CONTROLLABILITY RADII OF LINEAR NEUTRAL SYSTEMS UNDER STRUCTURED PERTURBATIONS

CONTROLLABILITY RADII OF LINEAR NEUTRAL
SYSTEMS UNDER STRUCTURED PERTURBATIONS
Do Duc Thuan†

∗

Nguyen Thi Hong‡

Dedicated to Professor Nguyen Khoa Son on the occassion of his 65th birthday

Abstract
In this paper we shall deal with the problem of calculation of the controllability radii of linear neutral systems of the form x (t) = A0 x(t) + A1 x(t − h) +
A−1 x (t − h) + Bu(t). We will derive the definition of exact controllability radius, approximate controllability radius and Euclidean controllability radius for
this system. By using multi-valued linear operators, the computable formulas
for these controllability radii are established in the case where the system’s coefficient matrices are subjected to structured perturbations. Some examples are
provided to illustrate the obtained results.

Keywords. Linear neutral systems, multi-valued linear operators, structured perturbations, controllability radius.

1

Introduction

In this paper, we investigate the robust controllability for linear neutral systems of the
form
x (t) = A0 x(t) + A1 x(t − h) + A−1 x (t − h) + Bu(t),
(1.1)
where A0 , A1 , A−1 ∈ Kn×n and B ∈ Kn×m .
Linear neutral systems play an important role in mathematical modeling arising
in physics, mechanics, biology, chemistry, etc., see [6, 16, 19]. It is well known that,
due to the fact that the dynamics of (1.1) is delay in both state and derivative, there

are many the notation of controllability for (1.1) such as exact controllability, approximate controllability and Euclidean controllability, see [1, 13, 18, 20]. The problem
of controllability for (1.1) leads to study of the abstract controllability problem in
infinite-dementional spaces.
∗
Mathematics Subject Classifications: 06B99, 34D99,47A10, 47A99, 65P99. Corresponding author:
D.D. Thuan, email:
†
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1
Dai Co Viet Str., Hanoi, Vietnam.
‡
Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Rd.,
Hanoi, Vietnam.

1


2
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 why we are interested in investigating the uncertain system subjected to general
structured perturbation of of the form
x (t) = A0 x(t) + A1 x(t − h) + A−1 x (t − h) + Bu(t),

(1.2)

with
[A0 , A1 , A−1 , B]

[A0 , A1 , A−1 , B] = [A0 , A1 , A−1 , B] + D∆E,

(1.3)

where ∆ is an unknown disturbance matrix; D, E are known scaling matrices defining
the “structure” of the perturbation. A natural question arises that under what condition the system (1.2) with perturbations (1.3) remains controllable, i.e., how robust
the controllability of the nominal system (1.1) is.
The so-called controllability radius is defined by the largest bound r such that the
controllability is preserved for all perturbations of norm strictly less than r. The problem of estimating and calculating controllability radii is of great interest in research
and application of control theory and has attracted a good deal of attention over last
decades (see, e.g. [2, 3, 8, 9, 11, 12, 17, 22, 23]). Earlier results for the controllability
radius of linear systems under unstructured perturbations is derived by Esing in [5].
After that, formulas for controllability radius of linear systems under structured perturbation has been employed in [7, 21]. Recently, the similar problem was considered in
[14, 24] for linear delay systems. Therefore, it is natural and meaningful to continuous
studying controllability radius for linear neutral systems.
The aim of this paper is to study the controllability robust for system (1.1). We will
derive formulas for the complex and real approximate controllability radii of system
(1.2) when this system is subjected to structured perturbations of the form (1.3). In
some particular cases, the main results yield new computable formulas of complex and
real structured controllability radii of linear neutral systems. The key technique is to
make use of some well-known facts from the theory of multi-valued linear operators
(see [4, 21]), the structure distance to non-surjectivity (see [22]) and Hautus test for
exact, approximate and Euclidean controllability (see [13, 18, 20]).
The organization of the paper is as follows. In the next section we shall present
the formulas for complex approximate controllability radii and some relationships with
complex Euclidean controllability radius. Section 2 will be devoted to study the real
controllability radii under structured perturbations and derive the computable formulas
in some special cases. In conclusion we summarize the obtained results and give some
remarks of further investigation.

