离散时间代数Riccati方程解矩阵的特征值分析

针对离散时间代数Riccati方程DTARE的唯一对称正定解X的特征值,通过矩阵的恒等变形,给出了一种新的分析方法.最后获得解X的极值特征值的上界和下界,以及解X的特征值的和———迹的一个下界.     

New complex solutions for some special nonlinear partial differential systems

In this present work, we explore new applications of direct algebraic method for some special nonlinear partial differential systems and equations. Then new types of complex travelling wave solutions are obtained to the Davey-Stewartson system, the coupled Higgs system and the perturbed nonlinear Schrodinger’s equation; the balance number of it is not a positive integer.     

摄动离散矩阵Lyapunov方程解的估计

研究摄动离散矩阵Lyapunov方程解的估计问题,利用矩阵运算性质及Lyapunov稳定性理论,给出在结构不确定性假设下方程解的存在条件及解的上下界估计,估计结果由一个线性矩阵不等式(LMI)和两个矩阵代数Riccati方程确定．针对几种不确定性假设,进一步给出矩阵代数Riccati方程的具体形式．最后通过一个算例说明了所得结果的有效性．     

On positive definite solutions to the algebraic Riccati equation

Complete necessary and sufficient conditions for the existence of a positive definite solution to the algebraic Riccati equation are given. It is also shown that when a positive definite solution exists, it is either unique, or else there are uncountably many such solutions.     

New soliton-like solutions and compacton-like periodic wave solutions of parametric type to a nonlinear dispersive equation

In this paper, by using the integral bifurcation method, we study a nonlinear dispersive equation. Some new soliton-like solutions and some compacton-like periodic wave solutions are obtained. Their dynamic characters are investigated and the profiles are given by the mathematical software Maple. From the graphs of some soliton-like solutions, we find that their profiles are transformable.     

On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function of four variables. In addition, the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found.     

Special types of the nonlinear dispersive Zakharov–Kuznetsov equation with compactons,solitons, and periodic solutions

In this article, we study nonlinear dispersive special types of the Zakharov–Kuznetsov equation with positive and negative exponents. The approach depends mainly on the sine–cosine algorithm. Compactons, solitary patterns, solitons, and periodic solutions are formally derived.     

Classification of traveling wave solutions to the Vakhnenko equations

The classification of all single traveling wave solutions to the Vakhnenko equation and its generalization are obtained by means of the complete discrimination system for the polynomial method.     

On the determination of the solutions and maximum admissible power of the load flow equation via LMIS

It is well‐known that the load flow equation plays a key role in power systems. This paper investigates the load flow equation addressing two fundamental problems, namely the determination of the solutions of this equation in a given region of interest, and the computation of the maximum admissible power for which this equation has a solution in such a region. For the first problem, it is shown that the sought solutions can be computed by solving an eigenvalue problem (EVP), i.e. a convex optimization problem with linear matrix inequality (LMI) constraints, and by looking for vectors with special structure into a linear subspace. For the second problem, it is shown that an upper bound of the sought maximum admissible power can be found by solving another EVP, moreover a sufficient and necessary condition for establishing the tightness of the found upper bound is provided. The proposed methodology is illustrated by some numerical examples. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Exact solutions of discrete complex cubic Ginzburg–Landau equation via extended tanh-function approach

In this paper, we obtained rich solutions for the discrete complex cubic Ginzburg–Landau equation by means of the extended tanh-function approach. These solutions include chirpless bright soliton, chirpless dark soliton, triangular function solutions and some solutions with alternating phases, and so on. Meanwhile, the range of parameters where some exact solution exists is given.     

A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation

This paper is concerned with semi-global stabilization of discrete-time linear periodic systems subject to actuator saturation. Provided that the open loop characteristic multipliers are within the closed unit circle, a low gain feedback design approach is proposed to solve the problem by state feedback. Our approach is based on the solution to a parametric discrete-time periodic Lyapunov equation. The proposed approaches not only generalize the corresponding results for time-invariant systems to periodic systems, but also reveal some important intrinsic properties of this class of periodic matrix equations. A numerical example is worked out to illustrate the effectiveness of the proposed approaches.     

