DSpace at VNU: Stability radius of implicit dynamic equations with constant coefficients on time scales

Systems & Control Letters 60 (2011) 596–603

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Systems & Control Letters
journal homepage: www.elsevier.com/locate/sysconle

Stability radius of implicit dynamic equations with constant coefficients
on time scales✩
Nguyen Huu Du a,∗ , Do Duc Thuan b , Nguyen Chi Liem a
a

Department of Mathematics, Mechanics and Informatics, Viet Nam National University, 334 Nguyen Trai, Hanoi, Viet Nam

b

Department of Applied Mathematics and Informatics, Hanoi University of Technology, 1 Dai Co Viet, Hanoi, Viet Nam

article

abstract

info

Article history:
Received 23 July 2010
Received in revised form
24 April 2011
Accepted 24 April 2011
Available online 24 May 2011

This paper deals with the stability radii of implicit dynamic equations on time scales when the structured
perturbations act on both the coefficient of derivative and the right-hand side. Formulas of the stability
radii are derived as a unification and generalization of some previous results. A special case where the real
stability radius and the complex stability radius are equal is studied. Examples are derived to illustrate
results.
© 2011 Elsevier B.V. All rights reserved.

Keywords:
Time scales
Implicit linear dynamic equation
Index of the pencil of matrices
Exponentially stable
Stability radius

1. Introduction

then we have a formula in [3] for computing the complex stability
radius

In the past decades, there have been extensive works on studying of robust measures, where one of the most powerful ideas is
the concept of the stability radii, introduced by Hinrichsen and
Pritchard [1]. The stability radius is defined as the smallest (in
norm) complex or real perturbations destabilizing the equation.
In [2], the authors consider the equation x′ = Bx and assume that
the perturbed equation can be represented in the form
x′ = (B + DΣ E )x,

(1.1)

where Σ is an unknown disturbance matrix and D, E are known

scaling matrices defining the ‘‘structure’’ of the perturbation. The
complex stability radius is then given by

[

max ‖E (tI − B)−1 D‖
t ∈iR

]−1

.

(1.2)

If the nominal equation is the difference equation xn+1 = Bxn with
a structured perturbation of the form
xn+1 = (B + DΣ E )xn ,

(1.3)

[
max

ω∈C:|ω|=1

‖E (ωI − B)−1 D‖

∗

Corresponding author. Fax: +84 4 8588817.

E-mail addresses: , (N.H. Du).

0167-6911/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.sysconle.2011.04.018

.

(1.4)

Moreover, in recent years, several technical problems in electronic
circuit theory and robotic designs lead to the problem of investigating the differential–algebraic equation f (x′ (t ), x(t )) = 0, where
the leading term x′ cannot be explicitly solved from x(t ). The linear
form of this equation is
Ax′ (t ) = Bx(t ),

(1.5)

with A and B denoting two constant matrices. Assume that Eq. (1.5)
is subjected to perturbations of the form
Ax′ (t ) = (B + DΣ E )x(t ).

(1.6)

Then the formula of the complex stability radius is given by
(see [4])

[

max ‖E (tA − B)−1 D‖

] −1

t ∈iR

✩ This work was done under the support of the Grant NAFOSTED 2011.

]−1

.

(1.7)

When the nominal equation is the difference equation Axn+1 = Bxn
with the structured perturbation of the form
Axn+1 = (B + DΣ E )xn ,

(1.8)


N.H. Du et al. / Systems & Control Letters 60 (2011) 596–603

we obtained an expression in [5] for the complex stability radius
given by

[
max

ω∈C:|ω|=1

‖E (ωA − B)−1 D‖

]−1

.

(1.9)

Earlier results of stability radii for time-varying equations can
be found, e.g., in [6,7]. The most successful attempt for finding a
formula for the stability radius was an elegant result given by Jacob [7]. Using this result, the notion and formula of the stability radius were extended to linear time-invariant differential–algebraic
equations [8,9,4]; and to linear time-varying differential and difference–algebraic equations [10,5].
On the other hand, in order to unify the continuous and discrete
analysis, the theory of the analysis on time scales was introduced
by Stefan Hilger in his Ph.D thesis in 1988 (supervised by Bernd
Aulbach) [11] and has received a lot of attention. By using the
notation of the analysis on time scales, Eqs. (1.1) and (1.3) can be
rewritten under the unified form

= (B + DΣ E )x,

x

(1.10)

and also Eqs. (1.6) and (1.8) become
Ax

= (B + DΣ E )x.

A formula of the stability radius for (1.10) is derived recently in [12]

and it is given by

Definition 2.1 (Delta Derivative). A function f : T → R is called
delta differentiable at t if there exists a scalar f (t ) such that for all
ϵ>0

|f (ς (t )) − f (s) − f (t )(ς (t ) − s)| ⩽ ϵ|ς (t ) − s|
for all s ∈ (t − δ, t + δ) ∩ T and for some δ > 0. The scalar f (t )
is called the delta derivative of f at t.
If T = R then delta derivative is f ′ (t ) from continuous calculus;
if T = Z then the delta derivative is the forward difference, f ,
from discrete calculus.
A point t ∈ T is said to be right-dense if ς (t ) = t, right-scattered
if ς (t ) > t, left-dense if ϱ(t ) = t and left-scattered if ϱ(t ) < t.
A function f defined on T is rd-continuous if it is continuous at
every right-dense point and if the left-sided limit exists at every
left-dense point. For any rd-continuous functions p(·) from T to R,
the solution of the dynamic equation x = p(t )x, with the initial
condition x(s) = 1, defines a so-called exponential function. We
denote this exponential function by ep (t , s). For the properties of
exponential function ep (t , s) the interested reader can see [13–15].
Denote T+ = [t0 , ∞) ∩ T. We consider the dynamic equation
on the time scale T
x

= f (t , x),

(2.14)

where f : T×R → R is a rd-continuous function and f (t , 0) = 0.

