Block Operator Matrices and Second-Order Systems - CiteSeerX

Block Operator Matrices and Second-Order Systems Birgit Jacob and Carsten Trunk

Abstract— In this paper, second-order equations of the form z¨(t) + Ao z(t) + Dz(t) ˙ = 0 are studied. We rewrite this second-order equation into the standard first-order equation x(t) ˙ = Ax(t), where A is a block operator matrix. The location of the spectrum and the essential spectrum of this semigroup generator A is described under various conditions on Ao and on the damping operator D. Keywords— Block operator matrices, spectrum, essential spectrum, second-order systems

I. INTRODUCTION The aim of this paper is to study the location of the spectrum of second-order equations of the form z¨(t) + Ao z(t) + Dz(t) ˙ = 0.

(I.1)

Here the stiffness operator Ao is a possibly unbounded positive operator on a Hilbert space H, which is assumed to be boundedly invertible, and D, the damping operator, −1/2 −1/2 is a is an unbounded operator, such that Ao DAo bounded non negative operator on H. The second-order equation (I.1) above is equivalent to the standard first-order 1/2 equation x(t) ˙ = Ax(t), where A : D(A) ⊂ D(Ao ) × 1/2 H → D(Ao ) × H, is given by · ¸ 0 I A= , −Ao −D o n 1/2 D(A) = [ wz ] ∈ D(A1/2 o ) × D(Ao ) | Ao z + Dw ∈ H . This block operator matrix has been studied in the literature for more than 20 years. Interest in this particular model is motivated by various problems such as stabilization, see for example [6], [19], [20], [18], solvability of the Riccati equations [10], and compensator problems with partial observations [11]. It is well-known that A generates a C0 -semigroup of contractions, and thus the spectrum of A is located in the closed left half plane. This goes back to [3] and [17], see also [4], [8]. Several authors have proved independently of each other that the condition −1/2

hAo

inf

1/2

z∈D(Ao

)\{0}

1/2

Dz, Ao ziH >0 kzk2H

(I.2)

is sufficient for exponential stability of the C0 -semigroup generated by A, see for example [3], [4], [5], [8], [12], [13], Birgit Jacob is with the Department of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands, [email protected] Carsten Trunk is with the Institut f¨ur Mathematik, Technische Universit¨at Berlin, Sekretariat MA 6-3, Straße des 17. Juni 136, D-10623 Berlin, Germany, [email protected]

[21], [23]. Other properties of the C0 -semigroup such as analyticity have been studied in [3], [4] and [8]. In this paper we are interested in a more detailed study of the location of the spectrum of A in the left half plane. Under the extra assumption that Ao has a compact resolvent some results in this direction were obtained in [8] and [16]. In particular, in this situation A has no nonreal essential spectrum. Under various conditions on Ao and on the damping operator D we describe the location of the spectrum and the essential spectrum of A. We proceed as follows. In Section II we introduce the framework and we present some preliminary results. Section III is devoted to the spectrum of the operator A. Sufficient conditions are given to guarantee that certain regions are contained in the resolvent set of A. In the case that Ao has a compact resolvent, it is shown as a main result of this paper that no point of the essential spectrum of A is an accumulation point of the non-real spectrum of A. The location of the essential spectrum is determined. Finally in Section IV we consider, as an application, an Euler-Bernoulli beam with Kelvin-Voigt damping. II. FRAMEWORK We study second-order systems of the following form z¨(t) + Ao z(t) + Dz(t) ˙ = 0. In common with [22], [23] we make the following assumptions throughout this paper. (A1) The stiffness operator Ao : D(Ao ) ⊂ H → H is a self-adjoint, positive definite linear operator on a Hilbert space H such that zero is in the resolvent set of Ao . A scale of Hilbert spaces Hα is defined as follows: For α ≥ 0, we define Hα = D(Aα o ) equipped with the norm k · kHα := ∗ kAα · k and H H −α = Hα . Here the duality is taken with o respect to the pivot space H, that is, equivalently H−α is the completion of H with respect to the norm kzkH−α = kA−α o zkH . Thus Ao extends (restricts) to Ao : Hα → Hα−1 for α ∈ R. We use the same notation Ao to denote this extension (restriction). We denote the inner product on H by h·, ·iH or h·, ·i, and the duality pairing on H−α × Hα by h·, ·iH−α ×Hα . Note that for (z 0 , z) ∈ H × Hα , α > 0, we have hz 0 , ziH−α ×Hα = hz 0 , ziH . (A2) The damping operator D : H 21 → H− 12 is a −1/2

