DSpace at VNU: On data dependence of stability domains, exponential stability and stability radii for implicit linear dynamic equations

DSpace at VNU: On data dependence of stability domains, exponential stability and stability radii for implicit linear dy...

DOI 10.1007/s00498-016-0164-7

O R I G I NA L A RT I C L E

On data dependence of stability domains, exponential

stability and stability radii for implicit linear dynamic

Abstract We shall deal with some problems concerning the stability domains, the

spectrum of matrix pairs, the exponential stability and its robustness measure for linearimplicit dynamic equations of arbitrary index First, some characterizations of thestability domains corresponding to a convergent sequence of time scales are derived.Then, we investigate how the spectrum of matrix pairs, the exponential stability andthe stability radii for implicit dynamic equations depend on the equation data whenthe structured perturbations act on both the coefficient of derivative and the right-handside

Keywords Implicit dynamic equations· Time scales · Convergence · Stabilitydomain· Spectrum · Exponential stability · Stability radius

Mathematics Subject Classification 06B99· 34D99 · 47A10 · 47A99 · 65P99

2 Department of Mathematics, Mechanics and Informatics, Vietnam National University,

334 Nguyen Trai Str., Hanoi, Vietnam

3 School of Applied Mathematics and Informatics, Hanoi University of Science and Technology,

1 Dai Co Viet Str., Hanoi, Vietnam

1 Introduction

In this paper, we study the stability domains, the spectrum of matrix pairs, the nential stability and the stability radii for sequence of implicit dynamic equations ontime scales of the form

where A n , B n ∈ Cm ×m , n ∈ N and t ∈ T n The leading coefficients A n , n ∈ N, are

allowed to be singular matrices

Implicit dynamic equations of the form (1.1) can be considered as a unified formbetween linear differential algebraic equations (DAEs) and linear implicit differenceequations Therefore, they play an important role in mathematical modeling arising inmultibody mechanics, electrical circuits, prescribed path control, chemical engineer-ing, 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 wellnumerical treatments of implicit dynamic equations These difficulties are typicallycharacterized 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 time systems, see [1,3,13,24,25] It enables us to analyze the stability of dynamicalsystems on non-uniform time domains [34] Based on this theory, stability analysis ontime scales has been studied for linear time-invariant systems [32], linear time-varyingdynamic 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 spectralcharacterizations for the exponential stability of dynamic systems and the stability

discrete-of numerical methods for approximating solutions relate to the stability domains discrete-ofscalar equations Therefore, it is meaningful to investigate the behavior of the stabilitydomains 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., stabil-ity) when the system comes under the effect of uncertain perturbations because thereare parameters which can be determined only by experiments or the remainder partignored during linearization process That is the reason why we are interested in inves-tigating the robust stability of the uncertain equations subjected to general structuredperturbations of the form



with

[A n ,  B n ]  [A n , B n ] + D n  n E n , (1.3)where  n , n ∈ N, are unknown disturbance matrices; D n , E n are known scalingmatrices defining the “structure” of the perturbations This leads to the notion ofstability 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-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 forimplicit dynamic equations on time scaleR (described by DAEs), see [5,7,9,16–18]

In [10], Chyan, Du and Linh have investigated the data dependence of the exponentialstability and the stability radii for linear time-varying DAEs of index 1 and withrespect to only the right-hand side perturbations Recently, Du et al [19] have derivedthe formula of stability radius for linear implicit dynamic equations with arbitraryindex subjected to general structured perturbation acting on both the coefficient ofderivative and the right-hand side Therefore, it is meaningful to continue studying thedata 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 domainscorresponding to a convergent sequence of time scales Then, we continue to analyzehow the spectrum of matrix pairs and the exponential stability of (1.1) depend on data

of the stability radii of Eq (1.2) with general structured perturbations of the form(1.3) when the data(A n , B n ; D n , 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 toapproximate them As a corollary, we will show that the stability radii of implicitdifference equations obtained from DAEs by the explicit Euler methods will tend tothe stability radius of DAEs when the mesh step tends to zero

The paper is organized as follows In the next section, we summarize some inary results on time scales and the exponential stability In Sect.3, we derive somecharacterizations of the stability domains corresponding to convergent sequences oftime scales Section4deals with the data dependence of the spectrum of matrix pairsand the exponential stability In Sect.5, the data dependence of the stability radii isanalyzed The last section gives some conclusions and open problems

prelim-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 thestandard topology onR 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,

right-scattered and left-scattered.T is said to be the time scale with bounded graininess

if supt∈Tμ(t) < ∞.

A function f defined on T is regulated if there exist a sided limit at every

left-dense point and a right-sided limit at every right-left-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 1 + μ(t) f (t) > 0 for every t ∈ T Use R+to denote the set of positivelyregressive functions fromT to R

A function f : T → Cd 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.

