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NETWORKS AND HETEROGENEOUS MEDIA c

American Institute of Mathematical Sciences Volume 5, Number 2, June 2010

doi:10.3934/nhm.2010.5.315 pp. 315–334

SPECTRUM AND DYNAMICAL BEHAVIOR OF A KIND OF PLANAR NETWORK OF NON-UNIFORM STRINGS WITH NON-COLLOCATED FEEDBACKS

Zhong-Jie Han and Gen-Qi Xu Department of Mathematics, Tianjin University Tianjin 300072, China

(Communicated by Benedetto Piccoli) Abstract. A kind of planar network of strings with non-collocated terms in boundary feedback controls is considered. Suppose that the network is constituted by n non-uniform strings, connected by one vibrating point mass. The displacements of these strings are continuous at the common vertex. The noncollocated terms are contained in feedback controls at exterior nodes. The well-posedness of the corresponding closed-loop system is proved. A complete spectral analysis is carried out and the asymptotic expression of the spectrum of this system operator is obtained, which implies that the asymptotic behavior of the spectrum is independent of these non-collocated terms. Then the Riesz basis property of the (generalized) eigenvectors of the system operator is proved. Thus, the spectrum determined growth condition holds. Finally, the exponential stability of a special case of this kind of network is gotten under certain conditions. In order to support these results, a numerical simulation is given.

1. Introduction. The dynamical behavior of multi-link flexible structures and their control problems (see e.g. [20],[13],[29]), are a lot of issues in engeneering and applied mathematics. What we shall consider in this paper is a planar network of non-uniform strings linked at its common vertex to a rigid body. This type of structure has been studied in [33],[9] for strings and [10],[11],[12] for beams. The dynamical behavior of such hybrid systems is complicated due to the dynamic coupling between the flexible strings and the rigid bodies. Here we consider the asymptotic behavior of this kind of flexible network system with non-collocated terms in boundary feedback controls. The relation between the distribution of the spectrum of the network system and these non-collocated feedbacks is considered. The Riesz basis property and the spectrum determined growth condition of this network are discussed. Based on these properties, the exponential stability is gotten under several conditions. In past decades, there have been many nice results on the spectrum of flexible networks. For instance, von Below in [21], [22] and Nicaise et al. in [35] studied the characteristic equation for networks of strings. Mercier et al. in [11] and Dekoninck et al. in [2] discussed the eigenvalue problem for networks of beams. Wang et al. in 2000 Mathematics Subject Classification. Primary: 35B40, 47A75; Secondary: 93D20, 93D15. Key words and phrases. network, non-uniform string, spectrum, non-collocated feedback, Riesz basis, exponential stability. This research is supported by the Natural Science Foundation of China grant NSFC-60874034.

315

316

ZHONG-JIE HAN AND GEN-QI XU

[23] and Han et al. in [37] and [39] studied the distribution of the spectrum by the method developed from [30]. Moreover, they discussed the dynamical behavior of the network when a control is applied. For example, [29], [12] and [3] studied the control of networks of Euler-Bernoulli beams and strings respectively; [25], [26] and [27] considered tree-shaped and star-shaped configurations of strings and got the asymptotical stability under certain conditions. Similar method was used to consider the energy decay of Euler-Bernoulli beams with star-shaped and tree-shaped network configurations (see [28]). Moreover, Xu et al. in [16] considered the Riesz basis property and stability of abstract second order hyperbolic systems and applied these results to controlled networks of strings. Wang et al. [23] obtained the Riesz basis property and exponential stability of a flexible structure of a symmetric treeshaped beam network, based on the asymptotic distribution of the spectrum of the system. [14], [37] and [38] studied the Riesz basis property of multi-connected networks of Timoshenko beams, and obtained that these systems satisfy the spectrum determined growth condition. Note that the papers mentioned above always consider the system with collocated feedback controls. Although there are some results studying the single string and beam with non-collocated feedbacks (see: [36] and [7], [8], [18], [19]), there are still few results on the dynamical behavior of networks of distributed parameter system with non-collocated feedback controls at present. In fact, it is a tough problem to consider the control problem with non-collocated feedback controls even for a single string or beam. Under collocated feedback controls, the well-posedness of systems can be obtained easily due to the dissipativity of such kinds of systems. However, the closed loop of non-collocated control system is always non-dissipative. It is a challenge to discuss the well-posedness of non-dissipative systems. Furthermore, the non-collocated control systems are always non-minimum-phase, which implies that their control performances are usually severely limited by the unstable zeros inherent in non-collocated sensors and actuators (see [31]). As a result, a small increment of feedback controller gains can easily make the closed-loop systems unstable (see [24], [7]). When the flexible networks systems are stabilized by collocated feedback controllers (see e.g. [23],[37],[38] and [15] for beams and [39] for strings), the corresponding closed-loop systems are always dissipative, whose spectrum are all located in the left hand side of the complex plane. Thus, the stability can be gotten as long as there is no spectrum of systems on the imaginary axis. However, when the noncollocated terms appear in the feedback controls, the controlled network systems are no longer dissipative. Therefore, in order to show the stability of systems, it needs to check that there is no spectrum not only on the imaginary axis but also in the right open complex plane, which is always difficult especially for network systems since the spectral property of flexible networks is very complicated. What we shall do in this paper is to consider a kind of network of non-uniform strings with non-collocated terms in the boundary feedback controls. The effect of these non-collocated terms on the distribution of the spectrum and the stability of this flexible network system are discussed. Let us describe this kind of network of non-uniform strings system on a planar graph (see Fig. 1). Let G = (V, E) be a tree-shaped planar graph with vertices V = {a0 , a1 , a2 , a3 , · · · , an } and edges E = {e1 , e2 , · · · , en }, where edge ej joins the vertices a0 and aj , and a0 is a common node. Suppose that the length of each edge

SPECTRUM AND DYNAMICAL BEHAVIOR OF A NETWORK

a4

317

a3

a2 A e A 3 e 2 .. e4A . A .. a0A• e1 a1 . @ en−2  @ en e n−1 @ @ a an−2  n an−1 Figure 1. Planar network ej is 1. The graph G describes an elastic structure. The function w(s, t) denotes the displacement from its equilibrium at position s ∈ G and time t. Then let us parameterize the edges of G according to [29]. We choose the common node a0 as initial point of each edge ej and node aj as end point. Then we parameterize ej by means of the bijective functions πj : [0, 1] → ej (that is πj (0) = a0 , πj (1) = aj ). Then set wj (x, t) := w(πj (x), t), x ∈ [0, 1], πj (x) ∈ ej . The motion of the elastic structure on each ej is governed as follows ρj (x) ∂

2

w j (x,t) ∂t2

−

∂w j (x,t) ∂ ] ∂x [kj (x) ∂x

= 0,

t > 0, x ∈ (0, 1),

(1)

where kj is the tension constant on the jth string and ρj is the mass density per unit length. In this paper, we always assume ρj (x), kj (x) ∈ C 2 [0, 1] and ρj (x), kj (x) > 0. If the length of ej is ℓj not 1, we always can use the transformation x = x eℓj , x e∈ (0, 1) to make the length of the string ℓj appear in system parameters, that is ρj (e x ℓj )

∂ 1 ∂wj (e xℓj , t) ∂ 2 wj (e xℓj , t) − [ k (e x ℓ ) ] = 0, j j 2 ∂t2 ∂e x ℓj ∂e x

t > 0, x e ∈ (0, 1).

