DSpace at VNU: Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions

DSpace at VNU: Stability criteria for differential-algebraic equations with multiple delays and their numerical solution...

Applied Mathematics and Computation 208 (2009) 397–415 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions Stephen L Campbell a, Vu Hoang Linh b,* a b Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam a r t i c l e i n f o a b s t r a c t This paper is concerned with the asymptotic stability of differential-algebraic equations with multiple delays and their numerical solutions First, we give a sufficient condition for delay-independent stability After characterizing the coefficient matrices that satisfy this stability condition, we propose some practical checkable criteria for asymptotic stability Then we investigate the stability of numerical solutions obtained by h-methods and BDF methods Finally, solvability and stability of a class of weakly regular delay differential-algebraic equations are analyzed Ó 2008 Elsevier Inc All rights reserved Keywords: Delay differential-algebraic equation Multiple delays Asymptotic stability Regular pencil Numerical solution Introduction In this paper we consider the linear differential-algebraic equation with multiple delays _ ỵ Bxtị ỵ Axtị M X _ si ị ỵ C i xt iẳ1 M X Di xt si ị ẳ 0; 1:1ị iẳ1 where A, B, Ci, Di (i = 1, 2, , M), are real (or complex) constant matrices of size m  m The time-delays are ordered increasingly, < s1 < s2 < Á Á Á < sM Matrix A is assumed to be singular with rank A = d < m We are also interested in a special subclass of (1.1) in the form, _ ỵ Bxtị ỵ Axtị M X iẳ1 _ isị ỵ C i xt M X Di xt isị ẳ 0: 1:2ị iẳ1 That is by si = is (i = 1, 2, , M), where s > is given From now on, if the unknown functions appear without argument and no confusion arises, we mean that they are evaluated at the actual time t For example, we write x instead of x(t) and x_ in_ stead of xðtÞ Differential-algebraic equations (DAEs) play important roles in mathematical modeling of real-life problems arising in a wide range of applications, for example, multibody mechanics, prescribed path control, electrical design, chemically reacting systems, biology and biomedicine See [3,16] and references therein In many problems, the systems in consideration contain time-delays, see [2,5–7,10,19,20,22,24–26] While the theory and the numerics for delay ordinary differential equations (DODEs) have been well known and discussed for decades in a wide range of literature, see [12] and references therein, there are very few results for the theory of delay differential-algebraic equations (DDAEs) The main reason is that even for linear DDAEs, their dynamics have not been well understood yet, in particular when the pencil {A, B} in (1.1) is not regular The * Corresponding author E-mail addresses: linhvh@vnu.edu.vn, vhlinh@hn.vnn.vn (V.H Linh) 0096-3003/$ - see front matter Ó 2008 Elsevier Inc All rights reserved doi:10.1016/j.amc.2008.12.008 398 S.L Campbell, V.H Linh / Applied Mathematics and Computation 208 (2009) 397–415 most difficult problem is that there exists no compressed form into which a tuple of more than two matrices can be simultaneously transformed Most of the existing results are only for linear time-invariant regular DDAEs [10,24] or DDAEs of special form [2,19,25,26] Until now there have been only two papers concerning nonregular DAEs [7,20] A general result for DDAEs’ solvability and stability is still missing The following examples illustrate some significant differences between delay ODEs, DAEs without delays, and delay DAEs Example Consider the system & x_ tị ỵ x1 tị x1 t 1ị x2 t 1ị ẳ 2x2 tị ỵ x1 t 1ị ỵ x2 t 1ị ẳ t P 0ị; where x1 and x2 are given by continuous functions on the initial interval (À1, 0] The dynamics of x1 is governed by a differential operator and continuity of x1 is expected The dynamics of x2 is determined by a difference operator and unlike x1, this component is piecewise continuous, in general Example [7] Consider the following inhomogenous system: & x_ tị ẳ f tị x1 tị x2 t 1ị ẳ gðtÞ ðt P 0Þ: The solution is given by x1 tị ẳ Z t f sịds ỵ c; x2 tị ẳ gt ỵ 1ị ỵ Z tỵ1 f sịds þ c ðt P 0Þ; where c is a constant The system dynamics is not causal Not only is x2 specified on (À1, 0], but the solution depends on future integrals of the input f(t) This interesting phenomenon should be noted in addition to the well-known fact in the DAE theory that the solution may depend on derivatives of the input