A New Semidiscretized Order Reduction Finite Difference

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. CONTROL OPTIM. © 2020 Society for Industrial and Applied MathematicsVol. 58, No. 4, pp. 2256--2287

A NEW SEMIDISCRETIZED ORDER REDUCTION FINITEDIFFERENCE SCHEME FOR UNIFORM APPROXIMATION OF

ONE-DIMENSIONAL WAVE EQUATION\ast

JIANKANG LIU\dagger AND BAO-ZHU GUO\ddagger

Abstract. In this paper, we propose a novel space semidiscretized finite difference schemefor approximation of the one-dimensional wave equation under boundary feedback. This scheme,referred to as the order reduction finite difference scheme, does not use numerical viscosity and yetpreserves the uniform exponential stability. The paper consists of four parts. In the first part, theoriginal wave equation is first transformed by order reduction into an equivalent system. A standardsemidiscretized finite difference scheme is then constructed for the equivalent system. It is shown thatthe semidiscretized scheme is second-order convergent and that the discretized energy converges tothe continuous energy. Very unexpectedly, the discretized energy also preserves uniformly exponentialdecay. In the second part, an order reduction finite difference scheme for the original system is deriveddirectly from the discrete scheme developed in the first part. The uniformly exponential decay,convergence of the solutions, as well as uniform convergence of the discretized energy are establishedfor the original system. In the third part, we develop the uniform observability of the semidiscretizedsystem and the uniform controllability of the Hilbert uniqueness method controls. Finally, in thelast part, under a different implicit finite difference scheme for time, two numerical experiments areconducted to show that the proposed implicit difference schemes preserve the uniformly exponentialdecay.

Key words. wave equation, finite difference discretization, stability, uniform approximation

AMS subject classifications. 39A12, 35L05, 34D20, 65L20

DOI. 10.1137/19M1246535

1. Introduction. For control systems described by partial differential equations(PDEs), numerical experiments and thus discretization PDEs are necessary for appli-cations in many situations where PDEs are to be approximated by finite-dimensionalsystems. Among many discretization methods, the finite difference method is themost often used approach due to its simplicity in theory and versatility in imple-mentation. It is also the one that is very familiar to most researchers. However, ithas been acknowledged for a long time that for PDEs, the finite difference method orother similar numerical discretization methods may suffer from certain approximationproblems. One such problem is that the exponential decay rate of an exponentiallystable PDE cannot be preserved uniformly in the process of the discretization. Letus study the following one-dimensional wave equation under boundary control:\left\{

wtt(x, t) - wxx(x, t) = 0, 0 < x < 1, t > 0,w(0, t) = 0, wx(1, t) = U(t),w(x, 0) = w0(x), wt(x, 0) = w1(x),y(t) = wt(1, t),

(1.1)

\ast Received by the editors February 25, 2019; accepted for publication (in revised form) June 12,2020; published electronically August 3, 2020.

https://doi.org/10.1137/19M1246535Funding: This work was partially supported by the National Natural Science Foundation of

China under projects 11901365, 61873260, and 11772177 and by the Project of the Department ofEducation of Guangdong Province, 2017KZDXM087.

\dagger School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China ([email protected]).\ddagger School of Mathematics and Physics, North China Electric Power University, Beijing 102206,

China, and School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China ([email protected]).

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https://doi.org/10.1137/19M1246535

mailto:[email protected]




FINITE DIFFERENCE OF WAVE EQUATION 2257

where (w0, w1) is the initial state, U(t) is the control, and y(t) is the output. System(1.1) arises in vibration control of a vibrating string where one end is fixed and thevertical force at other end is actuated [14]. Consider system (1.1) in the energy statespace \scrH = H1

L(0, 1)\times L2(0, 1), where H1L(0, 1) = \{ f \in H1(0, 1)| f(0) = 0\} . The energy

E(t) of system (1.1) is given by

E(t) =1

2

\int 1

0

\bigl[ w2

t (x, t) + w2x(x, t)

\bigr] dx.

Finding the derivative of E(t) along the solution of (1.1) gives

\.E(t) = U(t)y(t),

which means that system (1.1) is a passive system. A stabilizing output feedbackcontrol is then naturally designed as

U(t) = - \alpha y(t), \alpha > 0,

and the closed-loop system is therefore governed by\left\{ wtt(x, t) - wxx(x, t) = 0, 0 < x < 1, t > 0,

w(0, t) = 0, wx(1, t) = - \alpha wt(1, t),

w(x, 0) = w0(x), wt(x, 0) = w1(x).

(1.2)

Since the energy of the closed-loop satisfies

dE(t)

dt= - \alpha w2

t (1, t),(1.3)

the energy of the system is decreasing as time evolves. Actually, it has been knownfor a long time that for any given initial state (w0, w1), the solution of (1.2) decaysexponentially in time and the decay rate is uniform for all initial states (w0, w1) inthe state space \scrH :

E(t) \leq Ke - \omega tE(0) \forall t \in [0,\infty )(1.4)

for some K,\omega > 0 independent of the initial values; see, e.g., [5, 8, 16].In the past two decades, there has been extensive literature on approximation

of system (1.2) concerning the uniform decay rate, i.e., whether or not the exponen-tial decay rate of the discretized energy is preserving uniformly with respect to themesh size. Banks, Ito, and Wang in [3] first pointed out that the exponential decayof the discretized energy might not be uniform, with respect to the mesh size, forthe classical finite difference and finite element schemes. This is attributed highlyto the spurious oscillations of the high frequencies in discretization. To remedy thisproblem, the authors suggested to use mixed finite element methods or polynomialbased Galerkin methods to obtain the uniformly exponential decay [9, 12]. Someother remedies have also been proposed to damp out these high frequencies, whichinclude Tychonoff regularizations [15], two-grid algorithms [18, 22], and filtering tech-niques [2, 25, 26]. The numerical viscosity method is another effective technique todamp out the high frequencies. The method incorporates a numerical vanishing vis-cosity term effectively at the whole domain. By virtue of this viscosity term, one canobtain uniform observability of semidiscrete systems [7, 15, 25], uniform boundary

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2258 JIANKANG LIU AND BAO-ZHU GUO

controllability, convergence of controls [20, 21], and uniformly exponentially stableapproximations for damped systems [23]. Unfortunately, the semidiscretized finitedifference scheme using numerical viscosity has some problems in applications. First,the artificially introduced viscosity term is different from system to system, in partic-ular, the coefficient of the viscosity term specially selected for one system may not beeffective for different vanishing coefficients. Second, the proof of uniformly exponentialdecay relies on the analytic forms of the eigenpairs, which are available only for somespecial boundary conditions. In this situation, some natural schemes without usingnumerical viscosity have been proposed. A nonuniform numerical mesh was developedin [10] to preserve uniform observability. For some fully discretized finite differenceschemes, one can also obtain uniform observability [11, 22] and uniformly exponentialdecay [6] without using numerical viscosity, provided that the space step is identicalto the time step. To the best of our knowledge, there is no semidiscretized finitedifference scheme that is on a uniform numerical mesh and does not use numericalviscosity that can preserve the uniformly exponential decay of system (1.2).

This paper aims to propose for the first time a new semidiscretized finite differ-ence scheme for system (1.2) that enjoys uniformly exponential decay without usingnumerical viscosity. This new finite difference scheme provides a basic principle thatnot only is for numerical computation of the wave equation but is also potentiallyapplicable to other types of PDEs. In addition, it has the following advantages: (a) itnaturally keeps the finite difference discretization nature; (b) it can deal with variousboundary conditions; and (c) the exponential convergence can be proved similarly asin the case of continuous counterpart. To this end, we introduce the following twointermediate variables:

u(x, t) = wx(x, t), v(x, t) = wt(x, t).(1.5)

Then, system (1.2) can be reformulated as\left\{

ut(x, t) - vx(x, t) = 0, 0 < x < 1, t > 0,

vt(x, t) - ux(x, t) = 0,

v(0, t) = 0, u(1, t) + \alpha v(1, t) = 0,

u(x, 0) = u0(x) = w0x(x), v(x, 0) = v0(x) = w1(x),

(1.6)

where (u0, v0) \in L2(0, 1) \times L2(0, 1). Energy F (t) of system (1.6) now equivalentlyreads

F (t) =1

2

\int 1

0

\bigl[ u2(x, t) + v2(x, t)

\bigr] dx(1.7)

and satisfies the following energy dissipation law:

dF (t)

dt= - \alpha v2(1, t).(1.8)

Moreover, the exponential decay as in (1.4) holds true. The reason why we considersystem (1.6) instead of system (1.2) is that no partial derivative appears in the rightboundary condition in (1.6). It is well-known that how to discretize the partial deriv-atives at the right boundary condition in (1.2) is crucial to both the stability and theconvergence order of the finite difference schemes from the numerical analysis point of

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view. In the context of control theory, as described in [13, 4], an inadequate approxi-mation of boundary control often contributes to the poor numerical approximation ofthe normal derivative on the boundary for high eigenfunctions. The equivalent system(1.6) avoids this sort of problem.

