Liao Huiqing　Sun Leping　Huang Zhongwu

Abstract： Solving nonlinear problems through linearization.Although the linearization process is local，under certain conditions，linearization within the local neighborhood of some solution may not affect the original equations.Based on this idea，we consider the stability and asymptotic stability of a class of nonlinear delay differential-algebraic equations and numerical methods of implicit Euler methods by means of linearization process.Sufficient conditions for stability and asymptotic stability are obtained.

Key words： nonlinear delay differential-algebraic equations； stability； asymptotic stability

CLC number： O 241.81 Document code： A Article ID： 1000-5137（2018）04-0389-08

摘 要： 主要用線性化的方法處理解決非線性問題.雖然線性化的過程是局部的，但是在某些條件下，在某些解的局部鄰域內(nèi)的線性化不影響原方程的性質(zhì).基于這種思想，研究了一類非線性時(shí)滯微分代數(shù)方程解的穩(wěn)定性和漸進(jìn)穩(wěn)定性，并討論了隱式歐拉方法數(shù)值解穩(wěn)定性和漸進(jìn)穩(wěn)定性的充分條件.

關(guān)鍵詞： 時(shí)滯微分代數(shù)方程； 穩(wěn)定性； 漸進(jìn)穩(wěn)定性

1 Introduction

Delay differential-algebraic equations （DDAEs） have both delay and algebraic constraints.There are much work on numerical methods for linear DDAEs.In [1] Zhu and Petzold investigated the asymptotic stability of differential-algebraic equations （DAEs） and neutral delay differential-algebraic equations （NDDAEs） via characteristic equations of θ-methods，Runge-Kutta methods and linear multistep methods.

However，not much work has been done on numerical methods for nonlinear DDAEs.In [2-3] the authors concerned with the theory of asymptotic behavior of a class of nonlinear DDAEs，implicit Euler methods and backward differentiation formula （BDF） methods.They also gave sufficient conditions for the stability and asymptotic stability of the equations.

In this paper we consider the stability of nonlinear delay differential-algebraic equations and implicit Euler methods.We study the stability and asymptotic stability of DDAEs transformed by method of linearization.Some sufficient conditions of stability and asymptotic stability are obtained.

References：

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[3] Sun L P.Analysis of backward differentiation formula for nonlinear differential-algebraic equations with 2-delays [J].Springerplus，2016，5（1）：1-15.

[4] Zhu W J，Petzold L R.Asympototic stability of Hessenberg differential-algebraic equations of retarded or neutral type [J].Applied Nunerical Mathematics，1998，27：309-325.

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[7] Zhang X D.Matrix analysis and application [M].Beijing：Beijing Tsinghua University Press，2004.

（責(zé)任編輯：馮珍珍）