A sufficient condition for the delay-independent asymptotic stability of DAEs with single delay is proposed in [25] Under this condition, the asymptotic stability of h-methods, BDF methods, general linear multistep methods, as well as implicit Runge–Kutta methods are analyzed Unfortunately, it is very difficult to verify this condition in practice The main aim of the present paper is to give a complement to this result in the stability theory for DDAEs Namely, we intend to derive delay-independent stability criteria for DDAEs of the form (1.1) and (1.2) We focus on practical stability criteria that are easily checkable Our results extend those obtained for neutral DODEs [13,14] to neutral DDAEs Under these criteria, we will show that numerical solutions obtained by the h-methods and BDF methods preserve the asymptotic stability of the DDAE This result includes the single delay DAEs result of [25] as a special case Further, we also investigate the solvability and the stability of a special class of nonregular delay DAEs The paper is organized as follows In the next section we review basic notions and results from the theory of DAEs and regular delay DAEs The main results of the paper lie in Section 3, where we formulate sufficient conditions to provide the asymptotic stability of regular DDAEs We give a characterization of those coefficient matrices that satisfy the sufficient conditions We also propose some practical criteria for the asymptotic stability of DDAEs with multiple delays In Section 4, we analyze the stability of numerical solutions to (1.1) and (1.2) using h-methods and BDF methods Finally, in the last section, we discuss solvability and stability issues of a special class of weakly regular DDAEs Preliminary In this section, we give a brief summary of needed results on linear constant coefficient and delay DAEs We assume the reader is familiar with the basic theory of linear time invariant DAEs [3,11,16], such as Ax_ ỵ Bx ¼ 0: ð2:1Þ The matrix pencil {A, B} is said to be regular if there exists k C such that the determinant of kA + B, denoted by det(kA + B), is nonzero The system (2.1) is solvable if and only if {A, B} is regular If detkA ỵ Bị ẳ 8k C, we say that {A, B} is irregular or non-regular If {A, B} is regular, then k is a (generalized finite) eigenvalue of {A, B} if det(kA + B) = The set of all eigenvalues is called the spectrum of the pencil {A, B} and denoted by r{A, B} The maximum of the absolute values of the finite eigenvalues is called the spectral radius of the pencil {A, B} and denoted by q(A, B) These concepts are also extended to the case of a given tuple of matrices fAi gni¼0 (the generalized polynomial eigenvalue problem) by dening rfAi gniẳ0 ị ẳ fk C : det P P niẳ0 kni Ai ị ẳ 0g; and qfAi gniẳ0 ị ẳ maxfjkj : k Cand det niẳ0 kni Ai ị ẳ 0g: Thus, for a given matrix A CmÂm , the wellknown spectrum r(A) and the spectral radius q(A) are r(ÀI, A) and q(ÀI, A), respectively Suppose that A is singular and pencil {A, B} is regular Then there exist nonsingular matrices W, T such that WAT ¼  Id  ; N WBT ¼  B1 0 ImÀd  ; ð2:2Þ S.L Campbell, V.H Linh / Applied Mathematics and Computation 208 (2009) 397–415 399 where N is nilpotent of index k [3,11,16] If N is a zero matrix, then k = Furthermore, we may assume without loss of generality, that N and B1 are upper triangular If {A, B} is regular, the nilpotency index of N in (2.2) is called the index of matrix pencil {A, B} and we write index {A, B} = k If A is nonsingular, we set index {A, B} = Definition Suppose that {A, B} is regular Let Q be a projector onto the subspace of consistent initial conditions Let P = I À Q We say that the zero solution of (2.1) is stable if, for any e > there exists d > such that for an arbitrary vector x0 Cm satisfying kx0k < d, the solution of the initial value problem & Ax_ ỵ Bx ẳ 0; t ẵ0; 1ị; Px0ị x0 ị ẳ exists uniquely and the estimate kx(t)k < e holds for all t P The zero solution is said to be asymptotically stable if it is stable and limt?1kx(t)k = for solutions x of (2.1) If the zero solution of (2.1) is asymptotically stable, we say that system (2.1) is asymptotically stable If index {A, B} = one may choose Q as the projector onto (A) [11] A difference between ODE-s and DAE-s is that the equality x(0) = x0 is not expected here, in general That is, for DAEs, we need consistent initial value x0 such that (2.1) with the initial condition x(0) = x0 holds for a smooth solution We not consider impulsive solutions in this paper and for