Systems & Control Letters 60 (2011) 596–603 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Stability radius of implicit dynamic equations with constant coefficients on time scales✩ Nguyen Huu Du a,∗ , Do Duc Thuan b , Nguyen Chi Liem a a Department of Mathematics, Mechanics and Informatics, Viet Nam National University, 334 Nguyen Trai, Hanoi, Viet Nam b Department of Applied Mathematics and Informatics, Hanoi University of Technology, Dai Co Viet, Hanoi, Viet Nam article abstract info Article history: Received 23 July 2010 Received in revised form 24 April 2011 Accepted 24 April 2011 Available online 24 May 2011 This paper deals with the stability radii of implicit dynamic equations on time scales when the structured perturbations act on both the coefficient of derivative and the right-hand side Formulas of the stability radii are derived as a unification and generalization of some previous results A special case where the real stability radius and the complex stability radius are equal is studied Examples are derived to illustrate results © 2011 Elsevier B.V All rights reserved Keywords: Time scales Implicit linear dynamic equation Index of the pencil of matrices Exponentially stable Stability radius Introduction then we have a formula in [3] for computing the complex stability radius In the past decades, there have been extensive works on studying of robust measures, where one of the most powerful ideas is the concept of the stability radii, introduced by Hinrichsen and Pritchard [1] The stability radius is defined as the smallest (in norm) complex or real perturbations destabilizing the equation In [2], the authors consider the equation x′ = Bx and assume that the perturbed equation can be represented in the form x′ = (B + DΣ E )x, (1.1) where Σ is an unknown disturbance matrix and D, E are known scaling matrices defining the ‘‘structure’’ of the perturbation The complex stability radius is then given by [ max ‖E (tI − B)−1 D‖ t ∈iR ]−1 (1.2) If the nominal equation is the difference equation xn+1 = Bxn with a structured perturbation of the form xn+1 = (B + DΣ E )xn , (1.3) [ max ω∈C:|ω|=1 ‖E (ωI − B)−1 D‖ ∗ Corresponding author Fax: +84 8588817 E-mail addresses: dunh@vnu.edu.vn, nhdu2001@yahoo.com (N.H Du) 0167-6911/$ – see front matter © 2011 Elsevier B.V All rights reserved doi:10.1016/j.sysconle.2011.04.018 (1.4) Moreover, in recent years, several technical problems in electronic circuit theory and robotic designs lead to the problem of investigating the differential–algebraic equation f (x′ (t ), x(t )) = 0, where the leading term x′ cannot be explicitly solved from x(t ) The linear form of this equation is Ax′ (t ) = Bx(t ), (1.5) with A and B denoting two constant matrices Assume that Eq (1.5) is subjected to perturbations of the form Ax′ (t ) = (B + DΣ E )x(t ) (1.6) Then the formula of the complex stability radius is given by (see [4]) [ max ‖E (tA − B)−1 D‖ ] −1 t ∈iR ✩ This work was done under the support of the Grant NAFOSTED 2011 ]−1 (1.7) When the nominal equation is the difference equation Axn+1 = Bxn with the structured perturbation of the form Axn+1 = (B + DΣ E )xn , (1.8) N.H Du et al / Systems & Control Letters 60 (2011) 596–603 we obtained an expression in [5] for the complex stability radius given by [ max ω∈C:|ω|=1 ‖E (ωA − B)−1 D‖ ]−1 (1.9) Earlier results of stability radii for time-varying equations can be found, e.g., in [6,7] The most successful attempt for finding a formula for the stability radius was an elegant result given by Jacob [7] Using this result, the notion and formula of the stability radius were extended to linear time-invariant differential–algebraic equations [8,9,4]; and to linear time-varying differential and difference–algebraic equations [10,5] On the other hand, in order to unify the continuous