For the existence, uniqueness and extensibility of solution of
Eq. (2.14) we refer to [14]. A function f from T to R is positively
regressive if 1 + µ(t )f (t ) > 0 for every t ∈ T. We denote R+
the set of positively regressive functions from T to R. For any
τ ∈ T+ , let x(t ) = x(t , τ , x0 ) be a solution of (2.14) with the
initial condition x(τ , τ ) = x0 ∈ Rd . On the exponential stability of
dynamic equations on time scales, we use the following definition,
see, e.g. [16,11,17]:
d

[

max ‖E (tI − B)−1 D‖
t ∈Γus

]−1

,

(1.11)

where Γus is the stability domain of the time scale T.
The purpose of this paper is to present a unified formula for
(1.2), (1.4), (1.7), (1.9) and (1.11) and to generalize them by studying the stability radius of the implicit dynamic equations on time
scales
Ax (t ) = Bx(t ),

(1.12)

under the general structured perturbations of the form

[A, B]

[A˜ , B˜ ] = [A, B] + DΣ E .
(1.13)
When T = R (resp. T = N), we consider it as a generalization of
Eqs. (1.1) and (1.6) (resp. Eqs. (1.3) and (1.8)).
The difficulty we are faced when dealing with this problem is
that although A, B, D, E are constant matrices, the structure of time
scale (also the stability domain) is rather complicated and it can
make Eq. (1.12) become a time-varying equation. Moreover, the
disturbances affect not only the term on the right, but also the
coefficient of the derivative on the left-hand side and it seems that
we are working with an ill-posed problem.
This paper is organized as follows. In Section 2, we summarize
some preliminary results on time scales. In Section 3, by defining
the so-called domain of the uniformly exponential stability of time
scales, we give the formulas of the stability radii of Eq. (1.12), where
the general structured perturbations are considered. Section 4 is
concerned with special classes of {A, B} where the complex and
real stability radii are equal.
2. Preliminaries
A time scale is a nonempty closed subset of the real numbers
R and we usually denote it by the symbol T. The most popular
examples are T = R and T = Z. We assume throughout that a time
scale T inherits the topology from the standard topology of the real
numbers. We define the forward jump operator ς : T → T by
ς(t ) = inf{s ∈ T : s > t } (supplemented by inf ∅ = sup T) and the
backward jump operator ϱ : T → T by ϱ(t ) = sup{s ∈ T : s < t }
(supplemented by sup ∅ = inf T). The positively graininess function

µ : T → R+ ∪ {0} is given by µ(t ) = ς (t ) − t. For our
purpose, we will assume that the time scale T is unbounded above,
i.e., sup T = ∞.

597

d

Definition 2.2 (Exponential Stability). The dynamic equation (2.14)
is called exponentially stable if the condition

• for every τ ∈ T+ there exists an N = N (τ ) ⩾ 1 satisfying
‖x(t , τ , x0 )‖ ⩽ N (τ )‖x0 ‖e−α (t , τ )

(2.15)

for all t ⩾ τ , t ∈ T and x0 ∈ R , where x(t , τ , x0 ) is the
solution of (2.14) with the initial condition x(τ , τ ) = x0
+

d

holds for some α > 0 such that −α ∈ R+ . If the constant N can be
chosen independent of τ ∈ T+ then the dynamic equation (2.14)
is called uniformly exponentially stable.
Note that the condition −α ∈ R+ is equivalent to µ(t ) ⩽ α1 .
This means that we are working on time scales with bounded
graininess. Beside this definition one can find other definitions
of exponential stability in [18–20] where instead of using the
exponential function e−α (t , τ ) on time scale, one uses the classical

exponential function exp{−α(t − τ )} in (2.15). However, it is easy
to prove that these definitions are equivalent.
We now consider the condition of exponential stability for linear time-invariant equations
x

= Ax,

(2.16)

where A ∈ Kd×d (K = R or K = C). We denote the set of the
eigenvalues of A by σ (A).
The following theorem can be proved by a similar way as in [19],
although we use the exponential function on time scales to define
exponential stability.
Theorem 2.3 (See [19, Lemma 6.1]). The linear equation (2.16) is
uniformly exponentially stable if and only if for every λ ∈ σ (A), the
scalar equation x = λx is uniformly exponentially stable.


598

N.H. Du et al. / Systems & Control Letters 60 (2011) 596–603

It is easy to give an example where on the time scale T, the
scalar dynamic equation x = λx is exponentially stable but it is
not uniformly exponentially stable. Indeed, denote ((a, b)) = {n ∈
N : a < n < b}. Consider the time scale

T=

n

Let λ = −2 and τ ∈ T, say 2 ⩽ τ < 2
. We can choose α = −1
and N = 2m+1 to obtain |eλ (t , τ )| ⩽ Ne−1 (t , τ ). However, it is not
possible to choose N independent of τ .
Now, we denote
m

1
−1

Q A−
U ) T −1
1 A = T diag(0, (U − Im−r )

is a nilpotent matrix and



[22n , 22n+1 ] ((22n+1 , 22n+2 )).
n

Also by the decomposition (3.2) and the definition (3.3) of 
Q we
see that

m+1

S = {λ ∈ C, the scalar equation x

= λx is uniformly exponentially stable}.
The set S is called the domain of the uniform exponential
stability of the time scale T. By the definition, if λ ∈ S, there
exist α > 0 and N ⩾ 1 satisfying −α ∈ R+ and |eλ (t , τ )| ⩽
Ne−α (t , τ ) for all t ⩾ τ . As a corollary of proposition 3.1 in [21],
we have the following result.
Theorem 2.4. S is an open set in C.
For illustrating the domain of the uniform exponential stability
S of the time scale T, we consider some simple cases.

• When T = R then S = {λ ∈ C, Re λ < 0}.
• When T = hZ(
h > 0) then S = {λ ∈ C, |1 + λh| < 1}.
∞
• When T =
k=0 [2k, 2k + 1] then S = {λ ∈ C, Re λ
+ ln |1 + λ| < 0}.

1
−1 −1

Q A−
)T .
1 B = T diag(0, (U − Im−r )
 
y(t )
Denoting T −1 x(t ) = z (t ) where z (t ) ∈ Km−r , we obtain

Uz (t ) = z (t ).

