Adaptive Output Feedback Stabilization for One-Dimensional Wave ...

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SIAM J. CONTROL OPTIM. Vol. 51, No. 2, pp. 1679–1706

c 2013 Society for Industrial and Applied Mathematics

ADAPTIVE OUTPUT FEEDBACK STABILIZATION FOR ONE-DIMENSIONAL WAVE EQUATION WITH CORRUPTED OBSERVATION BY HARMONIC DISTURBANCE∗ WEI GUO† AND BAO-ZHU GUO‡ Abstract. In this paper, we are concerned with the output feedback stabilization of a onedimensional wave equation with an unstable term at one end, and the observation suffered by a general harmonic disturbance with unknown magnitudes at the other end. An adaptive observer is designed in terms of the corrupted observation. The backstepping method for infinite-dimensional systems is adopted in the design of the feedback law. It is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameters are shown to be convergent to the unknown parameters as time goes to infinity. Key words. vibrating string, disturbance rejection, boundary control, backstepping AMS subject classifications. 93C20, 93D15, 35B35, 35P10 DOI. 10.1137/120873212

1. Introduction and problem formulation. In the past several decades, collocated boundary feedback control for the second-order infinite-dimensional systems described by waves and flexible beams has been widely studied; see [1, 17, 22, 19], to name just a few. Most of the systems aforementioned are, when there is no boundary control imposed, conservative in the sense that the system energy remains constant. The main idea of feedback control design is to add dissipation to the system by means of boundary damping to make the energy of the systems decay polynomially or exponentially to zero as the time goes to infinity. Most often, the collocated control design is based on a passive principle and hence is almost trivial although the stability analysis is, due to its PDEs nature, very hard in many cases. This happens also for the stabilization of the most multidimensional PDEs ([15, 17, 18, 22]). However, when the boundary control system is noncollocated, or the system itself is unstable or even antistable, the collocated design is not enough to stabilize these systems. In this situation, the passive principle cannot be applied directly anymore. In the last few years, there have been a few works contributed to this aspect. An observer-based compensator which exponentially stabilizes the string system with a noncollocated actuator/sensor configuration is proposed in [3]. A dramatic change has taken place since the backstepping method was introduced in PDEs ([13]). In [11], the controller and observer are designed using both the displacement and velocity measurement via the method of backstepping to exponentially stabilize a one-dimensional wave equation that contains a destabilizing antistiffness boundary condition at its free end. Since the destabilizing term in [11] is proportional to the displacement, the system ∗ Received by the editors April 11, 2012; accepted for publication (in revised form) February 4, 2013; published electronically April 23, 2013. http://www.siam.org/journals/sicon/51-2/87321.html † Corresponding author. School of Statistics, University of International Business and Economics, Beijing 100029, China ([email protected]). This author was supported by Program for Innovative Research Team in UIBE. ‡ Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China, School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa, and King Abdulaziz University, Jeddah, Saudi Arabia ([email protected]). This author was supported by the National Natural Science Foundation of China, the National Basic Research Program of China (2011CB808002), and the National Research Foundation of South Africa.

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1680

WEI GUO AND BAO-ZHU GUO

is thereby unstable in the sense that the uncontrolled system has some eigenvalues on the right complex plane. An extension result is then presented in [4] where the controller and observer for a noncollocated wave equation are designed using the displacement measurement only. A recent breakthrough has been made in [23], where the antistable wave equation with an antidamping term on the uncontrolled boundary which is different from the destabilizing term in [11], is stabilized through a novel backstepping transformation method. The stabilization of the unstable shear beam equation can be found in [12] where noncollocated boundary stabilization is discussed by using a backstepping approach and observer-based feedback. However, the controllers and observers aforementioned are designed without consideration of the uncertainties suffered in the boundary input or boundary output. The earlier efforts for the design of adaptive controller and observer for PDEs, particularly for parabolic PDEs with boundary control and unknown parameters that may cause instability of the system and affect the interior of domain, are presented in [14, 20, 21]. Adaptive stabilization for the most challenging antistable wave equation system can be found in [10]. The first effort on the adaptive regulator design for undamped second-order hyperbolic systems with output disturbances and collocated control is made in [16]. Recent progress is made in [7] where an adaptive observer and controller is designed for a one-dimensional wave equation with simple corrupted periodic output disturbances θ¯1 sin t + θ¯2 cos t. An adaptive regulator for an unstable wave equation with simple periodic input disturbance is designed by using the backstepping method in [8] to achieve both parameter estimation and stabilization under the state feedback. A recent result on the stabilization of an unstable wave equation with general harmonic disturbance in boundary input is presented in [9]. The present paper is motivated by [9] where instead of boundary input, the general harmonic disturbance with unknown amplitudes appears in the boundary output. The problem that we are concerned with is a one-dimensional wave equation proposed in [11]: ⎧ wtt (x, t) − wxx (x, t) = 0, x ∈ (0, 1), t > 0, ⎪ ⎪ ⎪ ⎪ w ⎪ x (0, t) = −qw(0, t), t ≥ 0, ⎪ ⎪ ⎨ wx (1, t) = u(t), t ≥ 0, (1.1) w(x, 0) = w0 (x), wt (x, 0) = w1 (x), 0 ≤ x ≤ 1, ⎪ ⎪ m ⎪ ⎪ ⎪ ¯ ¯ ⎪ (θj sin αj t + ϑj cos αj t) , t ≥ 0, ⎪ ⎩ yout (t) = w(1, t), wt (1, t) + j=1

where and henceforth w or wx denotes the derivative of w with respect to x, and w˙ or wt the derivative with respect to t, u is the boundary control (input), yout is the boundary measurement (output), w0 and w1 are initial values, θ¯j , ϑ¯j , j ∈ J = {1, 2, . . . , m} are unknown amplitudes, and αj , j ∈ J are the known frequencies of the disturbance. Obviously, if some αj = 0, the term is reduced to the constant. This kind of disturbance contains the periodic disturbance as its special case. The major concern for this kind of output is that the velocity is relatively difficult to measure and the differentiation of amplitude that is easily measured amplifies the noise [2, pp. 17–18]. Obviously, the harmonic disturbance vector function (sin α1 t, cos α1 t, . . . , sin αm t, cos αm t) is a solution to the following homogeneous equation: ⎧ d ⎪ ⎪ ⎨ dt (ξ1 (t), η1 (t), . . . , ξm (t), ηm (t)) = B(ξ1 (t), η1 (t), . . . , ξm (t), ηm (t)) , (1.2) (ξ1 (0), η1 (0), . . . , ξm (0), ηm (0)) = (ξ10 , η10 , . . . , ξm0 , ηm0 ) ⎪ ⎪ ⎩ = (0, 1, 0, 1, . . . , 0, 1) ,


ADAPTIVE STABILIZATION FOR A WAVE EQUATION

⎛


where

⎜ ⎜ ⎜ B=⎜ ⎜ ⎝

0 −α1 .. .

α1 0 .. .

0 0

0 0

··· ··· ··· ··· ···

0 0 .. .

0 0 .. .

0 −αm

αm 0

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⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

For q = 0, (1.1) models a string which is free at the end x = 0. For q = 0, the free end of the string is subject to a force proportional to the displacement, which physically may be the result of various phenomena. We refer to [11] for the modeling in details. For q ≤ 0, system (1.1) with u = 0 is conservative. It is well known that if there is no uncertainty in the measurement, then system (1.1) with q ≤ 0 can be exponentially stablized under output feedback u(t) = −kwt (1, t), k > 0 and u(t) = −k1 w(1, t) − k2 wt (1, t), ki > 0, i = 1, 2, respectively. For the case q > 0, system (1.1) with u = 0 is unstable in the sense that there are finitely many eigenvalues located in the right complex plane. If there is no uncertainty in the measurement, system (1.1) can be exponentially stabilized by an observer-based infinite-dimensional controller using the backstepping method developed in [11]. A natural idea is to extract the real value of wt (1, t) by a notch filter and then use the controller u(t) = −kwt (1, t), k > 0 or u(t) = −k1 w(1, t) − k2 wt (1, t), ki > 0, i = 1, 2. However, in applying the notch filter, wt (1, t) cannot contain the frequency of the disturbed harmonic signal that is not known for our problem since our initial values are arbitrary. Another point that we want to make here is that based on passive principle, we can design an adaptive regulator for system (1.1) in the case q = 0 as ⎧ m ⎪ ⎪ u(t) = −k w(1, t) − k (1, t) + (θ¯j sin αj t + ϑ¯j cos αj t) w ⎪ 1 2 t ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ (θ (t) sin α t + ϑ (t) cos α t) , k1 , k2 > 0, − ⎪ j j j j ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ m m ⎪ ⎪ ⎪ ˙ ¯ ¯ ⎪ θ (t) = r (1, t) + ( θ sin α t + ϑ cos α t) − (θj (t) sin αj t w j t j j j j ⎪ ⎨ j (1.3)

j=1 j=1 ⎪ ⎪ ⎪ + ϑj (t) cos αj t) sin αj t, rj > 0, j ∈ J, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ˙ ⎪ ϑ w (t) = l (1, t) + (θ¯j sin αj t + ϑ¯j cos αj t) j j t ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ (θj (t) sin αj t + ϑj (t) cos αj t) cos αj t, lj > 0, j ∈ J. − ⎪ ⎩ j=1

The candidate Lyapunov function for q = 0 is as follows: m 1 1 2 k1 k2 θj2 (t) ϑ2j (t) Ew (t) = [wt (x, t) + wx2 (x, t)]dx + w2 (1, t) + + . 2 0 2 2 j=1 rj lj For the case q < 0, we can also design the same adaptive regulator for system (1.1) as (1.3) by removing the term −k1 w(1, t) in (1.3). In both q < 0 and q = 0, the resulting closed loop can be shown to be asymptotically stable and the estimated parameters are shown to be convergent to the unknown parameters as time goes to infinity but for some special known αj , j ∈ J only, for which the proof is similar to that in section 3.



