Stability analysis for an Euler-Bernoulli beam under local

J Control Theory Appl 2008 6 (4) 341–350DOI 10.1007/s11768-008-7217-5

Stability analysis for an Euler-Bernoulli beam underlocal internal control and boundary observation

Junmin WANG 1, Baozhu GUO 2, Kunyi YANG 2

(1. Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China;

2. Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100080, China)

Abstract: An Euler-Bernoulli beam system under the local internal distributed control and boundary point observationis studied. An infinite-dimensional observer for the open-loop system is designed. The closed-loop system that is non-dissipative is obtained by the estimated state feedback. By a detailed spectral analysis, it is shown that there is a set ofgeneralized eigenfunctions, which forms a Riesz basis for the state space. Consequently, both the spectrum-determinedgrowth condition and exponential stability are concluded.

Keywords: Euler-Bernoulli equation; Observer; Riesz basis; Controllability and observability; Stability

1 IntroductionMost output stabilization controls for systems described

by partial differential equations (PDEs) are designed in acollocated way, that is, the actuators and sensors are in thesame positions and designed in a “collocated” fashion. Thisis natural in the sense that the proportional output feedbackfor a collocated system results in a dissipative closed-loopsystem, and methods like the Lyapunov function methodsand the multiplier methods can be used to get the stabilityof the system.

On the other hand, it has been found long ago in engi-neering practice that the performance of the collocated con-trol design is not always good enough [1]. Although thenon-collocated control has been widely used in engineer-ing systems [2∼4], the theoretical studies on these systemsfrom the mathematical control point of view are quite few.The first difficulty is that the open-loops of non-collocatedsystems are usually not minimum-phase. This leads to theclosed-loop system being unstable for large feedback con-troller gains. Second, the closed-loops of non-collocatedsystems are usually non-dissipative, which gives rise tothe difficulty in applying the traditional Lyapunov functionmethods or the multiplier methods to the analysis of thestability. Recently, the estimated state feedback is designedthrough backstepping observers in [5] to stabilize a class ofone-dimensional parabolic PDEs. The abstract observer de-sign for a class of well-posed regular infinite-dimensionalsystems can be found in [6] but the stabilization is not ad-dressed.

Recently, some efforts have been made for non-collocated system control. In [7] and [8], the stabilizationof wave and beam equations under boundary control withnon-collocated observation has been handled respectively.

The objective of this paper is to study the stabilization ofan Euler-Bernoulli beam system under local distributed in-ternal control with point boundary observation. This resultsin a typical non-collocated system of PDEs. Such a distrib-

uted control is feasible in engineering practice due to theapplication of the smart materials. The pointwise measure-ment is the common observation for distributed parametersystems.

The system we are concerned with is the following Euler-Bernoulli beam with local internal distributed control andboundary point observation:

wtt(x, t) + wxxxx(x, t) + σ(x)u(x, t) = 0,

w(0, t) = wx(0, t) = 0,

wxx(1, t) = wxxx(1, t) = 0,

y(t) = wt(1, t),

(1)

where w(x, t) represents the transverse displacement of thebeam at position x ∈ [0, 1] and time t > 0, u(x, t) is thelocally distributed control (input), y(t) is the output (obser-vation), and

σ(x) =

1, x ∈ (a, b) ∈ (0,12), b > a,

0, otherwise.(2)

We choose the energy state space as H = H2E(0, 1) ×

L2(0, 1), H2E(0, 1) = {f | f ∈ H2(0, 1), f(0) = f ′(0) =

0}. H is equipped with the obvious inner product in-

duced norm |(f, g)|2H =w 1

0[|f ′′(x)|2 + |g(x)|2] dx for any

(f, g) ∈ H . The input and output spaces are U = L2(0, 1)and Y = C1, respectively.

Define the operator A : D(A)(⊂ H) → H as follows:

A(f, g) = (g,−f (4)),

D(A) = {(f, g) ∈ H| A(f, g) ∈ H,

f ′′(1) = f ′′′(1) = 0}.(3)

Received 17 September 2007; revised 30 January 2008.This work was supported by the National Natural Science Foundation of China and the Program for New Century Excellent Talents in University ofChina.

342 J. WANG et al. / J Control Theory Appl 2008 6 (4) 341–350

Then system (1) can be written as Σ(A,B, C) :

ddt

(w(x, t)wt(x, t)

)= A

(w(x, t)wt(x, t)

)+ Bu(x, t),

y(t) = C

(w(x, t)wt(x, t)

)= wt(1, t),

(4)

where

B =

(0

−σ(x)

), C = (0, 〈δ(x− 1), ·〉),

and δ(·) denotes the Dirac delta distribution. Obviously, Bis a bounded control operator while C is an unbounded ob-servation operator.

Theorem 1 For each u ∈ L2loc(0,∞) and initial da-

tum (w(·, 0), wt(·, 0)) ∈ H , there exists a unique solu-tion (w, wt) ∈ C(0,∞;H) to equation (1), and for eachT > 0, there exists a CT > 0 independent of u and(w(·, 0), wt(·, 0)) such that

|(w(·, T ), wt(·, T ))|2H +w T

0|y(τ)|2 dτ

6 CT [|(w(·, 0), wt(·, 0))|2H +w T

0|u(·, τ)|2L2 dτ ].

Proof Since B is bounded and C is admissible for eAt

(see [11]), by the well-posed linear infinite-dimensional sys-tem theory [9, 10], it is equivalent to showing that for anyT > 0 when w(x, 0) = wt(x, 0) ≡ 0, there exists a CT > 0such that

w T

0w2

t (1, t) dt 6 CT

w T

0

w 1

0u2(x, t) dxdt. (5)

In fact, let

E(t) =12

w 1

0[w2

t (x, t) + w2xx(x, t)] dx.