For convenience, we always assume that the length of each string is 1. At the common vertex a0 , the strings satisfy the continuity conditions of displacements, i.e., wi (0, t) = wj (0, t), i, j ∈ {1, 2, · · · , n},

t > 0.

Besides this, there is a vibrating point mass at the common vertex, i.e., n X 1 kj (0)wxj (0, t) = M wtt (0, t).

(2)

(3)

j=1

The boundary controllers are designed at the exterior vertices aj , j = 1, 2, · · · , n: kj (1)wxj (1, t) = uj (t), j = 1, 2, · · · , n,

(4)

where uj (t) are external exciting forces. Observing w(aj , t), wt (aj , t), j = 1, 2, · · · , n and wt (a0 , t), we adopt the following feedback control laws, which contain the noncollocated and collocated terms simultaneously:  uj (t) = −wj (1, t) − αj wtj (1, t) − βj wtj (0, t), (5) αj ≥ 0, βj ≥ 0, j = 1, 2, · · · , n,

where αj , βj , j = 1, 2, · · · , n are the collocated and non-collocated feedback gain coefficients, respectively. We shall discuss how these two kinds of terms in the

318

ZHONG-JIE HAN AND GEN-QI XU

boundary feedbacks affect the spectral distribution and dynamical behavior of this flexible network. In addition, we assume that the initial condition of the system is given by  j w (x, 0) = w(πj (x), 0) := w ¯0 (πj (x)) = w ¯0j (x), x ∈ [0, 1], j = 1, 2, · · · , n, wtj (x, 0) = wt (πj (x), 0) := w ¯1 (πj (x)) = w ¯1j (x), x ∈ [0, 1], j = 1, 2, · · · , n. (6) Thus, the network system (1)–(4) together with (5) and (6) becomes a closed-loop system:  j ∂ 2 w j (x,t) (x,t) ∂  − ∂x [kj (x) ∂w∂x ] = 0, t > 0, x ∈ (0, 1),  ρj (x) ∂t2   i j  w (0, t) = w (0, t), i, j ∈ {1, 2, · · · , n}, t > 0,    j j j   kj (1)wx (1, t) = −w (1, t) − αj wt (1, t) − βj wtj (0, t), t > 0, n P (7) 1 kj (0)wxj (0, t) = M wtt (0, t), t > 0,    j=1     wj (x, 0) = w ¯0j (x), wtj (x, 0) = w ¯1j (x), x ∈ [0, 1],    j = 1, 2, · · · , n.

We shall analyze the dynamical behavior of this hybrid system by the Riesz basis generation approach developed in non-harmonic Fourier analysis, which is a useful tool to discuss the dynamical behavior and control of flexible systems (see [20], [23], [15] and the referees therein). For distributed parameter systems, the Riesz basis generation of the (generalized) eigenvectors of the system operator is a very profound result. This yields not only the expansion of the solution in terms of the (generalized) eigenvectors of the system but also the spectrum determined growth condition. By a complete spectral analysis, we get the distribution of the spectrum and find that the asymptotic behavior of the spectrum of the network is independent of these non-collocated terms. What these non-collocated terms affect are only the low spectrum (at most finite number). Then by this spectral property of the system, we get the Riesz basis property of the (generalized) eigenvectors of the system operator. Thus, the spectrum determined growth condition holds (see [4]). Based on this property, we discuss the stability of this system and get the exponential stability under certain conditions. The content of this paper is organized as follows. In Section 2, we formulate our problem (7) in a Hilbert state space setting and then obtain the well-posedness of this system. In Section 3, by a complete asymptotic analysis of the spectrum of the system operator, we get the formulation of the asymptotic expression of the spectrum, which implies that the asymptotic spectrum is independent of the non-collocated feedback gains and in the left hand side of the complex plane. Furthermore, the spectrum is located in a strip parallel to the imaginary axis under a certain condition. In Section 4, we prove the completeness and Riesz basis property of the (generalized) eigenvectors of the system operator. Hence the spectrum determined growth condition holds. In Section 5, for a special case, we get the exponential stability for this kind of planar networks. Finally, a numerical simulation is also given to support these results. 2. Well-posedness of the system. In this section, we shall study the wellposedness of the closed-loop system (7). To that purpose, we reformulate this system in an appropriate Hilbert state space setting.

SPECTRUM AND DYNAMICAL BEHAVIOR OF A NETWORK

Set

319

  W (x, t) := (w1 (x, t), w2 (x, t), · · · , wn (x, t))τ , x ∈ [0, 1], t > 0, ¯ 0 (x) := (w¯1 (x), w¯2 (x), · · · , w W ¯0n (x))τ , x ∈ [0, 1], 0 0  ¯ 1 2 W1 (x) := (w¯1 (x), w¯1 (x), · · · , w ¯1n (x))τ , x ∈ [0, 1].

Define n × n matrices:

ρ(x) := diag(ρ1 (x), ρ2 (x), · · · , ρn (x)), β := diag(β1 , β2 , · · · , βn ),

k(x) := diag(k1 (x), k2 (x), · · · , kn (x)), α := diag(α1 , α2 , · · · , αn ).

Then equation (7) can be rewritten in matrix form:  2 (x,t) (x,t) ∂  ρ(x) ∂ W − ∂x [k(x) ∂W∂x ] = 0, t > 0, x ∈ (0, 1),  ∂t2    C(n−1)×nW (0, t) = 0, t > 0, k(1)Wx (1, t) = −W (1, t) − αWt (1, t) − βWt (0, t), t > 0,    I1×n k(0)Wx (0, t) = M δ1×n Wtt (0, t), t > 0,   ¯ 0 (x), Wt (x, 0) = W ¯ 1 (x), x ∈ [0, 1], W (x, 0) = W

where



   C(n−1)×n :=   

−1 1 0 −1 .. . ··· 0 ··· 0 ···

0 1 .. .

··· 0 .. .

··· ···

−1 0

I1×n := (1, 1, · · · , 1)1×n ,

0 ···

··· 1 −1

 0 0    ···   0  1 (n−1)×n

δ1×n := (1, 0, · · · , 0)1×n .

(8)

(9)

(10)

Set

VEk := {u = (uj )nj=1 ∈ Πnj=1 H k (0, 1) ui (0) = uj (0), ∀i, j = 1, 2, 3, · · · , n}.

