research article the applications of algebraic methods on stable analysis for general differential...

Research ArticleThe Applications of Algebraic Methods on Stable Analysis forGeneral Differential Dynamical Systems with Multidelays

Jian Ma1 and Baodong Zheng2

1Department of Mathematics, Northeast Forestry University, Harbin 150040, China2Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Jian Ma; [email protected]

Received 13 January 2015; Revised 31 March 2015; Accepted 3 April 2015

Academic Editor: Peng Shi

Copyright © 2015 J. Ma and B. Zheng. 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 distribution of purely imaginary eigenvalues and stabilities of generally singular or neutral differential dynamical systemswith multidelays are discussed. Choosing delays as parameters, firstly with commensurate case, we find new algebraic criteria todetermine the distribution of purely imaginary eigenvalues by using matrix pencil, linear operator, matrix polynomial eigenvaluesproblem, and the Kronecker product. Additionally, we get practical checkable conditions to verdict the asymptotic stability andHopf bifurcation of differential dynamical systems. At last, with more general case, the incommensurate, we mainly study criticaldelays when the system appears purely imaginary eigenvalue.

1. Introduction

Functional differential systems with multiple delays areimportantmathematicmodels to describe all kinds of naturaland society phenomena. So it is used in many fields, such aslossless transmission lines, partial element equivalent circuitsin electrical engineering, combustion systems, and controlledconstrained manipulators in mechanical engineering. Theasymptotic stabilization of differential systems with multipledelays is an important property in many applications. In thepast decades, many results have been derived. The bifur-cations and the stability analysis of functional differentialsystems with delays especially have received much attentionby researchers andmany excellent results have been obtained;see [1–7].The asymptotic stability of differential systems withmultiple delays can be established from the rightmost partof the spectrum. In the previous paper [8], we discusseda singular neutral linear differential system with a singledelay. We now consider more general classes of differentialsystem with multiple delays, that is, the general neutral lineardifferential system with multiple delays:

𝐵0�� (𝑡) +𝑚

∑

𝑘=1𝐵𝑘�� (𝑡 − 𝜏

𝑘)

= 𝐴0𝑋(𝑡) +𝑚

∑

𝑘=1𝐴𝑘𝑋(𝑡 − 𝜏

𝑘) ,

(1)

where coefficient matrix 𝐴𝑘, 𝐵𝑘∈ R𝑛×𝑛 (𝑘 = 0, 1, . . . , 𝑚).

𝑡 (𝑡 ≥ 0) is a time variant. 𝜏𝑘(𝜏𝑘≥ 0, 𝑘 = 1, 2, . . . , 𝑚) denote

the delayed parameter, which are ordered increasingly; thatis, 𝜏1 < 𝜏2 < ⋅ ⋅ ⋅ < 𝜏𝑚. 𝑋(𝑡) (𝑋(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡), . . . , 𝑥𝑛(𝑡))

𝑇

)

is a state variant, which is given by continuous functions onthe initial interval [−𝜏

𝑚, 0]; that is,

𝑋 (𝑡) = Φ (𝑡) , 𝑡 ∈ [−𝜏𝑚, 0] . (2)

Notice that (1) is quite general, which contains manysubclasses. For example, when the leading matrix 𝐵0 satisfiesRank𝐵0 < 𝑛, the system is called a singular neutral delayeddifferential system, which is also called a delayed differential-algebraic system. Besides, when Rank𝐵0 = 𝑛, a special sub-class of (1) is written as

�� (𝑡) +

𝑚

∑

𝑘=1𝐵𝑘�� (𝑡 − 𝜏

𝑘) = 𝐴0𝑋 (𝑡) +

𝑚

∑


𝑘) , (3)

which is a nonsingular neutral delayed differential system.For (1) especially, if coefficient matrix 𝐵

𝑘= 0 (𝑘 = 1, 2,

. . . , 𝑚), we have the singular retarded differential system

𝐵0�� (𝑡) = 𝐴0𝑋(𝑡) +𝑚

∑


𝑘) . (4)

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015, Article ID 984323, 8 pageshttp://dx.doi.org/10.1155/2015/984323

2 Discrete Dynamics in Nature and Society

At last, if Rank𝐵0 = 𝑛, we have the nonsingular retardeddelay-differential system

�� (𝑡) = 𝐴0𝑋 (𝑡) +𝑚

∑


𝑘) . (5)

For (4) and (5), dynamic behaviors have beenwell studiedby many researchers, and the theory of stability has beenwell known and discussed for decades in a wide range ofliterature; see [9–15]. But for linear neutral delayed system(4) or (5), the necessary and sufficient stability conditions arescarce and less efficient than their counterparts for retardedsystems (1) and (3). Besides, for the derived results, mainlyresearched methods can be divided into two cases, analyticmethods and numerical methods. The analytic methodscontain 𝑉-functional methods, Laplace transformation, cen-tral manifolds, normal forms, and so on. The numericalmethods mainly contain linear multistep methods, Runge-Kutta methods, Newton methods, 𝜃-methods, and so on.Those methods are main keys to solve problems of stabilityon functional differential equations with delays all the time.By the development of delayed systems, many new methodsappeared. In all of them, algebraic methods gradually growup and have become a new and effective tool. For researchon more complex systems especially, such as 𝑛-dimensionalsystems, algebraic methods are important tools to simplifythe forms of time-delay systems. Certainly some results havebeen derived, yet there are more different time-delay systemswaiting for study.