The paper is divided into four parts. In the first part, we show that the ordinarysemidiscretized finite difference scheme preserves uniformly exponential decay for sys-tem (1.6). To the best of our knowledge, this makes a first successful PDE examplefor such discretization. In the second part, we show that the ordinary semidiscretizedfinite difference scheme for system (1.6) leads to a semidiscretized order reductionfinite difference scheme for original system (1.2). It also turns out that the rightsemidiscretized finite difference scheme can preserve the uniformly exponential decay.In the third part, we show that other control properties like uniform observabilityand controllability can also be preserved by the semidiscretized order reduction finitedifference scheme. The numerical simulation verifies the result in the last part.

The rest of the paper is organized as follows. In section 2 we construct a spacesemidiscretized finite difference scheme for equivalent system (1.6). The uniformlyexponential decay of the discretized energy is proved in section 3 by the Lyapunovfunction method. The convergence of the discretized scheme to the original system isanalyzed in section 4, which includes the convergence of the discretized solutions andthe convergence of the discretized energy. In section 5, we derive the semidiscretizedorder reduction finite difference scheme for original system (1.2) directly from thediscretized scheme for equivalent system. The uniformly exponential decay and weakconvergence of the solutions are also proved. In section 6, the uniform observabilityof the semidiscrete system is proved. This leads to the uniformly exact controllabilityof control systems with the Hilbert uniqueness method (HUM) control. Numericalexperiments are conducted in section 7, followed by concluding remarks in section 8.

2. Derivation of finite difference scheme. Before we conduct the finite dif-ference semidiscretization for system (1.6), some notation is introduced. First of all,we denote sequences \{ uj(t)\} 0\leq j\leq N+1 by \{ uj(t)\} j for simplicity. For every N \in \BbbN +

we consider an equidistant partition of the interval [0, 1]:

0 = x0 < x1 < \cdot \cdot \cdot < xj = jh < \cdot \cdot \cdot < xN+1 = 1,

where the mesh size h = 1N+1 . Let \{ Uj(t)\} j , \{ Vj(t)\} j , and \{ Wj(t)\} j be grid functions

at grids \{ xj\} j , satisfying

Uj(t) = u(xj , t), Vj(t) = v(xj , t), Wj(t) = w(xj , t), j = 0, 1, . . . , N + 1.

Denote the first-order and second-order finite difference operators by

\delta xUj+ 12(t) =

Uj+1(t) - Uj(t)

h, \delta 2xUj(t) =

Uj+1(t) - 2Uj(t) + Uj - 1(t)

h2,

with the average operators being

Uj+ 12(t) =

Uj+1(t) + Uj(t)

2.

Now, we begin to discretize the space derivatives of system (1.6) by the finitedifference method. The first equation of system (1.6) is valid at (xj+ 1

2, t), i.e.,

u\prime \Bigl( xj+ 1

2, t\Bigr) - vx

\Bigl( xj+ 1

2, t\Bigr) = 0,

where xj+ 12= (j + 1

2 )h. Hereafter the prime \prime represents the derivative with respectto time. Replace the differential operator with the difference operator to get that

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U \prime j+ 1

2(t) - \delta xVj+ 1

2(t) = rj+ 1

2(t),(2.1)

where

rj+ 12(t) =

\biggl( - 1

8u\prime xx(\xi 1, t) +

1

24vxxx(\xi 2, t)

\biggr) h2, \xi 1, \xi 2 \in (xj , xj+1).(2.2)

Similarly, for the second equation of system (1.6), we have

V \prime j+ 1

2(t) - \delta xUj+ 1

2(t) = sj+ 1

2(t),(2.3)

where

sj+ 12(t) =

\biggl( - 1

8v\prime xx(\xi 3, t) +

1

24uxxx(\xi 4, t)

\biggr) h2, \xi 3, \xi 4 \in (xj , xj+1).(2.4)

By dropping the infinitesimal terms in (2.1) and (2.3), and replacing Uj(t) and Vj(t)by uj(t) and vj(t), respectively, we arrive at a semidiscretized finite difference schemeof system (1.6) as follows:\left\{

u\prime j+ 1

2

(t) - \delta xvj+ 12(t) = 0, j = 0, . . . , N,

v\prime j+ 1

2

(t) - \delta xuj+ 12(t) = 0,

v0(t) = 0, uN+1(t) + \alpha vN+1(t) = 0,

uj(0) = u0j , vj(0) = v0j ,

(2.5)

where uj(t) and vj(t) are approximations of u(xj , t) and v(xj , t), respectively, and u0j

and v0j are approximations of the initial values u0(xj) and v0(xj), respectively. If the

initial conditions are sufficiently smooth, we can set u0(xj) = u0j and v0(xj) = v0j .It is noted that (2.5) is an ordinary semidiscretized finite difference scheme widelydiscussed in the literature [15, 25, 7, 20, 21, 23].

Remark 2.1. From the construction of difference schemes we can see that thesemidiscretized finite difference schemes (2.5) is of second-order convergence to thetransformed system (1.6).

The following two lemmas are straightforward yet very useful in proving theconvergence of the semidiscritized finite difference scheme and the exponential decayof the discretized energy.

Lemma 2.1 (Gr\"onwall's inequality; see, e.g., [17]). Let y(t) be a nonnegative,absolutely continuous function over t \in [0,\infty ) that satisfies the differential inequality

y\prime (t) + f(t) \leq h(t) + g(t)y(t) \forall t \in (0,\infty ) a.e.,

where f(t), g(t), and h(t) are nonnegative, summable functions on [0,\infty ). Then,

y(t) +

\int t

0

f(s)ds \leq e\int t0g(s)ds

\biggl( y(0) +

\int t

0

h(s)ds

\biggr) \forall t \in [0,\infty ).

Lemma 2.2. For any grid functions \{ Uj\} j, \{ Vj\} j, and \{ Wj\} j at mesh grids \{ xj\} j,we have the following two formulas of summation by parts:

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h

N\sum j=0

\delta xUj+ 12Vj+ 1

2+ h

N\sum j=0

Uj+ 12\delta xVj+ 1

2= UN+1VN+1 - U0V0,(2.6)

h

N\sum j=0

\delta xUj+ 12Vj+ 1

2Wj+ 1

2+ h

N\sum j=0

Uj+ 12\delta xVj+ 1

2Wj+ 1

2+ h

N\sum j=0

Uj+ 12Vj+ 1

2\delta xWj+ 1

2

= UN+1VN+1WN+1 - U0V0W0 - 1

4

N\sum j=0

(Uj+1 - Uj)(Vj+1 - Vj)(Wj+1 - Wj).

(2.7)

Proof. The first formula (2.6) follows from

h

N\sum j=0

\delta xUj+ 12Vj+ 1

2+ h

N\sum j=0

Uj+ 12\delta xVj+ 1

2

=1

2

N\sum j=0

(Uj+1 - Uj)(Vj+1 + Vj) +1

2

N\sum j=0

(Uj+1 + Uj)(Vj+1 - Vj)

=

N\sum j=0

(Uj+1Vj+1 - UjVj) = UN+1VN+1 - U0V0,

and the second formula (2.7) can be similarly obtained with some elementaryoperations.

3. Uniformly exponential decay. In this section, we analyze the uniformlyexponential decay of the semidiscretized finite difference scheme (2.5) by the Lyapunovfunction method. Define the L2-norm \| \cdot \| of \~u(t) = \{ uj(t)\} j as

\| \~u(t)\| =

\sqrt{} h

N\sum j=0

u2j+ 1

2

(t).(3.1)

The discretized energy of the finite difference scheme (2.5) reads

Fh(t) =1

2

\bigl( \| \~u(t)\| 2 + \| \~v(t)\| 2

\bigr) , \~v(t) = \{ vj(t)\} j .(3.2)

Here, Fh(t) is a natural discretization of the continuous energy F (t) defined by (1.7).

