分数阶奇异系统的Lie对称性与守恒量

对称性与守恒量可以简化动力学问题从而进一步求出力学系统的精确解,这样更加有利于研究动力学行为.分数阶模型相比于整数阶模型,能够描述复杂系统的动力学过程,因此在分数阶模型下研究对称性与守恒量是不可或缺的.首先介绍两个分数阶奇异系统,一个系统包含混合整数和Caputo分数阶导数,另一个系统仅含Caputo分数阶导数.由两个分数阶奇异系统分别给出两个分数阶固有约束,并给出对应的分数阶约束Hamilton方程.然后,基于微分方程在无限小变换下的不变性,给出了分数阶约束Hamilton方程Lie对称性的定义,导出了相应的确定方程,限制方程和附加限制方程.第三,建立并证明了两个分数阶约束Hamilton系统的Lie对称性定理,得到了相应的分数阶约束Hamilton系统的Lie守恒量.在特定条件下,本文所得结果可以退化为整数阶约束Hamilton系统的Lie守恒量.最后通过两个算例来说明此结果的应用.     

基于积分因子方法研究Chaplygin非完整系统的守恒律

提出并研究了构建Chaplygin非完整系统守恒律的积分因子方法.基于正则形式的Chaplygin方程,定义了积分因子,给出了系统存在守恒量的必要条件,建立了Chaplygin非完整系统的守恒定理及其逆定理.研究表明:对应于必要条件的每一组非奇异函数解,系统存在一个守恒量;反之,对于一个已知守恒量,可找到相应的积分因子,且解是不唯一的.文末以匀质圆球在粗糙水平面上纯滚动为例,讨论了该方法的应用.     

微扰Kepler系统的守恒量与对称性

用直接积分法和Noether法研究微扰Kepler系统的守恒量,都得到了一个不同于Hamilton函数的守恒量,此守恒量与Runge—Lenz矢量有相同的量纲,可以称其为“类Runge-Lenz矢量守恒量”．文中还讨论了守恒量的Noether对称性、Lie对称性与Mei对称性,结果表明：与守恒量相应的无限小变换同时是Noether对称变换、Lie对称变换和Mei对称变换．     

高阶非完整系统的共形不变性与Noether守恒量

研究了高阶非完整系统的共形不变性与Noether守恒量,给出了与高阶非完整系统相应的完整系统的共形不变性的定义及其确定方程,通过系统共形不变性与Lie对称性的关系,推导出了系统运动方程具有共形不变性并且是Lie对称性的共形因子,利用限制方程和附加限制方程,给出了高阶非完整系统的弱Lie对称性和强Lie对称性的共形不变性,得到了共形不变性导致的Noether守恒量,举例说明了结果的应用.     

Poinearé-Chetaev变量下广义Routh方程的对称性与守恒量

根据Rumyantsev提出的Poincare-Chetaev变量下的广义Routh方程,用无限小变换的方法研究它的对称性与守恒量.得到守恒量存在的条件和形式.该结果比以往的Poincare-Chetaev方程的相关结论更一般. 最后,举例说明结果的应用.     

Chetaev型非完整系统Nielsen方程Lie对称性导致的一种守恒量

研究Chetaev型非完整系统Nielsen方程Lie对称性导致的一种守恒量,给出无限小群变换下Chetaev型非完整系统Nielsen方程Lie对称性的确定方程,得到Chetaev型非完整系统Nielsen方程Lie对称性直接导致的一种守恒量及其存在条件,并举例说明结果应用.     

Hamilton体系下中厚板静力弯曲和自由弯曲振动问题的一类模型及通解

对于中厚板的静力弯曲和自由弯曲振动问题,引入两个辅助函数,采用胡海昌在Reissner板理论基础上提出的中厚板微分方程及边界条件,将两类问题的控制方程引入Hamilton体系,分别得到Hamilton体系下中厚板静力弯曲和自由振动问题的微分方程组模型. 比较后得到了Hamilton体系下中厚板静力和振动问题的统一模型,其特点是: 微分方程组模型的统一形式中Hamilton矩阵在对角线位置有2个零子块矩阵. 对于中厚板静力和振动问题,比较了所得齐次微分方程组的特征根,给出齐次微分方程组的通解并进行了比较,从而使问题的求解更理性化和合理化,求解过程遵循一套统一的方法论,便于把这类解法推广到其它问题.     

Lagrange子流形理论在约束力学系统正则变换和勒让德变换中的应用

Lagrange方程与Hamilton方程之间的勒让德变换理论和Hamilton方程的正则变换理论在分析力学中具有重要的地位,从局域坐标的角度很难找到勒让德变换和正则变换之间的相关性. 本文主要基于辛流形的Lagrange子流形理论从全局上给出正则变换理论和勒让德变换理论的统一几何解释,进而在几何力学的角度清晰的描述Hamilton系统的正则变换和Lagrange方程与Hamilton方程之间的勒让德变换的几何结构.     

Poincaré-Chetaev变量下广义Routh方程的对称性与守恒量

根据Rumyantsev提出的Poincaré-Chetaev变量下的广义Routh方程,用无限小变换的方法研究它的对称性与守恒量.得到守恒量存在的条件和形式.该结果比以往的Poincaré-Chetaev方程的相关结论更一般.最后,举例说明结果的应用.     

