ON DATADEPENDENCE OF STABILITY DOMAINS, EXPONENTIAL STABILITY AND STABILITY RADII FOR IMPLICIT DYNAMIC EQUATIONS

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 righthand sid

ON DATA-DEPENDENCE OF STABILITY DOMAINS, EXPONENTIAL STABILITY AND STABILITY RADII ∗ FOR IMPLICIT DYNAMIC EQUATIONS Nguyen Thu Ha† Nguyen Huu Du‡ Do Duc Thuan§ 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. 1 Introduction In this paper, we study the stability domains, the spectrum of matrix pairs, the exponential stability and the stability radii for implicit dynamic equations on time scales of the form An x∆n (t) = Bn x(t), (1.1) ∗ Mathematics Subject Classifications: 06B99, 34D99,47A10, 47A99, 65P99. Department of Basic Science, Electric Power University, 235 Hoang Quoc Viet Str., Hanoi, Vietnam, email: ntha2009@yahoo.com ‡ Department of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam, email: dunh@vnu.edu.vn § School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet Str., Hanoi, Vietnam, email: thuan.doduc@hust.edu.vn † 1 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 (1.2) is constrained, 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]. The theory of dynamic systems on an arbitrary time scale was found promising because it demonstrates the interplay between the theories of continuous-time and discrete-time systems, see [1, 3, 13, 24, 25]. It enables us to analyze the stability of dynamical systems on non-uniform time domains which are a subset of real numbers [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 and the stability of numerical methods of dynamic systems relate the stability domains of scalar equations. Therefore, it is meaningful to investigate the behavior of the stability domains when the time scales of equations converge. On the other hand, many problems arising from real life contain uncertainty, because there are parameters which can be determined only by experiments or the remainder part ignored during linearization process can also be considered uncertainty. That is the reason why we are interested in investigating the uncertain equations subjected to general structured perturbations of the form An x∆n (t) = Bn x(t), (1.2) with [An , Bn ] [An , Bn ] + Dn Σn En , (1.3) where Σn , n ∈ N, are unknown disturbance matrices; Dn , En are known scaling matrices defining the “structure” of the perturbations. A natural question arises that under what condition equations (1.2) with perturbations (1.3) remain exponentially stable, i.e., how robust the stability of the nominal equations (1.1) is. 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. This measure of the robust stability was introduced by Hinrichsen and Pritchard [27] for linear 2 time-invariant systems of ordinary differential equations (ODEs) with respect to time- and output-invariant, i.e., static perturbations. See [15, 26, 27, 29] for results on stability radii of time-invariant linear systems. Earlier results for the robust stability of time-varying systems can be found, e.g., 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, 17, 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 1 and with respect to only the right-hand side perturbations. It is worth mentioning that the index notion, which plays a key role in the qualitative theory and in the numerical analysis of DAEs, should be taken into consideration in the robust stability analysis, too. Recently, Du, Thuan and Liem [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 equations (1.2) with general structured perturbations of the form (1.3) when the data (An , Bn ; Dn , En ; 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 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 Section 3, we derive some characterizations of the stability domains corresponding to the convergent time scales. Section 4 deals with the data-dependence of the spectrum of matrix pairs and the exponential stability. In Section 5, the data-dependence of the stability radii is analyzed. The last section gives some conclusions and open problems. 