DSpace at VNU: On data dependence of stability domains, exponential stability and stability radii for implicit linear dy...
Math Control Signals Syst (2016) 28:13 DOI 10.1007/s00498-016-0164-7 ORIGINAL ARTICLE On data dependence of stability domains, exponential stability and stability radii for implicit linear dynamic equations Nguyen Thu Ha1 · Nguyen Huu Du2 · Do Duc Thuan3 Received: 18 July 2015 / Accepted: April 2016 © Springer-Verlag London 2016 Abstract We shall deal with some problems concerning the stability domains, the spectrum of matrix pairs, the exponential stability and its robustness measure for linear implicit dynamic equations of arbitrary index First, some characterizations of the stability domains corresponding to a convergent sequence of time scales are derived Then, we investigate how the spectrum of matrix pairs, the exponential stability and the stability radii for implicit dynamic equations depend on the equation data when the structured perturbations act on both the coefficient of derivative and the right-hand side Keywords Implicit dynamic equations · Time scales · Convergence · Stability domain · Spectrum · Exponential stability · Stability radius Mathematics Subject Classification B 06B99 · 34D99 · 47A10 · 47A99 · 65P99 Do Duc Thuan thuan.doduc@hust.edu.vn Nguyen Thu Ha ntha2009@yahoo.com Nguyen Huu Du dunh@vnu.edu.vn Department of Basic Science, Electric Power University, 235 Hoang Quoc Viet Str., Hanoi, Vietnam Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Dai Co Viet Str., Hanoi, Vietnam 123 13 Page of 28 Math Control Signals Syst (2016) 28:13 Introduction In this paper, we study the stability domains, the spectrum of matrix pairs, the exponential stability and the stability radii for sequence of implicit dynamic equations on time scales of the form (1.1) An x n (t) = Bn x(t), where An , Bn ∈ Cm×m , n ∈ N and t ∈ Tn The leading coefficients An , n ∈ N, are allowed to be singular matrices Implicit dynamic equations of the form (1.1) can be considered as a unified form between linear differential algebraic equations (DAEs) and linear implicit difference equations Therefore, they play an important role in mathematical modeling arising in multibody mechanics, electrical circuits, prescribed path control, chemical engineering, etc., see [7,8,11,31] It is well known that, due to the fact that the dynamics of (1.1) are constrained, some extra difficulties appear in the analysis of stability as well numerical treatments of implicit dynamic equations These difficulties are typically characterized by index concepts, see [8,23,31] In recent years, the theory of dynamic systems on an arbitrary time scale, which is an nonempty closed subset of the real numbers, has been found promising because it demonstrates the interplay between the theories of continuous-time and discretetime systems, see [1,3,13,24,25] It enables us to analyze the stability of dynamical systems on non-uniform time domains [34] Based on this theory, stability analysis on time scales has been studied for linear time-invariant systems [32], linear time-varying dynamic equations [12], implicit dynamic equations [19,36], switched systems [34,35] and finite-dimensional control systems [3,4,14] It is well known that the spectral characterizations for the exponential stability of dynamic systems and the stability of numerical methods for approximating solutions relate to the stability domains of scalar equations Therefore, it is meaningful to investigate the behavior of the stability domains when a sequence of time scales of equations converges On the other hand, in a lot of applications there is a frequently arising question, namely, how robust is a characteristic qualitative property of a system (e.g., stability) when the system comes under the effect of uncertain perturbations because there are parameters which can be determined only by experiments or the remainder part ignored during linearization process That is the reason why we are interested in investigating the robust stability of the uncertain equations subjected to general structured perturbations of the form (1.2) An x n (t) = Bn x(t), with [ An , Bn ] [An , Bn ] + Dn n En , (1.3) where n , n ∈ N, are unknown disturbance matrices; Dn , E n are known scaling matrices defining the “structure” of the perturbations This leads to the notion of stability radius introduced firstly by Hinrichsen and Pritchard [26] The so-called stability radius is defined by the largest bound r such that the stability is preserved for all perturbations of norm strictly less than r Results on stability radii of time-invariant linear systems are derived in [15,26,27,29] Results for the robust stability of time- 123 Math Control Signals Syst (2016) 28:13 Page of 28 13 varying systems can be found in [28,30] Therefore, it is natural to extend the notion of the stability radius to implicit dynamic equations This problem has been solved for implicit dynamic equations on time scale R (described by DAEs), see [5,7,9,16–18] In [10], Chyan, Du and Linh have investigated the data dependence of the exponential stability and the stability radii for linear time-varying DAEs of index and with respect to only the right-hand side perturbations Recently, Du et al [19] have derived the formula of stability radius for linear implicit dynamic equations with arbitrary index subjected to general structured perturbation acting on both the coefficient of derivative and the right-hand side Therefore, it is meaningful to continue studying the data dependence of the exponential stability and the stability radii for these