2

Preliminary

For the readers’ convenience, we give a list of notations to be used in what follows.
Throughout the paper, K = C or R, the field of complex or real numbers, respectively.
Kn×m will stand for the set of all (n × m)− matrices, Kn (= Kn×1 ) is the n-dimensional
columns vector space equipped with the vector norm · and its dual space can


3
be identified with (Kn )∗ = (Kn×1 )∗ , the rows vector space equipped with the dual
norm. For A ∈ Kn×m , A∗ ∈ Km×n denotes its adjoint matrix and for Ai ∈ Kn×mi , i =
1, 2, . . . , k, [A1 , A2 , . . . , Ak ] will denote the n × (m1 + m2 + . . . mk )-matrix aggregated
by columns of Ai . A set-valued map F : Km ⇒ Kn is said to be multi-valued linear
operator if its graph gr F = {(x, y) : y ∈ F(x)} is a linear subspace of Km × Kn . The
readers are referred to [23] for the definitions and the properties of multi-valued linear
operators which are needed to derive the main results of this paper. In particular,
for each multi-valued linear operator F the adjoint F ∗ and the inverse F −1 are well
defined as multi-valued linear operators and we have the following useful relations
(F ∗ )−1 = (F −1 )∗ , (GF)∗ = F ∗ G ∗ ,

F = F∗ .

(2.1)

Here the norm of F is defined as
F = sup

inf

y : x ∈ dom F, x = 1 .

(2.2)

y∈F (x)

If we identify a matrix F ∈ Kn×m with a linear operator F : Km → Kn then its dual
operator F ∗ : (Kn )∗ → (Km )∗ is defined by F ∗ (y ∗ ) = y ∗ F and its inverse in terms of
multi-valued linear operators is defined as F −1 (y) = {x ∈ Km : F x = y}. Moreover,
m×n
the Moore-Penrose pseudo inverse matrix F † ∈ K
exists and if vector spaces Kn , Km
√
are equipped with Euclidean norms (i.e x = x∗ x) then F † defines a linear selector
of F −1 (i.e F † y ∈ F −1 (y), ∀y ∈ Kn ) satisfying F † y = inf{ x : x ∈ F −1 (y)}. This
implies, in particular, that
F † y ≤ x , for all x ∈ F −1 (y).

(2.3)

Now, consider the linear neutral systems
x (t) = A0 x(t) + A1 x(t − h) + A−1 x (t − h) + Bu(t),
x(0) = x0 , x(t) = g(t), ∀t ∈ [−h, 0],

(2.4)

where h is positive constant A1 , E0 , E1 ∈ Cn×n , B ∈ Cn×m , and g(t) : [−h, 0] → Cn is
a squared integral function.
Definition 2.1. System (2.4) is called exactly controllable if for any given initial conditions φ0 (.) ∈ W21 ([−h, 0], Cn ), disired final φ1 (.) ∈ W21 ([−h, 0], Cn ) and arbitrary
> 0, there exists T > 0 and a control function u(t) ∈ L2 ([−h, 0], Cn ) such that the
corresponding solution x(t) satisfies
x(θ) = φ0 (θ), ∀θ ∈ [−h, 0],

xT (θ) = φ1 (θ), ∀θ ∈ [−h, 0],
where xT (θ) = x(T + θ), for all θ ∈ [−h, 0].
Definition 2.2. System (2.4) is called approximately controllable if for any given initial
conditions φ0 (.) ∈ W21 ([−h, 0], Rn ), disired final φ1 (.) ∈ W21 ([−h, 0], Rn ) and arbitrary
> 0, there exists T > 0 and a control function u(t) ∈ L2 ([−h, 0], Cn ) such that the
corresponding solution x(t) satisfies
x(θ) = φ0 (θ), ∀θ ∈ [−h, 0],
xT (.) − φ1 (.) W21 < ,
where xT (θ) = x(T + θ), for all θ ∈ [−h, 0].


4
Definition 2.3. System (2.4) is called Euclidean controllable if for any given initial
conditions x0 , g(t) and desired final state x1 , there exists a time t1 , 0 < t1 < ∞, and a
measurable control function u(t) for t ∈ [0, t1 ] such that x t1 ; x0 , g(t), u(t) = x1 .
It is well known that controllability of linear systems has been derived by Hautus
in [10]. Let
P (λ) = A0 + e−hλ A1 + λe−hλ A−1 − λIn
(2.5)
be the characteristic polynomial of system (2.4). The following propositions give necessary and sufficient conditions in the form of Hautus test for controllability of linear
neutral systems, see [13, 18, 20].
Proposition 2.4. System (2.4) is exactly controllable if and only if
(i) rank[P (λ), B] = n, for all λ ∈ C,
(ii) rank[B, A−1 B, . . . , An−1
−1 B] = n.
Proposition 2.5. System (2.4) is approximately controllable if and only if
(i) rank[P (λ), B] = n, for all λ ∈ C,
(ii) rank[λA−1 + A1 , B] = n, for all λ ∈ C.
Proposition 2.6. System (2.4) is Euclidean controllable if and only if
rank[P (λ), B] = n, for all λ ∈ C.