3 2 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. A function f defined on T is regulated if there exist a left-sided limit at every left-dense point and a right-sided limit at every right-dense point. A regulated function is called rd-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 1 + µ(t)f (t) > 0 for every t ∈ T. Use R+ to denote the set of positively regressive functions from T to R. A function f : T → Rd is called delta differentiable at t if there exists a vector f ∆ (t) such that for all > 0 f (σ(t)) − f (s) − f ∆ (t)(σ(t) − s) |σ(t) − s| for all s ∈ (t − δ, t + δ) ∩ T and for some δ > 0. 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. Let f be a rd-continuous function and a, b ∈ T. Then, the Riemann integral b b b f (s)∆T s exists (see [22]). In case a, b ∈ T, writing a f (s)∆T s means a f (s)∆T s, a where a = min{t > a : t ∈ T}; b = max{t < b : t ∈ T}. If there is no confusion, we b b b b simply 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 s t0 , (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 t eλ (t, s) = exp Ln(1 + hλ) ∆τ , µ(τ ) h lim s h 4 where Ln a is the principal logarithm of the number a. Since t |eλ (t, s)| = exp ln |1 + hλ| ∆τ , µ(τ ) h lim s h we can rewrite t |eλ (t, s)| = exp ζλ (µ(τ ))∆τ (2.2) s with ln|1 + hλ| = s h ζλ (s) = lim h λ if s = 0 if s = 0. ln |1+sλ| s Note that |eλ (t, s)| = |eλ (t, s)| for any λ ∈ C. Further, it is easy to see that ζλ (x) |λ| for all x 0, and hence |eλ (t, s)| e|λ|(t−s) . 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 > t0 . (2.3) We assume that the function f : Tt0 × Rm → Rm satisfies conditions such that equation (2.3) has a unique solution x(t, s, x0 ), t > s with the initial condition x(s, s, x0 ) = x0 for any s ∈ Tt0 and x0 ∈ Rm . Definition 2.1 (Exponential stability). The dynamic equation (2.3) is called uniformly exponentially stable if there exists a constant α > 0 with −α ∈ R+ and K > 0 such that for every s < t, s, t ∈ Tt0 , the inequality x(t, s, x0 ) K x0 e−α (t, s) (2.4) holds for any x0 ∈ Rm . Beside this definition, one can use the classical exponential function exp{−α(t − τ )} in (2.4). 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 equation (2.3) is uniformly exponentially stable if and only if the scalar equation (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 dH (T1 , T2 ) := max{ sup d(t1 , T2 ), sup d(t2 , T1 )}, t1 ∈T1 5 t2 ∈T2 (2.5) where d(t1 , T2 ) = inf |t1 − t2 | and d(t2 , T1 ) = inf |t2 − t1 |. t1 ∈T1 t2 ∈T2 For properties of the Hausdorff distance, we refer the interested readers to [2, 33]. 3 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. [19, 32]). Since if |b1 | |b2 | then ζa+ib1 (x) ζa+ib2 (x) for any a ∈ R and x 0, 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 0 then ζλ (x) 0 for all x 0 and hence UT ⊂ C− = {λ ∈ C : λ < 0}. For each λ ∈ C, we define the Lyapunov exponent of the scalar function t 1 L(λ, T) := lim sup t−s→∞ t − s ζλ (µ(τ ))∆τ. (3.1) s By virtue of the inequality ζλ (x) |λ| for any x only if L(λ, T) < 0. Moreover, we have 0, it follows that λ ∈ UT if and Lemma 3.1. Let T ∈ T and λ ∈ C \ R. Then, λ ∈ UT if and only if L(λ, T) 0. 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 = 0 and λn = an + ibn . n→∞ Using the Lagrange finite increment formula, for all x > 0, we have x(|λn |2 − |λ|2 ) + 2(an − a) ζλn (x) − ζλ (x) = , θ ∈ (0, 1). 2(1 + 2x(a + θ(an − a)) + x2 (|λ|2 + θ(|λn |2 − |λ|2 ))) Since 0 < 1 + 2xa + x2 |λ|2 for all x 0, we can choose an n0 ∈ N and a constant c1 > 0 such that c1 < 1 + 2x(a + θ(an − a)) + x2 (|λ|2 + θ(|λn |2 − |λ|2 )) for all 0 x µ∗ and n > n0 . Thus, for any > 0, there exists n1 > n0 satisfying ζλ (x) − ζλn (x) < , ∀ 0 6 x µ∗ , ∀ n > n1 . This implies that t t ζλn (µ(τ ))∆τ + (t − s) < (t − s), ζλ (µ(τ ))∆τ < s ∀ t0 t, ∀ n > n1 . s s Hence lim sup t−s→∞ 1 t−s t ζλ (µ(τ ))∆τ < , ∀ > 0. s Thus L(λ, T) = lim sup t−s→∞ 1 t−s t ζλ (µ(τ ))∆τ Conversely, let λ = a + ib ∈ C \ R such that L(λ, T) lim sup t−s→∞ 1 t−s 0. s 0, i.e., t ζλ (µ(τ ))∆τ 0. s For any > 0, let λ = a + ib be chosen such that 0 < |b | < |b|; a < a and |λ| > |λ | > |λ| − . Since 0 < 1 + 2xa + x2 |λ|2 , we can choose a and b such that 0 < 1 + 2x(a + θ(a − a)) + x2 (|λ|2 + θ(|λ |2 − |λ|2 )) < c2 for all 