Then E(0) = 0 and

E(t) = −w b

awt(x, t)u(x, t) dx 6 E(t) +

w b

au2(x, t) dx.

Hence

E(t) 6 eTw T

0

w b

au2(x, t) dxdt,∀ t ∈ [0, T ]. (6)

On the other hand, set

ρ(t) =w 1

0xwx(x, t)wt(x, t) dx.

Then |ρ(t)| 6 E(t), and

ρ(t) =12y2

t (1, t)− 12

w 1

0[w2

t (x, t) + 3w2xx(x, t)] dx

−w b

axwx(x, t)u(x, t) dx.

This together with (6) shows that12

w T

0y2

t (1, t) dt

6 ρ(T ) + 3w T

0E(t) dt +

w T

0

w b

axwx(x, t)u(x, t) dxdt

6 E(T ) + 4w T

0E(t) dt +

w T

0

w b

au2(x, t) dxdt

6 [(1 + 4T )eT + 1]w T

0

w b

au2(x, t) dxdt.

This is (5), proving the result.Remark 1 The significance of Theorem 1 is that it not

only gives the well-posedness of the open-loop system (1)but also shows that for any L2 control, the output y makessense and is also in L2. This fact is very important to thedesign of the observer because for the observer, y becomesinput. The L2 property of y plays a key role in the solv-ability of the observer. This is attributed to the well-posedinfinite-dimensional system theory [9].

The remaining part of this paper is organized as follows.In Section 2, we construct an observer for the system (1) andshow that this observer is exponentially convergent. Section3 is devoted to the estimated state feedback design and somepreliminary results. In Section 4, we analyze the asymptoticbehavior of the eigenpairs. The Riesz basis property and ex-ponential stability are developed in Section 5.

2 Observer designWe design an observer of system (1) as follows: for

0 < x < 1, t > 0,

wtt(x, t) + wxxxx(x, t) + σ(x)u(x, t) = 0,

w(0, t) = wx(0, t) = wxx(1, t) = 0,

wxxx(1, t) = αwt(1, t)− αy(t),

(7)

where α ∈ R+ is a positive constant. System (7) can bewritten as [11]

wtt(x, t)+wxxxx(x, t)=−σ(x)u(x, t)+αδ(x−1)y(t),

w(0, t) = wx(0, t) = wxx(1, t) = 0,

wxxx(1, t) = αwt(1, t),(8)

or in H ,

wtt+Aw+αbb∗wt = −σ(x)u(x, t)+αδ(x−1)y(t), (9)

where A is given by (3) and b is given by

b = −δ(x− 1). (10)

The following result comes directly from [11].Theorem 2 System (9) is well-posed, that is to say, for

any (w(·, 0), wt(·, 0)) ∈ H and u, y ∈ L2loc(0,∞), there

exists a unique solution to (9) such that (w(·, t), wt(·, t)) ∈C(0,∞;H), and for any T > 0, there exists a constant CT

such that|(w(·, T ), wt(·, T ))|2H 6 CT [|(w(·, 0), wt(·, 0))|2H

+w T

0|u(·, t)|2L2 dt

w T

0y2(t) dt].

(11)

Let e(x, t) be the error of solutions of (1) and (7):

e(x, t) = w(x, t)− w(x, t). (12)

J. WANG et al. / J Control Theory Appl 2008 6 (4) 341–350 343

Then e(x, t) satisfies

ett(x, t) + exxxx(x, t) = 0,

e(0, t) = ex(0, t) = exx(1, t) = 0,

exxx(1, t) = αet(1, t).

(13)

It is well-known that the above system is exponentially sta-ble in H [12, 13]. So (7) is indeed an observer for system(1).

3 Estimated state feedback control designHaving obtained the estimated state through the observer,

we can now naturally design the following output feedbackbased on the estimated state as what we have done for col-located systems: u(x, t) = γwt(x, t), γ > 0. The closed-loop system now becomes, for 0<x<1, t>0,

wtt(x, t) + wxxxx(x, t) + γσ(x)wt(x, t) = 0,

w(0, t) = wx(0, t) = wxx(1, t) = 0,

wxxx(1, t) = αwt(1, t)− αwt(1, t),wtt(x, t) + wxxxx(x, t) + γσ(x)wt(x, t) = 0,

w(0, t) = wx(0, t) = 0,

wxx(1, t) = wxxx(1, t) = 0.

(14)

By e(x, t) = w(x, t) − w(x, t) defined in (12), we get theequivalent system of (14):

ett(x, t) + exxxx(x, t) = 0,

e(0, t) = ex(0, t) = exx(1, t) = 0,

exxx(1, t) = αet(1, t),wtt(x, t) + wxxxx(x, t) + γσ(x)wt(x, t) = 0,

w(0, t) = wx(0, t) = wxx(1, t) = 0,

wxxx(1, t) = exxx(1, t).

(15)

We consider system (15) in the state space X = H ×H with the obvious inner product induced norm: for∀ (f, g, φ, ψ) ∈ X ,

|(f, g, φ, ψ)|2

=w 1

0

[|f ′′(x)|2 + |g(x)|2 + |φ′′(x)|2 + |ψ(x)|2] dx.

The system operator A : D(A)(⊂ X) → X for (15) isdefined by

A(f, g, φ, ψ) = (g,−f (4), ψ,−φ(4) − γσψ),

∀ (f, g, φ, ψ) ∈ D(A),

D(A) = {(f, g, φ, ψ) ∈ X | A(f, g, φ, ψ) ∈ X,

f ′′(1) = φ′′(1) = 0,

f ′′′(1) = φ′′′(1), f ′′′(1) = αg(1)}.

(16)

With the operatorA at hand, we can write (15) as an evo-lution equation in X:

ddt

(e(·, t), et(·, t), w(·, t), wt(·, t))= A(e(·, t), et(·, t), w(·, t), wt(·, t)). (17)

We observe that A is not dissipative. Actually, let(f, g, φ, ψ) ∈ D(A). We have〈A(f, g, φ, ψ), (f, g, φ, ψ)〉X= 〈(g,−f (4), ψ,−φ(4) − γσψ), (f, g, φ, ψ)〉X=w 1

0[g′′f ′′ − f (4)g + ψ′′φ′′ − φ(4)ψ − γσ|ψ|2] dx

= −α|g(1)|2 − αg(1)ψ(1)

+w 1

0[g′′f ′′ − f ′′g′′ + ψ′′φ′′ − φ′′ψ′′ − γσ|ψ|2] dx.

Hence,Re 〈A(f, g, φ, ψ), (f, g, φ, ψ)〉X= −α|g(1)|2 − αRe(g(1)ψ(1))− γ

w 1

0σ|ψ|2 dx, (18)

which shows that A is not dissipative.Lemma 1 A−1 is compact on X and hence σ(A), the

spectrum of A, consists of isolated eigenvalues only.Proof For any (p1, q1, p2, q2) ∈ X , solve

A(f, g, φ, ψ) = (p1, q1, p2, q2)

to obtain

g(x) = p1(x), ψ(x) = p2(x),

−f (4)(x) = q1(x), −φ(4)(x)− γσ(x)ψ(x) = q2(x),

f(0) = f ′(0) = f ′′(1) = φ(0) = φ′(0) = φ′′(1) = 0,

f ′′′(1) = αg(1), φ′′′(1) = f ′′′(1).

This gives

g(x) = p1(x), ψ(x) = p2(x),

f(x) = αp1(1)[16x3 − 1

2x2]

+w x

0q1(ξ)(

16ξ3 − 1

2ξ2x) dξ

+12

w 1

xq1(ξ)(

13x3 − x2ξ) dξ,

φ(x) = αp1(1)[16x3 − 1

2x2]

+w x

0q3(ξ)(

16ξ3 − 1

2ξ2x) dξ

+12

w 1

xq3(ξ)(

13x3 − x2ξ) dξ,

q3(x) = γσ(x)p2(x) + q2(x).

Hence A−1 is defined everywhere on X and A−1 maps Xinto a subset of the space (H4(0, 1) × H2(0, 1))2, whichis compact in X . By the Sobolev embedding theorem [14],A−1 is compact on X , proving the required result.

Lemma 2 Re(λ) < 0 for any λ ∈ σ(A).Proof Let λ ∈ σ(A) and (f, g, φ, ψ) 6= 0 be a corre-

sponding eigenfunction. Then

A(f, g, φ, ψ) = λ(f, g, φ, ψ)

means thatg = λf, ψ = λφ,


and (f, φ) satisfies the following eigenvalue problem:

λ2f + f (4) = 0,

λ2φ + φ(4) + γλσφ = 0,

f(0) = f ′(0) = f ′′(1) = 0,

φ(0) = φ′(0) = φ′′(1) = 0,

f ′′′(1) = αλf(1), f ′′′(1) = φ′′′(1).

(19)

Multiply the first and second equations above respectivelyby f and φ and integrate over [0, 1] by parts with respect tox, to obtain

λ2w 1

0|f |2 dx + f ′′′f

∣∣∣1

0− f ′′f ′

∣∣∣1

0+w 1

0|f ′′|2 dx = 0,

λ2w 1

0|φ|2 dx + λγ

w 1

0σ|φ|2 dx

+ φ′′′φ∣∣∣1

0− φ′′φ′

∣∣∣1

0+w 1

0|φ′′|2 dx = 0.

By the boundary conditions of (19), it has

λ2w 1

0|f |2 dx + αλ|f(1)|2 +

w 1

0|f ′′|2 dx = 0,

λ2w 1

0|φ|2 dx + λγ

w 1

0σ|φ|2 dx

+ αλf(1)φ(1) +w 1

0|φ′′|2 dx = 0.

(20)

There are two cases:Case 1 f 6= 0. Let λ = Re(λ) + iIm(λ). The first

equation of (20) is equivalent to

((Re(λ))2 − (Im(λ))2)w 1

0|f |2 dx

+ αRe(λ)|f(1)|2 +w 1

0|f ′′|2 dx = 0,

Im(λ)(2Re(λ)w 1

0|f |2 dx + α|f(1)|2) = 0.

When Im(λ) 6= 0, it has

Re(λ) = − α|f(1)|2

2w 1

0|f |2 dx

6 0.

While Im(λ) = 0,

Re(λ)α|f(1)|2 + (Re(λ))2w 1

0|f |2 dx +

w 1

0|f ′′|2 dx = 0.

So, Re(λ) 6 0. Now we claim that Re(λ) < 0. If not, thereis an eigenvalue on the imaginary axis. Due to the symme-try of eigenvalues on the real axis, we only consider suchan eigenvalue λ = b2i with real b 6= 0. It is noted that thegeneral solutions of f in (19) is given by:

f(x) = a1(sin bx− sinh bx) + a2(cos bx− cosh bx),

where constants ai, i = 1, 2 are to be determined by theboundary conditions of (19) that

a1(sin b + sinh b) + a2(cos b + cosh b) = 0,

a1b(cos b + cosh b)− a2b(sin b− sinh b)

+iαa1(sin b− sinh b) + iαa2(cos b− cosh b) = 0.

