Automatica 45 (2009) 1468–1475

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Dynamic stabilization of an Euler–Bernoulli beam equation with time delay inboundary observationI

Bao-Zhu Guo a,c,∗, Kun-Yi Yang a,ba Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR Chinab Graduate School of the Chinese Academy of Sciences, Beijing 100039, PR Chinac School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

a r t i c l e i n f o

Article history:Received 6 May 2008Received in revised form17 November 2008Accepted 1 February 2009Available online 19 March 2009

Keywords:Euler–Bernoulli beam equationTime delayObserverFeedback controlExponential stability

a b s t r a c t

An Euler–Bernoulli beam equation under boundary control and delayed observation is considered. Anobserver–predictor based scheme is developed to stabilize the equation. On a time interval where theobservation is available, the state is tracked by the observer; when the observation is not available due tothe delay, the state is estimated by the predictor. The estimation is then used in proportional feedback. Itis shown that the state of closed-loop system decays exponentially for smooth initial values.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Timedelays are common in practical control systems, especiallyin networked systems, where it takes time for the observationto be available for use in control. These delays often createstability problem or cause periodic oscillations (Gumowski &Mira,1968). Indeed, as first noticed in Datko (1988) for one-dimensionalEuler–Bernoulli beam equation, even a small amount of timedelay in the stabilizing boundary output feedback schemes coulddestabilize the system. See also Datko (1991) for many otherhyperbolic systems.According to Fiagbedzi (1988, p. 69), stabilization of distributed

parameter control systems with time delay in observation andcontrol represents a difficult mathematical challenge. This islargely still the case today. Inspired by the work in Guo andXu (2007), we recently solved the stabilization problem for one-dimensional wave equation with boundary control and delayedobservation (Guo & Xu, 2008). The method we used in Guo and Xu

I This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Andrea Serraniunder the direction of Editor Miroslav Krstic.∗ Corresponding author at: Academy of Mathematics and Systems Science,Academia Sinica, Beijing 100190, PR China. Tel.: +86 10 62651443; fax: +86 1062587343.E-mail address: bzguo@iss.ac.cn (B.-Z. Guo).

0005-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.02.004

(2008) relies heavily on the explicit expression of analytic solutionof the system.In this paper, we consider the following Euler–Bernoulli beam

equation under boundary control and delayed observation:wtt(x, t)+ wxxxx(x, t) = 0, 0 < x < 1, t > 0,w(0, t) = wx(0, t) = wxx(1, t) = 0,wxxx(1, t) = u(t),y(t) = wt(1, t − τ), t ≥ τ ,w(x, 0) = w0(x), wt(x, 0) = w1(x),

(1)

where u is the control (or input), (w0, w1) is the initial state, τ > 0is a (known) constant time delay, and y is the delayed observation(or output).It is well known that if system (1) is time delay free (i.e., τ = 0),

then the static proportional output feedback u(t) = ky(t), k > 0exponentially stabilizes the system. In the presence of time delay(i.e., τ > 0), however, the closed-loop system with a staticfeedback is no more stable (Datko, 1988). Moreover, no dynamicstabilizing controller is robust to time delay in general for thoseplants like (1) with infinitely many unstable poles (Logemann,Rebarber, & Weiss, 1996).The above results lead to the search for control schemes in the

presence of a given time delay. We refer to Weiss and Curtain(1997) for a general method of designing stabilizing controllers forregular systems. In this paper, we propose an observer–predictorbased stabilization scheme. The state is dynamically estimated in

B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475 1469

two steps: On a time interval where the observation is available,the state is tracked by the observer; when the observation is notavailable due to the delay, the state is predicted in the scheme.We show that with the estimated state feedback, the state of theclosed-loop system decays exponentially for any smooth initialvalue.The idea of estimation can be explained for a finite-dimensional

time-invariant system described byX(t) = AX(t)+ Bu(t), y(t) = CX(t − τ),where τ > 0 is the time delay, A, B, C are matrices of appropriatedimensions. Let t > τ be a given time instant. For every s ∈[0, t − τ ] the state X(s) can be tracked by a standard Luenbergerobserver˙X(s) = AX(s)+ Bu(s)+ LCX(s)− Ly(s+ τ), s ∈ [0, t − τ ],where L is a matrix such that A + LC is stable. To determine X(s)for s ∈ [t − τ , t], where the observation is not available due to thedelay, we design a predictor as{˙X t(s) = AXt(s)+ Bu(s), s ∈ [t − τ , t]Xt(t − τ) = X(t − τ).