(3.6)

It is easy to see that this equation has a unique solution z (t ) ≡
0. Therefore, the solution x(t ) of (3.1) with the initial condition

P (x(t0 ) − x0 ) = 0 exists in T+ = [t0 , ∞) ∩ T, and it satisfies



y(t )
−1

Q x(t ) = T diag(0, Im−r )T x(t ) = T diag(0, Im−r )
= 0,
0

for all t ∈ T .
+

(3.7)

In particular, the initial condition x(t0 ) = 
Px0 must hold. Let
x( t , τ , 
Px0 ) be the solution of (3.1) with the initial value x(τ , τ ) =

Px0 . According to Definition 2.2, we get the following definition of
exponential stability:
Definition 3.1. The implicit dynamic equation (3.1) is called

exponentially stable if the condition

• for every τ ∈ T+ and x0 ∈ Rm there exists an N = N (τ ) ⩾ 1
satisfying

3. Stability radii of implicit dynamic equations on time scales

‖x(t , τ , 
Px0 )‖ ⩽ N (τ )‖
Px0 ‖e−α (t , τ )
(3.8)
+
for all t ⩾ τ , t ∈ T where x(t , τ , 
Px0 ) is the solution of (3.1)
with the initial value x(τ , τ ) = 
Px0

Consider the implicit dynamic equation on time scale T
Ax (t ) = Bx(t ),

(3.1)

where x(t ) ∈ K , and {A, B} ∈ K
are constant matrices;
underlying field K is either real or complex. We assume that the
pencil of matrices {A, B} is regular (that is, det(λA − B) ̸≡ 0) and the
index of {A, B} is k ⩾ 1. The Kronecker decomposition of the pencil
of matrices {A, B} indicates that there exists a pair of nonsingular
matrices W , T such that
m×m

m

A = W diag(Ir , U )T −1 ,

B = W diag(B1 , Im−r )T −1 ,

(3.2)

where Ir is the unit matrix in Kr ×r and B1 is a matrix in Kr ×r .
Further, U ∈ K(m−r )×(m−r ) is a nilpotent matrix whose nilpotency
degree is exactly k. Denote

holds for some α > 0 such that −α ∈ R+ . If the constant N can be
chosen independent of τ then the implicit dynamic equation (3.1)
is called uniformly exponentially stable.
We denote by σ (C , D) the spectrum of the pencil {C , D}, i.e., the
set of all solutions of the equation det(λC − D) = 0. When C = I,
we write simply σ (D) for σ (I , D).
Theorem 3.2. The implicit dynamic equation (3.1) is uniformly
exponentially stable if and only if σ (A, B) ⊂ S, where S is the domain
of the uniformly exponential stability of the time scale T.



y(t )
z (t )




Q = T diag(0r , Im−r )T −1 ,

Proof. Let x(t ) = T


P = Im − 
Q = T diag(Ir , 0m−r )T −1 .

decomposition (3.2) and (3.3) we get

(3.3)

It is known that for any α ∈ K such that α A + B is nonsingular, one
has

Km = ker[(α A + B)−1 A]k ⊕ im[(α A + B)−1 A]k ,

A1 = A − B
Q = W diag(Ir , U − Im−r )T −1 .

k

(3.4)

Since U is a nilpotent matrix, it is clear that A1 is invertible. Further,
1
−1 

by using (3.2) and (3.3) it follows that 

PA−
1 A = A1 AP = P and
−
1
−
1
−
1
1

PA1 B = A1 B
P =
PA1 B
P. Multiplying both sides of (3.1) by 
PA−
1
−
1
and 
Q A1 respectively we obtain



(
Px) (t ) = 
PA1 B(
Px)(t ),
(
Q A−1 Ax) (t ) = 
Q A−1 Bx(t ).

−1

1

1

(3.5)



y(t )
0



. By


PA−1 B = T diag(B1 , 0m−r )T −1 .
1

From (3.5) and (3.6) it follows that Eq. (3.1) is equivalent to



and 
Q is the projection onto ker[(α A + B) A] along the space
im[(α A + B)−1 A]k . In particular, 
Q does not depend on the choice
of W and T . Let

−1

. Then, we have 
Px(t ) = T

y (t ) = B1 y(t ),
z (t ) ≡ 0.

(3.9)

Therefore, Eq. (3.1) is uniformly exponentially stable if and only if
the linear equation y (t ) = B1 y(t ) is so. By Theorem 2.3, this is
equivalent to σ (B1 ) ⊂ S. On the other hand,

λA − B = W diag(λIr − B1 , λU − Im−r )T −1 .
This implies that
det(λA − B) = 0 ⇐⇒ det(λIr − B1 ) = 0.
Thus, σ (A, B) = σ (B1 ) and the uniformly exponential stability of
Eq. (3.1) is equivalent to σ (A, B) ⊂ S. The proof is complete.


N.H. Du et al. / Systems & Control Letters 60 (2011) 596–603

599

Now, we consider Eq. (3.1) subjected to general structured perturbations of the form

Conversely, take ϵ > 0 and a λ0 ∈ C \ S satisfying



Ax (t ) = 
Bx(t ),

(3.10)

−1

⩽
‖Eλ0 (λ0 A − B)−1 D‖

(3.11)

Following the same argument as in [15], we find a vector u ∈
Cl satisfying ‖u‖ = 1 and

with

[
A, 
B] = [A, B] + DΣ E ,



sup ‖Eλ (λA − B)−1 D‖

λ∈C\S

where D ∈ Km×l , E ∈ Kq×2m , the perturbation Σ ∈ Kl×q . The matrix DΣ E is called a structured perturbation of the Eq. (3.1). If we
let E = [E1 , E2 ] with E1 , E2 ∈ Kq×m then (3.11) is equivalent to

‖Eλ0 (λ0 A − B)−1 Du‖ = ‖Eλ0 (λ0 A − B)−1 D‖.