Lemma 3.1. The discretized energy Fh(t) defined by (3.2) satisfies

d

dtFh(t) = - \alpha v2N+1(t) \forall t \in [0,\infty ).

Proof. Multiply both sides of the first equation of (2.5) by huj+ 12(t) and sum up

j from 0 to N to obtain

h

N\sum j=0

u\prime j+ 12(t)uj+ 1

2(t) - h

N\sum j=0

\delta xvj+ 12(t)uj+ 1

2(t) = 0.

By virtue of (2.6) in Lemma 2.2, we obtain

1

2

d

dth

N\sum j=0

u2j+ 12(t) + h

N\sum j=0

\delta xuj+ 12(t)vj+ 1

2(t) - uN+1(t)vN+1(t) = 0.(3.3)

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Similarly, multiplying both sides of the second equation of (2.5) by hvj+ 12(t) and

summing up j from 0 to N , we obtain

1

2

d

dth

N\sum j=0

v2j+ 12(t) - h

N\sum j=0

\delta xuj+ 12(t)vj+ 1

2(t) = 0.(3.4)

Equations (3.3) and (3.4) together with the right boundary condition in (2.5) completethe proof.

Lemma 3.1 shows that discretized energy Fh(t) of system (2.5) is decreasing intime as long as the damping coefficient \alpha > 0. In order to establish the uniformlyexponential decay of Fh(t) we construct the following Lyapunov function:

Gh(t) = Fh(t) + \varepsilon \Phi h(t), 0 < \varepsilon < 1,(3.5)

where the auxiliary function \Phi h(t) is defined by

\Phi h(t) = h

N\sum j=0

xj+ 12uj+ 1

2(t)vj+ 1

2(t).(3.6)

Lemma 3.2. The Lyapunov function Gh(t) defined by (3.5) is equivalent to thediscretized energy Fh(t), that is, there exist two constants C1 and C2 such that

C1Fh(t) \leq Gh(t) \leq C2Fh(t),

where C1 = 1 - \varepsilon and C2 = 1 + \varepsilon for some scalar \varepsilon \in (0, 1).

Proof. A direct use of Cauchy's inequality and the triangle inequality shows thatthe function \Phi h(t) defined by (3.6) satisfies

| \Phi h(t)| \leq h

N\sum j=0

| xj+ 12| | uj+ 1

2(t)| | vj+ 1

2(t)| \leq h

2

N\sum j=0

u2j+ 12(t) +

h

2

N\sum j=0

v2j+ 12(t) = Fh(t).

The proof is completed by taking C1 = 1 - \varepsilon and C2 = 1+\varepsilon , according to the definitionof Gh(t).

Lemma 3.3. The auxiliary function \Phi h(t) defined by (3.6) satisfies

d

dt\Phi h(t) \leq - Fh(t) +

1

2

\bigl( 1 + \alpha 2

\bigr) v2N+1(t).

Proof. Finding the derivative of \Phi h(t) along the solution of (2.5) yields

d

dt\Phi h(t) = h

N\sum j=0

xj+ 12u\prime j+ 1

2(t)vj+ 1

2(t) + h

N\sum j=0

xj+ 12uj+ 1

2(t)v\prime j+ 1

2(t)

= h

N\sum j=0

xj+ 12\delta xvj+ 1

2(t)vj+ 1

2(t) + h

N\sum j=0

xj+ 12uj+ 1

2(t)\delta xuj+ 1

2(t).

By (2.7) in Lemma 2.2, we further have

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d

dt\Phi h(t) = - h

N\sum j=0

v2j+ 12(t) - h

N\sum j=0

xj+ 12vj+ 1

2(t)\delta xvj+ 1

2(t) + v2N+1(t) -

h

4

N\sum j=0

(vi+1 - vi)2

- h

N\sum j=0

u2j+ 1

2(t) - h

N\sum j=0

xj+ 12\delta xuj+ 1

2(t)uj+ 1

2(t) + u2

N+1(t) - h

4

N\sum j=0

(ui+1 - ui)2

= - 2Fh(t) - d

dt\Phi h(t) + v2N+1(t) + u2

N+1(t) - h

4

N\sum j=0

(vi+1 - vi)2 - h

4

N\sum j=0

(ui+1 - ui)2

\leq - 2Fh(t) - d

dt\Phi h(t) + (1 + \alpha 2)v2N+1(t).

This gives the required result.

Theorem 3.4. For any damped coefficient \alpha > 0, the discretized energy Fh(t)defined by (3.2) decays uniformly exponentially:

Fh(t) \leq 1 + \varepsilon

1 - \varepsilon e -

\varepsilon 1+\varepsilon tFh(0),(3.7)

where \varepsilon is independent of h and satisfies 0 < \varepsilon < 2\alpha /(1 + \alpha 2) \leq 1.

Proof. Taking the derivative of Gh(t) defined by (3.5) and combining with Lem-mas 3.1, 3.2, and 3.3, we obtain

d

dtGh(t) =

d

dtFh(t) + \varepsilon

d

dt\Phi h(t)

\leq - \alpha v2N+1(t) + \varepsilon 1

2

\bigl( 1 + \alpha 2

\bigr) v2N+1(t) - \varepsilon Fh(t)

\leq - \Bigl( \alpha - \varepsilon

2(1 + \alpha 2)

\Bigr) v2N+1(t) -

\varepsilon

1 + \varepsilon Gh(t)

\leq - \varepsilon

1 + \varepsilon Gh(t),

where 0 < \varepsilon < 2\alpha /(1 + \alpha 2) implied \alpha - \varepsilon 2 (1 + \alpha 2) > 0. A direct use of Gr\"onwall's

inequality in Lemma 2.1 leads to

Gh(t) \leq e - \varepsilon

1+\varepsilon tGh(0).

Again according to Lemma 3.2, we have

Fh(t) \leq 1 + \varepsilon

1 - \varepsilon e -

\varepsilon 1+\varepsilon tFh(0).

This completes the proof of the theorem.

4. Convergence of semidiscretized scheme. In this section, we consider theconvergence of the semidiscretized finite difference scheme (2.5). It presents bothconvergence of the solution and convergence of the energy. For convergence of thesolution, we show that the solutions of difference scheme (2.5) are of second-orderconvergence to the solutions of equivalent system (1.6) as h \rightarrow 0 for any t \in [0,\infty ).For convergence of the energy, we show that discretized energy Fh(t) converges tothe continuous energy F (t) as h \rightarrow 0 uniformly over t \in (0,\infty ) as long as the initialenergy is convergent.

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Let \theta i(t) = Ui(t) - ui(t) and \eta i(t) = Vi(t) - vi(t) be the error functions. From(2.1), (2.3), and (2.5), we have the following error equation:\left\{

\theta \prime j+ 1

2

(t) - \delta x\eta j+ 12(t) = rj+ 1

2(t), j = 0, . . . , N,

\eta \prime j+ 1

2

(t) - \delta x\theta j+ 12(t) = sj+ 1

2(t),

\eta 0(t) = 0, \theta N+1(t) + \alpha \eta N+1(t) = 0,

\theta j(0) = u0(xj) - u0j , \eta j(0) = v0(xj) - v0j ,

(4.1)

where rj+ 12(t) and sj+ 1

2(t) are defined by (2.2) and (2.4), respectively. If the initial

values are sufficiently smooth, we can set \theta j(0) = \eta j(0) = 0 for j = 0, 1, . . . , N + 1.Denote the energy eh(t) of error by

eh(t) =1

2\| \~\theta (t)\| 2 + 1

2\| \~\eta (t)\| 2,(4.2)

where \~\theta (t) = \{ \theta j(t)\} j , \~\eta (t) = \{ \eta j(t)\} j and the auxiliary function \varphi h(t) is given by

\varphi h(t) = h

N\sum j=0

xj+ 12\theta j+ 1

2(t)\eta j+ 1

2(t).(4.3)

Define the Lyapunov function lh(t) similarly with (3.5) as

lh(t) = eh(t) + \varepsilon \varphi h(t), 0 < \varepsilon < 1.(4.4)

Similarly to Lemma 3.2, we have

(1 - \varepsilon )eh(t) \leq lh(t) \leq (1 + \varepsilon )eh(t).(4.5)

Lemma 4.1. The energy of the error defined by (3.1) satisfies

d

dteh(t) = - \alpha \eta 2N+1(t) + h

N\sum j=0

rj+ 12(t)\theta j+ 1

2(t) + h

N\sum j=0

sj+ 12(t)\eta j+ 1

2(t) \forall t \geq 0.