0 x µ∗ . Thus, ζλ (x) − ζλ (x) < a −a , c2 ∀0 x µ∗ . This implies that t t ζλ (µ(τ ))∆τ + ζλ (µ(τ ))∆τ < s s a −a (t − s), c2 ∀ t0 s t. Hence, 1 L(λ , T) = lim sup t−s→∞ t − s t ζλ (µ(τ ))∆τ s which follows that λ ∈ UT. Since λ → λ as complete. a −a 0, there is a δ > 0 and n0 ∈ N such that t t ζλ (µn (h))∆n h − s ζλ (µ(h))∆h < 2 (t − s) + 8M s t−s dH (T, Tn ), δ (3.2) for all n > n0 , λ ∈ K, t > s, where M = supλ∈K,x∈[0,µ∗ ] |ζλ (x)|. Moreover |L(λ, Tn ) − L(λ, T)| 2 + 8M dH (T, Tn ) δ (3.3) for all n > n0 , 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+x2 |λ|2 )x 1 |λ|2 − ( λ)2 2 − ln(1+2x λ+x2 |λ|2 ) 2x2 if x > 0, if x = 0, is continuous in (x, λ), provided λ = 0. Therefore, the family of functions (ζλ (u))λ∈K is equi-continuous in variable u on [0, µ∗ ], i.e., for any > 0, there exists δ = δ( ) > 0 such that if |u−v| < δ then |ζλ (u)−ζλ (v)| < for any λ ∈ K. Since limn→∞ Tn = T, we can choose n0 such that dH (T, Tn ) < 2δ when n > n0 . Fix t0 s < t; s, t ∈ [0, ∞) and n > n0 . Denote A1 = {h ∈ Tn ∩ [s, t] : µn (h) δ}, A2 = {h ∈ Tn ∩ [s, t] : µn (h) < δ}. The assumption Tn ⊂ T implies that 0 µ(h) µn (h) for all h ∈ Tn . If h ∈ A2 then µ(h) µn (h) < δ, which implies |ζλ (µ(h)) − ζλ (µn (h))| < . On the other hand, the cardinal of A1 , say r, is finite ]. Thus, we can write A1 = {s1 < s2 < . . . < sr }. and r [ t−s δ Denote sequence τi by τi = max h ∈ T : h si + σn (si ) ; and τi = σ(τi ), i = 1, 2, . . . , r. 2 It follows that dH (T, Tn ) max{|τi −si |, |σn (si )−τi |} and |τi −si | < 2δ , |σn (si )−τi | < δ . Therefore, |µ(τi ) − µn (si )| = |σn (si ) − si + τi − τi | = |τi − si | + |σn (si ) − τi | < δ, 2 which implies |ζλ (µ(τi )) − ζλ (µn (si ))| < . 8 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 rd-continuous on T. By the definition of integral on time scales (see [22]), we have t t ζλ (µn (γ T,Tn (h)))∆h. ζλ (µn (h))∆n h = s s Therefore, t t t ζλ (µn (h))∆n h − |ζλ (µn (γ T,Tn (h))) − ζλ (µ(h))|∆h ζλ (µ(h))∆h s s s r s1 t∧τi |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h + = s si i=1 t∧τ i t∧σn (si ) |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h + + t∧τ i t∧τi r−1 |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h t si+1 |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h, |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h + + i=1 t∧σn (sr ) σn (si ) 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 |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h (s1 − s), s si+1 |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h (si+1 − σn (si )), σn (si ) t |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h (t − t ∧ σn (sr )). t∧σn (sr ) Since τi = σ(τi ) for i = 1, 2, ..., r, we have t∧τ i |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∧τi = (t ∧ τ i − t ∧ τi )|ζλ (µ(τi )) − ζλ (µn (si ))| (t ∧ τ i − t ∧ τi ). 9 On the other hand, for i = 1, 2, ..., r t∧τi 2M (t ∧ τi − si ) |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h 2M dH (T, Tn ) si t∧σn (si ) |ζλ (µ(h)) − ζλ (µn (γ T,Tn (h)))|∆h 2M (t ∧ σn (si ) − t ∧ τ i ) 2M dH (T, Tn ). t∧τ i Thus, we obtain r t |ζλ (µ(h)) − ζλ (µn (γ T,Tn (s1 − s) + (h)))|∆h s (t ∧ τ i − t ∧ τi ) i=1 r−1 r (si+1 − σn (si )) + (t − t ∧ σn (sr )) + 4M + i=1 dH (T, Tn ) i=1 t−s dH (T, Tn ). < (t − s) + 4M rdH (T, Tn ) < (t − s) + 4M δ Therefore, t t ζλ (µn (h))∆n h − s ζλ (µ(h))∆h < (t − s) + 4M s t−s dH (T, Tn ). δ If Tn ⊂ T we put Tn = Tn ∪ T. It is easy to see that dH (T, Tn ) = max{dH (Tn , T), dH (Tn , Tn )}. By the above proof, we have t t t−s dH (T, Tn ), δ s s t t t−s ζλ (µ(h))∆h − ζλ (µ(h))∆Tn h < (t − s) + 4M dH (T, Tn ). δ s s ζλ (µn (h))∆n h − ζλ (µ(h))∆Tn h < (t − s) + 4M This implies that t t ζλ (µn (h))∆n h − s ζλ (µ(h))∆h < 2 (t − s) + 8M s t−s dH (T, Tn ), δ and hence 1 t−s t t ζλ (µn (h))∆n h − s ζλ (µ(h))∆h < 2 + s By (3.1), we obtain (3.3). The proof is complete. 