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Journal of Applied Mathematics
Volume 2012, Article ID 749847, 11 pages
Research Article

A Matrix Method for Determining Eigenvalues and Stability of Singular Neutral Delay-Differential Systems

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Received 29 November 2011; Revised 6 April 2012; Accepted 8 April 2012

Academic Editor: Zhiwei Gao

Copyright © 2012 Jian Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The eigenvalues and the stability of a singular neutral differential system with single delay are considered. Firstly, by applying the matrix pencil and the linear operator methods, new algebraic criteria for the imaginary axis eigenvalue are derived. Second, practical checkable criteria for the asymptotic stability are introduced.

1. Introduction

Nowadays, the time-delay systems have became an important natural models in physics, engineering, multibody mechanics, computer-aided design, and economic systems. The theory on ordinary differential equations with delays have been discussed for decades in a wide range, so there are very many results for them. Especially, the eigenvalues and the stability analysis of time-delay systems have received much attention of researchers and many excellent results have been obtained, see [16]. Certainly most of them had been focused on the analytical methods or numerical methods, such as V-functional methods, Laplace transformation, Runge-Kutta methods, and linear multistep methods. In [79], the numerical techniques for the computation of the eigenvalues were discussed. In [10], Zhu and Petzold researched the asymptotic stability of delay-differential-algebraic equations by applying the 𝜃-methods, Runge-Kutta methods, and linear multistep methods. These methods play the key roles at last. But in recent years, algebraic methods are developing fast, especially for the research on the more complex systems, such as the n-dimensional systems. Though the algebraic methods as a new and effective tool is also applied to analyze the time-delay systems [2, 3], the results are very few.

In this paper, we will discuss the differential-algebraic equations by the algebraic methods. Their dynamics have not been well understood yet.

Example 1.1. Consider the simple differential-algebraic system: ̇𝑥1(𝑡)=𝑓1𝑥(𝑡),1(𝑡)𝑥2(𝑡𝜏)=𝑓2(𝑡),(1.1) where 𝑡0, 𝜏0, and 𝑥1 and 𝑥2 are given by continuous functions on the initial interval (𝜏,0]. So we have the solution: 𝑥1(𝑡)=𝑡0𝑓1(𝑥𝑠)𝑑𝑠+𝑐,2(𝑡)=𝑓2(𝑡+𝜏)+0𝑡+𝜏𝑓1(𝑠)𝑑𝑠+𝑐,(1.2) where 𝑐 is a constant. From the solution, we find that the solution depends on future integrals of the input 𝑓(𝑡). This interesting phenomenon arrested many scholars to research. For a general 𝑛-dimensional differential equations with delay, we can note by 𝐵̇𝑋(𝑡)=𝐴0𝑋(𝑡)+𝐴1𝑋(𝑡𝜏),(1.3) where 𝐵,𝐴0,𝐴1𝑅𝑛×𝑛, Rank𝐵𝑛, 𝑡0, 𝜏0, and 𝑋(𝑡)𝑅𝑛 is given by continuous functions on the initial interval (𝜏,0]. When Rank𝐵=𝑛, we called it the retarded differential equations. It can be improved as ̇𝑋(𝑡)=𝐴0𝑋(𝑡)+𝐴1𝑋(𝑡𝜏).(1.4) Many scholars have widely researched the delay-independent or delay-dependent stability and asymptotic stability by analytic methods or numerical methods. When Rank𝐵<𝑛, it is called a singular (or degenerated) delay-differential equations. The imaginary axis eigenvalues are discussed by using matrix pencil, see [2]. But because of the complex nature of the singular differential systems with delay, the research is very difficult by using the analytical treatment. So few studies on the stability and the bifurcations have been conducted so far. Particulary, for the singular neutral differential systems with delays, there are hardly flexible and efficient verdicts.
In this paper, we will apply the algebraic methods to discuss the stability of a singular neutral differential system with a single delay, as follows: 𝐵0̇𝑥(𝑡)+𝐵1̇𝑥(𝑡𝜏)=𝐴0𝑥(𝑡)+𝐴1𝑥(𝑡𝜏),(1.5) where 𝐵0,𝐵1,𝐴0,𝐴1𝑛×𝑛, 𝑥𝑛, 𝜏0. For the system (1.5), if det𝐵00, we can improve it as the form ̇𝑥(𝑡)+𝐵1̇𝑥(𝑡𝜏)=𝐴0𝑥(𝑡)+𝐴1𝑥(𝑡𝜏).(1.6) It is a neutral differential equation with a single delay. The problem of computing imaginary axis eigenvalues on the system (1.6) has been previously studied in [11]. Here, we consider the state rank𝐵0𝑛. The solvability of the system (1.5), which is essentially the existence and the uniqueness of the solution, is determined by the regularity. The matrix pencil (𝐵0,𝐴0) is said to be regular if 𝑠𝐵0+𝐴0 is not identically singular for any complex 𝑠. If (𝐵0,𝐴0) is regular, the zero 𝑠 of det(𝑠𝐵0+𝐴0) is called the eigenvalue of the matrix pencil (𝐵0,𝐴0). From [12], we know that the system (1.5) is solvable if and only if (𝐵0,A0) is regular. So, in this paper, we suppose that (𝐵0,𝐴0) is regular. In the following, we will analyze the eigenvalues and the stability of the system (1.5).

