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Generalized Riesz basis property in the analysis of neutral type systems Rabah Rabah, Grigory Sklyar, Aleksandr Rezounenko To cite this version: Rabah Rabah, Grigory Sklyar, Aleksandr Rezounenko. Generalized Riesz basis property in the analysis of neutral type systems. Comptes Rendus. Mathématique, Académie des sciences (Paris), 2003, I 337 (1), pp.19–24. ฀10.1016/S1631073X(03)00251-6฀. ฀hal-00819154฀ HAL Id: hal-00819154 https://hal.archives-ouvertes.fr/hal-00819154 Submitted on 30 Apr 2013 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Équations différentielles-Differential equations Generalized Riesz basis property in the analysis of neutral type systems Rabah Rabah1 , Grigory Sklyar2 and Alexander Rezounenko3 3 1 IRCCyN UMR 6597, 1 rue de la Noë, PB 92101, F-44321 Nantes Cedex 3, France Tél. : 02 40 37 69 15, Fax : 02 40 37 69 30 2 Institute of Mathematics, University of Szczecin, 70–451 Szczecin, Wielkopolska 15, Poland Department of Mechanics and Mathematics, Kharkov University, 4 Svobody sqr., Kharkov, 61077, Ukraine E–mails: rabah@irccyn.ec-nantes.fr, sklar@sus.univ.szczecin.pl, rezounenko@univer.kharkov.ua Abstract The functional differential equation of neutral type is studied. We consider the corresponding operator model in Hilbert space M2 = Cn × L2 (−1, 0; Cn ) and prove that there exists a sequence of invariant finite-dimensional subspaces which constitute a Riesz basis in M2 . We also give an example emphasizing that the generalized eigenspaces do not form a Riesz basis. Bases généralisées de Riesz pour les systèmes de type neutre Résumé On étudie une équation différentielle fonctionnelle de type neutre. Nous considérons le modèle opérationnel dans l’espace de Hilbert M2 = Cn × L2 (−1, 0; Cn ) et montrons qu’il existe dans cet espace une base de Riesz de sous-espaces de dimensions finies invariants par l’opérateur générateur infinitésimal du système. Nous donnons également un exemple précisant qu’il n’existe pas de base de Riesz de sous-espaces propres. Key words. Neutral type systems, Riesz basis, Hilbert space, stability. Mots clés. Systèmes de type neutre, base de Riesz, espace de Hilbert, stabilité. 2000 Mathematics Subject Classification. Primary 34K40; 47D06, 93C23. Version française abrégée Dans ce papier nous étudions l’équation différentielle fonctionnelle de type neutre (1), où A−1 est une matrice constante telle que det A−1 6= 0, A2 , A3 sont des matrices dont les éléments sont dans L2 (−1, 0). C’est une forme assez générale d’équation de type neutre qui apparaı̂t, en particulier, quand on stabilise un système de type neutre, même si le système original est à retards ponctuels. Le problème de la stabilité exponentielle de ce type de système est bien connu et étudié [7, 9]. Afin d’analyser des propriétés plus fines de stabilité asymptotique de (1), 1 la stabilité dite forte (cf. par exemple [3]), nous proposons de considérer (1) sous un forme opérationnelle abstraite et d’étudier ses propriétés spectrales dans des espaces de dimension infinie. Dans [7], l’approche est basée sur une description des systèmes de type neutre dans l’espace de fonctions C([−1, 0]; Cn ). Une autre méthode [13] consiste à traiter le système dans l’espace M2 = Cn × L2 (−1, 0; Cn ), qui est un espace de Hilbert. On peut ainsi utiliser les techniques fines de la théorie des operateurs dans les espaces de Hilbert [1, 