and discrete analysis, the theory of the analysis on time scales was introduced by Stefan Hilger in his Ph.D thesis in 1988 (supervised by Bernd Aulbach) [11] and has received a lot of attention By using the notation of the analysis on time scales, Eqs (1.1) and (1.3) can be rewritten under the unified form = (B + DΣ E )x, x (1.10) and also Eqs (1.6) and (1.8) become Ax = (B + DΣ E )x A formula of the stability radius for (1.10) is derived recently in [12] and it is given by Definition 2.1 (Delta Derivative) A function f : T → R is called delta differentiable at t if there exists a scalar f (t ) such that for all ϵ>0 |f (ς (t )) − f (s) − f (t )(ς (t ) − s)| ⩽ ϵ|ς (t ) − s| for all s ∈ (t − δ, t + δ) ∩ T and for some δ > The scalar f (t ) is called the delta derivative of f at t If T = R then delta derivative is f ′ (t ) from continuous calculus; if T = Z then the delta derivative is the forward difference, f , from discrete calculus A point t ∈ T is said to be right-dense if ς (t ) = t, right-scattered if ς (t ) > t, left-dense if ϱ(t ) = t and left-scattered if ϱ(t ) < t A function f defined on T is rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists at every left-dense point For any rd-continuous functions p(·) from T to R, the solution of the dynamic equation x = p(t )x, with the initial condition x(s) = 1, defines a so-called exponential function We denote this exponential function by ep (t , s) For the properties of exponential function ep (t , s) the interested reader can see [13–15] Denote T+ = [t0 , ∞) ∩ T We consider the dynamic equation on the time scale T x = f (t , x), (2.14) where f : T×R → R is a rd-continuous function and f (t , 0) = For the existence, uniqueness and extensibility of solution of Eq (2.14) we refer to [14] A function f from T to R is positively regressive if + µ(t )f (t ) > for every t ∈ T We denote R+ the set of positively regressive functions from T to R For any τ ∈ T+ , let x(t ) = x(t , τ , x0 ) be a solution of (2.14) with the initial condition x(τ , τ ) = x0 ∈ Rd On the exponential stability of dynamic equations on time scales, we use the following definition, see, e.g [16,11,17]: d [ max ‖E (tI − B)−1 D‖ t ∈Γus ]−1 , (1.11) where Γus is the stability domain of the time scale T The purpose of this paper is to present a unified formula for (1.2), (1.4), (1.7), (1.9) and (1.11) and to generalize them by studying the stability radius of the implicit dynamic equations on time scales Ax (t ) = Bx(t ), (1.12) under the general structured perturbations of the form [A, B] [A˜ , B˜ ] = [A, B] + DΣ E (1.13) When T = R (resp T = N), we consider it as a generalization of Eqs (1.1) and (1.6) (resp Eqs (1.3) and (1.8)) The difficulty we are faced when dealing with this problem is that although A, B, D, E are constant matrices, the structure of time scale (also the stability domain) is rather complicated and it can make Eq (1.12) become a time-varying equation Moreover, the disturbances affect not only the term on the right, but also the coefficient of the derivative on the left-hand side and it seems that we are working with an ill-posed problem This paper is organized as follows In Section 2, we summarize some preliminary results on time scales In Section 3, by defining the so-called domain of the uniformly exponential stability of time scales, we give the formulas of the stability radii of Eq (1.12), where the general structured perturbations are considered Section is concerned with special classes of {A, B} where the complex and real stability radii are equal Preliminaries A time scale is a nonempty closed subset of the real numbers R and we usually denote it by the symbol T The most popular examples are T = R and T = Z We assume throughout that a time scale T inherits the topology from the standard topology of the