Proof. Multiply both sides of the first equation of (4.1) by h\theta j+ 12(t) and sum up

j from 0 to N to obtain

1

2

d

dth

N\sum j=0

\theta 2j+ 12(t) + h

N\sum j=0

\delta x\theta j+ 12(t)\eta j+ 1

2(t) - \theta N+1(t)\eta N+1(t) = h

N\sum j=0


2(t).

(4.6)

Similarly, multiplying both sides of the second equation of (4.1) by h\eta j+ 12(t) and

summing up j from 0 to N , we obtain

1

2

d

dth

N\sum j=0

\eta 2j+ 12(t) - h

N\sum j=0

\delta x\theta j+ 12(t)\eta j+ 1

2(t) = h

N\sum j=0

sj+ 12(t)\eta j+ 1

2(t).(4.7)

Adding (4.6) to (4.7) gives the required result.

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Lemma 4.2. The auxiliary function \varphi h(t) defined by (2.2) satisfies

d

dt\varphi h(t) \leq - eh(t) +

1

2

\bigl( 1 + \alpha 2

\bigr) \eta 2N+1(t) + h

N\sum j=0

xj+ 12rj+ 1

2(t)\eta j+ 1

2(t)

+ h

N\sum j=0

xj+ 12sj+ 1

2(t)\theta j+ 1

2(t).

Proof. Finding the derivative of \varphi h(t) along the solution of (4.1) gives

d

dt\varphi h(t) = h

N\sum j=0

xj+ 12\theta \prime j+ 1

2(t)\eta j+ 1

2(t) + h

N\sum j=0

xj+ 12\theta j+ 1

2(t)\eta \prime j+ 1

2(t)

= h

N\sum j=0

xj+ 12\delta x\eta j+ 1

2(t)\eta j+ 1

2(t) + h

N\sum j=0

xj+ 12\theta j+ 1

2(t)\delta x\theta j+ 1

2(t)

+ h

N\sum j=0

xj+ 12rj+ 1

2(t)\eta j+ 1

2(t) + h

N\sum j=0

xj+ 12\theta j+ 1

2(t)sj+ 1

2(t).

By (2.7) in Lemma 2.2, we further have

d

dt\varphi h(t) = - h

N\sum j=0

\eta 2j+ 1

2(t) - h

N\sum j=0

xj+ 12\eta j+ 1

2(t)\delta x\eta j+ 1

2(t) + \eta 2

N+1(t) - h

4

N\sum j=0

(\eta i+1 - \eta i)2

- h

N\sum j=0

\theta 2j+ 12(t) - h

N\sum j=0

xj+ 12\delta x\theta j+ 1

2(t)\theta j+ 1

2(t) + \theta 2N+1(t) -

h

4

N\sum j=0

(\theta i+1 - \theta i)2

+ h

N\sum j=0

xj+ 12rj+ 1

2(t)\eta j+ 1

2(t) + h

N\sum j=0

xj+ 12sj+ 1

2(t)\theta j+ 1

2(t)

= - 2eh(t) - d

dt\varphi h(t) + (1 + \alpha 2)\eta 2

N+1(t) - h

4

N\sum j=0

(\eta i+1 - \eta i)2 - h

4

N\sum j=0

(\theta i+1 - \theta i)2

+ 2h

N\sum j=0

xj+ 12rj+ 1

2(t)\eta j+ 1

2(t) + 2h

N\sum j=0

xj+ 12sj+ 1

2(t)\theta j+ 1

2(t)

\leq - 2eh(t) - d

dt\varphi h(t) + (1 + \alpha 2)\eta 2

N+1(t) + 2h

N\sum j=0

xj+ 12rj+ 1

2(t)\eta j+ 1

2(t)

+ 2h

N\sum j=0

xj+ 12sj+ 1

2(t)\theta j+ 1

2(t).

This gives the required result.

Theorem 4.3 (convergence of solutions). If u, v \in C1\bigl( [0,\infty );C3[0, 1]

\bigr) solve

system (1.6), and \{ uj(t)\} j, \{ vj(t)\} j solve difference equation (2.5), then the errorfunctions \{ \theta j(t)\} j and \{ \eta j(t)\} j satisfy (4.1) and the following error estimation holds:

eh(t) \leq e - \varepsilon

2(1+\varepsilon )t 1 + \varepsilon

1 - \varepsilon eh(0) +

2(1 + \varepsilon )3

\varepsilon 2(1 - \varepsilon )C2h4 \forall t \in [0,\infty ),

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where eh(t) is defined by (3.1) and \varepsilon and C are scalars independent of h satisfying0 < \varepsilon < 2\alpha /(1 + \alpha 2) \leq 1.

Proof. Since u, v \in C1\bigl( [0,\infty );C3[0, 1]

\bigr) , it follows from (2.2) and (2.4) that there

exists a constant C independent of h such that

| rj+ 12(t)| \leq Ch2, | sj+ 1

2(t)| \leq Ch2, j = 0, 1, . . . , N, \forall t \in [0,\infty ).(4.8)

Denote \~r(t) = \{ rj+ 12(t)\} 0\leq j\leq N and \~s(t) = \{ sj+ 1

2(t)\} 0\leq j\leq N . Finding the derivative of

lh(t) defined by (2.4) along the solution of (4.1) and combining with Lemmas 4.1 and4.2, we obtain

d

dtlh(t) =

d

dteh(t) + \varepsilon

d

dt\varphi h(t)

(4.9)

\leq - \varepsilon eh(t) - \Bigl( \alpha - \varepsilon

2

\bigl( 1 + \alpha 2

\bigr) \Bigr) \eta 2N+1(t) + h

N\sum j=0


2(t)

+ h

N\sum j=0

sj+ 12(t)\eta j+ 1

2(t)+\varepsilon h

N\sum j=0

xj+ 12rj+ 1

2(t)\eta j+ 1

2(t)+\varepsilon h

N\sum j=0

xj+ 12sj+ 1

2(t)\theta j+ 1

2(t)

\leq - \varepsilon eh(t) +\xi

4h

N\sum j=0

\theta 2j+ 12(t) +

1

\xi h

N\sum j=0

r2j+ 12(t) +

\xi

4h

N\sum j=0

\eta 2j+ 12(t) +

1

\xi h

N\sum j=0

s2j+ 12(t)

+\varepsilon \xi

4h

N\sum j=0

\eta 2j+ 12(t) +

\varepsilon

\xi h

N\sum j=0

r2j+ 12(t) +

\varepsilon \xi

4h

N\sum j=0

\theta 2j+ 12(t) +

\varepsilon \xi

\xi h

N\sum j=0

s2j+ 12(t)

= - \varepsilon eh(t) +\xi

2(1 + \varepsilon )eh(t) +

1

\xi (1 + \varepsilon )

\bigl( \| \~r(t)\| 2 + \| \~s(t)\| 2

\bigr) = - \varepsilon

2eh(t) +

(1 + \varepsilon )2

\varepsilon

\bigl( \| \~r(t)\| 2 + \| \~s(t)\| 2

\bigr) \leq - \varepsilon

2(1 + \varepsilon )lh(t) +

(1 + \varepsilon )2

\varepsilon

\bigl( \| \~r(t)\| 2 + \| \~s(t)\| 2

\bigr) ,

where we used the fact \alpha - \varepsilon (1 + \alpha 2)/2 > 0 whenever 0 < \varepsilon < 2\alpha /(1 + \alpha 2), Young'sinequality ab \leq a2\xi /4 + b2/\xi with \xi = \varepsilon /(1 + \varepsilon ), and the equivalence (4.5). By (4.9),

lh(t) \leq e - \varepsilon

2(1+\varepsilon )tlh(0) +

(1 + \varepsilon )2

\varepsilon

\int t

0

e - \varepsilon

2(1+\varepsilon )(t - s) \bigl( \| \~r(\tau )\| 2 + \| \~s(\tau )\| 2

\bigr) d\tau .

