DSpace at VNU: STABILITY AND ROBUST STABILITY OF LINEAR TIME-INVARIANT DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS

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Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php SIAM J MATRIX ANAL APPL Vol 34, No 4, pp 1631–1654 c 2013 Society for Industrial and Applied Mathematics STABILITY AND ROBUST STABILITY OF LINEAR TIME-INVARIANT DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS∗ NGUYEN HUU DU† , VU HOANG LINH† , VOLKER MEHRMANN‡ , AND DO DUC THUAN§ Abstract Necessary and sufficient conditions for exponential stability of linear time-invariant delay differential-algebraic equations are presented The robustness of this property is studied when the equation is subjected to structured perturbations and a computable formula for the structured stability radius is derived The results are illustrated by several examples Key words delay differential-algebraic equation, strangeness-free DAE, exponential stability, spectral condition, restricted perturbation, stability radius AMS subject classifications 06B99, 34D99, 47A10, 47A99, 65P99 DOI 10.1137/130926110 Introduction In this paper we present the stability analysis of homogeneous linear time-invariant delay differential-algebraic equations (DDAEs) of the form (1.1) E x(t) ˙ = Ax(t) + Dx(t − τ ), where E, A, D ∈ Kn,n , K = R or K = C, and τ > represents a time delay We study initial value problems with an initial function φ, so that (1.2) x(t) = φ(t) for − τ ≤ t ≤ While standard differential-algebraic equations (DAEs) without delay are today standard mathematical models for dynamical systems in many application areas, such as multibody systems, electrical circuit simulation, control theory, fluid dynamics, and chemical engineering (see, e.g., [1, 4, 19, 25, 27, 33]), the delay version is typically needed to model effects that not arise instantaneously; see, e.g., [3, 16, 42] Note that (1.1) is a special case of more general neutral delay DAEs (1.3) E x(t) ˙ + F x(t ˙ − τ ) = Ax(t) + Dx(t − τ ) However, by introducing a new variable, (1.3) can be rewritten into the form (1.1) with double dimension; see [10] For this reason, here we only consider (1.1) The stability and robust stability analyses for DAEs are quite different from that of ordinary differential equations (ODEs) (see, e.g., [23]), and has recently received ∗ Received by the editors June 24, 2013; accepted for publication (in revised form) by W.-W Lin September 24, 2013; published electronically December 5, 2013 The second and fourth authors were supported by IMU Berlin Einstein Foundation Program (EFP) http://www.siam.org/journals/simax/34-4/92611.html † Faculty of Mathematics, Mechanics, and Informatics, Vietnam National University, Thanh Xuan, Hanoi, Vietnam (dunh@vnu.edu.vn, linhvh@vnu.edu.vn) The first author was partially supported by the NAFOSTED grant 101.02–2011.21 ‡ Institut fă ur Mathematik, MA 4-5, TU Berlin, D-10623 Berlin, Germany (mehrmann@math tu-berlin.de) This author’s work was supported by Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 910 Control of self-organizing nonlinear systems: Theoretical methods and application concepts § School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam (ducthuank7@gmail.com) 1631 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1632 N H DU, V H LINH, V MEHRMANN, AND D D THUAN a lot of attention; see, e.g., [5, 6, 12, 26, 29, 32, 37, 38] and [11] for a recent survey In contrast to this, the stability and robust stability analyses for ODEs with delay (DDEs) is already well established; see, e.g., [20, 21, 22, 24, 35] As an extension of both these theories, in this paper, we discuss DDAEs Such equations, containing both algebraic constraints and delays arise, in particular, in the context of feedback control of DAE systems (where the feedback does not act instantaneously) or as the limiting case for singularly perturbed ordinary delay systems; see e.g., [1, 2, 7, 8, 10, 31, 34, 43] In sharp contrast to the situation for DDEs and DAEs, even the existence and uniqueness theory of DDAEs is much less well established; see [17, 18] for a recent analysis and the discussion of many of the difficulties This unsatisfactory situation is even more pronounced in the context of (robust) stability analysis for DDAEs Most of the existing results are only for linear time-invariant regular DDAEs [13, 41] or DDAEs of special form [1, 30, 44] Many of the results that are known for DDEs not carry over to the DDAE case Even the well-known spectral analysis for the exponential stability or the asymptotic stability of linear time-invariant DDAEs (1.1) is much more complex than that for DAEs and DDEs; see [10, 39, 43] for some special cases The stability analysis is usually based on the eigenvalues of the nonlinear function H(s) = sE − A − e−sτ D, (1.4) associated with the Laplace transform of (1.1), i.e., the roots of the characteristic function (1.5) pH (s) := det H(s) Let us define the spectral set σ(H) = {s : pH (s) = 0} and the spectral abscissa α(H) = sup{Re s : pH (s) = 0} For linear time-invariant DDEs, i.e., if E = In , the exponential stability is equivalent to α(H) < (see [20]) and the spectral set σ(H) is bounded from the right However, for linear time-invariant DDAEs, the spectral set σ(H) may not be bounded on the right as the following example shows Example 1.1 Consider the DDAE from [9] 0 1 x(t) ˙ = 0 0 x(t) + x(t − 1) −1 with H(s) = −1 −e−s s , pH (s) = −1 + se−s , and thus there exist infinitely many solutions of pH (s) = and their real part can be arbitrarily large, i.e., α(H) = ∞ The dynamics of this system is easily analyzed Obtaining x2 from the second equation and substituting the result into the first equation, we obtain the delay ODE (m) x˙ (t − 1) = x1 (t), which is of advanced type Thus, x1 (t) = x1 (t − m) for m − ≤ t < m, m ∈ N Therefore, the solution is discontinuous in general and cannot be extended on [0, ∞) unless the initial function is infinitely often differentiable In some special cases, [31, 40], it has been shown that the exponential stability of DDAEs is equivalent to the spectral condition that α(H) < In general, however this spectral condition is only