Delay-range-dependent Stability Criterion for Interval Time-delay Systems with Nonlinear Perturbations

In this paper, we consider the problem of robust stability for a class of linear systems with interval time-varying delay under nonlinear perturbations using Lyapunov-Krasovskii （LK） functional approach. By partitioning the delay-interval into two segments of equal length, and evaluating the time-derivative of a candidate LK functional in each segment of the delay-interval, a less conservative delay-dependent stability criterion is developed to compute the maximum allowable bound for the delay-range within which the system under consideration remains asymptotically stable. In addition to the delay-bi-segmentation analysis procedure, the reduction in conservatism of the proposed delay-dependent stability criterion over recently reported results is also attributed to the fact that the time-derivative of the LK functional is bounded tightly using a newly proposed bounding condition without neglecting any useful terms in the delay-dependent stability analysis. The analysis, subsequently, yields a stable condition in convex linear matrix inequality （LMI） framework that can be solved non-conservatively at boundary conditions using standard numerical packages. Furthermore, as the number of decision variables involved in the proposed stability criterion is less, the criterion is computationally more effective. The effectiveness of the proposed stability criterion is validated through some standard numerical examples.

Delay-dependent robust stabilization for a class ofneutral systems with nonlinear perturbations

This note deals with the problem of stabilization/stability for neutral systems with nonlinear perturbations. A new stabilization/stability scheme is presented. Using improved Lyapunov functionals, less conservative stabilization/stability conditions are derived for such systems based on linear matrix inequalities （LMI）. Numerical examples are provided to show that the proposed results significantly improve the allowed upper bounds of the delay size over some existing ones in the literature.

Adaptive H_∞ Synchronization of Master-slave Systems with Mixed Time-varying Delays and Nonlinear Perturbations: An LMI Approach

This paper proposes an adaptive synchronization problem for the master and slave structure of linear systems with nonlinear perturbations and mixed time-varying delays comprising different discrete and distributed time delays. Using an appropriate Lyapunov-Krasovskii functional, some delay-dependent sufficient conditions and an adaptation law including the master-slave parame- ters are established for designing a delayed synchronization law in terms of linear matrix inequalities（LMIs）. The time-varying controller guarantees the H ∞ synchronization of the two coupled master and slave systems regardless of their initial states. Particularly, it is shown that the synchronization speed can be controlled by adjusting the updated gain of the synchronization signal. Two numerical examples are given to demonstrate the effectiveness of the method.

Robust Stability Criteria for Neutral Systems with Interval Time-Varying Delays and Nonlinear Perturbations

This paper investigates the problem of delay-dependent robust stability analysis for a class of neutral systems with interval time-varying delays and nonlinear perturbations. Such nonlinear perturbations are with time-varying but norm-bounded characteristics. Based on a new Lyapunov-Krasovskii functional together with a free-weighting matrices technique,improved delay-dependent stability criteria are established. It is shown that less conservative results can be obtained in terms of linear matrix inequalities( LMIs). Numerical examples are provided to demonstrate the effectiveness and less conservatism of the proposed approach.

A Variant Constrained Genetic Algorithm for Solving Conditional Nonlinear Optimal Perturbations

A variant constrained genetic algorithm （VCGA） for effective tracking of conditional nonlinear optimal perturbations （CNOPs） is presented. Compared with traditional constraint handling methods, the treatment of the constraint condition in VCGA is relatively easy to implement. Moreover, it does not require adjustments to indefinite pararneters. Using a hybrid crossover operator and the newly developed multi-ply mutation operator, VCGA improves the performance of GAs. To demonstrate the capability of VCGA to catch CNOPS in non-smooth cases, a partial differential equation, which has ＂on off＂ switches in its forcing term, is employed as the nonlinear model. To search global CNOPs of the nonlinear model, numerical experiments using VCGA, the traditional gradient descent algorithm based on the adjoint method （ADJ）, and a GA using tournament selection operation and the niching technique （GA-DEB） were performed. The results with various initial reference states showed that, in smooth cases, all three optimization methods are able to catch global CNOPs. Nevertheless, in non-smooth situations, a large proportion of CNOPs captured by the ADJ are local. Compared with ADJ, the performance of GA-DEB shows considerable improvement, but it is far below VCGA. Further, the impacts of population sizes on both VCGA and GA-DEB were investigated. The results were used to estimate the computation time of ~CGA and GA-DEB in obtaining CNOPs. The computational costs for VCGA, GA-DEB and ADJ to catch CNOPs of the nonlinear model are also compared.