equations The first aim of this paper is to study the relationship between the stability domains corresponding to a convergent sequence of time scales Then, we continue to analyze how the spectrum of matrix pairs and the exponential stability of (1.1) depend on data when (An , Bn ; Tn ) tends to (A, B; T) Finally, we will investigate the convergence of the stability radii of Eq (1.2) with general structured perturbations of the form (1.3) when the data (An , Bn ; Dn , E n ; Tn ) tends to (A, B; D, E; T) This fact plays an important role in the calculation of stability radii because in practice we need to approximate them As a corollary, we will show that the stability radii of implicit difference equations obtained from DAEs by the explicit Euler methods will tend to the stability radius of DAEs when the mesh step tends to zero The paper is organized as follows In the next section, we summarize some preliminary results on time scales and the exponential stability In Sect 3, we derive some characterizations of the stability domains corresponding to convergent sequences of time scales Section deals with the data dependence of the spectrum of matrix pairs and the exponential stability In Sect 5, the data dependence of the stability radii is analyzed The last section gives some conclusions and open problems Preliminaries For the reader’s sake, in this section we recall some basic notations, main definitions as well as some well-known properties regarding time scale calculus, (see e.g., [1,13, 19,25,32]) Let T be a closed subset of R, enclosed with the topology inherited from the standard topology on R Let σ (t) = inf{s ∈ T : s > t}, μ(t) = σ (t) − t and ρ(t) = sup{s ∈ T : s < t}, ν(t) = t − ρ(t) (supplemented by sup ∅ = inf T, inf ∅ = sup T) 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, left-scattered if ρ(t) < t and isolated if t is simultaneously right-scattered and left-scattered T is said to be the time scale with bounded graininess if supt∈T μ(t) < ∞ A function f defined on T is regulated if there exist a left-sided limit at every leftdense point and a right-sided limit at every right-dense point A regulated function is called r d-continuous if it is continuous at every right-dense point, and ld-continuous if it is continuous at every left-dense point A function f from T to R is positively regressive if + μ(t) f (t) > for every t ∈ T Use R+ to denote the set of positively regressive functions from T to R 123 13 Page of 28 Math Control Signals Syst (2016) 28:13 A function f : T → Cd is called delta differentiable at t if there exists a vector |σ (t) − s| for f (t) such that for all > f (σ (t)) − f (s) − f (t)(σ (t) − s) all s ∈ (t − δ, t + δ) ∩ T and for some δ > The vector f (t) is called the delta derivative of f at t If T = R then the delta derivative is f (t) from continuous calculus If T = Z then the delta derivative is the forward difference, f (t) = f (t + 1) − f (t) from discrete calculus A function F : T → Cd is called antiderivative of f : T → Cd provided F (t) = f (t) for all t ∈ T Let a, b ∈ T It is known that if f is a r d-continuous function, its antiderivative exists Therefore, we can define the delta integral of f on [a, b)T by b a f (s) T s = f (s) T s = F(b) − F(a) [a,b)T b b (see [22]) In case a, b ∈ / T, writing a f (s) T s means a f (s) T s, where a = min{t > a : t ∈ T}; b = max{t < b : t ∈ T} If there is no confusion, we simply b b b b write a f (s) s (resp a f (s) n s) for a f (s) T s (resp a f (s) Tn s) Let T be an unbounded above time scale, that is sup T = ∞ For any λ ∈ C, the solution of the dynamic equation x (t) = λx(t), t t0 , s (2.1) with the initial condition x(s) = 1, defines a so-called exponential function with the parameter λ We denote this exponential function by eλ (t, s) The exponential function with parameter λ can be presented by the formula eλ (t, s) = exp t s h lim μ(τ ) Ln(1 + hλ) τ , h where Ln a is the principal logarithm of the number a Since |eλ (t, s)| = exp t s h ln |1 + hλ| τ , μ(τ ) h lim we can rewrite |eλ (t, s)| = exp t s ζλ (μ(τ )) τ (2.2) with ζλ (s) = lim h s ln|1 + hλ| = h λ if s = if s = ln |1+sλ| s Note that |eλ (t, s)| = |eλ (t, s)| for any λ ∈ C Since ζλ (x) |λ| for all x exp{|λ|(t − s)} Moreover, for α > with −α ∈ R+ , there exists a 0, |eλ (t, s)| 123 Math Control Signals Syst (2016) 28:13 Page of 28 constant c > such that ζ−α (x) Therefore, for all t s, t, s ∈ T, e−α (t, s) −α ζ−α/c (x) for all x : exp{−α(t − s)} 13 x supt∈T μ(t) e−α/c (t, s) (2.3) For the other properties of exponential function eλ (t, s) the interested readers can refer to [6] Let Tt0 = {t ∈ T : t t0 } Consider the dynamic equation x = f (t, x), t ∈ Tt0 , (2.4) where f : Tt0 ×Cm → Cm For each s ∈ Tt0 , x0 ∈ Cm , a function t → x(t, s, x0 ), t s, is called a solution of (2.4) if x(·, s, x0 ) is delta differentiable and satisfies (2.4) with the initial condition x(s, s, x0 ) = x0 It is known that possible smoothness requirements of f guarantee existence of a unique solution of Cauchy problem for (2.4), for example, f satisfies the Lipschitz condition (see, e.g., [1,6]) Definition 2.1 (Exponential stability) The dynamic Eq (2.4) is called uniformly exponentially stable if there exists a constant α > with −α ∈ R+ and K > such that for every s t, s, t ∈ Tt0 , the inequality x(t, s, x0 ) K x0 e−α (t, s) (2.5) holds for any x0 ∈ Rm Beside this definition, one can use the classical exponential function exp{−α(t − s)} in (2.5) However, it is easy to prove that they are equivalent In the linear homogeneous case, i.e., f (t, x) = Ax, it is known that Eq (2.4) is uniformly exponentially