According to (4.5), we have

eh(t) \leq e - \varepsilon



(1 + \varepsilon )2

\varepsilon (1 - \varepsilon )

\int t

0

e - \varepsilon

2(1+\varepsilon )(t - s) \bigl( \| \~r(\tau )\| 2 + \| \~s(\tau )\| 2

\bigr) d\tau

\leq e - \varepsilon



2(1 + \varepsilon )3

\varepsilon 2(1 - \varepsilon )C2h4 \forall t \in [0,\infty ).


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Remark 4.1. Theorem 4.3 shows that for any t \in [0,\infty ), if the initial conditionsare exact, the semidiscretized finite difference scheme (2.5) is of second-order conver-gence.

From Theorem 4.3 we can further obtain the convergence of the discretized energyFh(t) to the continuous counterpart F (t) immediately.

Theorem 4.4 (convergence of energy). Suppose that u, v \in C1\bigl( [0,\infty );C3[0, 1]

\bigr) solve system (1.6) and \{ uj(t)\} j, \{ vj(t)\} j solve difference equation (2.5). If the initialconditions \{ uj(0)\} j, \{ vj(0)\} j in system (2.5) are exact and the initial discretizedenergy Fh(0) converges to the initial continuous energy F (0) as h\rightarrow 0, then

| Fh(t) - F (t)| \leq | Fh(0) - F (0)| + \alpha \surd Ch

32

\surd t \forall t \geq 0,

where C is a scalar independent of h. Furthermore,

limh\rightarrow 0

\| Fh - F\| C[0,\infty ) = 0.(4.10)

Proof. In what follows, we use C to represent a positive constant independent ofh although it may have different values in different contexts. From Theorem 4.3, ifthe initial values \{ uj(0)\} j , \{ vj(0)\} j in system (2.5) are exact, then

h

2| vN+1(t) - v(1, t)| 2 =

h

2| \eta N+1(t)| 2 \leq \| \~\eta (t)\| 2 \leq 2eh(t) \leq Ch4(4.11)

for some C > 0. By dissipation law (1.8) and Lemma 3.1, it follows that

F (t) = F (0) - \int t

0

\alpha v2(1, \tau )d\tau , Fh(t) = Fh(0) - \int t

0

\alpha v2N+1(\tau )d\tau .(4.12)

Since both F (t) and Fh(t) decay exponentially to zero as t\rightarrow \infty , it holds that

F (0) =

\int \infty

0

\alpha v2(1, t)dt, Fh(0) =

\int \infty

0

\alpha v2N+1(t)dt.

In addition, since Fh(0) \rightarrow F (0) as h \rightarrow 0, both\int \infty 0v2(1, t)dt and

\int \infty 0v2N+1(t)dt are

bounded. Therefore for any t \in [0,\infty ), by (4.11) and (4.12), it follows that

| Fh(t) - F (t)| \leq | Fh(0) - F (0)| +\int t

0

\alpha \bigm| \bigm| v2N+1(\tau ) - v(1, \tau )2

\bigm| \bigm| d\tau \leq | Fh(0) - F (0)| + \alpha

\biggl( \int t

0

| vN+1(\tau ) - v(1, \tau )| 2 d\tau \biggr) 1

2\biggl( \int \infty

0

| vN+1(t) + v(1, t)| 2 dt\biggr) 1

2

\leq | Fh(0) - F (0)| + \alpha

\biggl( \int t

0

| vN+1(\tau ) - v(1, \tau )| 2 d\tau \biggr) 1

2 \surd 2

\biggl( \int \infty

0

\bigl( v2N+1(t) + v2(1, t)

\bigr) dt

\biggr) 12

\leq | Fh(0) - F (0)| + \alpha \surd Ch

32

\biggl( \int t

0

d\tau

\biggr) 12\biggl( \int \infty

0

v2N+1(t)dt+

\int \infty

0

v2(1, t)dt

\biggr) 12

\leq | Fh(0) - F (0)| + \alpha \surd Ch

32\surd t,

where H\"older's inequality, (4.11), and the boundedness of\int \infty 0v2(1, t)dt and\int \infty

0v2N+1(t)dt were employed. From (1.4) and (3.7), when t\prime > t, both Fh(t

\prime ) andF (t\prime ) can be sufficiently small for sufficiently large t. We thus arrive at uniformconvergence (4.10).

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5. Order reduction finite difference scheme and its convergence. In thissection, we show that the semidiscretized finite difference scheme (2.5) for equivalentsystem (1.6) can lead to a semidiscretized finite difference scheme for original system(1.2). It is this order reduction finite difference scheme that has universal significancefor using the finite difference method to discretize PDEs.

Lemma 5.1. Suppose that w(x, t) solves system (1.2). The semidiscretized finitedifference scheme (2.5) leads to the following semidiscretized finite difference schemeto the original system (1.2):\left\{

1

4

\bigl( w\prime \prime

j - 1(t) + 2w\prime \prime j (t) + w\prime \prime

j+1(t)\bigr) - \delta 2xwj(t) = 0, j = 1, 2, . . . , N,

h

4


N (t) + w\prime \prime N+1(t)

\bigr) + \delta xwN+ 1

2+ \alpha w\prime

N+1 = 0,

w0(t) = 0, wj(0) = w0(xj), w\prime j(0) = w1(xj), j = 0, 1, . . . , N + 1,

(5.1)

where wj(t) is an approximation of w(xj , t).

Proof. From transformation (1.5), it has

uj+ 12= \delta xwj+ 1

2, vj+ 1

2= w\prime

j+ 12, j = 0, . . . , N.(5.2)

Multiplying both sides of the second equation of (2.5) by h/2 gives

uj+1 - uj2

=h

2v\prime j+ 1

2, j = 0, . . . , N,(5.3)

which together with (5.2) yields

uj+1 = \delta xwj+ 12+h

2w\prime \prime

j+ 12, uj = \delta xwj+ 1

2 - h

2w\prime \prime

j+ 12, j = 0, . . . , N.(5.4)

After cancellation of variable uj , we obtain

h

2w\prime \prime

j+ 12+h

2w\prime \prime

j - 12 - \delta xwj+ 1

2+ \delta xwj - 1

2= 0, j = 1, . . . , N,(5.5)

which gives the first equation of (5.1). Letting j = N in the first equation of (5.4),we have

uN+1 = \delta xwN+ 12+h

2w\prime \prime

N+ 12,(5.6)

which, together with the right boundary condition of (2.5), leads to the second equa-tion of (5.1). The left boundary condition and initial conditions are trivial. This endsthe proof of the lemma.

The finite difference scheme (5.1) is indeed the right semidiscretization of originalsystem (1.2) that we are looking for. Also note that scheme (5.1) can be derivedwithout using (2.5); the widely used order reduction finite difference scheme (see,e.g., [19]) can also be used for this purpose as follows. Introduce an intermediatevariable

\zeta (x, t) = wx(x, t).

Then, the second-order differentiation in spatial variable in system (1.2) is reduced tothe first order as

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FINITE DIFFERENCE OF WAVE EQUATION 2269\left\{ wtt(x, t) = \zeta x(x, t),

\zeta (x, t) = wx(x, t),

w(0, t) = 0, \zeta (1, t) = - \alpha wt(1, t).

(5.7)

Construct a standard second-order finite difference scheme for (5.7) to obtain\left\{ 1

2[w\prime \prime (xi, t) + w\prime \prime (xi+1, t)] =

1

h[\zeta (xi+1, t) - \zeta (xi, t)] +\scrO (h2),

1

2[\zeta (xi, t) + \zeta (xi+1, t)] =

1

h[w(xi+1, t) - w(xi, t)] +\scrO (h2), i = 0, . . . , N,

w(0, t) = 0, \zeta (xN+1, t) = - \alpha w\prime (xN+1, t).

(5.8)

Ignoring the infinitesimal terms gives\left\{ 1

2(w\prime \prime

i + w\prime \prime i+1) =

1

h(\zeta i+1 - \zeta i),

1

2(\zeta i + \zeta i+1) =

1

h(wi+1 - wi), i = 0, . . . , N,

w0 = 0, \zeta N+1 = - \alpha w\prime N+1.

