Leader‐following consensus problem of heterogeneous multi‐agent systems with nonlinear dynamics using fuzzy disturbance observer

This article considers the leader‐following consensus problem of heterogeneous multi‐agent systems. The proposed multi‐agent system is consisted of heterogeneous agents where each agents have their own nonlinear dynamic behavior. To overcome difficulty from heterogeneous nonlinear intrinsic dynamics of agents, a fuzzy disturbance observer is adopted. In addition, based on the Lyapunov stability theory, an adaptive control method is used to compensate the observation error caused by the difference between the unknown factor and estimated values. Two numerical examples are given to illustrate the effectiveness of the proposed method. © 2013 Wiley Periodicals, Inc. Complexity 19: 20–31, 2014     

Local well‐posedness for a system of quadratic nonlinear Schrödinger equations in one or two dimensions

In this article, the local well‐posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space L^p( R ⁿ). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in H^s(s≥0) and ill posed in H^s(s < 0) in two‐space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in L^p( R ²) for 1 < p < 2, for which p can arbitrarily be close to the scaling limit p_c=1. In one‐dimensional case, we show that the problem is locally well posed in L¹( R ); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd.     

An application of exp‐function method to approximate general and explicit solutions for nonlinear Schrödinger equations

In this work, we implement a relatively new analytical technique, the exp‐function method, for solving nonlinear equations and absolutely a special form of generalized nonlinear Schrödinger equations, which may contain high‐nonlinear terms. This method can be used as an alternative to obtain analytical and approximate solutions of different types of fractional differential equations, which is applied in engineering mathematics. Some numerical examples are presented to illustrate the efficiency and the reliability of exp‐function method. It is predicted that exp‐function method can be found widely applicable in engineering. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1016–1025, 2011     

Construction of solutions of the defocusing nonlinear Schrödinger equation with asymptotically time‐periodic boundary values

We study the defocusing nonlinear Schrödinger equation in the quarter plane with asymptotically periodic boundary values. We use the unified transform method, also known as the Fokas method, and the Deift‐Zhou nonlinear steepest descent method to construct solutions in a sector close to the boundary whose leading behavior is described by a single exponential plane wave. Furthermore, we compute the subleading terms in the long‐time asymptotics of the constructed solutions.     

An iterative solver for the finite‐difference frequency‐domain (FDFD) method for the simulation of materials with negative permittivity

A stable iterative solver for the simulation of optical waves in metals using finite difference frequency domain (FDFD) method is presented. The corresponding discretization of Maxwell's equations enables simulating electromagnetic waves in structures when materials with negative permittivity are involved. Convergence of the iterative solver is proved for positive and negative permittivities. Numerical results are presented for a thin‐film silicon solar cell structure containing silver back contact. Copyright © 2010 John Wiley & Sons, Ltd.     

Two‐grid mixed finite‐element methods for nonlinear Schrödinger equations

Two‐grid mixed finite element schemes are developed for solving both steady state and unsteady state nonlinear Schrödinger equations. The schemes use discretizations based on a mixed finite‐element method. The two‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all of the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. Numerical tests are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 63‐73, 2012     

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

In this work, we discuss the coupled modified nonlinear Schrödinger (CMNLS) equation, which describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. By use of the Fokas approach, the initial‐boundary value problem for the CMNLS equation related to a 3×3 matrix Lax pair on the half‐line is to be analyzed. Assuming that the solution {u(x,t),v(x,t)} of CMNLS equation exists, we will prove that it can be expressed in terms of the unique solution of a 3×3 matrix Riemann‐Hilbert problem formulated in the plane of the complex spectral parameter λ. Moreover, we also get that some spectral functions s(λ) and S(λ) are not independent of each other but meet a global relationship.     

Robust consensus of nonlinear multi‐agent systems via reliable control with probabilistic time delay

In this paper, the consensus problem of uncertain nonlinear multi‐agent systems is investigated via reliable control in the presence of probabilistic time‐varying delay. First, the communication topology among the agents is assumed to be directed and fixed. Second, by introducing a stochastic variable which satisfies Bernoulli distribution, the information of probabilistic time‐varying delay is equivalently transformed into the deterministic time‐varying delay with stochastic parameters. Third, by using Laplacian matrix properties, the consensus problem is converted into the conventional stability problem of the closed‐loop system. The main objective of this paper is to design a state feedback reliable controller such that for all admissible uncertainties as well as actuator failure cases, the resulting closed‐loop system is robustly stable in the sense of mean‐square. For this purpose, through construction of a suitable Lyapunov–Krasovskii functional containing four integral terms and utilization of Kronecker product properties along with the matrix inequality techniques, a new set of delay‐dependent consensus stabilizability conditions for the closed‐loop system is obtained. Based on these conditions, the desired reliable controller is designed in terms of linear matrix inequalities which can be easily solved by using any of the effective optimization algorithms. Moreover, a numerical example and its simulations are included to demonstrate the feasibility and effectiveness of the proposed control design scheme. © 2016 Wiley Periodicals, Inc. Complexity 21: 138–150, 2016     

The Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation: Long‐time dynamics of nonzero boundary conditions

We consider the Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation on the line. The initial value q(x,0) is given and satisfies the symmetric, nonzero boundary conditions at infinity, that is, q(x,0)→q_± as x→±∞, and |q_±|=q₀>0. The goal of this paper is to study the asymptotic behavior of the solution of this initial value problem as t→∞. The main tool is the asymptotic analysis of an associated matrix Riemann‐Hilbert problem by using the steepest descent method and the so‐called g‐function mechanism. We show that the solution q(x,t) of this initial value problem has a different asymptotic behavior in different regions of the xt‐plane. In the regions and , the solution takes the form of a plane wave. In the region , the solution takes the form of a modulated elliptic wave.     

Spots and stripes in nonlinear Schrödinger‐type equations with nearly one‐dimensional potentials

We consider the existence of spots and stripes for a class of nonlinear Schrödinger‐type equations in the presence of nearly one‐dimensional localized potentials. Under suitable assumptions on the potential, we construct various types of waves that are localized in the direction of the potential and have single‐ or multihump, or periodic profile in the perpendicular direction. The analysis relies upon a spatial dynamics formulation of the existence problem, together with a center manifold reduction. This reduction procedure allows these waves to be realized as unipulse or multipulse homoclinic orbits, or periodic orbits in a reduced system of ordinary differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Long cycles through prescribed vertices have the Erdős‐Pósa property

We prove that for every graph, any vertex subset S, and given integers : there are k disjoint cycles of length at least ℓ that each contain at least one vertex from S, or a vertex set of size that meets all such cycles. This generalizes previous results of Fiorini and Herinckx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in time, where s denotes the cardinality of S.     

On well‐posedness for some thermo‐piezoelectric coupling models

There is an increasing reliance on mathematical modelling to assist in the design of piezoelectric ultrasonic transducers since this provides a cost‐effective and quick way to arrive at a first prototype. Given a desired operating envelope for the sensor, the inverse problem of obtaining the associated design parameters within the model can be considered. It is therefore of practical interest to examine the well‐posedness of such models. There is a need to extend the use of such sensors into high‐temperature environments, and so this paper shows, for a broad class of models, the well‐posedness of the magneto‐electro‐thermo‐elastic problem. Because of its widespread use in the literature, we also show the well‐posedness of the quasi‐electrostatic case. Copyright © 2016 John Wiley & Sons, Ltd.     

Cyclic bi‐embeddings of Steiner triple systems on 12s + 7 points

A cyclic face 2‐colourable triangulation of the complete graph K_n in an orientable surface exists for n ≡ 7 (mod 12). Such a triangulation corresponds to a cyclic bi‐embedding of a pair of Steiner triple systems of order n, the triples being defined by the faces in each of the two colour classes. We investigate in the general case the production of such bi‐embeddings from solutions to Heffter's first difference problem and appropriately labelled current graphs. For n = 19 and n = 31 we give a complete explanation for those pairs of Steiner triple systems which do not admit a cyclic bi‐embedding and we show how all non‐isomorphic solutions may be identified. For n = 43 we describe the structures of all possible current graphs and give a more detailed analysis in the case of the Heawood graph. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 92–110, 2002; DOI 10.1002/jcd.10001     

Max‐type system of difference equations with positive two‐periodic sequences

This paper deals with the form and the periodicity of the solutions of the max‐type system of difference equations where , and are positive two‐periodic sequences and initial values x₀, x _{? 1}, y₀, y _{? 1} ∈ (0, + ∞ ). Copyright © 2014 John Wiley & Sons, Ltd.     

Local energy‐ and momentum‐preserving schemes for Klein‐Gordon‐Schrödinger equations and convergence analysis

In this article, we obtain local energy and momentum conservation laws for the Klein‐Gordon‐Schrödinger equations, which are independent of the boundary condition and more essential than the global conservation laws. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose local energy‐ and momentum‐preserving schemes for the equations. The merit of the proposed schemes is that the local energy/momentum conservation law is conserved exactly in any time‐space region. With suitable boundary conditions, the schemes will be charge‐ and energy‐/momentum‐preserving. Nonlinear analysis shows LEP schemes are unconditionally stable and the numerical solutions converge to the exact solutions with order . The theoretical properties are verified by numerical experiments. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1329–1351, 2017