3

Controllability radii

Assume that system (2.4) is subjected to structured perturbations of the form
x (t) = A0 x(t) + A1 x(t − h) + A−1 x (t − h) + Bu(t),

(3.1)

with
[A0 , A1 , A−1 , B]

[A0 , A1 , A−1 , B] = [A0 , A1 , A−1 , B] + D∆E.

(3.2)

Here ∆ ∈ Cl×q is the perturbation matrix and D ∈ Cn×l , E ∈ Cq×(3×n+m) determine
structure of the perturbation D∆E. We denote A = (A0 , A1 , A−1 ).
Definition 3.1. Let system (2.4) be exactly controllable. Given a norm · on Cl×q ,
the exact controllability radius of system (2.4) with respect to structured perturbations
of the form (3.2) is defined by
rKex (A, B; D, E) = inf

∆ : ∆ ∈ Kl×q s.t. (3.1) not exactly controllable .

(3.3)

If system (3.1) under structured perturbations (3.2) is exactly controllable for all ∆ ∈
Cl×q then we set rKex (A, B; D, E) = +∞.

5
Definition 3.2. Let system (2.4) be approximately controllable. Given a norm · on
Cl×q , the approximative controllability radius of system (2.4) with respect to structured
perturbations of the form (3.2) is defined by
rKap (A, B; D, E) = inf

∆ : ∆ ∈ Kl×q s.t. (3.1) not approximately controllable .
(3.4)

If system (3.1) under structured perturbations (3.2) is approximately controllable for
all ∆ ∈ Cl×q then we set rKap (A, B; D, E) = +∞.
Definition 3.3. Let system (2.4) be Euclidean controllable. Given a norm · on
Cl×q , the Euclidean controllability radius of system (2.4) with respect to structured
perturbations of the form (3.2) is defined by
∆ : ∆ ∈ Kl×q s.t. (3.1) not Euclidean controllable .

rKeu (A, B; D, E) = inf

(3.5)

If system (3.1) under structured perturbations (3.2) is Euclidean controllable for all
∆ ∈ Cl×q then we set rKeu (A, B; D, E) = +∞.
We define
W1 (λ) = [P (λ), B],

In
 e−hλ In
H1 (λ) = 

λe−hλ In
0

W2 (λ) = [A−1 − λIn , B], W3 (λ) = [λA−1 + A1 , B]





0
0 0
0
0




0
 , H2 =  0 0  , H3 (λ) =  In 0  ,
In 0 
λIn 0 
0
Im
0 Im
0 Im

E1 (λ) = EH1 (λ),

E2 = EH2 ,

(3.6)

E3 (λ) = EH3 (λ).

To establish the formula for the controllability radii of system (2.4), we need to
derive the notion of structured distance to non-surjectivity of a matrix. Let W ∈ Cn×m
be a sujective matrix, then the structured distance of W to non-surjectivity is given by
distC (W ; D, E) = inf{ ∆ : ∆ ∈ Cl,q s.t. W + D∆E is non-surjective}
1
=
,
EW −1 D

(3.7)

where W −1 is the multi-valued inverse operator of W , see [22].
Now, we derive the formula for the exact controllability radius of system (2.4) in
the following theorem.
Theorem 3.4. Assume that system (2.4) is exactly controllable and subjected to structured perturbations of the form (3.2). Then the exact controllability radius of (2.4) is
given by the formula
rCex (A, B; D, E) = min

inf E1 (λ)W1 (λ)−1 D

λ∈C

−1

, inf E2 W2 (λ)−1 D
λ∈C

−1

,

(3.8)

where W1 (λ)−1 , W2 (λ)−1 : Cn ⇒ Cn+m are the multi-valued inverse operator of W1 (λ),
W2 (λ) (respectively).


6
Proof. Suppose that [A0 , A1 , A−1 , B] = [A0 , A1 , A−1 , B] + D∆E is not exactly controllable for ∆ ∈ Cl×q . It means, by Proposition 2.4, for some λ0 ∈ C the operator
W1 (λ0 ) = [P (λ0 ), B] is not surjective, where P (λ0 ) = A0 + e−hλ0 A1 + λe−hλ0 A−1 − λ0 In ,
or the operator W2 (λ0 ) = [A−1 − λ0 In , B] is not surjective. If W1 (λ0 ) is not surjective,
by definitions (2.5) and (3.6) we can deduce
W1 (λ0 ) = [P (λ0 ), B] = [A0 , A1 , A−1 , B]H1 (λ0 ) − λ0 [In , 0]
= ([A0 , A1 , A−1 , B] + D∆E)H1 (λ0 ) − λ0 [In , 0]
= [A0 , A1 , A−1 , B]H1 (λ0 ) − λ0 [In , 0] + D∆EH1 (λ0 )
= [P (λ0 ), B] + D∆E1 (λ0 ) = W1 (λ0 ) + D∆E1 (λ0 ).