The state for t > τ is then estimated as Xt(t) ≈ X(t).To comment on the proposed scheme with respect to early

work, let us compute the estimate analytically. Solving the abovetwo equations gives˙X t(t) =

(A+ eAτ LCe−Aτ

)Xt(t)+ Bu(t)

− eAτ Ly(t)− eAτ LC∫ t

t−τeA(t−τ−θ)Bu(θ)dθ. (2)

We see that the estimate above becomes those in (17)–(18),(20)–(21) of Watanabe and Ito (1981) with D = A + eAτ LCe−Aτ ,E = −eAτ L, T = I , B = B, C = Ce−Aτ (see also Eq. 101 of Krstic andSmyshlyaev (2008) and Zhong (2006) on p. 40).It is noted that our scheme that views the time delay problem

from a different way is closely related to the observer designs forfinite-dimensional systems in Watanabe and Ito (1981) by time-domain approach (see also Klamka (1982) and Watanabe (1986))and Zhong (2006) by frequency-domain approach, and can beeasier to generalize to PDEs. Some other controllers for finite-dimensional systems were presented in Curtain, Weiss, and Weiss(1996) near the end of page 1527, which are actually the sameas controllers in Artstein (1982), Manitius and Olbrot (1979) andOlbrot (1978) (see also Fiagbedzi and Pearson (1986) and Zhong(2006)). The difficultywith infinite-dimensional systems is that thedelay free open-loop system (τ = 0 in (1)) has infinitely manypoles on the imaginary axis.We proceed as follows. In the next section, Section 2, we show

the well-posedness of open-loop system. Section 3 is devoted tothe design of observer and predictor. The exponential decay ofclosed-loop system under the estimated state feedback control ispresented in Section 4. Section 5 gives some simulation results.

2. Well-posedness of the open-loop system

Introduce a new variablez(x, t) = wt(1, t − τx).Then system (1) becomes

wtt(x, t)+ wxxxx(x, t) = 0, 0 < x < 1, t > 0,w(0, t) = wx(0, t) = wxx(1, t) = 0,wxxx(1, t) = u(t),τ zt(x, t)+ zx(x, t) = 0,z(0, t) = wt(1, t),w(x, 0) = w0(x), wt(x, 0) = w1(x),z(x, 0) = z0(x),y(t) = z(1, t),

(3)

where z0 is the initial value of variable z.

We consider the system (3) in the energy state spaceH = H × L2(0, 1),H = H2E (0, 1) × L

2(0, 1), H2E (0, 1) ={f ∈ H2(0, 1)|f (0) = f ′(0) = 0} with state variable(w(·, t), wt(·, t), z(·, t)), for which the inner product inducednorm is defined by

‖(w(·, t), wt(·, t), z(·, t))‖2H

=

∫ 1

0[w2xx(x, t)+ w

2t (x, t)+ z

2(x, t)]dx.

The input and output spaces are the same U = Y = R.

Theorem 2.1. The system (3) is well-posed: For any (w0, w1, z0) ∈H and u ∈ L2loc(0,∞), there exists a unique solution to (3) such that(w(·, t), wt(·, t), z(·, t)) ∈ C(0,∞;H); and for any T > 0, thereexists a constant CT such that

‖(w(·, T ), wt(·, T ), z(·, T ))‖2H +∫ T

0y2(t)dt

≤ CT

[‖(w0, w1, z0)‖2H +

∫ T

0u2(t)dt

]. (4)

Proof. It is well known that the following system{wtt(x, t)+ wxxxx(x, t) = 0,w(0, t) = wx(0, t) = wxx(1, t) = 0,wxxx(1, t) = u(t), yw(t) = wt(1, t),

(5)

can be written as a second-order system inH :{wtt(·, t)+ Aw(·, t)+ Bu(t) = 0,yw(t) = B∗wt(·, t),

(6)

where A is a self-adjoint operator inH and B is the input operator,both are unbounded:Af = f

(4), ∀ f ∈ D(A) = {f ∈ H4 ∩ H2E |f ′′(1) = f ′′′(1) = 0},

B = δ(x− 1).(7)

δ(·) denotes the Dirac distribution. It was shown in Guo and Luo(2002) that system (6)–(7) is well-posed in the sense of D. Salamon(Curtain et al., 1996): For any u ∈ L2loc(0,∞) and (w0, w1) ∈ H ,there exists a unique solution (w(·, t), wt(·, t)) ∈ C(0,∞;H) to(6); and for any T > 0, there exists a constant DT > 0 such that

‖(w(·, T ), wt(·, T ))‖2H +∫ T

0|yw(t)|2dt

≤ DT

[‖(w0, w1)‖

2H +

∫ T

0|u(t)|2dt

]. (8)

Once the solution of (5) is determined, one can solve the ‘‘z’’ partin Eq. (3):{τ zt(x, t)+ zx(x, t) = 0,z(0, t) = wt(1, t), z(x, 0) = z0(x).