IfT = 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

function, its antiderivative exists Therefore, we can define the delta integral of f on [a, b)Tby

LetT be an unbounded above time scale, that is sup T = ∞ For any λ ∈ C, the

solution of the dynamic equation

with the initial condition x (s) = 1, defines a so-called exponential function with the

parameterλ We denote this exponential function by e λ (t, s) The exponential function

with parameterλ can be presented by the formula

e λ (t, s) = exp

 t s



(2.2)with

Note that|e λ (t, s)| = |e λ (t, s)| for any λ ∈ C Since ζ λ (x)  |λ| for all x 

constant c > 1 such that ζ −α (x)  −α  ζ −α/c (x) for all x : 0  x  sup t∈Tμ(t).

Therefore, for all t  s, t, s ∈ T,

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

require-ments 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 nentially stable if there exists a constantα > 0 with −α ∈ R+and K > 0 such that

expo-for every s  t, s, t ∈ T t0, the inequality

x(t, s, x0 )  K x0e−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) isuniformly exponentially stable if and only if the scalar Eq (2.1) is uniformly expo-nentially stable for anyλ ∈ 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 distancebetween two time scalesT1andT2, which is defined by

d H (T1, T2) := max

sup

expo-soUT is symmetric with respect to the real line on the complex plan and λ ∈ UT

By the definition, it follows thatλ ∈ UT if and only if L(λ, T) < 0 Indeed, if λ ∈ UT

then there exist K , α > 0 with −α ∈ R+such that|e λ (t, s)|  K e −α (t, s) and hence,

Therefore,|e λ (t, s)|  exp{−α(t − s)} for all t, s ∈ T : t − s > H If t − s  H

then by virtue of the inequalityζ λ (x)  |λ| for any x  0, we have |e λ (t, s)| 

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),|e λ (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)  0.

{λ n} ⊂ UT such that lim

n→∞λ n = λ Let λ = a + ib with b = 0 and λ n = a n + ib n

Using the Lagrange finite increment formula, for all x > 0, we have

ζ λ n (x) − ζ λ (x) = x (|λ n|2− |λ|2) + 2(a n − a)

2(1 + 2x(a + θ(a n − a)) + x2(|λ|2+ θ(|λ n|2− |λ|2))) , θ ∈ (0, 1).

Since 0< 1 + 2xa + x2|λ|2for all x  0, we can choose an n0∈ N and a constant

c1 > 0 such that c1 < 1 + 2x(a + θ(a n − a)) + x2(|λ|2+ θ(|λ n|2− |λ|2)) for all

0 x  μ∗ and n > n0 Thus, for any > 0, there exists n1> n0satisfying

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)) < c2for all 0 x  μ∗.Thus,

ζ λ  (x) − ζ λ (x) < a  − a

This implies that

 t

s

ζ λ  (μ(τ))τ <

 t s

T, sup{μ(t) : t ∈ T} < ∞ Therefore, it is easy to prove that if lim

n→∞Tn = T thensup{μn (t) : t ∈ T n , n ∈ N} < ∞ Define

ζ λ (μ(h))h

< 2(t − s) + 8M t − s δ d H (T, T n ), (3.2) for all n > n0, λ ∈ K, t > s, where M = sup λ∈K,x∈[0,μ∗ ]|ζ λ (x)| Moreover,

|L(λ, T n ) − L(λ, T)|  2 + 8M

for all n > n0, λ ∈ K

that the function

existsδ = δ() > 0 such that if |u − v| < δ then |ζ λ (u) − ζ λ (v)| <  for any λ ∈ K

Since limn→∞Tn = T, we can choose n0 such that d H (T, T n ) < 2δ when n > n0.

Fix t0  s < t; s, t ∈ [0, ∞) and n > n0 Denote A1 = {h ∈ T n ∩ [s, t] :

that 0  μ(h)  μ n (h) for all h ∈ T n If h ∈ A2thenμ(h)  μ n (h) < δ, which

implies|ζ λ (μ(h)) − ζ λ (μ n (h))| <  On the other hand, the cardinal of A1, say r , is

For any h ∈ T, there exists a unique u ∈ T n , say u = γ T,T n (h), such that either

h = u or h ∈ (u, σ n (u)) It is easy to check that the function γ T,T n (h) is rd-continuous

where a ∧ b = min{a, b} For h ∈ T ∩ ([s, s1 ) ∪ [σ n (s i ), s i+1) ∪ [t ∧ σ n (s r ), t)), i =

ζ λ (μ(h))h

< (t − s) + 4M t − s δ d H (T, T n ).