that reason will frequently make an index one assumption For linear time-invariant systems, the concepts of asymptotic stability and exponential stability are equivalent The system (2.1) is asymptotically stable if and only if the matrix pencil {A, B} is (asymptotically) stable, i.e., rðA; BÞ & CÀ ; where CÀ denotes the open left half complex plane [23] Clearly r(RAS, RBS) = r(A, B) for nonsingular R, S 2.1 Solvability of regular delay DAEs The theory of delay ordinary differential equations (DODEs), when the leading matrix A in (2.3) is the identity matrix, has _ À sịỵ dxt sị ẳ f tị, been widely discussed [12] These systems are classified by their type For a scalar DODE ax_ ỵ bx ỵ cxt the system is of retarded type if a – 0, c = 0, of neutral type if a – 0, c – 0, and of advanced type if a = 0, b – 0, and c – One important attribute of the type is that it classifies how DODEs propagate discontinuities to future delay intervals (assuming an initial value problem) Discontinuities in retarded systems become smoother in each successive interval, whereas discontinuities in advanced systems become less smooth in each successive interval Discontinuities in neutral systems are carried into successive delay intervals with the same degree of smoothness Hence, we wish to study separately DDAEs which include retarded and neutral DODEs, but to avoid altogether those which lead to DODEs of advanced type For some interesting examples of DDAEs and some DDAEs which ‘‘look like” they should be of retarded type but are actually neutral or advanced type, see [6,7] In this section we consider DAEs with single delay Ax_ ỵ Bx ỵ Dxt sị ẳ 0: 2:3ị The delay DAE (2.3) is called regular [7] if the pencil {A, B} is regular and weakly regular if there exist a; b; c C such that det(aA + bB + cD) – 0, i.e., the triplet {A, B, D} is regular We suppose initially that {A, B} is regular and has index k Note that neutral DAEs with single delay _ sị ỵ Dxt sị ẳ Ax_ ỵ Bx ỵ C xt 2:4ị can always be transformed to the form (2.3) Indeed, by defining a new variable y by y(t) = x(t À s), we obtain a new delay DAE e ~x_ ỵ B e ~x ỵ D e ~xt sị ẳ A 2:5ị with ~x ¼   x ; y e¼ A  A C  ; e¼ B  B D I  ; e¼ D  0 ÀI  : However, this transformation can increase the index of the DAE system Further, the dimension of the new transformed system becomes 2m, which is less advantageous in practical computation e Bg e Bg e is regular if and only if {A, B} is regular However, the index of f A; e is equal either k or k + 1, Proposition The pencil f A; where k is the index of {A, B} e Bg e Proof The equivalence between the regularity of the two pencils is clear We verify the statement on the index of f A; Without loss of generality, we assume that the pencil {A, B} is given in the Kronecker normal form (2.2) Correspondingly, C and D are given in block form C¼  C1 C2 C3 C4  ; D¼  D1 D2 D3 D4  : ð2:6Þ 400 S.L Campbell, V.H Linh / Applied Mathematics and Computation 208 (2009) 397–415 e Bg e can be assumed to be Thus f A; C1 C2 B0 N e ¼B A B @0 C3 C4 C C C; A 0 I 0 B1 D1 D2 I D4 C C C: A B0 e¼B B B @0 I 0 0 C3 C4 C C C; A 0 D3 I It is not difficult to verify that 80 I > > > @ > > 0 : 0 B1 B0 B B @0 19 > > > = I 0C C C : I A> > > ; 0 I 0 e Bg e ¼ index N e where Then indexf A; N C3 e ¼B N @0 0 C4 C A: Then Nk B N ¼@ 0 NkÀ1 C NkÀ1 C 0 0 ek C A: e ¼ k if NkÀ1C3 = and NkÀ1C4 = Otherwise index N e ẳ k ỵ h Hence, index N Corollary Suppose that the pencil {A, B} is regular and has index-1 Further, suppose the matrices are in block form as in the e Bg e has index-1 if and only if C3 = 0, C4 = proof of Proposition Then the new pencil f A; Corollary means that the transformed system (2.5) has index-1 if and only if the pencil {A, B} has index-1 and the derivative of x(t À s) does not appear in the ‘‘algebraic part” Now, we turn back to the regular delay DAE (2.3) with an initial condition x(t) = u(t), t [Às, 0], where u is a continuous function defined on [Às, 0] The solvability of regular delay DAEs was discussed in detail in [5,6] Using appropriate constant coordinate changes, first we transform the matrix triplet A,B,D into the block form (2.2), (2.6) Then system (2.3) is decomposed as follows: z0 ỵ B1 z ỵ D1 zt sị ỵ D2 wt sị ẳ 0; Nw0 ỵ w þ D3 zðt À sÞ þ D4 wðt À sÞ ¼ 0; ð2:7Þ where x is decomposed into ‘‘differential” variables z and ‘‘algebraic” variables w Using the nilpotency of N, wtị ẳ k1 X Nịi ẵD3 ziị t sị ỵ D4 wiị t sị: 2:8ị iẳ0 Setting t = in (2.8), we obtain the consistency condition for the initial condition