LIU Lingling, FAN Ni, SUN Leping

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

1 Introduction

f(s)=f(x,y)=u(x,y)+iv(x,y)

(1)

Theorem1.1[1]If for any(x,y)∈?W,the real partu(x,y)in (1)does not vanish,thenf(x,y)≠0 for any(x,y)∈W.

Theorem 1.2 is an extension of Theorem 1.1.

2 Delay independent stability of DDAEs

Now we deal with the asymptotic stability of DDAEs,

Ax′(t)=Bx(t)+Cx(t-τ),

(2)

whereA,B,C∈d×dare constant real matrices,Ais singular andτ>0 stands for a constant delay.For the stability of the system(2),we investigate its characteristic equation

det[λA-B-Ce-λτ]=0.

(3)

(4)

(5)

then(3)may be written as

(6)

wherez=x+iy.By the above assumption that Res<0?Reλ<0 is valid.

The following two lemma are well-known.

Lemma2.1[2]If the real parts of all the characteristic roots of(5)are less than zero,then the system(2)is asymptotically stable; That is,the solutionx(t)of (2)satisfiesx(t)→0 ast→∞.

Lemma2.2[3]LetA∈d×dandB∈d×d.If the inequality |A|≤Bholds,then the inequalityρ(A)≤ρ(B)is valid.Here the order relation of matrices of the same dimensions should be interpreted componentwise.|A| stands for the matrix whose component is replaced by the modulus of the corresponding component ofA,andρ(A)means the spectral radius ofA.

For a complex matrixW,letμ(W)be the logarithmic norm ofW.

Lemma2.3[3]For each eigenvalue of a matrixW∈d×d,the inequality

-μp(-W)≤Reλi(W)≤μp(W)

holds.

Lemma2.4[4]LetU,Vben-by-krectangular matrices withk≤nandAbe ann-by-nmatrix.Then

T=I+VTA-1U

is nonsingular if and only ifA+UVTis nonsingular.In this case,we have

(A+UVT)-1=A-1-A-1UT-1VTA-1.

The following lemma states a sufficient condition for the stability of (2).

(7)

holds,the system (2)is asymptotically stable.

Applying the properties of the logarithmic norm,Lemma 2.4 and Lemma 2.5,we have

This,however,contradicts the condition.Hence the proof is completed.

The following theorem gives a region including all the roots of (5)with nonnegative real parts when the condition of Lemma 2.6 fails.

(i) If we have the estimation

then the inequalities

and

hold.

(ii) If we have the estimation

define a positive numberβsatisfying

Then the inequalities

and

Proof(i) A discussion similar to that of Lemma 2.6 yields

Next,the imaginary part of an eigenvalue of a matrixAB-1is equal to the real part of an eigenvalue of -iAB-1.The second inequality holds.

(ii) By Lemma(2.3)

(8)

A derivation similar to that in (i) leads to

(9)

Here the truth of the last inequality is attained form the following.

Hence,taking (9)into consideration,we have

Iteration

and the monotonicity

ensure that the limit of the series {βj} is equal toβ,whereβis a positive number satisfying

Therefore the first inequality holds.In a similar manner we can get the second inequality.

|s|≤ρ(|A|·|B-1|·(I-|CB-1|)-1)

holds.

ProofBy the assumption above,there exists an integerj(1≤j≤d)such that

This implies the inequality

It is obvious that

Due to Lemma 2.2,we have the conclusion.

3 Delay-dependent stability of DDAEs:boundary criteria

Let

By virtue of Lemma 2.6,ifγ<0,the system(2)is Delay-independent asymptotically stable.Ifγ≥0,the system (2)may be stable or unstable.We consider the stability of (2)whenγ≥0.

(i) Ifβ0≤0,then we put

(ii) Ifβ0>0,then we put

whereβis a root of the equation

Under the above notation we turn our attention to the following three kinds of bounded regions in thes-plane.

Definition1Letl1,l2,l3andl4denote the segments {(E0,y):F0≤y≤F},{(x,F):E0≤x≤E},{(E,y):F0≤y≤F} and {(x,F0):E0≤x≤E},respectively.Furthermore,l=l1∪l2∪l3∪l4and letDbe the rectangular region surrounded byl.

Definition2LetR=ρ(|A|·|B-1|·(I-|CB-1|)-1).LetKdenote the circular region with radiusRcentered at the origin of the plane ofC.

K={(r,θ):r≤R,0≤θ≤2π}.

A necessary and sufficient condition for the delay-independent stability of the system(2)is presented in [5].The following two theorems give criteria for the delay-dependent stability of the system(2).We apply Theorem 1.1 and Theorem 1.2 to prove them respectively.

Theorem3.1If for any (x,y)∈?T,the real partU(x,y)in(6)does not vanish,then the system (2)is asymptotically stable.

Due to Theorem 2,we can further extend the above result as follows.

Theorem3.2Assume that for any (x,y)∈?T,there exists a real constantλsatisfying

U(x,y)+λV(x,y)≠0.

Then the system (2)is asymptotically stable.

The proof is analogous to Theorem 3.1.

We give two criteria for the delay-dependent stability of the linear delay system(2).Theorem 2.1 and Theorem 2.2 show that the unstable characteristic roots of the system (2)are located in some specified bounded region in the complex plane,while Theorem 3.1 and Theorem 3.2 show that it is sufficient to check certain conditions on its boundary to exclude the possibility of such roots from the region.Theorem 1.1 and Theorem 1.2 provide general and simple criteria for the nonexistence of zeros of an analytic function in any boundary region.

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