stable if and only if the scalar Eq (2.1) is uniformly exponentially stable for any λ ∈ σ (A) Fix t0 ∈ R Let T be the set of all time scales with bounded graininess such that t0 ∈ T for all T ∈ T We endow T with the Hausdorff distance, i.e., Hausdorff distance between two time scales T1 and T2 , which is defined by d H (T1 , T2 ) := max sup d(t1 , T2 ), sup d(t2 , T1 ) , t1 ∈T1 (2.6) t2 ∈T2 where d(t1 , T2 ) = inf |t1 − t2 | and d(t2 , T1 ) = inf |t2 − t1 | t2 ∈T2 t1 ∈T1 For properties of the Hausdorff distance, we refer the interested readers to [2,33] Let {Tn } be a sequence of time scales with Tn ∈ T and T ∈ T Then the limit lim Tn = T is understood that lim d H (Tn , T) = n→∞ n→∞ 123 13 Page of 28 Math Control Signals Syst (2016) 28:13 Stability domains In this section, we derive some characterizations of the stability domains and the Lyapunov exponents corresponding to a convergent sequence of time scales This is a preparation for investigation of the data dependence of the exponential stability and the stability radii for implicit dynamic equations in next sections Denote by UT the set of the complex values λ such that (2.1) is uniformly exponentially stable We call UT the domain of uniformly exponential stability (or stability domain for short) of the time scale T It is known that UT is an open set in C (see, e.g., 0, [19,32]) Since if |b1 | |b2 | then ζa+ib1 (x) ζa+ib2 (x) for any a ∈ R and x so UT is symmetric with respect to the real line on the complex plan and λ ∈ UT implies the segment [λ, λ] ⊂ UT Moreover, if λ then ζλ (x) for all x and hence UT ⊂ C− = {λ ∈ C : λ < 0} For each λ ∈ C, we define the Lyapunov exponent of the scalar function L(λ, T) := lim sup t−s→∞ t −s t s ζλ (μ(τ )) τ (3.1) By the definition, it follows that λ ∈ UT if and only if L(λ, T) < Indeed, if λ ∈ UT then there exist K , α > with −α ∈ R+ such that |eλ (t, s)| K e−α (t, s) and hence, by (2.3), |eλ (t, s)| K exp{−α(t − s)} Therefore, L(λ, T) = lim sup t−s→∞ ln |eλ (t, s)| t −s lim sup t−s→∞ ln(K ) − α(t − s) = −α < t −s To prove the inverse, if L(λ, T) < then there exist H, α > with −α ∈ R+ such that t −s t s ζλ (μ(τ )) τ −α, ∀t, s ∈ T : t − s > H exp{−α(t − s)} for all t, s ∈ T : t − s > H If t − s H Therefore, |eλ (t, s)| |λ| for any x 0, we have |eλ (t, s)| then by virtue of the inequality ζλ (x) exp{|λ|(t −s)} exp{|λ|H } Thus, we choose K = exp{(|λ|+α)H } then |eλ (t, s)| K exp{−α(t − s)} and hence, by (2.3), |eλ (t, s)| K e−α/c (t, s) for all t s, t, s ∈ Tt0 This implies that λ ∈ UT Moreover, we have Lemma 3.1 Let T ∈ T and λ ∈ C\R Then, λ ∈ UT if and only if L(λ, T) Proof Denote μ∗ = sup{μ(t) : t ∈ T} and let λ ∈ UT\R Then, there is a sequence {λn } ⊂ UT such that lim λn = λ Let λ = a + ib with b = and λn = an + ibn n→∞ Using the Lagrange finite increment formula, for all x > 0, we have ζλn (x) − ζλ (x) = 123 x(|λn |2 − |λ|2 ) + 2(an − a) , θ ∈ (0, 1) 2(1 + 2x(a + θ(an − a)) + x (|λ|2 + θ(|λn |2 − |λ|2 ))) Math Control Signals Syst (2016) 28:13 Page of 28 13 Since < + 2xa + x |λ|2 for all x 0, we can choose an n ∈ N and a constant c1 > such that c1 < + 2x(a + θ (an − a)) + x (|λ|2 + θ (|λn |2 − |λ|2 )) for all x μ∗ and n > n Thus, for any > 0, there exists n > n satisfying ζλ (x) − ζλn (x) < , ∀ x μ∗ , ∀ n > n This implies that t t ζλ (μ(τ )) τ < s s ζλn (μ(τ )) τ + (t − s), ∀t0 t, ∀ n > n s Hence, lim sup t−s→∞ t −s t s ζλ (μ(τ )) τ lim sup t−s→∞ t −s t s ζλn (μ(τ )) τ + Since λn ∈ UT, L(λn , T) = lim sup t−s→∞ t −s t s ζλn (μ(τ )) τ < 0, and it follows that lim sup t−s→∞ t −s t s ζλ (μ(τ )) τ < , ∀ > Thus, L(λ, T) = lim sup t−s→∞ t −s t s ζλ (μ(τ )) τ Conversely, let λ = a + ib ∈ C\R such that L(λ, T) lim sup t−s→∞ t −s t s 0, i.e., ζλ (μ(τ )) τ For any > 0, let λ = a + ib be chosen such that < |b | < |b|; a < a and |λ| > |λ | > |λ| − Since < + 2xa + x |λ|2 , we can choose a and b such that μ∗ < + 2x(a + θ (a − a)) + x (|λ|2 + θ (|λ |2 − |λ|2 )) < c2 for all x Thus, ζλ (x) − ζλ (x) < a −a , ∀0 c2 x μ∗ 123 13 Page of 28 Math Control Signals Syst (2016) 28:13 This implies that t s ζλ (μ(τ )) τ < t s ζλ (μ(τ )) τ + a −a (t − s), ∀t0 c2 s t Hence, L(λ , T) = lim sup t−s→∞ t −s t s ζλ (μ(τ )) τ which follows that λ ∈ UT Since λ → λ as complete a −a 0, there is a δ > and n ∈ N such that any t s ζλ (μn (h)) nh t − s ζλ (μ(h)) h < (t − s) + 8M t −s d H (T, Tn ), (3.2) δ for all n > n , λ ∈ K , t > s, where M = supλ∈K ,x∈[0,μ∗ ] |ζλ (x)| Moreover, |L(λ, Tn ) − L(λ, T)| + 8M d H (T, Tn ) δ (3.3) for all n > n , λ ∈ K Proof Since K ⊂ C\R is a compact set, M < ∞ First, assume that Tn ⊂ T We see that the function dζλ (x) = dx λ+x|λ|2 (1+2x λ+x |λ|2 )x 2 |λ| − ( λ) − ln(1+2x λ+x |λ|2 ) 2x if x > 0, if x = 0, is continuous in (x, λ), provided λ = Therefore, the family of functions (ζλ (u))λ∈K is equi-continuous in variable u on [0, μ∗ ], i.e., for any > 0, there 123 Math Control Signals Syst (2016) 28:13 Page of 28 13 exists δ = δ( ) > such that if |u − v| < δ then |ζλ (u) − ζλ (v)| < for any λ ∈ K Since limn→∞ Tn = T, we can choose n such that d H (T, Tn ) < 2δ when n > n s < t; s, t ∈ [0, ∞) and n > n Denote A1 = {h ∈ Tn ∩ [s, t] : Fix t0 μn (h) δ}, A2 = {h ∈ Tn ∩ [s, t] : μn (h) < δ} The assumption Tn ⊂ T implies μn (h) < δ, which that μ(h) μn (h) for all h ∈ Tn If h ∈ A2 then μ(h) implies |ζλ (μ(h)) − ζλ (μn (h))| < On the other hand, the cardinal of A1 , say r , is finite and r [ t−s δ ] Thus, we can write A1 = {s1 < s2 < · · · < sr } Denote sequence τi by si + σn (si ) ; and τi = σ (τi ), i = 1, 2, , r τi = max h ∈ T : h It follows that d H (T, Tn ) max{|τi − si |, |σn (si ) − τi |} and |τi − si | < 2δ , |σn (si ) − τi | < 2δ Therefore, |μ(τi )−μn (si )| = |σn (si )−si +τi −τi | = |τi −si |+|σn (si )−τi | < δ, which implies |ζλ (μ(τi )) − ζλ (μn (si ))| < For any h ∈ T, there exists a unique u ∈ Tn , say u = γ T,Tn (h), such that either h = u or h ∈ (u, σn (u)) It is easy to check that the function γ T,Tn (h) is r d-continuous on T By the definition of integral on time scales (see [22]), we have t s ζλ (μn (h)) nh t = s ζλ μn γ T,Tn (h) h Therefore, t ζλ (μn (h)) s s1 = s r s t∧τi i=1 t∧τ i t∧τi + si ζλ (μ(h)) h t∧σn (si ) r −1 + i=1 t T,Tn t s |ζλ (μn (γ T,Tn (h))) − ζλ (μ(h))| h (h)))| h |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h t∧τ i + t − |ζλ (μ(h)) − ζλ (μn (γ + + nh |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h si+1 σn (si ) t∧σn (sr ) |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h, 123 13 Page 10 of 28 Math Control Signals Syst (2016) 28:13 where a ∧ b = min{a, b} For h ∈ T ∩ ([s, s1 ) ∪ [σn (si ), si+1 ) ∪ [t ∧ σn (sr ), t)), i = 1, 2, , r − 1, we have γ T,Tn (h) ∈ Tn ∩ ([s, s1 ) ∪ [σn (si ), si+1 ) ∪ [t ∧ σn (sr ), t)) ⊂ A2 , μn (γ T,Tn (h)) < δ, |μ(h) − μn (γ T,Tn (h))| < δ Therefore, and hence μ(h) s1 s |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h si+1 σn (si ) t (s1 − s), |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h t∧σn (sr ) (si+1 − σn (si )), |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h (t − t ∧ σn (sr )) Since τi = σ (τi ) for i = 1, 2, , r , we have t∧τ i t∧τi |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h = (t ∧ τ i − t ∧ τi )|ζλ (μ(τi )) − ζλ (μn (γ T,Tn (τi )))| = (t ∧ τ i − t ∧ τi )|ζλ (μ(τi )) − ζλ (μn (si ))| (t ∧ τ i − t ∧ τi ) On the other hand, for i = 1, 2, , r t∧τi si |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h t∧σn (si ) t∧τ i |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h 2M(t ∧ τi − si ) 2Md H (T, Tn ) 2M(t ∧ σn (si ) − t ∧ τ i ) 2Md H (T, Tn ) Thus, we obtain t s |ζλ (μ(h)) − ζλ (μn (γ T,Tn (h)))| h r (s1 − s) + (t ∧ τ i − t ∧ τi ) i=1 r −1 r (si+1 − σn (si )) + (t − t ∧ σn (sr )) + 4M + i=1 d H (T, Tn ) i=1 < (t − s) + 4Mr d H (T, Tn ) < (t − s) + 4M t −s d H (T, Tn ) δ Therefore, t s ζλ (μn (h)) 123 nh t − s ζλ (μ(h)) h < (t − s) + 4M t −s d H (T, Tn ) δ 13 Page 14 of 28 Math Control Signals Syst (2016) 28:13 Example 4.3 Consider the matrix pairs (An , Bn ), (A, B) with An = 10 , n1 A= 10 , 00 Bn = B = −1 (4.6) Then, we have ind(A, B) = and limn→∞ (An , Bn ) = (A, B) However, σ (An , Bn ) = {−1, n} σ (A, B) = {−1} as n → ∞ Moreover, if Tn = T = R then Eq (4.1) is uniformly exponentially stable but (4.5) is not uniformly exponentially stable for each n ∈ N Example 4.4 Consider the matrix pairs (An , Bn ), (A, B) with ⎡ ⎡ ⎤ 200 A n = A = ⎣0 ⎦ , 000 ⎤ 100 Bn = ⎣0 0⎦ , n1 ⎡ ⎤ 100 B = ⎣0 0⎦ 001 (4.7) Then, we have ind(A, B) = and limn→∞ (An , Bn ) = (A, B) However, σ (An , Bn ) = {1/2, n} σ (A, B) = {1/2} as n → ∞ Moreover, if Tn = T = Z then Eq (4.1) be uniformly exponentially stable but (4.5) is not uniformly exponentially stable for each n ∈ N Two above examples show that the spectrum of matrix pairs and the exponential stability of implicit dynamic equations are very sensitive to change of the coefficients The reasons is that they contain not only ordinary dynamic equations and algebraic constraints, but also hidden constraints which involve derivatives of several solution components as well It is well know that an arbitrary small perturbation may destroy the index of equations as well as the stability of solutions, even in the case of linear constant-coefficient DAEs, see [9,20,21] That is why in the following we restrict the direction of the matrix pairs (An , Bn ), n ∈ N when (An , Bn ) tends to (A, B) To this end, we prove first Lemma 4.5 Let S be an open set such that σ (A, B) ⊂ S Then, we have sup | P(λA − B)−1 < ∞, sup |λ P(λA − B)−1 < ∞, λ∈S c λ∈S c (4.8) where S c = C\S Moreover, if ind(A, B) = then sup |(λA − B)−1 < ∞ λ∈S c Proof By the Weierstraß–Kronecker canonical form, λA − B = W 123 λIr − J 0 T −1 λN − Im−r (4.9) Math Control Signals Syst (2016) 28:13 Page 15 of 28 13 Therefore, σ (J ) = σ (A, B) ⊂ S and P(λA− B)−1 = T (λI − J )−1 0 λ(λI − J )−1 W −1 , λ P(λA− B)−1 = T 0 W −1 for all λ ∈ S c Since supλ∈S c (λI − J )−1 < ∞ and supλ∈S c λ(λI − J )−1 < ∞, we obtain (4.8) If ind(A, B) = then we have (λA − B)−1 = T (λI − J )−1 0 W −1 Im−r Thus, (4.9) holds The proof is complete The following proposition investigates the data dependence of the spectrum of the matrix pair (A, B) Proposition 4.6 Let ind(A, B) = Assume that limn→∞ (An , Bn ) = (A, B) and (An − A) Q = for all n ∈ N Then we have lim σ (An , Bn ) = σ (A, B) (4.10) n→∞ in the sense that for any open neighborhood O of σ (A, B), there exists n such σ (An , Bn ) ⊂ O for all n n Proof Define σ (A, B) = ∪λ∈σ (A,B) B(λ, ) Then σ (A, B) ↓ σ (A, B) as For all λ ∈ σ (A, B)c = C\σ (A, B), we have λAn − Bn = λ(An − A) − (Bn − B) + λA − B ↓ (4.11) = (I + (λ(An − A) − (Bn − B))(λA − B)−1 )(λA − B) Since (An − A) Q = 0, An − A = (An − A) P and (λ(An − A) − (Bn − B))(λA − B)−1 = (An − A)λ P(λA − B)−1 −(Bn − B)(λA − B)−1 Therefore, by Lemma 4.5, (λ(An − A) − (Bn − B))(λA − B)−1 C( An − A + Bn − B ), for all λ ∈ σ (A, B)c , where C = max sup λ∈σ (A,B)c λ P(λA − B)−1 , sup λ∈σ (A,B)c (λA − B)−1 123 13 Page 16 of 28 Math Control Signals Syst (2016) 28:13 Choose N > such that An − A + Bn − B < (λ(An − A) − (Bn − B))(λA − B)−1 < 21 , and hence 2C for all n > N Then I + (λ(An − A) − (Bn − B))(λA − B)−1 is invertible for all λ ∈ σ (A, B)c , n > N By (4.11), λAn − Bn is invertible for all λ ∈ σ (A, B)c , n > N Thus, σ (An , Bn ) ⊂ σ (A, B) for all n > N and hence (4.10) holds Remark 4.7 In the case ind(A, B) = 1, the assumption (An − A) Q = means that the matrix pairs (An , Bn ) and (A, B) have the same nilpotent structure in the Weierstraß– Kronecker canonical form (see, e.g., [9,20,21]) Moreover, it is a sufficient condition, but not necessary Using the data dependence of the spectrum of the matrix pair (A, B) in Proposition 4.6, the data dependence of the exponential stability of equation (4.1) is showed in the following theorem Theorem 4.8 Let ind(A, B) = and Eq (4.1) be uniformly exponentially stable Assume that limn→∞ (An , Bn ; Tn ) = (A, B; T) and (An − A) Q = for all n ∈ N Then there exists an integer number N > such that Eq (4.5) is uniformly exponentially stable for all n > N Proof Since Eq (4.1) be uniformly exponentially stable, by Theorem 4.2 we have σ (A, B) ⊂ UT By Proposition 3.3, for any λ ∈ σ (A, B) we can find δλ > and n(λ) > such that B(λ, δλ ) ⊂ UT and B(λ, δλ ) ⊂ UTn for all n > n(λ) Let S = ∪λ∈σ (A,B) B(λ, δλ ) and N1 = max n(λ) (4.12) Then S is open and S ⊂ UT ∩ UTn for all n > N1 By Proposition 4.6, there exists N > N1 such that σ (An , Bn ) ⊂ S ⊂ UTn for all n > N Thus, by Theorem 4.2, Eq (4.5) is uniformly exponentially stable for all n > N The proof is complete Now, we consider the data dependence of the spectrum of the matrix pairs and the exponential stability for the case of implicit dynamic equations with the index greater than Proposition 4.9 Let ind(A, B) > Assume that limn→∞ (An , Bn ) = (A, B) and (An − A) Q = (Bn − B) Q = for all n ∈ N Then we have lim σ (An , Bn ) = σ (A, B) n→∞ 123 (4.13) Math Control Signals Syst (2016) 28:13 Page 17 of 28 13 in the sense of Proposition 4.6 Proof Since (An − A) Q = (Bn − B) Q = 0, it implies that (An − A) = (An − A) P, (Bn − B) = (Bn − B) P Therefore, (λ(An − A) − (Bn − B))(λA − B)−1 = (An − A)λ P(λA − B)−1 −(Bn − B) P(λA − B)−1 By Lemma 4.5, we imply that (λ(An − A) − (Bn − B))(λA − B)−1 C( An − A + Bn − B ) for all λ ∈ σ (A, B)c , where C = max sup λ∈σ (A,B)c λ P(λA − B)−1 , sup λ∈σ (A,B)c P(λA − B)−1 Now, the proof is similar with Proposition 4.6 Remark 4.10 In the case ind(A, B) > 1, the assumption (An − A) Q = (Bn − B) Q = means that the matrix pairs (An , Bn ) and (A, B) have the same nilpotent structure in the Weierstraß–Kronecker canonical form (see, e.g., [9,20,21]) Moreover, it is a sufficient, but not necessary condition Theorem 4.11 Let ind(A, B) > and Eq (4.1) be uniformly exponentially stable Assume that limn→∞ (An , Bn ; Tn ) = (A, B; T) and (An − A) Q = (Bn − B) Q = for all n ∈ N Then there exists an integer number N > such that Eq (4.5) is uniformly exponentially stable for all n > N Proof Similarly with Theorem 4.8 In the rest of this section, we investigate the data dependence of resolvent functions R(λ) = (λA − B)−1 that is also necessary to study the data dependence of the stability radii in next section Proposition 4.12 Let ind(A, B) = and S be an open set such that σ (A, B) ⊂ S Assume that limn→∞ (An , Bn ) = (A, B) and (An − A) Q = for all n ∈ N Then, there exist the constants C1 , C2 , N > such that sup (λAn − Bn )−1 − (λA − B)−1 λ∈S c C1 ( An − A + Bn − B ), sup λ P (λAn − Bn )−1 − (λA − B)−1 λ∈S c C2 ( An − A + Bn − B ), (4.14) for all n > N 123 13 Page 18 of 28 Math Control Signals Syst (2016) 28:13 Proof By Proposition 4.6, there exists N > such that σ (An , Bn ) ⊂ S for all n > N Therefore, for all n > N , λ ∈ S c , we have (λAn − Bn )−1 − (λA − B)−1 = (λAn − Bn )−1 (λA − B − λAn + Bn )(λA − B)−1 , λ P (λAn − Bn )−1 − (λA − B)−1 = λ P(λAn − Bn )−1 (λA − B − λAn + Bn )(λA − B)−1 By Lemma 4.5, for all λ ∈ S c , (λA − B − λAn + Bn )(λA − B)−1 = (λ(A − An ) P(λA − B)−1 + (Bn − B)(λA − B)−1 M0 ( An − A + Bn − B ), where M0 = max{M1 , M2 } with M1 = sup λ P(λA − B)−1 , M2 = sup (λA − B)−1 λ∈S c λ∈S c By (4.11), we have (λAn − Bn )−1 = (λA − B)−1 (I + (λ(An − A) − (Bn − B))(λA − B)−1 )−1 , λ P(λAn − Bn )−1 = λ P(λA − B)−1 (I + (λ(An − A) − (Bn − B))(λA − B)−1 )−1 Choose N > such that An − A + Bn − B < it implies that 2M0 Then, for all n > N , λ ∈ Sc , (I + (λ(An − A) − (Bn − B))(λA − B)−1 )−1 < and (λAn − Bn )−1 < 2M1 , λ P(λAn − Bn )−1 < 2M2 Let C1 = 2M1 M0 , C2 = 2M2 M0 , we obtain (4.15) The proof is complete Similarly, we have Proposition 4.13 Let ind(A, B) > and S be an open set such that σ (A, B) ⊂ S Assume that limn→∞ (An , Bn ) = (A, B) and (An − A) Q = (Bn − B) Q = for all n ∈ N Then, there exist the constants C1 , C2 , N > such that sup | P((λAn − Bn )−1 − (λA − B)−1 ) C1 ( An − A + Bn − B ), sup λ P((λAn − Bn )−1 − (λA − B)−1 ) C2 ( An − A + Bn − B ), λ∈S c λ∈S c for all n > N 123 (4.15) Math Control Signals Syst (2016) 28:13 Page 19 of 28 13 Data dependence of stability radii Assume that unperturbed Eq (4.1) is uniformly exponentially stable We consider Eq (4.1) subjected to general structured perturbations of the form Ax (t) = Bx(t), (5.1) [ A, B] = [A, B] + D E, (5.2) with where D ∈ Cm×l , E ∈ Cq×2m , the perturbation ∈ Cl×q The matrix D E is called a structured perturbation of Eq (4.1) We define C = ∈ Cl×q : equation (5.1) is either irregular or not uniformly exponentially stable Definition 5.1 The stability radius of Eq (4.1) under structured perturbations of the form (5.2) is defined by r (A, B; D, E; T) = inf{ : ∈ C }, where · can be any vector-induced matrix norm Let E = [E , E ] with E , E ∈ Cq×m and E λ = λE − E We have the following theorem: Theorem 5.2 (see [19]) The complex stability radius of Eq (4.1) under structured perturbations of the form (5.2) is given by the formula −1 r (A, B; D, E; T) = sup G(λ) λ∈UTc , (5.3) where G(λ) = E λ (λA − B)−1 D Assume that