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The exponential stability problem of uncertain neutral complex system with Markovian switching is investigated in the presence of nonlinear perturbations and partial information on transition rates. The study begins to consider the related nominal systems and construct a novel augmented stochastic Lyapunov functional which contains some triple-integral terms to reduce the conservatism. Then the exponential stability criteria are developed by utilizing Lyapunov stability theory, reciprocally convex lemma and free-weighting matrices. The results are further extended to the corresponding uncertain case. Finally, numerical examples are given to illustrate the effectiveness of the proposed methods.

In this paper, we proposes and analyzes the mixed 4th-order Runge-Kutta scheme of conditional nonlinear perturbation (CNOP) approach for the EI Niño-Southern Oscillation (ENSO) model. This method consists of solving the ENSO model by using a mixed 4th-order Runge-Kutta method. Convergence, the local and global truncation error of this mixed 4th-order Runge-Kutta method are proved. Furthermore, optimal control problem is developed and the gradient of the cost function is determined.

The impulsive synchronisation of singular hybrid coupled systems with time-varying nonlinear perturbation is discussed in this paper. The synchronising impulsive effects are considered for singular hybrid coupled systems. A sufficient condition that ensures the synchronisation of singular impulsive dynamical networks is derived by applying a single pinning impulsive controller. Moreover, the convergence rate of the stability is estimated for the discussed dynamical network. Finally, simulation result is given to illustrate the effectiveness of the developed criteria.

In this note, we consider perturbed linear functional difference equations with discrete and distributed delays which play a fundamental role in several stability and stabilizability problems of time-delay systems. Linear and nonlinear perturbations are considered. Sufficient conditions for exponential stability of the perturbed solutions are given.

We consider a nonlinear system of equations governing the motion of a viscosity-layered fluid with a free surface in long-wave approximation. Using the semi-Lagrangian coordinates, we rewrite the governing equations in the integro-differential form and obtain necessary and sufficient hyperbolicity conditions. We approximate the integro-differential model by a finite-dimensional system of differential conservation laws and propose a model of propagation of nonlinear perturbations in a viscosity-stratified fluid.

In this paper, we study the solitary wave solution and numerical simulation for the generalized Rosenau–Kawahara-RLW equation with generalized Novikov type nonlinear perturbation, which is an extension of our recent work He and Pan (Appl Math Comput 271:323–336, 2015), He (Nonlinear Dyn 82:1177–1190, 2015). We first derive the exact solitary wave solution for the newly proposed perturbed Rosenau–Kawahara-RLW equation with power law nonlinearity and then develop a three-level linearly implicit difference scheme for solving the equation. We prove that the proposed scheme is energy-conserved, unconditionally stable and second-order convergent both in time and space variables. Finally, numerical experiments are carried out to confirm the energy conservation, the convergence rates of the scheme and effectiveness for long-time simulation.

Summary This paper investigates the problem of delay-dependent robust stochastic stabilization and H∞ control for uncertain stochastic nonlinear systems with time-varying delay. System uncertainties are assumed to be norm bounded. Firstly, by using novel method to deal with the integral terms, robustly stochastic stabilization results are obtained for stochastic uncertain systems with nonlinear perturbation, and an appropriate memoryless state feedback controller can be chosen. Compared with previous results, the new technique can sufficiently utilize more negative items information. Then, robust H∞ control for uncertain stochastic system with time-varying delay and nonlinear perturbation is considered, and the controller is designed, which will guarantee that closed-loop system is robustly stochastically stable with disturbance attenuation level. Finally, two numerical examples are listed to illustrate that our results are effective and less conservative than other reports in previous literature. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, the problem of finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations is investigated. By constructing a novel Lyapunov-Krasovskii functional and employing a new summation inequality named discrete Wirtinger-based inequality, reciprocally convex approach and zero equality, the improved finite-time stability criteria are derived to guarantee that the state of the system with time-varying delay does not exceed a given threshold when fixed time interval. Furthermore, the obtained conditions are formulated in forms of linear matrix inequalities which can be solved by using some standard numerical packages. Finally, three numerical examples are given to show the effectiveness and less conservatism of the proposed method.