2. The Algebraic Criteria for Determining Imaginary Axis Eigenvalues

Firstly, we research an ordinary differential equation, which will motivate our analysis. Consider𝐵0̇𝑋(𝑡)+𝐵1̇𝑌(𝑡)=𝐴0𝑋(𝑡)+𝐴1̇𝑌(𝑡),𝑋(𝑡)𝐵𝑇1+̇𝑌(𝑡)𝐵𝑇0=𝑋(𝑡)𝐴𝑇1𝑌(𝑡)𝐴𝑇0,(2.1) where 𝐵0,𝐵1,𝐴0,𝐴1𝑛×𝑛, 𝑋,𝑌𝑛×𝑛. Let 𝑉 denote the vector space 𝑉=𝑛×𝑛×𝑛×𝑛 and 𝐄,𝐅 denote the operators on 𝑉, given by𝐄𝑋𝑌=𝐵0𝑋+𝐵1𝑌𝑋𝐵𝑇1+𝑌𝐵𝑇0𝑋𝑌=𝐴,𝐅0𝑋+𝐴1𝑌𝑋𝐴𝑇1𝑌𝐴𝑇0,𝑋,𝑌𝑛×𝑛.(2.2) With 𝑍(t)=𝑋(𝑡)𝑌(𝑡), the system (2.1) can be written as𝐄̇𝑍(𝑡)=𝐅𝑍(𝑡).(2.3) Supposing 𝑋𝑌=𝑍(𝑡)=𝑋0𝑒𝑠𝑡𝑌0𝑒𝑠𝑡 is a matrix solution of the system (2.1), we have𝑠𝐵0𝐴0𝑋+𝑠𝐵1𝐴1𝑋𝑌=0,𝑠𝐵𝑇1+𝐴𝑇1+𝑌𝑠𝐵𝑇0+𝐴𝑇0=0.(2.4) For any complex 𝑠, let 𝐓=𝐓(𝑠) be the operator 𝐓=𝑠𝐄𝐅; then𝐓𝑋𝑌=𝑠𝐵0𝐴0𝑋+𝑠𝐵1𝐴1𝑌𝑋𝑠𝐵𝑇1+𝐴𝑇1+𝑌𝑠𝐵𝑇0+𝐴𝑇0,𝑋,𝑌𝑛×𝑛.(2.5) For any complex 𝑠, let 𝐓+=𝐓+(𝑠)𝑉𝑉, satisfying𝐓+𝑋𝑌=𝑋𝑠𝐵𝑇0+𝐴𝑇0𝑠𝐵1𝐴1𝑌𝑋𝑠𝐵𝑇1+𝐴𝑇1+𝑠𝐵0𝐴0𝑌,𝑋,𝑌𝑛×𝑛.(2.6) For any complex 𝑠, let Λ=Λ(𝑠), satisfying𝚲𝑋=𝑠𝐵0𝐴0𝑋𝑠𝐵𝑇0+𝐴𝑇0𝑠𝐵1𝐴1𝑋𝑠𝐵𝑇1+𝐴𝑇1,𝑋𝑛×𝑛.(2.7) By simple computations, we can get𝑠𝐵0𝐴0𝑋𝑠𝐵𝑇0+𝐴𝑇0𝑠𝐵1𝐴1𝑋𝑠𝐵𝑇1+𝐴𝑇1=𝑋0𝑠𝐵𝑇0+𝐴𝑇0𝑠𝐵1𝐴1𝑌0,𝑠𝐵0𝐴0𝑌𝑠𝐵𝑇0+𝐴𝑇0𝑠𝐵1𝐴1𝑌𝑠𝐵𝑇1+𝐴𝑇1=𝑋0𝑠𝐵𝑇1+𝐴𝑇1+𝑠𝐵0𝐴0𝑌0.(2.8) Expressing by the operator language, that is,𝐓+𝐓𝑋𝑌=𝚲𝑋𝚲𝑌.(2.9)

