用非线性方程组求解等式约束非线性规划问题的降维算法

本文研究线性和非线性等式约束非线性规划问题的降维算法.首先,利用一般等式约束问题的降维方法,将线性等式约束非线性规划问题转换成一个非线性方程组,解非线性方程组即得其解;然后,对线性和非线性等式约束非线性规划问题用Lagrange乘子法,将非线性约束部分和目标函数构成增广的Lagrange函数,并保留线性等式约束,这样便得到一个线性等式约束非线性规划序列,从而,又将问题转化为求解只含线性等式约束的非线性规划问题.     

Analysis of the nonlinear vibration of a two-mass–spring system with linear and nonlinear stiffness

An analytical approach is developed for areas of nonlinear science such as the nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of this research is twofold. First, it introduces the transformation of two nonlinear differential equations for a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment of a nonlinear differential system by linearization coupled with Newton’s method. Secondly, the major section is the solving of the governing nonlinear differential equation where the displacement of the two-mass system can be obtained directly from the linear second-order differential equation using a first-order variational approach. The aforementioned approach proposed by J.H. He, who actually developed the method, is exactly He’s variational method. This approach is an explicit method with high validity for resolving strong nonlinear oscillation system problems. Two examples of nonlinear two-degree-of-freedom mass–spring systems are analyzed, and verified with published results and exact solutions. The method can be easily extended to other nonlinear oscillations and so could be widely applicable in engineering and science.     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

NONLINEAR EXPECTATIONS AND NONLINEAR MARKOV CHAINS

§1.IntroductionLet(?,F)be a measurable space and let Lb(F)be the space of F-measurable andbounded real functions.A nonlinear expectation is a continuous functionalE[·]:Lb(F)?→Rthat is order preserving(i.e.,E[X1]≥E[X2],if X1≥X2)and constant preserving     

Qualitative difference between solutions of stationary model Boltzmann equations in the linear and nonlinear cases

We consider problems for the nonlinear Boltzmann equation in the framework of two models: a new nonlinear model and the Bhatnagar-Gross-Krook model. The corresponding transformations reduce these problems to nonlinear systems of integral equations. In the framework of the new nonlinear model, we prove the existence of a positive bounded solution of the nonlinear system of integral equations and present examples of functions describing the nonlinearity in this model. The obtained form of the Boltzmann equation in the framework of the Bhatnagar-Gross-Krook model allows analyzing the problem and indicates a method for solving it. We show that there is a qualitative difference between the solutions in the linear and nonlinear cases: the temperature is a bounded function in the nonlinear case, while it increases linearly at infinity in the linear approximation. We establish that in the framework of the new nonlinear model, equations describing the distributions of temperature, concentration, and mean-mass velocity are mutually consistent, which cannot be asserted in the case of the Bhatnagar-Gross-Krook model.     

Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling

We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. This paper studies a sequential semidefinite programming (SSP) method, which is a generalization of the well-known sequential quadratic programming method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class.     

Stability performance for primary frequency regulation of hydro-turbine governing system with surge tank

This paper aims to study the stability for primary frequency regulation of hydro-turbine governing system with surge tank. Firstly, a novel nonlinear mathematical model of hydro-turbine governing system considering the nonlinear characteristic of penstock head loss is introduced. The nonlinear state equations under opening control mode and power control mode are derived. Then, the nonlinear dynamic performance of nonlinear hydro-turbine governing system is investigated based on the stable domain for primary frequency regulation. New feature of the nonlinear hydro-turbine governing system caused by the nonlinear characteristic of penstock head loss is described by comparing with a linear model, and the effect mechanism of nonlinear characteristic of penstock head loss is revealed. Finally, the concept of critical stable sectional area of surge tank for primary frequency regulation is proposed and the analytical solution is derived. The combined tuning and optimization method of governor parameters and sectional area of surge tank is proposed. The results indicate that for the primary frequency regulation under opening control mode and power control mode, the nonlinear hydro-turbine governing system is absolutely stable and conditionally stable, respectively. The stability of the nonlinear hydro-turbine governing system and linear hydro-turbine governing system is the same under opening control model and different under power control model. The nonlinear characteristic of penstock head loss mainly affects the initial stage of dynamic response process of power output, and then changes the stability of the nonlinear system. The critical stable sectional area of surge tank makes the system reach critical stable state. The governor parameters and critical stable sectional area of surge tank jointly determine the distributions of stability states.     

具有吸收和非线性边界流的非线性扩散方程的爆破估计

本文研究一类具有非线性吸收和非线性边界流的非线性扩散方程，建立了解的爆破速率估计，所得结果依赖于模型中三种非线性机制之间的相互作用。     

Positive solutions for a nonlinear Kirchhoff type beam equation

The existence of two solutions, a positive and a negative, for a nonlinear fourth order equation with nonlinear boundary conditions is considered. The problem models the bending equilibrium of extensible beams on nonlinear elastic foundations.     

Multidimensional nonlinear wave equations with multivalued solutions

We present the theory of breaking waves in nonlinear systems whose dynamics and spatial structure are described by multidimensional nonlinear hyperbolic wave equations. We obtain a general relation between systems of first-order quasilinear equations and nonlinear hyperbolic equations of higher orders, which, in particular, describe electromagnetic waves in a medium with nonlinear polarization of an arbitrary form. We use this approach to construct exact multivalued solutions of such equations and to study their spatial structure and dynamics. The results are generalized to a wide class of multidimensional equations such as d’Alembert equations, nonlinear Klein-Gordon equations, and nonlinear telegraph equations.     