In this paper, we mainly discuss neutral delayed dif-ferential systems by algebraic methods. The main work isaimed at developing more efficient stability tests for neutraldelay-differential systems. Compared to the retarded case,the neutral case induces complications. It is well known thatfor the retarded differential systems, number of eigenvaluesin the right half plane is always finite. But for neutraldifferential systems, there exist some characteristic chains,when imaginary parts tend to infinity, real parts may havefinite limit.That is to say, for some given characteristic chains{𝑠𝑛},

lim𝑛→+∞

Re 𝑠𝑛= 𝑎, if lim

𝑛→+∞

Im 𝑠𝑛= ∞. (6)

In addition, spectrum of neutral differential systemsmay be sensitive to delay parameter changes, which exhibitssome discontinuity. Even though each eigenvalue path iscontinuous, an infinitesimal change on delays may also causethe stability of system shifting. This discontinuity is closlyrelated to the essential spectrum of system (1), that is, thespectrum of difference equation

𝐵0𝑋 (𝑡) +𝑚

∑

𝑘=1𝐵𝑘𝑋(𝑡 − 𝜏

𝑘) = 0. (7)

So the stable analysis on neutral differential systems oftenexhibits more complicated. In this paper, we will find criteriato find delays margin, in which the neutral system (1) or (3)is asymptotically stable or unstable.

The main contents in the following sections can be sum-marized as follows. In Section 2, we will introduce stable

notions and present a number of preliminary facts. InSection 3, we will discuss the special case that delayedparameters are commensurate. By algebraic methods, suchas matrix pencil, linear operators, and Kronecker product,delayed margin and stability of the neutral linear differentialsystem (1) are derived. In Section 4, we will research thegeneral case, that is, the neutral linear differential systemwithincommensurate delays. For both types of above systems,we derive the criteria of stability and the distribution ofeigenvalues or generalized eigenvalues of constant matrixpencil. At last, this paper concludes in Section 5.

2. Preliminary

In this section, we begin with the description of mainnotation. Generally, let R denote the set of real numbers, Cthe set of complex numbers, and R+ the set of nonnegativereal numbers. Besides, we denote the open left half plane ofthe complex by C− = {𝑠 | Re(𝑠) < 0}, the imaginary axisby 𝜕C, and the open circle by 𝜕𝐷. From [8], we know thatthe solvability of system (1) is determined by the regularityof matrix pencil (𝐵0, 𝐴0), which is regular if 𝑠𝐵0 − 𝐴0 is notidentically singular for any complex 𝑠. Meantime, the zero 𝑠of equation det(𝑠𝐵0 −𝐴0) = 0 is called the general eigenvalueof matrix pencil (𝐵0, 𝐴0). So in this paper, we suppose thatthe matrix pencil (𝐵0, 𝐴0) is always regular. In addition, thecharacteristic equation of system (1) is

𝑃 (𝑠) V = [−𝑠𝐵 (𝑠) +𝐴 (𝑠)] V = 0, ‖V‖ = 1, (8)

where

𝐵 (𝑠) = 𝐵0 +𝑚

∑

𝑘=1𝐵𝑘𝑒−𝜏𝑘𝑠,

𝐴 (𝑠) = 𝐴0 +𝑚

∑

𝑘=1𝐴𝑘𝑒−𝜏𝑘𝑠.

(9)

The spectrum of system (1) is denoted by

𝜎 (𝑃) = {𝑠 ∈C | det (𝑃 (𝑠)) = 0} . (10)

For the difference equation (7), its characteristic equation is

𝑄 (𝑠) V = [𝐵0 (𝑠) +𝑚

∑

𝑘=1𝐵𝑘𝑒−𝜏𝑘𝑠] V = 0, ‖V‖ = 1. (11)

The spectrum of (7) is denoted by

𝜎 (𝑄) = {𝑠 ∈C | det (𝑄 (𝑠)) = 0} , (12)

which is also called the essential spectrum of system (1).From the introduction, we know that neutral differential

systems have many unpleasant properties, which have beenspecifically presented in [16, 17]. Firstly, the real part ofspectrum can have finite clustering points. Even the origincan be a clustering point. In fact, clustering points arecontained in the closure of Δ(𝑄):

Δ (𝑄) = {Re 𝑠 | 𝑠 ∈ 𝜎 (𝑄)} . (13)

Discrete Dynamics in Nature and Society 3

So the real part of the spectrum is not always continuouslyrelated to parameter delays. This discontinuity is clearlyrelated to the neutral part of system (1). Certainly, for system(1), unlike retarded differential systems, it is not sufficientfor stability that the spectrum is absolutely contained in theopen left half plane of the complex plane. Fortunately, thereis theory available in the literature which gives sufficientconditions for these pessimistic properties. By [6], we havederived that system (1) is asymptotically stable when

Re 𝑠 ≤ − 𝛿 < 0, (14)

where 𝑠 is an eigenvalue and 𝛿 > 0 is a real number. Thenecessary condition on the stability of system (1) especiallyis that the difference equation (7) of neutral part of system(1) is stable, that is, the spectrum radius 𝜌(𝑄) < 1. From [8],we all know that condition Re 𝑠 ≤ −𝛿 < 0 can be improvedto Re 𝑠 < 0 if the neutral part of system (1) is stable. That isto say, the critical condition for stability switch of system (1)is that the rightmost eigenvalue goes from the left half planeof the complex plane into the right half plane by passing theimaginary axis. So appearance of imaginary eigenvalues is acritical condition. In the following we will compute criticalvalue of delayed parameters such that the stability switchoccurs.

