力学系统的任意阶分数维梯度表示

FRCTIONAL GRADIENT REPRESENTATION OF ARBITRATRY ORDER FOR MECHANICAL SYSTEM

梯度系统是一类重要的数学系统,其性质适用Lyapunov函数研究系统的平衡和稳定性.约束力学系统的稳定性研究具有重要应用价值.若约束力学系统可转化为梯度系统,则可利用梯度系统性质研究其稳定性.然而有些约束力学系统并没有梯度表示,如果可以将其拓展到分数阶情形,可为这类约束系统的稳定性研究提供新思路.目前关于约束力学系统的分数维梯度表示限于二阶梯度系统(分数阶导数的阶数α=2),对任意阶导数的梯度系统还未有报道,限制了其应用.不同以往限于力学系统的二阶分数维梯度表示,文章研究力学系统的一类新分数维梯度表示,即任意阶分数维梯度表示.首先建立系统的微分方程,进而给出了力学系统可以表示为任意阶(α为正整数或任意分数)的分数维梯度系统的条件及其任意阶分数维梯度形式,通常梯度系统和二阶分数维梯度系统是文章结果的特例,然后讨论结果的一般性,发现不同分数阶导数定义下同阶分数维梯度系统的阶数取值范围并不同.通过算例说明了结果的适用性.

Gradient system is an important type of mathematical system.When solving out its potential function and making it as Lyapunov function,one can use the characters of gradient system to study the equilibrium and stability of this system by Lyapunov function.Stability of constrained mechanical system is important and has broad applications.If constrained mechanical system can be transformed into gradient system,one can use the characters of gradient system to study stability of constrained mechanical system.However,not all the constrained mechanical system can be transformed into gradient system,but can be transformed into fractional gradient system under some conditions.The fractional gradient system can propose a new thinking for the stability research of these mechanical systems.Up to now the gradient representation of constrained mechanical system is limited into second order fractional gradient system(which fractional orderα=2),to our best knowledge,there is no arbitrary order gradient representation for constrained mechanical system,which limits its application in constrained mechanical system.Different from former studies about only second order gradient representation of constrained mechanical system,new fractional gradient representation of holonomic mechanical system is studied.Two new type of propositions for holonomic mechanical system transforming into arbitrary order fractional gradient system are given,the results can degenerate into classical gradient system and second order fractional gradient system.Two examples are given to illustrate the results.