In the following, we will convert matrix ordinary differential equation (2.3) to vector form. Let 𝜉 be the elementary transform, 𝜉𝑛×𝑛𝑛2, that is,𝑎𝜉𝐴=𝑇1𝑎𝑇𝑛𝑎,𝐴=1𝑎𝑛,𝑎𝑇𝑖𝑛,𝑖=1,2,,𝑛.(2.10) Let 𝑥=𝜉𝑋,𝑦=𝜉𝑌,𝑧=𝑥𝑦, and𝐸0=𝐵0𝐼𝐵1𝐼𝐼𝐵1𝐼𝐵0,𝐹0=𝐴0𝐼𝐴1𝐼𝐼𝐴1𝐼𝐴0.(2.11) Using the property of the Kronecker product, we have𝜉(𝐴𝑋𝐵)=𝐴𝐵𝑇𝜉𝑋,(2.12) where 𝐴,𝐵,𝑋𝑛×𝑛. So (2.3) can be written as𝐸0̇𝑧(𝑡)=𝐹0𝑧(𝑡).(2.13) Similarly, by denoting 𝑇0=𝑇0(𝑠),𝑇+0=𝑇+0(𝑠),Λ0=Λ0(𝑠) as follows𝑇0=𝑇0(𝑠)=𝑠𝐵0𝐴0𝐼𝑠𝐵1𝐴1𝐼𝐼𝑠𝐵1+𝐴1𝐼𝑠𝐵0+𝐴0=𝑠𝐸0𝐹0,𝑇+0=𝑇+0(𝑠)=𝐼𝑠𝐵0+𝐴0𝑠𝐵1𝐴1𝐼𝐼𝑠𝐵1+𝐴1𝑠𝐵0𝐴0,Λ𝐼0=Λ0(𝑠)=𝑠𝐵0𝐴0𝑠𝐵0+𝐴0𝑠𝐵1𝐴1𝑠𝐵1+𝐴1,(2.14) we have𝑇+0𝑇0𝑥𝑦=Λ0𝑥Λ0𝑦,thatis,𝑇+0𝑇0=Λ000Λ0.(2.15)

Lemma 2.1. For all complex 𝑠, det𝑇+0(𝑠)=det𝑇0(𝑠), and so det𝑇0(𝑠)=±detΛ0(𝑠).

Proof. Let 𝑇0=𝑎𝑏𝑐𝑑, where 𝑎=(𝑠𝐵0𝐴0)𝐼, 𝑏=(𝑠𝐵1𝐴1)𝐼, 𝑐=𝐼(𝑠𝐵1+𝐴1), and 𝑑=𝐼(𝑠𝐵0+𝐴0). Noting the Kronecker product identities (𝐼𝑀)(𝑁𝐼)=(𝑁𝐼)(𝐼𝑀)=𝑁𝑀, we have 𝑑𝑏=𝑏𝑑, 𝑏𝑐=𝑐𝑏. By regularity of (𝐵0,𝐴0), we know 𝑠𝐵0+𝐴0 is nonsingular for enough complex 𝑠. So by the property of the polynomial and easy computations, we can get that det𝑇+0(𝑠)=det𝑇0(𝑠) for any complex 𝑠. Elsewhere, 𝑇+0𝑇0=Λ000Λ0,(2.16) so det𝑇02=det𝑇+0det𝑇0=detΛ02.(2.17) Thus det𝑇0(𝑠)=±detΛ0(𝑠).(2.18)

Theorem 2.2. Any imaginary axis eigenvalue of the systems (1.5) is a zero point of detΛ0(𝑠) and thus also one of the eigenvalues of the matrix pencil (𝐸0,𝐹0).