Next we will discuss the stability of system (1) by twosections. In Section 3, the delays are commensurate, in whichphenomena appeared in many natural species. In Section 4,the delays are incommensurate. These dynamics have notbeen well understood yet. By algebraic methods, we candemonstrate the dynamical property more compactly andintuitionally.

3. The Commensurate Case

In this section, wewill consider the neutral differential systemwith commensurate delays

𝐵0�� (𝑡) +𝑚

∑

𝑘=1𝐵𝑘�� (𝑡 − 𝑘𝜏)

= 𝐴0𝑋 (𝑡) +𝑚

∑

𝑘=1𝐴𝑘𝑋 (𝑡 − 𝑘𝜏) ,

(15)

where matrix 𝐴𝑘, 𝐵𝑘∈ R𝑛×𝑛 (𝑘 = 0, 1, . . . , 𝑚). When

Rank𝐵0 < 𝑛, system (15) is singular. When Rank𝐵0 = 𝑛,it can be rewritten as

�� (𝑡) +

𝑚

∑

𝑘=1𝐵𝑘�� (𝑡 − 𝑘𝜏)

= 𝐴0𝑋 (𝑡) +𝑚

∑

𝑘=1𝐴𝑘𝑋 (𝑡 − 𝑘𝜏) ,

(16)

which is nonsingular. For system (15) and (16), some scholarshave widely researched the delay-independent or delay-dependent stability and asymptotic stability by analyticmeth-ods or numerical methods. Because of the complex natureof singular differential systems with delays, research is very

difficult by using the analytical treatment. So few studies onstability and bifurcations have been conducted so far. Forthe singular neutral differential system (15) especially, thereare hardly flexible and efficient verdicts. In the following, wewill find algebraic criteria of the distribution of imaginaryeigenvalues and stability for system (15).

3.1. Criteria for Determining Imaginary Eigenvalues. Firstly,an ordinary differential equation is considered, which moti-vates future research. Consider matrix equation as follows:

𝐵�� (𝑡) = 𝐴𝑋 (𝑡) ,

��𝑇

(𝑡) 𝐵𝑇

= −𝑋𝑇

(𝑡) 𝐴𝑇

,

(17)

where 𝐵 = (𝐵0, 𝐵1, . . . , 𝐵𝑚), 𝐴 = (𝐴0, 𝐴1, . . . , 𝐴𝑚), 𝐵 = (𝐵𝑚,𝐵𝑚−1, . . . , 𝐵0), 𝐴 = (𝐴

𝑚, 𝐴𝑚−1, . . . , 𝐴0), 𝐵𝑘, 𝐴𝑘 ∈ R𝑛×𝑛,

𝑋(𝑡) = (𝑋0(𝑡), 𝑋1(𝑡), . . . , 𝑋𝑚(𝑡))𝑇, 𝑋𝑘(𝑡) ∈ C𝑛×𝑛, 𝑘 = 0, 1,

. . . , 𝑚. Let 𝑉 denote a vector space,

𝑉 = C((𝑚+1)𝑛)×𝑛

×C𝑛×((𝑚+1)𝑛)

. (18)

𝐿1, 𝐿2 denote operators on 𝑉,

𝐿1𝑋 (𝑡) = (𝐵𝑋 (𝑡)

𝑋𝑇

(𝑡) 𝐵𝑇) ,

𝐿2𝑋 (𝑡) = (𝐴𝑋 (𝑡)

−𝑋𝑇

(𝑡) 𝐴𝑇) ,

∀𝑋 (𝑡) ∈ C((𝑚+1)𝑛)×𝑛

.

(19)

Then system (17) can be rewritten as

𝐿1�� (𝑡) = 𝐿2𝑋 (𝑡) . (20)

Supposing 𝑋(𝑡) = (𝑌𝑒𝑠𝑡, 𝑌𝑒𝑠(𝑡−𝜏), 𝑌𝑒𝑠(𝑡−2𝜏), . . . , 𝑌𝑒𝑠(𝑡−𝑚𝜏))𝑇 is amatrix solution of system (17), we have

[(𝑠

𝑚

∑

𝑘=0𝐵𝑘𝑧𝑘

)−(

𝑚

∑

𝑘=0𝐴𝑘𝑧𝑘

)]𝑌 = 0,

𝑌𝑇

[(𝑠

𝑚

∑

𝑘=0𝐵𝑇

𝑚−𝑘𝑧𝑘

)+(

𝑚

∑

𝑘=0𝐴𝑇

𝑚−𝑘𝑧𝑘

)] = 0,

(21)


where 𝑧 = 𝑒−𝑠𝜏. For ∀𝑠 ∈ C, T = T(𝑠), Λ = Λ(𝑠) denote

operators on 𝑉,

T𝑋 =([(𝑠

𝑚

∑


) − (

𝑚

∑


)]𝑋

𝑋𝑇

[(𝑠

𝑚

∑

𝑘=0𝐵𝑇

𝑚−𝑘𝑧𝑘

) + (

𝑚

∑

𝑘=0𝐴𝑇

𝑚−𝑘𝑧𝑘

)]