10 8M dH (T, Tn ). δ (3.4) Denote by UTn (resp.UT) the domain of stability of the time scale Tn (resp.T). Now, we will study relationship between the stability domains UTn and UT when Tn tends to T. Proposition 3.3. Suppose that lim Tn = T. Then, for any λ ∈ UT we can find a n→∞ neighborhood B(λ, δ) of λ and nλ > 0 such that B(λ, δ) ⊂ UT UTn . n>nλ Proof. First, we prove the proposition with λ ∈ UT \ R. Following the proof of Lemma 3.1, there exists a δ1 > 0 satisfying B(λ, δ1 ) ⊂ UT and ζλ (x) − ζλ (x) < −L(λ, T) ; ∀0 4 x µ∗ , ∀ λ ∈ B(λ, δ1 ). Hence, by (3.1), L(λ, T) L(λ, T) + 3L(λ, T) −L(λ, T) = 4 4 for any λ ∈ B(λ, δ1 ). (3.5) By choosing δλ := min{δ1 , | 3λ| } > 0 we see that B(λ, δλ ) ⊂ UT \ R. Using Lemma 3.2 with K = B(λ, δλ ) and = −L(λ) we can find a δ2 > 0 and n0 such that 8 L(λ, Tn ) < L(λ, T) + −L(λ, T) 8M + dH (T, Tn ), 4 δ2 for all n > n0 and λ ∈ B(λ, δλ ). We choose nλ > n0 such that dH (T, Tn ) < for any n > nλ . From (3.5) and (3.6) we get (3.6) −δ2 L(λ,T) 32M L(λ, T) 3L(λ, T) L(λ, T) L(λ, T) − − = < 0, ∀n > nλ , ∀λ ∈ B(λ, δλ ). 4 4 4 4 This means that B(λ, δλ ) ⊂ UTn for all n > nλ . We now consider the case λ ∈ UT∩R. Since UT is an open set, there exists δ3 > 0 such that B(λ, δ3 ) ⊂ UT. Let λ1 = λ + i δ23 . Following the above argument, there exist nλ1 > 0 and 0 < δλ1 < δ3 /2 such that B(λ1 , δλ1 ) ⊂ UT ∩ UTn for all n > nλ1 . Since UTn is symmetric with respect to the real axis, the segment [λ , λ ] ⊂ UTn , for all λ ∈ B(λ1 , δλ1 ). Thus B(λ, δλ1 ) ⊂ UT ∩ UTn for all n > nλ1 . The proposition is proved. L(λ, Tn ) Theorem 3.4. If lim Tn = T then n→∞ ∞ UT ⊂ ∞ UTm and n=1 m n UTm \ R ⊂ UT \ R. n=1 m n 11 (3.7) Proof. The first relation follows immediately from Proposition 3.3. To prove the second one, let λ ∈ ∞ n=1 m n UTm \ R. By the definition, there is a sequence {nk }∞ , n → ∞ such that λ ∈ UT k nk \R for all k. Using again inequality k=1 (3.3), for any > 0, there exist δ > 0, n0 ∈ N such that L(λ, T) L(λ, Tnk ) + 2 + 8M dH (Tnk , T), ∀nk > n0 . δ (3.8) Since λ ∈ UTnk , L(λ, Tnk ) < 0 and taking the limit as nk → ∞ we obtain L(λ, T) < 2 , ∀ > 0. This implies that L(λ, T) complete. 4 0. Thus, λ ∈ UT \ R by Lemma 3.1. The proof is Data-dependence of spectrum and exponential stability Consider the implicit dynamic equation on time scale T Ax∆ (t) = Bx(t), (4.1) where x(t) ∈ Cm , and A, B ∈ Cm×m are constant matrices. We assume that the pencil of matrices (A, B) is regular (that is, det(λA − B) ≡ 0). Then, the pair (A, B) can be transformed to Weierstraß-Kronecker canonical form, see [8, 11, 31], i.e., there exist nonsingular matrices W, T ∈ Cn,n such that A=W Ir 0 0 N T −1 , B = W J 0 0 Im−r T −1 , (4.2) where Ir , Im−r are identity matrices, J ∈ Cr×r and N ∈ C(m−r)×(m−r) is nilpotent of degree k (i.e., N k = 0, N k−1 = 0). The integer k is called the index of the pair (A, B) and we write ind(A, B) = k. Denote Q=T 0 0 T −1 , 0 Im−r P = Im − Q = T Ir 0 −1 T . 0 0 It is known that for any α ∈ C such that αA + B is nonsingular, one has Km = ker[(αA + B)−1 A]k ⊕ Im [(αA + B)−1 A]k , 12 (4.3) and Q is the projection onto ker[(αA + B)−1 A]k along the space Im [(αA + B)−1 A]k . In particular, Q does not depend on the choice of W and T. Moreover, the solution x(t) satisfies Qx(t) = 0 for all t ∈ Tt0 and the initial condition x(t0 ) = P x0 must hold (see [19, 23]). Let x(t, s, P x0 ) be the solution of (4.1) with the initial value x(s, s) = P x0 . According to Definition 2.1, we get the following definition of the exponential stability: Definition 4.1. The implicit dynamic equation (4.1) is called uniformly exponentially stable if there exist constants α > 0 with −α ∈ R+ and K > 0 such that for every s < t, s, t ∈ Tt0 , the inequality x(t, s, P x0 ) K P x0 e−α (t, s) (4.4) holds for any x0 ∈ Rm . A complex number λ is called a finite eigenvalue of the pencil (A, B) if det(λA − B) = 0. The set of all finite eigenvalues of (A, B) is called the finite spectrum of the pair (A, B) and denoted by σ(A, B). When A = I, we write simply σ(B) for σ(I, B). Theorem 4.2 (See [19, Theorem 3.2]). The implicit dynamic equation (4.1) is uniformly exponentially stable if and only if σ(A, B) ⊂ UT. Now, consider the implicit dynamic equations An x∆n (t) = Bn x(t), (4.5) where An , Bn ∈ Cn×n and t ∈ Tn . By Theorem 4.2, the exponential stability depends on the spectrum of the matrix pair (An , Bn ) and the stability domain UTn . Example 4.3. Consider the matrix pairs (An , Bn ), (A, B) with An = 1 0 −1 0 1 0 , Bn = B = . 