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But this design is not applied for arbitrarily given frequencies αj , j ∈ J and q > 0. We also point out that the infinite-dimensional second-order system developed in [16] does not cover the case q > 0 which results in finitely many eigenvalues for the open-loop system in the right complex plane. Compared to the existing works [7, 16] and (1.3), the main contribution of this paper is that for any arbitrarily given frequencies αj , j ∈ J and any real number q, one can always construct an observer-based adaptive controller to achieve both the parameter convergent and the closed-loop system stable. We proceed as follows. In section 2, we design an adaptive observer and an observer-based feedback controller by the backstepping method. The main results are stated in section 3. Section 4 is devoted to the proofs of the main results. In section 5, some numerical simulations are presented to illustrate the theoretical results. Some concluding remarks are presented in section 6. 2. Adaptive observer and controller design. Since our observation is boundary pointwise measurement, we need to recover the state through the observer. This section is dedicated to the design of the adaptive observer and the observer-based feedback controller. It is a crucial part of the paper. We design the following adaptive observer for the system (1.1): (2.1) ⎧ w tt (x, t) − w xx (x, t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = q(c0 + q)eq(1−x) [w(1, t) − w(1, t)] ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ q(1−x) ⎪ + q)e k w (1, t) + (θ¯j sin αj t + ϑ¯j cos αj t) + (c ⎪ 0 t ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ −w t (1, t) − (θj (t) sin αj t + ϑj (t) cos αj t) , ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ w x (0, t) = −q w(0, t), ⎪ ⎪ ⎪ ⎪ w (1, t) = u(t) + (c0 + q)[w(1, t) − w(1, t)] ⎪ x ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ + k wt (1, t) + (θ¯j sin αj t ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ m ⎪ ⎨ ¯ + ϑj cos αj t) − w t (1, t) − (θj (t) sin αj t + ϑj (t) cos αj t) , ⎪ j=1 ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ˙ ⎪ (θ¯j sin αj t + ϑ¯j cos αj t) − w t (1, t) θj (t) = rj wt (1, t) + ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ − (θj (t) sin αj t + ϑj (t) cos αj t) sin αj t, j ∈ J, ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ˙ j (t) = lj wt (1, t) + ϑ (θ¯j sin αj t + ϑ¯j cos αj t) − w t (1, t) ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ (θj (t) sin αj t + ϑj (t) cos αj t) cos αj t, j ∈ J, − ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ θj (0) = θj0 , ϑj (0) = ϑj0 ∈ R, j ∈ J, ⎪ ⎪ ⎪ ⎩ w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x), where and henceforth k, ci , i = 0, 1, 2, rj and lj , j ∈ J are positive design parameters. Here and in the rest of the paper, we omit the (obvious) domains for t and x.




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Let ε(x, t) = w(x, t)− w(x, t) denote the observer error and θi (t) = θ¯j −θj (t), ϑj (t) ¯ = ϑj − ϑj (t), j ∈ J, the parameter estimation errors, respectively. Then, it is easy to see that ε is governed by (2.2) ⎧ m ⎪ q(1−x) ⎪ ⎪ ε (x, t) = ε (x, t) − (c + q)e (1, t) + (θj (t) sin αj t k ε tt xx 0 t ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ j (t) cos αj t) + qε(1, t) , + ϑ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ εx (0, t) = −qε(0, t), ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎨ εx (1, t) = −k εt (1, t) + (θj (t) sin αj t + ϑj (t) cos αj t) − (c0 + q)ε(1, t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

j=1

˙ θj (t) = −rj εt (1, t) +

m

(θj (t) sin αj t + ϑj (t) cos αj t) sin αj t, j ∈ J,

j=1 m ˙ (θj (t) sin αj t + ϑj (t) cos αj t) cos αj t, j ∈ J, ϑj (t) = −lj εt (1, t) + j=1

θj (0) = θ¯j − θj0 = θj0 , ϑj (0) = ϑ¯j − ϑj0 = ϑj0 ∈ R, j ∈ J, ε(x, 0) = ε0 (x), εt (x, 0) = ε1 (x),

where εi (x) = wi (x) − w i (x), i = 0, 1. We propose the following feedback controller based on the estimated state: 1 t (1, t) − (c1 + q)w(1, t) − (c1 + q) eq(1−ξ) [c4 w t (ξ, t) + q w(ξ, t)]dξ. (2.3) u(t) = −c2 w 0

Then the closed loop of system (2.1) corresponding to the controller (2.3) becomes ⎧ w tt (x, t) − w xx (x, t) = q(c0 + q)eq(1−x) ε(1, t) ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ q(1−x) ⎪ ⎪ + (c0 + q)e k εt (1, t) + (θj (t) sin αj t ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ + ϑj (t) cos αj t) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ w x (0, t) = −q w(0, t), ⎪ ⎪ ⎪ ⎪ ⎪ t (1, t) − (c1 + q)w(1, t) + (c0 + q)ε(1, t) w x (1, t) = −c2 w ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎨ + k εt (1, t) + (θj (t) sin αj t + ϑj (t) cos αj t) (2.4) j=1 1 ⎪ ⎪ ⎪ ⎪ ⎪ − (c + q) eq(1−ξ) [c2 w t (ξ, t) + q w(ξ, t)]dξ, 1 ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ m ⎪ ⎪ ˙ ⎪ ⎪ (θj (t) sin αj t + ϑj (t) cos αj t) sin αj t, j ∈ J, θj (t) = −rj εt (1, t) + ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ m ⎪ ˙ ⎪ ⎪ ⎪ ϑ (t) = −l (1, t) + ( θ (t) sin α t + ϑ (t) cos α t) cos αj t, j ∈ J, ε ⎪ 2 j t j j j j ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ θj (0) = θj0 , θj (0) = θj0 , j ∈ J, ⎪ ⎪ ⎩ t (x, 0) = w 1 (x). w(x, 0) = w 0 (x), w


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Make the invertible change of variables (2.5)

ε(x, t) = [(I + P1 )ε](x, t) = ε(x, t) − (c0 + q)

1

x

(2.6)

w(x, t) = [(I + P2 )w](x, t) = w(x, t) + (c1 + q)

ec0 (x−ξ) ε(ξ, t)dξ, x

0

eq(x−ξ) w(ξ, t)dξ.

Both P1 and P2 are Volterra transformations (see [11]). The inverse transformations (I + P1 )−1 and (I + P2 )−1 are given, respectively, by (2.7)

ε(x, t) = [(I + P1 )−1 ε](x, t) = ε(x, t) + (c0 + q) −1

t) = [(I + P2 ) (2.8) w(x,

1

x

w](x, t) = w(x, t) − (c1 + q)

e−q(x−ξ) ε(ξ, t)dξ, x

0

e−c1 (x−ξ) w(ξ, t)dξ.

Under transformation (2.5), (2.2) becomes

(2.9)

⎧ εtt (x, t) = εxx (x, t), ⎪ ⎪ ⎪ ⎪ ⎪ εx (0, t) = c0 ε(0, t), ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ (1, t) = −k ε (1, t) + ( θ (t) sin α t + ϑ (t) cos α t) , ε x t j j j j ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ m ⎨ ˙ θj (t) = −rj εt (1, t) + (θj (t) sin αj t + ϑj (t) cos αj t) sin αj t, j ∈ J, ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ m ⎪ ⎪ ˙ ⎪ ⎪ (θj (t) sin αj t + ϑj (t) cos αj t) cos αj t, j ∈ J, ϑj (t) = −lj εt (1, t) + ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ θj (0) = θj0 , ϑj (0) = θj0 , j ∈ J, ⎪ ⎪ ⎪ ⎩ ε(x, 0) = ε0 (x), εt (x, 0) = ε1 (x),

where (2.10)

εi (x) = εi (x) − (c0 + q)

1 x

ec0 (x−ξ) εi (ξ)dξ, i = 0, 1.

The motivation for the design of θj (t) and ϑj (t), j ∈ J is that the transformed error system (2.9) has the following Lyapunov function: 1 (2.11) Eε(t) = 2

0

1

c0 ε(0, t)]2 +k [ ε2t (x, t)+ ε2x (x, t)]dx+ [ 2

m 1 2 1 2 θ (t) + ϑ (t) . 2rj j 2lj j j=1

A formal computation along the solution of (2.9) gives ⎡

(2.12)

⎤2 m E˙ε(t) = −k ⎣εt (1, t) + θj (t) sin αj t + ϑj (t) cos αj t ⎦ ≤ 0. j=1



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Also under the transformation (2.6) and with (2.7), (2.4) becomes

(2.13)

⎧ w tt (x, t) = w xx (x, t) ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ + (c0 + q)m(x) q ε(1, t) + k εt (1, t) + (θj (t) sin αj t ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ + ϑj (t) cos αj t) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ t), w x (0, t) = c1 w(0, ⎪ ⎪ ⎪ ⎪ ⎪ w (1, t) = −c t (1, t) + (c0 + q) ε(1, t) ⎪ 2w ⎪ ⎨ x m + k ε (1, t) + ( θ (t) sin α t + ϑ (t) cos α t) , ⎪ t j j j j ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ m ⎪ ˙ ⎪ ⎪ ⎪ (t) = −r (1, t) + ( θ (t) sin α t + ϑ (t) cos α t) sin αj t, j ∈ J, θ ε j t j j j j ⎪ j ⎪ ⎪ ⎪ j=1 ⎪ ⎪ m ⎪ ⎪ ⎪ ˙ ⎪ ⎪ ϑ (t) = −l (1, t) + ( θ (t) sin α t + ϑ (t) cos α t) cos αj t, j ∈ J, ε j j t j j j j ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ θj (0) = θj0 , ϑj (0) = ϑj0 , j ∈ J, ⎪ ⎪ ⎪ ⎩ w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x),

where m(x) = eq(1−x) + (c1 + q)eq(x+1) (2.14)

x

e−2qξ dξ

0

⎧ ⎨ 1 + c1 x, q = 0, q = ⎩ e [q cosh(qx) + c1 sinh(qx)], q = 0, q

and (2.15)

i (x) + (c1 + q) w i (x) = w

0

x

eq(x−ξ) w i (ξ)dξ, i = 0, 1.