),

∀𝑋 ∈ C((𝑚+1)𝑛)×𝑛

,

Λ𝑋 = (

𝑚

∑


)𝑋(

𝑚

∑

𝑘=0𝐵𝑇

𝑚−𝑘𝑧𝑘

)

+(

𝑚

∑


)𝑋(

𝑚

∑

𝑘=0𝐴𝑇

𝑚−𝑘𝑧𝑘

) ,

∀𝑋 ∈ C((𝑚+1)𝑛)×𝑛

.

(22)

We have 𝑇𝑌 = 0; that is,

Λ ⋅ 𝑌𝑌𝑇

= (

𝑚

∑


)𝑌𝑌𝑇

(

𝑚

∑

𝑘=0𝐵𝑇

𝑚−𝑘𝑧𝑘

)

+(

𝑚

∑


)𝑌𝑌𝑇

(

𝑚

∑

𝑘=0𝐴𝑇

𝑚−𝑘𝑧𝑘

) = 0.

(23)

By the property of Kronecker product, we have

[(

𝑚

∑


)⊗(

𝑚

∑

𝑘=0𝐵𝑚−𝑘𝑧𝑘

)+(

𝑚

∑


)

⊗(

𝑚

∑

𝑘=0𝐴𝑚−𝑘𝑧𝑘

)] ⋅ 𝜉 (𝑌𝑌𝑇

) = 0.

(24)

So

det[(𝑚

∑


)⊗(

𝑚

∑

𝑘=0𝐵𝑚−𝑘𝑧𝑘

)+(

𝑚

∑


)

⊗(

𝑚

∑

𝑘=0𝐴𝑚−𝑘𝑧𝑘

)] = 0.

(25)

Theorem 1. Any imaginary eigenvalue of system (15) is one ofroots of (25).

Proof. The characteristic equation of system (15) is

𝑝 (𝑠, 𝑧) V = [−𝑠𝑚

∑


+

𝑚

∑


] V = 0, ‖V‖ = 1, (26)

where 𝑧 = 𝑒−𝜏𝑠. Suppose 𝑠 = 𝑖𝑤; by conjugating and trans-forming we have

V𝑇 [𝑠𝑚

∑

𝑘=0𝐵𝑇

𝑚−𝑘𝑧𝑘

+

𝑚

∑

𝑘=0𝐴𝑇

𝑚−𝑘𝑧𝑘

] = 0. (27)

By (26) and (27),

𝑠

𝑚

∑

𝑘=0𝐵𝑘𝑧𝑘V =

𝑚

∑

𝑘=0𝐴𝑘𝑧𝑘V,

− 𝑠V𝑇𝑚

∑

𝑘=0𝐵𝑇

𝑚−𝑘𝑧𝑘

= V𝑇𝑚

∑

𝑘=0𝐴𝑇

𝑚−𝑘𝑧𝑘

.

(28)

By simplistically computing,

𝑠 [

𝑚

∑


] VV𝑇 [𝑚

∑

𝑘=0𝐵𝑇

𝑚−𝑘𝑧𝑘

]

= [

𝑚

∑


] VV𝑇 [𝑚

∑

𝑘=0𝐵𝑇

𝑚−𝑘𝑧𝑘

] ,

− 𝑠 [

𝑚

∑


] VV𝑇 [𝑚

∑

𝑘=0𝐵𝑇

𝑚−𝑘𝑧𝑘

]

= [

𝑚

∑


] VV𝑇 [𝑚

∑

𝑘=0𝐴𝑇

𝑚−𝑘𝑧𝑘

] .

(29)

So

[

𝑚

∑


] VV𝑇 [𝑚

∑

𝑘=0𝐵𝑇

𝑚−𝑘𝑧𝑘

]

+[

𝑚

∑


] VV𝑇 [𝑚

∑

𝑘=0𝐴𝑇

𝑚−𝑘𝑧𝑘

] = 0.

(30)

By above operator, the result can be derived.

From [18], (25) is a polynomial eigenvalue problem. Bylinearization, we have the following lemma.

Lemma 2. All of 𝑧 = 𝑒−𝑠𝜏 of the characteristic equation of

system (15) are general eigenvalues of the matrix pencil (𝐸, 𝐹),where

𝐸 =(

𝐶2𝑚 0 ⋅ ⋅ ⋅ 00 𝐼𝑛⋅ ⋅ ⋅ 0

.

.

.... d

.

.

.

0 0 ⋅ ⋅ ⋅ 𝐼𝑛

),

𝐹 =

(((

(

−𝐶2𝑚−1 −𝐶2𝑚−2 ⋅ ⋅ ⋅ −𝐶1 −𝐶0

𝐼𝑛

0 ⋅ ⋅ ⋅ 0 00 𝐼

𝑛⋅ ⋅ ⋅ 0 0

.