Improved Exponential Stability Criteria for Uncertain Neutral System with Nonlinear Parameter Perturbations

This paper investigates the problem of robust exponential stability for neutral systems with time-varying delays and nonlinear perturbations. Based on a novel Lyapunov functional approach and linear matrix inequality technique, a new delay-dependent stability condition is derived. Since the model transformation and bounding techniques for cross terms are avoided, the criteria proposed in this paper are less conservative than some previous approaches by using the free-weighting matrices. One numerical example is presented to illustrate the effectiveness of the proposed results.

Inducing Unstable Grassland Equilibrium States Due to Nonlinear Optimal Patterns of Initial and Parameter Perturbations:Theoretical Models

Due to uncertainties in initial conditions and parameters, the stability and uncertainty of grassland ecosystem simulations using ecosystem models are issues of concern. Our objective is to determine the types and patterns of initial and parameter perturbations that yield the greatest instability and uncertainty in simulated grassland ecosystems using theoretical models. We used a nonlinear optimization approach, i.e., a conditional nonlinear optimal perturbation related to initial and parameter perturbations （CNOP） approach, in our work. Numerical results indicated that the CNOP showed a special and nonlinear optimal pattern when the initial state variables and multiple parameters were considered simultaneously. A visibly different complex optimal pattern characterizing the CNOPs was obtained by choosing different combinations of initial state variables and multiple parameters in different physical processes. We propose that the grassland modeled ecosystem caused by the CNOP-type perturbation is unstable and exhibits two aspects： abrupt change and the time needed for the abrupt change from a grassland equilibrium state to a desert equilibrium state when the initial state variables and multiple parameters are considered simultaneously. We compared these findings with results affected by the CNOPs obtained by considering only uncertainties in initial state variables and in a single parameter. The numerical results imply that the nonlinear optimal pattern of initial perturbations and parameter perturbations, especially for more parameters or when special parameters are involved, plays a key role in determining stabilities and uncertainties associated with a simulated or predicted grassland ecosystem.

Asymptotical solutions of coupled nonlinear Schrodinger equations with perturbations

In this paper Lou＇s direct perturbation method is applied to the perturbed coupled nonlinear Schrodinger equations to obtain their asymptotical solutions, which include not only the zero-order solutions but also the first-order modifications. Based on the asymptotical solutions, the effects of perturbations on soliton parameters and the collision between two solitons are then discussed in brief. Furthermore, we directly simulate the perturbed coupled nonlinear SchrSdinger equations by split-step Fourier method to check the validity of the direct perturbation method. It turns out that our analytical results are well supported by the numerical calculations.

Adaptive Stabilization for a Class of Dynamical Systems with Nonlinear Delayed State Perturbations

The problem of adaptive stabilization for a class of systems with nonlinear delayed state perturbations is considered. The bound of the perturbations is assumed to be unknown, by using the adaptive control method, an adaptive controller is designed. Based on the LyapunovKarasovskii functional, it is shown that the dynamical system can be stabilized by the adaptive controller. The effectiveness of the proposed controller is demonstrated by some simulations.

Stability of nonlinearly-perturbed systems with time varying delay using LMIs

This paper studies delay dependent robust stability and the stabilization problem of nonlinear perturbed systems with time varying delay. A new set of sufficient conditions for the stability of open as well as close loop systems are obtained in the sense of Lyapunov-Krasovskii. To reduce the conservatism, the work exploits the idea of splitting the delay interval into multiple equal regions so that less information on the time delay can be imposed to derive the results. The derived criterion not only improves the upper bounds of the time delay but also does not require the derivative of the delay to be known at prior. Easily testable sufficient criteria are presented in terms of linear matrix inequalities. It is shown that the derived conditions are very less conservative while comparing the maximum allowable upper bound of delay with the existing results in literature.