IfTn⊂ T we put Tn= Tn ∪ T It is easy to see that

ζ λ (μ(h))h

< 2 + 8M δ d H (T, T n ).

By (3.1), we obtain (3.3) The proof is complete Denote byUTn (resp.UT) the domain of stability of the time scale Tn (resp.T).Now, we will study relationship between the stability domainsUTnandUT when Tn

tends toT

Proposition 3.3 Suppose that lim

n→∞Tn = T Then, for any λ ∈ UT we can find a

n >n λ

UTn

3.1, there exists aδ1> 0 satisfying B(λ, δ1) ⊂ UT and

By choosingδ λ := min{δ1 , |λ|3 } > 0 we see that B(λ, δ λ ) ⊂ UT\R Using Lemma

3.2with K = B(λ, δ λ ) and  = −L(λ)8 we can find aδ2> 0 and n0such that

L (λ, T n ) < L(λ, T) + −L(λ, T)

δ2

for all n > n0andλ ∈ B(λ, δ λ ) We choose n λ > n0such that d H (T, T n ) < −δ2L (λ, T)

32M for any n > n λ From (3.5) and (3.6) we get

This means that B (λ, δ λ ) ⊂ UT n for all n > n λ

We now consider the caseλ ∈ UT ∩ R Since UT is an open set, there exists δ3> 0

such that B (λ, δ3) ⊂ UT Let λ1 = λ + i δ3

exist n λ1 > 0 and 0 < δ λ1 < δ3/2 such that B(λ1, δ λ1) ⊂ UT ∩ UT n for all n > n λ1.

SinceUTnis symmetric with respect to the real axis, the segment[λ, λ] ⊂ UTn, forallλ∈ B(λ1 , δ λ1) Thus, B(λ, δ λ1) ⊂ UT ∩ UT n for all n > n λ1 The proposition is

To prove the second one, letλ ∈∞n=1

m nUTm \R By the definition, there is

a sequence{n k}∞

k=1, n k → ∞ such that λ ∈ UT n k \R for all k Using again inequality

(3.3), for any > 0, there exist δ > 0, n0∈ N such that

4 Data dependence of spectrum and exponential stability

Consider the implicit dynamic equation on time scaleT



where x (t) ∈ C m , t  s, t ∈ T t0, and A, B ∈ C m ×mare 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 ∈ C n ,n such that

where I r , I m −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

and Q is the projection onto ker [(α A+ B)−1A]kalong the space Im[(α A+ B)−1A]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  s, t ∈ T t0 and the initial condition x (s) = x0= P x0must hold (see [19,23]) Thus, for each s ∈ Tt0, x0 = P x0 ∈ Cm , a function t →

x(t, s,  P x0), t  s, is called a solution of (4.1) if x (·, s,  P x0) is delta differentiable

and satisfies (4.1) with the initial condition x (s, s,  P x0) =  P x0 The existence and

uniqueness of solutions of Eq (4.1) have been shown in detail in [19] According toDefinition2.1, we get the following definition of the exponential stability:

Definition 4.1 The implicit dynamic Eq (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 ∈ T t0, the inequality

x(t, s,  P x0)  K  P x0e−α(t, s) (4.4)

holds for any x0∈ Rm

A complex numberλ 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 Eq (4.1) is uniformly

Now, consider the sequence of implicit dynamic equations

where A n , B n∈ Cn ×n and t∈ Tn By Theorem4.2, the exponential stability depends

on the spectrum of the matrix pair(A n , B n ) and the stability domain UT n

Example 4.3 Consider the matrix pairs (A n , B n ), (A, B) with

Moreover, ifTn = T = R then Eq (4.1) is uniformly exponentially stable but (4.5)

is not uniformly exponentially stable for each n∈ N

Moreover, ifTn= 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 exponentialstability of implicit dynamic equations are very sensitive to change of the coefficients.The reasons is that they contain not only ordinary dynamic equations and algebraicconstraints, but also hidden constraints which involve derivatives of several solutioncomponents as well It is well know that an arbitrary small perturbation may destroythe index of equations as well as the stability of solutions, even in the case of linearconstant-coefficient DAEs, see [9,20,21] That is why in the following we restrict thedirection of the matrix pairs(A n , B n ), n ∈ N when (A n , B n ) 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

is not uniformly exponentially stable for each n∈ N

Two above examples show that the spectrum of matrix pairs and the exponentialstability of implicit dynamic equations are very... change of the coefficients.The reasons is that they contain not only ordinary dynamic equations and algebraicconstraints, but also hidden constraints which involve derivatives of several solutioncomponents... solutioncomponents as well It is well know that an arbitrary small perturbation may destroythe index of equations as well as the stability of solutions, even in the case of linearconstant-coefficient