1 ,A = 0 0 0 1 0 n (4.6) Then, we have ind(A, B) = 1 and limn→∞ (An , Bn ) = (A, B). However, σ(An , Bn ) = {−1, n} → σ(A, B) = {−1} as n → ∞. Moreover, if Tn = T = R then equation (4.1) is uniformly exponentially stable but (4.5) is not uniformly exponentially stable for all n ∈ N. 13 Example 4.4. Consider the matrix pairs (An , Bn ), (A, B) with       2 0 0 1 0 0 1 0 0 An = A = 0 0 1 , Bn = 0 1 0 , B = 0 1 0 . 0 0 0 0 0 1 0 n1 1 (4.7) Then, we have ind(A, B) = 2 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 equation (4.1) be uniformly exponentially stable but (4.5) is not uniformly exponentially stable for all 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 |λP (λA − B)−1 < ∞, λ∈S c λ∈S c (4.8) where S c = C \ S. Moreover, if ind(A, B) = 1 then sup |(λA − B)−1 < ∞. (4.9) λ∈S c Proof. By the Weierstraß-Kronecker canonical form, λA − B = W λIr − J 0 T −1 . 0 λN − Im−r Therefore, σ(J) = σ(A, B) ⊂ S and P (λA−B)−1 = T (λI − J)−1 0 λ(λI − J)−1 0 W −1 , λP (λA−B)−1 = T W −1 0 0 0 0 14 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) = 1 then we have (λA − B)−1 = T (λI − J)−1 0 W −1 . 0 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) = 1. Assume that limn→∞ (An , Bn ) = (A, B) and (An − A)Q = 0 for all n ∈ N. Then we have lim σ(An , Bn ) = σ(A, B) (4.10) n→∞ in the Hausdorff distance. Proof. Define σ (A, B) = ∪λ∈σ(A,B) B(λ, ). Then σ (A, B) ↓ σ(A, B) as ↓ 0. For all λ ∈ σ (A, B)c = C \ σ (A, B), we have λAn − Bn = λ(An − A) − (Bn − B) + λA − B = (I + (λ(An − A) − (Bn − B))(λA − B)−1 )(λA − B). (4.11) Since (An − A)Q = 0, An − A = (An − A)P and (λ(An − A) − (Bn − B))(λA − B)−1 = (An − A)λP (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 λP (λA − B)−1 , (A,B)c sup λ∈σ Choose N > 0 such that An − A + Bn − B < A) − (Bn − B))(λA − B)−1 < 21 , and hence (λA − B)−1 . (A,B)c 1 2C for all n > N. Then (λ(An − 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. 15 Remark 4.7. In the case ind(A, B) = 1, the assumption (An − A)Q = 0 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]). By 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) = 1 and equation (4.1) be uniformly exponentially stable. Assume that limn→∞ (An , Bn ; Tn ) = (A, B; T) and (An − A)Q = 0 for all n ∈ N. Then there exists an integer number N > 0 such that equation (4.5) is uniformly exponentially stable for all n > N . Proof. Since equation (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 > 0 and n(λ) > 0 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, equation (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 1. Proposition 4.9. Let ind(A, B) > 1. Assume that limn→∞ (An , Bn ) = (A, B) and (An − A)Q = (Bn − B)Q = 0 for all n ∈ N. Then we have lim σ(An , Bn ) = σ(A, B) n→∞ in the Hausdorff distance. 16 (4.13) 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)λP (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 λP (λA − B)−1 , (A,B)c sup λ∈σ P (λA − B)−1 . (A,B)c 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 = 0 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]). Theorem 4.11. Let ind(A, B) > 1 and equation (4.1) be uniformly exponentially stable. Assume that limn→∞ (An , Bn ; Tn ) = (A, B; T) and (An −A)Q = (Bn −B)Q = 0 for all n ∈ N. Then there exists an integer number N > 0 such that equation (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) = 1 and S be an open set such that σ(A, B) ⊂ S. Assume that limn→∞ (An , Bn ) = (A, B) and (An − A)Q = 0 for all n ∈ N. Then, there exist the constants C1 , C2 , N > 0 such that sup (λAn − Bn )−1 − (λA − B)−1 C1 ( An − A + Bn − B ), λ∈S c sup λP (λAn − Bn )−1 − (λA − B)−1 λ∈S c for all n > N . 17 C2 ( An − A + Bn − B ), (4.14) Proof. By Proposition 4.6, there exists N > 0 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 , λP (λAn −Bn )−1 −(λA−B)−1 = λP (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 λP (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 , λP (λAn − Bn )−1 = λP (λA − B)−1 (I + (λ(An − A) − (Bn − B))(λA − B)−1 )−1 . Choose N > 0 such that An − A + Bn − B < it implies that 1 . 