Proof. The matrix polynomial for the system (1.5) is 𝑝(𝑠,𝑒𝑠𝜏)=𝑠𝐵0𝐴0+𝑠𝐵1𝐴1𝑒𝑠𝜏.(2.19) Let 𝑠=𝑖𝑤 be an imaginary axis eigenvalue of the system (1.5) and 𝑣 is associated eigenvector, 𝑣=1. We have 𝑝(𝑠,𝑒𝑠𝜏)𝑣=0. By conjugating and transforming, we can get 𝑠𝐵0𝐴0𝑣𝑣𝑠𝐵𝑇0+𝐴𝑇0𝑠𝐵1𝐴1𝑣𝑣𝑠𝐵𝑇1+𝐴𝑇1=0.(2.20) Via the elementary transform 𝜉, we get 𝑠𝐵0𝐴0𝑠𝐵0+𝐴0𝑠𝐵1𝐴1𝑠𝐵1+𝐴1𝜉𝑣𝑣=0,(2.21) that is, Λ0(𝑠)𝑢=0,𝑢=𝜉(𝑣𝑣). We know that detΛ0(𝑠)=0, and so det𝑠𝐸0𝐹0=±detΛ0(𝑠)=0.(2.22)
From Theorem 2.2, we know that all of the imaginary axis eigenvalues of the system (1.5) are zero points of the algebraic equation det𝑠𝐵0𝐴0𝑠𝐵0+𝐴0𝑠𝐵1𝐴1𝑠𝐵1+𝐴1=0.(2.23)

Corollary 2.3. If det(𝐵0𝐵0𝐵1𝐵1)0, then 𝐸0 is invertible, and any imaginary axis eigenvalue of the systems (1.5) is the eigenvalue of 𝐹0𝐸01.

Proof. By proof of Lemma 2.1, we have det𝐸0=||||||𝐵0𝐼𝐵1𝐼𝐼𝐵1𝐼𝐵0||||||𝐵=det0𝐵0𝐵1𝐵1.(2.24) The corollary follows immediately from Theorem 2.2.

Corollary 2.4. Any imaginary axis eigenvalue of the system with single delay ̇𝑥(𝑡)=𝐴0𝑥(𝑡)+𝐴1𝑥(𝑡𝜏)(2.25) is an eigenvalue of 𝐹0=𝐴0𝐼𝐴1𝐼𝐼𝐴1𝐼𝐴0.

Proof. It follows immediately from Theorem 2.2.

In fact, the above result contains the system (1.6). For the system (1.6), we also have the following corollary.

Corollary 2.5 (see Jarlebring and Hochstenbach [2, Theorem 1]). For the system (1.6), the imaginary axis eigenvalues are the roots of the equation det𝑠𝐼𝐴0𝑠𝐼+𝐴0𝑠𝐵1𝐴1𝑠𝐵1+𝐴1=0.(2.26)

Remark 2.6. In fact, (2.23) or (2.26) is usually called a polynomial eigenvalue problem. The classical and most widely used approach to research the polynomial eigenvalue problems is linearization, where the polynomial is converted into a larger matrix pencil with the same eigenvalues. There are many forms for the linearization: the companion form is most typically commission. The linearization method is also an important tool to research the characteristic equations in algebraic methods, see [2, 13].

From the above results, we find that the imaginary axis eigenvalues of the system (1.5) or (1.6) can be computed via the algebraic equation (2.23) and (2.26). The imaginary eigenvalues play an important role in the stability. Next, we will use the results to discuss the stability of the systems. At first, we will give the condition of the delay-independent stability on the system (1.5). Secondly, we will address the problem of finding the critical delays of the system (1.6), that is, the delay such that the system (1.6) has purely imaginary eigenvalues.