(9)

Actually, for any initial z0 ∈ L2(0, 1), integrating along thecharacteristic line gives the unique (weak) solution to (9):

z(x, t) =

z0

(x−

tτ

), x ≥

tτ,

wt (1, t − xτ) , x <tτ.

(10)

1470 B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475

Therefore, for any T > 0,∫ 1

0z2(x, T )dx

=

∫ 1− Tτ

0z20(x)dx+

1τ

∫ T

0w2t (1, t)dt, T ≤ τ ,

1τ

∫ T

T−τw2t (1, t)dt, T > τ,

(11)

∫ T

0y2(t)dt

=

τ

∫ 1

1− Tτ

z20(x)dx, T ≤ τ ,

τ

∫ 1

0z20(x)dx+

∫ T−τ

0w2t (1, t)dt, T > τ.

(12)

Collecting (8), (11) and (12) gives (4). �

The importance of Theorem 2.1 lies in that for any initial valuein the state space, the output belongs to L2loc(τ ,∞) as long as theinput u belongs to L2loc(0,∞). This fact is particularly important tothe solvability of observer designed in the next section.

3. Observer and predictor design

In this section, we go back to system (1). For any given t > τ ,we proceed two steps to estimate the state of (1) via the observerand predictor.

Step 1: Construct an observer to estimate the state {(w(x, s),ws(x, s)), s ∈ [0, t− τ ]} from the known observation {y(s+ τ)|s ∈[0, t − τ ], t > τ }.Since the observation {y(s + τ), s ∈ [0, t − τ ]} is known and

{(w(x, s), ws(x, s)), s ∈ [0, t − τ ]} satisfieswss(x, s)+ wxxxx(x, s) = 0, 0 < s < t − τ ,w(0, s) = wx(0, s) = wxx(1, s) = 0,wxxx(1, s) = u(s),y(s+ τ) = ws(1, s),

(13)

we can construct naturally a Luenberger observer for system (13)as following:wss(x, s)+ wxxxx(x, s) = 0, 0 < s < t − τ ,w(0, s) = wx(0, s) = wxx(1, s) = 0,wxxx(1, s) = u(s)+ k1 [ws(1, s)− y(s+ τ)] ,w(x, 0) = w0(x), ws(x, 0) = w1(x),

(14)

where (w0, w1) is the (arbitrarily assigned) initial state of observer.We first state the solvability and convergence of observer (14).

Proposition 3.1. The system (14) is well-posed: For any t > τ , and(w0, w1) ∈ H , (w0, w1) ∈ H , u ∈ L2loc(0,∞), there exists a uniquesolution to (14) such that (w(·, s), ws(·, s)) ∈ C(0, t − τ ;H); andfor any 0 < T ≤ t − τ , there exists a positive constant CT such that

‖(w(·, T ), ws(·, T ))‖2H

≤ CT

[‖(w0, w1)‖

2H + ‖(w0, w1)‖

2H +

∫ T+τ

0u2(s)ds

].

Proof. Notice that (14) can be written as

wss(·, s)+ Aw(·, s)+ k1BB∗ws(·, s)+ B[u(s)− k1y(s+ τ)]= 0 for all 0 < s < t − τ , t > τ, (15)

where A, B are defined in (7). By Corollary 1 of Guo and Luo (2002),(15) is well-posed: For any t > τ and (w0, w1) ∈ H , u ∈L2loc(0,∞), y(s+τ) ∈ L

2(0, t−τ)which is assured by Theorem 2.1,

there exists a unique solution to (15) such that (w(·, s), ws(·, s)) ∈C(0, t − τ ;H); and for any 0 < T ≤ t − τ , there exists a constantC1T such that

‖(w(·, T ), ws(·, T ))‖2H ≤ C1T

[‖(w0, w1)‖

2H

+

∫ T+τ

τ

y2(s)ds+∫ T

0u2(s)ds

]. (16)

From (8), it has∫ T+τ

τ

|y(s)|2ds ≤ DT

[‖(w0, w1)‖

2H +

∫ T+τ

0|u(s)|2ds

]. (17)

Combing (16) and (17) gives the required inequality by settingCT = C1T (1+ DT ). �

In order for (14) to be an observer for (13), we have to show itsconvergence. To do this, let

ε(x, s) = w(x, s)− w(x, s), 0 ≤ s ≤ t − τ . (18)

Then by (13) and (14), ε satisfies{εss(x, s)+ εxxxx(x, s) = 0, 0 < s < t − τ ,ε(0, s) = εx(0, s) = εxx(1, s) = 0,εxxx(1, s) = k1εs(1, s).