.

.... d

.

.

....

0 0 ⋅ ⋅ ⋅ 𝐼𝑛

0

)))

)

.

(31)


Proof. By (25) and property of Kronecker product, we have

det[

[

𝐴0 ⊗𝐵𝑚 +𝐵0 ⊗𝐴𝑚 +𝑚

∑

𝑘=1𝑧𝑘

(𝐴0 ⊗𝐵𝑚−𝑘 +𝐴𝐾

⊗𝐵𝑚+𝐵0 ⊗𝐴𝑚−𝑘 +𝐵𝐾 ⊗𝐴𝑚) +

𝑚

∑

𝑘=1,𝑗=1𝑧𝑘+𝑗

(𝐴𝑘

⊗𝐵𝑚−𝑗

+𝐵𝐾⊗𝐴𝑚−𝑗)]

]

= 0,

(32)

which is a polynomial equationwith 2𝑚degrees about variant𝑧,

det [𝑃2𝑚 (𝑧)]

= det [𝐶2𝑚𝑧2𝑚+𝐶2𝑚−1𝑧

2𝑚−1+ ⋅ ⋅ ⋅ + 𝐶1𝑧 +𝐶0]

= 0;

(33)

that is

𝑃2𝑚 (𝑧) 𝑌 = 0, 𝑌 = 𝜉 (VV𝑇) . (34)

Let

𝑈 =(

𝑢1

𝑢2

.

.

.

𝑢2𝑚

)=

((((

(

𝑧2𝑚−1

𝑌

𝑧2𝑚−2

𝑌

.

.

.

𝑧𝑌

𝑌

))))

)

. (35)

We can get

𝑃2𝑚 (𝑧) 𝑌 = 𝐶2𝑚𝑧𝑧2𝑚−1

𝑌+𝐶2𝑚−1𝑧2𝑚−1

𝑌+ ⋅ ⋅ ⋅

+ 𝐶1𝑧𝑌+𝐶0𝑌,(36)

which can be written as

𝑃2𝑚 (𝑧) 𝑌 = 𝐶2𝑚𝑧𝑧2𝑚−1

𝑌+𝐶2𝑚−1𝑧2𝑚−1

𝑌+ ⋅ ⋅ ⋅

+ 𝐶1𝑧𝑌+𝐶0𝑌.(37)

By the companion form, the linearization of (25) is

𝑃2𝑚 (𝑧) 𝑈 = [𝑧𝐸−𝐹]𝑈 = 0. (38)

So 𝑧 is the general eigenvalue of matrix pencil (𝐸, 𝐹).

3.2. Criteria for the Asymptotic Stability. Stability of delay-differential equations has been widely researched [19–21]. Weall know that the Lyapunov-Kraeovskii functional approach isan important analytic method to discuss delay-independentstability. Results for singular neutral differential systems arestill very few, especially by algebraic methods. Next we firstresearch the delay-independent stability of system (15).

Theorem 3. If coefficient matrices of system (15) satisfy thefollowing,

(1) for matrix pencil (−𝐵0, 𝐴0), 𝜎(−𝐵0, 𝐴0) ∈ 𝐶−,

(2) supRe(𝑠)=0 𝜌(∑𝑚

𝑖=1 |(−𝑠𝐵0 + 𝐴0)−1(−𝑠𝐵𝑖+ 𝐴𝑖)|) < 1,

then the stability of system (15) is delay-independent; that is,system (15) is asymptotically stable for all 𝜏 ≥ 0.

Proof. See [2].

Besides, it is well known that if the neutral part ofsystem (15) is stable, then system (15) is also stable when alleigenvalues pose in the half plane of the complex plane. Forthe difference equation:

𝑚

∑

𝑘=0𝐵𝑘𝑋 (𝑡 − 𝑘𝜏) = 0. (39)

The characteristic equation is

𝑄 (𝑠, 𝑧) V = (𝑚

∑


) V = 0, ‖V‖ = 1. (40)

Lemma 4. Variant 𝑧 is contained in the spectrum of matrixpencil (𝑀,𝑁), where

𝑀 =(

𝐵𝑚

0 ⋅ ⋅ ⋅ 00 𝐼𝑛⋅ ⋅ ⋅ 0

.

.

.... d

.

.

.

0 0 ⋅ ⋅ ⋅ 𝐼𝑛

),

𝑁 =

(((

(

−𝐵𝑚−1 −𝐵

𝑚−2 ⋅ ⋅ ⋅ −𝐵2 −𝐵1

𝐼𝑛

0 ⋅ ⋅ ⋅ 0 00 𝐼

𝑛⋅ ⋅ ⋅ 0 0

.

.

.... d

.

.

....

0 0 ⋅ ⋅ ⋅ 𝐼𝑛

0

)))

)

.

(41)

Proof. Comparing with Lemma 2, Let

𝑈 =(

(

V

𝑧V...

𝑧𝑚−1V

)

)

; (42)

we can get 𝜎(𝑀,𝑁) = {𝑧 | det(𝑧𝑀 − 𝑁) = 0}.