necessary, but not sufficient, as the following example shows Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1633 Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILITY OF DELAY DAES Example 1.2 ⎡ ⎢0 E=⎢ ⎣0 0 and Consider (1.1) with ⎡ ⎤ −1 0 ⎢0 0⎥ ⎥, A = ⎢ ⎣0 1⎦ 0 0 0 0 ⎤ 0⎥ ⎥, 0⎦ ⎡ ⎢ D=⎢ ⎣ −1/2 ⎡ 1+s −e−sτ ⎢ −1 s H(s) = sE − A − e−sτ D = ⎢ ⎣ 0 −1 e−sτ /2 0 0 0 0 ⎤ 0⎥ ⎥, 0⎦ ⎤ −e−sτ ⎥ ⎥ s ⎦ −1 Therefore, pH (s) = det H(s) = −(1 + s)(1 − e−2sτ /2), the eigenvalues are s = −1 and s = (− ln + 2kπi)/2τ, k ∈ Z, and hence all eigenvalues are in the open left half complex plane, which would suggest the exponential stability of the system, i.e., that all nontrivial solutions would be exponentially decaying However, we will see that the asymptotic behavior (and even the existence) of the solutions depends strongly on the smoothness and the behavior of the initial function φ Setting x = [x1 , x2 , x3 , x4 ]T , the system reads x˙ (t) = −x1 (t) + x3 (t − τ ) + x4 (t − τ ), x˙ (t) = x2 (t), x˙ (t) = x3 (t), = x4 (t) − x1 (t − τ )/2 Solving for x4 in the last equation and substituting this and x3 obtained from the third equation into the first equation, we arrive at x˙ (t) = −x1 (t) + x˙ (t − 2τ )/2 + x1 (t − 2τ )/2 This underlying neutral delay ODE has the characteristic function −pH (s), so its spectral set is the same as that of the original system The spectral condition ensures the exponential stability of the underlying equation for x1 ; see [20] However, x2 and x3 are just the second and the first derivatives of x4 (t) = x1 (t − τ )/2 Thus, if the first component of φ is not differentiable on (−τ, 0) or it is differentiable (almost everywhere) but the derivative is unbounded, then the solution does not exist or is unbounded For example, the function φ1 (t) = t sin(1/t) is continuous on [−τ, 0], differentiable on (−τ, 0), but the derivative is obviously unbounded Example 1.2 shows that linear time-invariant DDAEs may not be exponentially stable although all roots of the characteristic function are in the open left half complex plane To characterize when the roots of the characteristic function allow the classification of stability, in this paper we derive necessary and sufficient conditions that guarantee that for time-invariant DDAEs exponential stability is equivalent to the condition that all eigenvalues of H have a negative real part and thus extend recent results of [31] With a characterization of exponential stability at hand we also study the question of robust stability for linear time-invariant DDAEs, i.e., we discuss the structured stability radius of maximal perturbations that are allowed to the coefficients so that the system keeps its exponential stability These results extend previous results on DDEs and DAEs in [5, 6, 12, 11, 24, 35] Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1634 N H DU, V H LINH, V MEHRMANN, AND D D THUAN The paper is organized as follows In the next section we introduce the basic notation and present some preliminary results Then, in section 3, we characterize exponential stability for general linear time-invariant DDAEs In section 4, we will introduce allowable perturbations for two different classes of systems (1.1) and present a formula for the structured stability radius for DDAEs In section 5, some conclusions and open problems close the paper Preliminaries In the following, we denote by In ∈ Cn,n the identity matrix, by ∈ Cn,n the zero matrix, by AC(I,Cn ) the space of absolutely continuous functions, k (I,Cn ) the space of k-times piecewise continuously differentiable functions and by Cpw from I ⊂ [0, ∞) to Cn Definition 2.1 A function x(·, φ) : [0, ∞) → Cn is a called solution of the initial value problem (1.1)–(1.2) if x ∈ AC([0, ∞), Cn ) and x(·, φ) satisfies (1.1) almost everywhere An initial function φ is called consistent with (1.1) if the associated initial value problem (1.1) has at least one solution System (1.1) is called solvable if for every consistent initial function φ, the associated initial value problem (1.1)–(1.2) has a solution It is called regular if it is solvable and the solution is unique Note that instead of seeking solutions in AC([0, ∞), Cn ), alternatively we often consider the space Cpw ([0, ∞), Cn ) In fact, (1.1) may not be satisfied at (countably many) points, which usually arise at multiples of the delay time τ Definition 2.2 System (1.1)–(1.2) is called exponentially stable if there exist constants K > 0, ω > such that x(t, φ) ≤ Ke−ωt φ (2.1) ∞ for all t ≥ and all consistent initial functions φ, where φ ∞ = sup−τ ≥t≥0 φ(t) Note that one can transform (1.1) in such a way that a given solution x(t; φ) is mapped to the trivial solution by simply shifting the arguments Definition 2.3 A matrix pair (E, A), E, A ∈ Cn,n is called regular if there exists s ∈ C such that det(sE−A) is different from zero Otherwise, if det(sE−A) = for all s ∈ C, then we say that (E, A) is singular If (E, A) is regular, then a complex number λ is called a (generalized finite) eigenvalue of (E, A) if det(λE − A) = The set of all (finite) eigenvalues of (E, A) is called the (finite) spectrum of the pencil (E, A) and denoted by σ(E, A) If E is singular and the pair is regular, then we say that (E, A) has the eigenvalue ∞ Regular pairs (E, A) can be transformed to Weierstraß–Kronecker canonical form (see [4, 14, 15]), i.e., there exist nonsingular matrices W, T ∈ Cn,n such that (2.2) E=W Ir 0 N T −1 , A = W J 0 In−r T −1 , where Ir , In−r are identity matrices of indicated size, J ∈ Cr,r and N ∈ C(n−r),(n−r) are matrices in Jordan canonical form and N is nilpotent If E is invertible, then r = n, i.e., the second diagonal block does not occur Definition 2.4 Consider a regular pair (E, A) with E, A ∈ Cn,n in Weierstraß— Kronecker form (2.2) If r < n and N has nilpotency index ν ∈ {1, 2, }, i.e., N ν = 0, N i = for i = 1, 2, , ν − 1, then ν is called the index of the pair (E, A) and we write ind(E, A) = ν If r = n, then the pair has index ν = For system (1.1) with a regular pair (E, A), the existence and uniqueness of solutions has been studied in [7, 8, 9] and for the general case in [17] It follows from Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILITY OF DELAY DAES 1635 Corollary 4.12 in [17] that (1.1)–(1.2) has a unique solution if and only if the initial condition φ is consistent and pH (s) = det(H(s)) ≡ For a matrix triple (E, A, D) ∈ Cn,n × Cn,n × Cn,n , there always exists a nonsingular matrix W ∈ Cn,n such that ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ E1 A1 D1 (2.3) W −1 E = ⎣ ⎦ , W −1 D = ⎣D2 ⎦ , W −1 A = ⎣A2 ⎦ , D3 where E1 , A1 , D1 ∈ Cd,n , A2 , D2 ∈ Ca,n , D3 ∈ Ch,n with d + a + h = n, rank E1 = rank E = d, and rank A2 = a Then, system (1.1) can be scaled by W −1 to obtain ˙ = A1 x(t) + D1 x(t − τ ), E1 x(t) (2.4) = A2 x(t) + D2 x(t − τ ), = D3 x(t − τ ) In practice, the scaling matrix W and the transformed coefficient matrices can be easily constructed as follows Let U be the left unitary factor of the singular value decomposition (SVD) of E, i.e., U consists of the left singular vectors of E Assuming ˜ = [U ˜2 , U ˜3 ] that rank E = d, we decompose U = [U1 , U2 ] accordingly Then let U ∗ ∗ be the left unitary factor of the SVD of U2 A with rank U2 A = a Then, we define ˜ ) It is easy to check that multiplying by W −1 = diag(Id , U ˜ ∗ )U ∗ , W = U diag(Id , U the form (2.3) is obtained with ˜ ∗ U ∗ A, D2 = U ˜ ∗ U ∗ D, D3 = U ˜ ∗ U ∗ D E1 = U1∗ E, A1 = U1∗ A, D1 = U1∗ D, A2 = U 2 2 We immediately see that to obtain solvability of the equation, the initial function has to be in the set S := {φ : φ ∈ AC([−τ, 0], Cn ), A2 φ(0)+D2 φ(−τ ) = 0, D3 φ(t) = for all t ∈ [−τ, 0]} Shifting the time in the last equation of (2.4) by τ , we obtain (2.5) E1 x(t) ˙ = A1 x(t) + D1 x(t − τ ), A2 x(t) = −D2 x(t − τ ), = D3 x(t) Differentiating the second and third equations of (2.5), we get ˙ = A1 x(t) + D1 x(t − τ ), E1 x(t) (2.6) A2 x(t) ˙ = −D2 x(t ˙ − τ ), ˙ = D3 x(t) Following the concept of strangeness index in [25] we make the following definition; see also [17] Definition 2.5 Equation (1.1) is called strangeness free if there exists a nonsingular matrix W ∈ Cn,n that transforms the triple (E, A, D) to the form (2.3) and ⎡ ⎤ E1 rank ⎣ A2 ⎦ = n D3 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1636 N H DU, V H LINH, V MEHRMANN, AND D D THUAN It is easy to show that, although the transformed form (2.3) is not unique (any nonsingular matrix that operates block-wise in the three block rows can be applied), the strangeness-free property is invariant with respect to the choice of W If (1.1) is strangeness free, then, setting ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ E1 A1 D1 E = ⎣ A2 ⎦ , A = ⎣ ⎦ , D = ⎣ ⎦ , F = ⎣−D2 ⎦ , D3 0 the implicit system of (2.6) is equivalent to the neutral linear time-invariant DDE (2.7) ˙ − τ ), x(t) ˙ = E −1 Ax(t) + E −1 Dx(t − τ ) + E −1 F x(t in which any further factor cancels out and which admits a unique solution that satisfies the consistent initial condition (1.2) We conclude this section with two remarks The first one gives a characterization of the class of strangeness-free equations In the second one, since the matrix W in (2.3) is not unique, the relation between different such transformation matrices is established Remark 2.6 Consider a strangeness-free equation (1.1) together with its transformed coefficients (2.3) Only two cases are possible with the pair (E, A) If h = 0, i.e., the third block row of E vanishes, then the pair (E, A) is regular and of index at most Otherwise, the pair (E, A) is singular Consequently, the class of strangenessfree equations and the class of equations with regular higher-index pair (E, A) are complementary Remark 2.7 Suppose that (1.1) is strangeness free and W and W are two nonsingular matrices that both transform the coefficients of the equation to the form (2.3) Let Ei , Ai , and Di be the transformed blocks corresponding to W Define P = W W −1 and let ⎡ ⎤ P11 P12 P13 P = ⎣P21 P22 P23 ⎦ P31 P32 P33 Then, we have ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ A1 A1 E1 E1 P ⎣ ⎦ = ⎣ ⎦ , P ⎣A2 ⎦ = ⎣A2 ⎦ 0 0 Due to the assumptions on the form (2.3), it is easy to verify that P is a block lowertriangular matrix, i.e., P21 , P31 , and P32 are zero blocks Since P is nonsingular, the diagonal blocks Pii , i = 1, 2, 3, are nonsingular Thus, W = P W with ⎡ ⎤ P11 P12 P13 P = ⎣ P22 P23 ⎦ 0 P33 In the next section we present necessary and sufficient conditions such that the exponential stability for linear time-invariant DDAEs is characterized by the spectral function Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILITY OF DELAY DAES 1637 Exponential stability of linear DDAEs In this section we show that for strangeness-free systems the spectral condition characterizes exponential stability Theorem 3.1 Suppose that equation (1.1) is strangeness free Then (1.1) is exponentially stable if and only if α(H) < Proof Necessity Suppose that (1.1) is exponentially stable, i.e., inequality (2.1) holds with positive constants K and ω, but α(H) ≥ Then there exists an eigenvalue λ ∈ σ(H) with Re λ > −ω Let v = be an eigenvector associated with λ, i.e., (λE − A − e−λτ D)v = 0, then obviously x(t) = eλt v is a solution of (1.1), but it does not satisfy (2.1) This is a contradiction and thus α(H) < Sufficiency Suppose that α(H) < and consider a solution x of (1.1) As seen in the previous section, x also satisfies the neutral delay ODE system (2.7), whose spectral function is H(s) = sI − E −1 A − e−sτ E −1 D − se−sτ E −1 F = E −1 (sE − A − e−sτ D − se−sτ F ) It is easy to see that σ(H) = σ(H) ∪ {0} We have α(H) = 0, but because α(H) < 0, is an isolated (and semisimple) eigenvalue It has been shown in [20, Chapter 12] that the solutions of (2.7) can be represented in the form x(t) = v + x∗ (t), (3.1) where x∗ (t) satisfies (2.1) and either v = or v is an eigenvector associated with the eigenvalue λ = of H(λ) Hence, we have (3.2) A1 v + D1 v = Moreover, since limt→∞ x∗ (t) = 0, from the second and the third equation of (2.5), it follows that A2 v + D2 v = D3 v = (3.3) From (3.2) and (3.3), it follows that H(0)v = But since ∈ σ(H), this implies that v = and hence x(t) = x∗ (t) Thus, (1.1) is exponentially stable Remark 3.2 In the proof of Theorem 3.1, we see that α(H) ≤ α(H) always holds Thus, if system (1.1) is strangeness free, then the spectral set σ(H) is bounded from the