(3.9)

In this case, by (3.7), we get
∆ ≥ dist(W1 (λ0 ); D, E1 (λ0 )) = E1 (λ0 )W1 (λ0 )−1 D

−1

≥ inf E1 (λ)W1 (λ)−1 D

−1

λ∈C

.

If W2 (λ0 ) is not surjective with some λ0 , by definition (3.6) we can deduce
W2 (λ0 ) = [A−1 − λ0 In , B] = [A0 , A1 , A−1 , B]H2 − λ0 [In , 0]
= ([A0 , A1 , A−1 , B] + D∆E)H2 − λ0 [In , 0]
= W2 (λ0 ) + D∆E2
In this case, by (3.7), we get
∆ ≥ dist(W2 (λ0 ); D, E2 ) = E2 W2 (λ0 )−1 D

−1

≥ inf E2 W2 (λ)−1 D

−1

λ∈C

.

Therefore, we imply that
∆ ≥ min

inf E1 (λ)W1 (λ)−1 D

−1

λ∈C

, inf E2 W2 (λ)−1 D

−1

λ∈C

.

Since the above inequality holds for any disturbance matrix ∆ ∈ Cl×q such that D∆E
destroys controllability of (2.4), we obtain by definition,
rCex (A, B; D, E) ≥ min

inf E1 (λ)W1 (λ)−1 D

λ∈C

−1

, inf E2 W2 (λ)−1 D

−1

λ∈C

.

To prove the converse inequality, for any small enough > 0, there exists λ ∈ C such
that

E1 (λ )W1 (λ )−1 D ≥ sup E1 (λ)W1 (λ)−1 D − > 0.
λ∈C

By the definition of the structured distance to singularity, it follows that there exists
a perturbation ∆ such that
∆

≤

E1 (λ )W1 (λ )−1 D −

−1

and the perturbed matrix W1 (λ ) = W1 (λ ) + D∆ E1 (λ ) is not surjective. Hence,
equation (3.1) is not exactly controllable with the perturbation ∆ . Thus, by definition,
rCex (A, B; D, E)

≤

−1

E1 (λ )W1 (λ ) D −

−1

−1

≤

−1

sup E1 (λ)W1 (λ) D − 2
λ∈C

.


7
Letting → 0, we get
rCex (A, B; D, E) ≤ inf E1 (λ)W1 (λ)−1 D

−1

rCex (A, B; D, E) ≤ inf E2 W2 (λ)−1 D

,

λ∈C

.

Similarly, we imply
−1

λ∈C

and hence
rCex (A, B; D, E) ≤ min

inf E1 (λ)W1 (λ)−1 D

−1

λ∈C

, inf E2 W2 (λ)−1 D

−1

λ∈C

.

The proof is complete.
The above theorem have been proved for the case when the norms of matrices under
consideration are operator norms induced by arbitrary vector norms in corresponding
vector spaces.
Similarly, by using Propositions 2.5, 2.6 and formula (3.7), we obtain
Theorem 3.5. Assume that system (2.4) is approximately controllable and subjected to
structured perturbations of the form (3.2). Then the approximately controllable radius
of (2.4) is given by the formula
rCap (A, B; D, E) = min

inf E1 (λ)W1 (λ)−1 D

λ∈C

−1

, inf E3 (λ)W3 (λ)−1 D

−1

λ∈C

, (3.10)

where W1 (λ)−1 , W3 (λ)−1 : Cn ⇒ Cn+m are the multi-valued inverse operator of W1 (λ),
W3 (λ) (respectively).
Theorem 3.6. Assume that system (2.4) is Euclide controllable and subjected to structured perturbations of the form (3.2). Then the Euclidean controllable radius of (2.4)
is given by the formula
rCeu (A, B; D, E) = inf E1 (λ)W1 (λ)−1 D
λ∈C

−1

,

(3.11)

where W1 (λ)−1 : Cn ⇒ Cn+m are the multi-valued inverse operator of W1 (λ).
Example 3.7. Let us consider the linear neutral system
x (t) = A0 x(t) + A1 x(t − 1) + A−1 x (t − 1) + Bu(t),
where A0 =

1 1
2 2
2 2
2
, A1 =

, A−1 =
,B =
. We see that
1 1
−2 2
2 −2
2

W1 (λ) = [P (λ), B] =

1 + 2e−λ + 2λe−λ − λ
1 + 2e−λ + 2λe−λ
2
,
−λ
−λ
1 − 2e + 2λe
1 + 2e−λ − 2λe−λ − λ 2