(19)

System (19) can be written as:

dds

(ε(x, s)εs(x, s)

)= B

(ε(x, s)εs(x, s)

), (20)

where the operator B is defined as follows:B(f , g)> = (g,−f (4))>,D(B) = {(f , g)> ∈ (H4 ∩ H2E )× H

2E |

f ′′(1) = 0, f ′′′(1) = k1g(1)}.(21)

B generates an exponential stable C0-semigroup on H . For any(w0, w1) ∈ H , (w0, w1) ∈ H , there exists a unique solution εto (19) which satisfies

‖(ε(·, s), εs(·, s))‖H ≤ Me−ωs

‖(w0 − w0, w1 − w1)‖H , ∀ s ∈ [0, t − τ ], t > τ(22)

for some positive constantsM and ω.Step 2: Predict {(w(x, s), ws(x, s)), s ∈ (t − τ , t]} by {(w(x, s),

ws(x, s)), s ∈ [0, t − τ ]}.This is done by solving (1) with estimated initial value

(w(x, t − τ), ws(x, t − τ)) obtained from (14):wss(x, s, t)+ wxxxx(x, s, t) = 0, t − τ < s < t,w(0, s, t) = wx(0, s, t) = wxx(1, s, t) = 0,wxxx(1, s, t) = u(s),w(x, t − τ , t) = w(x, t − τ),ws(x, t − τ , t) = ws(x, t − τ).

(23)

By Theorem 2.1, for any t > τ and u ∈ L2loc(0,∞), there exists aunique solution to (23) such that (w(·, s, t), ws(·, s, t)) ∈ C(t −τ , t;H); and for any s ∈ (t − τ , t], there exists a constant Ct,s > 0such that

‖(w(·, s, t), ws(·, s, t))‖2H

≤ Ct,s

[‖(w(·, t − τ), ws(·, t − τ))‖2H +

∫ t

t−τ|u(s)|2ds

].

This together with Proposition 3.1 leads to (23).

B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475 1471

Proposition 3.2. The system (23) is well-posed: For any t > τ ,(w0, w1) ∈ H , (w0, w1) ∈ H , and u ∈ L2loc(0,∞), there exists aunique solution to (23) such that

(w(·, s, t), ws(·, s, t)) ∈ C(t − τ , t;H);

and for any s ∈ (t − τ , t], there exists a constant Ct,s such that

‖(w(·, s, t), ws(·, s, t))‖2H

≤ Ct,s

[‖(w0, w1)‖

2H + ‖(w0, w1)‖

2H +

∫ t

0u2(s)ds

].

We finally get the estimated state variable by

w(x, t) = w(x, t, t), wt(x, t) = ws(x, t, t), t > τ. (24)

Theorem 3.1. For any t > τ , we have

‖(w(·, t)− w(·, t), wt(·, t)− wt(·, t))‖H

≤ Me−ω(t−τ)‖(w0 − w0, w1 − w1)‖H , (25)

where (w0, w1) is the initial state of observer (14), (w0, w1) is theinitial state of original system (1), and M, ω are constants in (22).

Proof. Let

ε(x, s, t) = w(x, s, t)− w(x, s), t − τ ≤ s ≤ t. (26)

Then ε(x, s, t) satisfiesεss(x, s, t)+ εxxxx(x, s, t) = 0, t − τ < s ≤ t,ε(0, s, t) = εx(0, s, t) = 0,εxx(1, s, t) = εxxx(1, s, t) = 0,ε(x, t − τ , t) = ε(x, t − τ),εs(x, t − τ , t) = εt(x, t − τ),

(27)

which is a conservative system

‖(ε(·, t, t), εs(·, t, t))‖H = ‖(ε(·, t − τ), εt(·, t − τ))‖H . (28)

This together with (22) and (24) gives (25). �

4. Stabilization by the estimated state feedback

Since the feedback u(t) = k2wt(1, t) stabilizes exponentiallythe system (1), and we have the estimation wt(1, t) of wt(1, t), itis natural to design the estimated state feedback control law of thefollowing (with k2 > 0):

u∗(t) ={k2wt(1, t) = k2ws(1, t, t), t > τ,0, t ∈ [0, τ ], (29)

under which, the closed-loop system becomes a system of partialdifferential equations (30)–(32):wtt(x, t)+ wxxxx(x, t) = 0, t > τ,w(0, t) = wx(0, t) = wxx(1, t) = 0,wxxx(1, t) = k2ws(1, t, t),w(x, 0) = w0(x), wt(x, 0) = w1(x),

(30)

wss(x, s)+ wxxxx(x, s) = 0, 0 < s < t − τ ,w(0, s) = wx(0, s) = wxx(1, s) = 0,wxxx(1, s) = u∗(s)+ k1 [ws(1, s)− ws(1, s)] ,w(x, 0) = w0(x), ws(x, 0) = w1(x),

(31)

wss(x, s, t)+ wxxxx(x, s, t) = 0, t − τ < s < t,w(0, s, t) = wx(0, s, t) = wxx(1, s, t) = 0,wxxx(1, s, t) = u∗(s),w(x, t − τ , t) = w(x, t − τ),ws(x, t − τ , t) = ws(x, t − τ).