right, or equivalently the spectral abscissa satisfies α(H) < ∞ Now we consider the case when the pair (E, A) (1.1) is regular and it is transformed into the Weierstraß–Kronecker canonical form (2.2) Setting (3.4) W −1 DT = D11 D21 x (t) φ (t) D12 , T −1 φ(t) = , T −1 x(t) = D22 x2 (t) φ2 (t) with D11 ∈ Cr,r , D12 ∈ Cr,n−r , D21 ∈ Cn−r,r , D22 ∈ Cn−r,n−r , and x1 , x2 , φ1 , φ2 partitioned analogously Then (1.1) is equivalent to the system (3.5) x˙ (t) = A11 x1 (t) + D11 x1 (t − τ ) + D12 x2 (t − τ ), N x˙ (t) = x2 (t) + D21 x1 (t − τ ) + D22 x2 (t − τ ), with initial conditions (3.6) xi (t) = φi (t) for t ∈ [−τ, 0], i = 1, Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1638 N H DU, V H LINH, V MEHRMANN, AND D D THUAN From the explicit solution formula for linear time-invariant DAEs (see [7, 25]), the second equation of (3.5) implies that (3.7) ν−1 x2 (t) = −D21 x1 (t − τ ) − D22 x2 (t − τ ) − (i) (i) (i) (i) N i D21 x1 (t − τ ) + N i D22 x2 (t − τ ) , i=1 and for t ∈ [0, τ ), we get (3.8) ν−1 x2 (t) = −D21 φ1 (t − τ ) − D22 φ2 (t − τ ) − N i D21 φ1 (t − τ ) + N i D22 φ2 (t − τ ) i=1 It follows that φ needs to be differentiable at least ν times if the coefficients D21 and D22 not satisfy further conditions Extending this argument to t ∈ [τ, 2τ ), [2τ, 3τ ), etc., the solution cannot be extended to the full real half-line unless the initial function φ is infinitely often differentiable or the coefficient associated with the delay is highly structured Corollary 3.3 Consider the DDAE (1.1)–(1.2) with a regular pair (E, A), ind(E, A) ≤ 1, and its associated spectral function H Then (1.1) is exponentially stable if and only if α(H) < Proof If ind(E, A) ≤ 1, then the system is obviously strangeness free in the sense of Definition 2.5 with d + a = n and h = Thus, by Theorem 3.1, the system is exponentially stable if and only if α(H) < We note that the result of Corollary 3.3 is obtained in [31] by a direct proof Let us now consider exponential stability for the case that ind(E, A) > In order to avoid an infinite number of differentiations of φ induced by (3.8), it is reasonable to assume that for a system in Weierstraß–Kronecker form (2.2) with transformed matrices as in (3.4) the allowable delay condition N D2i = 0, i = 1, 2, holds Note that this condition is trivially true for the index-1 case, since then we have N = In terms of the original coefficients for (1.1) for a regular pair (E, A) with arbitrary index this allowable delay condition can be described as follows Choose any fixed sˆ ∈ C such that det(ˆ sE − A) = and set ˆ = (ˆ E sE − A)−1 E, (3.9) ˆ = (ˆ D sE − A)−1 D Proposition 3.4 Consider a DDAE of the form (1.1) with a regular pair (E, A) of arbitrary index, let sˆ ∈ C be such that det(sE − A) = 0, and consider the system (2.2) after transformation to Weierstraß–Kronecker canonical form Then the allowable delay conditions N D21 = and N D22 = are simultaneously satisfied if and only if ˆ E ˆD ˆ = 0, (I − Eˆ D E) (3.10) ˆ where Eˆ D denotes the Drazin inverse of E Proof From (2.2) it follows that sIr − J)−1 ˆ = T (ˆ E 0 T −1 (ˆ sN − In−r )−1 N and ˆ =T D (ˆ sIr − J)−1 D11 (ˆ sN − In−r )−1 D21 (ˆ sIr − J)−1 D12 T −1 (ˆ sN − In−r )−1 D22 Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1639 STABILITY OF DELAY DAES Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Therefore, sIr − J)−1 ˆ D = T (ˆ E 0 −1 T , and by elementary calculations we get (ˆ sN − In−r )−2 N D21 ˆ E ˆD ˆ =T (I − Eˆ D E) T −1 (ˆ sN − In−r )−2 N D22 ˆ D E) ˆ E ˆD ˆ = if and only if N D21 = and N D22 = Thus we have that (I − E Using Proposition 3.4, we have the following characterization of exponential stability for DDAEs with regular pair (E, A) Theorem 3.5 Consider the DDAE (1.1)–(1.2) with a regular pair (E, A) satisfying (3.10) Then (1.1) is exponentially stable if and only if α(H) < Proof Necessity The proof is analogous to that of Theorem 3.1 and we conclude that if (1.1) is exponentially stable, then α(H) < Sufficiency Suppose that α(H) < Since the pair (E, A) is regular, it follows that (1.1)–(1.2) is equivalent to the system in canonical form (3.5) Under the assumption (3.10), we have N D2i = 0, i = 1, 2, and thus (3.7) is reduced to = x2 (t) + D21 x1 (t − τ ) + D22 x2 (t − τ ) (3.11) This implies that xT1 (3.12) Ir 0 xT2 T is also a solution to the index-1 DDAE A11 x˙ (t) = x˙ (t) 0 In−r D11 x1 (t) + D21 x2 (t) x1 (t − τ ) , x2 (t − τ ) D12 D22 with the characteristic function I ˜ H(s) =s r 0 A − 11 0 D11 − e−sτ In−r D21 D12 D22 Using the Weierstraß–Kronecker canonical form (2.2), we have that W −1 H(s)T = sIr − A11 D11 − e−sτ D21 sN − In−r Since N D2i = 0, i = 1, 2, and −(sN − In−r )−1 = Ir ν−1 i i=0 (sN ) , D12 D22 it follows that W −1 H(s)T −(sN − In−r )−1 = sIr − A11 = sIr − A11 D11 D12 − e−sτ −(sN − In−r )−1 D21 −(sN − In−r )−1 D22 −In−r ⎡ ⎤ D11 D12 ν−1 ⎦ − e−sτ ⎣ν−1 −In−r (sN )i D21 (sN )i D22 i=0 sIr − A11 = ˜ = H(s) −Im−r −e −sτ D11 D21 i=0 D12 D22 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1640 N H DU, V H LINH, V MEHRMANN, AND D D THUAN ˜ ˜ = α(H) < This implies that det H(s) = if and only if det H(s) = 0, and hence α(H) Thus, by Corollary 3.3, system (3.12) with initial condition (3.6) is exponentially stable and hence system (1.1)–(1.2) is exponentially stable For the system in Example 1.2 which has a regular pair (E, A) that is already in Weierstraß–Kronecker form, we have N D21 = but N D22 = and the system has α(H) < but the system is not exponentially stable The following example presents the same observation for the case N D21 = but N D22 = Example 3.6 Consider (1.1) with ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 −1 0 0 0 ⎢ 0 0⎥ ⎢ 0⎥ ⎢0 0⎥ ⎥ ⎢ ⎥ ⎢ ⎥ E=⎢ ⎣0 0 1⎦ , A = ⎣ 0 0⎦ , D = ⎣0 0⎦ 0 0 0 0 We then have ⎡ 1+s ⎢ H(s) = sE − A − e−sτ D = ⎢ ⎣ 0 −2 − e−sτ 0 s −2 − e−sτ ⎤ −e−sτ ⎥ ⎥ ⎦ s −2 − e−sτ Therefore, det H(s) = −(1 + s)(2 + e−sτ )3 , the eigenvalues are λ = −1 and s = (− ln + (2k + 1)π)/τ, k ∈ Z, and hence all eigenvalues are in the open left half complex plane However, the system can be written as x˙ (t) = −x1 (t) + x4 (t − τ ), x˙ (t) = 2x2 (t) + x2 (t − τ ), x˙ (t) = 2x3 (t) + x3 (t − τ ), = 2x4 (t) + x4 (t − τ ) It is clear that if φ4 is not sufficiently smooth or its