(5.9)

Now we eliminate all \zeta i related terms in (5.9). First, multiplying by - h/2 the firstequation of (5.9) and adding to the second one gives

\zeta i+1 =h

4(w\prime \prime

i + w\prime \prime i+1) +

1

h(wi+1 - wi), i = 0, . . . , N.(5.10)

Second, multiplying by h/2 the first equation of (5.9) and adding to the second oneyields

\zeta i = - h4(w\prime \prime

i + w\prime \prime i+1) +

1

h(wi+1 - wi), i = 0, . . . , N.(5.11)

Shifting the subscript of (5.10) and combining with (5.11), we derive the first equationof (5.1). Finally, setting i = N in (5.10) and substituting into the last equation\zeta N+1 = - \alpha w\prime

N+1 of (5.9) we obtain the last equation of (5.1). We have thus derived(5.1) by the standard order reduction finite difference method which turns out to bethe right difference scheme for original system (1.2).

In this section, we denote the discretized energy of system (5.1) by

Eh(t) =h

2

N\sum j=0

\Biggl[ \bigm| \bigm| \bigm| \bigm| w\prime j+1(t) + w\prime

j(t)

2

\bigm| \bigm| \bigm| \bigm| 2 + \bigm| \bigm| \bigm| \bigm| wj+1(t) - wj(t)

h

\bigm| \bigm| \bigm| \bigm| 2\Biggr] .(5.12)

In subsection 5.1, we show that the discretized energy of system (5.1) preserves theuniformly exponential decay.

5.1. Uniformly exponential decay. In this subsection, we show that the en-ergy Eh(t) of semidiscretized system (5.1) defined by (5.12) decays exponentiallyuniformly via the Lyapunov function method. Lemma 5.2 states that the discreteenergy Eh(t) is nonincreasing with respect to time.

Lemma 5.2. The discretized energy Eh(t) defined by (5.12) satisfies

dEh(t)

dt= - \alpha

\bigl( w\prime

N+1

\bigr) 2.(5.13)

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Proof. Multiplying both sides of the first equation of (5.1) by hw\prime j and summing

for j from 1 to N , we obtain

h

4

N\sum j=1


j+1 + 2w\prime \prime j + w\prime \prime

j - 1

\bigr) w\prime

j - h

N\sum j=1

\biggl( wj+1 - 2wj + wj - 1

h2

\biggr) w\prime

j = 0.(5.14)

The first term of (5.14) can be expanded as

h

4

N\sum j=1


j+1 + 2w\prime \prime j + w\prime \prime

j - 1

\bigr) w\prime

j =h

4

N\sum j=1

\bigl[ (w\prime \prime

j+1 + w\prime \prime j )w

\prime j + (w\prime

j + w\prime \prime j - 1)w

\prime j

\bigr] =h

4

N\sum j=0

(w\prime \prime j+1 + w\prime \prime

j )(w\prime j + w\prime

j+1) - h

4w\prime \prime

N+1w\prime N+1 -

h

4w\prime \prime

Nw\prime N+1,(5.15)

and the second term of (5.14) can be expanded as

h

N\sum j=1

\biggl( wj+1 - 2wj + wj - 1

h2

\biggr) w\prime

j =1

h

N\sum j=0

(wj+1 - wj)(w\prime j - w\prime

j+1)(5.16)

+1

h(wN+1 - wN )w\prime

N+1.

Multiplying both sides of the second equation of (5.1) by w\prime N+1 gives

h

4(w\prime \prime

N + w\prime \prime N+1)w

\prime N+1 +

wN+1 - wN

hw\prime

N+1 + \alpha (w\prime N+1)

2 = 0.(5.17)

Plugging (5.15) and (5.16) into (5.14) and combining with (5.17) produce

h

4

N\sum j=0

(w\prime \prime j+1 + w\prime \prime

j )(w\prime j + w\prime

j+1) +1

h

N\sum j=0

(wj+1 - wj)(w\prime j+1 - w\prime

j) + \alpha (w\prime N+1)

2 = 0,

(5.18)

which gives the required result that

dEh(t)

dt= - \alpha (w\prime

N+1)2.

This ends the proof.

In order to acquire the uniformly exponential decay of Eh(t), we define a Lyapunovfunction Lh(t) similarly with (3.5) as

Lh(t) = Eh(t) + \varepsilon \psi h(t), 0 < \varepsilon < 1,(5.19)

where the auxiliary function \psi h(t) is defined by

\psi h(t) = h

N\sum j=1

w\prime j+1 + 2w\prime

j + w\prime j - 1

4jwj+1 - wj - 1

2+

1

4(w\prime

N + w\prime N+1)(wN+1 - wN ).

(5.20)

Lemma 5.3 shows that the Lyapunov function Lh(t) is equivalent to the discretizedenergy Eh(t).

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Lemma 5.3. Suppose that \alpha > 0 in (5.1). For 0 < \varepsilon < 1, the Lyapunov functionLh(t) defined by (5.19) is equivalent to the discretized energy Eh(t):

(1 - \varepsilon )Eh(t) \leq Lh(t) \leq (1 + \varepsilon )Eh(t).

Proof. For the auxiliary function \psi h(t) defined by (5.20), we have, by Cauchy'sinequality, that

| \psi h(t)| \leq h

2

N\sum j=1

\biggl( wj+1 - wj - 1

2h

\biggr) 2

+h

2

N\sum j=1

\biggl( w\prime

j+1 + 2w\prime j + w\prime

j - 1

4

\biggr) 2

+h

4

\biggl( wN+1 - wN

h

\biggr) 2

+h

4

\biggl( w\prime

N + w\prime N+1

2

\biggr) 2

\leq h

4

N\sum j=1

\biggl( wj+1 - wj

h

\biggr) 2

+h

4

N\sum j=1

\biggl( wj - wj - 1

h

\biggr) 2

+h

4

N\sum j=1

\biggl( w\prime

j+1 + w\prime j

2

\biggr) 2

+h

4

N\sum j=1

\biggl( w\prime

j + w\prime j - 1

2

\biggr) 2

+h

4

\biggl( wN+1 - wN

h

\biggr) 2

+h

4

\biggl( w\prime

N+1 + w\prime N

2

\biggr) 2

\leq h

2

N\sum j=0

\biggl( w\prime

j+1 + w\prime j

2

\biggr) 2

+h

2

N\sum j=0

\biggl( wj+1 - wj

h

\biggr) 2

= Eh(t).

(5.21)

Therefore,(1 - \varepsilon )Eh(t) \leq Lh(t) \leq (1 + \varepsilon )Eh(t).

Lemma 5.4. The auxiliary function \psi h(t) defined by (5.20) satisfies

d\psi h(t)

dt\leq - Eh(t) +

1

2(1 + \alpha 2)

\bigl( w\prime

N+1

\bigr) 2.

Proof. Finding the derivative of \psi h(t) along the solution of (5.1) gives

d\psi h(t)

dt= h

N\sum j=1


j + w\prime j - 1

4jw\prime

j+1 - w\prime j - 1

2+

1

4(w\prime

N + w\prime N+1)(w

\prime N+1 - w\prime

N )

+ h

N\sum j=1

w\prime \prime j+1 + 2w\prime \prime

j + w\prime \prime j - 1

4jwj+1 - wj - 1

2+

1

4(w\prime \prime

N + w\prime \prime N+1)(wN+1 - wN ).

(5.22)

The first term on the right-hand side of (5.22) implies

h

N\sum j=1


j + w\prime j - 1

4jw\prime

j+1 - w\prime j - 1

2=h

8

N\sum j=1

j\bigl( (w\prime

j+1 + w\prime j)

2 - (w\prime j + w\prime

j - 1)2\bigr) (5.23)

=h

8

N\sum j=1

j(w\prime j+1 + w\prime

j)2 - h

8

N - 1\sum j=0

(j + 1)(w\prime j+1 + w\prime

j)2

= - h2

N\sum j=0

\biggl( w\prime

j+1 + w\prime j

2

\biggr) 2

+1

8(w\prime

N+1 + w\prime N )2.