W2 (λ) = [A−1 − λI2 , B] =

2−λ
2
2
,
2
−2 − λ 2

W3 (λ) = [λA−1 + A1 , B] =

2 + 2λ 2 + 2λ 2
.
−2 + 2λ 2 − 2λ 2

(3.12)


8
It follows that rank W1 (λ) = 2 for all λ ∈ C and rank[A−1 , B] = 2. Therefore, by
Proposition 2.4, the system is exactly controllable. Assume that the control matrix
[A0 , A1 , A−1 , B] is subjected to structured perturbation of the form
1 1 2 2 2 2 2
1 1 −2 2 2 −2 2

1 + δ1 1 + δ1
2 2 + δ2 2 + δ2
2 2 + δ2
,
1 + 2δ1 1 + 2δ1 −2 2 + 2δ2 2 + 2δ2 −2 2 + 2δ2

where δi ∈ C, i ∈ 1, 2 are disturbance parameters. The above perturbed model can be
represented in the form
[A0 , A1 , A−1 , B]

[A0 , A1 , A−1 , B] + D∆E

1
1 1 0 0 0 0 0
,E =
and ∆ = [δ1 δ2 ]. It implies that E1 (λ) =

2
0 0 0 1 1 0 1
1 0
0 0 0
1 1 0
, E2 =
, E3 (λ) =
. We have, for v ∈ C,
e−λ 1
1 0 1
λ 1 1

with D =
1
λe−λ

v
E1 (λ)W1 (λ)−1 D(v) = E1 (λ)W1 (λ)−1
2v
 
p
(1 + 2e−λ + 2λe−λ − λ)p + (1 + 2e−λ + 2λe−λ )q + 2r = v,

= E1 (λ) q 
(1 − 2e−λ + 2λe−λ )p + (1 + 2e−λ − 2λe−λ − λ)q + 2r = 2v
r
=

p+q
−λ

λe p + e−λ q + r

(1 + 2e−λ + 2λe−λ − λ)p + (1 + 2e−λ + 2λe−λ )q + 2r = v,
.
(2 + 4λe−λ − λ)p + (2 + 4e−λ − λ)q + 4r = 3v

Thus, for each v ∈ C, the problem of computing d(0, E1 (λ)W1 (λ)−1 D(v)) is reduced to
the calculation of the distance from the origin to the straight line in C2 whose equation
can be rewritten in the form (2 − λ)x1 + 4x2 = 3v with
x2 = λe−λ p + e−λ q + r.

x1 = p + q,

Let C2 be endowed with the vector norms

·

∞,

then we can deduce,

3|v| ≤ |2 − λ||x1 | + 4|x2 | ≤ (|2 − λ| + 4) max{|x1 |, |x2 |} = (|2 − λ| + 4)

x1
x2

∞.

This implies
x1

x2

∞

≥

3|v|
,
|2 − λ| + 4

3v
and x1 = eiϕ x2 , where ϕ is chosen such
|2 − λ| + 4
= |2 − λ|. Therefore,

which yields the equality if x2 =
that (2 − λ)eiϕ

E1 (λ)W1 (λ)−1 D = sup d 0, E1 (λ)W1 (λ)−1 D(v)
|v|=1

=

3
,
|2 − λ| + 4


9
3

and hence supλ∈C E1 (λ)W1 (λ)−1 D = . Moreover, it is easy to see that
4
E2 W2 (λ)−1 D(v) =

0
p+r

(2 − λ)p + 2q + 2r = v,
2p − (2 + λ)q + 2r = 2v

E3 (λ)W3 (λ)−1 D(v) =

p+q
λp + q + r

(2 + 2λ)p + (2 + 2λ)q + 2r = v,
(−2 + 2λ)p + (2 − 2λ)q + 2r = 2v

=

p+q
λp + q + r

(2 + 2λ)p + (2 + 2λ)q + 2r = v,
4λp + 4q + 4r = 3v

Similarly, this implies that
3
sup E3 (λ)W3 (λ)−1 D = .
4

λ∈C

sup E2 W2 (λ)−1 D = 1,
λ∈C

Thus, by Theorems 3.4, 3.5, 3.6, we obtain
rCex (A, B; D, E) = 1,

4

4
rCap (A, B; D, E) = rCeu (A, B; D, E) = .
3

Some particular cases

Formulas (3.8), (3.10), (3.11) gives us a unified framework for computation of controllability radii, however, it is not easy to be used because this formula involves calculation of the norm of the multi-valued linear operators E1 (λ)W1 (λ)−1 D, E2 (λ)W2 (λ)−1 D,
E3 W3−1 D which do not have an explicit representation. We now derive from this result more computable formulas for the particular case, where the norm of the matrices
under consideration is the spectral
√ norm (i.e. the operator norm induced by Euclidean
vector norms of the form x = x∗ x). To this end, we need the following lemmas.
Lemma 4.1. Assume that Q ∈ Cn×(n+m) has full row rank and M ∈ Cq×(n+m) has full
column rank and the operator norms are induced by Euclidean vector norms. Then we
have
M Q−1 D = (Q(M ∗ M )−1/2 )† D ,
(4.1)
where

†

denotes the Moore-Penrose pseudoinverse.