derivatives are unbounded, then the second and the third component solutions cannot be extended or they are unbounded If the solution is defined for all t ≥ 0, it depends on the derivatives of the initial function in general Thus, the system is not exponentially stable We have seen that the spectral condition α(H) < is necessary for the exponential stability of (1.1), but in general it is not sufficient Introducing further restrictions on the delays, we get that exponential stability is equivalent to the spectral condition Robust exponential stability We have seen in the previous section that under some extra conditions the exponential stability of a linear time-invariant DDAE can be characterized by the spectral properties of the matrix function H(s) Typically, however, the coefficient functions are not exactly known, since they arise, e.g., from a modeling, or system identification process, or as coefficient matrices from a discretization process Thus, a more realistic scenario for the stability analysis is to analyze the robustness of the exponential stability under small perturbations To perform this analysis, in this section we study the behavior of the spectrum of the triple of coefficient matrices (E, A, D) under structured perturbations in the matrices E, A, D Suppose that system (1.1) is exponentially stable and consider a perturbed system (4.1) ˙ = (A + B2 Δ2 C)x(t) + (D + B3 Δ3 C)x(t − τ ), (E + B1 Δ1 C)x(t) Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1641 Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILITY OF DELAY DAES where Δi ∈ Cpi ,q , i = 1, 2, 3, are perturbations and Bi ∈ Cn,pi , i = 1, 2, 3, C ∈ Cq,n , are matrices that restrict the structure of the perturbations We could also consider different matrices Ci in each of the coefficients but for simplicity, see Remark 4.9 below, we assume that the column structure in the perturbations is the same for all coefficients Set ⎡ ⎤ Δ1 (4.2) Δ = ⎣Δ2 ⎦ , B = B1 B2 B3 , Δ3 and p = p1 + p2 + p3 and consider the set of destabilizing perturbations VC (E, A, D; B, C) = {Δ ∈ Cp×q : (4.1) is not exponentially stable} Then we define the structured complex stability radius of (1.1) subject to structured perturbations as in (4.1) as (4.3) rC (E, A, D; B, C) = inf{ Δ : Δ ∈ VC (E, A, D; B, C)}, where · is a matrix norm induced by a vector norm If only real perturbations Δ are considered, then we use the term structured real stability radius but here we focus on the complex stability radius With H as in (1.4), we introduce the transfer functions G1 (λ) = −λCH(λ)−1 B1 , G2 (λ) = CH(λ)−1 B2 , G3 (λ) = e−λτ CH(λ)−1 B3 , and with G(λ) = G1 (λ) (4.4) G2 (λ G3 (λ) , we obtain an explicit formula for the structured stability radius Theorem 4.1 Suppose that system (1.1) is exponentially stable Then the structured stability radius of (1.1) subject to structured perturbations as in (4.1) satisfies the inequality −1 (4.5) rC (E, A, D; B, C) ≤ sup Re λ≥0 G(λ) ¯ + , where C ¯ + = {λ ∈ Proof Let be an arbitrary positive number and let λ0 ∈ C C, Re λ ≥ 0} is the closed right half-plane, be such that G(λ0 ) −1 −1 ≤ sup Re λ≥0 G(λ) + , and let u ∈ Cn be such that u = and G(λ0 )u = G(λ0 ) Furthermore, let y ∈ Cq be such that y = and y H (G(λ0 )u) = G(λ0 )u = G(λ0 ) , and set (4.6) Δ = G(λ0 ) −1 uy H , x = H(λ0 )−1 −λ0 B1 B2 e−λ0 τ B3 u Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1642 N H DU, V H LINH, V MEHRMANN, AND D D THUAN Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Then Δ ≤ G(λ0 ) −1 u y = G(λ0 ) −1 and (4.7) u G(λ0 ) ΔG(λ0 )u = Since u = 0, it follows that Δ ≥ G(λ0 ) G(λ0 ) = u −1 G(λ0 )u = CH(λ0 )−1 −λ0 B1 and thus Δ = G(λ0 ) −1 Since e−λ0 τ B3 u = 0, B2 it follows that −λ0 B1 B2 e−λ0 τ B3 u = 0, and hence x = On the other hand, by (4.6) and (4.7) we have H(λ0 )x = −λ0 B1 B2 e−λ0 τ B3 u = −λ0 B1 = −λ0 B1 B2 e−λ0 τ B3 ΔCH(λ0 )−1 −λ0 B1 = −λ0 B1 B2 e−λ0 τ B3 ΔCx B2 e−λ0 τ B3 ΔG(λ0 )u B2 e−λ0 τ B3 u = (−λ0 B1 Δ1 C1 + B2 Δ2 C2 + e−λτ B3 Δ3 C3 )x, and thus, λ0 (E + B1 Δ1 C1 ) − (A + B2 Δ2 C2 ) − e−λ0 τ (D + B3 Δ3 C3 ) x = This relation implies that λ0 is a root of the characteristic function associated with (4.1) Since Re λ0 ≥ 0, it follows that (4.1) is not exponentially stable Thus, Δ ∈ VC (E, A, D; B, C), which implies that rC (E, A, D; B, C) ≤ Δ = G(λ0 ) Since −1 −1 ≤ sup Re λ≥0 G(λ) + is arbitrary, it follows that −1 rC (E, A, D; B, C) ≤ sup Re λ≥0 G(λ) , and the proof is complete For every perturbation Δ as in (4.2) we define (4.8) HΔ (λ) = λ(E + B1 Δ1 C) − (A + B2 Δ2 C) − e−λτ (D + B3 Δ3 C) and have the following proposition Proposition 4.2 Consider system (1.1) and the perturbed system (4.1) If the associated spectral abscissa satisfy α(H) < and α(HΔ ) ≥ 0, then we have −1 (4.9) Δ ≥ sup Re λ≥0 G(λ) Proof If supRe λ≥0 G(λ) = ∞, then (4.9) holds trivially Therefore, we may assume that sup Re λ≥0 G(λ) < ∞ Since α(HΔ ) ≥ 0, we have two cases Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1643 Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILITY OF DELAY DAES Case 1: There exists λ0 ∈ σ(HΔ ) such that Re λ0 ≥ Then, there exists a nonzero x ∈ Cn such that HΔ (λ0 )x = 0, and we have = HΔ (λ0 )x = H(λ0 )x − −λ0 B1 B2 e−λ0 τ B3 ΔCx Since H(λ0 ) is invertible, we have that H(λ0 )x = and thus x = H(λ0 )−1 −λ0 B1 (4.10) e−λ0 τ B3 ΔCx, B2 and also Cx = By multiplying C from the left on both sides of (4.10), we obtain Cx = CH(λ0 )−1 −λ0 B1 B2 e−λ0 τ B3 ΔCx = G(λ0 )ΔCx, and hence, Cx ≤ G(λ0 ) Δ Cx It follows that −1 Δ ≥ G(λ0 ) −1 ≥ sup G(λ) ¯+ λ∈C Case 2: There exists a sequence {λj }∞ j=1 such that λj ∈ σ(HΔ ) and Re λj < for all j but limn→∞ Re λj = Then, for all sufficiently large j, we have that Re λj > α(H), which implies λj ∈ σ(H) Similarly to the proof of Case 1, it follows that Δ ≥ G(λj ) −1 , and thus, −1 Δ ≥ sup Re λ≥Re λj G(λ) Since G(λ) is continuous and supRe λ≥0 G(λ) < ∞, letting j → ∞, we obtain −1 Δ ≥ lim sup j→∞ Re λ≥Re λj G(λ) −1 = sup Re λ≥0 G(λ) , and the proof is complete It is already known for the case of perturbed nondelay DAEs [6] (see also [11]), that it is necessary to restrict the perturbations in order to get a meaningful concept of the structured stability radius, since a DAE system may lose its regularity and/or stability