3. The Algebraic Criteria of the Asymptotic Stability

The stability of the delay ordinary differential equations has been widely discussed [14, 15]. It is well known that the Lyapunov-Krasovskii functional approach is the important analytic method to find the delay-independent stability criteria, which do not include any information on the size of delay. The main ideas for developing algebraic criteria of the stability analysis on the systems can be found in many works, such as “The Degenerate Differential Systems with Delay” (W. Jiang, 1998, [12]). The results for singular neutral differential equations are still very few, especially by algebraic methods.

Next we first research the delay-independent stability of the system (1.5). The characteristic equation for the system(1.5) is denoted again by𝑃𝑠𝐵(𝑠,𝑧)=det0+𝐵1𝑧𝐴0+𝐴1𝑧,𝑠,𝑧=𝑒𝑠𝜏.(3.1)

From [12], we known that the solution of the neutral time-delay systems is asymptotically stable if all roots of (3.1) have negative real part bounded away from 0, that is, there exists a number 𝛿>0, such that Re(𝑠)𝛿<0 for any root 𝑠 of (3.1). Especially for the system (1.5), Zhu and Petzold had found that there must exist the 𝛿, if the condition |𝑢𝑇𝐵0𝑢||𝑢𝑇𝐵1𝑢| for all 𝑢𝑛 holds, see [10]. So from [3, 10], we can get the following theorem.

Theorem 3.1. Let 𝑠 be the zeros of (2.23). If the coefficient matrices of the system (1.5) satisfies the following conditions:(i)Re𝜆<0,𝜆𝜎(𝐵0,𝐴0), (ii)maxRe𝑠=0𝜌((𝑠𝐵0𝐴0)1(𝑠𝐵1𝐴1))<1,then the system (1.5) is asymptotically stable for all 𝜏0, that is, the stability of the systems (1.5) is delay independent.

Proof. It follows immediately from [10].
In the following, we consider the neutral system (1.6), whose characteristic equation is denoted by 𝑃𝑠(𝑠,𝑧)=det𝐼+𝐵1𝑧𝐴0+𝐴1𝑧,𝑠,𝑧=𝑒𝑠𝜏.(3.2) It is well known that the spectrum of the neutral delay systems exhibits some discontinuity properties, that is to say, an infinitesimal change of the delay parameter may cause the stability of the system to shift. These discontinuity properties are closely related to the essential spectrum of the system. The critical condition for a stability switch of a neutral delay system is that the rightmost eigenvalue goes from the left complex half-plane into the right complex half-plane by passing the imaginary axis. So the appearance of the imaginary axis eigenvalue is the critical condition. In Section 2, we find all of the imaginary axis eigenvalues. Let the delay be a parameter. In the following, we will find the critical value of the delay parameter such that the stability switch occurs. It is known that, if a neutral delay system is stable, it is necessary that its neutral part must be stable. For the system (1.6), this requirement concerns the stability of the difference equation: 𝑥(𝑡)+𝐵1𝑥(𝑡𝜏)=0.(3.3) The eigenvalues of (3.3) are called the essential spectrum of the system (1.6). We know that (3.3) is stable if 𝜌(𝐵1)<1. It is important to point out that, under this assumption, the condition Re(𝑠)𝛿<0 can be improved to Re(𝑠)<0. Our task is to find the critical delay where the system (1.6) becomes unstable. So we have the following theorem.

Theorem 3.2. Supposing all of the eigenvalues of matrix 𝐴0 have negative real part and 𝜌(𝐵1)<1, one has the following.(i)If for any root 𝑠 of (3.2), Re(𝑠)<0, else if for any root 𝑠=𝑖𝑤,𝑤>0, 𝑧=𝑒𝑠𝜏,|𝑧|<1, then system (1.6) is asymptotic stability for any 𝜏0, that is, stability is delay independent. (ii) Otherwise, for each root 𝑠=𝑖𝑤,𝑤>0, |𝑧|=1, one can get the minimal critical value of delay parameter 𝜏, such that if 𝜏[0,𝜏), then system (1.6) is asymptotic stability, and when 𝜏𝜏, the stability of system (1.6) changes, that is, the system (1.6) is delay-dependent stable and generates bifurcation in 𝜏=𝜏.