The initial value problem for (2.3) with a consistent initial condition admits a unique solution, see [5,6,10] which can be obtained by solving the system (2.7) for z,w recursively on each interval ((l À 1)s,ls], l = 1, 2, The definition of the asymptotic stability for DDAEs of the form (2.3) is similar to that for DODEs Definition [10,24,25] The trivial solution of the DDAE (2.3) is said to be stable if for any e > theres exists d = d(e) such that for all contiuous functions u satisfying the consistent condition and supt2[s,0]ku(t)k < d, the solution x = x(t, u) of the initial value problem for (2.3) satifies kx(t, u)k < e for all t P The trivial solution of the DDAE (2.3) is said to be asymptotically stable if it is stable and furthermore limt?1 kx(t, u)k = For higher-index problems (k > 1), the formula for w involves derivatives of the solution taken in the past It was shown in [5] that solutions for (2.3) can be continuous on only finite intervals even if the initial function u (or the input, if there is an input function) is infinitely differentiable Further, discontinuities not necessarily get smoothed out as with the nonsingular problem Example [6] Consider a two dimensional delay DAE system  0  x0 tị ỵ     0 xtị ỵ xt 1ị ẳ 0; À1 À1 0 S.L Campbell, V.H Linh / Applied Mathematics and Computation 208 (2009) 397–415 401 with x = (x1, x2)T This system has index-2 It is easy to see that x1 satisfies an advanced type equation x1 tị ẳ x01 t 1ị, so that mị x1 tị ẳ x1 t mị That is, x1 (and x2, too) becomes progressively less smooth The system behaves like those of advanced type For the simplest case k = 1, the situation is somewhat better The evolution of z is given by a delay differential equation, meanwhile a difference operator defines the dynamics of w If a continuous initial function u is given, then z is continuous and w is piecewise continuous in general Furthermore, z is differentiable and w is continuous except possibly at integer multiples of s The system (2.3) behaves like a neutral delay system Extending all the results in this section to multiple-delay DAEs of the form (1.1) or (1.2) is straightforward We note that the smoothness of solutions now may be even worse Even in index-1 problems, the distance between the jump (or break) points can become arbitrarily small as t is increasing except for the case when all the ratios si/sj, i – j are rational numbers This fact gives rise to practical difficulties for numerical methods 2.2 Delay DAEs of Hessenberg form Delay DAEs arising in applications frequently have special structure One of the most important class of systems is that of Hessenberg forms which generalizes non-delay DAEs of Hessenberg form [3] Definition Linear delay DAEs of the form x_ ỵ B1 x1 ỵ B2 x2 ỵ D1 x1 t sị ỵ D2 x2 t sị ẳ 0; B3 x1 ỵ B4 x2 ỵ D3 x1 t sị ẳ 0; 2:9ị 2:10ị where B4 is nonsingular, is called semi-explicit index-1 linear DDAEs or index-1 linear DDAEs in Hessenberg form Note {A, B} is an index one pencil and D4 = Linear delay DAEs of the form x_ ỵ B1 x1 ỵ B2 x2 ỵ D1 x1 t sị ẳ 0; 2:11ị B3 x1 ¼ 0; ð2:12Þ where B3B2 is nonsingular, is called semi-explicit index-2 linear DDAEs or index-2 linear DDAEs in Hessenberg form Here {A, B} is an index two Hessenberg pencil and D2 = 0, D3 = 0, and D4 = Delay DAEs of the form (2.9)-(2.10) come from the linearization of index-1 nonlinear DDAEs in Hessenberg form f ðt; x_ ðtÞ; x1 ðtÞ; x1 ðt À sÞ; x2 ðtÞ; x2 t sịị ẳ 0; gt; x1 tị; x1 t sị; x2 tịị ẳ 2:13ị 2:14ị along a particular solution, where the Jacobian g x2 is assumed nonsingular Similarly, by linearizing index-2 nonlinear DDAEs f ðt; x_ ðtÞ; x1 ðtÞ; x1 ðt À sÞ; x2 ðtÞ; x2 t sịị ẳ 0; 2:15ị gt; x1 tịị ¼ 0; ð2:16Þ where g x1 fx2 is assumed nonsingular, one obtains DDAEs of the form (2.11) and (2.12), see [2,26] The derivative of the unknown function at a delayed time may appear in (2.9), (2.10) and (2.11), (2.12), as well Namely, DDAEs of the form x_ ỵ B1 x1 ỵ B2 x2 ỵ C x_ t sị ỵ C x_ t sị þ D1 x1 ðt À sÞ þ D2 x2 ðt sị ẳ 0; 2:17ị B3 x1 ỵ B4 x2 ỵ D3 x1 t sị ẳ 0; 2:18ị where B4 is nonsingular, are called index-1 linear neutral DDAEs in Hessenberg form Further, DDAEs of the form x_ ỵ B1 x1 ỵ B2 x2 ỵ C x_ t sị ỵ D1 x1 t sị ẳ 0; 2:19ị B3 x1 ẳ 0; 2:20ị where B3B2 is nonsingular, are called index-2 linear neutral DDAEs in Hessenberg form Neutral delay DAEs of the forms (2.17), (2.18) and (2.19), (2.20) can be transformed to delay DAEs of the forms (2.9), (2.10) and (2.11), (2.12) by introducing new auxiliary variables as discussed