under infinitesimal perturbations We therefore introduce the following set of allowable perturbations Definition 4.3 Consider a strangeness-free system (1.1) and let W ∈ Cn,n be such that (2.3) holds A structured perturbation as in (4.1) is called allowable if (4.1) is still strangeness free with the same triple (d, a, h), i.e., there exists a nonsingular ˜ ∈ Cn,n such that W ⎡ ⎤ ⎡ ⎤ ˜1 A˜1 E −1 −1 ˜ ˜ ⎦ ⎣ ⎣ W (E + B1 Δ1 C) = , W (A + B2 Δ2 C) = A˜2 ⎦ , 0 ⎡ ⎤ ˜ D1 ˜ −1 (D + B3 Δ3 C) = ⎣D ˜ 2⎦ , (4.11) W ˜3 D Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1644 N H DU, V H LINH, V MEHRMANN, AND D D THUAN ˜ ∈ Cd,n , A˜2 , D ˜ ∈ Ca,n , D ˜ ∈ Ch,n , such that where E˜1 , A˜1 , D ⎤ ⎡ ˜1 E ⎣ A˜2 ⎦ ˜3 D is invertible Assume that the matrices Bi , i = 1, 2, 3, that are restricting the structure have the form ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ B11 B21 B31 (4.12) W −1 B1 = ⎣B12 ⎦ , W −1 B2 = ⎣B22 ⎦ , W −1 B3 = ⎣B32 ⎦ , B13 B23 B33 where Bj1 ∈ Cd,pj , B2j ∈ Ca,pj , and B3,j ∈ Ch,pj , j = 1, 2, According to [6, Lemma 3.3], if the structured perturbation is allowable, then B12 Δ1 C = 0, B13 Δ1 C = 0, and B23 Δ2 C = Thus, without loss of generality, we assume that B12 = 0, B13 = 0, and B23 = (4.13) Note that, by Remark 2.7, condition (4.13) is invariant with respect to the choice of the transformation matrix W Furthermore, it is easy to see that with all structured perturbations with Bi , i = 1, 2, 3, satisfying (4.13), if the perturbation Δ is sufficiently small, then the strangeness-free property is preserved with the same sizes of the blocks We denote the infimum of the norm of all perturbations Δ such that (4.1) is no longer strangeness free or the sizes of the blocks d, a, h change, by dsC (E, A, D; B, C), and immediately have the following proposition Proposition 4.4 Suppose that (1.1) is strangeness free and subjected to structured perturbations with Bi , i = 1, 2, satisfying (4.13) Then ⎤−1 ⎡ B11 E1 dsC (E, A, D; B, C) = C ⎣ A2 ⎦ ⎣ 0 D3 ⎡ B22 ⎤ 0 ⎦ B33 −1 Proof With restriction matrices Bi , i = 1, 2, 3, satisfying (4.13), the perturbed ˜ ∈ Cd,n , A˜2 , D ˜ ∈ Ca,n , D ˜ ∈ Ch,n ˜1 , A˜1 , D system (4.1) is still strangeness free with E (as in (4.11)) if and only if ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ E1 + B11 Δ1 C 0 B11 E1 ⎣ A2 + B22 Δ2 C ⎦ = ⎣ A2 ⎦ + ⎣ B22 ⎦ ΔC D3 D3 + B33 Δ3 C 0 B33 is nonsingular Thus the distance problem is that of the distance of a nonsingular matrix to the nearest singular matrix For this problem it has been shown, see, e.g., [36], that the matrix ⎡ ⎤ ⎡ ⎤ ˜1 E E1 + B11 Δ1 C1 ⎣ A˜2 ⎦ = ⎣ A2 + B22 Δ2 C2 ⎦ ˜3 D3 + B33 Δ3 C3 D is nonsingular if ⎤−1 ⎡ B11 E1 Δ < C ⎣ A2 ⎦ ⎣ 0 D3 ⎡ B22 ⎤ 0 ⎦ B33 −1 , Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1645 STABILITY OF DELAY DAES Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php and the distance to singularity is given by ⎤−1 ⎡ B11 E1 dsC (E, A, D; B, C) = C ⎣ A2 ⎦ ⎣ D3 ⎡ ⎤ 0 ⎦ B33 B22 −1 Remark 4.5 Again following from Remark 2.7, it is not difficult to show that in fact the formula in Proposition 4.4 is independent of the choice of the transformation matrix W Proposition 4.6 Consider system (1.1) with α(H) < If the system is strangeness free and subjected to structured perturbations as in (4.1) with structure matrices B1 , B2 , B3 satisfying (4.13) and if the perturbation Δ satisfies −1 Δ < sup Re λ≥0 G(λ) , then the structured perturbation is allowable, i.e., the perturbed equation (4.1) is strangeness free with the same block sizes d, a, and h Proof To prove the assertion, we will show that −1 (4.14) sup Re λ≥0 G(λ) ⎡ ⎤−1 ⎡ E1 B11 ⎣ ⎦ ⎣ A ≤ C D3 0 B22 ⎤ 0 ⎦ B33 −1 We can rewrite G as G(λ) = CH(λ)−1 −λB1 B2 e−λτ B3 ⎡ ⎤−1 ⎡ λE1 − A1 − e−λτ D1 −λB11 = C ⎣ −A2 − e−λτ D2 ⎦ ⎣ −e−λτ D3 B21 B22 ⎤ e−λτ B31 e−λτ B32 ⎦ e−λτ B33 =: CF (λ), and thus it follows that ⎡ ⎤ ⎡ λE1 − A1 − e−λτ D1 −λB11 ⎣ −A2 − e−λτ D2 ⎦ F (λ) = ⎣ −e−λτ D3 If λ = 0, then this is equivalent to ⎤ ⎡ ⎡ B11 −E1 + A1 /λ + e−λτ D1 /λ ⎦ F (λ) = ⎣ ⎣ −A2 − e−λτ D2 −D3 B21 B22 ⎤ e−λτ B31 e−λτ B32 ⎦ e−λτ B33 ⎤ −B21 /λ −e−λτ B31 /λ B22 e−λτ B32 ⎦ B33 and, since ⎡ ⎤ ⎡ ⎤ −E1 + A1 /λ + e−λτ D1 /λ E1 ⎦ = − ⎣ A2 ⎦ −A2 − e−λτ D2 lim ⎣ Re λ→+∞ D3 −D3 Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1646 N H DU, V H LINH, V MEHRMANN, AND D D THUAN Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php and ⎡ B11 lim ⎣ Re λ→+∞ ⎤ ⎡ −B21 /λ −e−λτ B31 /λ B11 B22 e−λτ B32 ⎦ = ⎣ 0 B33 B22 ⎤ 0 ⎦, B33 it follows that limRe λ→+∞ F (λ) exists and ⎤−1 ⎡ B11 E1 lim F (λ) = − ⎣ A2 ⎦ ⎣ Re λ→+∞ D3 ⎡ B22 ⎤ 0 ⎦ B33 Thus, it follows that ⎡ ⎤−1 ⎡ E1 B11 lim G(λ) = C lim F (λ) = −C ⎣ A2 ⎦ ⎣ Re λ→+∞ Re λ→+∞ D3 0 B22 ⎤ 0 ⎦, B33 and hence (4.14) holds It is obvious that −1 sup Re λ≥0 G(λ) −1 ≤ lim Re λ→+∞ G(λ) By Proposition 4.4, it follows that if −1 Δ < sup Re λ≥0 G(λ) , then the perturbed equation (4.1) is strangeness free with the same block sizes d, a, and h as for (1.1) We combine these results to characterize the stability radius for strangeness-free DDAEs under suitable structured perturbations Theorem 4.7 Suppose that (1.1) is exponentially stable and strangeness free and subjected to structured perturbations as in (4.1) with structure matrices B1 , B2 , B3 satisfying (4.13) Then −1 (4.15) rC (E, A, D; B, C) = sup Re λ≥0 G(λ) Furthermore, if Δ < rC (E, A, D; B, C) then (4.1) is strangeness free with the same block sizes d, a, and h as for (1.1) Proof By Proposition 4.1, we have −1 rC (E, A, D; B, C) ≤ sup Re λ≥0 G(λ) To prove the reverse inequality, let Δ be an arbitrary perturbation that destroys the exponential stability of (1.1) Assume that −1 Δ < sup Re λ≥0 G(λ) Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1647 Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILITY OF DELAY DAES Since (1.1) is strangeness free and exponentially stable, we have α(H) < and by Proposition 4.2, we have also that α(HΔ ) < By Proposition 4.6 the perturbed equation (4.1) is strangeness free, and hence, by Theorem 3.1 we obtain that the perturbed equation (4.1) is exponentially stable, which is a contradiction Thus, −1 Δ ≥ sup Re λ≥0 G(λ) , and hence, −1 rC (E, A, D; B, C) ≥ sup Re λ≥0 G(λ) , which implies (4.15) Finally, by Proposition 4.6 we have that (4.1) is strangeness