−1/2

bounded operator such that Ao DAo self-adjoint operator in H satisfying hDz, ziH− 1 ×H 1 ≥ 0, 2

2

is a bounded

z ∈ H 12 .

(II.1)

The system (I.1) is equivalent to the following standard first-order equation x(t) ˙

= Ax(t)

additional assumptions, it is shown in [14] that systems of the form z¨(t) + Ao z(t) + Dz(t) ˙ = Bo u(t),

(II.2)

where A : D(A) ⊂ H 12 × H → H 12 × H, is given by · ¸ 0 I A= , −Ao −D n o D(A) = [ wz ] ∈ H 12 × H 12 | Ao z + Dw ∈ H .

equipped either with velocity measurements

The operator A itself is not self-adjoint in the Hilbert space H 21 × H. However, in [23, Proof of Lemma 4.5] it is shown that · ¸ I 0 A∗ = JAJ, with J = . 0 −I

are well-posed and have a minimum-phase transfer function.

In particular, JA is a self-adjoint operator in H 12 × H. For ( xy11 ) , ( xy22 ) ∈ H 12 × H we define an inner product on H 12 × H by [( xy11 ) , ( xy22 )] := hJ ( xy11 ) , ( xy22 )i = hx1 , x2 iH 1 − hy1 , y2 i 2 (II.3) Then (H 21 × H, [., .]) is a Krein space (for the basic theory of Krein spaces and operators acting in such spaces we refer to [7] and [1]) and A is a self-adjoint operator with respect to [., .]. Moreover, see [23, Proof of Lemma 4.5], A has a bounded inverse in H 12 × H, with · ¸ −1 −A−1 −1 o D −Ao A = , (II.4) I 0 where A−1 o D is considered as an operator acting in H 12 . This and the self-adjointness of A in the Krein space (H 12 × H, [., .]) imply the well-known (cf. [22] and [23, Proof of Lemma 4.5]) Lemma 2.1: The operator A has a bounded inverse and the spectrum of A is symmetric with respect to the real line. Moreover, the following theorem is well known, see [3], [4], [8] and [17]. Theorem 2.2: The operator A is the generator of a strongly continuous semigroup T (t) of contractions on the state space H 12 × H. This guarantees that the spectrum of A is contained in the closed left half plane Re(s) ≤ 0. If there exists a constant β > 0 such that hDz, ziH− 1 ×H 1 ≥ βkzk2H , 2

2

z ∈ H 12 ,

(II.5)

which is equivalent to (I.2), then it is well known that A generates an exponentially stable semigroup on H 12 × H, in particular there is no spectrum on the imaginary axis. If condition (II.5) is relaxed to hDz, ziH− 1 ×H 1 > 0 for any eigenvector z ∈ H1 of Ao 2 2 (II.6) and if, in addition, A−1 is a compact operator then it is o shown in [8, Lemma 4.1 and Theorem 4.4] that there is no spectrum on the imaginary axis. Moreover, under some

y(t) = Bo∗ z(t). ˙ or position measurements y(t) = Bo∗ z(t).