in the previous section Proposition shows the index of the transformed delay DAEs have the same index as the original neutral delay DAEs One of the most important features of delay DAEs in Hessenberg form is that one can easily get the so-called underlying DODEs For example, for the system 2.9,2.10, one can solve x2 from (2.10), then insert into (2.9), and get the underlying DODE À1 1 x_ ỵ B1 B2 B1 B3 ịx1 ỵ D1 D2 B4 B3 B2 B4 D3 Þx1 ðt À sÞ À D2 B4 D3 x1 t 2sị ẳ 0: 2:21ị Note that the UDODE (2.21) now has double delays while the original DDAE (2.9) and (2.10) has a single delay Further, it is easy to see that the index-1 DDAE (2.9) and (2.10) is asymptotically stable if and only if its UDODE (2.21) is asymptotically stable Similarly, we can derive the underlying neutral DODE for the index-1 neutral DDAE of the form (2.17) and (2.18) 402 S.L Campbell, V.H Linh / Applied Mathematics and Computation 208 (2009) 397–415 Obtaining the underlying DODE for semi-explicit index-2 DDAE of the form (2.11) and (2.12) is a little bit more complicated than the index-1 case First, observe that by differentiating (2.12) and inserting the result into (2.11), we obtain a hidden constraint B3 B1 x1 ỵ B3 B2 x2 ỵ B3 D1 x1 t sị ¼ 0: ð2:22Þ Since B3B2 is invertible, one can calculate the index-2 algebraic variable x2 from x1 Next, we proceed as follows (see ÞÂm1 [1,8]) Denote the row number and the column number of B2 by m1 and m2, respectively Take a matrix R Rðm1 Àm2 R T is whose linearly independent normalized rows form a basis for the null space of B2 Then RB2 = and the matrix B3 invertible Defining new variables u = Rx1, we can calculate x1 from u by  x1 ¼ R 1   u B3 2:23ị ẳ Su; where S is defined by RS = I, B3S = The underlying DODE is u_ ỵ RB1 Su ỵ RD1 Sut sị ẳ 0: 2:24ị From (2.22) and (2.23), it is clearly seen that the semi-explicit index-2 DDAE (2.11) and (2.12) is asymptotically stable if and only if the UDODE (2.24) is We obtain analogously the underlying neutral DODE for the index-2 neutral DDAE of the form (2.19) and (2.20) From the above introduction of DDAE in Hessenberg form, we conclude that if one wants to investigate the stability of DDAEs in Hessenberg form, it makes sense to consider their underlying DODEs Stability criteria Now consider the DAE of multiple delays of the form (1.1) or (1.2) The characteristic equation for (1.1) is dened by Psị ẳ det sA ỵ B ỵ s M X C i essi ỵ iẳ1 M X ! Di essi ẳ 0: 3:1ị iẳ1 For a given s C, we denote its real and imaginary parts by Re(s) and Im(s), respectively It is well known, see [25], that the system (1.1) is asymptotically stable if all the roots of (3.1) have negative real part and they are bounded away from the imaginary axis, i.e., for all root ki of (3.1) (i = 1, 2, ) and for some positive l, the inequalities Reðki Þ Àl < ð3:2Þ hold Note that (3.1) may have infinitely roots and they may accumulate at a finite point on the complex plane or at infinity In this section, we will derive some sufficient conditions for (3.2) We will need the following definition and an auxiliary result, which are well known in the theory of nonnegative matrices [17] Definition Let W CnÂn with elements wij and jWj denote the nonnegative matrix in RnÂn with element jwijj For two matrices U; V RnÂn , we write U V if and only if uij vij for each i,j {1, 2, , n} In particular, q(W) q(jWj) Lemma Let W CnÂn and V RnÂn If jWj V, then q(W) q(V) 3.1 Delay-independent asymptotic stability We introduce the following two-variable polynomials: ! M X Qs; zị ẳ det sA ỵ B ỵ sC i ỵ Di ịz ; iẳ1 Rs; zị ẳ det sA ỵ B ỵ 3:3ị ! M X sC i ỵ Di ịzi : 3:4ị iẳ1 Lemma Suppose that ðiÞ ðiiÞ rðA; BÞ CÀ ; sup q ResịP0 M X 3:5ị ! jsA ỵ Bị1 sC i ỵ Di ịj < 1: iẳ1 Then Q(s,z)0 for all s; z C such that Re(s) P 0, jzj ð3:6Þ S.L Campbell, V.H Linh / Applied Mathematics and Computation 208 (2009) 397–415 403 Proof Suppose that the contrary happens, i.e., there exist s, Re(s) P and z, jzj such that Q(s,z) = This implies that there exist a vector v – such that " # M X sA ỵ B ỵ sC i ỵ Di ịz v ẳ iẳ1 or equivalently (since (sA + B) is invertible) sA ỵ Bị1 M X sC i ỵ Di ịzv ẳ v : i¼1 This means that À1 is an eigenvalue of ðsA ỵ Bị1 q sA ỵ Bị1 PM iẳ1 sC i ! M X sC i ỵ Di ịz ỵ Di Þz, which implies P 1: i¼1 But q M X ! P q sA ỵ Bị jsA ỵ Bị sC i ỵ Di ịj iẳ1 ! M X sC i ỵ Di ịz ; iẳ1 which contradicts (3.6) h Note that in the single delay case, i.e M = 1, the statement holds without the need of taking the absolute value in (3.6), see also [25] Further, due to the maximum