两个耦合Van der Pol振子系统的一阶近似守恒量

用直接积分法计算两个耦合Van der Pol振子系统的一阶近似守恒量,将两个耦合Van der Pol振子系统看成是未受微扰系统与微扰项的迭加,先通过坐标变换将未受微扰系统解耦,并对解耦系统的3种可能状态进行讨论,得到未受微扰系统的13个精确守恒量,再考虑微扰项对精确守恒量的影响,运用一阶近似守恒量的性质,得到1个稳定的一阶近似守恒量.另外,由13个精确守恒量直接得到13个平凡的一阶近似守恒量.     

Delay optimal control and viscosity solutions to associated Hamilton–Jacobi–Bellman equations

In this article, optimal control problems of differential equations with delays are investigated for which the associated Hamilton–Jacobi–Bellman (HJB) equations are nonlinear partial differential equations with delays. This type of HJB equation has not been previously studied and is difficult to solve because the state equations do not possess smoothing properties. We introduce a new notion of viscosity solutions and identify the value functional of the optimal control problems as the unique solution to the associated HJB equations. An analytical example is given as application.     

Optimal control problems for stochastic delay evolution equations in Banach spaces

We consider the optimal control for a Banach space valued stochastic delay evolution equation. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of backward stochastic differential equations. An application to optimal control of stochastic delay partial differential equations is also given.     

非线性动力学方程的李级数解法及其应用

分别从推广的微分方程幂级数解的理论和线性算子半群理论等不同的角度研究了非线性动力学方程的求解问题,得到了所谓的李级数解法．并进一步讨论了算法的具体实施过程,它可以用于构造非线性动力学方程任意高阶的显式积分格式．最后,把李级数解法应用于求解广义Hamilton系统,它能保持广义Hamilton系统真解的典则性．数值算例显示该方法是有效的。     

Multiple stochastic integrals with Mathematica

In the construction of numerical methods for solving stochastic differential equations it becomes necessary to calculate the expectation of products of multiple stochastic integrals. Well-known recursive relationships between these multiple integrals make it possible to express any product of them as a linear combination of integrals of the same type. This article describes how, exploiting the symbolic character of Mathematica, main recursive properties and rules of Itô and Stratonovich multiple integrals can be implemented. From here, a routine that calculates the expectation of any polynomial in multiple stochastic integrals is obtained. In addition, some new relations between integrals, found with the aid of the program, are shown and proved.     

Optimal control of a stochastic delay heat equation with boundary-noise and boundary-control

In this paper, a controlled stochastic delay heat equation with Neumann boundary-noise and boundary-control is considered. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of the backward stochastic differential equations, which is applied to the optimal control problem.     

Observable state space realizations for multivariable systems

This paper derives two canonical state space forms (i.e., the observer canonical form and the observability canonical form) from multiple-input multiple-output systems described by difference equations. The state space model is expressed by the first-order difference equation and is equivalent to the input–output representation. More specifically, by setting the different state variables, the difference equations or the input–output representations can be transformed into two observable canonical forms and the canonical state space model can be also transformed into the difference equations. Finally, two examples are given.     

Bayesian inference for differential equations

Nonlinear dynamic systems such as biochemical pathways can be represented in abstract form using a number of modelling formalisms. In particular differential equations provide a highly expressive mathematical framework with which to model dynamic systems, and a very natural way to model the dynamics of a biochemical pathway in a deterministic manner is through the use of nonlinear ordinary or time delay differential equations. However if, for example, we consider a biochemical pathway the constituent chemical species and hence the pathway structure are seldom fully characterised. In addition it is often impossible to obtain values of the rates of activation or decay which form the free parameters of the mathematical model. The system model in many cases is therefore not fully characterised either in terms of structure or the values which parameters take. This uncertainty must be accounted for in a systematic manner when the model is used in simulation or predictive mode to safeguard against reaching conclusions about system characteristics that are unwarranted, or in making predictions that are unjustifiably optimistic given the uncertainty about the model. The Bayesian inferential methodology provides a coherent framework with which to characterise and propagate uncertainty in such mechanistic models and this paper provides an introduction to Bayesian methodology as applied to system models represented as differential equations.     

Quadrature imposition of compatibility conditions in Chebyshev methods

Often, in solving an elliptic equation with Neumann boundary conditions, a compatibility condition has to be imposed for well-posedness. This condition involves integrals of the forcing function. When pseudospectral Chebyshev methods are used to discretize the partial differential equation, these integrals have to be approximated by an appropriate quadrature formula. The Gauss-Chebyshev (or any variant of it, like the Gauss-Lobatto) formula cannot be used here since the integrals under consideration do not include the weight function. A natural candidate to be used in approximating the integrals is the Clenshaw-Curtis formula; however, we show in this article that this is the wrong choice and it may lead to divergence if time-dependent methods are used to march the solution to steady state. We develop, in this paper, the correct quadrature formula for these problems. This formula takes into account the degree of the polynomials involved. We show that this formula leads to a well-conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.     

Hamilton系统的保辛 守恒积分算法

给出了Hamilton系统基于辛矩阵乘法的显式时不变正则变换和时变正则变换．引入含参变量的近似Hamilton系统,并以近似Hamilton系统为基础进行辛矩阵乘法的正则变换．正则变换保证了数值积分的保辛性质,而通过调整引入的参变量可保证能量在积分格点上守恒．实现了Hamilton系统即保辛又保能量的算法．