The recommended choices for the control gains are c2 around one, and c0 , c1 relatively large. 3. Main results. First, we make use of the Galerkin method to solve the existence and the uniqueness of the solution of (2.9). To do this, we need a basis to construct the Galerkin approximation, which can be realized by the operator A in L2 (0, 1) as follows: (3.1) Aφ = −φ ∀ φ ∈ D(A) = φ ∈ L2 (0, 1), φ (0) = c0 φ(0), φ (1) = 0 . Then A is an unbounded self-adjoint positive definite operator in L2 (0, 1) with compact resolvent. A simple computation shows that the eigenpairs {(λn , φn )}∞ n=1 are (3.2)

⎧ ⎨ λn = ωn2 , ωn = nπ + O(n−1 ), c ⎩ φn (x) = 0 sin ωn x + cos ωn x = cos nπx + O(n−1 ). ωn



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Since {φn }∞ n=1 defined by (3.2) is approximately normalized, that is 0 < l1 < φn L2 (0,1) < l2 for some constants l1 , l2 independent of n, it forms an (orthogonal) Riesz basis for L2 (0, 1). Let V = H 3 (0, 1) ∩ D(A). We can follow the same steps as that in [8] to construct a Galerkin scheme to prove the existence and uniqueness of the classical solution to error system (2.9). Theorem 3.1. Suppose that ( ε0 , ε1 , θ10 , ϑ10 , . . . , θm0 , ϑm0 ) ∈ V × V × R2m and satisfies the following compatible conditions: ε1 (1) +

(3.3)

m

ϑj0 = 0

j=1

and ε0 (1) +

(3.4)

m

αj θj0 = 0.

j=1

Then there exists a unique (smoother) classical solution to (2.9) in the sense that for any time T > 0, ⎧ ε ∈ L∞ (0, T ; H 3(0, 1)), εt ∈ L∞ (0, T ; H 2 (0, 1)), εtt ∈ L∞ (0, T ; H 1 (0, 1)), ⎪ ⎪ ⎪ ⎪ 1 ⎪ j ∈ C 1 [0, T ], j ∈ J, ⎪ ⎪ θj ∈ C [0, T ], ϑ ⎪ ⎪ ⎪ ⎪ εtt (x, t) − εxx (x, t) = 0 in L∞ (0, T ; L2(0, 1)), ⎪ ⎪ ⎪ ⎪ ⎪ εx (0, t) = c0 ε(0, t), ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ ε (1, t) = −k ε (1, t) + (t) sin α t + ϑ (t) cos α t , θ t j j j j ⎪ ⎨ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

j=1

m ˙ θj (t) = −rj εt (1, t) + θj (t) sin αj t + ϑj (t) cos αj t sin αj t, j ∈ J,

j=1

m ˙ ϑj (t) = −lj εt (1, t) + θj (t) sin αj t + ϑj (t) cos αj t cos αj t, j ∈ J,

j=1

θj (0) = θj0 , ϑj (0) = ϑj0 , j ∈ J, ε(x, 0) = ε0 (x), εt (x, 0) = ε1 (x).

By the Sobolev embedding theorem, it follows that ε ∈ C([0, 1] × [0, T ]). Remark 3.1. Let us remark why the Galerkin method is necessary for the proof of Theorem 3.1. Actually, let L2 (0, 1) be the usual Hilbert space with the inner product ·, · , and inner product induced norm · . We consider (2.9) and (1.2) in the energy space H = H 1 (0, 1) × L2 (0, 1) × R4m = V × R2m with the inner product (f, g, θ1 , ϑ1 , . . . , θm , ϑm , ξ1 , η1 , . . . , ξm , ηm ), (f, g, θ1 , ϑ1 , . . . , θm , ϑm , ξ1 , η1 , . . . , ξm , ηm ) H 1 1 m ϑj ϑj θj θj = f (x)f (x)dx + g(x) g (x)dx + c0 f (0)f(0) + k + rj lj 0 0 j=1 +

m j=1

(ξj ξj + ηj ηj ) ∀ (f, g, θ1 , ϑ1 , . . . , θm , ϑm , ξ1 , η1 , . . . , ξm , ηm ), (f, g, θ1 , ϑ1 , . . . , θm , ϑm , ξ1 , η1 , . . . , ξm , ηm ) ∈ H.




1687

Define the operator A : D(A)(⊂ H) → H as follows: (3.5) ⎧ A(u, v, θ1 , ϑ1 , . . . , θm , ϑm , ξ1 , η1 , . . . , ξm , ηm ) ⎪ ⎪ ⎪ ⎛ ⎤ ⎤ ⎡ ⎡ ⎪ ⎪ m m ⎪ ⎪ ⎪ ⎪ = ⎝v, u , −r1 ⎣v(1) + (θj ξj + ϑj ηj )⎦ ξ1 , −l1 ⎣v(1) + (θj ξj + ϑj ηj )⎦ η1 , . . . , ⎪ ⎪ ⎪ ⎪ j=1 j=1 ⎪ ⎪ ⎪ ⎡ ⎡ ⎤ ⎤ ⎪ ⎪ m m ⎪ ⎪ ⎪ ⎪ − rm ⎣v(1) + (θj ξj + ϑj ηj )⎦ ξm , −lm ⎣v(1) + (θj ξj + ϑj ηj )⎦ ηm , ⎪ ⎪ ⎪ ⎪ j=1 j=1 ⎪ ⎨ ⎞ ⎪ α1 η1 , −α1 ξ1 , . . . , αm ξm , −αm ηm ⎠ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ D(A) = (u, v, θ1 , ϑ1 , . . . , θm , ϑm , ξ1 , η1 , . . . , ξm , ηm ) ∈ (H 2 (0, 1) ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎡ ⎤⎫ ⎪ ⎪ m ⎪ ⎬ ⎪ ⎪ ⎪ (θj ξj + ϑj ηj )⎦ . × H 1 (0, 1) × R4m | f (0) = c0 f (0), f (1) = −k ⎣v(1) + ⎪ ⎪ ⎩ ⎭ j=1

Then the systems (1.2) and (2.9) can be written as a nonlinear evolution equation d z(·, t) = Az(·, t), z(·, 0) = z0 (·) ∈ H, dt

(3.6)

where ! z(x, t) = ε(x, t), εt (x, t), θ1 (t), ϑ1 (t), . . . , θm (t), ϑm (t), ξ1 (t), η1 (t), . . . , ξm (t), ηm (t) , ε0 (x), ε1 (x), θ10 , ϑ10 , . . . , θm0 , ϑm0 , ξ10 , η10 , . . . , ξ10 , η10 ). z0 (x) = ( Obviously, (3.6) is an autonomous revolution equation. However, same as [7], it seems hard to use a nonlinear semigroup to prove its well-posedness due to lack of dissipativity of A defined by (3.5) or A + ωI for any constant ω ∈ R. Next, we establish the convergence of the transformed error system (2.9). To do this, we need the weak solution of (2.9). Definition 3.2. For any initial data ( ε0 , ε1 , θ10 , ϑ10 , . . . , θm0 , ϑm0 ) ∈ V , the weak solution ( ε, εt , θ1 , ϑ1 , . . . , θm , ϑm ) of (2.9) is defined as the limit of any convergent n n subsequence of ( εn , εnt , θ1n , ϑn1 , . . . , θm , ϑm ) in the space L∞ (0, ∞; H 1 (0, 1) × L2(0, 1) × 2m n n n n n n R ), where ( ε , εt , θ1 , ϑ1 , . . . , θm , ϑm ) is the classical solution ensured by Theorem 3.1 with the initial condition n (0), ϑnm (0)) ( εn (x, 0), εnt (x, 0), θ1n (0), ϑn1 (0), . . . , θm n n n , ϑ10 , . . . , θm0 , ϑnm0 ) ∈ V × V × R2m ∀x ∈ (0, 1) = ( εn0 , εn1 , θ10

satisfying n n n lim ( εn0 , εn1 , θ10 , ϑ10 , . . . , θm0 , ϑnm0 ) − ( ε0 , ε1 , θ10 , ϑ10 , . . . , θm0 , ϑm0 )V = 0.

n→∞

Remark 3.2. Definition 3.2 makes sense because from (2.11) and (2.12) we n n , ϑm )} must be a Cauchy sequence in L∞ (0, ∞; V) know that {( εn , εnt , θ1n , ϑn1 , . . . , θm and its limit does not depend on the choice of the initial values. Consequently,



1688


( ε, εt , θ1 , ϑ1 , . . . , θm , ϑm ) ∈ C(0, ∞; V). Moreover, by (2.12), this solution depends continuously on its initial value. Theorem 3.3. Suppose that c0 > max{αj / cot αj , j ∈ J}