principle in complex analysis, it suffices to take the supremum on the imaginary axis Re(s) = in the assumption (3.6) Theorem Suppose that the assumptions (3.5) and (3.6) in Lemma hold Then the system (1.1) is asymptotically stable for all sets of the delays fsi gM i¼1 , i.e., the asymptotic stability of (1.1) is delay-independent Proof Similarly to the proof of Lemma 2, it is not difficult to show that the equation P(s) = has only roots with negative real part Next, we prove that the real parts of the roots are bounded away from Suppose that this statement is not true Then, there exists a sequence {sn} such that limn?1Re(sn) = À0 meanwhile P(sn) = Choose a positive number e such that e < sup q Resịẳ0 M X ! jsA ỵ Bị sC i ỵ Di ịj : iẳ1  , where It is obvious that there exists a sufficiently large N0 such that for n P N0, we have Reðsn Þ P 12 l l ẳ maxfRekị; k rA; Bịg < is the spectral abscissa of the pencil {A, B}, and jeÀsMsn j ð1 À e=2ÞÀ1 For each sn, there exists a vector – such that " M X sn A ỵ B ỵ sn C i ỵ Di ịesi sn # v n ẳ 0; iẳ1 which implies q M X ! jsn A ỵ Bị1 sn C i ỵ Di ịj P e=2: iẳ1 Now we observe that the entries of the matrix functions sA ỵ Bị1 sC i ỵ Di ị; i ẳ 1; 2; ; M are rational function of s Only the finite eigenvalues of {A, B} may be poles of these functions Thus, each element of the (nonnegative) matrix function M X jsA ỵ Bị1 sC i ỵ Di ịj iẳ1 has the form jsjapq apq ỵ Oð1=jsjÞ, where apq are some integers and apq are nonnegative numbers (1 p; q m) Hence, for an arbitrarily small  > 0, there exists a bound s1 > such that AðsÞð1 À Þ M X jsA ỵ Bị1 sC i ỵ Di ịj Asị1 ỵ ị iẳ1 for all jsj P s1, where the elements of the matrix function AðsÞ are defined by ð3:7Þ 404 S.L Campbell, V.H Linh / Applied Mathematics and Computation 208 (2009) 397415 Apq sị ẳ jsjapq apq : Let a ẳ maxp;q apq and the matrix function Asị be decomposed such that Asị ẳ jsja A0ị ỵ Að1Þ ðsÞ ; where Að0Þ is a nonnegative constant matrix and each entry of Að1Þ ðsÞ is either zero or negative power of jsj Next, we investigate the asymptotic behavior of the spectral radius of AðsÞ as jsj tends to infinity The following cases are possible:  If a 0, then qðAðsÞÞ has a finite limit as jsj ? 1;  If a > and qðAð0Þ Þ > 0, then qðAðsÞÞ tends to infinity as jsj ?1;  If a > and qA0ị ị ẳ 0, then due to the Puisseux series of the eigenvalues of (Að0Þ þ Að1Þ ðsÞÞ, see [15], we have qðAð0Þ þ Að1Þ sịị ẳ jsjb c ỵ o1ịị; where c > is a constant and b is a negative fractional number In other words, we use the fact that the eigenvalues can be expanded into fractional power series of 1/jsj Depending on the sign of a + b, the spectral radius of AðsÞ either converges to a finite number or tends to infinity as jsj ? Summarizing the above cases, the spectral radius of AðsÞ either converges to a finite number or tends to infinity as jsj ? Since  in (3.7) is arbitrarily chosen, the same statement holds for the spectral radius of M X jsA ỵ Bị1 sC i ỵ Di ịj; iẳ1 which is a function of s The assumption (3.6) implies that the latter function must converge to a finite limit as jsj ? On  g Consequently, it is uniformly continuous in the other hand, this function is continuous in the domain fs C; ReðsÞ P 12 l the considered domain Finally, due to the verified uniform continuity, there exists sn sufficiently close to the imaginary axis such that  ! !   X M M X   À1 À1 q js A ỵ Bị s C ỵ D ịj q jIms ịA ỵ Bị Ims ịC ỵ D Þj  e=2:  n n i n n i i i   i¼1 i¼1 We obtain q M X ! jsn A ỵ Bị1 sn C i ỵ Di ịj 6q iẳ1 M X ! jImsn ịA þ BÞÀ1 ðImðsn ÞC i þ Di Þj þ e=2 iẳ1 sup q Resịẳ0 M X ! jsA þ BÞ ðsC i þ Di Þj þ e=2 < À e=2; i¼1 which yields contradiction The proof is complete h Assumptions (3.5) and (3.6) come from the straightforward generalization of the corresponding stability conditions for neutral DAEs with single delay given in [25] In that paper, a third assumption juT Auj > juT Cuj; 8u Cm , was needed to ensure that all the roots of the characteristic equations are bounded away from the imaginary axis From the proof of Theorem 1, we see that such an assumption is redundant and can be ignored Sometimes it is more convenient to check the assumptions by using an operator norm instead of the spectral radius Corollary Suppose that the assumption (3.5) holds and    X M   sup  jsA ỵ Bị sC i ỵ Di ịj < 1:   Resịẳ0 iẳ1 Then the system (1.1) is delay-independently asymptotically stable We have similar statements for the system (1.2) Lemma Suppose that ðiÞrðA; BÞ CÀ ;   ðiiÞ sup q fsC i ỵ Di gM iẳ0 < 1; ResịP0 where