A = A + D Σ E1 ,

y∗ (Eλ0 (λ0 A − B)−1 Du) = ‖Eλ0 (λ0 A − B)−1 Du‖


B = B + D Σ E2 .

A + D A ΣA E A ,

B

= ‖Eλ0 (λ0 A − B)−1 D‖.
Consider

B + DB ΣB EB ,

where EA ∈ Cq1 ×m , EB ∈ Cq2 ×m , DA = DB ∈ Cm×l , can be rewritten in the form (3.11) with D = DA = DB , Σ = [ΣA , ΣB ], E =
diag(EA , EB ).
We define

−1 ∗

uy ,
Σ = − ‖Eλ0 (λ0 A − B)−1 D‖

(3.13)

x = (λ0 A − B)−1 Du.
It is clear that

ΞK = {Σ ∈ K


−1
‖u‖‖y∗ ‖
‖Σ ‖ ⩽ ‖Eλ0 (λ0 A − B)−1 D‖

−1
,
= ‖Eλ0 (λ0 A − B)−1 D‖

Definition 3.3. The stability radius of Eq. (3.1) under structured
perturbations of the form (3.11) is defined by

and

l× q

: Eq. (3.10) is either irregular
or not uniformly exponentially stable}.

rK (A, B; D, E ) = inf{‖Σ ‖ : Σ ∈ ΞK },

λI



Let us use the notation Eλ = E −Im . We have the following
m
theorem.
Theorem 3.4. The complex stability radius of Eq. (3.1) under structured perturbations of the form (3.11) is given by the formula
rC (A, B; D, E ) =



sup ‖Eλ (λA − B)−1 D‖

 −1

λ∈∞∪∂ S

.

(3.12)

Proof. Let Σ ∈ Cl×q be such that the perturbed equation (3.10) is
irregular or it is regular but not uniformly exponentially stable. In
both cases, we can always choose an eigenvalue λ0 ∈ σ (
A, 
B)∩(C \
S ) and an eigenvector x ̸= 0 corresponding to λ0 , i.e., (λ0
A −
B)x =
0. From (3.11) this yields

[
]

[
]
λ I
λ I
λ0
A −
B = [
A, 
B] 0 m = ([A, B] + DΣ E ) 0 m
−I m
−I m
= λ0 A − B + DΣ Eλ0 .

−u
‖Eλ (λ0 A − B)−1 D‖ = −u.
‖Eλ0 (λ0 A − B)−1 D‖ 0

(3.14)

Since u ̸= 0,


 −1
.
‖Σ ‖ ⩾ ‖Eλ0 (λ0 A − B)−1 D‖
Combining these inequalities we obtain


 −1
‖Σ ‖ = ‖Eλ0 (λ0 A − B)−1 D‖

.
Furthermore, from (3.13) and (3.14) it follows that (λ0 A − B +
DΣ Eλ0 )x = 0, i.e., λ0 ∈ σ (
A, 
B), with [
A, 
B] = [A, B] + DΣ E,
which implies that the equation


Ax (t ) = 
Bx(t )
is either irregular or not uniformly exponentially stable. This
means that Σ ∈ ΞC which implies
rC (A, B; D, E ) ⩽ ‖Σ ‖ = ‖Eλ0 (λ0 A − B)−1 D‖




⩽

sup ‖Eλ (λA − B)−1 D‖

−1

−1

λ∈C\S

+ ϵ.

(3.15)

Since ϵ is arbitrary,

Therefore,

(λ0 A − B)x = −DΣ Eλ0 x.

rC (A, B; D, E ) =

This relation implies
Eλ0 x = −Eλ0 (λ0 A − B)−1 DΣ Eλ0 x.



sup ‖Eλ (λA − B)

−1

λ∈C\S

 −1
D‖

.

Note that the function G(λ) = Eλ (λA − B)−1 D is analytic on C \ S. By
the maximum principle, ‖G(·)‖ either reaches its maximum value
on the boundary ∂ S of S or supλ∈C\S ‖G(λ)‖ = limλ→∞ ‖G(λ)‖.

Thus, we obtain

Since Eλ0 x ̸= 0,


−1
‖Σ ‖ ⩾ ‖Eλ0 (λ0 A − B)−1 D‖

 −1
−1
⩾
sup ‖Eλ (λA − B) D‖
.

rC (A, B; D, E ) =

λ∈C\S



sup ‖Eλ (λA − B)−1 D‖

λ∈∞∪∂ S

 −1

.

The proof is complete.

Thus,
rC (A, B; D, E ) ⩾

Σ Eλ0 (λ0 A − B)−1 Du

=

where ‖ · ‖ can be any vector-induced matrix norm.



+ ϵ.

Let y∗ be a linear functional defined on Cl such that ‖y∗ ‖ = 1 and

It is easy to see that the perturbed model of the form
A

−1



sup ‖Eλ (λA − B)−1 D‖

λ∈C\S

 −1

.

Corollary 3.5. The complex stability radius of Eq. (3.1) under the
structured perturbation of the form


600

N.H. Du et al. / Systems & Control Letters 60 (2011) 596–603

Ax (t ) = (B + DB Σ EB )x(t ),

(3.16)

rC (B; DB , EB ) =



A=

 −1
sup ‖EB (λA − B) DB ‖
,
−1

(3.17)

λ∈∞∪∂ S

(A + DA Σ EA )x (t ) = Bx(t ),

(3.18)

is given by



sup ‖λEA (λA − B)

λ∈∞∪∂ S

−1

 −1
DA ‖

.

k=0

0
1
0



2

2

−1

−1

3

3


D=

and under the structured perturbation of the form

rC (A; DA , EA ) =

1
0
0



is given by

[2k, 2k + 1],

1
−2
1 ,
B = −1
0
−1


∞

where T =

0
0 ,
−1



0

2

−1 ,
3
1
1
0

0
−1
1


E = [E1 , E2 ] =

1
0

1

0
0
−2

−3
0
0

0
1 .
0



Since T = k=0 [2k, 2k + 1], S = {λ ∈ C : Re λ+ ln |λ+ 1| < 0}. It
is easy to see that ind(A, B) = 2 and σ (A, B) = − 13 . Therefore, the
pencil {A, B} is exponentially stable. When λ ∈ ∂ S, by the direct
computations, we obtain