Let{

E = b(cos b + cosh b) + iα(sin b− sinh b),

F = iα(cos b− cosh b)− b(sin b− sinh b).

Since∣∣∣∣∣sin b + sinh b cos b + cosh b

E F

∣∣∣∣∣

= F (sin b + sinh b)− E(cos b + cosh b)

= iα(sin b + sinh b)(cos b− cosh b)

− b(sin2 b− sinh2 b)

− iα(sin b− sinh b)(cos b + cosh b)

− b(cos2 b + cosh2 b + 2 cos b cosh b)

= −2(1 + cos b cosh b)− i2α(sin b cosh b− cos b sinh b)

6= 0,

we conclude that a1 = a2 ≡ 0. Hence f ≡ 0 and there isno eigenvalue on the imaginary axis, that is Re(λ) < 0.

Case 2 f ≡ 0. In this case, g = f ≡ 0 and hence it fol-lows from (18) that A is dissipative. Therefore, Re(λ) 6 0for λ ∈ σ(A). Let λ = bi with b ∈ R. Then (18) changes

0 ≡ Re 〈A(0, 0, φ, ψ), (0, 0, φ, ψ)〉X = −γw b

a|ψ|2 dx.

Thus, ψ ≡ 0 in [a, b] ⊂ [0, 1] and so ψ ≡ 0 in [0, 1] (see[15]). Hence φ = ψ ≡ 0 and there is no eigenvalue on theimaginary axis. Therefore, Re(λ) < 0 for any λ ∈ σ(A).The proof is completed.

4 Spectral analysisNow, we solve the eigenvalue problem (19). By Lemma

2 and the fact that the eigenvalues are symmetric about thereal axis again, we consider only those λ that are located inthe second quadrant of the complex plane:

λ = iρ2, ρ ∈ S = {ρ ∈ C | 0 6 arg ρ 6 π

4}. (21)

Note that for any ρ ∈ S , it has

Re(−ρ) 6 Re(iρ) 6 Re(−iρ) 6 Re(ρ), (22)

and

Re(−ρ) = −|ρ| cos(arg ρ) 6 −√

22|ρ| < 0,

Re(iρ) = −|ρ| sin(arg ρ) 6 0.

(23)

Lemma 3 For ρ ∈ S and x ∈ [0, 1],

eiρx, e−ρx, e−iρx, eρx (24)

are linearly independent fundamental solutions to f (4)(x)−ρ4f(x) = 0, and as |ρ| is large enough,

φ(4)(x)− ρ4φ(x) + iγρ2σ(x)φ(x) = 0,

where σ(x) is given by (2), has the following asymptotic


fundamental solutions:

φ1(x) = eiρx[1 + σ(x)ρ−1

(1 + ρ−1

)],

φ2(x) = e−ρx[1− iσ(x)ρ−1

(1 + ρ−1

)],

φ3(x) = e−iρx[1− σ(x)ρ−1

(1 + ρ−1

)],

φ4(x) = eρx[1 + iσ(x)ρ−1

(1 + ρ−1

)],

(25)

where

σ(x) =14γw x

0σ(ξ) dξ =

0, x ∈ [0, a),

γ(x− a)4

, x ∈ [a, b),

γ(b− a)4

, x ∈ [b, 1].

(26)

Proof The first claim is trivial. We only need to showthat (25) is a fundamental solution to φ(4)(x) − ρ4φ(x) +iγρ2σ(x)φ(x) = 0. This can be done along the way of Birk-hoff [16] and Naimark [17]. Here we present briefly a sim-ple calculation of (25) in sector S.

Note that −1 = i2,−i = i3, 1 = i4. Let

φs(x, ρ) := eisρx[1 +φs1(x)

ρ], s = 1, 2, 3, 4

and

D(φ) := φ(4)(x) + iγρ2σ(x)φ(x)− ρ4φ(x).

Then, substitute φs(x, ρ) into the expression of e−isρxD(g),to yield, for s = 1, 2, 3, 4, that

e−isρxD(φs(x, ρ))

= ρ4 + ρ3φs1(x) + 4(−i)sρ2φ′s1(x) + 6(−1)sρφ′′s1(x)

+ 4isφ′′′s1(x) + φ(4)s1 (x)ρ−1 + (iγρ2σ(x)− ρ4)

+ (iγρσ(x)− ρ3)φs1(x)

= ρ2 [4(−i)sφ′s1(x) + iγσ(x)] + ρFs(x, ρ),where

Fs(x, ρ) = 6(−1)sφ′′s1(x) + iγσ(x)φs1(x)

+4isφ′′′s1(x)ρ−1 + φ(4)s1 (x)ρ−2,

for which|Fi(z, ρ)| 6 M, x ∈ [0, 1]

for some positive constant M . Thus, letting the coefficientof ρ2 be zero, we obtain

4(−i)sφ′s1(x) + iγσ(x) = 0,

which yields

φs1(x) = (−i)s−1σ(x).

Hence,

φ11(x) = σ(x), φ21(x) = −iσ(x),

φ31(x) = −σ(x), φ41(x) = iσ(x).

Thus, we obtain a linearly independent asymptotic funda-mental solution to φ(4)(x) − ρ4φ(x) + iγρ2σ(x)φ(x) = 0given by (25) (see [16]):

φs(x) = φs(x, ρ) + eisρxO(ρ−2).

