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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： 主要用线性化的方法处理解决非线性问题.虽然线性化的过程是局部的，但是在某些条件下，在某些解的局部邻域内的线性化不影响原方程的性质.基于这种思想，研究了一类非线性时滞微分代数方程解的稳定性和渐进稳定性，并讨论了隐式欧拉方法数值解稳定性和渐进稳定性的充分条件.
关键词： 时滞微分代数方程； 稳定性； 漸进稳定性
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：
[1] Zhu W J，Petzold L R.Asympototic stability of linear delay differential-algebraic equations and numberical methods [J].Applied Nunerical Mathematics，1997，24：247-264.
[2] Sun L P，Cong J X，Kuang J X.Asympototic behavior of nonlinear delay differential-algebraic equation and implicit Euler methods [J].Applied Mathematics and Computation，2014，228：395-405.
[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.
[5] Kuang J X，Cong Y H.Stability of numberical Methods of Delay Differential Equations [M].Beijing：Science Press，2005.
[6] Tian H J，Kuang J X.The stability of methods for delay differential equations [J].Journal of Computational and Applied Mathmatics，1996，14：203-212.
[7] Zhang X D.Matrix analysis and application [M].Beijing：Beijing Tsinghua University Press，2004.
（责任编辑：冯珍珍）
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： 主要用线性化的方法处理解决非线性问题.虽然线性化的过程是局部的，但是在某些条件下，在某些解的局部邻域内的线性化不影响原方程的性质.基于这种思想，研究了一类非线性时滞微分代数方程解的稳定性和渐进稳定性，并讨论了隐式欧拉方法数值解稳定性和渐进稳定性的充分条件.
关键词： 时滞微分代数方程； 稳定性； 漸进稳定性
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：
[1] Zhu W J，Petzold L R.Asympototic stability of linear delay differential-algebraic equations and numberical methods [J].Applied Nunerical Mathematics，1997，24：247-264.
[2] Sun L P，Cong J X，Kuang J X.Asympototic behavior of nonlinear delay differential-algebraic equation and implicit Euler methods [J].Applied Mathematics and Computation，2014，228：395-405.
[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.
[5] Kuang J X，Cong Y H.Stability of numberical Methods of Delay Differential Equations [M].Beijing：Science Press，2005.
[6] Tian H J，Kuang J X.The stability of methods for delay differential equations [J].Journal of Computational and Applied Mathmatics，1996，14：203-212.
[7] Zhang X D.Matrix analysis and application [M].Beijing：Beijing Tsinghua University Press，2004.
（责任编辑：冯珍珍）