2M0 Then, for all n > N, λ ∈ S c , (I + (λ(An − A) − (Bn − B))(λA − B)−1 )−1 < 2 and (λAn − Bn )−1 < 2M1 , λP (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) > 1 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 = 0 for all n ∈ N. Then, there exist the constants C1 , C2 , N > 0 such that sup |P ((λAn − Bn )−1 − (λA − B)−1 ) C1 ( An − A + Bn − B ), λ∈S c sup λP ((λAn − Bn )−1 − (λA − B)−1 ) λ∈S c for all n > N . 18 C2 ( An − A + Bn − B ), (4.15) 5 Data-dependence of stability radii We consider equation (4.1) subjected to general structured perturbations of the form Ax∆ (t) = Bx(t), (5.1) [A, B] = [A, B] + DΣE, (5.2) with where D ∈ Cm×l , E ∈ Cq×2m , the perturbation Σ ∈ Cl×q . The matrix DΣE is called a structured perturbation of equation (4.1). We define ΞC = Σ ∈ Cl×q : equation (5.1) is either irregular or not uniformly exponentially stable . Definition 5.1. The stability radius of equation (4.1) under structured perturbations of the form (5.2) is defined by r(A, B; D, E; T) = inf{ Σ : Σ ∈ ΞC }, where · can be any vector-induced matrix norm. Let E = [E 1 , E 2 ] with E 1 , E 2 ∈ Cq×m and E λ = λE 1 −E 2 . We have the following theorem: Theorem 5.2 (see [19]). The complex stability radius of equation (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) , (5.3) λ∈UTc where G(λ) = E λ (λA − B)−1 D. Similarly, the stability radii of An x∆n (t) = Bn x(t), (5.4) [An , Bn ] = [An , Bn ] + Dn Σn En , (5.5) with where Dn ∈ Cm×l , En ∈ Cq×2m , the perturbation Σn ∈ Cl×q , are given by r(An , Bn ; Dn , En ; Tn ) = ( sup λ∈UTcn 19 Gn (λ) )−1 . (5.6) Here, Gn (λ) = Enλ (λAn − Bn )−1 Dn , En = [En1 , En2 ], Enλ = λEn1 − En2 with En1 , En2 ∈ 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 equation (4.1) be uniformly exponentially stable and lim (An , Bn ; Dn , En ; Tn ) = (A, B; D, E; T). Then, n→∞ lim sup r(An , Bn ; Dn , En ; Tn ) r(A, B; D, E; T). n→∞ Proof. Denote (5.7) ∞ UTcm . UTcm , Un = U= m n n=1 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) > 0. Since r−1 (A, B; D, E; T) = supλ∈U G(λ) , for any > 0 there exists λ0 = λ0 ( ) ∈ U such that G(λ0 ) r−1 (A, B; D, E; T)− . By the definition of U , there exits an n0 such that λ0 ∈ UTcn for all n n0 . Moreover, Gn (λ0 ) − G(λ0 ) = Enλ0 (λ0 An − Bn )−1 Dn − E λ0 (λ0 A − B)−1 D = Enλ0 ((λ0 An − Bn )−1 − (λ0 A − B)−1 )Dn + (Enλ0 − E λ0 )(λ0 A − B)−1 Dn + E λ0 (λ0 A − B)−1 (Dn − D). Since lim (An , Bn ; Dn , En ) = (A, B; D, E), there exists an n1 = n1 ( ) > n0 such n→∞ that Gn (λ0 ) G(λ0 ) − for all n r−1 (An , Bn ; Dn , En ; Tn ) = sup n1 . Therefore, λ∈UTcn Gn (λ) G(λ0 ) − for all n Gn (λ0 ) r−1 (A, B; D, E; T) − 2 n1 and > 0. Thus r−1 (A, B; D, E; T) lim inf r−1 (An , Bn ; Dn , En ; 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 . By using the same argument we have r−1 (An , Bn ; Dn , En ; Tn ) = sup λ∈UTcn Gn (λ) Gn (λ0 ) 20 G(λ0 ) −1 > N −1, ∀ n n1 . This implies that lim inf r−1 (An , Bn ; Dn , En ; Tn ) n→∞ N − 1 for any N > 0, and hence lim inf r−1 (An , Bn ; Dn , En ; Tn ) = ∞. n→∞ Thus we always have r−1 (A, B; D, E; T) lim inf r−1 (An , Bn ; Dn , En ; Tn ) n→∞ −1 = lim sup r(An , Bn ; Dn , En ; Tn ) . 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 equation (4.1) be uniformly exponentially stable and lim Tn = T. Then lim r(A, B; D, E; Tn ) = r(A, B; D, E; T). n→∞ 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 G(λ) = sup G(λ) . λ∈UTc Since Vn ⊃ Vn+1 for all n, gn λ∈V gn+1 . Therefore, there exists lim gn . n→∞ For each n we can find λn ∈ Vn such that gn G(λn ) + n1 . If the sequence {λn } is not bounded then there exists {nk } such that λnk → ∞ as k → ∞. Since G(λ) is a rational function, there exists the limit G(∞) = lim G(λ) . Moreover, λ→∞ if n > 0 then n ∈ UTc . Thus, r−1 (A, B; D, E; T) lim G(n) = G(∞) = lim G(λnk ) = lim gnk = lim gn . n→∞ k→∞ k→∞ n→∞ In the case the sequence {λn } is bounded, there exist λ ∈ C and a sequence {nk } such that λnk → λ as k → ∞. We will prove that λ ∈ UTc . Indeed, assume that 21 λ ∈ UT. Then, by Proposition 3.3, there is a δ > 0 such that B(λ, δ) ⊂ UTn for all n large enough. This contradicts to λnk ∈ UTcnk for all k and lim λnk = λ. Thus, k→∞ r−1 (A, B; D, E; T) G(λ) = lim G(λnk ) = lim gnk = lim gn . k→∞ k→∞ n→∞ 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 ) n→∞ n→∞ −1 , 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 , En ; Tn ) tends to (A, B; D, E; T) but r(A, B; D, E; T) > lim inf r(An , Bn ; Dn , En ; Tn ), n→∞ even only for the right-hand side perturbation (En1 = 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     1 0 0 −1 2 + n1 0 −1 1 0 0 0  1 1 0 1 −1  ; Bn =  n ; An = A =  −1 0 1  −2 0 1 0 0 1 0 −1 0 2 + n1 −1 0 1 22     2 0 −1 1 1 0 0  0 1 −1 0 2 1  ; Dn = D =  0  B= −2 1 −1 1  ; En = 0; En = I, 0 0 2 −1 0 1 1 −1 for all n ∈ N. It is easy to see that ind{A, B} = 3 and   1  1  0 0 t−2 t−2   t2  0 −1  n(t−2) −1   G(t) =  ; Gn (t) =  t . 0 0  n(t−2) 0  1 0 0 0 n(t−2) Thus, r(An , Bn ; Dn , En ; Tn ) = (sup Gn (t) )−1 = 0 for any n ∈ N meanwhile t∈iR −1 r(A, B; D, E; T) = = 1. sup G(t) t∈iR Even in the case ind(A, B) = 1, the inverse relation of (5.7) also may not hold if equation (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       0 1 −1 0 0 0 0 1 −1   1   1  A =  1 0 0 ; Bn =   −2 n − 1  ; B = −2 −1 0 ; −1 0 0 2 1 0 2 − n1 + 1 0   0 0 0 1 −1  Dn =  1 −1 1  ; En1 =   0 0 −1 1 0 0 − n1  0  1 1 n 0     2  . 0  0 1 0 = ; E n    1 0 0 1 n It is easy to verify that ind{A, B} = 1 and  −1   −1 0 0 t+2 t+2      G(t) =   0 1 0  ; Gn (t) =  0 0 0 1 0 23  3+t n(t+2) 0  1 0 t n −t+n n  .  Thus, r(An , Bn ; Dn , En ; Tn ) = (supt∈iR Gn (t) )−1 = 0 does not tend to −1 r(A, B; D, E; T) = sup G(t) = 1 as n → ∞. t∈iR We see in two above examples that r(An , Bn ; Dn , En ; Tn ) = 0 for all n ∈ N. Thus, for the lower semi-continuity of the stability radii, we need to restrict the structure matrices Dn , En for the stability radius r(An , Bn ; Dn , En ; Tn ) > 0. By formula (5.6), this is equivalent to (5.8) sup Gn (λ) < ∞. λ∈UTcn On the other hand, Gn (λ) = Enλ (λAn − Bn )−1 Dn = Enλ P (λAn − Bn )−1 Dn + Enλ Q(λAn − Bn )−1 Dn = Enλ P (λAn − Bn )−1 Dn + (λEn1 − En2 )Q(λAn − Bn )−1 Dn Thus, by Propositions 4.12, 4.13, (5.8) holds if En1 Q = 0 in the case ind(A, B) = 1, and En1 Q = En2 Q = 0 in the case ind(A, B) > 1. Theorem 5.7. Let ind(A, B) = 1 and equation (4.1) be uniformly exponentially stable. Assume that limn→∞ (An , Bn ; Dn , En ; Tn ) = (A, B; D, E; T) and (An −A)Q = En1 Q = 0 for all n ∈ N. Then, we have r(A, B; D, E; T) = lim inf r(An , Bn ; Dn , En ; Tn ). n→∞ (5.9) Proof. Let S be defined in (4.12). Then S is open and there exists an integer number N > 0 such that S ⊂ UT ∩ UTn and σ(An , Bn ) ⊂ S for all n > N. Since En1 Q = 0 for all n ∈ N and limn→∞ En = E, it implies that En1 = En1 P , E 1 = E 1P , for all n ∈ N. Therefore, we have Gn (λ) − G(λ) =Enλ ((λAn − Bn )−1 − (λA − B)−1 )Dn + (Enλ − E λ )(λA − B)−1 Dn + E λ (λA − B)−1 (Dn − D) =En1 λP ((λAn − Bn )−1 − (λA − B)−1 )Dn + (En1 − E 1 )λP (λA − B)−1 Dn + E 1 λP (λA − B)−1 (Dn − D) − En2 ((λAn − Bn )−1 − (λA − B)−1 )Dn − (En2 − E 2 )(λA − B)−1 Dn − E 2 (λA − B)−1 (Dn − D). 24 Since limn→∞ (En , Dn ) = (E, D), sup En1 < ∞, sup En2 < ∞, 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 > 0 such that G(λ) + C( An − A + Bn − B + Dn − D + En − E ) Gn (λ) for all λ ∈ S c , n > N. Since UTn ⊂ S c , r−1 (An , Bn ; Dn , En ; Tn ) = sup λ∈UTcn sup λ∈UTcn Gn (λ) G(λ) + C( An − A + Bn − B + Dn − D + En − E ) = r−1 (A, B; D, E; Tn ) + C( An − A + Bn − B + Dn − D + En − E ), for all n > N. This implies that lim sup r−1 (An , Bn ; Dn , En ; Tn ) n→∞ lim sup r−1 (A, B; D, E; Tn ). n→∞ 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, −1 lim inf r(An , Bn ; Dn , En ; Tn ) n→∞ = lim sup r−1 (An , Bn ; Dn , En ; Tn ) n→∞ −1 r (A, B; D, E; T), or equivalently lim inf r(An , Bn ; Dn , En ; 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, by using Proposition 4.13 we obtain 25 Theorem 5.8. Let ind(A, B) > 1. Assume that limn→∞ (An , Bn ; Dn , En ; Tn ) = (A, B; D, E; T) and (An − A)Q = (Bn − B)Q = En1 Q = En2 Q = 0 for all n ∈ N. Then, we have r(A, B; D, E; T) = lim inf r(An , Bn ; Dn , En ; Tn ). n→∞ With respect to only structured perturbations on the right-hand side (Bn = Bn + Dn Σn En ), we have the following corollary. Corollary 5.9. Let ind(A, B) = 1 and equation (4.1) be uniformly