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Our main goal is to explore the existence of positive solutions for a class of nonlinear fractional Schrödinger equation involving supercritical growth given by $$ (- \Delta)^{\alpha} u + V(x)u=p(u),\quad x\in \mathbb{R^N},\ N \geq 1. $$ We analyze two types of problems, with $V$ being periodic and asymptotically periodic; for this we use a variational method, a truncation argument and a concentration compactness principle.

In this paper, we aim at addressing the problem of robust quantized control for singular system with time delay, actuator saturation and nonlinear perturbations. Based on Lyapunov Krasovskii approach, a novel delay-dependent sufficient condition is given to guarantee the corresponding system is admissible in terms of Linear Matrix Inequalities (LMIs). The submitted results are derived by exploiting three integral inequalities. A robust quantized controller are further developed as well as the domain of attraction. Finally, examples are utilized to demonstrate the merit of the proposed results.

Background and objective: The stability problem in double delay differential diabetes system is affected by delay term and meal perturbation. This paper is addressed to assess the stability of such double diabetes system with unknown nonlinear perturbation and bounded time varying delay. Method: The stability of a double diabetes system following a meal function as an unknown nonlinear perturbation function has been analyzed by using a Lyapunov–Krasovskii function. The stability matrices according to linear matrix inequality (LMI) were constructed and solved using YALMIP. Results: The solution of the stability theorem gives semi definite matrices which fulfilled the stability criteria. Simulations observed such oscillation in the early meal intake. Conclusion: Thus, it shows that for given values (30–51 min delay and unknown time varying perturbation) the double diabetes system is stable.

This paper focuses on the problem of h-stability of certain classes of dynamic perturbed systems on time scales using time scale versions of some Gronwall type inequalities. We prove under certain conditions on the nonlinear perturbations that the resulting perturbed nonlinear initial value problem still acquire h-stable, if the associated abstract dynamic equation has already owned this property. The paper ends up with two illustrative examples to highlight the utility of our results.

The long time effect of nonlinear perturbation to oscillatory linear systems can be characterized by the averaging method, and we consider first-order averaging for its simplest applicability to high-dimensional problems. Instead of the classical approach, in which one uses the pullback of linear flow to isolate slow variables and then approximate the effective dynamics by averaging, we propose an alternative coordinate transform that better approximates the mean of oscillations. This leads to a simple improvement of the averaged system, which will be shown both theoretically and numerically to provide a more accurate approximation. Three examples are then provided: in the first, a new device for wireless energy transfer modeled by two coupled oscillators was analyzed, and the results provide design guidance and performance quantification for the device; the second is a classical coupled oscillator problem (Fermi-Pasta-Ulam), for which we numerically observed improved accuracy beyond the theoretically justified timescale; the third is a nonlinearly perturbed first-order wave equation, which demonstrates the efficacy of improved averaging in an infinite dimensional setting., Comment: Comments are welcomed

We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown nonlinearities can be uniquely determined from exterior measurements under suitable settings., 17 pages

We review the theoretical efforts to develop models to analyze fiber-optic coherent systems using perturbation analysis. We start with models for the nonlinear signal-signal distortions and continue to address nonlinear signal-noise interactions and SOA-induced distortions.

We study nonlinear gravitational perturbations of vacuum Einstein equations, with $��2$, generalizing previous studies for $n=2$. We follow the formalism by Ishibashi, Kodama and Seto to decompose the metric perturbations into tensor, vector and scalar sectors, and simplify the Einstein equations. The tensor perturbations are the new feature of higher dimensions. We render the metric perturbations asymptotically anti-de Sitter by employing a suitable gauge choice for each of the sectors. Finally, we analyze the resonant structure of the perturbed equations at second order for the five dimensional case, by a partial study of single mode tensor-type perturbations at the linear level. For the cases we studied, resonant terms vanish at second order., 43 pages, discussion expanded, numerical factor missed in a Mathematica file related to previous version corrected, thanks to referees for their comments that led to correction. Resonant terms now vanish at second order for cases studied

The aim of this paper is to study a boundary value problem of the hybrid differential equation with linear and nonlinear perturbations. It generalizes the existing problem of second type. The existence result is constructed using the Leray–Schauder alternative, and the uniqueness is guaranteed by Banach’s fixed-point theorem. Towards the end of this paper, an example is provided to illustrate the obtained results.