On some explicit non-standard methods to approximate nonnegative solutions of a weakly hyperbolic equation with logistic nonlinearity

We introduce non-standard, finite-difference schemes to approximate nonnegative solutions of a weakly hyperbolic (that is, a hyperbolic partial differential equation in which the second-order time-derivative is multiplied by a relatively small positive constant), nonlinear partial differential equation that generalizes the well-known equation of Fisher-KPP from mathematical biology. The methods are consistent of order 𝒪(Δ t+(Δ x)²). As a means to verify the validity of the techniques, we compare our numerical simulations with known exact solutions of particular cases of our model. The results show that there is an excellent agreement between the theory and the computational outcomes.     

Continuous dependence of solutions to the Lyapunov equation relative to an elliptic differential operator of order 2

We study a Lyapunov-type equationXL -BX = C, whereL, B, and C are given linear operators acting in infinite-dimensional Hilbert spaces, andL is a general elliptic differential operator of order 2 in a bounded domain of a Euclidean space. The equation is derived from a specific parabolic boundary feedback control system. When the coefficients ofL, especially those in the principal part and in the boundary condition, are perturbed, it is shown that the solutionX is strongly continuous relative to the coefficients ofL. The result is applied to the robustness analysis of a feedback stabilization scheme against such perturbations.Dedicated to Professor Yoshiyuki Sakawa on his 60th birthday     

Four positive periodic solutions of a discrete time delayed predator–prey system with nonmonotonic functional response and harvesting

In this paper, by employing the continuation theorem of coincidence degree theory, we establish an easily verifiable criteria for the existence of at least four positive periodic solutions for a discrete time delayed predator–prey system with nonmonotonic functional response and harvesting.     

Probabilistic solutions of some multi-degree-of-freedom nonlinear stochastic dynamical systems excited by filtered Gaussian white noise

In this paper, the state-space-split method is extended for the dimension reduction of some high-dimensional Fokker–Planck–Kolmogorov equations or the nonlinear stochastic dynamical systems in high dimensions subject to external excitation which is the filtered Gaussian white noise governed by the second order stochastic differential equation. The selection of sub state variables and then the dimension-reduction procedure for a class of nonlinear stochastic dynamical systems is given when the external excitation is the filtered Gaussian white noise. The stretched Euler–Bernoulli beam with hinge support at two ends, point-spring supports, and excited by uniformly distributed load being filtered Gaussian white noise governed by the second-order stochastic differential equation is analyzed and numerical results are presented. The results obtained with the presented procedure are compared with those obtained with the Monte Carlo simulation and equivalent linearization method to show the effectiveness and advantage of the state-space-split method and exponential polynomial closure method in analyzing the stationary probabilistic solutions of the multi-degree-of-freedom nonlinear stochastic dynamical systems excited by filtered Gaussian white noise.     

On the numerical solution of the Burgers's equation

In this paper, we consider the linear heat equation arisen from the Burgers's equation using the Hopf–Cole transformation. Discretization of this equation with respect to the space variable results in a linear system of ordinary differential equations. The solution of this system involves in computing exp(α A)y for some vector y, where A is a large special tridiagonal matrix and α is a positive real number. We give an explicit expression for computing exp(α A)y. Finally, some numerical experiments are given to show the efficiency of the method.     

A new existence theory for positive periodic solutions to functional differential equations with impulse effects

The principle of this paper is to deal with a new existence theory for positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments. Easily verifiable sufficient criteria are established. The approach is based on the fixed-point theorem in cones. The paper extends some previous results and obtains some new results.     

Methodology to resolve the transport equation with the discrete ordinates code TORT into the IPEN/MB-01 reactor

Resolution of the steady-state Neutron Transport Equation in a nuclear pool reactor is usually achieved by means of two different numerical methods: Monte Carlo (stochastic) and Discrete Ordinates (deterministic). The Discrete Ordinates method solves the Neutron Transport Equation for a set of selected directions, obtaining a set of directional equations and solutions for each equation which are the angular flux. In order to deal with the energy dependence, an energy multi-group approximation is commonly performed, obtaining a set of equations depending on the number of energy groups. In addition, spatial discretization is also required and the problem is solved by sweeping the geometry mesh. However, special cross-sections are required due to the energy and directional discretization, thus a methodology based on NJOY99 code capabilities has been used. Finally, in order to demonstrate the capability of this method, the 3D discrete ordinates code TORT has been applied to resolve the IPEN/MB-01 reactor.