III. LOCATION OF THE SPECTRUM Our aim is to describe the spectrum of A more precisely under some additional assumptions. Recall that the approximate point spectrum, σap (A), consists of all λ ∈ C for which there is a sequence {xn }n∈N in D(A) such that kxn k = 1 and k(A − λI)xn k → 0

as n → ∞

(see for example [9, page 242]). We point out that the point and continuous spectrum are subsets of the approximate point spectrum. Theorem 3.1: Assume that there exists γ > 0 such that for all z ∈ H 21 we have hAo z, ziH− 1 ×H 1 ≤ γhDz, ziH− 1 ×H 1 . 2

2

2

(III.1)

2

Then we have σ(A) ⊂ (−∞, 0)∪{λ ∈ C | |Im λ|2 < −2γRe λ−(Re λ)2 }. (III.2) Proof: By Lemma 2.1 we have 0 ∈ ρ(A). Let λ = µ + iσ with µ ≤ 0 and σ 6= 0. Assume that λ belongs to the spectrum of A. Then, by Lemma 2.1, λ ∈ σ(A). Since A is a self-adjoint operator in the Krein space (H 21 × H, [., .]), see (II.3), it follows from [7, Theorem VI.6.1] that at least one of the points λ, λ belongs to σap (A). Let us assume λ ∈ σap (A). Then there exists a sequence {( xynn )}n∈N in D(A) which satisfies k( xynn )kH 1 ×H = 1 2

and

lim k(λI − A) ( xynn )kH 1 ×H = 0.

n→∞

2

Therefore we have kyn − λxn kH 1 → 0 and

(III.3)

kAo xn + Dyn + λyn k → 0 as n → ∞.

(III.4)

2

It follows from (III.3) that {xn } has no subsequence which converges to zero in H 12 , that is lim inf kxn kH 1 > 0. n→∞

2

Combining (III.3) and (III.4) we get hAo xn , xn iH− 1 ×H 1 + (µ + iσ)hDxn , xn iH− 1 ×H 1 + 2

2

2

+ (µ + iσ)2 hxn , xn i → 0,

2

as n → ∞. Then the imaginary part tends to zero, i.e. hDxn , xn iH− 1 ×H 1 + 2µhxn , xn i → 0, 2

(III.5)

2

as n → ∞. Further, the real part tends to zero, i.e. hAo xn , xn iH− 1 ×H 1 + µhDxn , xn iH− 1 ×H 1 + 2

2

2

2

+ (µ2 − σ 2 )hxn , xn i → 0,

(III.6)

as n → ∞. Finally, combining (III.5) and (III.6) we obtain hAo xn , xn iH− 1 ×H 1 − (µ2 + σ 2 )kxn k2 → 0, 2

n → ∞, (III.7)

2

Then with (III.1) and (III.5) we have 0 = lim h(Ao − µ2 − σ 2 )xn , xn iH− 1 ×H 1 ≤ n→∞

2

2

≤ lim h(γD − µ2 − σ 2 )xn , xn iH− 1 ×H 1 = n→∞

2

2

= lim (−2µγ − µ2 − σ 2 )kxn k2 .

assertions of Theorem 3.2 follows from the fact that 0 ∈ ρ(A) and that by (II.1) −A−1 o D is a non-negative selfadjoint operator in H 12 . From the above theorem we obtain a criterion for the emptyness of the essential spectrum of A. Proposition 3.3: Assume that A−1 o is a compact operator. Assume that the operator D is also a bounded operator acting from H 12 into Hα for some α > − 12 . Then σess (A) = ∅. Moreover, it turns out that the non-real spectrum of A can only accumulate to ∞. Theorem 3.4: Assume that A−1 o is a compact operator. Then no point from σess (A) is an accumulation point of the non-real spectrum of A. Proof: Let µ < 0 and set λ = µ+iσ for some σ 6= 0. Then

n→∞

1

From (III.7) it follows that {xn } has no subsequence which converges to zero in H, hence 2

