Wick类型的随机广义Kdv方程的精确解

在Kondratiev分布空间(S)-1中通过埃尔米特变换和Painleve′分析导出了Wick-类型的随机广义Kdv方程的Backlund变换,并且把Wick-类型的随机广义Kdv方程变成广义系数Kdv-方程,再利用Backlund变换求出广义系数Kdv方程的精确解,最后通过埃尔米特逆变换求出随机广义Kdv方程在系数取不同白色噪音泛函条件下的精确解.     

强非线性中立型随机时滞系统的间歇性反馈控制

本文研究的是一类系数是强非线性的混合中立型随机时滞系统的间歇性反馈控制问题. 与现有的间歇性控制器不同, 该控制器可以随意改变有效时间域的长短, 并根据有效时间域的不同, 去接受不同的输入形式, 如: 周期性时间输入、有界非周期性时间输入、指数非周期性时间输入和全时间输入等. 最后, 利用Lyapunov函数, M–矩阵和Itˇo公式, 证明了其系统的指数稳定和几乎确定的稳定, 并通过算例仿真验证了该控制器的有效性和正确性     

Some investigation on Hermitian positive-definite solutions of a nonlinear matrix equation

In this paper, the Hermitian positive-definite solutions of the matrix equation X^s+A*X^?tA=Q are considered. New necessary and sufficient conditions for the equation to have a Hermitian positive-definite solution are derived. In particular, when A is singular, a new estimate of Hermitian positive-definite solutions is obtained. In the end, based on the fixed point theorem, an iterative algorithm for obtaining the positive-definite solutions of the equation with Q=I is discussed. The error estimations are found.     

Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations

The nonlinear delay differential equation with exponential and quadratic nonlinearities is considered. It is assumed that the equation is exposed to stochastic perturbations of the white noise type, which are directly proportional to the deviation of the system state from the equilibrium point. Sufficient conditions for stability in probability of the zero and positive equilibriums of the considered system under stochastic perturbations are obtained. The research results are illustrated by numerical simulations. The proposed investigation procedure can be applied for arbitrary nonlinear stochastic delay differential equations with an order of nonlinearity higher than one. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Burgers方程的精确解

引入一个变换,将二阶非线性偏微分方程—Burgers方程降阶为一阶的非线性方程,再直接求解该方程,得出了Burgers方程精确解的新形式,并与已有结果完全吻合.这种方法也适合于求解其他非线性偏微分方程.     

STWS approach for Hammerstein system of nonlinear Volterra integral equations of the second kind

In this paper, single-term Walsh series (STWS) method is applied to obtain the numerical solutions of Hammerstein systems of nonlinear Volterra integral equations of second kind (HSNVIES). Using the properties of the STWS method, HSNVIES can be easily converted into solvable recursive system of algebraic equations. Solutions obtained from the recursive system of algebraic equations are the solutions of the HSNVIES. Illustrative examples are provided with numerical solutions and the efficiency of this STWS method is also compared with the existing methods.     

Existence of solutions for a class of nonlinear Volterra singular integral equations

In this paper we prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space C[0,1] consisting of real functions defined and continuous on the interval [0,1]. The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra-Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.     

On solutions to fuzzy stochastic differential equations with local martingales

This paper deals with a class of fuzzy stochastic differential equations (FSDEs) driven by a continuous local martingale under the Lipschitzian condition. Such equations can be useful in modeling hybrid systems, where the phenomena are simultaneously subjected to two kinds of uncertainties: randomness and fuzziness. The solutions of the FSDEs are the fuzzy stochastic processes, and their uniqueness is considered to be in a strong sense. Thus, the existence and uniqueness of solutions to the FSDEs under the Lipschitzian condition is first proven. Moreover, some asymptotic properties of the solutions to the FSDEs are investigated. Finally, an illustrating example on the interest term model is provided.     

基于随机障碍验证的随机连续系统安全性验证

针对一类同时具有随机初始状态和随机微分方程的随机连续系统的安全性验证问题,提出一种基于随机障碍验证以及初始集选择的计算方法。首先,介绍了随机连续系统及其安全性验证的相关知识及概念;然后,讨论了如何对于服从几种不同分布的初始变量确定初始状态集,并根据选定的初始状态集使用随机障碍验证的方法将安全性验证问题转化为多项式优化问题;最后,运用平方和松弛方法将问题转化为平方和规划问题,并利用SOSTOOLS工具求得安全性概率的下界。理论分析以及实验结果表明,所提方法具有多项式时间的复杂度,能有效地给出随机连续系统在无界时间内的安全性概率的下界。     

Output feedback for stochastic nonlinear systems with unmeasurable inverse dynamics

This paper considers a concrete stochastic nonlinear system with stochastic unmeasurable inverse dynamics. Motivated by the concept of integral input-to-state stability （iISS） in deterministic systems and stochastic input-to-state stability （SISS） in stochastic systems, a concept of stochastic integral input-to-state stability （SiISS） using Lyapunov functions is first introduced. A constructive strategy is proposed to design a dynamic output feedback control law, which drives the state to the origin almost surely while keeping all other closed-loop signals almost surely bounded. At last, a simulation is given to verify the effectiveness of the control law.     