This paper deals with the following Choquard equation with a local nonlinear perturbation: − Δ u + u = I α ∗ | u | α N + 1 | u | α N − 1 u + f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where I α : R N → R is the Riesz potential, N ≥ 3 , α ∈ ( 0 , N ) , the exponent α N + 1 is critical with respect to the Hardy–Littlewood–Sobolev inequality, and the nonlinear perturbation f is only required to satisfy some weak assumptions near 0 and ∞ . Our results improve the previous related ones in the literature.

The novel delay-dependent asymptotic stability of a differential and Riemann-Liouville fractional differential neutral system with constant delays and nonlinear perturbation is studied. We describe the new asymptotic stability criterion in the form of linear matrix inequalities (LMIs), using the application of zero equations, model transformation and other inequalities. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we not only present the improved delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with single constant delay but also the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral equation with constant delays. Numerical examples are exploited to represent the improvement and capability of results over another research as compared with the least upper bounds of delay and nonlinear perturbation.

In this paper, the problem of robust stability analysis for a class of uncertain systems with interval time-varying delay and nonlinear perturbations is studied. In order to develop a less conservative stability condition, a Lyapunov-Krasovskii functional (LKF) comprising quadruple-integral term is introduced. A novel delay dependent stability criterion in terms of linear matrix inequalities (LMIs) is given by using a new delay-partitioning approach and reciprocally convex combination technique, which is derived by integral inequality approach (IIA). Compared with the existing literature, this criterion can greatly reduce the complexity of theoretical derivation and computation. Finally, three well-known numerical comparative examples are given to verify the superiority of the proposed approach in reducing the conservation of conclusion.

This paper considers the problem of event-triggered filtering for Markovian jump systems with time delay and nonlinear perturbation. The purpose is to design a filter such that the filtering error system is stochastically stable with a prescribed mixed passivity and $H_{\infty}$ performance level. The existence condition for such filters is proposed, and the filter coefficients are found by solving a set of linear matrix inequalities. The usefulness of the filter design method is demonstrated by a numerical example.

The solution, obtained in the environment of finite precision machine arithmetic must be allays accompanied by an analysis of the conditioning of the problem solved. The perturbation analysis derives measures for the sensitivity of the solution to perturbations in the matrix coefficients. Motivated by these, in order to ascertain the accuracy of an iteratively calculated solution to a nonlinear matrix equation arising in Tree-like stochastic processes, in this paper norm-wise, mixed and component-wise condition numbers, as well as local perturbation bounds are formulated and norm-wise non-local residual bounds are derived using the methods of nonlinear perturbation analysis (Frechet derivatives, Lyapunov majorants, fixed-point principles). The residual bounds are formulated in terms of the computed approximate solution to the equation and can be used as a stop criteria of the iterations, when solving the considered nonlinear matrix equation by a numerically stable iterative algorithm.

This chapter is concerned with the exponential stability problem for a class of uncertain neutral switched systems with interval time-varying mixed delays and nonlinear perturbations. By adopting the piece-wise Lyapunov functional technique and the average dwell time approach, some sufficient conditions are first proposed, in terms of a set of LMIs that guarantee the robustly exponential stability for the uncertain neutral switched systems, where the decay estimate is explicitly given to quantify the convergence rate. Finally, there numerical examples are introduced to show the effectiveness of the main results.

We consider a class of the second-order quasilinear differential equations. By deriving relations between certain types of monotonic solutions of the quasilinear equation and corresponding reciprocal half-linear equation on a finite interval (a, b), we obtain criteria for all solutions of the main equation, which do not change sign in (a, b), to be non-monotonic in (a, b). This work is also extended to a perturbed half-linear equation as well as to the half-line $$(a,\infty )$$ .

Asymptotic expansions for stationary and conditional quasi-stationary distributions of nonlinearly perturbed birth-death-type semi-Markov models are presented. Applications to models of population ...