Denote
G1 (λ) = (W1 (λ)(E1 (λ)∗ E1 (λ))−1/2 )† D,
G2 (λ) = (W2 (λ)(E2∗ E2 )−1/2 )† D,
G3 (λ) = (W3 (λ)(E3 (λ)∗ E3 (λ))−1/2 )† D.
Theorem 4.2. Assume that E has full column rank and the operator norms are induced
by Euclidean vector norms. Then we have
inf G1 (λ)

−1

inf G1 (λ)

−1

rCex (A, B; D, E) = min

λ∈C

rCap (A, B; D, E) = min

λ∈C

rCeu (A, B; D, E) = inf G1 (λ)
λ∈C

−1

.

, inf G2 (λ)

−1

, inf G3 (λ)

−1

λ∈C

λ∈C

(4.2)


10
Proof. We note that if system (2.4) if E has full column rank, then E(λ) have full
column rank for all λ ∈ C. Now, formula (4.2) follows from Theorems 3.4, 3.5, 3.6 and
Lemma 4.1.
For Q ∈ Cn×(m+n) , let U = {u∗ ∈ (Cn )∗ : Q∗ (u∗ ) ∈ range(M ∗ )}
Lemma 4.3. Assume that Q is surjective and the operator norms are induced by
Euclidean vector norms. Then,
1
M ∗† Q∗ (u∗ )
=
inf
,
0=u∗ ∈U
M Q−1 D

D∗ (u∗ )

(4.3)

Moreover, if M has full column rank then
1
= σmin (M ∗† Q∗ , D∗ ),
M Q−1 D

(4.4)

where σmin denotes the smallest generalized singular value of the matrix pair.
Proof. Since Q is surjective, Q∗−1 is single-valued (see section Preliminary in [23]).
Thus, we have, by (3.7),
1
x∗
1
=
=
inf
.
0=x∗ D ∗ Q∗−1 M ∗ (x∗ )
M Q−1 D
D∗ Q∗−1 M ∗
For each x∗ = 0 such that M ∗ (x∗ ) ∈ dom Q∗−1 , we put Q∗−1 M ∗ (x∗ ) = u∗ . It follows
that M ∗ (x∗ ) = Q∗ (u∗ ), u∗ ∈ U and x∗ ∈ M ∗−1 Q∗ (u∗ ). It follows, by (2.3), that
x∗ ≥ M ∗† Q∗ (u∗ ) . Therefore, we obtain
inf∗

0=x

x∗
M ∗† Q∗ (u∗ )
≥
inf
.
0=u∗ ∈U
D∗ Q∗−1 M ∗ (x∗ )
D∗ (u∗ )

On the other hand, if x∗ = M ∗† Q∗ (u∗ ) with 0 = u∗ ∈ U then x∗ = 0, u∗ = Q∗−1 M ∗ (x∗ )
and D∗ (u∗ ) = D∗ Q∗−1 M ∗ (x∗ ). Thus,
inf
∗

0=u ∈U

M ∗† Q∗ (u∗ )
x∗
x∗
≥
inf
≥
inf
,
0=x∗ D ∗ Q∗−1 M ∗ (x∗ )
D∗ (u∗ )
x∗ =M ∗† Q∗ (u∗ ),0=u∗ ∈U D ∗ Q∗−1 M ∗ (x∗ )

and we obtain

1
M ∗† Q∗ (u∗ )
=
inf
.
0=u∗ ∈U
M Q−1 D
D∗ (u∗ )

If M has full column rank then U = (Cn )∗ and hence
1
M ∗† Q∗ (u∗ )
=
inf
= σmin (M ∗† Q∗ , D∗ ),
u∗ =1
M Q−1 D
D∗ (u∗ )
the last equality being just the definition of the smallest generalized singular value,
provided that the vector norms are Euclidean norms (see [25]). The proof is complete.


11
Denote
S1λ = {u∗ ∈ (Cn )∗ : u∗ = 1, W1 (λ)∗ (u∗ ) ∈ range(E1 (λ)∗ )},
S2λ = {u∗ ∈ (Cn )∗ : u∗ = 1, W2 (λ)∗ (u∗ ) ∈ range(E2∗ )},
S3λ = {u∗ ∈ (Cn )∗ : u∗ = 1, W3 (λ)∗ (u∗ ) ∈ range(E3 (λ)∗ )}.
By Theorems 3.4, 3.5, 3.6 and Lemma 4.3, we obtain
Theorem 4.4. Assume that the operator norms are induced by Euclidean vector norms.