(3.7)

in observer (2.2), Then for any initial value ( ε0 , ε1 , θ10 , ϑ10 , . . . , θm0 , ϑm0 ) ∈ V, the solution of the system (2.9) is asymptotically stable in the sense that 1 1 c0 [ ε2t (x, t) + ε2x (x, t)]dx + ε2 (0, t) = 0. lim t→∞ 2 0 2 Moreover, for any j ∈ J, lim θj (t) = θ¯j , lim ϑj (t) = ϑ¯j .

t→∞

t→∞

Now, we consider the transformed system (2.13) without dynamic equations for θj (t) and ϑj (t), j ∈ J since they have been determined by the transformed error system (2.9) already. The system now reads ⎧ w tt (x, t) = w xx (x, t) ⎪ ⎪ ⎪ ⎪ "m ⎪ ⎪ ⎪ ε(1, t) + k εt (1, t) + j=1 (θj (t) sin αj t + (c0 + q)m(x) q ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ϑj (t) cos αj t) , ⎨ (3.8) ⎪ ⎪ w x (0, t) = c1 w(0, t), ⎪ ⎪ ⎪ ⎪ ⎪ t (1, t) + (c0 + q) ε(1, t) w x (1, t) = −c2 w ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ + k ε (1, t) + ( θ (t) sin α t + ϑ (t) cos α t) . ⎪ t j j j j ⎩ j=1

We consider system (3.8) in the energy space H = H 1 (0, 1) × L2 (0, 1). The norm of H is induced by the inner product 1 1 (p, q)2H = [p (x)]2 dx + [q(x)]2 dx + c1 [p(0)]2 ∀ (p, q) ∈ H. 0

0

Define the operate A : D(A)(⊂ H) → H as follows: A(p, q) = (q, p ) ∀ (p, q) ∈ D(A), (3.9) D(A) = {(p, q) ∈ H| A(p, q) ∈ H, p (0) = c1 p(0), p (1) = −c2 q(1)}. It is well known that A generates a C0 -semigroup of contractions eAt on H ([5]). From (3.9), it is readily found that ! ∗ A (φ, ψ) = (−ψ, −φ ) ∀ (φ, ψ) ∈ D(A∗ ), (3.10) D(A∗ ) = {(φ, ψ) ∈ H| A∗ (φ, ψ) ∈ H, φ (0) = c1 φ(0), φ (1) = c2 ψ(1)}. Take the inner product of (φ, ψ) ∈ D(A∗ ) with (3.8) to obtain % # % # % # d φ w φ 0 φ w ∗ , = ,A + , ψ w $t ψ f (x, t) ψ w $t dt % # 0 φ + g(t), , −δ(x − 1) ψ


1689



where ⎧ ⎪ ⎪ f (x, t) = (c + q)m(x) q ε(1, t) + k( εt (1, t) 0 ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎨ + θj (t) sin αj t + ϑj (t) cos αj t , (3.11) j=1 ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ε(1, t) + k εt (1, t) + θj (t) sin αj t + ϑj (t) cos αj t , ⎪ ⎩ g(t) = (c0 + q) j=1

and δ(·) denotes the Dirac distribution. Hence (3.8) is equivalent to ⎧ xx (x, t) = f (x, t) − δ(x − 1)g(t), w tt (x, t) − w ⎪ ⎪ ⎪ ⎨ w x (0, t) = c1 w(0, t), ⎪ w t (1, t), x (1, t) = −c2 w ⎪ ⎪ ⎩ w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x),

(3.12)

or in abstract form:

(3.13)

d dt

w(·, t) w $t (·, t)

w(·, t) w $t (·, t)

=A

w(·, t) w $t (·, t)

=A

+

+

0 f (·, t) 0 f (·, t)

+

0 −δ(x − 1)

g(t)

+ Bg(t), t > 0,

where B = (0, −δ(x − 1)) . Remark 3.3. In the above setting, we understand the solution of (3.13) in D(A∗ ) ˜ which is an extension of A defined by by identifying A with A ˜ = H, AF, ˜ G D(A∗ ) ×D(A∗ ) = F, A∗ G H ∀ G ∈ D(A∗ ), D(A) where D(A∗ ) is the dual space of D(A∗ ) with respective to the pivot space H. Theorem 3.4. For any initial value (w 0 , w 1 ) ∈ H, there exists a unique solution (w, w t ) ∈ C(0, ∞; H) to (3.8), and for all T > 0, there exists a DT > 0 depending on T only such that ! (w(·, T ), w t (·, T ))2H

≤ DT

1 )2H (w 0 , w

T

+ 0

& 2

[|g(τ )| + f (·, τ )]dτ

.

1 ) ∈ D(A) and initial condition ( ε0 , ε1 , θ10 , ϑ10 , . . . , θm0 , Moreover, for each (w 0 , w 2m ϑm0 ) ∈ V × V × R with ε0 (1) = 0 given by Theorem 3.1, there exists a classical solution (w, w t ) ∈ C 1 (0, ∞; H) to (3.8). Theorem 3.5. The transformed system (3.8) is asymptotically stable, that is, for 1 ) ∈ H, the (weak) solution of (3.8) justified by Theorem 3.4 satisfies any (w 0 , w 1 t→∞ 2

lim Ew (t) = lim

t→∞

0

1

[w x2 (x, t) + w t2 (x, t)]dx + c1 [w(0, t)]2 = 0.


1690



We go back to the closed-loop system of (1.1) under the feedback (2.3): (3.14) ⎧ wtt (x, t) − wxx (x, t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ wx (0, t) = −qw(0, t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ wx (1, t) = −c2 w t (1, t) − (c1 + q)w(1, t) ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ − (c1 + q) eq(1−ξ) [c2 w t (ξ, t) + q w(ξ, t)]dξ, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ w(x, 0) = w0 (x), wt (x, 0) = w1 (x), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ tt (x, t) − w xx (x, t) = q(c0 + q)eq(1−x) [w(1, t) − w(1, t)] ⎪ w ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (c0 + q)eq(1−x) k wt (1, t) − w t (1, t) ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ ⎪ + (θj (t) sin αj t + ϑj (t) cos αj t) , ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ ⎪ ⎪ ⎪ w (0, t) = −q w(0, t), ⎪ ⎨ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

w x (1, t) = −c2 w t (1, t) − (c1 + q)w(1, t) + (c0 + q)[w(1, t) − w(1, t)] m t (1, t) + (θj (t) sin αj t + ϑj (t) cos αj t) + k wt (1, t) − w − (c1 + q)

0

j=1 1

eq(1−ξ) [c2 w t (ξ, t) + q w(ξ, t)]dξ,

⎡ ⎤ m ˙ θj (t) = −rj ⎣wt (1, t) − w t (1, t) + (θj (t) sin αj t + ϑj (t) cos αj t)⎦ sin αj t, j=1

⎡ ⎤ m ˙ t (1, t) + (θj (t) sin αj t + ϑj (t) cos αj t)⎦ cos αj t, ϑj (t) = −lj ⎣wt (1, t) − w j=1

θj (0) = θj0 , ϑj (0) = ϑj0 , j ∈ J, w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x).

We consider system (3.14) in the state space X = (H 1 (0, 1) × L2 (0, 1))2 × R2m . Theorem 3.6. For any initial value (w0 , w1 , w 0 , w 1 , θ10 , ϑ10 , . . . , θm0 , ϑm0 ) ∈ X , t), w t (·, t), there exists a unique (weak) solution to (3.14) such that (w(·, t), wt (·, t), w(·, θ1 (t), ϑ1 (t), . . . , θm (t), ϑm (t)) ∈ C([0, ∞); X ). Moreover, the closed-loop solution (w, w, θ1 , ϑ1 , . . . , θm , ϑm ) of (3.14) is asymptotically stable in the sense that 1 lim [wx2 (x, t) + wt2 (x, t) + w x2 (x, t) + w t2 (x, t)]dx + c0 w2 (0, t) + c0 w 2 (0, t) t→∞

0

+

m

[θj2 (t) + ϑ2j (t)] = 0.

j=1



1691


Moreover, lim θj (t) = θ¯j , lim ϑj (t) = ϑ¯j , j = 1, 2 . . . , m.

t→∞

t→∞

4. Proof of the main results. This section is devoted to the proof of the main results stated in the previous section. Proof of Theorem 3.3. By the density argument, we may regard without loss of generality that the initial value ( ε0 , ε1 , θ10 , ϑ10 , . . . , θm0 , ϑm0 ) belongs to V × V × R2m . Construct Lyapunov functional Vε(t) for the system (3.6) following:

Vε(t) =

1 2 +

1

0

[ ε2t (x, t) + ε2x (x, t)]dx +

m j=1

m c0 k 2 θj (t) [ ε(0, t)]2 + 2 2r j j=1

m k 2 2 [ξj (t) + ηj2 (t)], ϑj (t) + 2lj j=1

where ξj (t) = sin αj t, ηj (t) = cos αj t, j ∈ J. The time derivative of Vε(t) along the solution of system (3.6) is found to be ⎡

⎤2 m V˙ ε(t) = −k ⎣εt (1, t) + θj (t) sin αj t + ϑj (t) cos αj t ⎦ ≤ 0. j=1

This shows that Vε(t) ≤ Vε(0) and hence ⎡ (4.1)

1 sup ⎣ 2 t≥0

0

1

⎤ m c 0 ε(0, t)]2 + [ ε2t (x, t) + ε2x (x, t)]dx + [ (|θj (t)| + |ϑj (t)|)⎦ < ∞. 2 j=1

Hence each trajectory is bounded and in particular

(4.2)

h(t) = εt (1, t) +

m

[θj (t) sin αj t + ϑj (t) cos αj t] ∈ L2 (0, ∞).

j=1

Similarly, let 1 U (t) = 2

0

1

[ ε2xx (x, t) + ε2tx (x, t)]dx +

c0 [ εt (0, t)]2 . 2

It is found that the time derivative of U (t) along the solution of error system (2.9)


1692



can be estimated as U˙ (t) = −k[ εtt (1, t)]2 − k

m

αj θj (t) cos αj t − ϑj (t) sin αj t εtt (1, t)

j=1

(4.3) +k

m

(rj sin2 αj t + lj cos2 αj t) εtt (1, t)h(t).

j=1

Let (t) = 0

1

x εxx (x, t) εxt (x, t)dx.