∞

(3.19)

Proof. With D = DB and E = [E1 , E2 ] = [0, EB ], the perturbation
(3.16), we can write

[
A, 

B] = [A, B] + DΣ E ,
 
λI
Further, Eλ = E −Im = −EB and by Theorem 3.4, we get (3.17).
m
For the perturbation (3.18), we choose D =
 DA and E =
λIm
[E1 , E2 ] = [EA , 0]. By seeing that Eλ = E −Im = λEA we get

λ+1
 λ−1
=
3λ + 1
−λ − 1


(λA − B)−1

1 7

4
Q =
9

3
3
−3

2

and therefore,

Theorem 3.6. (a) rC (B; DB , EB ) > 0 if and only if the polynomial
p(λ) = EB 
Q (λA − B)−1 DB is constant.
(b) rC (A; DA , EA ) > 0 if and only if the polynomial q(λ) = λEA 
Q
(λA − B)−1 DA is constant.
(c) Let E = [E1 , E2 ]. Then rC (A, B; D, E ) > 0 if and only if the
polynomial s(λ) = (λE1 − E2 )
Q (λA − B)−1 D is constant.

Eλ (λA − B)

Proof. (a) We have
EB (λA − B)−1 DB = EB
P (λA − B)−1 DB + EB 
Q (λA − B)−1 DB .
−1

It is easy to prove that limλ→∞
 −‖1EB P (λA − B) DB ‖ = limλ→∞
−1
‖EB T diag (λI − B1 ) , 0m−r W DB ‖ = 0. Moreover, since
U k = 0,

−1

= EB T diag 0r , (λU − Im−r )

W −1 D B


k−1
−
= EB T diag 0r , −
(λU )i W −1 DB ,

−1

1
−2 ,
8

D=


−12

1

0
6

3λ + 1

−1

−12

−12

0
6

0
6

‖Eλ (λA − B)−1 D‖∞ =

36

|3λ + 1|

Corollary 3.7. Let ind(A, B) = 1. Then, rC (A, B; D, E ) > 0 if and
only if rC (A; D, E1 ) > 0 where E = [E1 , E2 ].


.

,

where ‖ · ‖∞ is the operator’s norm induced by ‖ · ‖v∞ . This implies
that
sup ‖Eλ (λA − B)−1 D‖∞ = ‖E0 (−B)−1 D‖∞ = 36.

λ∈∞∪∂ S

Thus, we get
1

36

.

Moreover, we see that

[A, B]

A˜ = A + DΣ E1 ,
B˜ = B + DΣ E2 ,


A
[A˜ , B˜ ] = [A, B] + DΣ E ⇐⇒
B

i =0

where T , W and U as mentioned in (3.2) and 
Q = T diag(0r , Im−r )
T −1 , 
P = Im − 
Q = T diag(Ir , 0m−r )T −1 . Hence, limλ→∞ ‖EB (λA −
B)−1 DB ‖ exists and it equals ∞ if p(λ) is not constant. Thus, we get
(a).
A similar argument can be applied to prove (b) and (c).
The proof is complete.


λ

−1 ,
λ

Let ‖ · ‖v∞ be the maximum norm of C3 . We have

rC (A, B; D, E ) =

p(λ) = EB 
Q (λA − B)−1 DB


−λ2
2
−λ2 − λ  ,
1
λ2 + 3λ + 1

λ 3
Eλ = λE1 − E2 = λ −λ
2
λ



1

(3.19).



−1
−1

and the polynomials


q(λ) = λE1 
Q (λA − B)

−1

D=

p(λ) = E2 
Q (λA − B)−1 D

3λ
3λ
λ+2
= λ+2
2λ − 2 2λ − 2

3λ

3λ

3λ

2λ

2λ

2λ

λ

3λ

λ+2
2λ − 2

λ



λ

̸= constant,


̸= constant.

Therefore

When A = I, (3.17) has been proved in [12] in order to unify
the continuous and discrete stability radii of the linear dynamic
equations. In case T = R, it is seen that S = C− and we get (1.7).
If T = N, S = {ω : |1 + ω| < 1} and (1.9) is deduced.

rC (A; D, E1 ) = rC (B; D, E2 ) = 0.

Example 3.8. Let us calculate the stability radius of the equation
Ax (t ) = Bx(t ) under the structured perturbation of the form

Now, we consider the problem when the real and complex
stability radii are equal. It seems that this is a difficult problem
in the implicit dynamic equations on time scales because in this
case, the positive cone Rm
+ is no longer invariant under the action of

[A, B]

[A˜ , B˜ ] = [A, B] + DΣ E ,

4. The equality of real and complex radii


N.H. Du et al. / Systems & Control Letters 60 (2011) 596–603

the pencil of matrices {A, B}; even when both A and B are positive.
Moreover, the domain of uniformly exponential stability S has
the property that although λ ∈ ∂ S, Re λ ∈ S which means we
cannot use the approach in [4]. However, we are able to answer
this question under some assumptions.
Let us consider Eq. (3.1) subjected to structured perturbations
(3.10). Firstly, we prove the following result which provides
a difference between ordinary dynamic equations and implicit
dynamic equations. Denoting G(λ) = Eλ (λA − B)−1 D, we have:
Theorem 4.1. If ‖G(λ)‖ does not reach its maximum at a finite point
on ∂ S then

601

When ‖G(λ)‖ attains its maximum value at a finite point in ∂ S,
we need further assumptions. A matrix M = (mij ) ∈ Rk×p is said
to be positive, written as M ⩾ 0, if mij ⩾ 0 for any i, j. We define
a partial order relation in Rk×p by M ⩾ N ⇔ M − N ⩾ 0. We
define the absolute value of a matrix M = (mij ) as the matrix
|M | = (|mij |); similarly for a vector x we use the notation |x| =
(|x1 |, |x2 |, . . . , |xp |). Let ρ(C , D) be the spectral radius of the pencil
of matrices {C , D}, i.e., ρ(C , D) := max{|λ| : λ ∈ σ (C , D)}.
Consider the implicit dynamic equation on time scale with
structured perturbations of the form
Ax (t ) = (B + DB Σ EB )x(t ),

(4.2)

rC (A, B; D, E ) = rR (A, B; D, E ).

where A, B ∈ Rm×m , DB ∈ Rm×l , and EB ∈ Rq×m . Suppose that
ind(A, B) = 1. Then, 
Q is the projection onto ker A along the space

Proof. It is clear that rC (A, B; D, E ) ⩽ rR (A, B; D, E ). Therefore, it
is sufficient to prove that there exists a sequence of disturbances
{Σn } ⊂ ΞR such that

S = {y ∈ Rm : By ∈ im A}.