unperturbed Eq (4.5) are uniformly exponentially stable Similarly, the stability radii of (5.4) An x n (t) = Bn x(t), with [ An , Bn ] = [An , Bn ] + Dn where Dn ∈ Cm×l , E n ∈ Cq×2m , the perturbation n n En , (5.5) ∈ Cl×q , are given by −1 r (An , Bn ; Dn , E n ; Tn ) = sup λ∈UTcn G n (λ) (5.6) 123 13 Page 20 of 28 Math Control Signals Syst (2016) 28:13 Here, G n (λ) = E nλ (λAn − Bn )−1 Dn , E n = [E n1 , E n2 ], E nλ = λE n1 − E n2 with E n1 , E n2 ∈ Cq×m The following proposition shows that the stability radius r is upper semi-continuous on both the coefficients and time scales Proposition 5.3 Assume that Eq (4.1) is uniformly exponentially stable and lim (An , Bn ; Dn , E n ; Tn ) = (A, B; D, E; T) Then, n→∞ lim sup r (An , Bn ; Dn , E n ; Tn ) r (A, B; D, E; T) n→∞ (5.7) Proof Denote ∞ U= UTcm , Un = n=1 m n UTcm m n c Then, Un−1 ⊂ Un ⊂ U and Un ↑ U By Theorem 3.4 we have UT \R ⊂ U ⊂ UTc c c Since the set UT \R is dense in UT , it follows that r −1 (A, B; D, E; T) = sup G(λ) λ∈U First, we suppose that r (A, B; D, E; T) > Since r −1 (A, B; D, E; T) = supλ∈U G(λ) , for any > there exists λ0 = λ0 ( ) ∈ U such that G(λ0 ) r −1 (A, B; D, E; T) − By the definition of U , there exits an n such that λ0 ∈ UTcn for all n n Moreover, G n (λ0 ) − G(λ0 ) = E nλ0 (λ0 An − Bn )−1 Dn − E λ0 (λ0 A − B)−1 D = E nλ0 ((λ0 An − Bn )−1 − (λ0 A − B)−1 )Dn + (E nλ0 − E λ0 )(λ0 A − B)−1 Dn + E λ0 (λ0 A − B)−1 (Dn − D) Since lim (An , Bn ; Dn , E n ) = (A, B; D, E), there exists an n = n ( ) > n n→∞ such that G n (λ0 ) G(λ0 ) − for all n r −1 (An , Bn ; Dn , E n ; Tn ) = sup λ∈UTcn n Therefore, G n (λ) G(λ0 ) − for all n n and > Thus, r −1 (A, B; D, E; T) G n (λ0 ) r −1 (A, B; D, E; T) − lim inf r −1 (An , Bn ; Dn , E n ; Tn ) n→∞ In the case r (A, B; D, E; T) = 0, for any N > 0, we can choose λ0 such that G(λ0 ) > N Using the same argument we have r −1 (An , Bn ; Dn , E n ; Tn ) = sup λ∈UTcn 123 G n (λ) G n (λ0 ) G(λ0 ) − > N − 1, ∀ n n1 Math Control Signals Syst (2016) 28:13 Page 21 of 28 This implies that lim inf r −1 (An , Bn ; Dn , E n ; Tn ) n→∞ 13 N −1 for any N > 0, and hence lim inf r −1 (An , Bn ; Dn , E n ; Tn ) = ∞ n→∞ Thus, we always have r −1 (A, B; D, E; T) lim inf r −1 (An , Bn ; Dn , E n ; Tn ) n→∞ = lim sup r (An , Bn ; Dn , E n ; Tn ) −1 n→∞ This implies (5.7) The proof is complete If we restrict on the time scale variable, then the stability radius is continuous, which is given in the following proposition Proposition 5.4 Assume that Eq (4.1) is uniformly exponentially stable and lim n→∞ Tn = T Then lim r (A, B; D, E; Tn ) = r (A, B; D, E; T) n→∞ Proof Let ∞ UTcm , Vn = V = n=1 m n UTcm , gn = sup G(λ) λ∈Vn m n c By Theorem 3.4, UT \R ⊂ V ⊂ UTc , which follows r −1 (A, B; D, E; T) = sup λ∈UTc Since Vn ⊃ Vn+1 for all n, gn G(λ) = sup G(λ) λ∈V gn+1 Therefore, there exists lim gn n→∞ G(λn ) + n1 If the sequence {λn } For each n we can find λn ∈ Vn such that gn is not bounded then there exists {n k } such that λn k → ∞ as k → ∞ Since G(λ) is a rational function, there exists the limit G(∞) = lim G(λ) Moreover, if n > λ→∞ then n ∈ UTc Thus, r −1 (A, B; D, E; T) = lim k→∞ lim n→∞ G(n) = G(∞) G(λn k ) = lim gn k = lim gn k→∞ n→∞ In the case the sequence {λn } is bounded, there exist λ ∈ C and a sequence {n k } such that λn k → λ as k → ∞ We will prove that λ ∈ UTc Indeed, assume that λ ∈ UT Then, by Proposition 3.3, there is a δ > such that B(λ, δ) ⊂ UTn for all n large enough This contradicts to λn k ∈ UTcn k for all k and lim λn k = λ Thus, k→∞ r −1 (A, B; D, E; T) G(λ) = lim k→∞ G(λn k ) = lim gn k = lim gn k→∞ n→∞ 123 13 Page 22 of 28 Math Control Signals Syst (2016) 28:13 On the other hand, since UTcn ⊂ Vn , gn sup λ∈UTcn G(λ) = r −1 (A, B; D, E; Tn ) for all n ∈ N Thus, we get r −1 (A, B; D, E; T) lim sup r −1 (A, B; D, E; Tn ) = lim inf r (A, B; D, E; Tn ) −1 n→∞ n→∞ , or equivalently r (A, B; D, E; T) lim inf r (A, B; D, E; Tn ) n→∞ Combining with (5.7), we obtain lim r (A, B; D, E; Tn ) = r (A, B; D, E; T) n→∞ The proof is complete In general case, the inverse relation of (5.7) is not true, i.e., the stability radius may not be lower semi-continuous The following example shows that (An , Bn ; Dn , E n ; Tn ) tends to (A, B; D, E; T) but r (A, B; D, E; T) > lim inf r (An , Bn ; Dn , E n ; Tn ), n→∞ even only for the right-hand side perturbation (E n1 = 0) Example 5.5 Let us consider the stability radii of (5.4) under structured perturbations of the form (5.5) with Tn = R for all n ∈ N and ⎛ ⎛ ⎞ + n1 −1 −1 0 ⎜ 0 −1⎟ ⎜ 1 ⎟ ; Bn = ⎜ n An = A = ⎜ ⎝ −1 ⎠ ⎝ −2 −1 + n1 −1 ⎛ ⎛ ⎞ ⎞ −1 1 ⎜ 0 −1⎟ ⎜0 0⎟ ⎜ ⎟ ⎟ B=⎜ ⎝−2 0 ⎠ ; Dn = D = ⎝−1 ⎠ ; −1 1 −1 ⎞ −1⎟ ⎟; 0⎠ E n1 = 0; for all n ∈ N It is easy to see that ind(A, B) = and ⎛ −1 t+2 ⎜0 G(t) = ⎜ ⎝0 123 ⎞ ⎛ −1 ⎞ 0 t+2 ⎜ −t −1 ⎟ ⎟ −1 ⎟ ⎟ ; G n (t) = ⎜ ⎟ ⎜ n(t+2) −t ⎠ ⎝ n(t+2) ⎠ −1 0 n(t+2) E n2 = I, Math Control Signals Syst (2016) 28:13 Page 23 of 28 13 Thus, r (An , Bn ; Dn , E n ; Tn ) = ( sup G n (t) )−1 = for any n ∈ N meanwhile t∈i R r (A, B; D, E; T) = −1 sup G(t) t∈i R = Even in the case ind(A, B) = 1, the inverse relation of (5.7) also may not hold if Eq (5.4) is subjected to structured perturbations (5.5) The following example illustrates this point Example 5.6 Let Tn = R for all n, we consider equation (5.4) under structured perturbations of the form (5.5) with ⎛ A=⎝1 −1 ⎛ Dn = ⎝ −1 0 ⎞ 0⎠ ; ⎛ ⎞ ⎛ ⎞ −1 −1 ⎠ ; B = ⎝−2 −1 ⎠ ; Bn = ⎝ −2 n1 − 1 2 −n + ⎞ ⎞ ⎛ ⎛ ⎞ 0 −1 n 0 ⎠ E n2 = ⎝ ⎠ ⎠ ; E n1 = ⎝ 0 0 − n1 n1 0 1 −1 It is easy to verify that ind{A, B} = and ⎛ −1 t+2 G(t) = ⎝ 0 ⎞ ⎛ −1 t+2 ⎠ ; G n (t) = ⎝ Thus, r (An , Bn ; Dn , E n ; Tn ) = supt∈i R G n (t) r (A, B; D, E; T) = t∈i