A robust neurocontroller incorporating a priori knowledge of plant dynamics

This paper presents a nonlinear controller design method that integrates linear optimal control techniques and nonlinear neural networks. The multilayered neural networks (MNN's) are incorporated into a model-based linear optimal controller (LOR) to add nonlinear effects on the LOR. The proposed controller can tolerate a wider range of uncertainties than the LOR alone, because the MNN can compensate nonlinear system uncertainties that are not considered in the LOR design. The control performance is improved by using a priori knowledge of the plant dynamics as the system equation and the corresponding LOR. Using the similar technique, a nonlinear servo controller is designed by combining the MNN-based controller and the linear optimal servo controller. Computer simulations are performed to show the applicability and the limitation of the new nonlinear controllers.     

非线性时变广义分布参数系统的适定性问题

本文的主要目的是讨论非线性时变广义分布参数系统的适定性问题.首先,在Banach空间中引入由连续(可能是非线性的)算子引导的非线性广义发展算子,该非线性广义发展算子是广义发展算子的推广,并研究非线性广义发展算子的性质;然后,应用非线性广义发展算子讨论非线性时变广义分布参数系统的适定性问题,并给出解的构造性表达式.     

The inverse scattering transform: Tools for the nonlinear fourier analysis and filtering of ocean surface waves

Surface wave data from the Adriatic Sea are analysed in the light of new data analysis techniques which may be viewed as a nonlinear generalization of the ordinary Fourier transform. Nonlinear Fourier analysis as applied herein arises from the exact spectral solution to large classes of nonlinear wave equations which are integrable by the inverse scattering transform (IST). Numerical methods are discussed which allow for implementation of the approach as a tool for the time series analysis of oceanic wave data. The case for unidirectional propagation in shallow water, where integrable nonlinear wave motion is governed by the Korteweg-deVries (KdV) equation with periodic/quasi-periodic boundary conditions, is considered. Numerical procedures given herein can be used to compute a nonlinear Fourier representation for a given measured time series. The nonlinear oscillation modes (the IST ‘basis functions’) of KdV obey a linear superposition law, just as do the sine waves of a linear Fourier series. However, the KdV basis functions themselves are highly nonlinear, undergo nonlinear interactions with each other and are distinctly non-sinusoidal. Numerical IST is used to analyse Adriatic Sea data and the concept of nonlinear filtering is applied to improve understanding of the dominant nonlinear interactions in the measured wavetrains.     

Time-discretization of nonlinear control systems with state-delay via Taylor–Lie series

The state-delay is always existent in the practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computer. In this paper, we propose a new scheme for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption. This scheme is applied to the sample-data representation of a nonlinear system with constant state time-delay. The mathematical expressions of the discretization scheme are presented and the effect of the time-discretization method on equilibrium properties of nonlinear control system with state time-delay is examined. The proposed scheme provides a finite-dimensional representation for nonlinear systems with state time-delay enabling existing controller design techniques to be applied to them. The performance of the proposed discretization procedure is evaluated using a nonlinear system. For this nonlinear system, various sampling rates and time-delay values are considered.     

有限变形弹性圆杆中的孤波

在同时引入横向惯性和横向剪切应变的情况下,导出了有限变形弹性圆杆的非线性纵向波动方程,方程中包含了二次和三次的非线性项以及由横向剪切与横向惯性导致的两种几何弥散效应．借助Mathematica软件,利用双曲正割函数的有限展开法,对该方程和对应的截断的非线性方程进行求解,得到了非线性波动方程的孤波解,同时给出了这些解存在的必要条件．     

A new nonlinear dynamic analysis method of rotor system supported by oil-film journal bearings

The strong nonlinear behavior usually exists in rotor systems supported by oil-film journal bearings. In this paper, the partial derivative method is extended to the second-order approximate extent to predict the nonlinear dynamic stiffness and damping coefficients of finite-long journal bearings. And the nonlinear oil-film forces approximately represented by dynamic coefficients are used to analyze nonlinear dynamic performance of a symmetrical flexible rotor-bearing system via the journal orbit, phase portrait and Poincaré map. The effects of mass eccentricity on dynamic behaviors of rotor system are mainly investigated. Moreover, the computational method of nonlinear dynamic coefficients of infinite-short bearing is presented. The nonlinear oil-film forces model of finite-long bearing is validated by comparing the numerical results with those obtained by an infinite-short bearing-rotor system model. The results show that the representation method of nonlinear oil-film forces by dynamic coefficients has universal applicability and allows one easily to conduct the nonlinear dynamic analysis of rotor systems.     

Blow-up and blow-up rate for a reaction–diffusion model with multiple nonlinearities

This paper deals with interactions among three kinds of nonlinear mechanisms: nonlinear diffusion, nonlinear reaction and nonlinear boundary flux in a parabolic model with multiple nonlinearities. The necessary and sufficient blow-up conditions are established together with blow-up rate estimates for the positive solutions of the problem.     

非线性系统运动稳定性的一种判据

对于自治的非线性系统来说,只要其线性部分系数矩阵的特征值不属于临界情形,其无扰运动在其足够小的邻域内的稳定性完全可以由其线性部分的特征值确定.关于线性系统的稳定性,已有不少简单易行的判别方法,而关于非线性系统的稳定性,很多数学家和力学家作了大量的研究工作;但大都是针对特殊类型的非线性系统解决了一些问题,直到现在为止,还没有普遍适用于任何的非线性系统的简单易行的判别方法.本文所给的是判别非线性系统稳定性的充要条件,常用的克拉索夫斯基方法只是这一方法的一个特例[1],[2].     

Nonlinear Lagrangians for Nonlinear Programming Based on Modified Fischer-Burmeister NCP Functions

This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. It is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Moreover, the paper develops the dual algorithm associated with the proposed nonlinear Lagrangians. Numerical results reported suggest that the dual algorithm based on proposed nonlinear Lagrangians is effective for solving some nonlinear optimization problems.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.