Let H be the following space

H = VE1 × Πni=1 L2 (0, 1) × C

equipped with an inner product, for  τ (Wi , Zi , pi )τ := (wij )nj=1 , (zij )nj=1 , pi ∈ H, i = 1, 2, via

=

((W1 , Z1 , p1 )τ , (W2 , Z2 , p2 )τ )H n Z 1 n Z X X j j kj (x)w1,x (x)w2,x (x)dx + j=1

+

0

n X

j=1

0

1

ρj (x)z1j (x)z2j (x)dx + M p1 p2

w1j (1)w2j (1).

j=1

Here the third components pi are introduced for describing the state of the point mass at the common vertex of the network.
τ We define the operator A in H: for (W, Z, p)τ := (wj )nj=1 , (z j )nj=1 , p ∈ D(A),     W Z A  Z  =  ρ−1 (x)[k(x)Wx (x)]x  , (11) p M −1 I1×n k(0)Wx (0)

320

ZHONG-JIE HAN AND GEN-QI XU

with domain

 

 W ∈ VE2 , Z ∈ VE1  j D(A) = (W, Z, p) ∈ H p = z (0) ,  k(1)Wx (1) = −W (1) − αZ(1) − βZ(0) 

(12)

where the notation p denotes the velocity of the point mass in the network system. Then, (7) can be rewritten as an evolutionary equation in H  dU(t) t > 0, dt = AU (t), (13) U (0) = U0 ,

¯ 0, W ¯ 1, w where U (t) = (W (·, t), Wt (·, t), wt1 (0, t))τ and U (0) = (W ¯11 (0))τ ∈ H is given. We have the following result. Lemma 2.1. Let H and A be defined as before, γj (j = 1, 2, · · · , n) be positive con2α β n max{ γjj } is dissipative stants which satisfy γj ≤ βjj , j = 1, 2, · · · , n. Then A − 2M j

in H. Proof. For any Y = (wj (x))nj=1 , (z j (x))nj=1 , p 2ℜ(AY, Y )H

= =


τ

∈ D(A),

(AY, Y )H + (Y, AY )H n n X X [−wj (1) − αj z j (1) − βj z j (0)]z j (1) + z j (1)wj (1) j=1

+ =

i=1

n X

z j (1)[−wj (1) − αj z j (1) − βj z j (0)] +

j=1 n X

−2

=

−2

αj |z j (1)|2 −

n X

βj (z j (0)z j (1) + z j (1)z j (0))

j=1 n X

αj |z j (1)|2 − 2

j=1

βj ℜ(z j (0)z j (1)).

j=1

Using Cauchy-Schwarz inequality, we have 1 1 ℜ(z j (0)z j (1)) ≤ |z j (0)||z j (1)| ≤ (γj |z j (1)|2 + |z j (0)|2 ), 2 γj and hence ℜ(AY, Y )H

Since γj ≤

2αj βj ,

≤

wj (1)z j (1)

i=1

j=1

n X

n X

j = 1, 2, · · · , n,

  n 1X 1 j 2 2 − αj |z (1)| + βj γj |zj (1)| + |z (0)| 2 j=1 γj j=1 n X

≤

−

≤

−

j

2

n X

1 n βj [αj − βj γj ]|z j (1)|2 + max{ }|p|2 j 2 2 γj j=1

n X

1 n βj [αj − βj γj ]|z j (1)|2 + max{ }kY k2H . j 2 2M γj j=1

j = 1, 2, · · · , n, we have

ℜ((A −

n X n βj 1 max{ })Y, Y )H ≤ − [αj − βj γj ]|z j (1)|2 ≤ 0, 2M j γj 2 j=1

SPECTRUM AND DYNAMICAL BEHAVIOR OF A NETWORK

which yields that A −

n 2M

321

β

max{ γjj } is dissipative in H. j

Lemma 2.2. Let A and H be defined as before. Then 0 ∈ ρ(A). Furthermore, A−1 is compact on H. Therefore, the spectrum of A consists only of the isolated eigenvalues with finite multiplicity, i.e., σ(A) = σp (A). Proof. Clearly, D(A) is dense in H. For any fixed F = (F1 , F2 , F3 )τ ∈ H where F1 = (f11 , f12 , · · · , f1n )τ , F2 = (f21 , f22 , · · · , f2n )τ , we consider the solvability of AY = F, Y = (W, Z, p)τ ∈ D(A), where W = (w1 , w2 , · · · , wn )τ , Z = (z 1 , z 2 , · · · , z n )τ , that is   Z = F1 , ρ−1 (x)[k(x)Wx (x)]x = F2 ,  M −1 I1×n k(0)Wx (0) = F3

with boundary conditions   p = z j (0), j = 1, 2 · · · , n, k(1)Wx (1) = −W (1) − αZ(1) − βZ(0),  j w (0) = wi (0) = w(a0 ), f1j (0) = f1i (0), i, j = 1, 2, · · · , n.

Integrating the second equation in (14) from 0 to x yields Z x k(x)Wx (x) = k(0)Wx (0) + ρ(s)F2 ds.

(14)

(15)

(16)

0

Then integrating (16) from 0 to x, we get

W (x) = k(x)−1 [k(0)W (0) + k(0)Wx (0)x +

Z

0

x

Z

s

ρ(r)F2 drds].

(17)

0

According to the continuity condition of displacements: wj (0) = w(a0 ), we have τ τ W (0) = I1×n w(a0 ), where I1×n is the transpose of I1×n , and thus Z 1Z s τ ρ(r)F2 drds]. W (1) = k(1)−1 [k(0)I1×n w(a0 ) + k(0)Wx (0) + 0

0

Then by substituting the expression of W (1) into the boundary condition: k(1)Wx (1) = −W (1) − αZ(1) − βZ(0), together with (16), we get τ k(0)Wx (0) = (1 + k −1 (1))−1 [−k −1 (1)k(0)I1×n w(a0 ) + G1 ],

where G1 := −

Z

0

1

ρ(s)F2 (s)ds − k −1 (1)

Z

0

1

Z

(18)

s

ρ(r)F2 (r)drds − αF1 (1) − βF1 (0).

0

According to the boundary condition: M −1 I1×n k(0)Wx (0) = F3 , we have τ M −1 I1×n (1 + k −1 (1))−1 [−k −1 (1)k(0)I1×n w(a0 ) + G1 ] = F3 .

Hence, we derive w(a0 ) from the above equation as follows  −1 −1 −1 −1 −1 τ w(a0 ) = − M I1×n (1 + k (1)) k (1)k(0)I1×n

322

ZHONG-JIE HAN AND GEN-QI XU

  · F3 − M −1 I1×n (1 + k −1 (1))−1 G1 .

(19)

Then substituting w(a0 ) into (18) yields k(0)Wx (0)

 τ τ = (1 + k −1 (1))−1 k −1 (1)k(0)I1×n [M −1 I1×n (1 + k −1 (1))−1 k −1 (1)k(0)I1×n ]−1    · F3 − M −1 I1×n (1 + k −1 (1))−1 G1 + G1 .

(20) Summarizing the calculations above, we get the expression of W (x), Z(x), p as follows  RxRs  W (x) = k(x)−1 [k(0)w(a0 ) + k(0)Wx (0)x + 0 0 ρ(r)F2 drds], Z(x) = F1 (x), (21)  p = f11 (0),

where w(a0 ) and k(0)Wx (0) are given by (19) and (20). Hence, (W, Z, p)τ ⊂ D(A) ⊂ H. Therefore, there exists a nontrivial element Y ∈ D(A) such that AY = F . The Inverse Operator Theorem implies that 0 ∈ ρ(A). Since D(A) is a compact subspace of H due to the Sobolev Embedding Theorem, we have that A−1 is compact on H. Thus, A is a discrete operator in H. So the spectrum of A consists only of isolated eigenvalues with finite multiplicity(see [5]). The proof is complete.