C0 :¼ A, D0 :¼ B Then R(s,z) – for all s; z C such that Re(s) P 0, jzj ð3:8Þ ð3:9Þ S.L Campbell, V.H Linh / Applied Mathematics and Computation 208 (2009) 397–415 405 Proof Recall that  (  q fsC i ỵ Di gMiẳ0 ẳ max jkj; det M X sC i ỵ Di ÞkMÀi ! ) ¼0 for a given fixed s: i¼0 Hence, for a given s, Re(s) P 0, if z C is such that R(s, z) = 0, then 1/z is an eigenvalue of the polynomial eigenvalue problem with data fsC i ỵ Di gM iẳ0 Note that z cannot be zero because det(sA + B) – for all s, Re(s) P Therefore, assumption (3.9) implies that for a given s, Re(s) P 0, if R(s, z) = 0, then jzj > h Theorem Suppose that the assumptions (3.8) and (3.9) in Lemma hold Then the system (1.2) is asymptotically stable for all s P 0, i.e., the asymptotic stability of (1.2) is delay-independent Proof Similar to the proof of Theorem h Next, we attempt to characterize the set of admissible coefficient matrices which satisfy the assumptions of Theorem and analyze the effect of the index of the pencil {A, B} with second assumption (3.6) For sake of simplicity, and due to Proposition 1, we consider the single delay Eq (2.3) The assumption (3.6) now becomes supRe(s)=0q((sA + B)À1D) < Assume the coefficient matrices are transformed again in block form We have sA ỵ Bị1 D ẳ sI ỵ B1 ị1 D1 sI ỵ B1 ị1 D2 sN ỵ Iị1 D3 sN ỵ Iị1 D4 ! : Using the nilpotency of N, it is obvious that the spectral radius of the matrix sI ỵ B1 ị1 D1 Pk1 i i iẳ0 Nị s D3 sI ỵ B1 ị1 D2 Pk1 i i iẳ0 Nị s D4 ! is necessarily bounded for s, Re(s) = Since all the entries of the first block row tend to as jsj tends to infinity, we get some consequences on D4 Namely, if k = 1, then q(D4) < must be satisfied For higher index cases, we have q(D4) < and q(Ni D 4) = for i = 1, 2, , k À 1, otherwise the spectral radius in question is unbounded That is, for higher index pencil {A, B}, the block D4 (and D3 as well) must be of special structure Taking into account the result of Proposition 1, the same statement holds for higher index neutral delay DAEs (2.4) and for higher index DAEs systems with multiple delays of the form (1.1) and (1.2) Note that these necessary conditions on D4 are trivially satisfied by the delay and neutral delay DAEs of Hessenberg forms With the assumption (3.5), the problem of finding sufficient condition for asymptotic stability for delay DAEs is closely related to the robust stability question of DAE, see [4,21,9,18] We have a nominal DAE system without delays which is assumed to be asymptotically stable The delay terms can be considered uncertain perturbations From this point of view, a somewhat simpler condition can be given instead of (3.6) Proposition Consider the delay DAE of the form (1.1) Suppose that Ci = 0, for i = 1, 2, , M and assumption (3.5) holds Then if !À1 kð D1 D2 sup ksA ỵ Bị1 k DM ịk < ; 3:10ị Resịẳ0 the delay DAE system is asymptotically stable Proof Eq (3.10) implies (3.6) See also [18] Note that in (3.10) we can take any matrix norm induced by a vector norm h Unfortunately, if the index of {A, B} is greater than 1, then the right hand side of (3.10) is simply zero, and the proposition does not apply This once again confirms that for higher-index problems, the coefficient matrices Ci, Di, (i = 1, 2, , M) must be highly structured so that the asymptotic stability would be preserved 3.2 Practical algebraic stability criteria Theorems and give us sufficient conditions for the asymptotic stability of delay DAEs of the form (1.1) and (1.2), respectively Unfortunately, checking the conditions (3.6) or (3.9) is rather difficult because computing the supremum of the spectral radius of a matrix function or a polynomial matrix function over an unbounded domain is very costly In this section, we propose some checkable algebraic criteria for the asymptotic stability Our results extend some recent results for neutral delay ODEs, see [13,14], to neutral delay DAEs (a) The index-1 case We first restrict the investigation to index-1 problems Since the matrices A and B can easily be transformed to the upper block-triangular form using QZ or QR decompositions, we assume that  A¼ A1 A2 0  ;  B¼ B1 B2 B4  : 406 S.L Campbell, V.H Linh / Applied Mathematics and Computation 208 (2009) 397–415 The index-1 assumption on the pencil {A, B} implies that the submatrix B4 is invertible Furthermore, due to Corollary 1, we assume that Ci ¼  C i1 C i2 0  Di ¼ ;  Di1 Di2 Di3 Di4  ; i ¼ 1; 2; ; M: We introduce some auxiliary matrix sequences Li ¼ A ỵ Bị1 Di ỵ C i ị; M i ẳ A ỵ Bị1 Di C i ị; E ẳ A ỵ Bị1 A Bị 3:11ị for i = 1, 2, , M Lemma Let the assumption (3.5) hold Then sA ỵ Bị1 sC i ỵ Di ị ẳ I zEị1 zMi ỵ Li ị for all Re(s) P 0, where z ẳ 1s (which implies jzj 1, z 1) 1ỵs Proof The proof is similar to the proof of Theorem 2.2 [13] It is easy to derive I À zE ẳ I ! 