(32)

We consider system (30)–(32) in the state space X = H3. It isobvious that system (30)–(32) is equivalent to (33)–(35) for t > τprovided that u∗ ∈ L2loc(0,∞)which will be clarified in (51).wtt(x, t)+ wxxxx(x, t) = 0, t > τ,w(0, t) = wx(0, t) = wxx(1, t) = 0,wxxx(1, t) = k2wt(1, t)+ k2εs(1, t, t),w(x, 0) = w0(x), wt(x, 0) = w1(x),

(33)

εss(x, s)+ εxxxx(x, s) = 0, 0 < s < t − τ ,ε(0, s) = εx(0, s) = εxx(1, s) = 0,εxxx(1, s) = k1εs(1, s),ε(x, 0) = w0(x)− w0(x),εs(x, 0) = w1(x)− w1(x),

(34)

εss(x, s, t)+ εxxxx(x, s, t) = 0, t − τ < s < t,ε(0, s, t) = εx(0, s, t) = 0,εxx(1, s, t) = εxxx(1, s, t) = 0,ε(x, t − τ , t) = ε(x, t − τ),εs(x, t − τ , t) = εs(x, t − τ),

(35)

where ε(x, s) and ε(x, s, t) are given by (18) and (26), respectively.System (34) is the same as system (19).

Theorem 4.1. Let k1 > 0, k2 > 0 and t > τ . Then for any(w0, w1) ∈ H , (w0, w1) ∈ H , there exists a unique solutionto system (33)–(35) such that (w(·, t), wt(·, t)) ∈ C(τ ,∞;H),(ε(·, s), εs(·, s)) ∈ C(0, t − τ ;H), (ε(·, s, t), εs(·, s, t)) ∈ C([t −τ , t] × [τ ,∞);H) and for any t > τ, s ∈ [0, t − τ ], q ∈ (t − τ , t],there exists a constant Ctsq > 0 such that

‖(w(·, t), wt(·, t))‖H + ‖(ε(·, s), εs(·, s))‖H+‖(ε(·, q, t), εs(·, q, t))‖H≤ Ctsq[‖(w0, w1)‖H + ‖(w0, w1)‖H ].

Moreover, for any (w0 − w0, w1 − w1) ∈ D(B), where B is definedby (21), system (33) decays exponentially in the sense that

‖(w(·, t), wt(·, t))‖2H ≤ [‖(w0, w1)‖2H

+M0‖B(w0 − w0, w1 − w1)‖2H ]e−2ω0t , ∀ t ≥ 0 (36)

for some M0, ω0 > 0 that are independent of initial value.

Proof. For any (w0, w1) ∈ H, (w0, w1) ∈ H , since B defined by(21) generates an exponential stable C0-semigroup onH , there isa unique solution (ε(·, s), εs(·, s)) ∈ C(0, t − τ ;H) to (34) suchthat (22) holds true. Now, for any given time t > τ , write (35) as:

dds

(ε(x, s, t)εs(x, s, t)

)= A

(ε(x, s, t)εs(x, s, t)

),

where A is defined by

A(f , g)> = (g,−f (4))>,D(A) = {(f , g)> ∈ (H4 ∩ H2E )× H

2E | f′′(1) = f ′′′(1) = 0}.

Then A is skew-adjoint in H and hence generates a conservativeC0-semigroup onH . For any (ε(·, t − τ), εt(·, t − τ)) ∈ H that isdetermined by (34), there exists a unique solution to (35) such that

‖(ε(·, s, t), εs(·, s, t))‖H= ‖(ε(·, t − τ), εt(·, t − τ))‖H , ∀ s ∈ [t − τ , t]. (37)

So, (ε(·, s, t), εs(·, s, t)) ∈ C([t − τ , t] × [τ ,∞);H). Moreover,sinceA is skew-adjointwith compact resolvent, the solution of (35)can be, in terms of s, represented as(ε(x, s, t)εs(x, s, t)

)=

∞∑n=1

an(t)eiω2ns(φn(x)iω2nφn(x)

)

+

∞∑n=1

bn(t)e−iω2ns(

φn(x)−iω2nφn(x)

), (38)

1472 B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475

where {(φn(x),±iω2nφn(x))>}∞

n=1 is a sequence of all ω-linearlyindependent approximated normalized orthogonal eigenfunctionsof A corresponding to eigenvalues±iω2n , ωn > 0 and φn satisfies{φ(4)n (x) = ω

4nφn(x), 0 < x < 1,

φn(0) = φ′n(0) = φ′′

n (1) = φ′′′

n (1) = 0.(39)