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The third term on the right-hand side of (5.22) together with the first equation of(5.1) yields

h

N\sum j=1

w\prime \prime j+1+2w\prime \prime

j +w\prime \prime j - 1

4jwj+1 - wj - 1

2=

1

2h

N\sum j=1

j(wj+1 - wj - 1)(wj+1 - 2wj+wj - 1)

(5.24)

=1

2h

N\sum j=1

j(wj+1 - wj)2 - 1

2h

N\sum j=1

j(wj - wj - 1)2

= - h2

N\sum j=0

\biggl( wj+1 - wj

h

\biggr) 2

+1

2

\biggl( wN+1 - wN

h

\biggr) 2

.

The fourth term on the right-hand side of (5.22) together with the second equationin (5.1) implies

1

4(w\prime \prime

N + w\prime \prime N+1)(wN+1 - wN ) = -

\biggl( wN+1 - wN

h

\biggr) 2

- \alpha w\prime N+1

wN+1 - wN

h.(5.25)

Plugging (5.23), (5.24), and (5.25) into (5.22), we obtain

d\psi h(t)

dt= - Eh(t) +

1

8

\bigl( 3(w\prime

N+1)2 + 2w\prime

Nw\prime N+1 - (w\prime

N )2\bigr)

- 1

2

\biggl( wN+1 - wN

h

\biggr) 2

- \alpha w\prime N+1

wN+1 - wN

h

\leq - Eh(t) +1

2(1 + \alpha 2)

\bigl( w\prime

N+1

\bigr) 2,

where in the last inequality, the Cauchy's inequality was applied twice.

Theorem 5.5. Suppose that \alpha > 0 in (5.1). There exists a constant \varepsilon , indepen-dent of h, satisfying 0 < \varepsilon < 2\alpha /(1 + \alpha 2), such that for all initial conditions \{ w0

j\} j,\{ w1

j\} j \in \BbbR N+2, the energy Eh(t) defined by (5.12) for semidiscretized finite differencescheme (5.1) satisfies

Eh(t) \leq 1 + \varepsilon

1 - \varepsilon e -

\varepsilon 1+\varepsilon tEh(0) \forall t > 0.

Proof. Finding the derivative of the Lyapunov function Lh(t) defined by (5.19)gives

dLh(t)

dt=

dEh(t)

dt+ \varepsilon

d\psi h(t)

dt.

According to Lemmas 5.2 and 5.4, there is

dLh(t)

dt\leq -

\Bigl( \alpha - \varepsilon

2(1 + \alpha 2)

\Bigr) \bigl( w\prime

N+1

\bigr) 2 - \varepsilon Eh(t).

Since \alpha - \varepsilon (1 + \alpha 2)/2 > 0 for 0 < \varepsilon < 2\alpha /(1 + \alpha 2) it follows from the equivalence inLemma 3.2 that

dLh(t)

dt\leq - \varepsilon Eh(t) \leq - \varepsilon

1 + \varepsilon Lh(t).

A direct application of Gronwall's inequality in Lemma 2.1 leads to

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Lh(t) \leq e - \varepsilon

1+\varepsilon tLh(0),

which together with Lemma 5.3 shows that

Eh(t) \leq 1 + \varepsilon

1 - \varepsilon e -

\varepsilon 1+\varepsilon tEh(0).

5.2. Convergence of the solution. In this subsection, we show that the solu-tions of the semidiscretized finite difference scheme (5.1) converge to the solutions ofthe continuous counterpart (1.2). To this purpose, set vectors wh = (wj)j , w

0h = (w0

j )jand w1

h = (w1j )j . Introduce the extension operators defined by

(5.26)

\Biggl\{ q - h vh(x) = vi, x \in [xi, xi+1], i = 0, 1, . . . , N,

q+h vh(x) = vi+1, x \in [xi, xi+1], i = 0, 1, . . . , N.

Define qhvh = 12 (q

+h + q - h )vh. It is a routine task to check that\int 1

0

(qhuh)(qhvh)dx = h

N\sum j=0

uj+ 12vj+ 1

2.(5.27)

Introduce an extension operator

phvh(x) = vix - xi+1

xi - xi+1+ vi+1

x - xixi+1 - xi

, x \in [xi, xi+1], i = 0, . . . , N.(5.28)

It verifies that \int 1

0

(phuh)x(phvh)xdx = h

N\sum j=0

\delta xuj+ 12\delta xvj+ 1

2.(5.29)

For every function v \in C[0, 1], if we set vh = (vj)j = (v(xj))j , it follows easilythat \left\{

q - h vh \rightarrow v strongly in L\infty (0, 1),

q+h vh \rightarrow v strongly in L\infty (0, 1),

qhvh \rightarrow v strongly in L\infty (0, 1).

(5.30)

Moreover, if v \in H1(0, 1), then

\| phvh - qhvh\| 2L2(0,1) =h3

12

N\sum j=0

\Bigl( \delta xvj+ 1

2

\Bigr) 2= \scrO (h2)(5.31)

and

\| (phvh)x\| 2L2(0,1) = h

N\sum j=0

\Bigl( \delta xvj+ 1

2

\Bigr) 2= \scrO (1),(5.32)

which, together with (5.30), imply that

phvh \rightarrow v weakly in H1(0, 1).(5.33)

Now we are in a position to state the convergence of the discrete solution to thecontinuous counterpart.

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Theorem 5.6. Let wh(t) be the solution of (5.1). Assume that w0h and w1

h satisfyfor h\rightarrow 0 that \Biggl\{

phw0h \rightarrow w0 weakly in H1

L(0, 1),

qhw1h \rightarrow w1 weakly in L2(0, 1).

(5.34)

Then, as h\rightarrow 0, \Biggl\{ phwh \rightarrow w weakly\ast in L\infty \bigl( 0,\infty ;H1

L(0, 1)\bigr) ,

qhw\prime h \rightarrow w\prime weakly\ast in L\infty \bigl( 0,\infty ;L2(0, 1)

\bigr) ,

(5.35)

where w(x, t) is the solution of system (1.2). In addition, if Eh(0) \rightarrow E(0) as h\rightarrow 0,then

limh\rightarrow 0

\| Eh - E\| C[0,\infty ) = 0.

Proof. By definitions of ph and qh, for every t \geq 0 and the energy Eh(t) definedby (5.12), it has

Eh(t) =1

2

\Bigl( \| qhw\prime

h(t)\| 2L2(0,1) + \| phwh(t)\| 2H1L(0,1)

\Bigr) .(5.36)

Since Eh(t) is nonincreasing with respect to time t, by the convergence of the ini-tial state (5.34), we know that phwh is bounded in L\infty \bigl( 0,\infty ;H1

L(0, 1)\bigr) and qhw

\prime h is

bounded in L\infty \bigl( 0,\infty ;L2(0, 1)\bigr) . Moreover, it follows from Lemma 5.2 that for all t > 0

Eh(0) = Eh(t) + \alpha

\int t

0

\bigl( w\prime

N+1

\bigr) 2dt,(5.37)

from which we can see that w\prime N+1(t) is bounded in L2(0,\infty ). By the Sobolev embed-

ding theorem and up to extraction of a subsequence if necessary, we have, by Lemma4 of [1], that \left\{

phwh \rightarrow w weakly\ast in L\infty \bigl( 0,\infty ;H1L(0, 1)

\bigr) ,

w\prime N+1 \rightarrow w\prime (1, \cdot ) weakly in L2(0,\infty ),

qhwh \rightarrow w weakly\ast in L\infty \bigl( 0,\infty ;L2(0, 1)\bigr) ,

qhw\prime h \rightarrow w\prime weakly\ast in L\infty \bigl( 0,\infty ;L2(0, 1)

\bigr) ,

(5.38)

where we implicitly assumed that the limits of phwh and qhwh are the same, becausefor every t \geq 0,\int 1

0

| (phwh - qhwh) (x, t)| 2 dx =h3

12

N\sum j=0

\Bigl( \delta xwj+ 1

2

\Bigr) 2\leq h2

6Eh(t) = \scrO (h2).

The second convergence in (5.38) can be proved as follows. Suppose that w\prime N+1 \rightarrow

g weakly in L2(0,\infty ), i.e., for all \phi \in \scrD (0,\infty ), it has\int \infty

0

w\prime N+1\phi dt\rightarrow

\int \infty

0

g\phi dt.

Then, \int \infty

0

w\prime N+1\phi dt = -

\int \infty

0

wN+1\phi \prime dt\rightarrow -

\int \infty

0

w(1, t)\phi \prime dt.