Then we have
rCex (A, B; D, E) = min

E1 (λ)∗† W1 (λ)∗ (u∗ )
E2∗† W2 (λ)∗ (u∗ )
,
inf
D∗ (u∗ )
D∗ (u∗ )
λ∈C,u∗ ∈S2λ

inf

E1 (λ)∗† W1 (λ)∗ (u∗ )
E3 (λ)∗† W3 (λ)∗ (u∗ )
,
inf
D∗ (u∗ )
D∗ (u∗ )
λ∈C,u∗ ∈S3λ

λ∈C,u∗ ∈S1λ

rCap (A, B; D, E) = min
rCeu (A, B; D, E) =

inf

λ∈C,u∗ ∈S1λ

inf

λ∈C,u∗ ∈S1λ

E1 (λ)∗† W1 (λ)∗ (u∗ )
D∗ (u∗ )

Moreover if E has full column rank then
inf σmin (E1 (λ)∗† W1 (λ)∗ , D∗ ), inf σmin (E2∗† W2 (λ)∗ , D∗ )

rCex (A, B; D, E) = min

λ∈C

rCap (A, B; D, E) = min

λ∈C

rCeu (A, B; D, E)

λ∈C

inf σmin (E1 (λ)∗† W1 (λ)∗ , D∗ ), inf σmin (E3 (λ)∗† W3 (λ)∗ , D∗ )
λ∈C

∗†

∗

∗

= inf σmin (E1 (λ) W1 (λ) , D )
λ∈C

Now, for two matrices L ∈ Cn×p and Q ∈ Cq×p with rank L = n, the generalized
real perturbation value of the matrix pair (L, Q), denoted τn (L, Q), is defined as
τn (L, Q) = inf{ ∆

2

: ∆ ∈ Rn×q , rank(L − ∆Q) < n}.

(4.5)

It is easy to see that τn (L, Q) = τn (L, −Q). It have been known in [15]
Re L −γ Im L
,
Im L
Re L

τn (L, Q) = sup σ2n−1

1
γ

γ∈(0,1]

Re Q −γ Im Q
Im Q
Re Q

1
γ

,

(4.6)

where σi (H1 , H2 ) is the i-th generalized singular value of the matrix pair (H1 , H2 ).
Theorem 4.5. Assume that D is inverse and the vector spaces are endowed with the
Euclidean norm. Then, we have
inf τn (D−1 W1 (λ), E1 (λ)), inf τn (D−1 W2 (λ), E2 )

rRex (A, B; D, E) = min

λ∈C

rRap (A, B; D, E) = min

λ∈C

rReu (A, B; D, E)

inf τn (D−1 W1 (λ), E1 (λ)), inf τn (D−1 W3 (λ), E3 (λ))
−1

= inf τn (D W1 (λ), E1 (λ)).
λ∈C

λ∈C

λ∈C


12
Proof. Since D is inverse, we imply that
rank(W1 (λ)) = rank(W1 (λ) + D∆E1 (λ)) = rank(D−1 W1 (λ) + ∆E1 (λ)).
rank(W2 (λ)) = rank(W2 (λ) + D∆E2 ) = rank(D−1 W2 (λ) + ∆E2 ).
Similarly with Theorem 3.4, the proof now follows from (4.5),(4.6) and characterization
of controllability.
In the rest of this section, we consider a particular case of separate structured
perturbations, which can be covered by the model (3.2) and thus the above result are
applicable. Assume that system (2.4) is subjected to separate perturbations of the
form
B

Ai = Ai + DAi ∆Ai EAi , for all i ∈ {0, 1, −1}, (4.7)

B = B + DB ∆B EB , Ai

where DAi = DB ∈ Cn×l , EAi ∈ CqAi ×n , EB ∈ CqB ×m , for all i ∈ {0, 1, −1}, are given
matrices and ∆B ∈ Cl×qB , ∆Ai ∈ Cl×qAi , for all i ∈ {0, 1, −1}, are the perturbation
matrices. It is easy to see that the perturbation model (4.7) can be rewritten in the
form
[A0 , A1 , A−1 , B]
[A0 , A1 , A−1 , B] = [A0 , A1 , A−1 , B] + D∆E,
where D = DB , E = diag(EA0 , EA1 , EA−1 , EB ) and the perturbation
∆ = [∆A0 , ∆A1 , ∆A−1 , ∆B ].
In this situation, we define