The proof is completed.From Lemma 3, we get that f(x) and φ(x) have the fol-

lowing asymptotic forms in S respectively:{

f(x) = c1eiρx + c2e−ρx + c3e−iρx + c4eρx,

φ(x) = d1φ1(x) + d2φ2(x) + d3φ3(x) + d4φ4(x),

where ci and di (i = 1, 2, 3, 4) are constants. Substitute theabove into the boundary conditions of (19), to obtain

∆(ρ)[c1, c2, c3, c4, d2, d2, d3, d4]T = 0, (27)

where

∆(ρ) =

[∆11(ρ) ∆12(ρ) 04×2 04×2

∆21(ρ) ∆22(ρ) ∆23(ρ) ∆24(ρ)

], (28)

∆11(ρ) =

1 1i −1

−eiρ e−ρ

−ieiρ(1 + αρ−1) − e−ρ(1 + iαρ−1)

,

(29)

∆12(ρ) =

1 1−i 1

−e−iρ eρ

ie−iρ(1− αρ−1) eρ(1− iαρ−1)

, (30)

∆21(ρ) =

0 00 00 0

−ieiρ − e−ρ

,∆21(ρ) =

0 00 00 0

ie−iρ eρ

,

∆23(ρ) =

1 1i −1

−eiρ

[1 +

σ(1)ρ

]

2

e−ρ

[1− i

σ(1)ρ

]

2

ieiρ

[1 +

σ(1)ρ

]

2

e−ρ

[1− i

σ(1)ρ

]

2

,

(31)

∆24(ρ) =

1 1−i 1

−e−iρ

[1− σ(1)

ρ

]

2

eρ

[1 + i

σ(1)ρ

]

2

−ie−iρ

[1− σ(1)

ρ

]

2

− eρ

[1 + i

σ(1)ρ

]

2

,

(32)

where [a]2 = a +O(ρ−2). Hence, (27) has a nontrivial so-lution if and only if

det(∆(ρ)) = 0, (33)


which equals

det(∆(ρ))

= det[∆11(ρ),∆12(ρ)

]det

[∆23(ρ),∆24(ρ)

]

= 0. (34)

Due to inequalities (22) and (23) for ρ ∈ S, after a directcomputation, we obtain

det[∆11(ρ),∆12(ρ)]

= eρ(e−iρ(1 + i)(1 + i− i2αρ−1)

− eiρ(1− i)(1− i− i2αρ−1) +O(ρ−2))

= 2eρ(e−iρ(i + (1− i)αρ−1)

+ eiρ(i + (1 + i)αρ−1) +O(ρ−2)),

and

det[∆23(ρ),∆24(ρ)]

= eρ((1 + iσ(1)ρ−1)[(i− 1)2eiρ(1 + σ(1)ρ−1)

− (i + 1)2e−iρ(1− σ(1)ρ−1)] +O(ρ−2))

= −2ieρ((1 + iσ(1)ρ−1)[eiρ(1 + σ(1)ρ−1)

+ e−iρ(1− σ(1)ρ−1)] +O(ρ−2))

= −2ieρ(eiρ(1 + (1 + i)σ(1)ρ−1)

+ e−iρ(1− (1− i)σ(1)ρ−1) +O(ρ−2)).

From the above computations, we can prove the follow-ing Theorem 3.

Theorem 3 Let λ = iρ2 and let ∆(ρ) be given by(28). Then det∆(ρ) is the characteristic determinant of theeigenvalue problem (19) and it has the asymptotic expres-sion in S:

det(∆(ρ)) = det[∆11(ρ),∆12(ρ)] det[∆23(ρ),∆24(ρ)],

where

det[∆11(ρ),∆12(ρ)]

= 2eρ(e−iρ(i + (1− i)αρ−1)

+eiρ(i + (1 + i)αρ−1) +O(ρ−2)), (35)

and

det[∆23(ρ),∆24(ρ)]

= −2ieρ(eiρ(1 + (1 + i)σ(1)ρ−1

)

+ e−iρ(1− (1− i)σ(1)ρ−1

)+O(ρ−2)). (36)

Moreover, the eigenvalue of (19) has the following asymp-totic forms:

λ1n = −2α + i(12

+ n)2π2 +O(n−1),

λ2n = −2σ(1) + i(12

+ n)2π2 +O(n−1),(37)

as n →∞, where n′s are positive integers and

σ(1) =γ(b− a)

4

given by (26). Therefore,

Re{λ1n, λ1n} → −2α,

Re{λ2n, λ2n} → −γ(b− a)2

,(38)

as n →∞.Proof We only need to show the asymptotic expression

of eigenvalues. Let ρ ∈ S and let det(∆(ρ)) = 0. Thendet(∆11(ρ)) = 0 or det(∆22(ρ)) = 0. From (35), we get

e−iρ(i + (1− i)αρ−1)

+eiρ(i + (1 + i)αρ−1) +O(ρ−2) = 0. (39)This leads to

e−iρ + eiρ +O(ρ−1) = 0. (40)

Notice that in the first quadrant of the complex plane, thesolutions to the equation eiρ + e−iρ = 0 are given by

ρ1n = (12

+ n)π, n = 0, 1, 2, · · · .

Apply the Rouche’s theorem to (40) to obtain the solutionsto (40):

ρ1n = ρ1n + α1n = (12

+ n)π + α1n,

α1n = O(n−1),(41)

as n → ∞. Substitute ρ1n into (39) and use the facteiρ1n = −e−iρ1n , to obtain

(1− (i− 1)αρ1n

)eiα1n−(1− (i + 1)αρ1n

)e−iα1n +O(ρ−21n ) = 0.

Expand the exponential functions above into Taylor seriesto yield

α1n = − α

i(12

+ n)π+O(n−2).

Substitute the above into (41) to produce, as n →∞,

ρ1n = (12

+ n)π − α

i(12

+ n)π+O(n−2). (42)

Since λ1n = iρ21n, we get eventually that

λ1n = −2α + i(12

+ n)2π2 +O(n−1) as n →∞.