free if Δ < rC (E, A, D; B, C) Remark 4.8 By the maximum principle [28], the supremum of G(λ) over the right half-plane is attained at a finite point on the imaginary axis or at infinity For strangeness-free DDAEs, it can be shown that it suffices to take the supremum of G(λ) over the imaginary axis instead of the whole right half-plane, i.e., we have −1 rC (E, A, D; B, C) = sup Re λ=0 G(λ) ; see Lemma A.1 in the appendix Remark 4.9 Perturbed systems of the form (4.1) represent a subclass of the class of systems with more general structured perturbations (4.16) ˙ = (A + B2 Δ2 C2 )x(t) + (D + B3 Δ3 C3 )x(t − τ ), (E + B1 Δ1 C1 )x(t) where Δi ∈ Cpi ,qi , i = 1, 2, 3, are perturbations, Bi ∈ Cn,pi and Ci ∈ Cqi ,n , i = 1, 2, 3, are different matrices One may formulate a structured stability radius subject to (4.16), as well, but an exact formula for it could not be expected as in the case of (4.1) For another special case that B1 = B2 = B3 = B and Ci are different, an analogous formulation and similar results for the structured stability radius can be obtained However, due to the special row structure of the strangeness-free form and of allowable perturbations, the consideration of perturbed systems of the form (4.1) is more reasonable As a corollary we obtain the corresponding result for a special case of strangenessfree systems where already the pair (E, A) is regular with ind(E, A) ≤ Corollary 4.10 Consider system (1.1) with a regular pair (E, A) satisfying ind(E, A) ≤ and suppose that the system is exponentially stable and has Weierstraß– Kronecker canonical form (2.2) If the system is subjected to structured perturbations as in (4.1), where the structure matrix B1 satisfies W −1 B1 = B11 , with B11 ∈ Cd×p1 , then the structured stability radius is given by −1 rC (E, A, D; B, C) = sup Re λ=0 G(λ) Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1648 N H DU, V H LINH, V MEHRMANN, AND D D THUAN For nondelayed DAEs it has been shown [11] that if the perturbation is such that the nilpotent structure in the Weierstraß–Kronecker canonical form is preserved, then one can also characterize the structured stability radius in the case that the pair (E, A) is regular and ind(E, A) > We have seen in section that exponential stability is characterized by the spectrum of H if we assume that N D21 = and N D22 = In the following we assume that this property is preserved and that in the perturbed equation (4.1), the structure matrices B1 , B2 , B3 satisfy (4.17) W −1 B1 = B11 B21 B31 , W −1 B2 = , W −1 B3 = , N B32 = 0, 0 B32 where Bj,1 ∈ Cd,pj , j = 1, 2, 3, B32 ∈ Cn−d,p3 , and W ∈ Cn,n , N ∈ Cn−d,n−d are as in (2.2) In the following we consider structured perturbations that not alter the nilpotent structure of the Weierstraß–Kronecker form (2.2) of (E, A), i.e., the nilpotent matrix N and the corresponding left invariant subspace associated with eigenvalue ∞ is preserved; see [6] for the case that ind(E, A) = and D = Similarly to the approach in [6], we now introduce the distance to the nearest pair with a different nilpotent structure dnC (E, A, D; B, C) = inf{ Δ : (4.1) does not preserve the nilpotent structure} Under assumption (4.17), we obtain the following result; see [11] for the case of nondelay DAEs Proposition 4.11 Consider (1.1) with regular (E, A) and ind(E, A) > 1, subjected to transformed perturbations satisfying (4.17) Let us decompose CT = C11 C12 with C11 ∈ Cq,r , C12 ∈ Cq,n−r Then the distance to the nearest system with a different nilpotent structure is given by dnC (E, A, D; B, C) = C11 B11 −1 Proof With regard to (4.17), the nilpotent structure of the perturbed equation (4.1) is preserved if and only if the perturbed matrix Ir + B11 Δ1 C11 is nonsingular Thus using again the distance of a nonsingular matrix to singularity (see again [36]), we obtain dnC (E, A, D; B, C) = C11 B11 −1 Remark 4.12 By their definition, the blocks B11 and C11 depend on the transformation matrices W −1 and T , respectively It is known that the Weierstraß–Kronecker canonical form (2.2) is not unique However, [25, Lemma 2.10] implies that neither the product C11 B11 nor the condition (4.17) depends on the choice of pair (W, T ) Thus, the distance formula for dnC (E, A, D; B, C) obtained in Proposition 4.11 is indeed independent of the choice of the transformations Theorem 4.13 Consider an exponentially stable equation (1.1) with regular pair (E, A) and ind(E, A) > and assume that (1.1) is subjected to transformed perturbations satisfying (4.17) Then the stability radius is given by the formula −1 rC (E, A, D; B, C) = sup Re λ=0 G(λ) Moreover, if Δ < rC (E, A, D; B, C), then the perturbed equation (4.1) has a regular pair (E + B1 Δ1 C, A + B2 Δ2 C) with the same nilpotent structure in the Weierstraß– Kronecker canonical form and it is exponentially stable Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1649 STABILITY OF DELAY DAES Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Proof Under the assumption (4.17), elementary calculations yield lim Re λ→+∞ G1 (λ) = C11 B11 , lim Re λ→+∞ G2 (λ) = lim Re λ→+∞ G3 (λ) = Therefore, lim Re λ→+∞ G(λ) = C11 B11 Using the fact that sup Re λ≥0 G(λ) ≥ lim Re λ→+∞ G(λ) and Proposition 4.11, the remainder of the proof is analogous to that of Theorem 4.7 Again by using the maximum principle, it suffices to take the supremum of G(λ) on the imaginary axis instead of the whole right half-plane To illustrate the results of this section consider the following example Example 4.14 Consider the strangeness-free linear DDAE ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 −1 0 ⎣0 0⎦ x(t) (4.18) ˙ = ⎣ 0⎦ x(t) + ⎣0 1⎦ x(t − 1) 0 0 0 0 with singular pair (E, A) subjected to the structured perturbations ⎤ ⎡ ⎡ ⎤ 0 + δ11 δ12 δ13 0 ⎦, E=⎣ E = ⎣0 0⎦ 0 0 0 ⎤ ⎡ ⎡ ⎤ −1 + 3δ21 −1 3δ22 3δ23 δ21 + δ22 δ23 ⎦ , A=⎣ A = ⎣ 0⎦ 0 0 0 ⎡ ⎤ ⎡ ⎤ 2δ31 + 2δ32 2δ33 D = ⎣0 1⎦ D = ⎣2δ31 + 2δ32 + 2δ33 ⎦ , −1 δ31 δ32 + δ33 which can be represented in the form (4.1) with ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ δ11 B1 = ⎣0⎦ , B2 = ⎣1⎦ , B3 = ⎣2⎦ , C = I3 , Δ = ⎣ δ21 0 δ31 We have H(λ) = λE − A − e−λ D = 1+λ −2(2+e−λ ) 0 −2−e−λ −e−λ 0 −e−λ δ12 δ22 δ32 ⎤ δ13 δ23 ⎦ δ33 , and it is easy to check that α(H) < and