EA0
0
 e−hλ EA1
0 

E(λ) = EH(λ) = 
−hλ
e EA−1 0  ,
0
EB




0
0
 0
0 

M =
EAk 0  .
0 EB

(4.8)

Therefore if EAk , EB has full column rank then E(λ), B has full column rank for all λ ∈
C. Applying Theorems 3.4, 3.5,3.6 we get the controllability radii rCex (A, B; D, E), rCap (A,
B; D, E), rCeu (A, B; D, E) of system (2.4) under these structured perturbations. As a

special case, we consider system (2.4) subjected to perturbations of the form
B

B = B + ∆B , A i

Ai = Ai + αi ∆Ai , for all i ∈ {0, 1, −1},

(4.9)

where αi ∈ C are given scalar parameters and ∆Ai ∈ Cn×n , ∆B ∈ Cn×m are unknown
matrices, i ∈ {0, 1, −1}. Define
µ1 (λ) = |α0 |2 +α1 |2 |e−2hλ |+|α−1 |2 |λ|2 |e−2hλ |, µ2 (λ) = |α−1 |2 +1, µ3 (λ) = |α1 |2 +|α−1 |2 |λ|2 .
(4.10)
Now, we will derive the formula of the controllability radius for linear delay systems
under affine perturbations (4.9).


13
Corollary 4.6. Assume that the exactly controllable system (2.4) is subjected to perturbations of the form (4.9) and the vector spaces are endowed with the Euclidean norm.
Then,
rCex (A, B; D, E) = min

λ∈C

rCap (A, B; D, E) = min

inf σmin

P (λ)

inf σmin

µ1 (λ)
P (λ)
µ1 (λ)

λ∈C

P (λ)

rCeu (A, B; D, E) = inf σmin

µ1 (λ)

λ∈C

,B

,B

, inf σmin

,B

, inf σmin

A−1 − λIn
µ2 (λ)

λ∈C

,B

A1 − λA−1

λ∈C

µ3 (λ)

,B

.
(4.11)

Proof. We see in model (4.9) that
DAi = DB = In , EAi = αi In , EB = Im , for all i ∈ 0, k.
We get D = In , E = diag(α0 In , α1 In , . . . , αk In , Im ), and by (4.8)




α0 In
0
µ1 (λ)−1 α0 In
0
 α1 e−hλ In 0 
 µ1 (λ)−1 α1 e−hλ In
0
∗†
∗

−1



E(λ) = 
,
E(λ)
=
E(λ)(
E(λ)
E(λ))
=
α−1 e−hλ In 0 
µ1 (λ)−1 α−1 λe−hλ In 0  .
0
Im
0
Im
It follows that
E(λ)∗†

x
y

µ1 (λ)−1/2 In 0
0
Im

=

x
y

, for all x ∈ Cn , y ∈ Cm .

Therefore
σmin (Eλ∗† W (λ)∗ , D∗ ) = σmin

µ1 (λ)−1/2 In 0
W1 (λ)∗ , In
0
Im

= σmin

P (λ)
µ(λ)

,B

.

Similarly, we have Therefore, by Theorem 4.4, we obtain formula (4.11).
Corollary 4.7. Assume that the exactly controllable system (2.4) is subjected to perturbations of the form (4.9) and the vector spaces are endowed with the Euclidean norm.
Then,
rRex (A, B; D, E) = min

λ∈C

rRap (A, B; D, E) = min

λ∈C

rReu (A, B; D, E)

inf τn (W1 (λ), E1 (λ)), inf τn (W2 (λ), E2 )
λ∈C

inf τn (W1 (λ), E1 (λ)), inf τn (W3 (λ), E3 (λ))
λ∈C

= inf τn (W1 (λ), E1 (λ)),
λ∈C






α0 In
0
0
0
0
0
 α1 e−hλ In
 0
 α1 In
0
0

0





where E1 (λ) = 
,
E
=
,
E
(λ)
=
2
3
α−1 λe−hλ In 0 
α−1 In 0 
α−1 λIn 0  .
0
Im
0
Im
0
Im



14

5

Conclusion

In this paper, we obtained some general formulas of approximate controllability radius
of linear delay systems under the assumption that the system coefficient matrices are
subjected to structured perturbations. These results unify and extend many existing
results to more general cases. Moreover, it has been shown that from our general results,
some easily computable formulas can be derived. However, numerical algorithms must
be elaborated further to solve the global optimization problems which are involved in
the formulas we have established in the previous sections. These problems are the
topics of our further study.
Acknowledgments
The first author would like to thank Vietnam Institute for Advance Study in Mathematics (VIASM) for support and providing a fruitful research environment and hospitality.
This work was supported financially by Vietnam National Foundation for Science and
Technology Development (NAFOSTED) under grant number 101.01-2014.27.

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