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
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: 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 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. 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(2012), ‘The structured distance to non-surjectivity and its applications to calculating the controllability radius of descriptor systems’, J. Math. Anal. Appl., 388, 272–281. [23] Son, N.K., Thuan, D.D. (2013), ‘The structured controllability radii of higher order systems’, Linear Algebra Appl., 438, 2701–2716. [24] Thuan, D.D. (2013), ‘The structured controllability radius of linear delay systems’, International Journal of Control, 86, 512–518. [25] Van Loan, C.F. (1976), ‘Generalizing the singular value decomposition’, SIAM J. Numer. Anal., 13, 76–83. [...]... D., Pritchard, A.J (1986), ‘Stability radii of linear systems , Systems Control Lett., 7, 1–10 [12] Hinrichsen, D., Pritchard, A.J (1986), ‘Stability radius for structured perturbations and the algebraic Riccati equation’, Systems Control Lett., 8, 105–113 [13] Jacobs, M.Q., Langenhop, C.E (1976), ‘Criteria for function space controllability of linear neutral systems , SIAM J Control Optim., 14, 1009–1048... controllabilty of linear retarded systems: Derivation from abstract operator conditions’, SIAM J Control Optim., 16, 590–645 [17] Mei, Z.D and Peng, J.G (2010), ‘On robustness of exact controllability and exact observability under cross perturbations on the generator in Banach spaces’, Proceedings of the AMS, 138, 4445–4468 [18] O’connor, D.A., Tarn, T.J (1983), ‘On the function space controllability of linear neutral. .. (2010), Structured distance to uncontrollability under multi -perturbations: an approach using multi-valued linear operators’, Systems Control Lett., 59, 476–483 [22] Son, N.K., Thuan, D.D (2012), ‘The structured distance to non-surjectivity and its applications to calculating the controllability radius of descriptor systems , J Math Anal Appl., 388, 272–281 [23] Son, N.K., Thuan, D.D (2013), ‘The structured. .. descriptor systems , J Math Anal Appl., 388, 272–281 [23] Son, N.K., Thuan, D.D (2013), ‘The structured controllability radii of higher order systems , Linear Algebra Appl., 438, 2701–2716 [24] Thuan, D.D (2013), ‘The structured controllability radius of linear delay systems , International Journal of Control, 86, 512–518 [25] Van Loan, C.F (1976), ‘Generalizing the singular value decomposition’, SIAM... space controllability of linear neutral systems , SIAM J Control Optim., 21, 306–329 [19] Rabah, R., Sklyar, G.M (2007), ‘The analysis of exact controllability of neutraltype systems by the moment problem approach’, SIAM J Control Optim., 46, 2148–2181 [20] Rodas, H.R., Langenhop, C.E (1978), ‘A sufficient condition for function space controllability of a linear neutral system’, SIAM J Control Optim.,... Lam, S and Davison, E.J (2014), ‘Computation of the Real Controllability Radius and Minimum-Norm Perturbations of Higher-Order, Descriptor, and Time-Delay LTI Systems , IEEE Trans Automatic Control, 59, 3354–3359 [15] Lam, S and Davison, E.J (2009), ‘Generalized real perturbation values with applications to the structured real controllability radius of LTI systems , in Pro Amer Control Conf., St Louis,... |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... 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... (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... 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 ... 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. .. 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... [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