therefore (4.18) is exponentially stable By simple computations, we get ⎤ ⎡ −λ ⎥ ⎢1 + λ 1+λ ⎢ −1 −e−λ ⎥ G(λ) = ⎢ ⎥ ⎦ ⎣ + e−λ + e−λ 0 −1 Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1650 Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php and with N H DU, V H LINH, V MEHRMANN, AND D D THUAN · being the maximum norm of C3 , it follows that sup G(λ) λ∈iR ∞ = G(iπ) ∞ = Thus, by Theorem 4.7, we obtain rC (E, A, D; B, C) = supλ∈iR G(λ) ∞ = We note that by using (4.6), a destabilizing perturbation is easily constructed as ⎡ ⎤ 0 Δ = ⎣ −1/2 ⎦ , 1/2 with norm 1/2 Further, one can easily check that with this Δ the perturbed system remains strangeness free, but α(HΔ ) = 0, which means that the perturbed system is not asymptotically stable Conclusion Characterizations for exponential stability and robust exponential stability of DDAEs have been derived under the assumption that the coefficient matrices are subjected to structured perturbations The spectral condition for exponential stability has been investigated in the class of strangeness-free DDAEs as well as higher-index DDAEs Formulas for the complex stability radius and the class of allowable perturbations for DDAEs have been derived in both cases However, the validity of a spectral condition for the exponential stability of DDAEs in the general case and formulas for the real stability radius of DDAEs are still open problems Appendix A In this appendix we give a proof for the statement in Remark 4.8 which is stated as the following lemma Lemma A.1 Under the conditions of Theorem 4.7 or Corollary 4.10, we have (A.1) sup Re λ≥0 G(λ) = sup Re λ=0 G(λ) , where G is defined in (4.4) Proof Since G(λ) is analytic in the right half of the complex plane, by the maximum principle, the supremum of G(λ) is attained on the boundary, that is either on the imaginary axis or somewhere at infinity It remains to show that the supremum is indeed attained on the imaginary axis (either at a finite point or at infinity) (i) Let us first consider the case that ind(E, A) ≤ and that the system is in Weierstraß–Kronecker canonical form (2.2) We then have H(λ)−1 = T λI − J − D11 e−λτ −D21 e−λτ −D12 e−λτ −I − D22 e−λτ −1 W −1 Since for sufficiently large |λ|, λI −J −D11 e−λτ is invertible, we can apply the inversion formula for block matrices M of the form M= M11 M21 M12 M22 Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1651 Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php STABILITY OF DELAY DAES with M11 = λI − J − D11 e−λτ , M12 = −D12 e−λτ , M21 = −D21 e−λτ , and M22 = −I − D22 e−λτ , and the inverse is given by (A.2) −1 −1 I 0 M12 M11 I −M11 M −1 = −1 −1 I I (M22 − M21 M11 M12 )−1 −M21 M11 Moreover, lim|λ|→∞ (λI − J − D11 e−λτ )−1 = and lim|λ|→∞ λ(λI − J − D11 e−λτ )−1 = I Therefore, for all ε > 0, there exists L > such that for λ satisfying |λ| ≥ L and Re λ ≥ 0, we have ˜ G(λ) − G(λ) ≤ ε, (A.3) where ˜ ˜ (λ) G(λ) = G ˜ (λ) G ˜ (λ) G with ˜ (λ) = CT I G 0 B11 ˜ (λ) = CT ,G 0 −(I + D22 e−λτ )−1 B21 , B22 and ˜ (λ) = e−λτ CT G 0 −(I + D22 e−λτ )−1 B31 B32 By introducing a new variable z = e−λτ , since Re λ ≥ 0, we have |z| ≤ By the ˜ as a function of z over the disk |z| ≤ is maximum principle, the supremum of G attained on the boundary |z| = 1, or equivalently, sup Re λ≥0,|λ|≥L ˜ G(λ) = sup Re λ=0,|λ|≥L ˜ G(λ) Because of (A.3), we have sup Re λ≥0,|λ|≥L G(λ) ≤ sup Re λ≥0,|λ|≥L = sup Re λ=0,|λ|≥L ≤ sup Re λ=0,|λ|≥L ˜ G(λ) +ε ˜ G(λ) +ε G(λ) + 2ε Analogously, we have sup Re λ≥0,|λ|≥L G(λ) ≥ sup Re λ=0,|λ|≥L G(λ) − 2ε On the other hand, sup Re λ≥0,|λ|≤L G(λ) = max sup Re λ=0,|λ|≤L G(λ) , sup Re λ≥0,|λ|=L G(λ) Hence, sup Re λ=0 G(λ) − 2ε ≤ sup Re λ≥0 G(λ) ≤ sup Re λ=0 G(λ) + 2ε Since ε is arbitrary, the identity (A.1) holds Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1652 N H DU, V H LINH, V MEHRMANN, AND D D THUAN (ii) For the general case of a strangeness-free system of the form (1.1) that is transformed into the form (2.3) and that satisfies (4.13) we have ⎤−1 λE1 − A1 − e−λτ D1 = ⎣ −A2 − e−λτ D2 ⎦ W −1 −e−λτ D3 ⎤−1 ⎡ ⎡ I λE1 − A1 − e−λτ D1 = ⎣ −A2 − e−λτ D2 ⎦ ⎣0 I 0 −D3 ⎡ H(λ)−1 ⎤ 0 ⎦ W −1 eλτ Taking into account (4.13), we obtain ⎤ ⎤−1 ⎡ ⎡ B11 λE1 − A1 − e−λτ D1 G1 (λ) = λ ⎣ −A2 − e−λτ D2 ⎦ ⎣ ⎦ , −D3 ⎤ ⎤ ⎡ ⎡ −1 B21 λE1 − A1 − e−λτ D1 G2 (λ) = ⎣ −A2 − e−λτ D2 ⎦ ⎣B22 ⎦ , −D3 and ⎡ ⎤−1 ⎡ −λτ ⎤ B31 λE1 − A1 − e−λτ D1 e G3 (λ) = ⎣ −A2 − e−λτ D2 ⎦ ⎣e−λτ B32 ⎦ −D3 B33 The assumption that the system is strangeness free implies that the matrix pair ⎡ ⎤ ⎡ ⎤ E1 A1 ⎣ ⎦ , ⎣ A2 ⎦ D3 is of index and thus the claim follows by analogous arguments as in part (i) Acknowledgment We thank the anonymous referees for their useful suggestions that led to an improvement of the paper REFERENCES [1] U M Ascher and L R Petzold, The numerical solution of delay-differential algebraic equations of retarded and neutral type, SIAM J Numer Anal., 32 (1995), pp 1635–1657 [2] C T H Baker, C A H Paul, and H Tian, Differential algebraic equations with after-effect, J Comput Appl Math., 140 (2002), pp 63–80 [3] A Bellen and M Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, UK, 2003 [4] K E Brenan, S L Campbell, and L R Petzold, Numerical Solution of Initial-Value Problems in Differential Algebraic Equations, 2nd ed., Classics Appl Math., SIAM, Philadelphia, 1996 [5] R Byers, C He, and V Mehrmann, Where is the nearest non-regular pencil, Linear Algebra Appl., 285 (1998), pp 81–105 [6] R Byers and N K Nichols, On the stability radius of a generalized state-space system, Linear Algebra Appl., 188–189 (1993), pp 113–134 [7] S L Campbell, Singular Systems of Differential Equations I, Pitman, San Francisco, CA, 1980 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 06/30/14 to 130.63.180.147 Redistribution subject to SIAM license or copyright; 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Asymptotic stability of linear delay differential-algebraic equations and numerical methods, Appl Numer Math., 24 (1997), pp 247–264 [44] W Zhu and L R Petzold, Asymptotic stability of Hessenberg delay. .. and V Mehrmann, Analysis and reformulation of linear delay differential-algebraic equations, Elect J Linear Algebra, 23 (2012), pp 703–730 [18] P Ha, V Mehrmann, and A Steinbrecher, Analysis of