We consider a system described by the linear heat equation with adiabatic boundary conditions. We impose a nonlinear perturbation determined by a family of interval maps characterized by a certain set of parameters. The time instants of the perturbation are determined by an additional dynamical system, seen here as part of the external interacting system. We analyse the complex behaviour of the system, through the scope of symbolic dynamics, and the dependence of the behaviour on the time pattern of the perturbation, comparing it with previous results in the periodic case.

Summary This paper deals with the fault detection and isolation (FDI) problem for uncertain closed-loop systems with external disturbances and nonlinear perturbations. To address the system uncertainties and the nonlinear perturbations in different faulty models, adaptive and switching techniques are introduced to construct a bank of FDI observers, such that one of them can match the current system, and the corresponding observer estimate errors can converge asymptotically to zero. An effective FDI scheme is then presented by introducing some model-matching indexes. Moreover, the introduced switching laws liberate the equality constraints often used in the existing FDI approaches, which are hard to satisfy if the system matrices include uncertainties. Finally, a simulation example of F/A-18A automatic carrier landing system is used to illustrate the effectiveness of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

This paper investigates the finite-time stability problem for a class of time-delay switched systems with nonlinear perturbation and delayed impulse effects. Based on the Lyapunov function method and the technique of inequalities, stability criteria are established to guarantee that the state trajectory of the system does not exceed a certain threshold over a pre-specified finite time interval. Compared with existing results for related problems, the obtained results can be applied to a larger class of hybrid delayed systems, including those in which all of the subsystems are stable and unstable. Two examples are given to demonstrate the validity of the main results.

In this paper, a sliding mode control (SMC) with passivity of uncertain singular Markovian jump systems (SMJS) with time-varying delay and nonlinear perturbations is investigated. Firstly, a new integral-type switching surface is designed. Secondly, by introducing free matrix method and Lyapunov functionals, a delay-dependent sufficient condition in terms of strict linear matrix inequalities (LMI) is derived for the resulting sliding mode dynamics such that the closed-loop system is stochastically admissible and robustly passive for all admissible uncertainties. Thirdly, a SMC law is synthesized to drive the system trajectories onto the predefined switching surface in a finite time. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed method.

In this paper we study the existence of solutions for fractional Schrodinger equations of the form where V is a potential bounded and the nonlinear term has the critical exponential growth. We prove the existence of at least one weak solution by combining the mountain-pass theorem with the Trudinger–Moser inequality and a version of a result due to Lions for critical growth in .

Let Ω o and Ω i be open bounded regular subsets of R n such that the closure of Ω i is contained in Ω o . Let f o be a regular function on ∂ Ω o and let F and G be continuous functions from ∂ Ω i × R to R . By exploiting an argument based on potential theory and on the Leray–Schauder principle we show that under suitable and completely explicit conditions on F and G there exists at least one pair of continuous functions ( u o , u i ) such that { Δ u o = 0 in Ω o ∖ cl Ω i , Δ u i = 0 in Ω i , u o ( x ) = f o ( x ) for all x ∈ ∂ Ω o , u o ( x ) = F ( x , u i ( x ) ) for all x ∈ ∂ Ω i , ν Ω i ⋅ ∇ u o ( x ) − ν Ω i ⋅ ∇ u i ( x ) = G ( x , u i ( x ) ) for all x ∈ ∂ Ω i , where the last equality is attained in certain weak sense. A simple example shows that such a pair of functions ( u o , u i ) is in general neither unique nor locally unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we prove the existence of at least one classical solution which is in addition locally unique.