real numbers We define the forward jump operator ς : T → T by ς(t ) = inf{s ∈ T : s > t } (supplemented by inf ∅ = sup T) and the backward jump operator ϱ : T → T by ϱ(t ) = sup{s ∈ T : s < t } (supplemented by sup ∅ = inf T) The positively graininess function µ : T → R+ ∪ {0} is given by µ(t ) = ς (t ) − t For our purpose, we will assume that the time scale T is unbounded above, i.e., sup T = ∞ 597 d Definition 2.2 (Exponential Stability) The dynamic equation (2.14) is called exponentially stable if the condition • for every τ ∈ T+ there exists an N = N (τ ) ⩾ satisfying ‖x(t , τ , x0 )‖ ⩽ N (τ )‖x0 ‖e−α (t , τ ) (2.15) for all t ⩾ τ , t ∈ T and x0 ∈ R , where x(t , τ , x0 ) is the solution of (2.14) with the initial condition x(τ , τ ) = x0 + d holds for some α > such that −α ∈ R+ If the constant N can be chosen independent of τ ∈ T+ then the dynamic equation (2.14) is called uniformly exponentially stable Note that the condition −α ∈ R+ is equivalent to µ(t ) ⩽ α1 This means that we are working on time scales with bounded graininess Beside this definition one can find other definitions of exponential stability in [18–20] where instead of using the exponential function e−α (t , τ ) on time scale, one uses the classical exponential function exp{−α(t − τ )} in (2.15) However, it is easy to prove that these definitions are equivalent We now consider the condition of exponential stability for linear time-invariant equations x = Ax, (2.16) where A ∈ Kd×d (K = R or K = C) We denote the set of the eigenvalues of A by σ (A) The following theorem can be proved by a similar way as in [19], although we use the exponential function on time scales to define exponential stability Theorem 2.3 (See [19, Lemma 6.1]) The linear equation (2.16) is uniformly exponentially stable if and only if for every λ ∈ σ (A), the scalar equation x = λx is uniformly exponentially stable 598 N.H Du et al / Systems & Control Letters 60 (2011) 596–603 It is easy to give an example where on the time scale T, the scalar dynamic equation x = λx is exponentially stable but it is not uniformly exponentially stable Indeed, denote ((a, b)) = {n ∈ N : a < n < b} Consider the time scale T= n Let λ = −2 and τ ∈ T, say ⩽ τ < We can choose α = −1 and N = 2m+1 to obtain |eλ (t , τ )| ⩽ Ne−1 (t , τ ) However, it is not possible to choose N independent of τ Now, we denote m −1 Q A− U ) T −1 A = T diag(0, (U − Im−r ) is a nilpotent matrix and [22n , 22n+1 ] ((22n+1 , 22n+2 )) n Also by the decomposition (3.2) and the definition (3.3) of Q we see that m+1 S = {λ ∈ C, the scalar equation x = λx is uniformly exponentially stable} The set S is called the domain of the uniform exponential stability of the time scale T By the definition, if λ ∈ S, there exist α > and N ⩾ satisfying −α ∈ R+ and |eλ (t , τ )| ⩽ Ne−α (t , τ ) for all t ⩾ τ As a corollary of proposition 3.1 in [21], we have the following result Theorem 2.4 S is an open set in C For illustrating the domain of the uniform exponential stability S of the time scale T, we consider some simple cases • When T = R then S = {λ ∈ C, Re λ < 0} • When T = hZ( h > 0) then S = {λ ∈ C, |1 + λh| < 1} ∞ • When T = k=0 [2k, 2k + 1] then S = {λ ∈ C, Re λ + ln |1 + λ| < 0} −1 −1 Q A− )T B = T diag(0, (U − Im−r ) y(t ) Denoting T −1 x(t ) = z (t ) where z (t ) ∈ Km−r , we obtain Uz (t ) = z (t ) (3.6) It is easy to see that this equation has a unique solution z (t ) ≡ Therefore, the solution x(t ) of (3.1) with the initial condition P (x(t0 ) − x0 ) = exists in T+ = [t0 , ∞) ∩ T, and it satisfies y(t ) −1 Q x(t ) = T diag(0, Im−r )T x(t ) = T diag(0, Im−r ) = 0, for all t ∈ T + (3.7) In particular, the initial condition x(t0 ) = Px0 must hold Let x( t , τ , Px0 ) be the solution of (3.1) with the initial value x(τ , τ ) = Px0 According to Definition 2.2, we get the