Global attractivity and oscillation in a nonlinear periodic delay difference equation

The existence of a periodic solution for a class of nonlinear delay difference equations with periodic coefficients is established. Some sufficient conditions for the global attractivity and oscillation about the periodic solutions are also obtained.     

Active queue management for alleviating Internet congestion via a nonlinear differential equation with a variable delay

Active Queue Management (AQM) for mitigating Internet congestion has been addressed via various feedback control syntheses, especially P, PI, and PID regulators, by using a linear approximation where the “round trip time,” i.e., the delay, is assumed to be constant. This constraint is lifted here by using a nonlinear modeling with a variable delay, introduced more than 20 years ago. This delay, intimately linked to the congestion phenomenon, may be viewed as “ a flat output.” All other system variables, especially the control variable, i.e., the packet loss ratio, are expressed as a function of the delay and its derivatives: they are frozen if the delay is kept constant. This flatness-like property, which demonstrates the mathematical discrepancy of the linear approximation adopted until today, yields also our control strategy in two steps: Firstly, designing an open-loop control, thanks to straightforward Flatness-Based Control (FBC) techniques, and secondly, closing the loop via Model-Free Control (MFC) in order to take into account severe model mismatches, like, here, the number of TCP sessions. Several convincing computer simulations, which are easily implementable, are presented and discussed.     

具有变时滞阶段结构捕食系统的全局吸引性

研究了一类具有变时滞相互干扰阶段结构捕食系统。利用微分不等式比较原理和迭代法得到了系统平衡态全局吸引的充分条件,通过举例并进行数值模拟验证了结论的可行性。     

Stabilisation of second-order nonlinear equations with variable delay

For a wide class of second-order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal allows to stabilise unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilisation by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points.     

Global satisfactory control for nonlinear integrator processes with long delay

Integrator processes with long delay are difficult to control. Nonlinear characteristics of actuators make the control problem more challenging. A technique is proposed in this paper for global satisfactory control (GSC) of such processes with relay-type nonlinearity. An oscillatory control signal is injected into the nonlinear process; the amplitude and frequency of the oscillatory signal are designed to linearise the nonlinear process in the sense of harmonic analysis; and a state feedback controller is configured to implement GSC over the linearised process. An illustrative example is given to demonstrate the effectiveness of the proposed method.     

Global satisfactory control for nonlinear integrator processes with long delay

Integrator processes with long delay are difficult to control. Nonlinear characteristics of actuators make the control problem more challenging. A technique is proposed in this paper for global satisfactory control （GSC） of such processes with relay-type nonlinearity. An oscillatory control signal is injected into the nonlinear process; the amplitude and frequency of the oscillatory signal are designed to linearise the nonlinear process in the sense of harmonic analysis; and a state feedback controller is configured to implement GSC over the linearised process. An illustrative example is given to demonstrate the effectiveness of     

Monotone methods for a finite difference system of reaction diffusion equation with time delay

The aim of this paper is to present some monotone iterative schemes for computing the solution of a system of nonlinear difference equations which arise from a class of nonlinear reaction-diffusion equations with time delays. The iterative schemes lead to computational algorithms as well as existence, uniqueness, and upper and lower bounds of the solution. An application to a diffusive logistic equation with time delay is given.     

Global stability for separable nonlinear delay differential equations

Consider the following separable nonlinear delay differential equation

, where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that

In this paper, to the above separable nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f(x) which contains a case of f(x) = e^x−1, we give more explicit conditions. Applying these, we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays.     

Global stabilisation of nonlinear delay systems with a compact absorbing set

Predictor-based stabilisation results are provided for nonlinear systems with input delays and a compact absorbing set. The control scheme consists of an inter-sample predictor, a global observer, an approximate predictor, and a nominal controller for the delay-free case. The control scheme is applicable even to the case where the measurement is sampled and possibly delayed. The input and measurement delays can be arbitrarily large but both of them must be constant and accurately known. The closed-loop system is shown to have the properties of global asymptotic stability and exponential convergence in the disturbance-free case, robustness with respect to perturbations of the sampling schedule, and robustness with respect to measurement errors. In contrast to existing predictor feedback laws, the proposed control scheme utilises an approximate predictor of a dynamic type that is expressed by a system described by integral delay equations. Additional results are provided for systems that can be transformed to systems with a compact absorbing set by means of a preliminary predictor feedback.     

Global asymptotic stability of a nonlinear parabolic equation

Necessary and sufficient conditions are derived for the asymptotic stability of a nonlinear parabolic equation. Spatial Fourier transformation converts the stability determination to that for an infinite system of ordinary differential equations, and facilitates approximate to exact solutions, depending on the boundary conditions, the nonlinearities, and the harmonic content of the steady-state disturbances.     

New criteria for exponential stability of nonlinear difference systems with time-varying delay

General nonlinear time-varying difference systems with time-varying delay are considered. Some new explicit criteria for global exponential stability are given. Two examples are given to illustrate the obtained results.     

Global attractivity in a competition system with feedback controls

Sufficient conditions are derived for the existence of a globally attracting positive equilibrium of a two species competition system with feedback controls; the indirect controls can act instantaneously or with a fixed discrete delay.     

Global stability analysis scheme for a class of nonlinear time delay systems

We consider a class of nonlinear time delay systems created by generalizing the model for FAST TCP, an Internet congestion control algorithm. We achieve a time delay independent sufficient condition for the global asymptotic stability of the class of systems. The sufficient condition is verified by constructing two sequences that represent the lower and upper bound variations of the system trajectory in time, and showing that the two sequences converge to the equilibrium point of the system. The simulation results exemplify that the sufficient condition is valid for global asymptotic stability, and that the sufficient condition is a close approximation to the unknown necessary and sufficient condition for global asymptotic stability.     

A finite difference method for korteweg-de vries like equation with nonlinear dispersion

The Korteweg-de Vries (KdV) equation has been generalized by Rosenau and Hyman [3] to a class of partial differential equations (PDEs) which has solitary wave solution with compact support. These solitary wave solutions are called compactons. Compactons are solitary waves with the remarkable soliton property, that after colliding with other compactons, they reemerge with the same coherent shape. These particle like waves exhibit elastic collision that are similar to the soliton interaction associated with completely integrable systems. The point where two compactons collide are marked by a creation of low amplitude compacton-anticompacton pair. These equations have only a finite number of local consevation laws. In this paper, an implicit numerical method has been developed to solve the K(2,3) equation. Accuracy and stability of the method have been studied. The analytical solution and the conserved quantities are used to assess the accuracy of the suggested method. The numerical results have shown that this compacton exhibits true soliton behavior.

     

Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations

The nonlinear delay differential equation with exponential and quadratic nonlinearities is considered. It is assumed that the equation is exposed to stochastic perturbations of the white noise type, which are directly proportional to the deviation of the system state from the equilibrium point. Sufficient conditions for stability in probability of the zero and positive equilibriums of the considered system under stochastic perturbations are obtained. The research results are illustrated by numerical simulations. The proposed investigation procedure can be applied for arbitrary nonlinear stochastic delay differential equations with an order of nonlinearity higher than one. Copyright © 2016 John Wiley & Sons, Ltd.     

General finite difference approximation for the wave equation with variable coefficients using a cubic spline technique

General implicit finite difference approximations for the wave equation with variable coefficients in one and two space variables are derived by using a cubic spline function and continuous parameter approach. An optimal scheme of the family and high accuracy schemes due to McKee are obtained for special cases of the parameters. The stability of the schemes are discussed and one test example is solved to illustrate the theory.