It is seen

1
Recall that A1 = A − B
Q is nonsingular, A
Q = 0 and 
P = A−
1 A.
Moreover, from Perron–Frobenius extension theorem (see [22]) we
1 
−1 
have ρ(A, B) = ρ(A−
1 BP ) and if A1 BP ⩾ 0 then ρ(A, B) is an
eigenvalue of the pencil of matrices {A, B}.
1 
 −1
Let 
B = A−
1 BP = PA1 B. From (3.2)–(3.4), it follows that

G(λ) = Eλ
P (λA − B)−1 D + Eλ 
Q (λA − B)−1 D.



(λA − B)−1 = T diag (λIr − B1 )−1 , −Im−r W −1

lim ‖Σn ‖ ⩽ rC (A, B; D, E ).

n→∞

1
 −1
= (λI − 
B)−1
PA−
1 + Q A1 .

By using the Kronecker decomposition (3.2), we get





(4.3)

With the perturbed equation (4.2) then G(λ) = EB (λA − B)
The relation (4.3) implies that

k−1
−
Eλ 
Q (λA − B)−1 D = Eλ T diag 0r , −
(λU )i W −1 D,

−1

DB .

i=0

1
Q A−
G(∞) = lim G(λ) = EB 
1 DB .

and

λ→∞

Eλ
P (λA − B)−1 D = Eλ T diag (λIr − B1 )−1 , 0m−r W −1 D.





Assume that E = [E1 , E2 ] with E1 , E2 ∈ Rm×m . Then, Eλ = λE1 − E2 ,
and it follows that
lim Eλ
P (λA − B)−1 D = E1 T diag (Ir , 0m−r ) W −1 D

λ→∞

1
= E1
PA−
1 D,

(4.1)

and




k−1
−
i
Eλ T diag 0r , −
(λU ) W −1 D
= (λE1 − E2 )T diag 0r , −



k−1
−
(λU )i W −1 D
i=0

is a polynomial in λ. Therefore the limit limλ→∞ ‖G(λ)‖ exists
(possibly +∞). Since ‖G(λ)‖ does not reach its maximum at a
finite point on ∂ S, it follows that
rC (A, B; D, E )−1 =

sup ‖G(λ)‖ = lim ‖G(λ)‖.

λ∈∞∪∂ S

For α ⩾ 0, we define the ball Bα (−α) = {z ∈ C : |z + α| < α}.
For a time scale with bounded graininess several essential features

are captured by an associated characteristic ball. The analysis of
positive linear equations show that Bη (−η) ⊂ S, with S is the
domain of uniform exponential stability of the time scale T, and
η is defined by

η=

1
sup{µ(t ) : t ∈ T}

,

(4.5)

see, e.g. [12].

i=0



(4.4)

λ→∞

This implies that rC (A, B; D, E )−1 = limn∈N;n→∞ ‖G(n)‖. For
any n ∈ N, let un ∈ Rl be a vector with ‖un ‖ = 1 :
‖G(n)un ‖ = ‖G(n)‖; let y∗n be a linear functional defined on Rq
with ‖y∗n ‖ = 1 and y∗n (G(n)un ) = ‖G(n)un ‖ as in Theorem 3.4.
By denoting Σn = ‖G(n)‖−1 un y∗n we see that n is an eigenvalue
of the pencil {

A, 
B} with [
A, 
B] = [A, B] + DΣn E and the
corresponding eigenvector xn = (nA − B)−1 Dun . Note that Σn
is indeed a real perturbation. Therefore, Σn ∈ ΞR . Further,
‖Σn ‖ = ‖G(n)‖−1 ‖un y∗n ‖ ⩽ ‖G(n)‖−1 and ‖Σn (G(n)un )‖ =
‖G(n)‖−1 ‖un y∗n (G(n)un )‖ = ‖un ‖ = 1 which implies that
‖Σn ‖ = ‖G(n)‖−1 and limn→∞ ‖Σn ‖ = limn→∞ ‖G(n)‖−1 =
rC (A, B; D, E ). This relation says that rR (A, B; D, E ) ⩽ rC (A, B; D, E ).
The proof is complete.

1
1
Hypotheses 4.2. (i) A−
⩾
0, EB
P
⩾
0 and EB 
Q A−
1 DB
1
DB ⩾ 0.
(ii) There exists α ⩾ 0 with Bα (−α) ⊂ S such that 
B + α
P ⩾ 0.

We need the following simple lemma.
Lemma 4.3. Suppose that the bounded linear operator triplet: M :

X → Y , N : Y → Z , P : Z → X is given, where X , Y , Z are Banach
spaces. Then the operator I − MNP is invertible if and only if I − NPM
is invertible, moreover,

(I − NPM)−1 = I + NP(I − MNP)−1 M.
Proof. Suppose that I − MNP is invertible. By direct calculation, it
is easy to verify that the inverse of I − NPM is

(I − NPM)−1 = I + NP(I − MNP)−1 M.

(4.6)

Furthermore, if (I − MNP) is bounded then so is (I − NPM)−1 .
The converse is proved similarly.
−1

Theorem 4.4. Assume that the pencil of matrices {A, B} has ind(A, B)
= 1 and satisfies Hypotheses 4.2. Then, we have
rC (B; DB , EB ) = rR (B; DB , EB ).