It is seen that |(t)| ≤ M U (t) for some positive constant M . A simple computation shows that 1 1 1 εtt (1, t)]2 + [ εtx (1, t)]2 − (t) ˙ = [ 2 2 2

0

1

[ ε2xx (x, t) + ε2tx (x, t)]dx

⎡ ⎤ 2 m 1 1 εtt (1, t)]2 + k 2 ⎣ αj θj (t) cos αj t − ϑj (t) sin αj t ⎦ = (1 + k 2 )[ 2 2 j=1 ⎡ ⎤2 m 1 2 ⎣ rj sin2 αj t + lj cos2 αj t ⎦ h2 (t) + k 2 j=1

(4.4)

2

+ k εtt (1, t)

m

αj θj (t) cos αj t − ϑj (t) sin αj t

j=1

m 2 2 − k h(t) εtt (1, t) rj sin αj t + lj cos αj t 2

j=1

− k 2 h(t) ×

m j=1

m

αj θj (t) cos αj t − ϑj (t) sin αj t

j=1

1 rj sin αj t + lj cos αj t − 2 2

2

1 0

[ ε2xx (x, t) + ε2tx (x, t)]dx.

Define U ∗ (t) = U (t) + α(t), α > 0. Obviously, (4.5)

(1 − αM )U (t) ≤ U ∗ (t) ≤ (1 + αM )U (t).




1693

By (4.3) and (4.4), we have the estimate for U˙ ∗ (t): (4.6) m U˙ ∗ (t) = −k[ εtt (1, t)]2 − k αj θj (t) cos αj t − ϑj (t) sin αj t εtt (1, t) j=1 m

+k

(rj sin2 αj t + lj cos2 αj t) εtt (1, t)h(t) +

j=1

α (1 + k 2 )[ εtt (1, t)]2 2

⎡ ⎤ 2 m α 2 ⎣ + k αj θj (t) cos αj t − ϑj (t) sin αj t ⎦ 2 j=1 ⎡ ⎤ 2 m α 2 ⎣ 2 + k εtt (1, t) rj sin αj t + lj cos2 αj t ⎦ h2 (t) + k 2 α 2 j=1 ×

m

αj θj (t) cos αj t − ϑj (t) sin αj t − k 2 αh(t) εtt (1, t)

j=1 m × rj sin2 αj t + lj cos2 αj t j=1 2

− k αh(t) −

α 2

0

m

m 2 2 αj θj (t) cos αj t − ϑj (t) sin αj t rj sin αj t + lj cos αj t

j=1 1

j=1

[ ε2xx (x, t) + ε2tx (x, t)]dx

1 1 γ α 1 2 εxx (x, t)dx − (α − 2γc0 ) ε2xt (x, t)dx − c0 ε2t (0, t) 2 0 2 2 0 (1 + k 2 )α − k − 2ηk − [ εtt (1, t)]2 2

≤−

⎤ ⎡ 2 m η α 2 η 2 2 k 2 α ⎣ + k+ k + k α + αj θj (t) cos αj t − ϑj (t) sin αj t ⎦ 2 2 2 2 j=1 ⎧⎡ ⎫ ⎤2 ⎪ ⎪ m ⎨ ⎬ 2 2 k α 2 k α k α ⎣ ⎦ + k + + + (rj + lj ) + γc0 h2 (t) ⎪ ⎪ 2η 2 2η 2 ⎩ j=1 ⎭ ⎡ ⎤2 m + γc0 ⎣ (θj (t) sin αj t + ϑj (t) cos αj t)⎦ .

j=1

Take three positive constants η, γ, and α in (4.6) so that η < 12 , and

(4.7)

α < α < min γ< 1 + 2c0

1 2k(1 − 2η) , M 1 + k2

.


1694



Then we have

η α 2 η 2 2 k2 α k+ k + k α + 2 2 2 2 ⎤ ⎡ 2 m αj θj (t) cos αj t − ϑj (t) sin αj t ⎦ ×⎣

U˙ ∗ (t) ≤ −γU (t) +

j=1

(4.8)

⎧⎡ ⎫ ⎤2 ⎪ ⎪ m ⎨ ⎬ 2 2 α 2 k α k α k ⎣ ⎦ + k + + + (rj + lj ) + γc0 h2 (t) ⎪ ⎪ 2η 2 2η 2 ⎩ j=1 ⎭ ⎡ ⎤2 m + γc0 ⎣ (θj (t) sin αj t + ϑj (t) cos αj t)⎦ . j=1

Applying Gronwall’s inequality to (4.8) and in view of (4.5), we have the estimate U ∗ (t) ≤ e

γ − 1+αM t

U ∗ (0) η α 2 η 2 2 k2 α k+ k + k α + + 2 2 2 2 ⎡ ⎤ 2 t m γ(t−s) − e 1+αM ⎣ αj θj (s) cos αj s − ϑj (s) sin αj s ⎦ ds ×

0

(4.9)

j=1

⎧⎡ ⎫ ⎤2 ⎪ ⎪ m ⎨ ⎬ 2 2 k α 2 k α k α ⎣ ⎦ + k + + + (rj + lj ) + γc0 ⎪ ⎪ 2η 2 2η 2 ⎩ j=1 ⎭ t γ(t−s) − × e 1+αM h2 (s)ds 0 ⎡ ⎤2 t m γ(t−s) − + γc0 e 1+αM ⎣ (θj (s) sin αj s + ϑj (s) cos αj s)⎦ ds. 0

j=1

It is found from (4.1), (4.2), (4.5), (4.7), and (4.9) that sup U (t) < ∞, t≥0

which implies that the trajectory of system (3.6) ε, εt , θ1 (t), ϑ1 (t), . . . , θm (t), ϑm (t), ξ1 (t), η1 (t), . . . , ξm (t), ηm (t))|t ≥ 0} γ(z0 ) = {( is precompact in H 1 (0, 1) × L2 (0, 1) × R4m . By Remark 3.2, (3.6) produces a dynamic system. In light of Lasalle’s invariance principle ([25, p. 168]), any solution of system (3.6) tends to, as time goes to infinity, the maximal invariant set of the following: S = {( ε, εt , θ1 (t), ϑ1 (t), . . . , θm (t), ϑm (t), sin α1 t, cos α1 t, . . . , sin αm t, cos αm t) ∈ H 1 (0, 1) × L2 (0, 1) × R4m |V˙ ε(t) = 0}.




1695

By V˙ ε(t) = 0, it follows that h(t) = 0, θj ≡ θj0 , and ϑj ≡ ϑj0 , j ∈ J. Hence we have, in this case, that ⎧ εtt (x, t) = εxx (x, t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ εx (0, t) = c0 ε(0, t), εx (1, t) = 0, (4.10) m ⎪ ⎪ ⎪ ⎪ [θj0 sin αj t + ϑj0 cos αj t] = 0. εt (1, t) + ⎪ ⎩ j=1

Now we show that (4.10) admits zero solution only. To this end, we first consider the equation ⎧ ⎨ εtt (x, t) = εxx (x, t), εx (0, t) = c0 ε(0, t), (4.11) ⎩ εx (1, t) = 0. System (4.11) is a conservative system in Hilbert space H0 = H 1 (0, 1) × L2 (0, 1) with the inner product 1 (y1 , z1 ), (y2 , z2 ) H0 = [y1 (x)y2 (x) + z1 (x)z2 (x)]dx + c0 y1 (0)y2 (0). 0

Define a linear operator A0 associated with system (4.11), A0 (y, z) = (z, y ), (4.12) D(A0 ) = {(y, z) ∈ H 2 (0, 1) × H 1 (0, 1)|y (0) = c0 y(0), y (1) = 0}. It is a simple exercise to show that A−1 is compact on H0 . That is, A0 is a skew0 adjoint operator with compact resolvent on H0 . Consequently, the spectrum of A0 consists of isolated eigenvalues on the imaginary axis only, and from a general result of functional analysis, the algebraic multiplicity of each eigenvalue of A0 is equal to its geometric multiplicity. For any λ ∈ σp (A0 ), solving the eigenvalue problem A0 (φ, ψ) = λ(φ, ψ), one has ψ = λφ with φ = 0 satisfying 2 λ φ(x) − φ (x) = 0, (4.13) φ (0) = c0 φ(0), φ (1) = 0. Solving (4.13) gives (4.14)

φ(x) =

c0 + λ λx e + e−λx λ − c0

with (4.15)

c0 + λ λ e − e−λ = 0 λ − c0

or (4.16)

e2λ = −

c0 − λ . c0 + λ

So λ is geometrically simple. So each λ is algebraically simple.