Lemma 2.1, 2.2 together with the Lumer-Phillips Theorem (see Pazy [1]) assert the following result. β

n max{ γjj } generates Theorem 2.3. Let A and H be defined as before. Then A − 2M j

2α

a C0 semigroup of contractions on H for γj ≤ βjj , j = 1, 2, · · · , n. Hence A generates a C0 semigroup on H. Therefore, system (13) is well-posed. 3. Asymptotic behavior of eigenvalues. In this section, we shall discuss the asymptotic distribution of the spectrum of A. From Lemma 2.2, it suffices to discuss the distribution of the eigenvalues of the system (13). τ For any λ ∈ σ(A), let U = (W, Z, p) be a corresponding eigenvector. Then (λI − A)U = 0,

(22)

which yields Z(x) = λW (x). Let us consider the boundary eigenvalue problem:  ρ(x)λ2 W (x) − [k(x)Wx (x)]x = 0, x ∈ (0, 1),    C(n−1)×nW (0) = 0, (23) k(1)W  x (1) = −W (1) − αλW (1) − βλW (0),   2 I1×n k(0)Wx (0) = M λ δ1×n W (0).

We shall discuss equation (23) to get the distribution of σ(A). Firstly, let us consider the differential equation ρ(x)λ2 W (x) − [k(x)Wx (x)]x = 0. A direct calculation yields ρ(x)λ2 W (x) − kx (x)Wx (x) − k(x)Wxx (x) = 0.

SPECTRUM AND DYNAMICAL BEHAVIOR OF A NETWORK

323


n 1 c (x) = w Set W bj (x) j=1 := [k(0)−1 k(x)] 2 W (x). Then (23) is equivalent to   1 1 −1 2 −1 −1 2 c c Wxx (x) + [kx (x)k(x) ] − kxx (x)k(x) − ρ(x)k(x) λ W (x) = 0, (24) 4 2

with the boundary conditions:

 c (0) = 0, C(n−1)×n W  
  − 1 I1×n kx (0) − M λ2 δ1×n W c (0) + I1×n k(0)W cx (0) = 0, 2   c (1) c (0) + − 1 (k(0)k(1)−1 ) 21 kx (1) + (k(0)k(1)−1 ) 21 + αλ(k(0)k(1)−1 ) 12 W  βλ W  2   1 cx (1) = 0. +(k(1)k(0)) 2 W

Set

c , Z2 := Z1 := W

Then (24) is equivalent to

1c Wx , Z := λ

dZ 1 = λ(A1 (x) + 2 A−1 (x))Z, dx λ where A1 (x) :=



0 ρk −1

1 0



, A−1 (x) :=

2n×2n





Z1 Z2



(25)

(26)

.

x ∈ (0, 1).

(27)

0 1 −1 − 14 (kx k −1 )2 2 kxx k

0 0



. 2n×2n

The boundary conditions (25) can be reformulated as B0 (λ)Z(0) + B1 (λ)Z(1) = 0, where

B1 (λ) :=

"

 C(n−1)×n B0 (λ) :=  − 12 I1×n kx (0) − Mλ2 δ1×n λβ

(28)

 0 , I1×n k(0)λ  0 2n×2n

0 1 − 12 (k(0)k(1)−1 ) 2

0

1 kx (1)+(k(0)k(1)−1 ) 2

+αλ(k(0)k(1)

−1

1 )2

1 [k(0)k(1)] 2

λ

#

Therefore, the system (24)–(25) can be rewritten as the following form:  dZ 1 x ∈ (0, 1), dx = λ(A1 (x) + λ2 A−1 (x))Z, B0 (λ)Z(0) + B1 (λ)Z(1) = 0. Set the transformation Z(x) := T0 (x)U (x), where   1 1 T0 (x) := . 1 1 (ρ(x)k(x)−1 ) 2 −(ρ(x)k(x)−1 ) 2 2n×2n

. 2n×2n

(29)

(30)

Then under this transformation, (29) can be transformed into 

dU(x) −1 −1 −1 T0 (x)A−1 (x)T0 (x)]U (x), dx = [λΛ(x) − T0 (x)T0,x (x) + λ B0 (λ)T0 (0)U (0) + B1 (λ)T0 (1)U (1) = 0,

x ∈ (0, 1), (31)

where Λ(x) :=



1

(ρ(x)k(x)−1 ) 2 0

0 1 −(ρ(x)k(x)−1 ) 2



. 2n×2n

b λ) be the fundamental solution matrix of the equation (31). We have Let U(x, the following result which can be gotten directly from [36].

324

ZHONG-JIE HAN AND GEN-QI XU

Lemma 3.1. The fundamental solution matrix of the differential equation (31) has the following expression: ∞ X P−k (x) b (x, λ) = U E(x, λ), (32) λk k=0

where

E(x, λ) :=

P0 :=

"

"

eλ

R

1 x (ρ(s)k(s)−1 ) 2 0

ds

e−λ

0

ρ(0)k(x)(k(0)ρ(x))−1


41

Rx 0

0 1

(ρ(s)k(s)−1 ) 2 ds

0

#

,


1 0 ρ(0)k(x)(k(0)ρ(x))−1 4 and all entries of P−k , k = 1, 2 · · · , are uniformly bounded in [0, 1].

#

,

This lemma gives us a formula for the fundamental solution matrix of the differential equation in (31). Based on the formula (32), together with the boundary conditions in (31), we can further determine the asymptotic spectral distribution of the system (13). Set b (1, λ), H := B0 (λ)T0 (0) + B1 (λ)T0 (1)U b λ) is given by (30) and (32), respectively. where T0 (x) and U(x, We have the following result on the spectral property of system (13). Lemma 3.2. Let A and H be defined as before. Then λ ∈ σ(A) if and only if λ satisfies ∆(λ) := det H(λ) = 0. (33) Proof. Since the necessary and sufficient condition for λ ∈ σ(A) is that the equation (23) has a nonzero solution, so does the equation (29) or equation (31). From b (x, λ) is the fundamental solution matrix of (31). Then Y (x) = Lemma 3.1, we get U b U(x, λ)η is a solution to (31) and hence when the algebraic equation b λ)]η = 0 [B0 (λ)T0 (0) + B1 (λ)T0 (1)U(1,

b (1, λ)) = 0. has a nonzero solution η, it must have det(B0 (λ)T0 (0) + B1 (λ)T0 (1)U The desired result follows. Obviously, we have the following result.

Corollary 1. Let A and H be defined as before. Then the spectrum of A distributes in conjugate pairs on the complex plane, i.e., σ(A) = σ(A). From Lemma 3.2, in order to get the spectrum of A, we only need to identify the zeros of ∆(λ). Firstly, let us get the asymptotic expression of ∆(λ). Set [B]1 := B + O(λ−1 ). b (1, λ) can be rewritten as follows Then U b (1, λ) = P0 (1) U

"

Rx 1 [1]1 exp(λ 0 (ρ(s)k(s)−1 ) 2 ds) Rx 1 [0]1 exp(λ 0 (ρ(s)k(s)−1 ) 2 ds)