Solve (39) to obtain

φ′′n (x) = −sin(ωnx)+ cos(ωnx)+ e−ωnx

+eωn(x−1)(sinωn + cosωn)+e−ωn [2 cos(ωn(1− x))+ eωn(x−1)

+e−ωnx(cosωn − sinωn)]+e−2ωn [sin(ωnx)+ cos(ωnx)] ,iω2nφn(x) = i{sin(ωnx)− cos(ωnx)+ e

−ωnx

+eωn(x−1)(sinωn + cosωn)+e−ωn [−2 cos(ωn(1− x))+ eωn(x−1)

+e−ωnx(cosωn − sinωn)]−e−2ωn [sin(ωnx)+ cos(ωnx)]},

(40)

where (see e.g., Guo and Luo (2002))

ωn =

(n−

12

)π + O(n−1) as n→∞. (41)

Therefore

εs(1, t, t) = i∞∑n=0

ω2n

[an(t)eiω

2n t − bn(t)e−iω

2n t]φn(1), (42)

where

c1 ≤ `n = ‖(φn(x), iω2nφn(x))>‖2H ≤ c2, ∀ n ≥ 1 (43)

with some constants c1, c2 > 0. Now suppose (ε(x, 0), εs(x, 0)) ∈D(B). Then by C0-semigroup theory, (ε(x, t − τ), εs(x, t − τ)) ∈D(B). This means, by (20), that εs(·, t − τ) ∈ H2E (0, 1), εxxxx(·, t −τ) ∈ L2(0, 1) for any t > τ . From (38), it follows that

`nan(t)eiω2n(t−τ)

=

⟨(ε(x, t − τ)εs(x, t − τ)

),

(φn(x)iω2nφn(x)

)⟩H

=k1εs(1, t − τ)

ω2n

[−2 sinωn + 2e−2ωn sinωn

]+iω2nεsx(1, t − τ)

[−2 cosωn − 4e−ωn

− 2e−2ωn cosωn]+1ω2n

∫ 1

0εxxxx(x, t − τ)

×[sin(ωnx)− cos(ωnx)+ e−ωnx

+ (sinωn + cosωn)eωn(x−1) − 2e−ωn cosωn(1− x)+ eωn(x−2) − (sinωn − cosωn)e−ωn(x+1)

− e−2ωn [sin(ωnx)+ cos(ωnx)]]dx

+iω2n

∫ 1

0εsxx(x, t − τ)

[−sin(ωnx)+ cos(ωnx)

+ e−ωnx + (sinωn + cosωn)eωn(x−1)

+ 2e−ωn cosωn(1− x)+ eωn(x−2)

− (sinωn − cosωn)e−ωn(x+1)

+ e−2ωn [sin(ωnx)+ cos(ωnx)]]dx, (44)

`nbn(t)e−iω2n(t−τ)

=

⟨(ε(x, t − τ)εs(x, t − τ)

),

(φn(x)−iω2nφn(x)

)⟩H

=εxxx(1, t − τ)

ω2n

[−2 sinωn + 2e−2ωn sinωn

]+iω2nεsx(1, t − τ)

[2 cosωn + 4e−ωn

+ 2e−2ωn cosωn]+1ω2n

∫ 1

0εxxxx(x, t − τ)

×[sin(ωnx)− cos(ωnx)+ e−ωnx

+ (sinωn + cosωn)eωn(x−1)

− 2e−ωn cosωn(1− x)+ eωn(x−2)

− (sinωn − cosωn)e−ωn(x+1)

− e−2ωn [sin(ωnx)+ cos(ωnx)]]dx

−iω2n

∫ 1

0εsxx(x, t − τ) [−sin(ωnx)

+ cos(ωnx)+ e−ωnx + (sinωn+ cosωn)eωn(x−1) + 2e−ωn cosωn(1− x)+ eωn(x−2) − (sinωn − cosωn)e−ωn(x+1)

+ e−2ωn [sin(ωnx)+ cos(ωnx)]]dx. (45)

By (40)

|ω2nφn(1)| ≤ 12, ∀ n ≥ 1. (46)

It then follows from (44) that

|`nan(t)| ≤1ω2n

[4k1|εs(1, t − τ)|

+ 8|εsx(1, t − τ)| + 12(∫ 1

0ε2xxxx(x, t − τ)dx

)1/2+ 12

(∫ 1

0ε2sxx(x, t − τ)dx

)1/2]

≤32+ 4k1ω2n

‖B(ε(·, t − τ), εs(·, t − τ))‖H . (47)

By (45) and boundary condition εxxx(1, t − τ) = k1εs(1, t − τ),it has

|`nbn(t)| ≤1ω2n

[4k1|εs(1, t − τ)|

+ 8|εsx(1, t − τ)| + 12(∫ 1

0ε2xxxx(x, t − τ)dx

)1/2+ 12

(∫ 1

0ε2sxx(x, t − τ)dx

)1/2]

≤32+ 4k1ω2n

[‖B(ε(·, t − τ), εs(·, t − τ))‖]H . (48)

Notice that in the last steps of (47) and (48), we used the facts that

|εs(1, t − τ)| =∣∣∣∣∫ 1

0εsx(x, t − τ)dx

∣∣∣∣≤

(∫ 1

0ε2sx(x, t − τ)dx

)1/2≤

(∫ 1

0ε2sxx(x, t − τ)dx

)1/2,

|εsx(1, t − τ)| =∣∣∣∣∫ 1

0εsxx(x, t − τ)dx

∣∣∣∣≤

(∫ 1

0ε2sxx(x, t − τ)dx

)1/2.

