Stability analysis of nonlinear delaydifferential-algebraic equations and of theimplicit euler methods

JIANG Lanlan, JIN Xiangying, SUN Leping

(College of Mathematics and Science,Shanghai Normal University,Shanghai 200234,China)



Stability analysis of nonlinear delaydifferential-algebraic equations and of theimplicit euler methods

JIANG Lanlan, JIN Xiangying, SUN Leping

(College of Mathematics and Science,Shanghai Normal University,Shanghai 200234,China)

We consider the stability and asymptotic stability of a class of nonlinear delay differential-algebraic equations and of the implicit Euler methods.Some sufficient conditions for the stability and asymptotic stability of the equations are given.These conditions can be applied conveniently to nonlinear equations.We also show that the implicit Euler methods are stable and asymptotically stable.

nonlinear differential-algebraic equation; delay; implicit Euler method

1　Introduction

In recent years,much research has been focused on numerical solutions of systems of differential-algebraic equations (DAEs).These systems can be found in a wide variety of scientific and engineering applications,including circuit analysis,computer-aided design and real-time simulation of mechanical systems,power-systems,chemical process simulation,and optimal control.In some cases,time delays appear in variables of unknown functions so that the Differential-Algebraic Equations (DAEs) are converted to Delay Differential-Algebraic Equations (DDAEs).Delay-DAEs (DDAEs),which have both delays and algebraic constraints,arise frequently in circuit simulation and power system,Among numerous results on DDAE systems,there are few achievements on nonlinear systems.The solution of a nonlinear system depends on a nonlinear manifold of a product space as well as on consistent initial valued-vectors over a space of continuous functions.It is pointed in [1-2] that research on nonlinear DDAEs is more complicated and still remains investigated.

In this paper,we investigate a class of nonlinear DDAE systems,and show the conditions under which the analytical solutions are stable and asymptotically stable.Similarly,the implicit Euler methods retain the asymptotic behaviors.

2　Asymptotic behavior of a class of nonlinear DDAEs

2.1Stability of analytical solutions of nonlinear DDAEs

In this subsection,we consider the following nonlinear system of delay differential-algebraic equations,

(1)

(2)

According to [3] the assumption thatφvis nonsingular allows one to solve the constraint equations (2) forv(t) (using the implicit function theorem),yielding

(3)

By substituting (3) into (1) we obtain the DODE

(4)

Thus,the DDAE (1),(2) are stable if the DODE (4) is stable.Note that if all the delay terms are present in this retarded DODE,then the initial conditions need to be defined forton [-2τ,0],In this paper we investigate the following nonlinear DDAEs

(5)

(6)

(7)

(8)

and its perturbed equations

(9)

(10)

(11)

(12)

Definition 2.1[4]The system (5)-(8) are said to be stable.if the following inequalities are satisfied:

(13)

(14)

whereM>0 is a constant,

(15)

To study the stability of the DDAE (5)-(8),it is necessary to introduce several lemmas.

Lemma 2.1[4-5]Consider the following initial value problem

(16)

wherea(t),η(t) are continuous functions oftwhent≥0,Re(a(t))<0.Then the solution of the initial value problem (16) satisfies

(17)

for allt≥0.

Now we require thatf,φsatisfy the following Lipschitz conditions (1)-(4):

(3)φvis nonsingular,so that forg(u,v) in (3) there existsL>0 andK>0 such that

(4)σ(t)<0,γ1(t)+(L+K)γ2(t)+(L+K)γ3(t)≤-σ(t),∀t≥0.

Theorem 2.1Iffandφin (5)-(6) satisfy the conditions (1)-(4),then (5)-(6) is stable.

ProofLetu=u(t),uτ=u(t-τ),v=v(t),vτ=v(t-τ).Then

(18)

(19)

From the condition (3),we get

Thus

(20)

According to (20),(19) becomes

Consider the following initial value problem of differential equation:

(21)

(22)

Using the Lemma 2.1,the solution of (21)-(22) satisfies

LetΥ(t)=γ1(t)+Kγ2(t)+Lγ3(t).It becomes

(23)

(24)

According to the condition (4),we get

(25)

It is easy to verify that

Thus

(26)

Ift∈[τ,2τ]⟹s-τ∈[0,τ],s-2τ∈[-τ,0].Similarly,we also get

(27)

Applying mathematical induction,we conclude that

(28)

Therefore

By (20)

2.2Asymptotic stability of analytical solutions of nonlinear DDAEs

In order to study the asymptotic stability of (5)-(8),the following Lemma is needed.

Lemma 2.2[4-5]Suppose that a non-negative functionZ(t) satisfies

whereφ(t)≥0,ω(t),γ1(t),γ2(t),are given functions,and

ThenZ(t)→0(t→∞).

Apply (20) to (19),we have

(29)

Theorem 2.2If (5)-(8) satisfies conditions (1),(2),(3) and (4)′

then (5)-(8) is asymptotically stable.

ProofConsider the initial value problem of the delay differential equations

where

(30)

From (4)′,all the conditions of Lemma (2.2) are satisfied,so

Note (29).It is easy to verify

Therefore,

and the theorem is proved.

3　The stability and asymptotic stability Applying Implicit Euler Methods

Consider the initial value problem of the ordinary differential equations

(31)

(32)

The implicit Euler methods can be written as:

(33)

(34)

wherexn～x(tn),h>0 is the step size.Note (1)-(3),to solve (5)-(8) by (33)-(34),we get

(35)

(36)

(37)

(38)

The Perturbations of (35)-(38) are

(39)

(40)

(41)

(42)

Theorem 3.1The implicit Euler methods are stable for DDAEs.

(43)

Applying the Schwartz theorem and the condition (1)-(2),we obtain

(44)

(45)

(46)

(47)

In (45),let 0≤n≤m-1,and note the initial value function.We have

(48)

In the rest of this section,we study the asymptotic stability of Euler methods.First,we have the following definition.

wheref,gsatisfy conditions (1),(2),(3)′,(4).

Theorem 3.2The implicit Euler methods are asymptotically stable for DDAEs.

ProofNoting (47) and (26),we have

Let 0≤n≤m-1 in the above inequality,we get

Note condition (4)′,

Therefore,when 0≤n≤m-1,

For the casen=m

For the caserm≤n≤(r+1)m-1,it can be shown by induction that

Whenn→∞,r→∞,

Thus,

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(责任编辑:冯珍珍)

10.3969/J.ISSN.1000-5137.2016.04.002

非线性延时微分代数方程和隐式欧拉方法的稳定性分析

姜兰兰, 金香英, 孙乐平

(上海师范大学 数理学院,上海 200234)

考虑了一类非线性延时微分代数方程隐式欧拉方法的稳定性和渐近稳定性,给出了稳定和渐近稳定的一些充分条件.这些条件便于应用到非线性方程.也证明了隐式欧拉方法是稳定和渐近稳定的.

非线性微分代数方程; 延迟; 隐式欧拉方法

date: 2014-06-20

Shanghai Natural Science Foundation (15ZR1431200)

SUN Leping,College of Mathematics and Science,Shanghai Normal University,No.100 Guiling Rd,Shanghai 200234,China,E-mail:sunleping@shnu.edu.cn

O 241.81Document code: AArticle ID: 1000-5137(2016)04-0395-07