6]. Dans ce contexte nous mettons en valeur l’importance du concept remarquable de base de Riesz dans l’analyse du modèle fonctionnel. Nous utilisons le modèle proposé dans [13] pour écrire le système (1) sous la forme (2), où l’operateur A est le générateur d’un C0 -semigroupe dans M2 . Nous montrons que A est un opérateur discret spectral [5] et nous étudions les conditions d’existence d’une base spectrale de Riesz. Traditionnellement par base spectrale de Riesz on entend une base de vecteurs propres, de vecteurs propres généralisés ou de sous-espaces propres (cf. par exemple [4]). Il se trouve que l’operateur A ne possède pas une telle base dans le cas général (voir Exemple 8). Dans un premier temps nous étudions le cas de l’opérateur Ā correspondant au système (1) avec A2 (θ) = A3 (θ) = 0. On montre que les vecteurs propres et les vecteurs propres généralisés de cet opérateur forment une base de Riesz et donc une base de Riesz de sous-espaces propres généralisés. Ceci permet, par une technique d’approximation basée sur la théorie des perturbations des opérateurs linéaires [8], d’approximer une famille de sous-espaces invariants de A par ceux de Ā. Il vient alors le résultat suivant: Théorème. L’opérateur A donné en (2), correspondant au système de type neutre (1), possède une suite de sous-espaces invariants, de dimensions finies, qui forment une base de Riesz de M2 . Ce résultat est d’une grande importance dans l’étude de la stabilité et de la stabilisabilité fortes des systèmes de type neutre. 1. Introduction and operator model In this paper we analize the functional differential equation of neutral type ẋ(t) = A−1 ẋ(t − 1) + Z 0 −1 A2 (θ)ẋ(t + θ)dθ + Z 0 −1 A3 (θ)x(t + θ)dθ (1) where A−1 is constant n × n-matrix, det A−1 6= 0, A2 , A3 are n × n-matrix whose elements belong to L2 (−1, 0). This equation occurs, for example, when a system of neutral type is stabilized. Even if the initial system contains pointwise delays only, then the set of natural feedback laws contains distributed delays (see e.g., [11, 10]), so the corresponding closed-loop system takes the form (1). The problem of exponential stability of systems like (1) is well studied [7, 9]. In order to analize more subtle properties of stability (and stabilizability), namely strong asymptotic stability (see e.g. [3]), one needs to consider an operator model generated by the system (1) in some infinite dimensional space. Such an approach is also helpfull in studying of controllability and other problems of control theory. In [7], the framework is based on the description of the neutral type system in the space of continuous functions C([−1, 0]; Cn ). Another way [13] is to treat the system in the space M2 = Cn × L2 (−1, 0; Cn ), which is a Hilbert space. That gives the possibility to use deep ideas and technique of the operator theory in Hilbert space [1, 6]. 2 In this context, we emphasize the importance of the remarkable concept of Riesz basis in the analysis of operator models. We use the model [13] to write the system (1) in the form d dt y(t) zt (·) ! ! y(t) , =A zt (·) y A z(·) ! R0 0 A2 (θ)ż(θ)dθ + −1 A3 (θ)z(θ)dθ , dz(θ)/dθ = ! R −1 (2) where the domain of A is given by D(A) = {(y, z(·)) : z ∈ H 1 (−1, 0; C n ), y = z(0) − A−1 z(−1)} ⊂ M2 and the operator A is the generator of a C0 -semigroup. The relation between the solutions of the system (1) and the system (2) is, as usual, given by zt (θ) = x(t + θ) for θ ∈ [−1, 0]. This operator model is an extension of the model introduced in [13, 14] in the case of pointwise delays. We show that A is a discrete spectral operator [5] and examine its spectral Riesz basis properties. Traditionally, when studying the concept of spectral Riesz basis, one means the basis of eigen- and generalized eigenvectors, or generally, eigenspaces (see e.g., [4] and references therein). However, it turns out, that the operator A does not possess such a basis in general case (see Example 8). At the same time, we prove our main result: Theorem The operator A (see (2)) corresponding to neutral type system (1) possesses the following generalized Riesz basis property: There exists a sequence of invariant for A finite-dimensional subspaces which constitute a Riesz basis in M2 . This result is crucial in the investigation of strong stability (stabilizability) of neutral type systems which is to be given in our forthcoming paper. 2. Preliminary results In the sequel we will consider the matrix A−1 in a Jordan basis and change the norm in Cn such that the corresponding eigen- and rootvectors of A−1 form an orthogonal basis. Let us denote by µ1 , ..., µℓ , µi 6= µj if i 6= j, the eigenvalues of A−1 and the diP mensions of their rootspaces by p1 , ..., pℓ , ℓk=1 pk = n. Consider the points λ(k) ≡ m ln |µm | + i(arg µm + 2πk), m = 1, .., ℓ; k ∈ Z and the circles Lkm of fixed radius r ≤ r0 ≡ (j) 1 (k) min{|λ(k) m − λi |, (m, k) 6= (i, j)} centered at λm . 3 Proposition 1. The spectrum of A consists of the eigenvalues only which are the roots of the equation det ∆(λ) = 0, where ∆A (λ) = ∆(λ) ≡ −λI + λe−λ A−1 + λ  Z 0 −1 eλs A2 (s)ds + −λ Z 0 −1 eλs A3 (s)ds. (3)  The corresponding eigenvectors of A are ϕ = C−eeλθAC−1 C , with C ∈ Ker∆(λ). There exists N1 such that for any k, such that |k| ≥ N1 , the total multiplicity of the roots of the equation det ∆(λ) = 0, contained in the circle Lkm , equals pm . Proposition 2. The resolvent of A is given by x R(A, λ) ψ(·) ! ≡ A−1 e−λ R0 −λs ψ(s)ds + (I − A−1 e−λ )∆−1 (λ)D −1 e Rθ λ(θ−s) ψ(s)ds + eλθ ∆−1 (λ)D 0 e where D = D(x, ψ) ≡ x + λe−λ A−1 R A3 (θ)}eλθ { 0θ e−λs ψ(s)ds}dθ. R0 −1 e−λs ψ(s)ds − 3 R0 −1 A2 (s)ψ(s)ds − ! R0 , (4) −1 {λA2 (θ) + The proof of Proposition 2 is straightforward. Let us highlight the main steps of the proof of Proposition 1. It is easy to see from the explicit form of R(A, λ) and the compactness of embedding of H 1 (−1, 0; Cn ) into L2 (−1, 0; Cn ), that R(A, λ) is compact. It gives that A has point spectrum only. Each eigenvalue is a root of det ∆(λ) of finite multiplicity. Calculations give the form of eigenvectors of A, taking into account the explicit definition of D(A). To describe the location of the spectrum of A we use Rouche theorem. More precisely, for sufficiently large k and any m we show that |f1 (λ)| > |f2 (λ)| for any λ ∈ Lkm and f1 (λ) ≡ det(A−1 − e−λ I), f2 (λ) ≡ det(A−1 − e−λ I) − R 0 λs R 0 λs det A−1 − e−λ I + eλ −1 e A2 (s)ds + eλ λ−1 −1 e A3 (s)ds . Thus, f1 − f2 has the same k number of roots inside Lm as function f1 . Proposition 3. The operator A generates a C0 -semigroup in M2 . In the particular case when A2 (θ) = A3 (θ) ≡ 0, we will use the notation Ā for A. We will show that the properties of Ā can be expressed in terms of the properties of matrix A−1 only. The basis properties of the operator A will be deduced from the ones of Ā. 3. Basis property of eigen- and