following definition of exponential stability: Definition 3.1 The implicit dynamic equation (3.1) is called exponentially stable if the condition • for every τ ∈ T+ and x0 ∈ Rm there exists an N = N (τ ) ⩾ satisfying Stability radii of implicit dynamic equations on time scales ‖x(t , τ , Px0 )‖ ⩽ N (τ )‖ Px0 ‖e−α (t , τ ) (3.8) + for all t ⩾ τ , t ∈ T where x(t , τ , Px0 ) is the solution of (3.1) with the initial value x(τ , τ ) = Px0 Consider the implicit dynamic equation on time scale T Ax (t ) = Bx(t ), (3.1) where x(t ) ∈ K , and {A, B} ∈ K are constant matrices; underlying field K is either real or complex We assume that the pencil of matrices {A, B} is regular (that is, det(λA − B) ̸≡ 0) and the index of {A, B} is k ⩾ The Kronecker decomposition of the pencil of matrices {A, B} indicates that there exists a pair of nonsingular matrices W , T such that m×m m A = W diag(Ir , U )T −1 , B = W diag(B1 , Im−r )T −1 , (3.2) where Ir is the unit matrix in Kr ×r and B1 is a matrix in Kr ×r Further, U ∈ K(m−r )×(m−r ) is a nilpotent matrix whose nilpotency degree is exactly k Denote holds for some α > such that −α ∈ R+ If the constant N can be chosen independent of τ then the implicit dynamic equation (3.1) is called uniformly exponentially stable We denote by σ (C , D) the spectrum of the pencil {C , D}, i.e., the set of all solutions of the equation det(λC − D) = When C = I, we write simply σ (D) for σ (I , D) Theorem 3.2 The implicit dynamic equation (3.1) is uniformly exponentially stable if and only if σ (A, B) ⊂ S, where S is the domain of the uniformly exponential stability of the time scale T y(t ) z (t ) Q = T diag(0r , Im−r )T −1 , Proof Let x(t ) = T P = Im − Q = T diag(Ir , 0m−r )T −1 decomposition (3.2) and (3.3) we get (3.3) It is known that for any α ∈ K such that α A + B is nonsingular, one has Km = ker[(α A + B)−1 A]k ⊕ im[(α A + B)−1 A]k , A1 = A − B Q = W diag(Ir , U − Im−r )T −1 k (3.4) Since U is a nilpotent matrix, it is clear that A1 is invertible Further, −1 by using (3.2) and (3.3) it follows that PA− A = A1 AP = P and − − − 1 PA1 B = A1 B P = PA1 B P Multiplying both sides of (3.1) by PA− − and Q A1 respectively we obtain ( Px) (t ) = PA1 B( Px)(t ), ( Q A−1 Ax) (t ) = Q A−1 Bx(t ) −1 1 (3.5) y(t ) By PA−1 B = T diag(B1 , 0m−r )T −1 From (3.5) and (3.6) it follows that Eq (3.1) is equivalent to and Q is the projection onto ker[(α A + B) A] along the space im[(α A + B)−1 A]k In particular, Q does not depend on the choice of W and T Let −1 Then, we have Px(t ) = T y (t ) = B1 y(t ), z (t ) ≡ (3.9) Therefore, Eq (3.1) is uniformly exponentially stable if and only if the linear equation y (t ) = B1 y(t ) is so By Theorem 2.3, this is equivalent to σ (B1 ) ⊂ S On the other hand, λA − B = W diag(λIr − B1 , λU − Im−r )T −1 This implies that det(λA − B) = ⇐⇒ det(λIr − B1 ) = Thus, σ (A, B) = σ (B1 ) and the uniformly exponential stability of Eq (3.1) is equivalent to σ (A, B) ⊂ S The proof is complete N.H Du et al / Systems & Control Letters 60 (2011) 596–603 599 Now, we consider Eq (3.1) subjected to general structured perturbations of the form Conversely, take ϵ > and a λ0 ∈ C \ S satisfying Ax (t ) = Bx(t ), (3.10) −1 ⩽ ‖Eλ0 (λ0 A − B)−1 D‖ (3.11) Following the same argument as in [15], we find a vector u ∈ Cl satisfying ‖u‖ = and with [ A, B] = [A, B] + DΣ E , sup ‖Eλ (λA − B)−1 D‖ λ∈C\S where D ∈ Km×l , E ∈ Kq×2m , the perturbation Σ ∈ Kl×q The matrix DΣ E is called a structured perturbation of the Eq (3.1) If we let E = [E1 , E2 ] with E1 , E2 ∈ Kq×m then (3.11) is equivalent to ‖Eλ0 (λ0 A − B)−1 Du‖ = ‖Eλ0 (λ0 A − B)−1 D‖ A = A + D Σ E1 , y∗ (Eλ0 (λ0 A − B)−1 Du) = ‖Eλ0 (λ0 A − B)−1 Du‖ B = B + D Σ E2 A + D A ΣA E A , B = ‖Eλ0 (λ0 A − B)−1 D‖ Consider B + DB ΣB EB , where EA ∈ Cq1 ×m , EB ∈ Cq2 ×m , DA = DB ∈ Cm×l , can be rewritten