Now we investigate the second branch eigenvalues. Letdet(∆22(ρ)) = 0. From (36), it has

eiρ(1 + (1 + i)σ(1)

ρ) + e−iρ(1− (1− i)

σ(1)ρ

)

+O(ρ−2) = 0. (43)A similar analysis gives the asymptotic solutions for theabove equation: as n →∞,

ρ2n = ρ1n + α2n = (12

+ n)π + α2n, α2n = O(n−1).

Substitute the above into (43) and use the fact eiρ1n =−e−iρ1n again, to obtain

(1 + (1 + i)σ(1)ρ2n

)eiα2n


−(1− (1− i)σ(1)ρ2n

)e−iα2n +O(ρ−22n ) = 0.

Hence, as n →∞,

α2n = − σ(1)

i(12

+ n)π+O(n−2),

and

ρ2n = (12

+ n)π − σ(1)

i(12

+ n)π+O(n−2). (44)

Since λ2n = iρ22n, we get eventually that

λ2n = −2σ(1) + i(12

+ n)2π2 +O(n−1) as n →∞.

The proof is completed.Theorem 4 Let {λ1n, λ1n, λ2n, λ2n, n ∈ N} be the

eigenvalues of A with λin being given in (37). Then thecorresponding eigenfunctions

{[f ′′in(x), λinfin(x), φ′′in(x), λinφin(x)], i = 1, 2, n ∈ N}(45)

have the following asymptotic expressions:

f ′′1n(x) = 2e−iρ1ne−ρ1n(1−x) + (1− i)eiρ1nx + 2ie−ρ1nx

− (1 + i)e−iρ1nx +O(n−1),

λ1nf1n(x) = 2ie−iρ1ne−ρ1n(1−x)−2e−ρ1nx−(1+i)eiρ1nx

− (1− i)e−iρ1nx +O(n−1),

φ′′1n(x) = f ′′1n(x) +O(n−1),

λ1nφ1n(x) = λ1nf1n(x) +O(n−1),

f ′′2n(x) = λ2nf2n(x) = 0,

φ′′2n(x) = 2e−iρ2ne−ρ2n(1−x) + (1− i)eiρ2nx

+ 2ie−ρ2nx − (1 + i)e−iρ2nx +O(n−1),

λ2nφ2n(x) = 2ie−iρ2ne−ρ2n(1−x) − 2ie−ρ2nx

−(1+i)eiρ2nx−(1−i)e−iρ2nx+O(n−1),(46)

where ρ1n and ρ2n are given by (42) and (44), respectively.Moreover,

{[fin, λinfin, φin, λinφin], i = 1, 2, n ∈ N}are approximately normalized in X in the sense that thereexist positive constants c1 and c2, independent of n, suchthat for n ∈ N,

c1 6 |f ′′in|L2 , |λinfin|L2 , |φ′′in|L2 , |λinφin|L2 6 c2. (47)

Proof In view of (19), (27), (28) and some facts in lin-ear algebra, the eigenfunction (fin, φin) (i = 1, 2) corre-sponding to the eigenvalue λin = iρ2

in can be given by thedeterminant of (28) in which one of its rows are replacedby (24) and (25) for fin and φin, respectively, so that suchobtained (fin, φin) is not identical to zero.

For λ1n, f1n and φ1n can be determined by replacing thefourth row of (28) with (24) and (25) respectively.

Thus (for brevity in notation, we omit the subscript with-out confusion)f1(x, ρ)

= det[∆23(ρ),∆24(ρ)

]× det

1 1 1 1i −1 −i 1

−eiρ e−ρ −e−iρ eρ

eiρx e−ρx e−iρx eρx

= eρ det[∆23(ρ),∆24(ρ)

]

×

det

1 1 1 0i −1 −i 0

−eiρ 0 −e−iρ 1eiρx e−ρx e−iρx e−ρ(1−x)

+O(e−c|ρ|)

= eρ det[∆23(ρ),∆24(ρ)

]

×

e−ρ(1−x) det

1 1 1i −1 −i

−eiρ 0 −e−iρ

−det

1 1 1i −1 −i

eiρx e−ρx e−iρx

+O(e−c|ρ|)

= eρ det[∆23(ρ),∆24(ρ)

]

×

e−ρ(1−x) det

1 + i 0 1− i

i −1 −i

−eiρ 0 −e−iρ

−det

1 + i 0 1− i

i −1 −i

eiρx e−ρx e−iρx

+O(e−c|ρ|)

= eρ det[∆23(ρ),∆24(ρ)

]

× {e−ρ(1−x)[(1 + i)e−iρ − (1− i)eiρ

]

− [(1− i)eiρx − 2ie−ρx − (1 + i)e−iρx

]+O(e−c|ρ|)},

where c is an arbitrary small positive constant. By (40), ithas

f1(x, ρ) = eρ det[∆23(ρ),∆24(ρ)

]

× {2e−iρe−ρ(1−x) − (1− i)eiρx + 2ie−ρx

+ (1 + i)e−iρx +O(ρ−1)}. (48)Similarly,

f ′′1 (x, ρ)

= ρ2 det[∆23(ρ),∆24(ρ)

]

× det

1 1 1 1i −1 −i 1


−eiρx e−ρx −e−iρx eρx


= ρ2eρ det(∆22(ρ))× {2e−iρe−ρ(1−x) + 2ie−ρx

+ (1− i)eiρx − (1 + i)e−iρx +O(ρ−1)}. (49)

Also,

φ1(x, ρ) = det(∆1(ρ))× det[∆21(x, ρ), ∆22(x, ρ)

],

where

∆1(ρ) =

1 1 1 1i −1 −i 1


−ieiρ −e−ρ ie−iρ eρ

,

∆21(x, ρ) =

φ1(x) φ2(x)1 1i −1

−eiρ

[1 +

σ(1)ρ

]

2

e−ρ

[1− i

σ(1)ρ

]

2

,

∆22(x, ρ) =

φ3(x) φ4(x)1 1−i 1

−e−iρ

[1− σ(1)

ρ

]

2

eρ

[1 + i

σ(1)ρ

]

2

,

and

det(∆1(ρin)) 6= 0, i = 1, 2, n ∈ N,

det(∆1(ρ)) = det[∆11(ρ),∆12(ρ)

]+O(ρ−1)

= −det[∆23(ρ),∆24(ρ)

]+O(ρ−1).