1696


Obviously, β = 0 is not one solution of (4.16). If β = kπ + π2 , k ∈ Z, then π ei2β = −1 = − cc00 −iβ +iβ . Thus iβ = i(kπ + 2 ), k ∈ Z is not the solution of (4.16). Similarly, It is seen that iβ = ikπ, k ∈ Z is not the solutions of (4.16). For given β = 0, kπ + π2 , kπ, k ∈ Z, it is easy to show that iβ is a solution to (4.16) if and only if β satisfies (4.17)

cot β =

β . c0

In fact, assume β = 0, kπ + π2 , kπ, k ∈ Z and β is a solution to (4.16). Then 2β = arg

β iβ − c0 + 2kπ = −2 arctan + π + 2kπ. c0 + iβ c0

Hence (4.18)

β=

β π − arctan + kπ. 2 c0

From (4.18), one has cot β = If cot β =

β . c0

β , then c0 β=

π β π − arctan + kπ = − arg(c0 + iβ) + kπ. 2 c0 2

Thus ei2β = −e−2i arg(c0 +iβ) = −e−i arg(c0 +iβ) e−i arg(c0+iβ) i arg

c0 −iβ

0 −iβ) c0 +iβ = −e−i arg(c0 +iβ) ei arg(c ( = −e ( ' ( c0 − iβ ( c c0 − iβ − iβ 0 ( ( =− ( c0 + β ( = − c0 + iβ , c0 + iβ

which implies that iβ is a solution to (4.16). Hence, if we choose c0 satisfying condition (3.7), c0 > max{αj / cot αj , j ∈ J} in observer (2.2), then iαj + c0 iαj e − e−iαj = 0, j ∈ J. iαj − c0

(4.19)

Furthermore, from (4.14) and (4.15), we can obtain the following asymptotic expressions of eigenpairs of A0 : λn = nπi + O(n−1 ), φn (x) = cos nπx + O(n−1 ).

(4.20) Define ! (4.21)

λn = nπi + O(n−1 ), λ−n = λn , −1 Φn = (λ−1 n φn , φn ), Φ−n = (λ−n φn , φn ), n = 1, 2, . . . .



1697


By the general theory of functional analysis, {Φn }n∈Z forms an orthogonal basis for H0 . Therefore, the solution of (4.10) can be represented as ∞

( ε(·, t), εt (·, t)) =

∞

an eλn t Φn +

n=1

a−n eλ−n t Φ−n ,

n=1

where the constants {an }n∈Z are determined by the initial condition. That is, ε0 =

∞ ∞ ∞ ∞ an a−n φn + φn , ε1 = an φn + a−n φn . λ λ n=1 n n=1 −n n=1 n=1

It is easy to show that εt (x, t) converges in H 1 (0, 1) uniformly in t. By the continuity of the trace operator εt → εt (1, t) in H 1 (0, 1), the equation εt (1, t) +

m

[θj0 sin αj t + ϑj0 cos αj t] = 0

j=1

reads as ∞

an φn (1)eλn t +

n=1

∞

a−n φn (1)eλ−n t +

n=1

m

[θj0 sin αj t + ϑj0 cos αj t] = 0.

j=1

Therefore, ∞

an φn (1)eλn t +

n=1

∞

a−n φn (1)eλ−n t

n=1

* 1 ) (ϑj0 − iθj0 )eiαj t + (ϑj0 + iθj0 )e−iαj t = 0. + 2 j=1

(4.22)

m

We claim that a±n = 0 for all n ≥ 1. Since otherwise, if there exists an n0 such that |an0 φ" 0 ∈ H 1 (0, 1) guarantees n0 (1)| = 0, then the smoothness of the initial value that ε that n∈Z |an φn (1)| < ∞, which implies that there exists an integer N > n0 such that (4.23)

∞ n=N

∞ 1 1 |an φn (1)| < |an0 φn0 (1)| and |a−n φ−n (1)| < |an0 φn0 (1)| . 4 4 n=N

Since λn = λm for any n, m ∈ Z, n = m and limn→∞ |λn+1 − λn | = π, n ∈ Z, one has, for t > 0, ∞

an0 φn0 (1) +

(4.24) +

n=N +1

N

+

an φn (1)e(λn −λn0 )t

an φn (1)e(λn −λn0 )t

n=1,n =n0 ∞

a−n φ−n (1)e(λ−n −λn0 )t +

n=N +1 m )

1 + 2

N

a−n φ−n (1)e(λ−n −λn0 )t

n=1

* (ϑj0 − iθj0 )e(iαj −λn0 )t + (ϑj0 + iθj0 )e−(iαj +λn0 )t = 0.

j=1



1698


Integrating both sides of (4.24) and using the fact Reλn = 0 and (4.23), we obtain ( ( ( t ( N ( ( an φn (1)e(λn −λn0 )s ds(( |an0 φn0 (1)| t ≤ 2 (( ( 0 n=1,n =n0 ( ( N ( ( t ( ( (λ−n −λn0 )s ( + 2( a−n φ−n (1)e ds( ( 0 ( n=1 ( ( m ( t ( (iαj −λn0 )s −(iαj +λn0 )s ( + + (ϑj0 + iθj0 )e ]ds(( . ( [(ϑj0 − iθj0 )e j=1

0

Since the right side of the above inequality has an upper bound for all t ≥ 0, we get that an0 = 0, which is a contradiction. By (4.22), a±n = 0, n = 1, 2, . . . , θj0 = 0, and ϑj0 = 0, j ∈ J. The proof is complete. Proof of Theorem 3.4. We first prove the first part. By the well-posed linear ∗ infinite-dimensional systems theory, it suffices to show that B∗ is admissible for eA t ∗ ∗ −1 (see, e.g., Theorem 6.9 of [26]). This is equivalent to saying that B A is bounded, and for any T > 0, there exists an MT > 0 depending on T only such that the system ⎧ xx (x, t) = 0, w tt (x, t) − w ⎪ ⎪ ⎨ t), w x (0, t) = c1 w(0, (4.25) w (1, t) = −c t (1, t), ⎪ x 2w ⎪ ⎩ yw (t) = −w t (1, t), satisfies

T

0

where Ew (t) =

1 2

0

1

[w t (1, t)]2 dt ≤ MT Ew (0),

[w x2 (x, t) + w t2 (x, t)]dx +

c1 [w(0, t)]2 . 2

First, a simple computation shows that + 1 1 1 ∗ −1 (φ, ψ) = − c2 φ(1) x − ψ(τ )dτ A + c1 c1 0 , x 1 + τ ψ(τ )dτ + xψ(τ )dτ, −φ(x) , 0

∗

∗ −1

B A

x

(φ, ψ) = (0, φ(1)) ∀ (φ, ψ) ∈ H.

Hence B∗ A∗ −1 is bounded on H. Second, differentiate Ey(t) with respect to t along the solution of (4.25) to give E˙w (t) = −c2 [w t (1, t)]2 ≤ 0 -T and hence, Ew (T ) ≤ Ew (0) for any T > 0 and 0 [w t (1, t)]2 dt ≤ c12 Ew (0). The first part is proved. Now we show the classical solution. First, we claim that (4.26)

g ∈ H 1 (0, T ) ∀ T > 0.




1699

Actually, from the definition of g given by (3.11), one has m g(t) ˙ = (c0 + q) εt (1, t) + k εtt (1, t) + k αj (θj (t) sin αj t − ϑj (t) cos αj t) −k

m

j=1

m 2 2 (rj sin αj t + lj cos αj t) εt (1, t) + θj (t) sin αj t + ϑj (t) cos αj t .

j=1

j=1

-1

By Theorem 3.1, | εtt (1, t)| = | εtt (0, t) + 0 εttx (x, t)dx| ≤ ¨ εL∞ (0,T ;H 1 (0,1)) for any t ∈ [0, T ], and so are for lower-order terms of ε. Furthermore, by Theorem 3.3, limt→∞ θj (t) = θ¯j , limt→∞ ϑj (t) = ϑ¯j , θj (t) = θ¯j − θj (t), ϑj (t) = ϑ¯j − θj (t), j ∈ J. So (4.26) is valid. $ (·, t) = (w(·, Denote W t), w t (·, t)) and F (s) = (0, f (·, s)) . Then the solution of (3.13) can be written as t t At $ A(t−s) $ e F (s)ds + eA(t−s) Bg(s)ds. (4.27) W (·, t) = e W (·, 0) + 0

0

$ (·, 0) = (w(·, For W 0), w t (·, 0))T ∈ D(A), it follows from (4.27) that t d A(t−s) −1 $ (·, t) = eAt W $ (·, 0) − W [e ]A F (s)ds 0 ds t d A(t−s) −1 $ (·, 0) [e − ]A Bg(s)ds = eAt W ds 0 (4.28) t − A−1 F (t) + eAt A−1 F (0) + A−1 −1

−A

Bg(t) + e

At

−1

A

0 −1

eA(t−s) Fs (s)ds

Bg(0) + A

t

eA(t−s) Bg(s)ds. ˙

0

By the fact A−1 B is bounded and (4.26), the right-hand side of (4.28) makes sense. ε0 (1)− Notice that the compatible condition (3.3) and ε0 (1) = 0, so g(0) = (c0 +q) "m k[ ε1 (1) + j=1 θj0 ] = 0. Thus (4.28) becomes (4.29)

$ (·, t) = eAt W $ (·, 0) − A−1 F (t) − A−1 Bg(t) W t t −1 A(t−s) −1 +A e Fs (s)ds + A eA(t−s) Bg(s)ds. ˙ 0

0

Differentiate (4.29) with respect to t to give t t d$ At A(t−s) $ (4.30) W (·, t) = Ae W (·, 0) + e Fs (s)ds + eA(t−s) Bg(s)ds. ˙ dt 0 0 ˙ $ (·, t) ∈ H, In terms of (4.26) and the same arguments as the first part, we have W which implies that (4.27) is a classical solution. The result is thus proved. Proof of Theorem 3.5. We first assume that ( ε0 , ε1 , θ10 , ϑ10 , . . . , θm0 , ϑm0 ) ∈ V × V × R2m , (4.31)

ε0 (1) = 0, ε0 (1) = −

m j=1

αj θj0 , ε1 (1) +

m

0 , w 1 ) ∈ D(A). ϑj0 = 0, (w

j=1


1700



With (4.31), θj (t) and ϑj (t), j ∈ J in (3.8) are uniquely determined by ε. Now, Theorem 3.4 assures that the classical solution exists. Let ρw (t) =

0

1

(1 + x)w x (x, t)w t (x, t)dx.