Apparently, the difference equation is stable when spec-trum radius 𝜌(𝑀,𝑁) < 1. If this condition is satisfied, thensystem (15) is asymptotically stable when all eigenvalues havenegative real part. So we have the following result.

Theorem 5. Suppose system (15) is stable at 𝜏 = 0 and𝜌(𝑀,𝑁) < 1.


(1) If system (15) satisfies one of the following conditions,

(a) 𝜎(𝐸, 𝐹) ∩ 𝜕𝐷 = 0,(b) 𝜎(𝐸, 𝐹) ∩ 𝜕𝐷 = 0, for 𝑧

𝑘∈ 𝜎(𝐸, 𝐹) ∩ 𝜕𝐷, 𝜎(𝑝(𝑠,

𝑧𝑘)) = {0},

(c) 𝜎(𝐸, 𝐹) ∩ 𝜕𝐷 = 0, ∀𝑧𝑘∈ 𝜎(𝐸, 𝐹) ∩ 𝜕𝐷, 𝜎(𝑝(𝑠,

𝑧𝑘)) ∩ 𝜕𝐶

+

= 0,

then system (15) is asymptotically stable for 𝜏 ∈ [0,+∞).

(2) Otherwise, if 𝜎(𝑝(𝑠, 𝑧𝑘)) ∩ 𝜕𝐶

+

= 0, there is 𝑠 = 𝑖𝑤(𝑗)𝑘∈

𝑅+, 𝑗 = 1, 2, . . . , 𝑡, 𝑡 ≤ 𝑙. Let 𝑧

𝑘= 𝑒𝑖𝛽𝑘 , which satisfies

det[𝑝(𝑖𝑤(𝑗)𝑘, 𝑧𝑘)] = 0. Then there exists

𝜏∗

= min1≤𝑘≤𝑙

min1≤𝑗≤𝑡

𝛽𝑘

𝑤(𝑗)

𝑘

. (43)

System (15) is asymptotically stable at [0, 𝜏∗) andunstable when 𝜏 > 𝜏∗ and bifurcates at 𝜏 = 𝜏∗.

4. The Incommensurate Case

For the general case, system (1) can be rewritten as

𝑚

∑

𝑘=0𝐵𝑘�� (𝑡 − 𝜏

𝑘) =

𝑚

∑


𝑘) , (44)

where 𝜏0 = 0. For the simplicity, the characteristic equationcan also be rewritten as

𝑀(𝑠) V = [−𝑠𝐵 (𝑠) +𝐴 (𝑠)] V = 0, (45)

where

𝐵 (𝑠) =

𝑚

∑

𝑘=0𝐵𝑘𝑒−𝑠𝜏𝑘 ,

𝐴 (𝑠) =

𝑚

∑

𝑘=0𝐴𝑘𝑒−𝑠𝜏𝑘 .

(46)

Apparently, the stability of system (44) is similar to system(15). Firstly, for the delay-independent stability, we have thesame conclusion.

Theorem 6. If

(1) 𝜎(−𝐵0, 𝐴0) ∈ 𝐶−,

(2) supRe(𝑠)=0 𝜌(∑𝑚

𝑖=1 |(−𝑠𝐵0 + 𝐴0)−1(−𝑠𝐵𝑖+ 𝐴𝑖)|) < 1,

then the stability of system (44) is delay-independent; thatis, system (44) is asymptotically stable for delay parameters{𝜏𝑘} (𝜏𝑘> 0, 𝑘 = 1, 2, . . . , 𝑚).

Proof . See Theorem 3.

Next, we will focus on delay-dependent stability con-ditions. For delay parameters {𝜏

𝑘} (𝑘 = 1, 2, . . . , 𝑚), let

𝑟(𝜏1, 𝜏2, . . . , 𝜏𝑚) denote the real part of the rightmost eigen-value or the corresponding supremum. Unlike retarded

delay-differential systems, system (44) is asymptotically sta-ble if the real part of the spectrum lies in the left half plane ofcomplex plane and is bounded away from a negative number;that is,

𝑟 (𝜏1, 𝜏2, . . . , 𝜏𝑚) < − 𝛿 < 0. (47)

The associated difference equation is

𝑚

∑

𝑘=0𝐵𝑘𝑋(𝑡 − 𝜏

𝑘) = 0. (48)

As we all know if system (48) is strongly exponentiallystable and the origin is not a clustering point of real part ofspectrum, the spectral abscissa (the real part of eigenvalues)of system (44) will be continuous on delay parameters {𝜏

𝑘}.

Under above condition, system (44) is asymptotically stablewhen

𝑟 (𝜏1, 𝜏2, . . . , 𝜏𝑚) < 0. (49)

When 𝑟(𝜏1, 𝜏2, . . . , 𝜏𝑚) > 0, the system (44) is unstable. So thestability of system (44) could shift when 𝑟(𝜏1, 𝜏2, . . . , 𝜏𝑚) =0. The root (𝜏∗1 , 𝜏

∗

2 , . . . , 𝜏∗

𝑚) of 𝑟(𝜏1, 𝜏2, . . . , 𝜏𝑚) = 0 is called

critical delays. In the following, we will find algebraic criteriafor the critical delays.

