CONTROLLABILITY RADII OF LINEAR NEUTRAL SYSTEMS UNDER STRUCTURED PERTURBATIONS

In this paper we shall deal with the problem of calculation of the controllability radii of linear neutral systems of the form x 0 (t) = A0x(t) + A1x(t − h) + A−1x 0 (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 multivalued 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

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 controlla-bility radii of linear neutral systems of the form x0(t) = A0x(t) + A1x(t − h) +

A−1x0(t − h) + Bu(t) We will derive the definition of exact controllability ra-dius, 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 co-efficient matrices are subjected to structured perturbations Some examples are provided to illustrate the obtained results

Keywords Linear neutral systems, multi-valued linear operators, structured pertur-bations, controllability radius

1 Introduction

In this paper, we investigate the robust controllability for linear neutral systems of the form

x0(t) = A0x(t) + A1x(t − h) + A−1x0(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, approx-imate 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: ducthuank7@gmail.com.

† 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

On the other hand, many problems arising from real life contain uncertainty, be-cause there are parameters which can be determined only by experiments or the re-mainder 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

x0(t) = eA0x(t) + eA1x(t − h) + eA−1x0(t − h) + eBu(t), (1.2) with

[A0, A1, A−1, B] [ eA0, eA1, eA−1, eB] = [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 con-dition 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 prob-lem 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 per-turbation 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 k · k and its dual space can

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×m i, 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∗, kF k = kF∗k (2.1) Here the norm of F is defined as

kF k = sup

inf

y∈F (x)kyk : x ∈ dom F , kxk = 1 (2.2)

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, the Moore-Penrose pseudo inverse matrix F†∈ Km×nexists and if vector spaces Kn, Km

are equipped with Euclidean norms (i.e kxk =√

x∗x) then F† defines a linear selector

of F−1 (i.e F†y ∈ F−1(y), ∀y ∈ Kn) satisfying kF†yk = inf{kxk : x ∈ F−1(y)} This implies, in particular, that

kF†yk ≤ kxk, for all x ∈ F−1(y) (2.3) Now, consider the linear neutral systems

(

x0(t) = A0x(t) + A1x(t − h) + A−1x0(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 con-ditions φ0(.) ∈ W1

2([−h, 0], Cn), disired final φ1(.) ∈ W1

2([−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(.) ∈ W1

2([−h, 0], Rn), disired final φ1(.) ∈ W1

2([−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],

kxT(.) − φ1(.)kW1 < , where xT(θ) = x(T + θ), for all θ ∈ [−h, 0]

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−hλA1+ λe−hλA−1− λIn (2.5)

be the characteristic polynomial of system (2.4) The following propositions give nec-essary 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−1B, , 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

x0(t) = eA0x(t) + eA1x(t − h) + eA−1x0(t − h) + eBu(t), (3.1) with

[A0, A1, A−1, B] [ eA0, eA1, eA−1, eB] = [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 k · k on Cl×q, the exact controllability radius of system (2.4) with respect to structured perturbations

of the form (3.2) is defined by

rex

K (A, B; D, E) = infk∆k : ∆ ∈ 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 rex(A, B; D, E) = +∞

Definition 3.2 Let system (2.4) be approximately controllable Given a norm k · k on

Cl×q, the approximative controllability radius of system (2.4) with respect to structured perturbations of the form (3.2) is defined by

rap

K (A, B; D, E) = infk∆k : ∆ ∈ 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 rap

K(A, B; D, E) = +∞

Definition 3.3 Let system (2.4) be Euclidean controllable Given a norm k · k on

Cl×q, the Euclidean controllability radius of system (2.4) with respect to structured perturbations of the form (3.2) is defined by

rKeu(A, B; D, E) = infk∆k : ∆ ∈ Kl×q s.t (3.1) not Euclidean controllable (3.5)

If system (3.1) under structured perturbations (3.2) is Euclidean controllable for all

∆ ∈ Cl×q then we set reuK(A, B; D, E) = +∞

We define

W1(λ) = [P (λ), B], W2(λ) = [A−1− λIn, B], W3(λ) = [λA−1+ A1, B]

H1(λ) =









In 0

e−hλIn 0

λe−hλIn 0

0 Im







 , H2 =









0 0

0 0

In 0

0 Im







 , H3(λ) =









0 0

In 0

λIn 0

0 Im







 ,

E1(λ) = EH1(λ), E2 = EH2, E3(λ) = EH3(λ)