There are many models that have been used to simulate the rainfall-runoff relationship. The artificial neural network (ANN) model was selected to investigate an approach of improving daily runoff forecasting accuracy in terms of data preprocessing. Singular spectrum analysis (SSA) as one data preprocessing technique was adopted to deal with the model inputs and the SSA-ANN model was developed. The proposed model was compared with the original ANN model without data preprocessing and a nonlinear perturbation model (NLPM) based on ANN, i.e., the NLPM-ANN model. Eight watersheds were selected for calibrating and testing these models. Comparative study shows that the learning and training ability of ANN models can be improved by SSA and NLPM techniques significantly, and the performance of the SSA-ANN model is much better than the NLPM-ANN model, with high foresting accuracy. The SSA-ANN1 model, which only considers rainfall as model input, was compared with the SSA-ANN2 model, which considers both rainfall and previous runoff as model inputs. It is shown that the Nash-Sutcliffe criterion of the SSA-ANN2 model is much higher than that of the SSA-ANN1 model, which means that the proper selection of previous runoff data as rainfall-runoff model inputs can significantly improve model performance since they usually are highly auto-correlated.

This paper investigates the finite-time stability FTS analysis problem for switched systems with both nonlinear perturbation and time-varying delays. For the system to be finite-time stable, a sufficient condition is proposed based on some delay differential inequalities DDIs, rather than the Lyapunov-like functions which are commonly used in the FTS analysis of switched systems. Compared with the Lyapunov-like function method, the FTS conditions based on the DDI method are easier for checking and do not require FTS of each subsystem. Two examples are given to illustrate the effectiveness of the developed theory.

This paper studies the problem of the stability analysis of interval time-varying delay systems with nonlinear perturbations. Based on the Lyapunov–Krasovskii functional (LKF), a sufficient delay-range-dependent criterion for asymptotic stability is derived in terms of linear matrix inequality (LMI) and integral inequality approach (IIA) and delayed decomposition approach (DDA). Further, the delay range is divided into two equal segments for stability analysis. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the present method. Two well-known examples are given to show less conservatism of our obtained results and the effectiveness of the proposed method.

In this paper, we apply level set and nonlinear perturbation methods to obtain the asymptotic behavior of the solution to a kind of parabolic Monge–Ampere equation at infinity. The Jorgens–Calabi–Pogorelov theorem for parabolic and elliptic Monge–Ampere equation can be regarded as special cases of our result.

Alexseev``s formula generalizes the variation of constants formula and permits the study of a nonlinear perturbation of a system with certain stability properties. In this paper, we investigate bounds for solutions of the functional nonlinear perturbed di®erential systems using the two notion of h-stability and t1- similarity.

For cocycles defined by sequences of linear operators in a Banach space, we study the relation between the notions of nonuniform exponential stability and admissibility. The latter refers to the existence of bounded solutions for any bounded nonlinear perturbation of the original cocycle. In particular, using appropriate adapted norms to deal with the nonuniform behaviour, we show that if is admissible for a given cocycle, then the cocycle is a nonuniform exponential contraction. We also exhibit a collection of admissible Banach spaces for any given nonuniform exponential contraction. In addition, we obtain related results in the more general case of nonuniform exponential dichotomies. As an application of the ideas and methods discussed in the paper, we establish the robustness of nonuniform exponential contractions, in the sense that a sufficiently small linear perturbation of a nonuniform exponential contraction is again a nonuniform exponential contraction.

In this paper, we establish the existence of positive solutions for a class of quasilinear Schrodinger equations involving supercritical growth. By using a change of variables, the quasilinear equation is reduced to a semilinear equation. Then, variational method is used together with a truncation argument used in Del Pino and Felmer (Calc var Partial Differ Equ 4:121–137, 1996) and concentration compactness principle given in Zelati and Rabinowtiz (Commun Pure Appl Math 45:1217–1269, 1992.

The notion of an exponential contraction is only one among many possible rates of contraction of a nonautonomous system, while for an autonomous system all contractions are exponential. We consider the notion of an L1 contraction that includes exponential contractions as a very particular case and that is naturally adapted to the variation-of-parameters formula. Both for discrete and continuous time, we show that under very general assumptions the notion of an L1 contraction persists under sufficiently small linear and nonlinear perturbations, also maintaining the type of stability. As a natural development, we establish a version of the Grobman–Hartman theorem for nonlinear perturbations of an L1 contraction.