R t n −1 −1 sup G(t) 3+t n(t+2) 0 −t+n n ⎞ ⎠ = does not tend to = as n → ∞ We see in two above examples that r (An , Bn ; Dn , E n ; Tn ) = for all n ∈ N Thus, for the lower semi-continuity of the stability radii, we need to restrict the structure matrices Dn , E n for the stability radius r (An , Bn ; Dn , E n ; Tn ) > By formula (5.6), this is equivalent to (5.8) sup G n (λ) < ∞ λ∈UTcn On the other hand, G n (λ) = E nλ (λAn − Bn )−1 Dn = E nλ P(λAn − Bn )−1 Dn + E nλ Q(λAn − Bn )−1 Dn = E nλ P(λAn − Bn )−1 Dn + (λE n1 − E n2 ) Q(λAn − Bn )−1 Dn Thus, by Propositions 4.12, 4.13, (5.8) holds if E n1 Q = in the case ind(A, B) = 1, and E n1 Q = E n2 Q = in the case ind(A, B) > 123 13 Page 24 of 28 Math Control Signals Syst (2016) 28:13 Theorem 5.7 Let ind(A, B) = and Eq (4.1) be uniformly exponentially stable Assume that limn→∞ (An , Bn ; Dn , E n ; Tn ) = (A, B; D, E; T) and (An − A) Q = E n1 Q = for all n ∈ N Then, we have r (A, B; D, E; T) = lim inf r (An , Bn ; Dn , E n ; Tn ) n→∞ (5.9) Proof Let S be defined in (4.12) Then S is open and there exists an integer number N > such that S ⊂ UT ∩ UTn and σ (An , Bn ) ⊂ S for all n > N Since E n1 Q = for all n ∈ N and limn→∞ E n = E, it implies that E n1 = E n1 P, E = E P, for all n ∈ N Therefore, we have G n (λ) − G(λ) = E nλ ((λAn − Bn )−1 − (λA − B)−1 )Dn +(E nλ − E λ )(λA − B)−1 Dn + E λ (λA − B)−1 (Dn − D) = E n1 λ P((λAn − Bn )−1 − (λA − B)−1 )Dn +(E n1 − E )λ P(λA − B)−1 Dn + E λ P(λA − B)−1 (Dn − D) −E n2 ((λAn − Bn )−1 − (λA − B)−1 )Dn −(E n2 − E )(λA − B)−1 Dn − E (λA − B)−1 (Dn − D) Since limn→∞ (E n , Dn ) = (E, D), sup E n1 < ∞, sup E n2 < ∞, sup Dn < ∞ n∈N n∈N n∈N Therefore, by Lemma 4.5 and Proposition 4.12, we imply that there exists a constant C > such that G n (λ) G(λ) + C( An − A + Bn − B + Dn − D + E n − E ) for all λ ∈ S c , n > N Since UTn ⊂ S c , r −1 (An , Bn ; Dn , E n ; Tn ) = sup λ∈UTcn sup λ∈UTcn G n (λ) G(λ) + C( An − A + Bn − B + Dn − D + E n − E ) = r −1 (A, B; D, E; Tn ) + C( An − A + Bn − B + Dn − D + E n − E ), for all n > N This implies that lim sup r −1 (An , Bn ; Dn , E n ; Tn ) n→∞ 123 lim sup r −1 (A, B; D, E; Tn ) n→∞ Math Control Signals Syst (2016) 28:13 Page 25 of 28 13 By Proposition 5.4, we have lim sup r −1 (A, B; D, E; Tn ) = lim r −1 (A, B; D, E; Tn ) = r −1 (A, B; D, E; T) n→∞ n→∞ Thus, lim inf r (An , Bn ; Dn , E n ; Tn ) −1 n→∞ = lim sup r −1 (An , Bn ; Dn , E n ; Tn ) n→∞ r −1 (A, B; D, E; T), or equivalently lim inf r (An , Bn ; Dn , E n ; Tn ) n→∞ r (A, B; D, E; T) Now, by Proposition 5.3, we obtain (5.9) The proof is complete In the case ind(A, B) > 1, similarly with the above proof, using Proposition 4.13 we obtain Theorem 5.8 Let ind(A, B) > Assume that limn→∞ (An , Bn ; Dn , E n ; Tn ) = (A, B; D, E; T) and (An − A) Q = (Bn − B) Q = E n1 Q = E n2 Q = for all n ∈ N Then, we have r (A, B; D, E; T) = lim inf r (An , Bn ; Dn , E n ; Tn ) n→∞ With respect to only structured perturbations on the right-hand side ( Bn = Bn + Dn n E n ), we have the following corollary Corollary 5.9 Let ind(A, B) = and Eq (4.1) be uniformly exponentially stable Assume that limn→∞ (Bn ; Dn , E n ; Tn ) = (B; D, E; T) Then, we have lim r (A, Bn ; Dn , E n ; Tn ) = r (A, B; D, E; T) n→∞ Example 5.10 Consider the equation Ax (t) = Bx(t), t ∈ R (5.10) Applying the explicit Euler method with step size h > to this equation we have A x((m + 1)h) − x(mh) = Bx(mh), h m ∈ N This equation can be considered as a dynamic equation Ax (t) = Bx(t) (5.11) 123 13 Page 26 of 28 Math Control Signals Syst (2016) 28:13 on the time scale Th = hN It can be easily seen that lim Th = R Assume h→0 that Eq (5.10) is uniformly exponentially stable Then, by Proposition 5.4, we get lim r (A, B; D, E; Th ) = r (A, B; D, E; R) Moreover, if we use the explicit Euler h→0 method with the mesh step h = n to Ax (t) = Bn x(t), t ∈ R, n ∈ N, then we obtain a dynamic equation Ax (t) = Bn x(t) (5.12) on the time scale Tn = n1 N Let r (A, Bn ; Dn , E n ; Tn ) be the stability radius of equation (5.12) on Tn Assume that ind(A, B) = and lim (Bn , Dn , E n ) = (B, D, E) n→∞ By Corollary 5.9, we obtain lim r (A, Bn ; Dn , E n ; Tn ) = r (A, B; D, E; R) n→∞ Conclusion In this paper, we have analyzed the data dependence of the stability domains, spectra of matrix pair, exponential stability and stability radii for linear implicit dynamic equations of arbitrary index Relevant properties of the stability domains as well as the relation between the spectra of matrix pair have been investigated As the main result, under certain structure restriction, we have shown that the exponential stability and the stability radii depend continuously on the coefficient matrices and time scales As a practical consequence, the complex stability radius of DAEs can be approximated by one of the implicit difference equations As a future work, an analysis of the exponential stability and the stability radii for time-varying implicit dynamic equations on time scales with respect to structured perturbations acting on both the coefficient of the derivative and the right-hand side seems to be an interesting problem, for which more technical difficulties are expected Acknowledgments The second author was supported financially by Vietnam National Foundation for Science and Technology Development (NAFOSTED) 101.03-2014.58 The third author was supported financially by the Foundation for Science and Technology Development of Vietnam’s Ministry of Education and Training B2016-BKA-03 This work was done while the third author was visiting at Vietnam Institute for Advance Study in Mathematics (VIASM) The third author would