(50)A direct computation gives

det[∆21(x, ρ), ∆22(x, ρ)

]

= eρ

det

eiρx e−ρx e−iρx e−ρ(1−x)

1 1 1 0i −1 −i 0

−eiρ 0 −e−iρ 1

+O(ρ−1)

= −eρ{2e−iρe−ρ(1−x) − (1− i)eiρx

+ 2ie−ρx + (1 + i)e−iρx +O(ρ−1)},and hence,

e−ρφ1(x, ρ) = e−ρf1(x, ρ) +O(ρ−1). (51)

Similarly,

e−ρφ′′1(x, ρ) = e−ρf ′′1 (x, ρ) +O(ρ−1). (52)

Next we investigate the eigenfunction (f2n, φ2n) witheigenvalue λ2n in which f2n and φ2n can be determined byreplacing the eighth row of (28) with (24) and (25) respec-tively. Hence (for brevity in notation, we omit the subscriptwithout confusion again),

f2(x, ρ) = f ′′2 (x, ρ) = 0, (53)φ2(x, ρ) = det[∆11(x, ρ),∆12(x, ρ)]

× det[∆31(x, ρ), ∆32(x, ρ)], (54)

where

∆31(x, ρ) =

1 1i −1

−eiρ

[1 +

σ(1)ρ

]

2

e−ρ

[1− i

σ(1)ρ

]

2

φ1(x) φ2(x)

,

∆32(x, ρ) =

1 1−i 1

−e−iρ

[1− σ(1)

ρ

]

2

eρ

[1 + i

σ(1)ρ

]

2

φ3(x) φ4(x)

.

Thus, one hasφ2(x, ρ) = eρ det

[∆11(ρ),∆12(ρ)

]

×{2e−iρe−ρ(1−x) − (1− i)eiρx

+2ie−ρx + (1 + i)e−iρx +O(ρ−1)}. (55)Moreover,

φ′′2(x, ρ) = ρ2eρ det[∆11(ρ),∆12(ρ)

]

×{2e−iρe−ρ(1−x) + (1− i)eiρx

+2ie−ρx − (1 + i)e−iρx +O(ρ−1)}. (56)

Notice that det(∆11(ρ2n)) 6= 0, det(∆22(ρ1n)) 6= 0, andρ1n, ρ2n are given by (42) and (44), respectively. (46) canbe deduced from (48) to (56) by setting

f1n(x) =f1(x, ρ1n)

ρ2eρ∆22(ρ1n), φ1n(x) =

φ1(x, ρ1n)ρ2eρ∆22(ρ1n)

and

f2n(x) = 0, φ2n(x) =φ2(x, ρ2n)

ρ2eρ∆11(ρ2n).

Finally, it follows from (42) and (44) that for i = 1, 2,

|e−ρinx|L2 = O(n−1), |e−iρinx|L2 = 1 +O(n−1).

These together with (46) yield (47). The proof is completed.

5 Riesz basis generation and exponential sta-bility

This section is devoted to the Riesz basis property forsystem (17). The main result is the following Theorem 5.

Theorem 5 LetA be defined by (16). Then each eigen-value with large modulus is algebraically simple. Moreover,there is a set of generalized eigenfunctions of A, whichforms a Riesz basis for X .

Proof Define a linear operator A : D(A)(⊂ H) → H:

A(f, g) = (g,−f (4)), ∀ (f, g) ∈ D(A),

D(A) = {(f, g) ∈ H| A(f, g) ∈ H,

f ′′(1) = 0, f ′′′(1) = αg′(1)}.(57)

Let λ = iρ2. It is easy to check that det[∆11(ρ),∆12(ρ)

],

where[∆11(ρ),∆12(ρ)

]is given by (29) and (30), is also

the characteristic determinant of A, which has the asymp-totic expression (35). The eigenvalues of A, which is de-


noted by {λ1n, λ1n, n ∈ N} and corresponding general-ized eigenfunctions

{(f1n, λ1nf1n), (f1n, λ1nf1n), n ∈ N}have the asymptotic expressions (37) and (46), respectively.In terms of regular theory of the second-order partial differ-ential equations (see e.g., [18]), we know that each eigen-value with sufficiently large module is algebraically simpleand there is a set of generalized eigenfunctions of A, whichforms a Riesz basis for H .

Now define the operator A : D(A)(⊂ H) → H as fol-lows:

A(f, g) = (g,−f (4) − γσg),

D(A) = {(f, g) ∈ H| A(f, g) ∈ H,

f ′′(1) = f ′′′(1) = 0}.(58)

Then A is a bounded perturbation of A and the followingresults hold true (see [19]): for λ = iρ2,

1) det[∆23(ρ),∆24(ρ)

], where

[∆23(ρ),∆24(ρ)

]is

given by (31) and (32), is the characteristic determinant ofeigenvalues of A, which has the asymptotic expression (36);

2) the eigenvalues {λ2n, λ2n, n ∈ N} have the asymp-totic expansions (37) and each eigenvalue is algebraicallysimple when its modulus is large enough;

3) the corresponding eigenfunctions

{(φ2n, λ2nφ2n), (φ2n, λ2nφ2n), n ∈ N}have the asymptotic expressions (46);

4) there is a set of generalized eigenfunctions of A, whichforms a Riesz basis for H .