1s A ỵ Bị1 A Bị 1ỵs ẳ A ỵ Bị1 ẵA ỵ Bị1 ỵ sị sịA Bị1 ỵ sị1 ẳ 2A ỵ Bị1 sA ỵ Bị1 ỵ sị1 : In the same way, we get zMi ỵ Li ẳ 2A þ BÞÀ1 ðsC i þ Di Þð1 þ sÞÀ1 for all i ¼ 1; 2; ; which yields the statement h Now, let S = (A1 + B1)À1 By direct calculations, we have  E¼ SðA1 À B1 ị 2SA2 Li ẳ  I  ẳ: E1 E2 I  ; SẵDi1 ỵ C i1 A2 ỵ B2 ịB1 SẵDi2 ỵ C i2 A2 ỵ B2 ịB1 Di3 Di4 Mi ẳ B1 Di3 SẵDi1 C i1 A2 ỵ ! B2 ịB1 Di3 SẵDi2 C i2 A2 ỵ B1 Di3  ẳ: BÀ1 Di4 B2 ÞBÀ1 Di4 Š ! BÀ1 Di4 ¼: Li1 Li2  ; Li3 Li4   Mi1 M i2 Mi3 M i4 Matrix E always has an eigenvalue k = À1, which makes the straightforward extension of the results in [13,14] impossible since q(jEj) < would be required We can still give estimates for the left hand-side of (3.6) and (3.9) by estimating separately the ‘‘differential” part and the ‘‘algebraic” one Furthermore, in order to ease the matrix calculations, we may transform A1, B1, and B4 into upper triangular form prior to the calculations We have I À zE ¼  I À zE1 zE2 ỵ zịI  ; zMi ỵ Li ẳ  zMi1 ỵ Li1 zM i2 ỵ Li2 þ zÞLi3 ð1 þ zÞLi4  : ð3:12Þ Note that Li3 = Mi3, Li4 = Mi4 We introduce the following auxiliary matrices Definition Assume q(jE1j) < For an integer l P 0, and for i = 1, , M; j = 1, 2, let Gij lị ẳ l X e ij jg ỵ I jE1 jị1 jElỵ1 Lij j ỵ jElỵ1 M e ij jị; fjEm1 Lij j ỵ jEm1 M 1 3:13ị mẳ0 e i1 ẳ Mi1 ỵ E2 Li3 ; M e i2 ẳ Mi2 ỵ E2 Li4 ; i ẳ 1; 2; ; M Further, let where M Gi lị ẳ  Gi1 lị Gi2 lị jLi3 j jLi4 j  : ð3:14Þ The following estimate will be very useful (see also [13, Theorem 3.1]) Proposition Assume that the assumption (3.5) holds and the pencil {A, B} has index Further, assume q(jE1j) < Then for any z satisfying jzj 1, we have e ij Þj Gij ðlÞ Gij ð0Þ: jðI À zE1 ị1 Lij ỵ z M S.L Campbell, V.H Linh / Applied Mathematics and Computation 208 (2009) 397–415 407 Furthermore Gij(l) Gij(l À 1) for all l P and i = 1, 2, , M; j = 1, Proof The required inequalities can be verified by the same arguments as in the proof of [13, Theorem 3.1] By defining T = zE1, for jzj 1, we have e ij ị ẳ I ỵ T ỵ T ỵ ịLij ỵ z M e ij ị ẳ I zE1 ị1 Lij þ z M l X e ij g þ ðI þ T þ T þ Á Á ÁÞðT lþ1 Lij ỵ zT lỵ1 M e ij ị: fT m Lij ỵ zT m M mẳ0 Since jzj and jTj jE1j, the inequality e ij Þj jI zE1 ị1 Lij ỵ z M l X e ij jg ỵ I ỵ jE1 j ỵ jE1 j2 ỵ ịjElỵ1 Lij j ỵ jElỵ1 M e ij jị fjEm1 Lij j ỵ jEm1 M 1 mẳ0 e ij ịj Gij lị comes immediately Next, we show Gij(l) Gij(l À 1) for l P holds, from which the estimate jðI À zE1 ị1 Lij ỵ z M Indeed, we have Gij lị ẳ l X e ij jg ỵ I ỵ jE1 j ỵ jE1 j2 ỵ ịjElỵ1 Lij j þ jElþ1 M e ij jÞ fjEm1 Lij j þ jEm1 M 1 m¼0 l X e ij jg ỵ jE1 j ỵ jE1 j2 ỵ ịjEl Lij j ỵ jEl M e ij jị fjEm1 Lij j ỵ jEm1 M 1 mẳ0 ẳ l1 X e ij jg ỵ I ỵ jE1 j ỵ jE1 j2 ỵ ịjEl Lij j ỵ jEl M e ij jị ẳ Gij l 1ị: fjEm1 Lij j ỵ jEm1 M 1 mẳ0 As a consequence, the inequality Gij(l) Gij(0) holds for any positive integer l h Note that transforming A1 and B1 into upper triangular form has an additional advantage If A1, B1 are upper triangular, then E1 is of upper triangular form, too In this case the eigenvalues of E1 are (1 À ki)/(1 + ki), where ki, i = 1, 2, are finite eigenvalues of the pencil {A, B} Thus (3.5) implies that q(E1) < and q(jE1j) < as well Hence, the condition q(jE1j) < is obviously fulfilled and it does not mean an extra assumption for the existence of the inverse of (I À jE1j) is required Using this result, it is easy to get estimates for the left-hand side of (3.5) Proposition Assume that the assumptions of Proposition hold Then q M X ! jI zEị Li ỵ zMi Þj i¼1 6q M X i¼1 ! Gi ðlÞ 6q M X ! Gi 0ị iẳ1 holds for any z satisfying jzj 1, z – À 1, where the parametrized matrices Gi(Á) are defined in (3.14) Proof Using the formulas in (3.12), it is easy to verify that À1 I zEị Li ỵ zMi ị ẳ e i1 ị I zE1 ị1 Li2 ỵ M e i2 ị I zE1 ị1 Li1 ỵ M Li3 Li4 ! : Then, using Proposition and the definition of matrices Gi(l) given in (3.14), we have j(I À zE)À1 (Li + zMi)j Gi(l) Gi(0), i = 1, , M, for any positive integer l Summing up the inequalities and then invoking Lemma 1, the proof is complete h Now, we are in position to state a practical algebraic criterion for the asymptotic stability of the delay DAE system of the form (1.1) Theorem Assume the assumptions of Proposition hold If there exists an integer l P such that q M X ! Gi lị