switching-output-reachability is proposed for SBNs, based on which, a necessary and sufficient condition is presented for the output tracking of SBNs under arbitrary open-loop switching signal. Secondly, a constructive procedure is proposed for the design of closed-loop switching signals for SBNs to track a constant reference signal. The study of an illustrative example shows that the obtained new results are very effective.     

Robust μ-stability analysis of Markovian switching uncertain stochastic genetic regulatory networks with unbounded time-varying delays

This paper investigates the global robust stability problem of Markovian switching uncertain stochastic genetic regulatory networks with unbounded time-varying delays and norm bounded parameter uncertainties. The structure variations at discrete time instances during the process of gene regulations known as hybrid genetic regulatory networks based on Markov process is proposed. The jumping parameters considered here are generated from a continuous-time discrete-state homogeneous Markov process, which are governed by a Markov process with discrete and finite state space. The concept of global robust μ-stability in the mean square for genetic regulatory networks is given. Based on Lyapunov function, stochastic theory and Itô’s differential formula, the stability criteria are presented in the form of linear matrix inequalities (LMIs). Numerical examples are presented to demonstrate the effectiveness of the main result.     

Synchronization of drive–response singular Boolean networks

Network synchronization, explicating typical collective behaviors of coupled systems, plays a crucial role in social production and life. This paper addresses the synchronization problem of drive–response singular Boolean networks (SBNs). The solvability of drive–response SBNs is investigated based on the matrix representation. In view of the existence and uniqueness of the solutions to drive–response SBNs, three types of concepts, synchronization, strong synchronization and weak synchronization, are put forward for the first time. By two new systems, a restricted BN and a switched restricted BN, which are constructed from the considered systems, several synchronization conditions are provided to deal with the circumstances of unique solutions and multiple solutions, respectively. Besides, the synchronous ratio is defined to characterize the synchronization capability of drive–response SBNs for the case of multiple solutions. Finally, several examples are given to illustrate the effectiveness of the obtained results.     

Robust stability analysis of stochastic neural networks with Markovian jumping parameters and probabilistic time‐varying delays

This article discusses the issue of robust stability analysis for a class of Markovian jumping stochastic neural networks (NNs) with probabilistic time‐varying delays. The jumping parameters are represented as a continuous‐time discrete‐state Markov chain. Using the stochastic stability theory, properties of Brownian motion, the information of probabilistic time‐varying delay, the generalized Ito's formula, and linear matrix inequality (LMI) technique, some novel sufficient conditions are obtained to guarantee the stochastical stability of the given NNs. In particular, the activation functions considered in this article are reasonably general in view of the fact that they may depend on Markovian jump parameters and they are more general than those usual Lipschitz conditions. The main features of this article are described in the following: first one is that, based on generalized Finsler lemma, some improved delay‐dependent stability criteria are established and the second one is that the nonlinear stochastic perturbation acting on the system satisfies a class of Lipschitz linear growth conditions. By resorting to the Lyapunov–Krasovskii stability theory and the stochastic analysis tools, sufficient stability conditions are established using an efficient LMI approach. Finally, two numerical examples and its simulations are given to demonstrate the usefulness and effectiveness of the proposed results. © 2014 Wiley Periodicals, Inc. Complexity 21: 59–72, 2016     

Neural networks for solving second-order cone constrained variational inequality problem

In this paper, we consider using the neural networks to efficiently solve the second-order cone constrained variational inequality (SOCCVI) problem. More specifically, two kinds of neural networks are proposed to deal with the Karush-Kuhn-Tucker (KKT) conditions of the SOCCVI problem. The first neural network uses the Fischer-Burmeister (FB) function to achieve an unconstrained minimization which is a merit function of the Karush-Kuhn-Tucker equation. We show that the merit function is a Lyapunov function and this neural network is asymptotically stable. The second neural network is introduced for solving a projection formulation whose solutions coincide with the KKT triples of SOCCVI problem. Its Lyapunov stability and global convergence are proved under some conditions. Simulations are provided to show effectiveness of the proposed neural networks.     

Asymptotic stability and stabilizability of special classes of discrete-time positive switched systems

In this paper we consider discrete-time positive switched systems, switching among autonomous subsystems, characterized either by monomial matrices or by circulant matrices. Necessary and sufficient conditions are provided guaranteeing either (global uniform) asymptotic stability or stabilizability (i.e. the possibility of driving to zero the state trajectory corresponding to any initial state by resorting to some switching sequence). Such conditions lead to simple algorithms that allow to easily detect, under suitable conditions, whether a given positive switched system is not stabilizable.     

A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach

In this paper, global asymptotic stability is discussed for neural networks with time-varying delay. Several new criteria in matrix inequality form are given to ascertain the uniqueness and global asymptotic stability of equilibrium point for neural networks with time-varying delay based on Lyapunov method and Linear Matrix Inequality (LMI) technique. The proposed LMI approach has the advantage of considering the difference of neuronal excitatory and inhibitory efforts, which is also computationally efficient as it can be solved numerically using recently developed interior-point algorithm. In addition, the proposed results generalize and improve previous works. The obtained criteria also combine two existing conditions into one generalized condition in matrix form. An illustrative example is also given to demonstrate the effectiveness of the proposed results.     