A NOVEL STOCHASTIC HEPATITIS B VIRUS EPIDEMIC MODEL WITH SECOND-ORDER MULTIPLICATIVE α-STABLE NOISE AND REAL DATA

This work presents an advanced and detailed analysis of the mechanisms of hepatitis B virus(HBV)propagation in an environment characterized by variability and stochas-ticity.Based on some biological features of the virus and the assumptions,the corresponding deterministic model is formulated,which takes into consideration the effect of vaccination.This deterministic model is extended to a stochastic framework by considering a new form of disturbance which makes it possible to simulate strong and significant fluctuations.The long-term behaviors of the virus are predicted by using stochastic differential equations with second-order multiplicative α-stable jumps.By developing the assumptions and employing the novel theoretical tools,the threshold parameter responsible for ergodicity(persistence)and extinction is provided.The theoretical results of the current study are validated by numerical simulations and parameters estimation is also performed.Moreover,we obtain the following new interesting findings:(a)in each class,the average time depends on the value ofα;(b)the second-order noise has an inverse effect on the spread of the virus;(c)the shapes of population densities at stationary level quickly changes at certain values of α.The last three conclusions can provide a solid research base for further investigation in the field of biological and ecological modeling.

Stability Analysis of Cyber-Physical Micro Grid Load Frequency Control System with Time-Varying Delay and Non-Linear Load Perturbations

In a cyber-physical micro-grid system,wherein the control functions are executed through open communication channel,stability is an important issue owing to the factors related to the time-delay encountered in the data transfer.Transfer of feedback variable as discrete data packets in communication network invariably introduces inevitable time-delays in closed loop control systems.This delay,depending upon the network traffic condition,inherits a time-varying characteristic;nevertheless,it adversely impacts the system performance and stability.The load perturbations in a micro-grid system are considerably influenced by the presence of fluctuating power generators like wind and solar power.Since these non-conventional energy sources are integrated into the power grid through power electronic interface circuits that usually works at high switching frequency,noise signals are introduced into the micro-grid system and these signals gets super-imposed to the load variations.Based on this back ground,in this paper,the delay-dependent stability issue of networked micro-grid system combined with time-varying feedback loop delay and uncertain load perturbations is investigated,and a deeper insight has been presented to infer the impact of time-delay on the variations in the system frequency.The classical Lyapunov-Krasovskii method is employed to address the problem,and using a standard benchmark micro-grid system,and the proposed stability criterion is validated.

Application of the Conditional Nonlinear Optimal Perturbation Method to the Predictability Study of the Kuroshio Large Meander

A reduced-gravity barotropic shallow-water model was used to simulate the Kuroshio path variations. The results show that the model was able to capture the essential features of these path variations. We used one simulation of the model as the reference state and investigated the effects of errors in model parameters on the prediction of the transition to the Kuroshio large meander （KLM） state using the conditional nonlinear optimal parameter perturbation （CNOP-P） method. Because of their relatively large uncertainties, three model parameters were considered： the interracial friction coefficient, the wind-stress amplitude, and the lateral friction coefficient. We determined the CNOP-Ps optimized for each of these three parameters independently, and we optimized all three parameters simultaneously using the Spectral Projected Gradient 2 （SPG2） algorithm. Similarly, the impacts caused by errors in initial conditions were examined using the conditional nonlinear optimal initial perturbation （CNOP-I） method. Both the CNOP-I and CNOP-Ps can result in significant prediction errors of the KLM over a lead time of 240 days. But the prediction error caused by CNOP-I is greater than that caused by CNOP-P. The results of this study indicate not only that initial condition errors have greater effects on the prediction of the KLM than errors in model parameters but also that the latter cannot be ignored. Hence, to enhance the forecast skill of the KLM in this model, the initial conditions should first be improved, the model parameters should use the best possible estimates.

Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP)

The conditional nonlinear optimal perturbation （CNOP）, which is a nonlinear generalization of the linear singular vector （LSV）, is applied in important problems of atmospheric and oceanic sciences, including ENSO predictability, targeted observations, and ensemble forecast. In this study, we investigate the computational cost of obtaining the CNOP by several methods. Differences and similarities, in terms of the computational error and cost in obtaining the CNOP, are compared among the sequential quadratic programming （SQP） algorithm, the limited memory Broyden-Fletcher-Goldfarb-Shanno （L-BFGS） algorithm, and the spectral projected gradients （SPG2） algorithm. A theoretical grassland ecosystem model and the classical Lorenz model are used as examples. Numerical results demonstrate that the computational error is acceptable with all three algorithms. The computational cost to obtain the CNOP is reduced by using the SQP algorithm. The experimental results also reveal that the L-BFGS algorithm is the most effective algorithm among the three optimization algorithms for obtaining the CNOP. The numerical results suggest a new approach and algorithm for obtaining the CNOP for a large-scale optimization problem.

Applications of Conditional Nonlinear Optimal Perturbation to the Study of the Stability and Sensitivity of the Jovian Atmosphere

A two-layer quasi-geostrophic model is used to study the stability and sensitivity of motions on smallscale vortices in Jupiter＇s atmosphere. Conditional nonlinear optimal perturbations （CNOPs） and linear singular vectors （LSVs） are both obtained numerically and compared in this paper. The results show that CNOPs can capture the nonlinear characteristics of motions in small-scale vortices in Jupiter＇s atmosphere and show great difference from LSVs under the condition that the initial constraint condition is large or the optimization time is not very short or both. Besides, in some basic states, local CNOPs are found. The pattern of LSV is more similar to local CNOP than global CNOP in some cases. The elementary application of the method of CNOP to the Jovian atmosphere helps us to explore the stability of variousscale motions of Jupiter＇s atmosphere and to compare the stability of motions in Jupiter＇s atmosphere and Earth＇s atmosphere further.

ON THE SINGULAR PERTURBATION OF A NONLINEAR ORDINARY DIFFERENTIAL EQUATION WITH TWO PARAMETERS

In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and the remainders have been estimated.

NONLINEAR SINGULARLY PERTURBED PREDATOR-PREY REACTION DIFFUSION SYSTEMS

A class of nonlinear predator prey reaction diffusion systems for singularly pe rturbed problems are considered.Under suitable conditions, by using theory of di fferential inequalities the existence and asymptotic behavior of solution for in itial boundary value problems are studied.

THE PROBLEMS OF NONLINEAR BENDING FOR ORTHOTROPIC RECTANGULAR PLATE WITH FOUR CLAMPED EDGES

In this paper,under the non-uniformtransverse load,the problems of nonlinear bending for orthotropic rectangular plate are studied by using'the method of twovariable'[1]and 'the method of mixing perturbation'[2].The uniformly valid asymptotic solutions of Nth-order for ε1 and Mth-order for ε2 for ortholropic rectangular plale with four clamped edges are oblained.

Global Solution for Quasilinear Wave Equations with Viscosity and Nonlinear Perturbation

In this paper we prove the global existence and decay of solutions to the initial-boundary value problem for the quasilinear wave equations with viscosity and a nonlinear perturbation.

Connection between Volterra series and perturbation method in nonlinear systems analyses

This paper is concerned with the connection between the Volterra series and the regular perturbation method in nonlinear systems analyses. It is revealed for the first time that, for a forced polynomial nonlinear system, if its derived linear system is a damped dissipative system, the steady response obtained through the regular perturbation method is exactly identical to the response given by the Volterra series. On the other hand, if the derived linear system is an undamped conservative system, then the Volterra series is incapable of modeling the forced polynomial nonlinear system. Numerical examples are further presented to illustrate these points. The results provide a new criterion for quickly judging whether the Volterra series is applicable for modeling a given polynomial nonlinear system.