First, assuming that system (44) is hyperbolic (the spec-trumhas no zero eigenvalue) and zero is not a clustering pointof the real part of spectrum. Let 𝑠 = 𝑖𝑤, 𝑤 ∈ R be thepurely imaginary eigenvalue and V ∈ C𝑛 be the correspondingeigenvector. Meantime, let 𝜑

𝑘= 𝑤𝜏𝑘(𝑘 = 0, 1, 2, . . . , 𝑚 − 1)

denote 𝑚 free parameters (𝜑0 = 0) and 𝑧 = 𝑒−𝑖𝑤𝜏𝑚 ∈ 𝜕𝐷. Forthe characteristic equation (45), we have

𝑠 [𝑧𝐵𝑚+

𝑚−1∑

𝑘=0𝐵𝑘𝑒−𝑖𝜑𝑘] V = [𝑧𝐴

𝑚+

𝑚−1∑

𝑘=0𝐴𝑘𝑒−𝑖𝜑𝑘] V, (50)

where 𝐴(��) = ∑𝑚−1𝑘=0 𝐴𝑘𝑒

−𝑖𝜑𝑘 , 𝐵(��) = ∑𝑚−1𝑘=0 𝐵𝑘𝑒

−𝑖𝜑𝑘 . Then

𝑠 [𝑧𝐵𝑚+𝐵 (��)] V = [𝑧𝐴

𝑚+𝐴 (��)] V. (51)

By conjugating and transforming, we have

− 𝑠V∗ [1𝑧𝐵∗

𝑚+𝐵∗

(��)] = V∗ [1𝑧𝐴∗

𝑚+𝐴∗

(��)] . (52)

Multiplying with 𝑧,

− 𝑠V∗ [𝐵∗𝑚+ 𝑧𝐵∗

(��)] = V∗ [𝐴∗𝑚+ 𝑧𝐴∗

(��)] . (53)

That is

𝑠 [𝑧𝐵𝑚+𝐵 (��)] VV∗ [𝐵∗

𝑚+ 𝑧𝐵∗

(��)]

= [𝑧𝐴𝑚+𝐴 (��)] VV∗ [𝐵∗

𝑚+ 𝑧𝐵∗

(��)] ,

− 𝑠 [𝑧𝐵𝑚+𝐵 (��)] VV∗ [𝐵∗

𝑚+ 𝑧𝐵∗

(��)]

= [𝑧𝐵𝑚+𝐵 (��)] VV∗ [𝐴∗

𝑚+ 𝑧𝐴∗

(��)] ,

(54)


So

[𝑧𝐵𝑚+𝐵 (��)] VV∗ [𝐴∗

𝑚+ 𝑧𝐴∗

(��)]

+ [𝑧𝐴𝑚+𝐴 (��)] VV∗ [𝐵∗

𝑚+ 𝑧𝐵∗

(��)] = 0.(55)

Expanding (55)

𝑧2[𝐵𝑚VV∗𝐴∗ (��) +𝐴

𝑚VV∗𝐵∗ (��)] + 𝑧 [𝐵

𝑚VV∗𝐴∗𝑚

+𝐵 (��) VV∗𝐴∗ (��) +𝐴𝑚VV∗𝐵∗𝑚

+𝐴 (��) VV∗𝐵∗ (��)] + 𝐵 (��) VV∗𝐴∗𝑚+𝐴 (��) VV∗𝐵∗

𝑚

= 0.

(56)

By Kronecker property, we have

[𝑧2(𝐵𝑚⊗𝐴 (−��) +𝐴

𝑚⊗𝐵 (−��)) + 𝑧 (𝐵

𝑚⊗𝐴𝑚

+𝐵 (��) ⊗𝐴 (−��) +𝐴𝑚⊗𝐵𝑚+𝐴 (��) ⊗ 𝐵 (−��))

+ 𝐵 (��) ⊗𝐴𝑚+𝐴 (��) ⊗ 𝐵

𝑚] 𝜉 (VV∗) = 0.

(57)

Let

𝑋(��) = 𝐵𝑚⊗𝐴 (−��) +𝐴

𝑚⊗𝐵 (−��) ,

𝑌 (��) = 𝐵𝑚⊗𝐴𝑚+𝐵 (��) ⊗𝐴 (−��) +𝐴

𝑚⊗𝐵𝑚

+𝐴 (��) ⊗ 𝐵 (−��) ,

𝑍 (��) = 𝐵 (��) ⊗𝐴𝑚+𝐴 (��) ⊗ 𝐵

𝑚,

𝑢 = 𝜉 (VV∗) ,

(58)

so we have

𝑧2𝑋(��) + 𝑧𝑌 (��) +𝑍 (��) 𝑢 = 0. (59)

Lemma 7. All of 𝑧 = 𝑒−𝑠𝜏𝑚 of the characteristic equation of

system (44) are general eigenvalues of matrix pencil (𝐸, 𝐹),where

𝐸 = (

𝑋 (��) 00 𝐼

) ,

𝐹 = (

−𝑌 (��) −𝑍 (��)

𝐼 0) .

(60)

Proof. The proof is similar to Lemma 2.