Global stability of stage-structured predator–prey models with Beddington–DeAngelis functional response

Two stage-structured predator–prey systems with Beddington–DeAngelis functional response are proposed. The first one is deterministic. The Second one takes the random perturbation into account. For each system, sufficient conditions for global asymptotic stability are established. Some simulation figures are introduced to support the analytical findings.     

Global optimization in stabilizing controller design

In this paper, we develop a global optimization methodology to solve stabilization problems. We first formulate stabilization problems as bilevel programming problems. By invoking the Hurwitz stability conditions, we reformulate these bilevel programs to equivalent single-level nonconvex optimization programs. The branch-and-reduce global optimization algorithm is finally applied to these problems. Using the proposed methodology, we report improved solutions for two feedback stabilization problems from the literature. In addition, we improve the lower bound of the stabilizability parameter of the Belgian chocolate problem from the previous best known 0.96 to 0.973974.     

Dynamical Systems with Multiple Time-Varying Delays: Stability and Stabilizability

This paper deals with the class of uncertain systems with multiple time delays. The stability and stabilizability of this class of systems are considered. System robustness is also studied when the norm-bounded uncertainties are considered. LMI delay-dependent sufficient conditions for stability, stabilizability, and system robustness are established to check whether a system of this class is stable and/or stabilizable. Numerical examples are provided to show the usefulness of the proposed results.     

Stability properties for Hopfield neural networks with delays and impulsive perturbations

In this paper, we consider the uniform asymptotic stability, global asymptotic stability and global exponential stability of the equilibrium point of Hopfield neural networks with delays and impulsive perturbation. Some new stability criteria for such system are derived by using the Lyapunov functional method and the linear matrix inequality approach. The results are related to the size of delays and impulses. Our results are less restrictive and conservative than that given in some earlier references. Finally, two numerical examples showing the effectiveness of the present criteria are given.     

Stability analysis of impulsive delayed switched systems and applications

In this paper, a general class of impulsive delayed switched systems is considered. By employing the Lyapunov–Razumikhin method and some analysis techniques, we established several global asymptotic stability and global exponential stability criteria for the considered impulsive delayed switched systems, which improve and extend some recent works. As an application, the result of global exponential stability are used to study a class of uncertain linear switched systems with time‐varying delays. Several LMI‐based conditions are proposed to guarantee the global robust stability and global exponential stabilization. The designed memoryless state feedback controller can be easily checked by the LMI toolbox in Matlab. Moreover, the dwell time constraint is imposed for the switching law. Finally, two numerical examples and their simulations are given to show the effectiveness of our proposed results. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical conformal mapping based on the generalised conjugation operator

An iterative procedure for numerical conformal mapping is presented which imposes no restriction on the boundary complexity. The formulation involves two analytically equivalent boundary integral equations established by applying the conjugation operator to the real and the imaginary parts of an analytical function. The conventional approach is to use only one and ignore the other equation. However, the discrete version of the operator using the boundary element method (BEM) leads to two non-equivalent sets of linear equations forming an over-determined system. The generalised conjugation operator is introduced so that both sets of equations can be utilised and their least-square solution determined without any additional computational cost, a strategy largely responsible for the stability and efficiency of the proposed method. Numerical tests on various samples including problems with cracked domains suggest global convergence, although this cannot be proved theoretically. The computational efficiency appears significantly higher than that reported earlier by other investigators.

     

Lyapunov Stabilizability of Controlled Diffusions via a Superoptimality Principle for Viscosity Solutions

We prove optimality principles for semicontinuous bounded viscosity solutions of Hamilton-Jacobi-Bellman equations. In particular, we provide a representation formula for viscosity supersolutions as value functions of suitable obstacle control problems. This result is applied to extend the Lyapunov direct method for stability to controlled Ito stochastic differential equations. We define the appropriate concept of the Lyapunov function to study stochastic open loop stabilizability in probability and local and global asymptotic stabilizability (or asymptotic controllability). Finally, we illustrate the theory with some examples.     

Robust exponential stability and stabilizability of linear parameter dependent systems with delays

The robust exponential stability and stabilizability problems are addressed in this paper for a class of linear parameter dependent systems with interval time-varying and constant delays. In this paper, restrictions on the derivative of the time-varying delay is not required which allows the time-delay to be a fast time-varying function. Based on the Lyapunov-Krasovskii theory, we derive delay-dependent exponential stability and stabilizability conditions in terms of linear matrix inequalities (LMIs) which can be solved by various available algorithms. Numerical examples are given to illustrate the effectiveness of our theoretical results.     

Long-term dynamics of discrete-time predator–prey models: stability of equilibria,cycles and chaos

ABSTRACT

In [A.S. Ackleh, M.I. Hossain, A. Veprauskas, and A. Zhang, Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Differ. Equ. Appl. 25 (2019), pp. 1568–1603.], we established conditions for the persistence and local asymptotic stability of the interior equilibrium for two discrete-time predator–prey models (one without and with evolution to resist toxicants). In the current paper, we provide a more in-depth analysis of these models, including global stability of equilibria, existence of cycles and chaos. Our main focus is to examine how the speed of evolution ν may impact population dynamics. For both models, we establish conditions under which the interior equilibrium is global asymptotically stable using perturbation analysis together with the construction of Lyapunov functions. For small ν, we show that the global dynamics of the evolutionary system are nothing but a continuous perturbation of the non-evolutionary system. However, when the speed of evolution is increased, we perform numerical studies which demonstrate that evolution may introduce rich dynamics including cyclic and chaotic behaviour that are not observed when evolution is absent.