We introduce a versatile framework to model and study networked control systems (NCSs). An NCS is described as a feedback interconnection of a plant and a controller communicating through a bidirectional channel modelled by cascaded nonlinear two-port networks. This model is sufficiently rich to capture various properties of a real-world communication channel, such as distortion, interference, and nonlinearity. Uncertainties in the plant, controller and communication channels can be handled simultaneously in the framework. We provide a necessary and sufficient condition for the robust finite-gain stability of an NCS when the model uncertainties in the plant and controller are measured by the gap metric and those in the nonlinear communication channels are measured by operator norms of the uncertain elements. This condition is given by an inequality involving "arcsine" of the uncertainty bounds and is derived from novel geometric insights underlying the robustness of a standard closed-loop system in the presence of conelike nonlinear perturbations on the system graphs., Comment: paper submitted to Automatica

We deal with the problem of identifying a source term in the Rayleigh-Stokes type equation with a nonlinear perturbation, where the nonlinearity may have a superlinear growth and the additional measurement is given at final time and depends on the state. Our aim is to prove the unique solvability and stability of solution. Furthermore, we show that the obtained solution is differentiable and it is the strong one.

We study the generalized Rayleigh-Stokes problem involving a fractional derivative and nonlinear perturbation. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and asymptotic stability of solutions. In particular, if the nonlinearity is Lipschitzian then the mild solution of the mentioned problem becomes a classical one and its convergence to equilibrium point is proved.

We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -{\Delta}u+ u=\big(I_\alpha*|u|^{\frac{\alpha}{N}+1}\big)|u|^{\frac{\alpha}{N}-1}u+f(x,u)\qquad \text{ in } \mathbb{R}^N \] where $N\geq 1$, $I_\alpha$ is the Riesz potential of order $\alpha \in (0, N)$, the exponent $\frac{\alpha}{N}+1$ is critical with respect to the Hardy--Littlewood--Sobolev inequality and the nonlinear perturbation $f$ satisfies suitable growth and structural assumptions., Comment: 18 pages

The problem of fault detection for a class of nonlinear systems with randomly occurring nonlinear perturbations under the adaptive event-triggered scheme is investigated. The phenomena of randomly ...

We explore nonlinear perturbations of a flow generated by a linear nonautonomous impulsive differential equation x'=A(t)x,t≠τi,∆x|t=τi=Bix(τi),i ∈ Z in Banach spaces. Here we assume that the linear nonautonomous impulsive equation admits a more general dichotomy on R called the nonuniform (h,k,µ,ν)-dichotomy, which extends the existing uniform or nonuniform dichotomies and is related to the theory of nonuniform hyperbolicity. Under nonlinear perturbations, we establish a new version of the GrobmanHartman theorem and construct stable and unstable invariant manifolds for nonlinear nonautonomous impulsive differential equations x'=A(t)x+f(t,x), t≠τi,∆x|t=τi=Bix(τi) + gi(x(τi)),i ∈ Z with the help of nonuniform (h,k,µ,ν)-dichotomies.

We prove the existence of weak solutions of variational inequalities for general quasilinear parabolic operators of order m = 2 with strongly nonlinear perturbation term. The result is based on a priori bound for the time derivatives of the solutions.

This paper deals with the following Choquard equation with a local nonlinear perturbation: $$\begin{aligned} \left\{ \begin{array}{ll} - \Delta u+V(x)u=(I_{\alpha }*F(u))f(u)+g(u), &{}\quad x\in \mathbb {R}^N; \\ u\in H^1(\mathbb {R}^N), \end{array} \right. \end{aligned}$$where $$I_{\alpha }: \mathbb {R}^N\rightarrow \mathbb {R}$$ is the Riesz potential, $$N\ge 3$$, $$\alpha \in (0, N)$$, $$F(t)=\int _{0}^{t}f(s)\mathrm {d}s\ge 0\ (\not \equiv 0)$$, $$V\in {\mathcal {C}}^1(\mathbb {R}^N, [0, \infty ))$$ and $$f, g\in {\mathcal {C}}(\mathbb {R}, \mathbb {R})$$ satisfying the subcritical growth. Under some suitable conditions on V, we prove that the above problem admits ground state solutions without super-linear conditions near infinity or monotonicity properties on f and g. In particular, some new tricks are used to overcome the combined effects and the interaction of the nonlocal nonlinear term and the local nonlinear term. Our results improve and extends the previous related ones in the literature.