in the form (3.11) with D = DA = DB , Σ = [ΣA , ΣB ], E = diag(EA , EB ) We define −1 ∗ uy , Σ = − ‖Eλ0 (λ0 A − B)−1 D‖ (3.13) x = (λ0 A − B)−1 Du It is clear that ΞK = {Σ ∈ K −1 ‖u‖‖y∗ ‖ ‖Σ ‖ ⩽ ‖Eλ0 (λ0 A − B)−1 D‖ −1 , = ‖Eλ0 (λ0 A − B)−1 D‖ Definition 3.3 The stability radius of Eq (3.1) under structured perturbations of the form (3.11) is defined by and l× q : Eq (3.10) is either irregular or not uniformly exponentially stable} rK (A, B; D, E ) = inf{‖Σ ‖ : Σ ∈ ΞK }, λI Let us use the notation Eλ = E −Im We have the following m theorem Theorem 3.4 The complex stability radius of Eq (3.1) under structured perturbations of the form (3.11) is given by the formula rC (A, B; D, E ) = sup ‖Eλ (λA − B)−1 D‖ −1 λ∈∞∪∂ S (3.12) Proof Let Σ ∈ Cl×q be such that the perturbed equation (3.10) is irregular or it is regular but not uniformly exponentially stable In both cases, we can always choose an eigenvalue λ0 ∈ σ ( A, B)∩(C \ S ) and an eigenvector x ̸= corresponding to λ0 , i.e., (λ0 A − B)x = From (3.11) this yields [ ] [ ] λ I λ I λ0 A − B = [ A, B] m = ([A, B] + DΣ E ) m −I m −I m = λ0 A − B + DΣ Eλ0 −u ‖Eλ (λ0 A − B)−1 D‖ = −u ‖Eλ0 (λ0 A − B)−1 D‖ (3.14) Since u ̸= 0, −1 ‖Σ ‖ ⩾ ‖Eλ0 (λ0 A − B)−1 D‖ Combining these inequalities we obtain −1 ‖Σ ‖ = ‖Eλ0 (λ0 A − B)−1 D‖ Furthermore, from (3.13) and (3.14) it follows that (λ0 A − B + DΣ Eλ0 )x = 0, i.e., λ0 ∈ σ ( A, B), with [ A, B] = [A, B] + DΣ E, which implies that the equation Ax (t ) = Bx(t ) is either irregular or not uniformly exponentially stable This means that Σ ∈ ΞC which implies rC (A, B; D, E ) ⩽ ‖Σ ‖ = ‖Eλ0 (λ0 A − B)−1 D‖ ⩽ sup ‖Eλ (λA − B)−1 D‖ −1 −1 λ∈C\S + ϵ (3.15) Since ϵ is arbitrary, Therefore, (λ0 A − B)x = −DΣ Eλ0 x rC (A, B; D, E ) = This relation implies Eλ0 x = −Eλ0 (λ0 A − B)−1 DΣ Eλ0 x sup ‖Eλ (λA − B) −1 λ∈C\S −1 D‖ Note that the function G(λ) = Eλ (λA − B)−1 D is analytic on C \ S By the maximum principle, ‖G(·)‖ either reaches its maximum value on the boundary ∂ S of S or supλ∈C\S ‖G(λ)‖ = limλ→∞ ‖G(λ)‖ Thus, we obtain Since Eλ0 x ̸= 0, −1 ‖Σ ‖ ⩾ ‖Eλ0 (λ0 A − B)−1 D‖ −1 −1 ⩾ sup ‖Eλ (λA − B) D‖ rC (A, B; D, E ) = λ∈C\S sup ‖Eλ (λA − B)−1 D‖ λ∈∞∪∂ S −1 The proof is complete Thus, rC (A, B; D, E ) ⩾ Σ Eλ0 (λ0 A − B)−1 Du = where ‖ · ‖ can be any vector-induced matrix norm + ϵ Let y∗ be a linear functional defined on Cl such that ‖y∗ ‖ = and It is easy to see that the perturbed model of the form A −1 sup ‖Eλ (λA − B)−1 D‖ λ∈C\S −1 Corollary 3.5 The complex stability radius of Eq (3.1) under the structured perturbation of the form 600 N.H Du et al / Systems & Control Letters 60 (2011) 596–603 Ax (t ) = (B + DB Σ EB )x(t ), (3.16) rC (B; DB , EB ) = A= −1 sup ‖EB (λA − B) DB ‖ , −1 (3.17) λ∈∞∪∂ S (A + DA Σ EA )x (t ) = Bx(t ), (3.18) is given by sup ‖λEA (λA − B) λ∈∞∪∂ S −1 −1 DA ‖ k=0 2 −1 −1 3 D= and under the structured perturbation of the form rC (A; DA , EA ) = 0 is given by [2k, 2k + 1], −2 , B = −1 −1 ∞ where T = 0 , −1 −1 , 1 0 −1 E = [E1 , E2 ] = 1 0 −2 −3 0 Since T = k=0 [2k, 2k + 1], S = {λ ∈ C : Re λ+ ln |λ+ 1| < 0} It is easy to see that ind(A, B) = and σ (A, B) = − 13 Therefore, the pencil {A, B} is exponentially stable When λ ∈ ∂ S, by the direct computations, we obtain ∞ (3.19) Proof With D = DB and E = [E1 , E2 ] = [0, EB ], the perturbation (3.16), we can write [ A, B] = [A, B] + DΣ E , λI Further, Eλ = E −Im = −EB and by Theorem 3.4, we get (3.17) m For the perturbation (3.18), we choose D = DA and E = λIm [E1 , E2 ] = [EA , 0] By seeing that Eλ = E −Im = λEA we get λ+1 λ−1 = 3λ + −λ − (λA − B)−1 Q = 3 −3 and therefore, Theorem 3.6 (a) rC (B; DB , EB ) > if and only if the polynomial p(λ) = EB Q (λA − B)−1 DB is constant (b) rC (A; DA , EA ) > if and only if the polynomial q(λ) = λEA Q (λA − B)−1 DA is constant (c) Let E = [E1 , E2 ] Then rC (A, B; D, E ) > if