To sum up, we have obtained that

{((f1n, λ1nf1n), 0, 0), (0, 0, φ2n, λ2nφ2n, n ∈ N}and their conjugates

{(f1n, λ1nf1n, 0, 0), (0, 0, φ2n, λ2nφ2n, n ∈ N}form a Riesz basis for H ×H = X .

Furthermore, due to the fact(

I2×2 02×2

I2×2 I2×2

)(f1n, λ1nf1n, 0, 0)T

= (f1n, λ1nf1n, f1n, λ1nf1n)T

and (I2×2 02×2

I2×2 I2×2

)(0, 0, φ2n, λ2nφ2n)T

= (0, 0, φ2n, λ2nφ2n)T,

we conclude that

{(f1n, λ1nf1n, f1n, λ1nf1n), (0, 0, φ2n, λ2nφ2n), n ∈ N}together with their conjugates

{(f1n, λ1nf1n, f1n, λ1nf1n), (0, 0, φ2n, λ2nφ2n), n ∈ N}

form a Riesz basis for X . Now, by virtue of the Bari’stheorem (see [20, 21]) and the expressions (46), we knowthat each eigenvalue with sufficiently large module is alge-braically simple and there is a set of generalized eigenfunc-tions of A, which forms a Riesz basis for X . The proof iscompleted.

Theorem 6 Let A be defined by (16). Theni) A generates a C0-semigroup eAt on X .ii) The spectrum-determined growth condition holds true

for eAt, that is to say, S(A) = ω(A), where S(A) :=sup

λ∈σ(A)

Re λ is the spectral bound of A, and

ω(A) := inf{ω | ∃M > 0 such that ‖ eAt ‖6 Meωt

}

is the growth order of eAt.iii) System (17) is exponentially stable.Proof Since there is a set of generalized eigenfunctions

of A, which forms a Riesz basis for X , (i) and (ii) then fol-low from the asymptotic expansion (37) for eigenvalues. By(ii), the stability of (17) can be determined by the maxi-mal value of the real parts of eigenvalues of A. Now, byLemma 2, Re(λ) < 0 for any λ ∈ σ(A), and from Theorem3, the imaginary axis is not the asymptote of the eigenval-ues. Hence, system (17) is exponentially stable. The proofis completed.

6 ConclusionsIn this paper, we study the stabilization of an Euler-

Bernoulli beam equation with local internal distributed con-trol and boundary observation. The problem is a typicalnon-collocated control problem. The observer-based dy-namic feedback control is adopted in investigation: Aninfinite-dimensional observer is designed for the open-loopsystem; the estimated state feedback control is designed tostabilize the system. The closed-loop system is hence non-dissipative in the state space. We give a detailed spectralanalysis for the closed-loop system, from which we showthat the system is a Riesz spectral system. Some importantproperties like the spectrum determined growth conditionand exponential stability are thus concluded.

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Junmin WANG was born in Shanxi, China, in1973. He received the B.S. degree in mathemat-ics from Shanxi University in 1995 and the M.S.degree in applied mathematics from Beijing Insti-tute of Technology in 1998, respectively, and thePh.D. degree in applied mathematics from the Uni-versity of Hong Kong in 2004. From 1998 to 2000,he worked as a teaching assistant at the Beijing In-stitute of Technology. He was a postdoctoral fellow

at the University of the Witwatersrand, South Africa from 2004 to 2006.Since 2006, he has been with Beijing Institute of Technology as an asso-ciate professor in mathematics. His research interests focus on the controltheory of distributed parameter systems.

Dr. Wang is the recipient for new century excellent talents program fromMinistry of Education of China (2007).

Baozhu GUO received the B.S. degree in math-ematics from Shanxi University, China, in 1982,the M.S. degree from the Institute of Systems Sci-ence Academia Sinica in 1984, and the Ph.D. de-gree from the Chinese University of Hong Kongin 1991, both in applied mathematics. From 1985to 1987, he worked as a research assistant at Bei-jing Institute of Information and Control, China.He was a postdoctoral fellow at Institute of Sys-

tems Science, Academia Sinica from 1991 to 1993. From 1993 to 2000,he was with the Department of Applied Mathematics at Beijing Instituteof Technology, where he was employed as an associate professor and pro-fessor. Since 2000, he has been with Academy of Mathematics and Sys-tems Science, Academia Sinica, where he is a research professor in math-ematical control theory. He is the author or coauthor of over 100 jour-nal articles including 90 papers in international journals, and a coauthorof two books: “Population Distributed Parameter Control Theory” (1999,Huang-Zhong Institute of Technology Press) and “Stability and Stabiliza-tion of Infinite Dimensional Systems with Applications” (Springer-Verlag,1999). His research interests include mathematical biology and infinite-dimensional system control.

Dr. Guo is the member of one hundred-talent program, Academia Sinica(1999) and recipient of the national science fund for distinguished youngscholars, China (2003).

Kunyi YANG was born in Anhui, China, in 1984.She received the B.S. degree in mathematics atHefei University of Technology, China, in 2003,the M.S. degree from Beijing Institute of Technol-ogy in 2006. She is currently a Ph.D. student atAcademy of Mathematics and Systems Sciences,Academia Sinica. Her research interests focus ondistributed parameter systems control.

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Stability analysis for an Euler-Bernoulli beam under local