In the present paper, discussion of perturbations against a Q-ball solution is continued. It is shown that in order to correctly describe perturbations containing nonoscillation modes, it is also necessary to consider nonlinear equations of motion for the perturbations, like in the case of oscillation modes only. It is also shown that the additivity of the charge and the energy of different modes holds for the most general nonlinear perturbation consisting of oscillation and nonoscillation modes., 15 pages. v2: typos corrected, minor changes in the text

The Toda lattice is a nonlinear but integrable system. Here we study the thermalization problem in one-dimensional, perturbed Toda lattices in the thermodynamic limit. We show that the thermalization time, $T_{eq}$, follows a universal law; i.e., $T_{eq}\sim \epsilon^{-2}$, where the perturbation strength, $\epsilon$, characterizes the nonlinear perturbations added to the Toda potential. This universal law applies generally to weak nonlinear lattices due to their equivalence to perturbed Toda systems.

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.

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 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.

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.

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.

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 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 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, two new nonlinear difference inequalities are considered, where the inequalities consist of multiple iterated sums and composite function of nonlinear function and unknown function may be involved in each layer. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. Further, the derived results are applied to the stability problem of a class of linear control systems with nonlinear perturbations.

This paper investigates the stabilization of a class of nonlinear systems with distributed delays using impulsive control and switching control. Stabilizing impulsive forces as well as destabilizing disturbance impulses are considered. Verifiable sufficient conditions are established which guarantee the asymptotic or exponential stability of switched and impulsive systems with distributed delays. Results are found for when the impulses are applied at pre-specified times or at the switching instances. The criteria found are based on a special type of state-dependent switching rule which partitions the state space into stabilizing subregions. The main results are proved using a common Lyapunov functional.

In this work, the propagation of higher-dimensional nonlinear Rossby waves under the generalized beta effect is considered. Using the methods of weak nonlinear perturbation expansions and the multiple scales, we obtain a new (2 + 1)-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation for the first time. Furthermore, a new dispersion relation for the linear Rossby waves is given corresponding to the linearized Boussinesq equation. More importantly, based on the methods of the traveling wave setting and the Jacobi elliptic function expansions, several kinds of exact traveling wave solutions for the higher-dimensional nonlinear Rossby waves, including the periodic solutions, solitary solutions and others are obtained. Finally, we simulate the solitary solutions obtained by using the method of the Jacobi elliptic function. The numerical results show that the amplitude of the Rossby solitary waves is decreasing with the increase of generalized beta effect.

We prove the existence of solution for a class of p(x)-Laplacian equations where the nonlinearity has a critical growth. Here, we consider two cases: the first case involves the situation where the variable exponents are periodic functions. The second one involves the case where the variable exponents are nonperiodic perturbations.

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 recent years M. Pinto introduced the notion of -stability. S.K. Choi et al. investigated -stability for the nonlinear differential systems using the notion of -similarity. Applying these two notions, we study bounds for solutions of the perturbed differential systems.

This paper is concerned with nonlinear nonlocal differential inclusion of evolution type in Frechet spaces, defined on right half-line. The underlying feature of the inclusion under consideration is that it is non-autonomous. We obtain some compactness characterizations of integral solution sets for the inclusion without nonlinear perturbations. Then, making use of these characterizations, we derive a new existence result of global integral solutions for the original inclusion. No invariance condition on the nonlinearity is involved. The results we obtained here extend the semilinear case of the previous related ones such as [I.I. Vrabie, Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions, J. Funct. Anal. 262 (2012) 1363–1391] and are new even for the case of the nonlinearity being a single-valued function.

This paper studies the global asymptotic stability for uncertain systems with mixed delays. The mixed delays include constant delay in the leakage term (i.e., leakage delay) and time-varying delays. The uncertainties under consideration are nonlinear time-varying parameter perturbations and norm-bounded uncertainties. Based on a appropriate Lyapunov–Krasovskii functional with triple integral terms, some integral inequalities and convex combination technique, a novel delay-dependent stability criterion is derived in terms of linear matrix inequalities (LMIs). Finally, three numerical examples are included to show the superiority of the proposed method.

This paper is concerned with the finite-time stability problem for switched systems subject to both nonlinear perturbation and impulse effects. The average dwell time approach, combined with the algebraic matrix theory, is utilized to derive a criterion guaranteeing that the state trajectory does not exceed a certain threshold over a pre-specified finite-time interval. The requirement that at least one subsystem should be stable to ensure asymptotic stability is no longer necessary. Moreover, the finite-time stability degree could be positive, which is a relaxed condition for asymptotic stability. A numerical example is presented to illustrate the effectiveness of the proposed method.