Approximate controllability of stochastic nonlinear third‐order dispersion equation

In this work, we consider a class of control systems governed by the stochastic nonlinear third‐order dispersion equation in Hilbert spaces. We first prove the existence of mild solutions of stochastic nonlinear third‐order dispersion equation by using fixed point theory, infinite dimensional semigroup properties, stochastic analysis techniques, and then a new set of sufficient conditions are formulated which guarantees the approximate controllability of the main problem. Finally, an example is provided to illustrate the application of the obtained results.Copyright © 2012 John Wiley & Sons, Ltd.     

Continuous simultaneous stabilization of single‐input nonlinear stochastic systems

This paper investigates the simultaneous stabilization of a collection of continuous single‐input non‐linear stochastic systems, with coefficients that are not necessarily locally Lipschitz. A sufficient condition for the existence of a continuous simultaneously stabilizing feedback control is proposed — it is based on the generalized stochastic Lyapunov theorem and on the technique of stochastic control Lyapunov functions. This condition is also necessary, provided that the system's coefficients satisfy some regularity conditions. Moreover, the proposed feedback can be chosen to be bounded under the assumption that appropriate control Lyapunov functions are known. All the proposed simultaneously stabilizing state feedback controllers are explicitly constructed. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed approach.     

Almost sure exponential stability of stochastic fluid networks with nonlinear control

Due to the objective of controlling fluid flows, fluid networks are high-order nonlinear stochastic systems in essential. Taking the nonlinearities in pressure drop of fan branch, measurement error of inertia coefficients, and random factors into consideration, this paper presents a stochastic fluid networks model described by Itô stochastic differential equations. As drift term of the stochastic fluid networks does not satisfy the linear growth condition, the almost sure exponential stability cannot get from the moment exponential stability. We directly investigate the almost sure exponential stability of the tracking and regulation problems for fluid networks under decentralised and adaptive control. It shows that the fluid flows in all branches can reach their reference values if we set controller only in links of the networks. We illustrate the results with two examples.     

State feedback stabilization of stochastic nonlinear systems with SiISS inverse dynamics

This paper further considers a more general class of stochastic nonlinear systems with stochastic integral input‐to‐state stability (SiISS) inverse dynamics and drift and diffusion terms depending upon the other states besides stochastic inverse dynamics and the first state. By skillfully choosing the designed functions and the update laws of parameters, and using the important mathematical tools established in IEEE Trans. Automat. Contr. 2010; 55 (2):304–320, a unifying framework of state feedback controller is proposed to guarantee that all the signals of the closed‐loop system are bounded almost surely and the states can be regulated to zero almost surely. A simulation example demonstrates the effectiveness of the control scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

Exact solutions of a KdV equation hierarchy with variable coefficients

In this paper, a nonisospectral and variable-coefficient KdV equation hierarchy with self-consistent sources is derived from the related linear spectral problem. Exact solutions of the KdV equation hierarchy are obtained through the inverse scattering transformation (IST). It is shown that the IST is an effective mathematical tool for solving the whole hierarchy of nonisospectral nonlinear partial differential equations with self-consistent sources.     

Global stabilisation for a class of switched high-order stochastic nonlinear systems

This paper is devoted to the problem of global stabili sation for a class of switched high-order stochastic nonlinear systems under arbitrary switchings. By extending the adding -a -power -integrator technique and choosing an appropriate common Lyapunov function, a common state-feedback controller is explicitly constructed to render the closed-loop system globally asymptotically stable in probability. The effectiveness of the proposed approach is demonstrated by a simulation example.     

求一类非线性偏微分方程解析解的简洁构造算法

通过引入一个变换,利用齐次平衡原理和选准一个待定函数来构造求解一类非线性偏微分方程解析解的算法.作为实例,我们将该算法应用到了mKdV方程,KdV-Burgers方程和KdV-Burgers-Kuramoto方程.借助符号计算软件Mathematica获得了这些方程的解析解.不难看出,该方法不仅简洁,而且有望进一步扩展.     

一类高阶次随机非线性系统的输出反馈镇定

针对一类高阶次随机非线性系统,研究其输出反馈镇定问题．通过选择有效的观测器和李雅普诺夫函数,所设计的光滑输出反馈控制器保证了闭环系统的平衡点是依概率全局渐近稳定的,输出几乎处处调节到零．数值仿真验证了控制方案的有效性．