A direct calculation yields

B0 (λ)T0 (0) =



# Rx 1 [0]1 exp(−λ 0 (ρ(s)k(s)−1 ) 2 ds) Rx 1 [1]1 exp(−λ 0 (ρ(s)k(s)−1 ) 2 ds)

c1 n×n G λβ

c2 n×n G λβ



,

. 2n×2n

SPECTRUM AND DYNAMICAL BEHAVIOR OF A NETWORK

B1 (λ)T0 (1) = where

0 c3 n×n + λ(ρ(1)k(0)) 12 G



0 c3 n×n − λ(ρ(1)k(0)) 12 G

,

 C(n−1)×n , 1 k (0) − M λ2 δ1×n + I1×n k(0)λ(ρ(0)k(0)−1 ) 2 n×n −1I   2 1×n x C(n−1)×n := , 1 − 12 I1×n kx (0) − M λ2 δ1×n − I1×n k(0)λ(ρ(0)k(0)−1 ) 2 n×n 

c1 n×n := G

c2 n×n G



325

c3 n×n := − 1 (k(0)k(1)−1 ) 12 kx (1) + (αλ + 1)(k(0)k(1)−1 ) 12 . G 2

Thus,

=

b (1, λ) H = B0 (λ)T0 (0) + B1 (λ)T0 (1)U  1 R1 −1 2 c 1 ) ds)  G n×n + [0]1 exp(λ 0 (ρ(s)k(s) 1  1  1 R c [λβ +(G3 n×n +λ(ρ(1)k(0)) 2 ) ρ(0)k(1)(k(0)ρ(1))−1 4 ]1 exp(λ 01 (ρ(s)k(s)−1 ) 2 ds)

 1 R1 −1 2 c2 G ) ds) n×n + [0]1 exp(−λ 0 (ρ(s)k(s)  1  1  1 R −1 4 c3 [λβ +(G ]1 exp(−λ 01 (ρ(s)k(s)−1 ) 2 ds) n×n −λ(ρ(1)k(0)) 2 ) ρ(0)k(1)(k(0)ρ(1))

. 2n×2n

Since 0 ∈ ρ(A) by Lemma 2.2, we have the following approximation: ∆(λ) det H(λ) = λ2+n  λ2+n  " C (n−1)×n

=

det

−M δ1×n

+O(λ

).

n×n

1 1 1  1 R β +[α(k(0)k(1)−1 ) 2 +(ρ(1)k(0)) 2 ] ρ(0)k(1)(k(0)ρ(1))−1 4 exp(λ 01 (ρ(s)k(s)−1 ) 2 ds)   # C(n−1)×n −M δ1×n n×n 1 1  1 1 R β +[α(k(0)k(1)−1 ) 2 −(ρ(1)k(0)) 2 ] ρ(0)k(1)(k(0)ρ(1))−1 4 exp(−λ 01 (ρ(s)k(s)−1 ) 2 ds) 2n×2n −1

Furthermore, since



C(n−1)×n −M δ1×n



is inverted, we can eliminate β by left-handed

n×n



b :=  multiplying the above matrix by B

−β ·



0

In×n −1 C(n−1)×n −M δ1×n n×n

In×n

In×n is the identity matrix with order n. b = 1, a direction calculation yields Since det B ∆(λ)

λ2+n

=

 

2n×2n

  C(n−1)×n −M δ 1×n n×n 1 1 1  1 R [α(k(0)k(1)−1 ) 2 + (ρ(1)k(0)) 2 ] ρ(0)k(1)(k(0)ρ(1))−1 4 exp(λ 01 (ρ(s)k(s)−1 ) 2 ds)   C(n−1)×n −M δ1×n n×n 1 1 1  1 R [α(k(0)k(1)−1 ) 2 −(ρ(1)k(0)) 2 ] ρ(0)k(1)(k(0)ρ(1))−1 4 exp(−λ 01 (ρ(s)k(s)−1 ) 2 ds) −1

=

, where

+O(λ )  1 −1 4 M k(0)ρ(0)(k(1)ρ(1)) · det



det n×n

1 − (α + (ρ(1)k(1)) 2 ) exp(λ

Z 1





+(α − (ρ(1)k(1)) 2 ) exp(−λ

n×n

Z 1

(ρ(s)k(s)

0


1 Since k(0)ρ(0)(k(1)ρ(1))−1 4

2n×2n

1 −1 2 (ρ(s)k(s) ) ds)

0

1

C(n−1)×n −δ1×n



−1 1 ) 2 ds)





+ O(λ

−1

).

n×n



C(n−1)×n 6= 0, we deduce −δ1×n n×n that the asymptotic spectrum of A can be determined by the zeros of the following n×n

6= 0 and det

326

ZHONG-JIE HAN AND GEN-QI XU

equation: det



1

− (α + (ρ(1)k(1)) 2 ) exp(λ 1

R1 0

1

(ρ(s)k(s)−1 ) 2 ds)

+(α − (ρ(1)k(1)) 2 ) exp(−λ

A direct calculation yields

λjm =

          

1

1 R 2 01 (ρj (s)kj (s)−1 ) 2 ds

1 1 R 2 01 (ρj (s)kj (s)−1 ) 2 ds





R1

α −(ρ (1)kj (1)) 21 ln j j 1 α +(ρ (1)k (1)) 2 j

j

j

α −(ρ (1)kj (1)) 12 ln j j 1 α +(ρ (1)k (1)) 2 j

j

j

1

(ρ(s)k(s)−1 ) 2 ds) 0  + 2miπ ,



= 0. n×n

1

αj − (ρj (1)kj (1)) 2 > 0,

 +(2m + 1)iπ , αj − (ρj (1)kj (1)) 12 < 0,

where j = 1, 2, · · · , n, m = 1, 2 · · · . Applying the Rouch´e Theorem, we have the following result: 1

Theorem 3.3. Let A and H be defined as before. Then if αj − (ρj (1)kj (1)) 2 6= 0, j = 1, 2, · · · n, the asymptotic expression of the spectrum of A is given as follows

bj = λ m

              

1

1 R 2 01 (ρj (s)kj (s)−1 ) 2 ds 1

1 R 2 1 (ρj (s)kj (s)−1 ) 2 ds 0

"

"

α −(ρ (1)k (1)) 12 j j ln j α +(ρ (1)k (1)) 12 j

j

j

α −(ρ (1)k (1)) 21 j j ln j α +(ρ (1)k (1)) 12 j

j

j

# 1 1 ), αj − (ρj (1)kj (1)) 2 > 0, + 2miπ + O( m

# 1 1 ), αj − (ρj (1)kj (1)) 2 < 0, + (2m + 1)iπ + O( m (34)

where j = 1, 2, · · · , n, m = 1, 2 · · · . Furthermore, the spectrum is simple and separated when |λ|(λ ∈ σ(A)) is sufficiently large.

Remark 1. The asymptotic expression of the spectrum of A in (34) shows that the asymptotic spectrum of A is located in the left hand side of the complex plane. Moreover, the asymptotic spectrum only depends on the choice of collocated feedback gain coefficients αj and is independent of non-collocated feedback gain coefficients βj . Note that when βj = 0, j = 1, 2, · · · , n, the system (7) becomes a dissipative collocated feedback controlled system. Therefore, we always can find ˜ > 0 such that when |λ| > M ˜ , λ ∈ σ(A), the distribution of the a constant M spectrum of A only depends on αj , is in the left hand side of the complex plane and away from the imaginary axis. What βj affects are only the low spectrum of ˜ . The finite low spectrum may be A (at most finite number) which satisfy |λ| < M located in the right hand side of the complex plane by the choices of βj and αj . 4. Riesz basis property. In this section, we shall discuss the Riesz basis property of the (generalized) eigenvectors of A by the distribution of the spectrum of A. Similarly to [14], define the auxiliary operator A0 in H as follows     W h Z i   ∂ A0  Z  =  ρ−1 (x) ∂x [k(x) ∂W∂x(x) ]  , (35) −1 p M I1×n k(0)Wx (0) with domain

 