We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x′ = A(t)x + f(t, x, λ), t ≠ τi and x(τi+) = Bix(τi) + gi(x(τi), λ), i ∈ ℕ in Banach spaces, assuming that the linear impulsive differential equation x′ = A(t)x, t ≠ τi and x(τi+) = Bix(τi), i ∈ ℕ admits a nonuniform (µ, ν)-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter λ and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.

This paper deals with a class of uncertain Takagi–Sugeno fuzzy delayed systems with nonlinear perturbations. Based on the Gronwall–Bellman inequality, we present the norm equivalent conditions of robust exponential stability for such systems. Namely, if the delay terms and the nonlinear terms can be controlled by the norm of system states, then the robust exponential stability can be achieved. Finally, an example is provided to verify technical feasibility and operability of the developed results.

A new theoretical formulation is proposed to describe the propagation of large amplitude perturbations in an arbitrary flow of inviscid fluid. This formulation relies on the mixed Eulerian–Lagrangian description, for which exact nonlinear equations are derived. Using properties of Hamiltonian formalism, generalized expressions are found for the acoustic pseudo-energy density and flux. They verify a conservative energy balance. All these expressions only depend on the reference flow variables and the displacement vector of the fluid particles due to the perturbation. When the small perturbation assumption is added, the linear expressions from the literature are retrieved.

Robust stability of a class of model-based networked control systems (MB-NCSs, for short) with nonlinear perturbation is analyzed. Based on the model-based networked control algorithm, a linear model of the plant is used to estimate the plant state behavior between transmission times. The case that the nonlinear plant and the linear plant model are connected via a network channel with transmission times that are varying within a time interval is of particular interest. Sufficient condition on stability of MB-NCSs with nonlinear perturbation is given. One advantage of the proposed method is that the maximum transmission interval and the robustness bound on nonlinear perturbation can be computed. Finally, numerical simulation is worked out to show our main result.

This paper deals with the problems of robust stochastic stabilization and H"~ control for uncertain stochastic systems with time-varying delay and nonlinear perturbation. System uncertainties are assumed to be norm bounded and time delay is assumed to be bound and time varying with delay-derivative bounded by a constant, which may be greater than one. First, new delay-dependent criterion is proposed by exploiting delay-partitioned Lyapunov-krasovskii functional and by employing tighter integral equalities to estimate the upper bound of the stochastic differential of Lyapunov-krasovskii functional without ignoring some useful terms. Second, based on the criterion obtained, a delay-dependent criterion for the existence of a state feedback H"~ controller that ensures robust stochastic stability and a prescribed H"~ performance level of the closed-loop system for all admissible uncertainties is proposed. These developed results have advantages over some previous ones, in that they involve fewer matrix variables but have less conservatism and they also enlarge the application scope. New sufficient conditions are presented in terms of linear matrix inequality. Numerical examples are used to illustrate the effectiveness and feasibility of the proposed method.

For the nonlinear kinetic equation describing the one-dimensional motion of a quasineutral collisionless plasma, perturbation velocities are determined and conditions of generalized hyperbolicity are formulated. Exact (in particular, periodical) solutions of the model are constructed and interpreted physically for the class of traveling waves. Differential conservation laws approximating the basic integrodifferential equation are proposed. These laws are used to perform numerical calculations of wave propagation, which show the possibility of turnover of the kinetic distribution function.

We consider systems that can be described by a linear part with a nonlinear perturbation, where the perturbation is parameterized by a vector of unknown, constant parameters. Under a set of technical assumptions about the perturbation and its relationship to the outputs, we present a modular design technique for estimating the system states and the unknown parameters. The design consists of a high-gain observer that estimates the states of the system together with the full perturbation, and a parameter estimator constructed by the designer to invert a nonlinear equation. We illustrate the technique on a simulated dc motor with friction.

Using variational methods, we establish existence of positive solutions for a class of elliptic problems like $$-\Delta{u}+V(x)u=H(u-\beta)f(u)\,\,\,\, {\rm in}\,\,\,\mathbb{R}^{N},$$ where β > 0, V is a positive, continuous perturbations of a periodic function, H is the Heaviside function and f is a continuous function with subcritical growth.