2

−2µγ ≥ µ + σ , and thus (III.2) follows. Recall that a densely defined closed operator T in some Hilbert (or Banach) space is called Fredholm if the dimension of the kernel of T and the codimension of the range of T are finite. Then the set σess (T ) := {λ ∈ C | T − λI is not Fredholm} is called the essential spectrum of T . Moreover, by σp,norm (T ) we denote the set of all λ ∈ C which are isolated points of σ(T ) and normal eigenvalues of T , that is, the corresponding Riesz-Dunford projection is of finite rank. In Theorem 3.2 below we describe the location of the essential spectrum of A. Instead of (III.1) we assume now that A−1 o is a compact operator. Theorem 3.2: Assume that A−1 o is a compact operator. Then 1 σess (A) := {λ ∈ C | ∈ σess (−A−1 o D)}, λ where A−1 o D is considered as an operator in L(H 12 ). Moreover there exist α > 0 such that σess (A) ⊂ (−∞, −α] and σess (A) = σ(A) \ σp,norm (A). Proof: By Lemma 2.1 we have 0 ∈ ρ(A), hence σess (A) = {λ ∈ C\{0} | 1/λ ∈ σess (A−1 )}. By relation (II.4) it remains to show σess (A−1 )\{0} = σess (−A−1 o D)\{0}. The identity I is a compact linear operator from H 12 to H, and −A−1 o is a compact operator from H to H 12 . Since the essential spectrum of an operator remains unchanged under compact perturbations, we have σess (A−1 )\{0} = σess (−A−1 o D)\{0}. The remaining

Gλ := span {x ∈ H 12 | Ao2 x = νx, ν ≤ −µ}

(III.8)

is a finite dimensional subspace of H 21 . Assume that there exists a sequence {( xynn )}n∈N in D(A)∩(Gλ ×Gλ )⊥ which satisfies k( xynn )kH 1 ×H = 1 (III.9) 2

and lim k(λI − A) ( xynn )kH 1 ×H = 0.

n→∞

(III.10)

2

Then, similar as in the proof of Theorem 3.1, we have lim inf n→∞ kxn kH 1 > 0 and 2

hAo xn , xn iH− 1 ×H 1 −(µ2 +σ 2 )hxn , xn i → 0, as n → ∞ 2

2

(cf. III.7)). As xn belongs to G⊥ λ , n ∈ N, there exists a δ > 0 with 1

hAo xn , xn iH− 1 ×H 1 = kAo2 xn k2 ≥ (µ2 + δ 2 )kxn k2 2

2

and, if σ < δ, it follows lim hAo xn , xn iH− 1 ×H 1 = lim kxn kH 1 = 0,

n→∞

2

2

n→∞

2

a contradiction. Therefore there exists an open neighbourhood V in C of (−∞, 0) such that for every λ ∈ V \ (−∞, 0) there exists a finite dimensional subspace Gλ and a constant cλ > 0 such that for all ( xy ) in D(A) ∩ (Gλ × Gλ )⊥ the relation k(A − λI) ( xy )kH 1 ×H ≥ cλ k( xy )kH 1 ×H . 2

2

holds. Then by Lemma 2.1 and [15, IV.§5.6] there exists a discrete set Ξ in V with V \ Ξ ⊂ ρ(A) ∪ (−∞, 0). Hence each point of (−∞, 0) is an accumulation point of ρ(A). Let λ ∈ (−∞, 0) and choose Gλ as in (III.8). For every sequence {( xynn )}n∈N in D(A)∩(Gλ ×Gλ )⊥ which satisfies

(III.9) and (III.10) it follows that kyn − λxn kH 1 → 0 as 2 n → ∞. Therefore we have ³ ´ lim inf [( xynn ) , ( xynn )] = lim inf hxn , xn iH 1 − hyn , yn i = n→∞ n→∞ 2 ¡ ¢ = lim inf hAo xn , xn i − λ2 hxn , xn i n→∞ ³ 1 ´ = lim inf kAo2 xn k2 − λ2 kxn k2 >

same point: y(t) = z(ξ, t) u(t) = ¸ · ¸ · ∂4z ∂3z ∂4z ∂3z − E 3 + Cd 3 = E 3 + Cd 3 ∂r ∂r ∂t r=ξ+ ∂r ∂r ∂t r=ξ−

n→∞

> 0, as xn belongs to G⊥ λ , n ∈ N. This implies (−∞, 0] ∩ σ(A) ⊂ σπ+ (A) (for a definition of σπ+ (A) we refer to [2]) and, by [2, Theorem 18], no point of the interval (−∞, 0) is an accumulation point of non-real spectrum. We give an example such that the non-real spectrum of A accumulates to iR in the case A−1 o is a compact operator. Example 3.5: Let H be an infinite-dimensional Hilbert space with orthonormal basis {en }n∈N . We define the operators Ao : D(Ao ) ⊂ H → H and D ∈ L(H) by Ao z