(3.6)

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{k∆k : ∆ ∈ Cl,q s.t W + D∆E is non-surjective}

= 1 kEW−1Dk,

(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 struc-tured perturbations of the form (3.2) Then the exact controllability radius of (2.4) is given by the formula

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

 inf

λ∈CkE1(λ)W1(λ)−1Dk−1, inf

λ∈CkE2W2(λ)−1Dk−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)

Proof Suppose that [ eA0, eA1, eA−1, eB] = [A0, A1, A−1, B] + D∆E is not exactly con-trollable for ∆ ∈ Cl×q It means, by Proposition 2.4, for some λ0 ∈ C the operator f

W1(λ0) = [ eP (λ0), eB] is not surjective, where eP (λ0) = eA0+ e−hλ0

e

A1+ λe−hλ0

e

A−1− λ0In,

or the operator fW2(λ0) = [ eA−1− λ0In, eB] is not surjective If fW1(λ0) is not surjective,

by definitions (2.5) and (3.6) we can deduce

f

W1(λ0) = [ eP (λ0), eB] = [ eA0, eA1, eA−1, eB]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

k∆k ≥ dist(W1(λ0); D, E1(λ0)) = kE1(λ0)W1(λ0)−1Dk−1 ≥ inf

λ∈CkE1(λ)W1(λ)−1Dk−1

If fW2(λ0) is not surjective with some λ0, by definition (3.6) we can deduce

f

W2(λ0) = [ eA−1− λ0In, eB] = [ eA0, eA1, eA−1, eB]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

k∆k ≥ dist(W2(λ0); D, E2) = kE2W2(λ0)−1Dk−1 ≥ inf

λ∈CkE2W2(λ)−1Dk−1 Therefore, we imply that

k∆k ≥ min

 inf

λ∈CkE1(λ)W1(λ)−1Dk−1, inf

λ∈CkE2W2(λ)−1Dk−1



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

rexC(A, B; D, E) ≥ min

 inf

λ∈CkE1(λ)W1(λ)−1Dk−1, inf

λ∈CkE2W2(λ)−1Dk−1



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

kE1(λ)W1(λ)−1Dk ≥ sup

λ∈C

kE1(λ)W1(λ)−1Dk −  > 0

By the definition of the structured distance to singularity, it follows that there exists

a perturbation ∆ such that

k∆k ≤ kE1(λ)W1(λ)−1Dk − 
−1 and the perturbed matrix fW1(λ) = W1(λ) + D∆E1(λ) is not surjective Hence, equation (3.1) is not exactly controllable with the perturbation ∆ Thus, by definition,

rexC(A, B; D, E) ≤ kE1(λ)W1(λ)−1Dk − 
−1 ≤

 sup

λ∈C

kE1(λ)W1(λ)−1Dk − 2

−1

Letting  → 0, we get

rCex(A, B; D, E) ≤ inf

λ∈CkE1(λ)W1(λ)−1Dk−1 Similarly, we imply

rCex(A, B; D, E) ≤ inf

λ∈CkE2W2(λ)−1Dk−1, and hence

rexC(A, B; D, E) ≤ min

 inf

λ∈CkE1(λ)W1(λ)−1Dk−1, inf

λ∈CkE2W2(λ)−1Dk−1



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

rap

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

 inf

λ∈CkE1(λ)W1(λ)−1Dk−1, inf

λ∈CkE3(λ)W3(λ)−1Dk−1

 , (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 struc-tured 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

λ∈CkE1(λ)W1(λ)−1Dk−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

x0(t) = A0x(t) + A1x(t − 1) + A−1x0(t − 1) + Bu(t), (3.12)

where A0 =1 1

1 1

 , A1 = 2 2

−2 2

 , A−1 =2 2

2 −2

 , B =2

2

 We see that

W1(λ) = [P (λ), B] =1 + 2e−λ+ 2λe−λ− λ 1 + 2e−λ+ 2λe−λ 2

1 − 2e−λ+ 2λe−λ 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



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

with D = 1

2

 , E = 1 1 0 0 0 0 0

0 0 0 1 1 0 1

 and ∆ = [δ1 δ2] It implies that E1(λ) =



1 1 0

λe−λ e−λ 1

 , E2 =0 0 0

1 0 1

 , E3(λ) =1 1 0

λ 1 1

 We have, for v ∈ C,

E1(λ)W1(λ)−1D(v) = E1(λ)W1(λ)−1 v

2v



=



E1(λ)