Summary This paper addresses the stabilization problem of neutral systems. The neutral systems assumed to be subjected to nonlinear perturbations and mixed time-varying delays. Specifically, the adaptive control method is used to stabilize the unknown neutral systems. If the system states and the delayed states with its derivatives are available for measurement, a new and less conservative adaptive control algorithm is proposed, and sufficient conditions for the existence of the adaptive control are derived based on Lyapunov's stability theory. The main idea is to make all the stability conditions of the system depend on desired arbitrary matrices, not on the given system matrices. This is accomplished by the adaptation process. Illustrative examples with numerical simulations are studied. The simulation results show the effectiveness of the proposed design methods. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, the problem of exponential stability analysis for switched neutral systems with mixed time-delays and nonlinear perturbations based on descriptor system approach is addressed. Delay-dependent exponential stability results are derived in terms of linear matrix inequalities based on piecewise descriptor type Lyapunov–Krasovskii functional. Average dwell time approach is used to analyze switched neutral systems. In addition convex combination, delay decomposition and free-weighting matrix approaches have been employed to derive less conservative results. Further, numerical examples are illustrated to demonstrate the effectiveness and less conservativeness of the derived theoretical results.

The delay-range-dependent stochastic stability for uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations is investigated. The perturbations under consideration are time-varying and norm-bounded. By delay interval dividing, a novel augmented Lyapunov functional which contains triple-integral terms to reduce the conservativeness is introduced. Based on the Lyapunov functional approach and the nature of convex combination, some improved delay-range-dependent stochastic stability criteria are obtained in terms of linear matrix inequalities without introducing any free-weighting matrices. Finally, numerical examples are given to illustrate the effectiveness of the developed techniques.

In this paper, we investigate the problem of exponential stabilization of a class of nonlinear time varying systems. We use Lyapunov techniques to obtain exponential stability of the system in closed-loop with a controller in presence of nonlinear perturbation. The convergence of solutions are in the sense that they converge to a small neighborhood of the origin. An application to a D-C motor is given.

We investigate a nonlinear wave phenomenon described by the perturbationK(n,m)Rosenau-Hyman equation within the concept of derivative with fractional order. We used the Caputo fractional derivative and we proposed an iteration method in order to find a particular solution of the extended perturbation equation. We proved the stability and the convergence of the suggested method for solving the extended equation without any restriction on(m,n)and also on the perturbations terms. Using the inner product we proved the uniqueness of the special solution. By choosing randomly the fractional orders andm, we presented the numerical solutions.

A networked control system (NCS) consisting of cascaded two-port communication channels between the plant and controller is modeled and analyzed. Towards this end, the robust stability of a standard closed-loop system in the presence of conelike perturbations on the system graphs is investigated. The underlying geometric insights are then exploited to analyze the two-port NCS. It is shown that the robust stability of the two-port NCS can be guaranteed when the nonlinear uncertainties in the transmission matrices are sufficiently small in norm. The stability condition, given in the form of "arcsin" of the uncertainty bounds, is both necessary and sufficient., 8 pages, in preparation for journal submission

This paper is concerned with the problem of finite-time stability (FTS) of a class of switched systems with delayed arguments and nonlinear perturbations which are related not only with the current state and the delayed state but also with time $t$. Novel Lyapunov--Krasovskii functions are introduced, and a new finite-time stability criterion is derived by employing the average dwell time (ADT) approach and linear matrix inequality technique. An example is given to illustrate the effectiveness of the proposed method.

This study investigates the problem of finite-time control for uncertain systems with nonlinear perturbations. The aim is to design the state-feedback and output-feedback controller which ensure finite-time boundedness and with a desired $H_{\infty}$ performance index υ. Specifically, first, we divide the time-varying delay into non-uniformly subintervals and decompose the corresponding integral intervals to estimate the bounds of integral terms exactly. Second, the conditions obtained in this paper are formulated in terms of linear matrix inequalities (LMIs), which can be efficiently solved via standard numerical software. Finally, numerical examples are presented to demonstrate the effectiveness and advantages of the theoretical results.