rootvectors of the operator Ā. Let νm be the number of Jordan blocks, corresponding to µm ∈ spec(A−1 ). Denote by P m pm,j −1 0 pm,j , j = 1, .., νm , νj=1 pm,j = pm , the orders of these blocks and by {Cm,j , ..., Cm,j } 0 0 the orthonormal system of corresponding eigen- and rootvectors i.e., A−1 Cm,j = µm Cm,j , d−1 d d A−1 Cm,j = µm Cm,j + Cm,j , d = 1, .., pm,j − 1; j = 1, .., νm . If there exists k ∈ {1, . . . , ℓ} such that µk = 1 ∈ spec(A−1 ) we denote by K the rootspace of A−1 corresponding to d ,m ∈ the eigenvalue 1 and put K = {0} otherwise. Finally, let K1 = K ⊥ = Lin{Cm,j {0, . . . , ℓ} : µm 6= 1; j = 1, . . . , νm ; d = 0, . . . , pm,j − 1}. In order to describe eigen- and rootvectors of the operator Ā (see Theorem 5) we need the following lemma. Lemma 4. i) Let us put ν(1) = 0 if 1 6∈ spec(A−1 ) and ν(1) = νk if for some k ∈ {0, . . . , ℓ} : µk = 1. For any y ∈ Cn there exists an unique polynomial vector Py (θ) of the form ν(1) pk,j −1 Py (θ) = X X j=1 d=0 d Ck,j αy0,j θpk,j −d−1 θpk,j −d + αy1,j + · · · + αypk,j −d−1,j θ + γy , (pk,j − d)! (pk,j − d − 1)! ! such that γy ∈ K1 and y = Py (0) − A−1 Py (−1). D ii) The mapping y → Py (·) is a linear operator D : Cn → H 1 (−1, 0; Cn ). Remark. In particular case when 1 6∈ spec(A−1 ) Lemma 4 gives Py (θ) = (I − A−1 )−1 y. Theorem 5. The spectrum of Ā consists of points λ(k) m with corresponding eigen- and rootvectors 0 (k) 0 eλm θ Cm,j !  !  0 0 , , . . . ,  λ(k) θ 0 θpm,j −1 , λ(k) θ 0 pm,j −1 1 e m (Cm,j (pm,j −1)! + ... + Cm,j e m (Cm,j θ + Cm,j ) ) 4 (5) m = 1, . . . , ℓ, k ∈ Z, and also of the point 0 for which one can choose eigen- or rootvectors of the form ! ei , (6) Pei (θ) where {ei }ni=1 is an arbitrary orthogonal basis in Cn and the polynomial Py (θ) is described in Lemma 4. The collection (5) and (6) constitutes a Riesz basis in M2 which becomes an orthogonal basis if we choose the equivalent norm k(y, z(·)k21 2 = kyk + Z 0 −1 kT (Py (θ) − z(θ))k2 dθ, (7) where T is a bounded operator in L2 (−1, 0; Cn ) with bounded inverse. 4. Basis property of finite-dimensional invariant subspaces Theorem 6. There exists N0 large enough, such that for any N ≥ N0 , i) the sequence of subspaces {Vm(k) } |k|>N form a Riesz basis of the closure of their m=1,..,ℓ linear span, say LN . Here Vm(k) ≡ Pm(k) M2 and Pm(k) M2 = projectors; L(k) m are circles defined in Section 2; ii) codim LN = (2N + 1)n + n = 2(N + 1)n. 1 R 2πi L(k) m R(A, λ)dλ are spectral To prove item i) we estimate the norm of the difference Pm(k) − P̄m(k) using the explicit P P form of resolvent (see (4)). Then we choose N0 big enough such that ℓm=1 |k|>N0 ||Pm(k) − P̄m(k) ||2 < 1. The last estimate means that the sequences of subspaces {Vm(k) } |k|>N and m=1,..,ℓ {V̄m(k) } |k|>N m=1,..,ℓ are quadratically close. Theorem 5.2 [6] (see also Th.2.20 and Corollary 2.22 in [8]) and the basis property of {V̄m(k) } k∈Z m=1,..,ℓ (see Theorem 5) prove item i). To prove item ii) we use that the sequence {V̄m(k) } M2 (see Theorem 5) and {V̄m(k) } {V̄m(k) } k∈Z m=1,..,ℓ |k|≤N m=1,..,ℓ ∪ {Vm(k) } (see item i). Hence {V̄m(k) } |k|≤N m=1,..,ℓ |k|>N m=1,..,ℓ k∈Z m=1,..,ℓ form an orthogonal basis in is quadratically close to the sequence ∪ {Vm(k) } |k|>N m=1,..,ℓ form a Riesz basis in M2 and dim{V̄m(k) } |k|≤N = 2(N + 1)n. m=1,..,ℓ The proof of Theorem 6 is complete. The main result of our work is the following Theorem 7. There exists a sequence of invariant for A finite-dimensional