 W ∈ VE2 , Z ∈ VE1  1 D(A0 ) = (W, Z, p) ∈ H p = z (0) .   k(1)Wx (1) = 0

(36)

In fact, A0 is the corresponding system operator without feedback controls. We can easily check that

SPECTRUM AND DYNAMICAL BEHAVIOR OF A NETWORK

327

Lemma 4.1. A0 is a skew-adjoint operator in H. In order to obtain the Riesz basis property of (generalized) eigenvectors of A in H, we shall first show the completeness and Riesz basis property of (generalized) eigenvectors of A for the special case: β = 0, and then further deduce the same property of the (generalized) eigenvectors of A for any β by the asymptotic behavior of the spectrum of A. We have the following result: 1

Theorem 4.2. Let A and H be defined as before. If |α − (ρ(1)k(1)) 2 |n×n 6= 0 and β = 0, the system of the (generalized) eigenvectors of A is complete in H. Proof. Let A|β=0 be the operator A satisfying β = 0. In what following, we shall prove the completeness of the (generalized) eigenvectors of A|β=0 in H so as to yield the Riesz basis property of the (generalized) eigenvectors of A|β=0 . Define X Span(A|β=0 ) := { yk , yk ∈ E(λk , A|β=0 )H, λk ∈ σ(A|β=0 )}, k

where E(λk , A|β=0 ) is the Riesz projector corresponding to λk . Then the completeness of the (generalized) eigenvectors of A|β=0 is just Span(A|β=0 ) = H.

Assume U = (e u, ve, pe) ∈ H and U ⊥Span(A|β=0 ). Denote by R∗ (λ, A|β=0 ) the conjugate operator of the resolvent operator of A|β=0 . Since U ⊥Span(A|β=0 ), then R∗ (λ, A|β=0 )U is an entire function on C valued in H. Thus for any F = (F1 , F2 , F3 ) ∈ H, G(λ) := (F, R∗ (λ, A|β=0 )U )H , ∀λ ∈ C (37) is an entire function which satisfies

lim

ℜλ→+∞

G(λ) = 0, since A|β=0 generates a C0

semigroup. In particular, for λ ∈ ρ(A|β=0 ), it holds that G(λ) = (R(λ, A|β=0 )F, U )H . We consider the resolvent problem (λI − A|β=0 )Y1 = F,

(λI − A0 )Y2 = F,

λ ∈ ρ(A|β=0 ) ∩ ρ(A0 ) ∩ R− ,

(38)

where Y1 := (W1 , Z1 , p1 ) ∈ D(A|β=0 ), Y2 = (W2 , Z2 , p2 ) ∈ D(A0 ). Set Y3 (λ) := Y1 − Y2 = (W3 , Z3 , p3 ). Then W3 satisfies the following equations:  ∂ 3 (x)  ρ(x)λ2 W3 (x) − ∂x [k(x) ∂W∂x ] = 0, x ∈ (0, 1),   C(n−1)×nW3 (0) = 0, (39)  k(1)W3,x (1) + W3 (1) + αZ3 (1) = −W2 (1) − αZ2 (1),   I1×n k(0)W3,x (0) = M λ2 δ1×n W3 (0).

Similarly to [39], we shall prove that U = 0 via three steps below, and this implies Span(A|β=0 ) = H. f1 > 0, such Step 1). For λ ∈ ρ(A|β=0 ) ∩ ρ(A0 ) ∩ R− , there exists a constant M that f1 kZ2 (1)k. kZ3 (1)k ≤ M Step 2). For λ ∈ ρ(A|β=0 ) ∩ ρ(A0 ) ∩ R− , kY3 (λ)k satisfies f2 (|λ|kY2 k + kF k)2 , |λ|kY3 (λ)k2 ≤ M

f2 is a positive constant. where M

(40)

328

ZHONG-JIE HAN AND GEN-QI XU

Step 3). For λ ∈ ρ(A|β=0 ) ∩ ρ(A0 ) ∩ R− , q f2 |λ|− 21 kF k + |λ|−1 kF k. kR(λ, A|β=0 )F k = kY1 k ≤ 2 M

In fact, since A0 is a skew-adjoint operator in H, we have 1 , ∀λ ∈ ρ(A0 ), kR(λ, A0 )k ≤ |ℜλ|

1 which implies kY2 k = k(λ−A0 )−1 F k ≤ k(λ−A0 )−1 kkF k ≤ |ℜλ| kF k, ∀λ ∈ ρ(A0 ). Then for λ ∈ ρ(A|β=0 ) ∩ ρ(A0 ) ∩ R− ,   q 1 f2 |λ|− 12 . +2 M kR(λ, A|β=0 )F k = kY1 k ≤ kY2 k + kY3 k ≤ kF k |λ|

Therefore, when λ ∈ ρ(A|β=0 ) ∩ ρ(A0 ) ∩ R− for |λ| sufficiently large, we have lim kR(λ, A|β=0 )F k = kY2 + Y3 k = 0,

λ→−∞

which implies that lim |G(λ)| = 0.

λ→−∞

(41)

It is easy to show that A|β=0 is dissipative in H and 0 ∈ ρ(A|β=0 ). So |G(λ)| is bounded on the domain ℜλ ≥ cˆ, cˆ > 0. Since G(λ) is an entire function of finite exponential type, together with (41), the Phragm´en-Lindel¨of Theorem (see [32]) shows that |G(λ)| is bounded on the sector with boundary rays ℜλ = cˆ, ℑλ ≥ 0 and ℜλ ≤ cˆ, ℑλ = 0. Similarly, we can get |G(λ)| is bounded on the sector with boundary rays ℜλ = cˆ, ℑλ ≤ 0 and ℜλ ≤ cˆ, ℑλ = 0. Therefore, |G(λ)| is uniformly bounded on C, that is, |G(λ)| ≤ M, ∀λ ∈ C. Furthermore, by Liouville’s Theorem, we deduce that G(λ) is constant, since G(λ) is an entire function. Then lim G(λ) = 0 yields G(λ) ≡ 0. Note that G(λ) = λ→∞

(F, R∗ (λ, A|β=0 )U )H holds for any F ∈ H. It must be R∗ (λ, A|β=0 )U = 0, which means U = 0. Therefore, Span(A|β=0 ) = H. The desired result follows. To study the Riesz basis generation of the (generalized) eigenvectors of A|β=0 , we need the following result from [17]. Theorem 4.3. Let H be a separable Hilbert space, and A be the generator of a C0 semigroup T (t) on H. Suppose that: 1). σ(A) = σ1 (A) ∪ σ2 (A), where σ2 (A) = {λk }∞ k=1 consists of isolated eigenvalues of A with finite multiplicity; 2). sup ma (λk ) < ∞, where ma (λk ) = dim E(λk , A)H and E(λk , A) is the Riesz k≥1

projector associated with λk , 3). There is a constant α such that sup{Reλ|λ ∈ σ1 (A)} ≤ α ≤ inf{Reλ|λ ∈ σ2 (A)}; and inf |λn − λm | > 0. Then the following assertions are true. n6=m

i). There exist two T (t)-invariant closed subspaces H1 , H2 with the property that σ(A H1 ) = σ1 (A), σ(A H2 ) = σ2 (A), {E(λk , A)H2 }∞ k=1 forms a subspace Riesz L basis for H2 , and H = H1 H2 . ii). If sup ||E(λk , A)|| < ∞, then D(A) ⊂ H1 ⊕ H2 ⊂ H. k≥1