This is the first paper to present such a digital redesign method for the (conventional) OKID system and apply this novel technique for nonlinear system identification. First, the Observer/Kalman filter Identification (OKID) method is used to obtain the lower-order state-space model for a stochastic chaos system. Then, a digital redesign approach with the high-gain property is applied to improve and replace the observer identified by OKID. Therefore, the proposed OKID combined with an observer-based digital redesign novel tracker not only suppresses the uncertainties and the nonlinear perturbations, but also improves more accurate observation parameters of OKID for complex Multi-Input Multi-Output systems. In this research, Chen’s stochastic chaotic system is used as an illustrative example to demonstrate the effectiveness and excellence of the proposed methodology.

In this paper we consider the hyperbolic formulation of the constraints introduced by R\'acz. Using the numerical framework recently developed by us we construct initial data sets which can be interpreted as nonlinear perturbations of Schwarzschild data in Kerr-Schild coordinates and investigate their asymptotics. Our results suggest that, unless one finds a way to exploit the freedom to pick the free part of the initial data in some suitable way, generic initial data sets obtained by this method may violate fundamental asymptotic conditions., Comment: 26 pages, 10 figures

In this chapter we consider the topological structure of the solution set of non-autonomous parabolic evolution inclusions with time delay, defined on noncompact intervals. The result restricted to compact intervals is then extended to non-autonomous parabolic control problems with time delay . Moreover, as the applications of the information about the structure, we establish the existence result of global mild solutions for non-autonomous Cauchy problems subject to nonlocal condition , and prove the invariance of a reachability set for non-autonomous control problems under single-valued nonlinear perturbations. Finally, some illustrating examples are supplied.

This chapter is concerned with the design of energy-efficient and reliable distributed controllers for a class of nonlinear large-scale systems. Techniques such as reducing the packet size and the communication times are used to save the energy consumption of the sensors, and thereby extend the lifetime of the networks.

In this study, we focus on stability analysis for systems with time-varying delay and nonlinear perturbations. In order to cut down the conservatism of the existing stability criteria, we utilize the triple integral forms of Lyapunov-Krasovskii functional (LKF). In addition, by using single and double integral forms of Wirtinger-based inequality, we overcome some conservatism which come from Jensen’s inequality. Three well-known numerical examples are given at the end. Compared with some existing results, our results have less conservatism.

In this paper, classical small perturbations against a stationary solution of the nonlinear Schrodinger equation with the general form of nonlinearity are examined. It is shown that in order to obtain correct (in particular, conserved over time) nonzero expressions for the basic integrals of motion of a perturbation even in the quadratic order in the expansion parameter, it is necessary to consider nonlinear equations of motion for the perturbations. It is also shown that, despite the nonlinearity of the perturbations, the additivity property is valid for the integrals of motion of different nonlinear modes forming the perturbation (at least up to the second order in the expansion parameter)., Comment: 20 pages, v5: several minor typos corrected

In this paper we consider the topological structure of the solution set of non-autonomous parabolic evolution inclusions with time delay, defined on non-compact intervals. The result restricted to compact intervals is then extended to non-autonomous parabolic control problems with time delay. Moreover, as the applications of the information about the structure, we establish the existence result of global integral solutions for non-autonomous Cauchy problems subject to nonlocal condition, and prove the invariance of a reachability set for non-autonomous control problems under single-valued nonlinear perturbations. Finally, some illustrating examples are supplied.

This study proposes a novel approach to obtain less conservative conditions for the robust stability analysis for time-varying delay systems with parameter uncertainties and nonlinear perturbations. Specifically, first, we divide the time-varying delay into non-uniformly subintervals and decompose the corresponding integral intervals to estimate the bounds of integral terms more exactly. Second, novel delay-derivation-dependent stability criteria are derived by introducing a parameter @c in the Lyapunov-Krasovskii functional. Third, we define some new variables to deal with uncertain parameters. Finally, two numerical examples are presented to demonstrate the effectiveness and advantages of the theoretical results.