exponentially stable. Assume that limn→∞ (Bn ; Dn , En ; Tn ) = (B; D, E; T). Then, we have lim r(A, Bn ; Dn , En ; 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 > 0 to this equation we have A x((m + 1)h) − x(mh) = Bx(mh), m ∈ N. h This equation can be considered as a dynamic equation Ax∆ (t) = Bx(t) (5.11) on the time scale Th = hN. It can be easily seen that lim Th = R. Therefore, by h→0 Proposition 5.4, lim r(A, B; D, E; Th ) = r(A, B; D, E; R). Moreover, if we use the h→0 explicit Euler method with the mesh step h = 1 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 , En ; Tn ) be the stability radius of equation (5.12) on Tn . By Corollary 5.9 we obtain lim r(A, Bn ; Dn , En ; Tn ) = n→∞ r(A, B; D, E; R) provided ind(A, B) = 1 and lim (Bn , Dn , En ) = (B, D, E). n→∞ 26 6 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 a main result, 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 implicit difference equations which is more computable. 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Control Inform., to appear 30 [...]... σ(B) for σ(I, B) Theorem 4.2 (See [19, Theorem 3.2]) The implicit dynamic equation (4.1) is uniformly exponentially stable if and only if σ(A, B) ⊂ UT Now, consider the implicit dynamic equations An x∆n (t) = Bn x(t), (4.5) where An , Bn ∈ Cn×n and t ∈ Tn By Theorem 4.2, the exponential stability depends on the spectrum of the matrix pair (An , Bn ) and the stability domain UTn Example 4.3 Consider... ) be the stability radius of equation (5.12) on Tn By Corollary 5.9 we obtain lim r(A, Bn ; Dn , En ; Tn ) = n→∞ r(A, B; D, E; R) provided ind(A, B) = 1 and lim (Bn , Dn , En ) = (B, D, E) n→∞ 26 6 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... Bn ) ⊂ S ⊂ UTn for all n > N Thus, by Theorem 4.2, equation (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 1 Proposition 4.9 Let ind(A, B) > 1 Assume that limn→∞ (An , Bn ) = (A, B) and (An − A)Q =... depend on the choice of W and T Moreover, the solution x(t) satisfies Qx(t) = 0 for all t ∈ Tt0 and the initial condition x(t0 ) = P x0 must hold (see [19, 23]) Let x(t, s, P x0 ) be the solution of (4.1) with the initial value x(s, s) = P x0 According to Definition 2.1, we get the following definition of the exponential stability: Definition 4.1 The implicit dynamic equation (4.1) is called uniformly exponentially... 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: This work was supported financially by Vietnam National Foundation for. .. Differential Equations 230:579–599 [18] Du NH (2008) Stability radii of differential-algebraic equations with structured perturbations Systems Control Lett 57:546–553 28 [19] Du NH, Thuan DD, Liem NC (2011) Stability radius of implicit dynamic equations with constant coefficients on time scales Systems Control Lett 60:596–603 [20] Du NH, Linh VH, Mehrmann V, Thuan DD (2013) Stability and robust stability of. .. 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,... properties of the stability domains as well as the relation between the spectra of matrix pair have been investigated As a main result, 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 implicit difference equations which is... (1989) Robustness of stability of timevarying linear systems J Differential Equations 82:219–250 [29] Hinrichsen D, Son NK (1989) The complex stability radius of discrete-time systems and symplectic pencils Proceedings of the 28th IEEE Conference on Decision and Control 1-3:2265–2270 [30] Jacob B (1998) A formula for the stability radius of time-varying systems J Differential Equations 142:167–187 [31]... ind(A, B) = 2 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 equation (4.1) be uniformly exponentially stable but (4.5) is not uniformly exponentially stable for all 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 ... by one of implicit difference equations which is more computable As a future work, an analysis of the exponential stability and the stability radii for time-varying implicit dynamic equations on. .. 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. .. 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