:=

∞ X

n2 hz, en ien ,

n=1

D(Ao ) :=

{z ∈ H |

X

z ∈ D(Ao ),

n4 |hz, en i|2 < ∞},

The beam is pinned at each end so h 2 i 3 z(0, t) = 0, E ∂∂rz2 + Cd ∂r∂ 2 z∂t h 2 ir=0 3 z(1, t) = 0, E ∂∂rz2 + Cd ∂r∂ 2 z∂t r=1

= 0, = 0.

We will put this control system into the framework of d4 this paper. Here H is L2 (0, 1) and Ao = E dr 4 with D(Ao ) given by ½ z ∈ H 4 (0, 1) | z(0) =

¾ d2 z d2 z (0) = z(1) = (1) = 0 . dr2 dr2

Also, © ª H 12 = z ∈ H 2 (0, 1) | z(0) = z(1) = 0

n∈N

Dz

:=

∞ X 1 hz, en ien . n n=1

q 1 +i n2 − 4n1 2 , An easy calculation shows that λn := − 2n ¡ e ¢ n ∈ N, are eigenvalues of A with λnnen as corresponding eigenvectors. Remark 3.6: We note that A does not necessarily have a compact resolvent if Ao has a compact resolvent. Indeed, if we choose Ao = D then, by Theorem 3.2, σess (A) = {−1}, hence −1 ∈ σess (A−1 ) and A−1 is not a compact operator. IV. AN APPLICATION As an application we consider an Euler-Bernoulli beam with Kelvin-Voigt damping. Consider two rigidly joined beams with a force acting at the joint ξ. Let z(r, t) denote the deflection of the beam from its rigid body motion at time t and position r. Use of the Euler-Bernoulli model for the beam deflection an the Kelvin-Voigt damping model leads to the following description of the vibrations: · ¸ ∂2z ∂2 ∂2z ∂3z + E + C = 0, d ∂t2 ∂r2 ∂r2 ∂r2 ∂t for r ∈ (0, ξ) ∪ (ξ, 1) and t > 0. Here 0 < ξ < 1, and E and Cd are positive physical constants. The assumption that the beams are rigidly joined ensures that ∂z ∂z (ξ−) = (ξ+). ∂r ∂r We thus consider the system as a single beam. A force u is applied at the joint ξ, with position measurement at the z(ξ−) = z(ξ+),

with inner product hz, viH 1 = Ehz 00 , v 00 i. Let x(t) = 2 (z(·, t), z(·, ˙ t)). The damping operator D : H 12 → H− 12 is defined by hDz, φiH− 1 ×H 1 = 2

2

Cd hz, φiH 1 E 2

for z, φ ∈ H 21 . Hence D = CEd Ao . Assumptions (A1)-(A2) are satisfied. The operator A−1 is compact and (III.1) holds for o γ = CEd . Then the results of this paper holds, that is, Theorems 3.1, 3.2, 3.4 and Proposition 3.3 imply the following proposition. Proposition 4.1: We have ½ ¾ −E σess (A) = Cd and each point λ ∈ σ(A), λ 6= −E Cd , is an isolated normal eigenvalue. Moreover, the non-real spectrum consists of at most finitely many points. We mention that for this example the spectrum can be calculated directly. However, the main advantage of the results of this paper is that they can be applied to vibrations on general domains or to wave problems as in [22] and [23]. V. ACKNOWLEDGMENTS Carsten Trunk gratefully acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG), grant KON 921/2006.

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