subspaces which constitute a Riesz basis in M2 . More precisely, these subspaces are {Vm(k) } |k|>N (see Theorem 6) and a 2(N + 1)nm=1,..,ℓ dimensional subspace spaned by all eigen- and rootvectors, corresponding to all eigenvalues of A, which are outside of all circles L(k) m , |k| > N, m = 1, .., ℓ. Proof of Theorem 7. The proof consists of three steps. Step 1. Let us set X1 = LN , where LN is defined in Theorem 6. The subspace X1 is of finite co-dimension and invariant for A. 5 One can show that M2 can be split into the direct sum   M2 = X1 + X2 , and the operator A11 A12 A can be presented in the triangular form A = 0 A22 , where A11 ≡ P1 AP1 : X1 → X1 , A22 ≡ P2 AP2 : X2 → X2 , A12 ≡ P1 AP2 : X2 → X1 . Here Pi are projectors on Xi along Xj , j 6= i. Step 2. We prove that spec(A11 ) ∩ spec(A22 ) = ∅. Step 3. Let us show that M2 can be split into the direct sum M2 = X1 + X̂2 , with an invariant X̂2 . Consider finite-dimensional operator A22 . There exists spec(A22 ) = {µ̂1 , .., µ̂s } with the total multiplicity equals to dimX2 = codimX1 .   Consider h- an eigenvector of A22 corresponding to µ̂ and find y ∈ X1 such that A hy =   µ̂ hy . Such y is given by y = −(A11 − µ̂I)−1 A12 h (see the triangular form of A). This is due to the property spec(A11 ) ∩ spec(A22 ) = ∅, which implies µ̂ 6∈ spec(A11 ). Exactly in the same way one can find all rootvectors of A for all µ̂. Hence the number of eigenand rootvectors of A corresponding to spec(A22 ) is equal to dimX2 = 2(N + 1)n. By the construction, the linear span of these vectors gives an invariant X̂2 . Now Theorem 6 completes the proof of Theorem 7. We emphasize that the operator A does not possess in general case a Riesz basis of generalized eigenspaces. We illustrate this on the following Example 8. Consider the particular case of the system (1): ẋ(t) = A−1 ẋ(t − 1) + A0 x(t), A−1 11 = , A0 = 01   ! α0 . 0β (8) One can check that the characteristic equation is det∆(λ) = (α−λ+λe−λ )(β−λ+λe−λ ) = 0 and for α 6= β there are two sequences of eigenvectors, such that ||vn1 − vn2 || → 0, as n → ∞. By definition, such vectors can not form a Riesz basis. More precisely, assume opposite i.e., that {vni } form a Riesz basis. Then there exists [6] an orthonormal basis {ηni } and a bounded operator T such that T vni = ηni , i = 1, 2; n ∈ Z. We have that hT (vn1 −vn2 ), ηn1 i = hηn1 −ηn2 , ηn1 i = 1 for all n. On the other hand, one gets hT (vn1 −vn2 ), ηn1 i ≤ kT k · kvn1 − vn2 k · kηn1 k → 0, as n → ∞. We arrive to a contradiction. Acknowledgements. This work was realized with the financial support of Region Pays de la Loire, Ecole Centrale de Nantes (France) and grant 5 PO3A 030 21 (KBN, Poland). References 1. Akhiezer N.I. and Glazman I.M. Theory of linear operators in Hilbert space. Transl. from the Russian. Repr. of the 1961 and 1963 transl. (English) New York, NY: Dover Publications. 218 p. 2. Bellman R. and Cooke K.L. Differential-difference equations, in “Mathematics in Science and Engineering”. Vol. 6. New York-London: Academic Press, XVI, 1963. 3. Brumley W.E., On the asymptotic behavior of solutions of differential-difference equations of neutral type, J. Differential Equations 7(1970), 175-188. 4. Curtain R.F. and Zwart H., An introduction to infinite-dimensional linear systems theory, in “Texts in Applied Mathematics”, Vol. 21, Springer-Verlag, New York, 1995. 6 5. 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