Example 3.3. Consider a neutral neural networks with a single delay, see [16]:̇𝑥1(𝑡)=𝑥1(𝑥𝑡)+𝑎𝑓1(𝑥𝑡𝜏)+𝑏𝑔2(𝑥𝑡𝜏)+𝑏𝑔3(𝑡𝜏)+𝑎2𝑓2̇𝑥1(𝑡𝜏)+𝑏2𝑔2̇𝑥2(𝑡𝜏)+𝑏2𝑔2̇𝑥3,(𝑡𝜏)̇𝑥2(𝑡)=𝑥2𝑥(𝑡)+𝑏𝑔1𝑥(𝑡𝜏)+𝑎𝑓2𝑥(𝑡𝜏)+𝑏𝑔3(𝑡𝜏)+𝑏2𝑔2̇𝑥1(𝑡𝜏)+𝑎2𝑓2̇𝑥2(𝑡𝜏)+𝑏2𝑔2̇𝑥3,(𝑡𝜏)̇𝑥3(𝑡)=𝑥3𝑥(𝑡)+𝑏𝑔2𝑥(𝑡𝜏)+𝑎𝑓3(𝑡𝜏)+𝑏2𝑔2̇𝑥2(𝑡𝜏)+𝑎2𝑓2̇𝑥3.(𝑡𝜏)(3.4) Now we rewrite system (3.4) as the matrix equations: ̇̇𝑋(𝑡)+𝐵𝑋(𝑡𝜏)=𝐴𝑋(𝑡)+𝐶1𝑓(𝑋(𝑡𝜏))+𝐶2𝑔(𝑋(𝑡𝜏)).(3.5) The linearization of the system (3.4) around the origin is given by ̇𝑋(𝑡)+𝐵1̇𝑋(𝑡𝜏)=𝐴0𝑋(𝑡)+𝐴1𝑋(𝑡𝜏),(3.6) where 𝐴0=100010001,𝐴1=𝑎𝑏𝑏𝑏𝑎𝑏0𝑏𝑎,𝐵1=𝑎2𝑏2𝑏2𝑏2𝑎2𝑏20𝑏2𝑎2.(3.7) Assume that 𝑎=2, 𝑏=1.5, 𝑎2=0.3, 𝑏2=0.3, and 𝑓(𝑥)=𝑔(𝑥)=tanh(𝑥). We carry out the numerical simulations for system (3.4). From (2.26), by MATLAB computation, we can get the imaginary eigenvalue 𝑠±0.445𝑖,𝜏0.45. From Theorem (3.2), we know that the zero solution of the system (3.4) is delay-dependent stable. The direction of the Hopf bifurcation at 𝜏=𝜏 is supercritical and the bifurcating periodic solutions are asymptotically stable. The simulation results are shown in Figures 1 and 2.

Figure 1: For system (3.4), when 𝜏=0.44<𝜏, the equilibrium is asymptotically stable.
Figure 2: For system (3.4), when 𝜏=0.5>𝜏, the periodic solution bifurcates from the equilibrium.

4. Conclusion

In this paper, we consider a singular neutral differential system with a single delay. Via applying the algebraic method, that is, the matrix pencil and the linear operators, we discussed the eigenvalues and the stability of the time-delay systems (1.5) and (1.6). By using MATLAB, we could easily compute imaginary eigenvalues from the algebraic equation (2.23) or (2.26). In fact, we only find the imaginary axis eigenvalues, which are the small part of the infinite eigenvalues. So compared with the analytic methods and the numerical methods, the algebraic methods are more simple and more explicit for some time-delay system. Certainly, applying the algebraic methods to analyze the dynamical properties of the singular neutral differential systems with delays is a new and immature field. So we believe that the algebraic methods used to research the stability of the dynamical systems would be more interesting in the future.


The National Natural Science Foundation of China (10871056) and the Fundamental Research Funds for the Central Universities (DL12BB24) are greatly acknowledged.


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