Theorem 8. Assuming that system (44) is hyperbolic and zerois not root of 𝐵(𝑠)V = 0, then critical delays satisfy

𝜏𝑘=𝜑𝑘+ 2𝑛𝑘𝜋

𝑤, 𝑛𝑘∈ Z, 𝑘 = 1, 2, . . . , 𝑚 − 1,

𝜏𝑚=−Arg𝑧 + 2𝑛

𝑚𝜋

𝑤, 𝑛𝑚∈ Z,

(61)

where

𝑤 = − 𝑖V∗ [𝐵∗𝑚+ 𝑧𝐵∗

(��)] [𝑧𝐴𝑚+ 𝐴 (��)] V

V∗ [𝐵∗𝑚+ 𝑧𝐵∗ (��)] [𝑧𝐵

𝑚+ 𝐵 (��)] V

∈ R, (62)

𝑧 ∈ 𝜕𝐷 is general eigenvalue of matrix pencil (𝐸, 𝐹), 𝑠 =𝑖𝑤 (𝑤 ∈ R) is the purely imaginary eigenvalue, V ∈ C𝑛, andV = 0 is the corresponding eigenvector.

Proof. For the characteristic equation of system (44)

𝑠 [𝑧𝐵𝑚+𝐵 (��)] V = [𝑧𝐴

𝑚+𝐴 (��)] V. (63)

By multiplying with V∗[𝑧𝐵𝑚+ 𝐵(��)]

∗

𝑠V∗ [𝑧𝐵𝑚+𝐵 (��)]

∗

[𝑧𝐵𝑚+𝐵 (��)] V

= V∗ [𝑧𝐵𝑚+𝐵 (��)]

∗

[𝑧𝐴𝑚+𝐴 (��)] V.

(64)

Apparently, V∗[𝑧𝐵𝑚+ 𝐵(��)]

∗

[𝑧𝐵𝑚+ 𝐵(��)]V ∈ R, so

𝑠 =V∗ [𝑧𝐵

𝑚+ 𝐵 (��)]

∗

[𝑧𝐴𝑚+ 𝐴 (��)] V

V∗ [𝑧𝐵𝑚+ 𝐵 (��)]

∗

[𝑧𝐵𝑚+ 𝐵 (��)] V

. (65)

By 𝑠 = 𝑖𝑤, we have

𝑤 = − 𝑖V∗ [𝐵∗𝑚+ 𝑧𝐵∗

(��)] [𝑧𝐴𝑚+ 𝐴 (��)] V

V∗ [𝐵∗𝑚+ 𝑧𝐵∗ (��)] [𝑧𝐵

𝑚+ 𝐵 (��)] V

. (66)

For (55), define

𝑀 = 𝑧𝐴𝑚+𝐴 (��) ,

𝑁 = 𝑧𝐵𝑚+𝐵 (��) .

(67)

Then

𝑁VV∗𝑀∗ +𝑀VV∗𝑁∗ = 0. (68)

That is,

V∗𝑁∗𝑁VV∗𝑀∗𝑁V+ V∗𝑁∗𝑀VV∗𝑁∗𝑁V = 0. (69)

BecauseV∗ [𝑧𝐵

𝑚+𝐵 (��)]

∗

[𝑧𝐵𝑚+𝐵 (��)] V = 0,

V∗𝑁∗𝑁V = 0,(70)

we have

V∗𝑀∗𝑁V+ V∗𝑁∗𝑀V = 0. (71)

We have

V∗𝑀∗𝑁V+ [V∗𝑀∗𝑁V]∗ = 0. (72)

So

V∗𝑀∗𝑁V = V∗ [𝐵∗𝑚+ 𝑧𝐵∗

(��)] [𝑧𝐴𝑚+𝐴 (��)] V ∈ 𝑖R. (73)

So we get that 𝑤 ∈ R. Next, by 𝜑𝑘= 𝑤𝜏𝑘, 𝑧 = 𝑒−𝑖𝑤𝜏𝑚 , we can

get

𝜏𝑘=𝜑𝑘+ 2𝑛𝑘𝜋

𝑤, 𝑛𝑘∈ Z, 𝑘 = 1, 2, . . . , 𝑚 − 1,

𝜏𝑚=−𝐴𝑟𝑔𝑧 + 2𝑛

𝑚𝜋

𝑤, 𝑛𝑚∈ Z.

(74)

From above conclusion, we find critical delays of thegeneral neutral delay-differential equation with multipledelays. By stable theory, we can also discuss the stability ofsystem (44).


5. Conclusion

In this paper, we consider singular neutral differential systemswith multiple delays. By applying algebraic method, such asthe matrix pencil spectrum, general eigenvalues, Kroneckerproduct, and linear operators, we discussed eigenvalues andstability of delay-differential systems (1). For the commensu-rate case, we expatiate the distribution of purely imaginaryeigenvalue, the delay-independent, or delay-dependent sta-bility. For the incommensurate case, we also get the criticaldelays in which stability changed. Certainly, applying alge-braic methods to analyze dynamical properties of singularneutral differential systemswith delays is a new and immaturefield, so we believe that algebraic methods used to researchthe stability of dynamical systems would be more interest inthe future.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

Theauthors acknowledge theNational Natural Science Foun-dations of China (11426058) and the Fundamental ResearchFunds for the Central Universities (2572015CB25).

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