Then there exists an M1 > 0 such that |ρw (t)| ≤ M1 Ew (t).

(4.32)

The time derivatives of Ew and ρw (t) along the solution of (3.8) are, respectively, E˙w (t) =

1

0

[w t (x, t)w tt (x, t) + w x (x, t)w xt (x, t)]dx + c1 w(0, t)w t (0, t)

=w t (1, t)w x (1, t) − c3 w t (0, t)w(0, t) 1 + (c0 + q)[q ε(1, t) + kh(t)] w t (x, t)m(x)dx + c1 w(0, t)w t (0, t) 0

= −c2 w t2 (1, t) + (c0 + q) ε(1, t)w t (1, t) + kh(t)w t (1, t) 1 + (c0 + q)[q ε(1, t) + kh(t)] w t (x, t)m(x)dx, 0

where h(t) is given by (4.2), and ρ˙w (t) =

1

0

(1 + x)w xt (x, t)w t (x, t)dx +

0

1

(1 + x)w x (x, t)w tt (x, t)dx

1

1 2 (0, t) + w [w x2 (x, t) + w t2 (x, t)]dx − w t2 (1, t) + [w x (1, t)]2 2 t 0 1 1 2 2 − c1 w (0.t) + (c0 + q)[q ε(1, t) + kh(t)] (1 + x)m(x)w x (x, t)dx. 2 0

=−

1 2

Let Fδ (t) = Ew (t) + δρw (t), δ > 0. Then, F˙δ (t) = E˙w (t) + δ ρ˙w (t) δ 2 δ 1 2 [w (x, t) + w t2 (x, t)]dx − w t2 (1, t) =− (0, t) − [c2 − δ(1 + c22 )]w 2 0 x 2 t + δ(c0 + q)2 ε2 (1, t) + δk 2 h2 (t) + (1 − 2δc2 )(c0 + q) ε(1, t)w t (1, t) 2 δc 2 (0, t) + 2δ(c0 + q)kh(t) + (1 − 2δc2 )kh(t)w t (1, t) − 3 w ε(1, t) 2 1 + (c0 + q)[q ε(1, t) + kh(t)] m(x)[w t (x, t) + δ(1 + x)w x (x, t)]dx. 0




1701

By further performing two completions of squares, we get (4.33) δ 1 2 δc2 2 δ 2 ˙ Fδ (t) = − (0, t) − w (0, t) [w x (x, t) + w t2 (x, t)]dx − 1 w 4 0 2 2 t (1 − 2δ)2 c22 2 2 2 2 2 k h (t) t (1, t) + 2δk + − [c2 − δ(3 + c2 )]w 4δ (1 − 2δ)2 c22 + 2δ + (c0 + q)2 ε2 (1, t) 4δ 2 (1 − 2δ)c2 kh(t) −δ w t (1, t) − 2δ 2 (1 − 2δ)c2 (c0 + q) −δ w t (1, t) − ε(1, t) − δ[(c0 + q) ε(1, t) − kh(t)]2 2δ δ 1 2 − [w x (x, t) − 2(c0 + q)[q ε(1, t) + kh(t)](1 + x)m(x)] dx 4 0 2 δ 1 2(c0 + q) [q ε(1, t) + kh(t)]m(x) dx − w t (x, t) − 4 0 δ 1 1 2 + ε(1, t) + kh(t)] . δ(1 + x)2 + m2 (x)dx(c0 + q)2 [q δ 0 In view of (4.32), we have (1 − M1 δ)Ew (t) ≤ Fδ (t) ≤ (1 + M1 δ)Ew (t).

(4.34) Take δ ≤

c2 1 min{ 3+c 2, M } 1 2

in (4.33) , to get δ (1 − 2δ)2 c22 F˙δ (t) ≤ − Ew (t) + 2δ + (c0 + q)2 ε2 (1, t) 2 4δ (1 − 2δ)2 c22 2 2 2 + 2δk + k h (t) 4δ 1 1 2 ε(1, t) + kh(t)]2 δ(1 + x) + + m2 (x)dx(c0 + q)2 [q δ 0

(4.35)

δ ≤ − Ew (t) + M2 ε2 (1, t) + M3 h2 (t), 2 where

1 1 (1 − 2δ)2 c22 2 2 2 + 2q M2 = (c0 + q) 2δ + δ(1 + x) + m (x)dx , 4δ δ 0 1 1 (1 − 2δ)2 c22 + 2(c0 + q)2 M3 = k 2 2δ + δ(1 + x)2 + m2 (x)dx . 4δ δ 0 2

Now, in term of (4.34) and (4.35), we have F˙δ (t) ≤ −

(4.36)

δ 2(1+M1 δ)

and apply Gronwall’s inequality to (4.36) to conclude t t Fδ (t) ≤ e−μt Fδ (0) + M2 e−μ(t−τ ) ε2 (1, τ )dτ + M3 e−μ(t−τ ) h2 (τ )dτ.

Let μ = (4.37)

δ Fδ (t) + M2 ε2 (1, t) + M3 h2 (t). 2(1 + M1 δ)

0

0



1702


By the Poinc´ are inequality, we have 1 2 2 2 2 ε (0, t) + 2 εx (x, t)dx ≤ + 4 Eε(t) → 0 as t → 0. ε (1, t) ≤ 2 c0 0 Thus, ε(1, t) → 0 as t → ∞. For given κ > 0, we choose t0 such that for t > t0 , | ε(1, t)| < Therefore, t −μ(t−τ ) 2 e ε (1, τ )dτ ≤ 0

t0

e

−μ(t−τ ) 2

ε (1, τ )dτ +

0

≤

κ . 2μ

t

t0

e−μ(t−τ ) ε2 (1, τ )dτ

2 κ + 4 Eε(0)μ−1 e−μ(t−t0 ) + . c0 2

Choosing t > t0 large enough, the first - t term on the right-hand side above will be less than κ2 and thus for t large enough 0 e−μ(t−τ ) ε2 (1, τ )dτ < κ, which implies that

t

(4.38) 0

e−μ(t−τ ) ε2 (1, τ )dτ → 0 as t → ∞.

Next, we show 0

t

e−μ(t−τ ) h2 (τ )dτ → 0 as t → ∞.

In fact,

t

e

−μ(t−τ ) 2

t 2

h (τ )dτ =

0

e

= (4.39)

h (τ )dτ +

0 t

e

t 2

≤ e− ≤e

−μ(t−τ ) 2

μt 2

− μt 2

h (t − s)ds +

t t 2

t 2

−μs 2

t

t t 2

e−μ(t−τ ) h2 (τ )dτ

e−μ(t−τ ) h2 (τ )dτ

h2 (t − s)ds + tmax e−μ(t−τ )

∞

2

h (s)ds + 0

2 ≤τ ≤t

∞ t 2

t t 2

h2 (τ )dτ

h2 (τ )dτ.

This together with (4.2) gives t (4.40) e−μ(t−τ ) h2 (τ )dτ → 0 as t → ∞. 0

Combining (4.37), (4.38), and (4.40), we have limt→∞ Fδ (t) = 0, and by virtue of (4.34), we get lim Ew (t) = 0.

t→∞


1703



Finally, since V × V × R2m is dense in H 1 (0, 1) × L2(0, 1) × R2m and D(A) is dense in H, for any ( y0 , y1 ) ∈ H and ( ε0 , ε1 , θ10 , ϑ10 , . . . , θm0 , ϑm0 ) ∈ H 1 (0, 1)×L2 (0, 1)×R2m , n n n n n we can take (w 0 , w 1 ) ∈ D(A) and ( εn0 , εn1 , θ10 , ϑ10 , . . . , θm0 , ϑnm0 ) ∈ V × V × R2m such that ⎧ (w 0n , w 1n ) → (w 0 , w 1 ) in H, ⎪ ⎪ ⎪ ⎪ ⎪ n n n n n ⎨ ( , ϑnm0 ) → ( ε0 , ε1 , θ10 , ϑ10 , . . . , θm0 , ϑm0 ) in V, ε0 , ε1 , θ10 , ϑ10 , . . . , θm0 m m ⎪ ⎪ ⎪ n n n = 0, εn (1) + ⎪ (1) = 0, ( ε ) (1) + α θ ϑnj0 = 0. ε ⎪ j 0 0 j0 1 ⎩ j=1

j=1

The result then follows from the density argument and the conclusion just justified for the classical solution. Proof of Theorem 3.6. For any initial value (w0 , w1 , w 0 , w 1 , θ10 , ϑ10 , . . . , θm0 , θm0 ) ∈ X , from (2.9) and (2.10), it is easy to verify that ( ε0 (x), ε1 (x), θ10 , ϑ10 , . . . , θm0 , θm0 ) ∈ 1 2 2m H (0, 1) × L (0, 1) × R and (w 0 (x), w 1 (x)) ∈ H, which implies that there exists a unique solution to (2.2) and (3.8), respectively. Let ⎧ 1 ⎪ ⎪ ⎪ w(x, t) = w(x, t) + (c + q) e−q(x−ξ) ε(ξ, t)dξ + ε(x, t), 0 ⎨ x x ⎪ ⎪ ⎪ t) = w(x, t) − (c1 + q) e−c1 (x−ξ) w(ξ, t)dξ. ⎩ w(x, 0

Then a direct computation shows that such a defined (w, w) satisfies (3.14) with initial value (w0 , w1 , w 0 , w 1 ). This solution is unique by the invertible transformation (4.41)

w t , θ1 , ϑ1 , . . . , θm , ϑm ) = D(w, wt , w, w t , θ1 , ϑ1 , . . . , θm , ϑm ) , ( ε, εt , w,

where ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ D=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

I + P1 0 0 0 0 0 .. .