and only if the polynomial s(λ) = (λE1 − E2 ) Q (λA − B)−1 D is constant Eλ (λA − B) Proof (a) We have EB (λA − B)−1 DB = EB P (λA − B)−1 DB + EB Q (λA − B)−1 DB −1 It is easy to prove that limλ→∞ −‖1EB P (λA − B) DB ‖ = limλ→∞ −1 ‖EB T diag (λI − B1 ) , 0m−r W DB ‖ = Moreover, since U k = 0, −1 = EB T diag 0r , (λU − Im−r ) W −1 D B k−1 − = EB T diag 0r , − (λU )i W −1 DB , −1 −2 , D= −12 3λ + −1 −12 −12 6 ‖Eλ (λA − B)−1 D‖∞ = 36 |3λ + 1| Corollary 3.7 Let ind(A, B) = Then, rC (A, B; D, E ) > if and only if rC (A; D, E1 ) > where E = [E1 , E2 ] , where ‖ · ‖∞ is the operator’s norm induced by ‖ · ‖v∞ This implies that sup ‖Eλ (λA − B)−1 D‖∞ = ‖E0 (−B)−1 D‖∞ = 36 λ∈∞∪∂ S Thus, we get 36 Moreover, we see that [A, B] A˜ = A + DΣ E1 , B˜ = B + DΣ E2 , A [A˜ , B˜ ] = [A, B] + DΣ E ⇐⇒ B i =0 where T , W and U as mentioned in (3.2) and Q = T diag(0r , Im−r ) T −1 , P = Im − Q = T diag(Ir , 0m−r )T −1 Hence, limλ→∞ ‖EB (λA − B)−1 DB ‖ exists and it equals ∞ if p(λ) is not constant Thus, we get (a) A similar argument can be applied to prove (b) and (c) The proof is complete λ −1 , λ Let ‖ · ‖v∞ be the maximum norm of C3 We have rC (A, B; D, E ) = p(λ) = EB Q (λA − B)−1 DB −λ2 −λ2 − λ , λ2 + 3λ + λ Eλ = λE1 − E2 = λ −λ λ (3.19) −1 −1 and the polynomials q(λ) = λE1 Q (λA − B) −1 D= p(λ) = E2 Q (λA − B)−1 D 3λ 3λ λ+2 = λ+2 2λ − 2λ − 3λ 3λ 3λ 2λ 2λ 2λ λ 3λ λ+2 2λ − λ λ ̸= constant, ̸= constant Therefore When A = I, (3.17) has been proved in [12] in order to unify the continuous and discrete stability radii of the linear dynamic equations In case T = R, it is seen that S = C− and we get (1.7) If T = N, S = {ω : |1 + ω| < 1} and (1.9) is deduced rC (A; D, E1 ) = rC (B; D, E2 ) = Example 3.8 Let us calculate the stability radius of the equation Ax (t ) = Bx(t ) under the structured perturbation of the form Now, we consider the problem when the real and complex stability radii are equal It seems that this is a difficult problem in the implicit dynamic equations on time scales because in this case, the positive cone Rm + is no longer invariant under the action of [A, B] [A˜ , B˜ ] = [A, B] + DΣ E , The equality of real and complex radii N.H Du et al / Systems & Control Letters 60 (2011) 596–603 the pencil of matrices {A, B}; even when both A and B are positive Moreover, the domain of uniformly exponential stability S has the property that although λ ∈ ∂ S, Re λ ∈ S which means we cannot use the approach in [4] However, we are able to answer this question under some assumptions Let us consider Eq (3.1) subjected to structured perturbations (3.10) Firstly, we prove the following result which provides a difference between ordinary dynamic equations and implicit dynamic equations Denoting G(λ) = Eλ (λA − B)−1 D, we have: Theorem 4.1 If ‖G(λ)‖ does not reach its maximum at a finite point on ∂ S then 601 When ‖G(λ)‖ attains its maximum value at a finite point in ∂ S, we need further assumptions A matrix M = (mij ) ∈ Rk×p is said to be positive, written as M ⩾ 0, if mij ⩾ for any i, j We define a partial order relation in Rk×p by M ⩾ N ⇔ M − N ⩾ We define the absolute value of a matrix M = (mij ) as the matrix |M | = (|mij |); similarly for a vector x we use the notation |x| = (|x1 |, |x2 |, , |xp |) Let ρ(C , D) be the spectral radius of the pencil of matrices {C , D}, i.e., ρ(C , D) := max{|λ| : λ ∈ σ (C , D)} Consider the implicit dynamic equation on time scale with structured perturbations of the form Ax (t ) = (B + DB Σ EB )x(t ), (4.2) rC (A, B; D, E ) = rR (A, B; D, E ) where A, B ∈ Rm×m , DB ∈ Rm×l , and EB ∈ Rq×m Suppose that ind(A, B) = Then, Q is the projection onto ker A along the space Proof It is clear that rC (A, B; D, E ) ⩽ rR (A, B; D, E ) Therefore, it is sufficient to prove that there exists a sequence of disturbances {Σn } ⊂ ΞR such that S = {y ∈ Rm : By ∈ im A} It is seen Recall that A1 = A − B Q is nonsingular, A Q = and P = A− A Moreover, from Perron–Frobenius extension theorem (see [22]) we −1 have ρ(A, B) = ρ(A− BP ) and if A1 BP ⩾ then ρ(A, B) is an eigenvalue of the pencil of matrices {A, B} −1 Let B = A− BP = PA1 B From (3.2)–(3.4), it follows that G(λ) = Eλ P (λA − B)−1 D + Eλ Q (λA − B)−1 D (λA − B)−1 = T diag (λIr − B1 )−1 , −Im−r W −1 lim ‖Σn ‖ ⩽ rC (A, B; D, E ) n→∞ −1 = (λI − B)−1 PA− + Q A1 By using the Kronecker decomposition (3.2), we get (4.3) With the perturbed equation (4.2) then G(λ) = EB (λA − B) The relation (4.3) implies that k−1 − Eλ Q (λA − B)−1 D = Eλ T diag 0r , − (λU )i W −1 D, −1 DB i=0 Q A− G(∞) = lim G(λ) = EB DB and λ→∞ Eλ P (λA − B)−1 D = Eλ T diag (λIr − B1 )−1 , 0m−r W −1 D Assume that E = [E1 , E2 ] with E1 , E2 ∈ Rm×m Then, Eλ = λE1 − E2 , and it follows that lim Eλ P (λA − B)−1 D = E1 T diag (Ir , 0m−r ) W −1 D λ→∞ = E1 PA− D, (4.1) and k−1 − i Eλ T diag 0r , − (λU ) W −1 D = (λE1 − E2 )T diag 0r , − k−1 − (λU )i W −1 D i=0 is a polynomial in λ Therefore the limit limλ→∞ ‖G(λ)‖ exists (possibly +∞) Since ‖G(λ)‖ does not reach its maximum at a finite point on ∂ S, it follows that rC (A, B; D, E )−1 = sup ‖G(λ)‖ = lim ‖G(λ)‖ λ∈∞∪∂ S For α ⩾ 0, we define the ball Bα (−α) = {z ∈ C : |z + α| < α} For a time scale with bounded graininess several essential features are captured by an associated characteristic ball The analysis of positive linear equations show that Bη (−η) ⊂ S, with S is the domain of uniform exponential stability of the time scale T, and η is defined by η= sup{µ(t ) : t ∈ T} , (4.5) see, e.g [12] i=0 (4.4) λ→∞ This implies that rC (A, B; D, E )−1 = limn∈N;n→∞ ‖G(n)‖ For any n ∈ N, let un ∈ Rl be a vector with ‖un ‖ = : ‖G(n)un ‖ = ‖G(n)‖; let y∗n be a linear functional defined on Rq with ‖y∗n ‖ = and y∗n (G(n)un ) = ‖G(n)un ‖ as in Theorem 3.4 By denoting Σn = ‖G(n)‖−1 un y∗n we see that n is an eigenvalue of the pencil { A, B} with [ A, B] = [A, B] + DΣn E and the corresponding eigenvector xn = (nA − B)−1 Dun Note that Σn is indeed a real perturbation Therefore, Σn ∈ ΞR Further, ‖Σn ‖ = ‖G(n)‖−1 ‖un y∗n ‖ ⩽ ‖G(n)‖−1 and ‖Σn (G(n)un )‖ = ‖G(n)‖−1 ‖un y∗n (G(n)un )‖ = ‖un ‖ = which implies that ‖Σn ‖ = ‖G(n)‖−1 and limn→∞ ‖Σn ‖ = limn→∞ ‖G(n)‖−1 = rC (A, B; D, E ) This relation says that rR (A, B; D, E ) ⩽ rC (A, B; D, E ) The proof is complete 1 Hypotheses 4.2 (i) A− ⩾ 0, EB P ⩾ and EB Q A− DB DB ⩾ (ii) There exists α ⩾ with Bα (−α) ⊂ S such that B + α P ⩾ We need the following simple lemma Lemma 4.3 Suppose that the bounded linear operator triplet: M : X → Y , N : Y → Z , P : Z → X is given, where X , Y , Z are Banach spaces Then the operator I − MNP is invertible if and only if I − NPM is invertible, moreover, (I − NPM)−1 = I + NP(I − MNP)−1 M Proof Suppose that I − MNP is invertible By direct calculation, it is easy to verify that the inverse of I − NPM is (I − NPM)−1 = I + NP(I − MNP)−1 M (4.6) Furthermore, if (I − MNP) is bounded then so is (I − NPM)−1 The converse is proved similarly −1 Theorem 4.4 Assume that the pencil of matrices {A, B} has ind(A, B) = and satisfies Hypotheses 4.2 Then, we have rC (B; DB , EB ) = rR (B; DB , EB ) (4.7) 602 N.H Du et al / Systems & Control Letters 60 (2011) 596–603 Proof Clearly, it is sufficient to prove that rC (B; DB , EB ) ⩾ rR (B; DB , EB ) If ‖G(λ)‖ does not reach its maximum value at a finite point on ∂ S then from Theorem 4.1 it follows that rC (B; DB , EB ) = rR (B; DB , EB ) Else, let λ0 ∈ C satisfy ‖G(λ0 )‖ = sup∂ S ‖G(λ)‖ and the perturbation Σ , given by (3.13), destroy stability We aim to show • Σ can be the complex perturbation, but |Σ | is the real perturbation making the dynamic equation unstable, • furthermore ‖|Σ |‖ = ‖Σ ‖ A− 1,Σ (B + α A + DB Σ EB )P −1 (A−1 (B + α A + DB Σ EB ) P) = (I − A− 1 