SPECTRUM AND DYNAMICAL BEHAVIOR OF A NETWORK

iii). H has the decomposition H = H1 ⊕H2 if

329

(topological direct sum), if and only

n

X

sup E(λk , A) < ∞.

n≥1 k=1

From Theorem 3.3, we obtain that the asymptotic behavior of the spectrum of A is independent of β. Then applying Theorem 3.3, 4.2 and 4.3 to our problem, we get the following result: 1

Theorem 4.4. Let H and A be defined as before. If |α − (ρ(1)k(1)) 2 |n×n 6= 0 and β = 0, there is a sequence of (generalized) eigenvectors of A which forms a Riesz basis for H. Proof. Set σ1 (A|β=0 ) := ∅, σ2 (A|β=0 ) := σ(A|β=0 ). By Theorem 3.3, we get conditions 1), 2), 3) in Theorem 4.3 holds. Then there is a sequence of (generalized) eigenvectors of A|β=0 that forms a Riesz basis for H2 . From Theorem 4.2, we know that the system of (generalized) eigenvectors is complete in H. Therefore, the sequence is also a Riesz basis for H. The desired result follows. Thus, we have gotten the Riesz basis property of the (generalized) eigenvectors of A with β = 0. In order to show that the Riesz basis property also holds for any β, we give the following lemma from [6], which is a corollary of Bari’s Theorem. Lemma 4.5. Let A be a densely defined discrete operator ( that is, (λ − A)−1 is compact for some λ) in a Hilbert space H. Let {zn }∞ 1 be a Riesz basis for H. If there exist N ≥ 0 and a sequence of generalized eigenvectors {xn }∞ N +1 of A such that ∞ X kxn − zn k2 < ∞, (42) N +1

then there exist M > N and generalized eigenvectors {xn0 }M 1 of A such that ∞ {xn0 }M 1 ∪ {xn }M+1 forms a Riesz basis for H.

Let Φn , Ψn , n = 1, 2, 3 · · · be the eigenvector of A and A|β=0 respectively. Since the asymptotic eigenvalues of A are independent of β, Theorem 3.3 implies e > 0 such that kΦn − Ψn k = O( 1 ) when n ≥ N e . Then Lemma that there exists N n 4.5 together with Theorem 4.4 implies the following result: 1

Theorem 4.6. Let H and A be defined as before. If |α − (ρ(1)k(1)) 2 |n×n 6= 0, then there is a sequence of (generalized) eigenvectors of A that forms a Riesz basis for H.

From Lemma 2.2 and Theorem 3.3, we deduce that the multiplicities of the eigenvalues of A are uniformly bounded. Therefore, by the Riesz basis property, according to [4] we have 1

Corollary 2. Let H and A be defined as before. If |α − (ρ(1)k(1)) 2 |n×n 6= 0, then the system associated with A satisfies the spectrum determined growth condition, i.e., ω(A) = S(A), where ω(A) = lim 1t ln keAt k is the growth order of eAt and t→∞

S(A) = sup{ℜλ|λ ∈ σ(A)} is the spectral bound of A. 1

Remark 2. From the expressions (34) of the spectrum of A, if |α−(ρ(1)k(1)) 2 |n×n 6= 0, then when |λ|(λ ∈ σ(A)) is big enough, the spectrum of A is all located in the left

330

ZHONG-JIE HAN AND GEN-QI XU

hand side of the complex plane and away from the imaginary axis. Furthermore, the spectrum determined growth condition implies 1). when αj > 0, j = 1, 2, · · · , n, the system (7) is exponentially stable, if and only if this system is asymptotically stable. 2). if there exists at least one j ∈ {1, 2, · · · , n} such that αj = 0, the system (7) can not reach exponential stability. 5. Exponential stability. In this section, we shall consider the exponential stability of a special case for this kind of network. Assume that ρj (x) = kj (x) = 1, b j = 1, 2, · · · , n and the length of each string in this network is 1. αj = α b, βj = β, We shall discuss that how to choose the collocated gain α b and non-collocated gain βb to stabilize this network system exponentially. Under the choice of the system parameters, this kind of network becomes  2 j 2 j ∂ w (x,t)  − ∂ w∂x(x,t) = 0, t > 0, x ∈ (0, 1),  2 2 ∂t   i j  w (0, t) = w (0, t), i, j ∈ {1, 2, · · · , n}, t > 0,    j  b j (0, t), t > 0,  wx (1, t) = −wj (1, t) − α bwtj (1, t) − βw t n P j (43) 1 w (0, t) = M w (0, t), t > 0,  x tt   j=1     wj (x, 0) = w ¯0j (x), wtj (x, 0) = w ¯1j (x), x ∈ [0.1],    j = 1, 2, · · · , n. Obviously, all results in the sections above also hold for this special case. From the asymptotic expression (34) of the spectrum of A, we get that when |λ|(λ ∈ σ(A)) is big enough, the spectrum of A is in the left hand side of the complex plane and away from the imaginary axis. Since this system satisfies the spectrum determined growth condition (see Corollary 2), if the real parts of those finite low eigenvalues are all less than 0, the system is exponentially stable. In this Section, we shall find a sufficient condition on α b and βb to let the real parts of all those finite low eigenvalues of A be less than 0. To this end, let us consider the eigenvalue problem:  2 W (x)  λ2 W (x) − ∂ ∂x = 0, x ∈ (0, 1), 2   C(n−1)×nW (0) = 0, (44) b  bλW (1) − βλW (0),   Wx (1) = −W (1) −2 α I1×n Wx (0) = M λ δ1×n W (0). The solution of the differential equation in (44) have the following form W (x) = eλx D1 + e−λx D2 , τ

(45) τ

where D1 = (d11 , d12 , · · · , d1n ) , D2 = (d21 , d22 , · · · , d2n ) . Substituting (45) into the boundary conditions in (44), we get the characteristic determinant of this system as follows C(n−1)×n C(n−1)×n I1×n λ − M λ2 δ1×n −I1×n λ − M λ2 δ1×n ∆(λ) = [λeλ + (λb b n×n [−λe−λ + (λb b n×n α + 1)eλ + βλ]I α + 1)e−λ + βλ]I 2n×2n

(46)

where In×n is the identity matrix with order n. Thus, a direct calculation leads to

b ∆(λ) = ((λb α +λ+1)eλ − (λb α −λ+1)e−λ )n−1 ∆(λ),

SPECTRUM AND DYNAMICAL BEHAVIOR OF A NETWORK

where

b ∆(λ) :=

nλ − M λ2 b (λb α +λ+1)eλ + βλ

331

−nλ − M λ2 . −λ b (λb α −λ+1)e + βλ 2×2

Now let us find the condition such that there is no eigenvalue of A on the imaginary axis. Set λ := ix, x 6= 0, x ∈ R. It is easy to verify (λb α +λ+1)eλ − (λb α −λ+1)e−λ 6= 0. b Thus we only need to check that ∆(ix) 6= 0 for x ∈ R. Since 0 ∈ ρ(A), a direct calculation yields ∆(ix) = p(x) + iq(x) 2λ

(47)

where

Then we have

p(x) := −nx sin x + n cos x − M x2 cos x − M x sin x, b − M x2 α q(x) := nxb α cos x + nβx b sin x.

p′ (x)

=

(−2n + M x2 − M ) sin x + (−nx − 3M x) cos x,

q ′ (x)

=

b (−nb αx − 2M xb α) sin x + (nb α − M x2 α b) cos x + nβ,

and thus,

p(x)q ′ (x) − q(x)p′ (x)

b − nβM b x3 ) = α b(M x sin x − n cos x)2 + sin x(n2 βx b 2 + n2 β) b + (nx + M x)nb b x2 + n2 βx αx + M 2 x4 α b + cos x(2nβM

sin x 2 b 2 b x4 ) (n βx − nβM x b x2 + n2 βx b 2 + n2 β) b + (nx + M x)nb + cos x(2nβM αx + M 2 x4 α b.