B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475 1473

By (41), collecting (42)–(48) gives

|εs(1, t, t)| ≤ C‖B(ε(·, t − τ), εs(·, t − τ))‖H (49)

for some constant C > 0 independent of t . Now, by (20) and C0-semigroup theory, we have

‖B(ε(·, t − τ), εs(·, t − τ))‖H

≤ Me−ω(t−τ)‖B(ε(·, 0), εs(·, 0))‖H (50)

for any t ∈ [τ ,+∞), whereM, ω are given by (22). We finally get

|εs(1, t, t)| ≤ CMe−ω(t−τ)‖B(ε(·, 0), εs(·, 0))‖H . (51)

In particular u∗ ∈ L2loc(0,∞). Furthermore, like (15), Eq. (33) canbe written as

wtt(·, t)+ Aw(·, t)+ k2BB∗wt(·, t)+ k2Bεs(1, t, t) = 0 for all t > τ, (52)

where A, B are defined in (7). It then follows from Guo andLuo (2002) that there exists a unique solution to (52) such that(w(·, t), wt(·, t)) ∈ C(τ ,∞;H). In what follows, we may assumewithout loss of generality that the solution of (52) is a classicalsolution because εs(1, t, t) can be approximated by a smoothfunction (see e.g., Guo and Xu (2008)).Define

E(t) =12

∫ 1

0[w2t (x, t)+ w

2xx(x, t)]dx

+ δ

∫ 1

0xwx(x, t)wt(x, t)dx, (53)

where 0 < δ < 1 is a small constant. Then

E(t) �∫ 1

0[w2t (x, t)+ w

2xx(x, t)]dx.

E(t) = −(k2 −

δ

2

)w2t (1, t)− k2wt(1, t)εs(1, t, t)

− k2δwx(1, t)wt(1, t)− k2δwx(1, t)εs(1, t, t)

−δ

2

∫ 1

0

[w2t (x, t)+ w

2xx(x, t)

]dx− δ

∫ 1

0w2xx(x, t)dx

≤ −

(k2 −

δ

2

)w2t (1, t)+ k2

[η1

4w2t (1, t)

+1η1ε2s (1, t, t)

]+ k2δ

[η2

4w2x (1, t)+

1η2w2t (1, t)

]+ k2δ

[η3

4w2x (1, t)+

1η3ε2s (1, t, t)

]−

δ

1+ δE(t)

− δ

∫ 1

0w2xx(x, t)dx

= −δ

1+ δE(t)−

(k2 −

δ

2−k2η14−k2δη2

)w2t (1, t)

+ k2δ(η24+η3

4

)w2x (1, t)

− δ

∫ 1

0w2xx(x, t)dx+

(k2η1+k2δη3

)ε2s (1, t, t)

for some constants ηi, i = 1, 2, 3. This together with the factw2x (1, t) ≤

∫ 10 w

2xx(x, t)dx gives

E(t) ≤ −δ

1+ δE(t)−

(k2 −

δ

2−k2η14−k2δη2

)w2t (1, t)

− δ

(1−

k2η24−k2η34

)∫ 1

0w2xx(x, t)dx

+

(k2η1+k2δη3

)ε2s (1, t, t). (54)

Choose η1, η2, η3 > 0 and δ > 0 so that

k2 −δ

2−k2η14−k2δη2

> 0,

1−k2η24−k2η34

> 0, 2ω >δ

1+ δ

and set k2η1+k2δη3= M1 to get

E(t) ≤ −δ

1+ δE(t)+M1ε2s (1, t, t). (55)

Take (51) into account and apply Gronwall’s inequality to (55) toobtain

E(t) ≤ e−δ1+δ tE(0)+M1

∫ t

0e−

δ1+δ (t−ρ)ε2s (1, ρ, ρ)dρ

≤ e−δ1+δ tE(0)+M1C2M2 · ‖B(ε(·, 0), εs(·, 0))‖2H

×

∫ t

0e−

δ1+δ (t−ρ)e−2ω(ρ−τ)dρ

≤

[E(0)+M1C2M2‖B(ε(·, 0), εs(·, 0))‖2H

× e2ωτ1− e−(2ω−

δ1+δ )t

2ω − δ1+δ

]e−

δ1+δ t

≤

[E(0)+

M1C2M2‖B(ε(·, 0), εs(·, 0))‖2H · e2ωτ

2ω − δ1+δ

]e−

δ1+δ t .