DB Σ EB Q ) 1 −1 ( = (I − A− B + α P + A− DB Σ EB Q ) DB Σ EB P ) −1 Above we use the identities B = A− BP and A1 A = P Using (4.6) −1 with M = EB Q , N = I , P = A1 DB Σ , we get (4.8) −1 −1 EB Q = I + A− DB Σ (I − EB Q A1 DB Σ ) −1 = I + A− EB Q DB Σ (I − G(∞)Σ ) A− 1,Σ (B + α A + DB Σ EB )P = B + α P + A− DB Σ EB P + Φ (Σ ) Similarly, we also have A1,Σ = A − (B + DB Σ EB ) Q A− 1,|Σ | (B + α A + DB |Σ |EB )P = B + α P + A− DB |Σ |EB P + Φ (|Σ |) From (3.4) it follows that (4.9) Applying Lemma 4.3 with M = EB Q , N = I , P = A− DB Σ we get −1 I − A1 DB Σ EB Q is invertible This implies that A1,Σ is invertible as well Now, we will prove that σ (A, B + α A + DB Σ EB ) ∪ {0} = σ (A− 1,Σ (B + α A + DB Σ EB )P ) (4.10) Moreover, by Hypotheses 4.2 and the inequality ‖G(∞)‖‖Σ ‖ = ‖G(∞)‖‖|Σ |‖ < 1, it follows that −1 G(∞)Σ |EB P |Φ (Σ )| = A− DB |Σ (I − G(∞)Σ ) − ∞ = A− (G(∞)Σ )i+1 EB P DB Σ i =0 ∞ − −1 ⩽ A DB |Σ | (G(∞)|Σ |)i+1 EB P i =0 Indeed, for any λ ̸= 0, because of the properties A Q = P Q = 0, Q Q = Q , P + Q = I, we have det(λI + A− 1,Σ (B + α A + DB Σ EB )P ) = |A− 1,Σ (B + α A + DB Σ EB )P | − ⩽ | B + α P | + |A DB Σ EB P | + |Φ (Σ )| ⩽ B + α P + A1 DB |Σ |EB P + Φ (|Σ |) −1 ⇐⇒ det[(A − (B + DB Σ EB ) Q )(λ P − Q) + (B + α A + DB Σ EB ) P] = ⇐⇒ det[A(λ P − Q ) + (B + DB Σ EB ) Q + (B + α A + DB Σ EB ) P] = ⇐⇒ det[λA( P + Q ) + (B + α A + DB Σ EB )( P + Q) − (1 + α + λ)A Q] = ⇐⇒ det[λA + B + α A + DB Σ EB ] = = A− 1,|Σ | (B + α A + DB |Σ |EB )P ρ(A− 1,Σ (B + α A + DB Σ EB )P ) ⩽ ρ(A1,|Σ | (B + α A + DB |Σ |EB ) P ) := β −1 (4.17) Since A− 1,|Σ | (B+α A+DB |Σ |EB )P ⩾ 0, by Perron–Frobenius theorem, (4.11) Similarly, since ‖|Σ |‖ = ‖Σ ‖ < ‖G(∞)‖−1 , σ (A, B + α A + DB |Σ |EB ) ∪ {0} (4.12) and ρ(A, B + α A + DB |Σ |EB ) = ρ(A− 1,|Σ | (B + α A + DB |Σ |EB )P ) (4.16) From theory of nonnegative matrices, see, e.g [23], it follows that This implies that the spectral equality (4.10) holds In particular, = σ (A− 1,|Σ | (B + α A + DB |Σ |EB )P ), = A1 DB |Σ |(I − G(∞)|Σ |)−1 G(∞)|Σ |EB P = Φ (|Σ |), −1 and therefore, ⇐⇒ det(λA1,Σ + (B + α A + DB Σ EB ) P) = Q ⇐⇒ det λA1,Σ + (B + α A + DB Σ EB )P P − =0 λ ρ(A, B + α A + DB Σ EB ) = ρ(A− 1,Σ (B + α A + DB Σ EB )P ) (4.15) −1 Define Φ (Σ ) = A− G(∞)Σ EB P Then, from DB Σ (I − G(∞)Σ ) (4.14) and (4.15), because of the property Q B= Q P = 0, we obtain Q A− Further, the inequality ‖Σ ‖‖EB DB ‖ = ‖Σ ‖‖G(∞)‖ < − implies that the matrix I − EB Q A1 DB Σ is invertible Define A1,Σ = A1 (I − A− DB Σ EB Q ) (4.14) −1 (I − A− DB Σ EB Q ) It is seen that ‖Σ ‖ = ‖G(λ0 )‖−1 < ‖G(∞)‖−1 Moreover, Σ has rank one which implies that ‖|Σ |‖ = ‖Σ ‖ Since Bα (−α) ⊂ S and the pencil of matrices {A, B + DB Σ EB } is unstable, it follows that σ (A, B + DB Σ EB ) ̸⊂ S and ρ(A, B + α A + DB Σ EB ) = max {|z + α| : z ∈ σ (A, B + DB Σ EB )} ⩾ α From (4.9) it follows that β is an eigenvalue of the matrix A− 1,|Σ | (B + α A + DB |Σ |EB )P with maximum module and by (4.12) it follows that β ∈ σ (A, B + α A + DB |Σ |EB ) Therefore, by (4.11), (4.8) and (4.17), we obtain ⩽ β −α ∈ σ (A, B + DB |Σ |EB ) Thus the perturbation |Σ | ∈ Rl×q , with ‖|Σ |‖ = ‖Σ ‖, destroys stability which implies rC (B; DB , EB ) ⩾ rR (B; DB , EB ) The proof is complete We consider the case where Eq (3.1) is positive, i.e., for any x0 ∈ Rm + , the solution x(t ) of Eq (3.1) with P (x(t0 ) − x0 ) = satisfies the condition x(t ) ⩾ for all t ∈ T, t ⩾ t0 It is known that if x(t ) is a solution of (3.1) then Q x(t ) = which implies Px(t ) = x(t ) for all t ∈ T, t ⩾ t0 Therefore, (3.1) can be rewritten (4.13) x = Bx, x(t0 ) = Px0 , (4.18) N.H Du et al / Systems & Control Letters 60 (2011) 596–603 where B = (A − B Q )−1 B P = P (A − B Q )−1 B It is easy to see that Eq (4.18) has a general solution ∏ x(t ) = x(t , t0 , x0 ) = ( P + µ(s) B) exp{mes [t0 , t ) B} Px0 , t0 ⩽s