(4.7)


602

N.H. Du et al. / Systems & Control Letters 60 (2011) 596–603

Proof. Clearly, it is sufficient to prove that rC (B; DB , EB ) ⩾ rR (B;
DB , EB ). If ‖G(λ)‖ does not reach its maximum value at a finite
point on ∂ S then from Theorem 4.1 it follows that rC (B; DB , EB ) =

rR (B; DB , EB ). Else, let λ0 ∈ C satisfy ‖G(λ0 )‖ = sup∂ S ‖G(λ)‖ and
the perturbation Σ , given by (3.13), destroy stability. We aim to
show

• Σ can be the complex perturbation, but |Σ | is the real perturbation making the dynamic equation unstable,

• furthermore ‖|Σ |‖ = ‖Σ ‖.

1

A−
1,Σ (B + α A + DB Σ EB )P
1
 −1 (A−1 (B + α A + DB Σ EB )
P)
= (I − A−
1
1 DB Σ EB Q )
1
1
 −1 (

= (I − A−
B + α
P + A−
1 DB Σ EB Q )
1 DB Σ EB P ).

1 
−1


Above we use the identities 
B = A−
1 BP and A1 A = P. Using (4.6)
−1

with M = EB Q , N = I , P = A1 DB Σ , we get

(4.8)

1
−1 
 −1
EB Q
= I + A−
1 DB Σ (I − EB Q A1 DB Σ )
1
−1 
= I + A−
EB Q .
1 DB Σ (I − G(∞)Σ )

1

A−
1,Σ (B + α A + DB Σ EB )P
1

=
B + α

P + A−
1 DB Σ EB P + Φ (Σ ).

Similarly, we also have

A1,Σ = A − (B + DB Σ EB )
Q.

1

A−
1,|Σ | (B + α A + DB |Σ |EB )P
1

=
B + α
P + A−
1 DB |Σ |EB P + Φ (|Σ |).

From (3.4) it follows that
(4.9)

1
Applying Lemma 4.3 with M = EB 
Q , N = I , P = A−
1 DB Σ we get
−1

I − A1 DB Σ EB Q is invertible. This implies that A1,Σ is invertible as
well. Now, we will prove that

σ (A, B + α A + DB Σ EB ) ∪ {0}
1

= σ (A−
1,Σ (B + α A + DB Σ EB )P ).

(4.10)

Moreover, by Hypotheses 4.2 and the inequality ‖G(∞)‖‖Σ ‖ =
‖G(∞)‖‖|Σ |‖ < 1, it follows that
1
−1
G(∞)Σ |EB
P
|Φ (Σ )| = A−
1 DB |Σ (I − G(∞)Σ )
 −

∞


1

= A−
(G(∞)Σ )i+1 EB
P
1 DB  Σ
i =0



∞
−
−1
⩽ A DB |Σ |
(G(∞)|Σ |)i+1 EB
P
1

i =0

Indeed, for any λ ̸= 0, because of the properties A
Q = 
P
Q =





0, Q Q = Q , P + Q = I, we have
1

det(λI + A−
1,Σ (B + α A + DB Σ EB )P ) = 0

1

|A−
1,Σ (B + α A + DB Σ EB )P |

−
1
⩽ |
B + α
P | + |A DB Σ EB
P | + |Φ (Σ )|
1

⩽
B + α
P + A1 DB |Σ |EB
P + Φ (|Σ |)
−1

⇐⇒ det[(A − (B + DB Σ EB )
Q )(λ
P −
Q)
+ (B + α A + DB Σ EB )
P] = 0
⇐⇒ det[A(λ
P −
Q ) + (B + DB Σ EB )
Q
+ (B + α A + DB Σ EB )
P] = 0
⇐⇒ det[λA(
P +
Q ) + (B + α A + DB Σ EB )(
P +

Q)
− (1 + α + λ)A
Q] = 0
⇐⇒ det[λA + B + α A + DB Σ EB ] = 0.

1

= A−
1,|Σ | (B + α A + DB |Σ |EB )P .

1

ρ(A−
1,Σ (B + α A + DB Σ EB )P )

⩽ ρ(A1,|Σ | (B + α A + DB |Σ |EB )
P ) := β.
−1

(4.17)

1

Since A−
1,|Σ | (B+α A+DB |Σ |EB )P ⩾ 0, by Perron–Frobenius theorem,

(4.11)

Similarly, since ‖|Σ |‖ = ‖Σ ‖ < ‖G(∞)‖−1 ,

σ (A, B + α A + DB |Σ |EB ) ∪ {0}
(4.12)

and

ρ(A, B + α A + DB |Σ |EB )
1

= ρ(A−
1,|Σ | (B + α A + DB |Σ |EB )P ).

(4.16)

From theory of nonnegative matrices, see, e.g. [23], it follows that

This implies that the spectral equality (4.10) holds. In particular,

1

= σ (A−
1,|Σ | (B + α A + DB |Σ |EB )P ),

= A1 DB |Σ |(I − G(∞)|Σ |)−1 G(∞)|Σ |EB
P
= Φ (|Σ |),
−1

and therefore,

⇐⇒ det(λA1,Σ + (B + α A + DB Σ EB )

P) = 0






Q


⇐⇒ det λA1,Σ + (B + α A + DB Σ EB )P P −
=0
λ

1

ρ(A, B + α A + DB Σ EB ) = ρ(A−
1,Σ (B + α A + DB Σ EB )P ).

(4.15)

1
−1
Define Φ (Σ ) = A−
G(∞)Σ EB
P. Then, from
1 DB Σ (I − G(∞)Σ )

(4.14) and (4.15), because of the property Q
B=

Q
P = 0, we obtain

1
Q A−
Further, the inequality ‖Σ ‖‖EB 
1 DB ‖ = ‖Σ ‖‖G(∞)‖ < 1
−
1
implies that the matrix I − EB 
Q A1 DB Σ is invertible. Define

1

A1,Σ = A1 (I − A−
1 DB Σ EB Q ).