0 I + P1 0 0 0 0 .. .

−I − P1 0 I + P2 0 0 0 .. .

0 −I − P1 0 I + P2 0 0 .. .

0 0 0 0 1 0 .. .

0 0 0 0 0 1 .. .

0 0

0 0

0 0

0 0

0 0

0 0

··· ··· ··· ··· ··· ··· ··· ··· ···

0 0 0 0 0 0 .. . 1 0

0 0 0 0 0 0 .. .

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎠ 1

and the uniqueness of the solutions to (2.2) and (3.8). The asymptotic stability follows from (4.41) and Theorems 3.3 and 3.5. 5. Numerical simulations. In this section, we present some numerical simulations to illustrate the theoretical results. For simplicity, we just give numerical simulations for the case m = 1 and αj = 1. Notice that there is an invertible transform between the closed-loop system (w, wt , w, w t , θ1 (t), ϑ1 (t)) and the system


1704



(w, wt , ε, εt , θ1 (t), ϑ1 (t)) given by ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

w wt w w t θ1 (t) ϑ1 (t)

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝

1 0 1 0 0 0

0 1 0 1 0 0

0 0 0 0 0 0 −1 0 0 0 −1 0 0 0 1 0 0 0

0 0 0 0 0 1

⎞⎛ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝

w wt ε εt θ1 (t) ϑ1 (t)

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

We only need to give the numerical simulation results for the system (w, wt , ε, εt , θ1 (t), ϑ1 (t)) which is described by (5.1) ⎧ wtt (x, t) − wxx (x, t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ wx (0, t) = −qw(0, t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ wx (1, t) = −c2 [wt (1, t) − εt (1, t)] − (c1 + q)[w(1, t) − ε(1, t)] ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ −(c1 + q) eq(1−ξ) [c2 (wt (ξ, t) − εt (ξ, t)) + q(w(ξ, t) − ε(ξ, t))] dξ, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ . / ⎪ q(1−x) ⎪ εt (1, t) + θ1 (t) sin t + ϑ1 (t) cos t + qε(1, t) , ⎪ ⎨ εtt (x, t) = εxx (x, t) − (c0 + q)e ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

εx (0, t) = −qε(0, t), . / εx (1, t) = −k εt (1, t) + θ1 (t) sin t + ϑ1 (t) cos t − (c0 + q)ε(1, t), . / ˙ θ1 (t) = −r1 εt (1, t) + θ1 (t) sin t + ϑ1 (t) cos t sin t, . / ˙ ϑ1 (t) = −l1 εt (1, t) + θ1 (t) sin t + ϑ1 (t) cos t cos t, θ1 (0) = θ10 , ϑ1 (0) = ϑ10 , w(x, 0) = w0 (x), wt (x, 0) = w1 (x), ε(x, 0) = ε0 (x), εt (x, 0) = ε1 (x).

In the numerical simulation, we first convert the second-order equation in time into a system of two first-order equations in time and then the backward Euler method in time and Chebyshev spectral method in space are used. The numerical algorithm is programmed in MATLAB (see [24]). We take the grid size N = 20 and time step dt = 10−3 . The parameter values are q = 0.7, k = 0.5, r1 = 0.4, l1 = 0.5, c0 = c1 = 500, c2 = 0.9, θ¯1 = 0.8, and ϑ¯1 = 0.2. The initial values are taken to be w0 (x) = ε0 (x) = 0.2(1 − x), w1 (x) = ε1 (x) =

1 5,0

< x ≤ 12 , θ10 = 0.2, ϑ10 = 0.8. 0, otherwise,

Figure 1 shows that the system (5.1) is asymptotically stable. Figure 2 shows the tracking of the parameters. It is seen that the estimates θ1 (t) and ϑ1 (t) with initial values θ10 = 0.2 and ϑ10 = 0.8 approximate, respectively, the unknown parameter values θ¯1 = 0.8 and ϑ¯1 = 0.2 quite satisfactorily.




1705

Fig. 1. Amplitude w(x, t) (left) and error ε(x, t) (right).

Fig. 2. Parameter estimation.

6. Concluding remarks. In the present paper, the boundary output feedback stabilization of a one-dimensional wave equation is considered. The difficulty of the problem lies in that (a) the system in the case q > 0 contains instability at its free end, which results in finitely many eigenvalues for the open-loop system located on the right complex plane and (b) the boundary observation is suffered from a harmonic disturbance with unknown magnitude and any given frequencies αj , j ∈ J, which contains an unknown constant disturbance as its special case. An infinite-dimensional observer is designed to recover the state and an adaptive update law is designed to estimate the unknown parameters. The output feedback controller is designed by the backstepping method for infinite-dimensional systems. It is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameter is shown to be convergent to the unknown parameter as time goes to infinity. The numerical simulations validate the theoretical results. The result generalizes [7] from the simple period disturbance to the general harmonic disturbance. REFERENCES [1] G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a boundary domain, J. Math. Pures Appl. (9), 58 (1979), pp. 249–273. [2] J. L. Fason, An Experimental Investigation of Vibration Suppression in a Large Space Structures Using Positive Position Feedback, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1987.



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[3] B. Z. Guo and C. Z. Xu, The stabilization of a one-dimensional wave equation by boundary feedback with non-collocated observation, IEEE Trans. Automat. Control, 52 (2007), pp. 371–377. [4] B. Z. Guo and W. Guo, The strong stabilization of a one-dimensional wave equation by noncollocated dynamic boundary feedback control, Automatica J. IFAC, 45 (2009), pp. 790–797. [5] B. Z. Guo and R. Yu, The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control, IMA J. Math. Control Inform., 18 (2001), pp. 241–251. [6] B. Z. Guo and X. Zhang, The regularity of the wave equation with partial Dirichlet control and colocated observation, SIAM J. Control Optim., 44 (2005), pp. 1598–1613. [7] W. Guo, B. Z. Guo, and Z. C. Shao, Parameter estimation and stabilization for a wave equation with boundary output harmonic disturbance and non-collocated control, Internat. J. Robust Nonlinear Control, 21 (2011), pp. 1297–1321. [8] W. Guo and B. Z. Guo, Stabilization and regulator design for a one-dimensional unstable wave equation with input harmonic disturbance, Internat. J. Robust Nonlinear Control, 23 (2013), pp. 514–533. [9] W. Guo and B. Z. Guo, Parameter estimation and non-collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance, IEEE Trans. Automat. Control, in press. [10] M. Krstic, Adaptive control of an anti-stable wave PDE, Dynam. Contin. Discrete Impuls. Systems Ser. A Math. Anal., 17 (2010), pp. 853–882. [11] M. Krstic, B. Z. Guo, A. Balogh, and A. Smyshlyaev, Output-feedback stabilization of an unstable wave equation, Automatica J. IFAC, 44 (2008), pp. 63–74. [12] M. Krstic, B. Z. Guo, A. Balogh, and A. Smyshlyaev, Control of a tip-force destabilized shear beam by observer-based boundary feedback, SIAM J. Control Optim., 47 (2008), pp. 553–574. [13] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, Philadelphia, 2008. [14] M. Krstic and A. Smyshlyaev, Adaptive boundary control for unstable parabolic PDEs - Part I: Lyapunov design, IEEE Trans. Automat. Control, 53 (2008), pp. 1575–1591. [15] V. Komornok, Exact Controllability and Stabilization: The Multiplier Method, John Wiley Chichester, 1994. [16] T. Kobayashi, T. Sakamoto, and M. Oya, Adaptive regulator design for undamped secondorder hyperbolic systems with output disturbances, IMA J. Math. Control Inform., 25 (2008), pp. 205–219. [17] J. Lagnese, Decay of soultions of wave equations in a bounded domain with boundary dissipation, J. Differential Equations, 50 (1983), pp. 163–182. [18] J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), pp. 1–68. [19] Z. H. Luo, B. Z. Guo, and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag, London, 1999. [20] A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs - Part II: Estimation-based designs, Automatica J. IFAC, 43 (2007), pp. 1543–1556. [21] A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs - Part III: Output-feedback examples with swapping identifiers, Automatica J. IFAC, 43 (2007), pp. 1557–1564. [22] M. Slemrod, Stabilization of boundary control systems, J. Differential Equations, 22 (1976), pp. 402–415. [23] A. Smyshlyaev and M. Krstic, Boundary control of an anti-stable wave equation with antidamping on the uncontrolled boundary, Systems Control Lett., 58 (2009), pp. 617–623. [24] L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000. [25] J. A. Walker, Dynamical Systems and Evolution Equations: Theory and Applications, Plenum Press, New York, London, 1980. [26] G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), pp. 17–43.


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