like to thank VIASM for support and providing a fruitful research environment and hospitality References Agarwal R, Bohner M, O’Regan D, Peterson A (2002) Dynamic equations on time scales: a survey J Comput Appl Math 141:1–26 Attouch H, Lucchetti R, Wets RJ-B (1991) The topology of ρ-Hausdorff distance Ann Mat Pur Appl CLX 160:303–320 Bartosiewicz Z (2013) Linear positive control systems on time scales Math Control Signal Syst 25:327– 343 123 Math Control Signals Syst (2016) 28:13 Page 27 of 28 13 Bartosiewicz Z, Piotrowska E (2013) On stabilisability of nonlinear systems on time scales Int J Control 86(1):139–145 Berger T (2014) Robustness of stability of time-varying index-1 DAEs Math Control Signal Syst 26:403–433 Bohner M, Peterson A (2003) Advances in dynamic equations on time scales Birkhäuser, Boston Bracke M (2000) On stability radii of parametrized linear differential-algebraic systems PhD thesis, University of Kaiserslautern Brenan KE, Campbell SL, Petzold LR (1996) Numerical solution of initial value problems in differential algebraic equations SIAM, Philadelphia Byers R, Nichols N (1993) On the stability radius of a generalized state-space system Linear Algebra Appl 188(189):113–134 10 Chyan C-J, Du NH, Linh VH (2008) On data-dependence of exponential stability and stability radii for linear time-varying differential-algebraic systems J Differ Equ 245:2078–2102 11 Dai L (1989) Singular Control Systems Springer, Berlin 12 DaCuhna JJ (2005) Stability for time varying linear dynamic systems on time scales J Comput Appl Math 176(2):381–410 13 DaCunha JJ, Davis JM (2011) A unified floquet theory for discrete, continuous, and hybrid periodic linear systems J Differ Equ 251:2987–3027 14 Davis JM, Gravagne IA, Jackson B, Marks RJ (2009) Controllability, observability, realisability and stability of dynamic linear systems Electron J Differ Equ 37:1–32 15 Doan TS, Kalauch A, Siegmund S, Wirth F (2010) Stability radii for positive linear time-invariant systems on time scales Syst Control Lett 59:173–179 16 Du NH, Linh VH (2006) Robust stability of implicit linear systems containing a small parameter in the leading term IMA J Math Control Inf 23:67–84 17 Du NH, Linh VH (2006) Stability radii for linear time-invarying differential-equations with respect to dynamic perturbations J Differ Equ 230:579–599 18 Du NH (2008) Stability radii of differential-algebraic equations with structured perturbations Syst Control Lett 57:546–553 19 Du NH, Thuan DD, Liem NC (2011) Stability radius of implicit dynamic equations with constant coefficients on time scales Syst Control Lett 60:596–603 20 Du NH, Linh VH, Mehrmann V, Thuan DD (2013) Stability and robust stability of linear time-invariant delay differential-algebraic equations SIAM J Matrix Anal Appl 34:1631–1654 21 Du NH, Linh VH, Mehrmann V (2013) Robust stability of differential-algebraic equations In: Surveys in differential-algebraic equations I, DAE Forum, 63–95 22 Guseinov GS (2003) Integration on time scales J Math Anal Appl 285:107–127 23 Griepentrog E, März R (1986) Differential-algebraic equations and their numerical treatment TeubnerTexte zur Mathematik, Leipzig 24 Hilger S (1988) Ein Maβkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten PhD thesis, Universität Würzburg 25 Hilger S (1990) Analysis on measure chains: a unified approach to continuous and discrete calculus Results Math 18:18–56 26 Hinrichsen D, Pritchard AJ (1986) Stability radii of linear systems Syst Control Lett 7:1–10 27 Hinrichsen D, Pritchard AJ (1986) Stability radius for structured perturbations and the algebraic Riccati equation Syst Control Lett 8:105–113 28 Hinrichsen D, Ilchmann A, Pritchard AJ (1989) Robustness of stability of time-varying linear systems J Differ Equ 82:219–250 29 Hinrichsen D, Son NK (1989) The complex stability radius of discrete-time systems and symplectic pencils In: Proceedings of the 28th IEEE conference on decision and control, vol 1–3, pp 2265–2270 30 Jacob B (1998) A formula for the stability radius of time-varying systems J Differ Equ 142:167–187 31 Kunkel P, Mehrmann V (2006) Differential-algebraic equations, analysis and numerical solution EMS Publishing House, Zürich 32 Pötzsche C, Siegmund S, Wirth F (2003) A spectral characterization of exponential stability for linear time-invariant systems on time scales Discr Contin Dyn Syst 9:1123–1241 33 Memoli F (2012) Some properties of Gromov–Hausdorff distance Discr Comput Geom 48:416–440 34 Taousser FZ, Defoort M, Djemai M (2014) Stability analysis of a class of switched linear systems on non-uniform time domains Syst Control Lett 74:24–31 123 13 Page 28 of 28 Math Control Signals Syst (2016) 28:13 35 Taousser FZ, Defoort M, Djemai M (2015) Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions Nonlinear Anal Hybrid Syst 16:13–23 36 Thuan DD, Du NH, Liem NC (2016) Stabilizability and robust stabilizability of implicit dynamic equations with constant coefficients on time scales IMA J Math Control Inf 33:121–136 123 ... exponents corresponding to a convergent sequence of time scales This is a preparation for investigation of the data dependence of the exponential stability and the stability radii for implicit dynamic. .. Conclusion In this paper, we have analyzed the data dependence of the stability domains, spectra of matrix pair, exponential stability and stability radii for linear implicit dynamic equations. .. matrices Implicit dynamic equations of the form (1.1) can be considered as a unified form between linear differential algebraic equations (DAEs) and linear implicit difference equations Therefore,