= α b(M x sin x − n cos x)2 +

Since | sinx x | ≤ 1, it follows that

p(x)q ′ (x) − q(x)p′ (x)

b 2 − nβM b x4 − 2nβM b x2 − n2 βx b 2 ≥ α b(M x sin x − n cos x)2 − n2 βx +n2 βb cos x + n2 α bx2 + M nb αx2 + M 2 x4 α b

b 2 − nβM b x4 − 2nβM b x2 − n2 βx b 2 ≥ α b(M x sin x − n cos x)2 − n2 βx +n2 βb cos x + n2 α bx2 + M nb αx2 + M 2 x4 α b b + n2 α ≥ α b(M x sin x − n cos x)2 + x2 (−2n2 βb − 2nβM b + M nb α) 4 2 2b b +x (−nβM + M α b) + n β cos x.

As cos x ≥ 1 −

x2 2 , ′

we have

p(x)q (x) − q(x)p′ (x)

≥

b + n2 α α b(M x sin x − n cos x)2 + x2 (−2n2 βb − 2nβM b + M nb α− b + M 2α b +x4 (−nβM b) + n2 β.

Choosing α b such that

α b > max{

(2n2 + 2nM )βb + n2 + M n

b n2 β 2

,

nβb }, M

n2 βb ) 2

(48)

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ZHONG-JIE HAN AND GEN-QI XU

we can get p(x)q ′ (x) − q(x)p′ (x) > n2 βb ≥ 0. (49) (2n+2M)+ n n 2 b Set γ0 := max{ n+M , M }. When α b > γ0 β ≥ 0, there is no eigenvalue on the imaginary axis. Since when βb = 0 there is no eigenvalue of A on the imaginary axis and in the right hand side of the complex plane, according to Theorem 2.1 in [34], there is no eigenvalue in the right hand side of the complex plane when α b > γ0 βb ≥ 0. Therefore, since this system (43) satisfies the spectrum determined growth condition, we have the following result: Theorem 5.1. The system (43) is exponentially stable, if α b > γ0 βb ≥ 0, where

(2n + 2M ) + n2 n , }. n+M M Furthermore, the decay rate of system (43) is sup{ℜλ|λ ∈ σ(A)}. γ0 := max{

(50)

6. Simulation. In order to support our results in this paper, we shall give a numerical simulation for this kinds of networks. Let us consider the case where the system (43) is composed of 3 strings. Set α b = 3, βb = 1, M = 1. We shall get the distribution of the numerical eigenvalues of the system operator and then check the stability of the system (43). Using Matlab Scientific Calculation, we get the following figure on distribution of the spectrum of the system operator, in which “ * ” denotes the eigenvalues of A whose imaginary parts are limited to domain [−300, 300] (see Fig. 2). 300

200

100

0

−100

−200

−300 −0.7

−0.65

−0.6

−0.55

−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

Figure 2. Distribution of the spectrum From this figure, we find that all the spectrum of the system is located in the left hand side of the complex plane and away from the imaginary axis, which implies the exponential stability of the system according to the spectrum determined

SPECTRUM AND DYNAMICAL BEHAVIOR OF A NETWORK

333

growth condition. Furthermore, the decay rate of this network system is possibly b we can −0.27. The result shows that by suitable choice of parameters α b and β, stabilize exponentially the network system by the non-collocated feedback controls. Although this is one special case of system (7), this simulation implies that the idea of the design of our feedback controllers is reasonable. Remark 3. The condition α b > γ0 βb ≥ 0 is sufficient for system (43) to get the exponential stability, but not necessary. In fact, by simulations, we find that this system can still be exponentially stable even though the condition is not satisfied. Acknowledgments. The authors would like to thank the referees very much for their useful and helpful comments and suggestions. The authors also thank Professor Guang-Gui Ding in Nankai University for his helpful seminar. REFERENCES [1] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Springer-Verlag, Berlin, 1983. [2] B. Dekoninck and S. Nicaise, The eigenvalue problem for networks of beams, Linear Algebra Appl., 314 (2000), 165–189. [3] B. Dekoninck and S. Nicaise, Control of networks of Euler-Bernoulli beams, ESAIM Control Optim. Calc. Var., 4 (1999), 57–81. [4] B. Z. Guo and G. Q. Xu, Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition, Journal of Functional Analysis, 231 (2006), 245–268. [5] B. Z. Guo and Y. Xie, A sufficient condition on Riesz basis with parentheses of non-selfadjoint operator and application to a serially connected string system under joint feedbacks, SIAM J. Control Optim., 43 (2004), 1234–1252. [6] B. Z. Guo, Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients, SIAM J. Control Optim., 40 (2002), 1905–1923. [7] B. Z. Guo and C. Z. Xu, The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation, IEEE Trans. Automat. Control, 52 (2007), 371–377. [8] B. Z. Guo, J. M. Wang and K. Y. Yang, Dynamic stabilization of an Euler-Bernoulli beam under boundary control and non-collocated observation, Systems and Control Letters, 57 (2008), 740–749. [9] C. Castro, Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass, ESAIM Control Optim. Calc. Var., 2 (1997), 231–280. [10] C. Castro and E. Zuazua, Exact boundary controllability of two Euler-Bernoulli beams connected by a point mass, Math. Comput. Modelling, 32 (2000), 955–969. [11] D. Mercier and V. R´ egnier, Spectrum of a network of Euler-Bernoulli beams, Journal of Mathematical Analysis and Applications, 337 (2008), 174–196. [12] D. Mercier and V. R´ egnier, Control of a network of Euler-Bernoulli beams, Journal of Mathematical Analysis and Applications, 342 (2008), 874–894. [13] G. Chen, M. C. Delfour, A. M. Krall and G. Payres, Modeling, stabilization and control of serially connected beams, SIAM J. Control Optim, 25 (1987), 526–546. [14] G. Q. Xu, Z. J. Han, and S. P. Yung, Riesz basis property of serially connected Timoshenko beams, International Journal of Control, 80 (2007), 470–485. [15] G. Q. Xu and S. P. Yung, Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping, Networks and Heterogeneous Media, 3 (2008), 723–747. [16] G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled networks of strings, SIAM J. Control Optim., 47 (2008), 1762–1784. [17] G. Q. Xu and S. P. Yung, The expansion of a semigroup and a Riesz basis criterion, Journal of Differential Equations, 210 (2005), 1–24. [18] M. Krstic, A. A. Siranosian and A. Smyshlyaev, Backstepping boundary controllers and observers for the slender Timoshenko beam: Part I–design, Proceedings of the 2006 American Control Conference, 2006, Minneapolis, Minnesota, USA, 2412–2417.

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Received September 2009; revised April 2010. E-mail address: [email protected] E-mail address: [email protected]