(4.14)

1
 −1
(I − A−
1 DB Σ EB Q )

It is seen that ‖Σ ‖ = ‖G(λ0 )‖−1 < ‖G(∞)‖−1 . Moreover, Σ has
rank one which implies that ‖|Σ |‖ = ‖Σ ‖.
Since Bα (−α) ⊂ S and the pencil of matrices {A, B + DB Σ EB } is
unstable, it follows that σ (A, B + DB Σ EB ) ̸⊂ S and

ρ(A, B + α A + DB Σ EB )

= max {|z + α| : z ∈ σ (A, B + DB Σ EB )} ⩾ α.

From (4.9) it follows that

1

β is an eigenvalue of the matrix A−
1,|Σ | (B + α A + DB |Σ |EB )P with
maximum module and by (4.12) it follows that β ∈ σ (A, B +
α A + DB |Σ |EB ). Therefore, by (4.11), (4.8) and (4.17), we obtain 0 ⩽
β −α ∈ σ (A, B + DB |Σ |EB ). Thus the perturbation |Σ | ∈ Rl×q , with
‖|Σ |‖ = ‖Σ ‖, destroys stability which implies rC (B; DB , EB ) ⩾
rR (B; DB , EB ). The proof is complete.

We consider the case where Eq. (3.1) is positive, i.e., for any

x0 ∈ Rm
+ , the solution x(t ) of Eq. (3.1) with P (x(t0 ) − x0 ) = 0
satisfies the condition x(t ) ⩾ 0 for all t ∈ T, t ⩾ t0 . It is known
that if x(t ) is a solution of (3.1) then 
Q x(t ) = 0 which implies

Px(t ) = x(t ) for all t ∈ T, t ⩾ t0 . Therefore, (3.1) can be rewritten


(4.13)

x =
Bx,
x(t0 ) = 

Px0 ,

(4.18)


N.H. Du et al. / Systems & Control Letters 60 (2011) 596–603

where 
B = (A − B
Q )−1 B
P =
P (A − B
Q )−1 B. It is easy to see that
Eq. (4.18) has a general solution




∏

x(t ) = x(t , t0 , x0 ) =

(
P + µ(s)
B) exp{mes [t0 , t )
B} 
Px0 ,

t0 ⩽s

where mes(C ) is the Lebesgue measure of the set C . With t = t0 ,
we have x(t0 ) = 
Px0 ⩾ 0. If 
B is a 
P-Metzler matrix, i.e.,
there exists α ∈ R such that 
B + α
P ⩾ 0, then by paying attention that 
P and 
B are commutative, we see that exp{mes [t0 , t )
B} =
exp{mes [t0 , t )(
B + α
P )} exp{−mes [t0 , t )α
P }. Therefore exp{mes
[t0 , t )
B}
Px0 = exp{mes [t0 , t )(
B + α
P )} exp{−α mes [t0 , t )}
Px0 ⩾


0. Further, we need P + µ(t )B ⩾ 0 for all t ∈ T which implies
that η
P +
B ⩾ 0. Thus, the positiveness condition of Eq. (3.1) is
equivalent to η
P +
B ⩾ 0. This means that the condition (ii) of

Hypotheses 4.2 holds.
Corollary 4.5. Assume that the solution of the linear equation x =
Bx is positive and this equation is subjected to perturbations of the
form x = (B + DB Σ EB )x with DB ⩾ 0, EB ⩾ 0. Then, we have
rC (B; DB , EB ) = rR (B; DB , EB ).

(4.19)

Example 4.6. Let us consider the stability radius of the perturbed
equation
Ax (t ) − (B + DB Σ EB )x(t ) = 0,

(4.20)

with T = Z and


A=

1
0
0

0
1
0

1
0
0

DB =



0
1
0



1/2
−1
0

−1
B = 1/2

0
0 ,
0



0

1
−1 ,
1




EB =

1
1
1

0
1
1

0
1 ,
−1



0
0 .
1



It is seen that ind(A, B) = 1 and

σ (A, B) = {−1/2; −3/2} ⊂ S = {λ ∈ C : |1 + λ| < 1}.
Moreover,
1

0
0



P =

0
1
0

0
0 ,
0




Q =


−1

A1

= (A − B
Q)

−1

0
0
0

0
0
0



=

1
0
0

0
1
0

0
0 ,
1



0
1 .
1



Therefore, for α = 1, we have B1 (−1) = S and
0
1/2
0


−1


B +
P = A1 B
P +
P =

1/2
0
0

0
0
0


⩾ 0.

1

 −1

It is easy to see that A−
1 DB ⩾ 0, EB P ⩾ 0 and G(∞) = EB Q A1 DB ⩾
0. From Theorem 4.4, we have rC (B; DB , EB ) = rR (B; DB , EB ). By the
computations, we get

(λA − B)−1 =

1

(λ 
+ 1)2 − 1/4
λ + 1 1/2
×  1/2 λ + 1
0

G(λ) = EB (λA − B)

−1

0



DB

λ+1
λ + 3/2
=
2
(λ + 1) − 1/4 λ + 3/2



1

1 /2
,
λ+1
(λ + 1)2 − 1/4
1/2
λ + 3/2
λ + 3/2


λ+1
λ + 3/2  .
(λ + 3/2)2

603

Let ‖ · ‖3 be the maximum norm of C3 , it follows that
sup ‖G(λ)‖∞ = ‖G(0)‖∞ = 7.

λ∈∞∪∂ S

Thus, we obtain
rC (B; DB , EB ) = rR (B; DB , EB ) =



sup ‖G(λ)‖∞

λ∈∞∪∂ S

 −1
=

1
7

.

5. Conclusion
In this paper we have considered the uniformly exponential
stability and given the formulas for the stability radius of implicit
dynamic equations with general structured perturbations on time
scales and obtain the characterizations of these formulas. We
also provide some sufficient conditions for which the complex
stability radius and the real stability radius are the same. So far
we do not know whether the positive condition of the implicit
dynamic equations under general structured perturbations implies
the equality of the complex and real stability radii. An answer to
this problem would be of great interest.
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