p q r





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



=

 

p + q

λe−λp + e−λq + 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)−1D(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

x1 = p + q, x2 = λe−λp + e−λq + r

Let C2 be endowed with the vector norms k · k∞, then we can deduce,

3|v| ≤ |2 − λ||x1| + 4|x2| ≤ (|2 − λ| + 4) max{|x1|, |x2|} = (|2 − λ| + 4)kx1

x2



k∞

This implies

kx1

x2



k∞≥ 3|v|

|2 − λ| + 4, which yields the equality if x2 = 3v

|2 − λ| + 4 and x1 = e

iϕx2, where ϕ is chosen such that (2 − λ)eiϕ= |2 − λ| Therefore,

kE1(λ)W1(λ)−1Dk = sup

|v|=1

d 0, E1(λ)W1(λ)−1D(v)

= 3

|2 − λ| + 4,

and hence supλ∈CkE1(λ)W1(λ)−1Dk = 3

4 Moreover, it is easy to see that

E2W2(λ)−1D(v) =

  0

p + r



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



E3(λ)W3(λ)−1D(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

sup

λ∈C

kE2W2(λ)−1Dk = 1, sup

λ∈C

kE3(λ)W3(λ)−1Dk = 3

4. Thus, by Theorems 3.4, 3.5, 3.6, we obtain

rexC(A, B; D, E) = 1, rapC(A, B; D, E) = reuC(A, B; D, E) = 4

3.

4 Some particular cases

Formulas (3.8), (3.10), (3.11) gives us a unified framework for computation of control-lability radii, however, it is not easy to be used because this formula involves calcula-tion of the norm of the multi-valued linear operators E1(λ)W1(λ)−1D, E2(λ)W2(λ)−1D,

E3W3−1D which do not have an explicit representation We now derive from this re-sult 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 kxk =√

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

kM Q−1Dk = k(Q(M∗M )−1/2)†Dk, (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

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

 inf

λ∈CkG1(λ)k−1, inf

λ∈CkG2(λ)k−1



rap

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

 inf

λ∈CkG1(λ)k−1, inf

λ∈CkG3(λ)k−1



reu

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

λ∈CkG1(λ)k−1

(4.2)

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

kM Q−1Dk =06=uinf∗ ∈U

kM∗†Q∗(u∗)k

kD∗(u∗)k , (4.3) Moreover, if M has full column rank then

1

kM Q−1Dk = σmin(M

∗†

Q∗, 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

kM Q−1Dk =

1

kD∗Q∗−1M∗k = inf06=x ∗

kx∗k

kD∗Q∗−1M∗(x∗)k. For each x∗ 6= 0 such that M∗(x∗) ∈ dom Q∗−1, we put Q∗−1M∗(x∗) = u∗ It follows that M∗(x∗) = Q∗(u∗), u∗ ∈ U and x∗ ∈ M∗−1Q∗(u∗) It follows, by (2.3), that

kx∗k ≥ kM∗†Q∗(u∗)k Therefore, we obtain

inf

06=x ∗

kx∗k

kD∗Q∗−1M∗(x∗)k ≥ inf

06=u ∗ ∈U

kM∗†Q∗(u∗)k

kD∗(u∗)k .

On the other hand, if x∗ = M∗†Q∗(u∗) with 0 6= u∗ ∈ U then x∗ 6= 0, u∗ = Q∗−1M∗(x∗) and D∗(u∗) = D∗Q∗−1M∗(x∗) Thus,

inf

06=u ∗ ∈U

kM∗†Q∗(u∗)k

kD∗(u∗)k ≥ inf

x ∗ =M ∗† Q ∗ (u ∗ ),06=u ∗ ∈U

kx∗k

kD∗Q∗−1M∗(x∗)k ≥ inf

06=x ∗

kx∗k

kD∗Q∗−1M∗(x∗)k,

and we obtain

1

kM Q−1Dk =06=uinf∗ ∈U

kM∗†Q∗(u∗)k

kD∗(u∗)k .

If M has full column rank then U = (Cn)∗ and hence

1

kM Q−1Dk = infku ∗ k=1

kM∗†Q∗(u∗)k

kD∗(u∗)k = σmin(M

∗†

Q∗, D∗),

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

x∗x)...



Thus, for each v ∈ C, the problem of computing d(0, E1(λ)W1(λ)−1D(v)) is reduced to the calculation of the distance from the origin to the straight