The proof is complete by setting M0 = 2M1C2M2(1+δ)e2ωτ

2ω(1+δ)−δ , ω0 =δ

2(1+δ) . �

Remark 4.1. (1)Whenwe take (w0, w1) = (w0, w1), (36) reducesto the usual exponential stability:

‖(w(·, t), wt(·, t))‖H ≤ ‖(w0, w1)‖He−ω0t , ∀ t ≥ 0.

(ii) When (w0, w1) ∈ D(B), then we can take (w0, w1) = 0;and (36) reduces to

‖(w(·, t), wt(·, t))‖H ≤ (M0 + ‖B−1‖)‖B(w0, w1)‖He−ω0t .

5. Simulation results

In this section, we use finite difference method to give somenumerical simulation results for the closed-loop system (33)–(35).Here we choose the space grid size N = 30, time step dt = 0.0005and time span [0, 6]. Parameters are chosen to be τ = k1 = k2 = 1.For the initial value{w0(x) = x2,w1(x) = 1

,

{ε(x, 0) = x2,εs(x, 0) = 1,

∀ x ∈ [0, 1], (1)

the latter is in H but not in the domain of B, we plot in Fig. 1 thedisplacementw(x, t) and velocitywt(x, t).It is seen that the system is still stable. This is not surprise

because the asymptotical stability in Theorem 4.1 is expectedfor non-smooth initial value. Indeed, for a one-dimensional waveequation, it was shown in Guo and Xu (2008) that for non-smoothinitial values, the closed-loop system is asymptotically stable.Obviously, the predictor–observer based scheme developed in thispaper can be useful in designing stabilizing controller for any otherPDEs with observation delay. But the stability proof for the lattercase poses a big deal of mathematical challenge.

1474 B.-Z. Guo, K.-Y. Yang / Automatica 45 (2009) 1468–1475

Fig. 1. Displacement and velocity of the beam with initial value (1) and time delay τ = 1.

Fig. 2. Displacement and velocity of the beam with delay disturbance θ = 0.1, initial value (1) and time delay τ = 1.

In the end of this paper, we mention a few words about therobust to time delay for the present developed scheme. It should bepointed out that there is no positive result up to present since theproblemwas proposed in Datko (1988). Actually, as wementionedearlier, it was proved in Logemann et al. (1996) that for our system,because there are infinitely many poles on the imaginary axisfor delay free system, there does not exist generally a stabilizingcontroller that is robust to time delay. The problemwas addressedearlier in Georgiou and Smith (1989) by cascading a low-pass filterto the controller. A recent study for time delay system in Hilbertspacewith bounded control operator by LMI approach canbe foundin Fridman and Orlov (2009). For our scheme, if τ = 0, it issimply an observed based scheme. From the frequency-domainpoint of view, this case is the sameas the direct proportional outputfeedback control and hence is not robust to time delay by Theorem1.1 of Logemann et al. (1996). For τ > 0, we did a numericalexperiment with τ = 1 and time delay disturbance θ = 0.1.Using the same initial value (1), we get results plotted in Fig. 2.It is seen that both displacement and velocity are not convergent.We may expect at least based on numerical experiment that thescheme is not robust to time delay. The analytical proof needsfurther investigations.

Acknowledgements

The Authors would like to thank Professor Cheng-Zhong Xuof University of Lyon I, France for valuable discussions. Thesupports of the National Natural Science Foundation of China andthe National Research Foundation of South Africa are gratefullyacknowledged.

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Bao-Zhu Guo received his Ph.D. degree in AppliedMathematics from the Chinese University of Hong Kongin 1991. From 1993 to 2000, he had been with theDepartment of Applied Mathematics, Beijing Institute ofTechnology, China, where he was an associate professor(1993–1998) and then professor (1998–2000). Since2000, he has been a research professor in Academy ofMathematics and System Sciences, the Chinese Academyof Sciences.

Dr. Guo is the recipient of One Hundred Talent Program from the ChineseAcademy of Sciences (1999) and the National Natural Science Foundation ofChina for Distinguished Young Scholars (2003). His interests include mathematicalbiology and infinite-dimensional system control.

Kun-Yi Yang received her B.Sc. degree in Mathematicsfrom Hefei University of Technology, China, in 2003,the M.Sc. degree from Beijing Institute of Technology in2006. She is currently a Ph.D. student in Academy ofMathematics and System Sciences, Academia Sinica. Herresearch interests focus on distributed parameter systemscontrol.