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Thus, g(t) = w\prime (1, t) in the sense of distribution.Next we show that the limit w(x, t) is the (weak) solution of (1.2). Let \varphi \in

\scrD ([0, 1] \times (0,\infty )) with \varphi (0, \cdot ) = 0 and let \varphi h = (\varphi (xj , \cdot ))j . First, multiplying thefirst equation of (5.1) by h\varphi j , performing the integration by parts over (0,\infty ), andsumming up from j = 1, . . . , N , we obtain\int \infty

0

h

4

N\sum j=1

(wj - 1 + 2wj + wj+1)\varphi \prime \prime j dt -

\int \infty

0

h

N\sum j=1

\delta 2xwj\varphi jdt = 0.(5.39)

Second, multiplying the second equation of (5.1) by \varphi N+1 and performing integrationby parts over (0,\infty ), we have

\int \infty

0

h

4(wN + wN+1)\varphi

\prime \prime N+1dt+

\int \infty

0

\delta xwN+ 12\varphi N+1dt+ \alpha

\int \infty

0

w\prime N+1\varphi N+1dt = 0.

(5.40)

Adding (5.39) to (5.40) and performing summation by parts, we obtain\int \infty

0

h

N\sum j=0

wj+ 12\varphi \prime \prime j+ 1

2dt+

\int \infty

0

h

N\sum j=0

\delta xwj\delta x\varphi jdt+ \alpha

\int \infty

0

w\prime N+1\varphi N+1dt = 0.(5.41)

By definitions of qh and ph, it is easy to check that (5.41) is equivalent to

\int \infty

0

\int 1

0

(qhwh)(qh\varphi \prime \prime h)dxdt+

\int \infty

0

\int 1

0

(phwh)x(ph\varphi h)xdxdt+ \alpha

\int \infty

0

w\prime N+1\varphi (1, t)dt = 0.

(5.42)

If we can show that for every \varphi \in \scrD ([0, 1]\times (0,\infty )),\Biggl\{ ph\varphi h \rightarrow \varphi strongly in L2

\bigl( 0,\infty ;H1

L(0, 1)\bigr) ,

qh\varphi h \rightarrow \varphi strongly in L2\bigl( 0,\infty ;L2(0, 1)

\bigr) ,

(5.43)

then, with the convergence in (5.38), we can pass to the limit as h \rightarrow 0 for all termsin (5.42) to arrive at [1]

\int \infty

0

\int 1

0

w(x, t)\varphi \prime \prime (x, t)dxdt+

\int \infty

0

\int 1

0

wx(x, t)\varphi x(x, t)dxdt+ \alpha

\int \infty

0

w\prime (1, t)\varphi (1, t)dt = 0.

(5.44)

This shows that w(x, t) is a weak solution to (1.2).Now we show (5.43). The first convergence in (5.43) is similar to that in [24]. For

the second one, observe that

\int \infty

0

\int 1

0

\bigm| \bigm| q+h \varphi h - \varphi \bigm| \bigm| 2 dxdt = \int \infty

0

h

N\sum j=0

\varphi 2j+1dt

(5.45)

+

\int \infty

0

\int 1

0

\varphi 2dxdt - 2

\int \infty

0

h

N\sum j=0

\varphi j+1

\int 1

0

\varphi (xj + sh, t)dsdt.

By the Riemann sum nature, we have

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limh\rightarrow 0

\int \infty

0

\int 1

0

\bigm| \bigm| q+h \varphi h - \varphi \bigm| \bigm| 2 dxdt = 0.(5.46)

Similarly,

limh\rightarrow 0

\int \infty

0

\int 1

0

\bigm| \bigm| q - h \varphi h - \varphi \bigm| \bigm| 2 dxdt = 0.(5.47)

Since

\int \infty

0

\int 1

0

| qh\varphi h - \varphi | 2 dxdt \leq 1

2

\int \infty

0

\int 1

0

\bigm| \bigm| q+h \varphi h - \varphi \bigm| \bigm| 2 dxdt+ 1

2

\int \infty

0

\int 1

0

\bigm| \bigm| q - h \varphi h - \varphi \bigm| \bigm| 2 dxdt,

(5.48)

the second convergence follows from (5.46) and (5.47).It remains to show that w(x, t) satisfies the initial conditions w(\cdot , 0) = w0 and

w\prime (\cdot , 0) = w1. For this purpose, we set v \in \scrD (0, 1) and l \in \scrD [0,\infty ) and letvh = (v(xj))j . First, multiplying the first equation of (5.1) by hvj l, performingthe integration by parts over [0,\infty ), and summing up from j = 1, . . . , N , we obtain

- l(0)h4

N\sum j=1

\bigl( w1

j - 1 + 2w1j + w1

j+1

\bigr) vj + l\prime (0)

h

4

N\sum j=1

\bigl( w0

j - 1 + 2w0j + w0

j+1

\bigr) vj

+

\int \infty

0

h

4

N\sum j=1

(wj - 1 + 2wj + wj+1) vj l\prime \prime dt -

\int \infty

0

h

N\sum j=1

\delta 2xwjvj ldt = 0.

(5.49)

Second, multiplying the second equation of (5.1) by vN+1l and performing the inte-gration by parts over [0,\infty ), we have

- l(0)h

4(w1

N + w1N+1)vN+1 + l\prime (0)

h

4(w0

N + w0N+1)vN+1

+

\int \infty

0

h

4(wN + wN+1)vN+1l

\prime \prime dt+

\int \infty

0

\delta xwN+ 12vN+1ldt = 0.

(5.50)

Add (5.49) to (5.50) to obtain

- l(0)h

N\sum j=0

w1j+ 1

2vj+ 1

2+ l\prime (0)h

N\sum j=0

w0j+ 1

2vj+ 1

2

+

\int \infty

0

h

N\sum j=0

wj+ 12vj+ 1

2l\prime \prime dt+

\int \infty

0

h

N\sum j=0

\delta xwj\delta xvj ldt = 0.

(5.51)

Using the definition of ph and qh, it is easy to check that (5.51) is equivalent to

- l(0)

\int 1

0

qhw1hqhvhdx+ l\prime (0)

\int 1

0

qhw0hqhvhdx

+

\int \infty

0

\int 1

0

qhwhqhvhl\prime \prime dxdt+

\int \infty

0

\int 1

0

(phwh)x(phvh)xldxdt = 0.

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Passing to the limit as h\rightarrow 0 in (5.52) gives

- l(0)\int 1

0

w1vdx+ l\prime (0)

\int 1

0

w0vdx+

\int \infty

0

\int 1

0

wvl\prime \prime dxdt+

\int \infty

0

\int 1

0

wxvxldxdt = 0,

(5.53)

from which we can easily derive w(\cdot , 0) = w0, w\prime (\cdot , 0) = w1. Since system (1.2) admitsa unique solution, we can conclude that the convergence in (5.38) holds for the wholesequence \{ h\} , not only for an extracted subsequence.

Finally, by (1.3) and (5.13), it follows that

E(0) = E(t) + \alpha

\int t

0

(w\prime (1, t))2dt, Eh(0) = Eh(t) + \alpha

\int t

0

(w\prime N+1)

2dt.(5.54)

Since both E(t) and Eh(t) are exponentially stable, we have

E(0) = \alpha

\int \infty

0

(w\prime (1, t))2dt, Eh(0) = \alpha

\int \infty

0

(w\prime N+1)

2dt.(5.55)

The assumption Eh(0) \rightarrow E(0) implies that\int \infty

0

(w\prime N+1)

2dt\rightarrow \int \infty

0

(w\prime (1, t))2dt as h\rightarrow 0,(5.56)

which together with the weak convergence in (5.38) yields

w\prime N+1 \rightarrow w\prime (1, \cdot ) strongly in L2(0,\infty ).(5.57)

By (5.54) and (5.57), we have for all t > 0 that

| Eh(t) - E(t)| \leq | Eh(0) - E(0)| + \alpha

\bigm| \bigm| \bigm| \bigm| \int t

0

(w\prime N+1)

2dt - \int t

0

(w\prime (1, t))2dt

\bigm| \bigm| \bigm| \bigm| \leq | Eh(0) - E(0)| + \alpha

\biggl( \int t

0

\bigm| \bigm| w\prime N+1 + w\prime (1, t)

\bigm| \bigm| 2 dt\biggr) 1/2 \biggl( \int t

0

\bigm| \bigm| w\prime N+1 - w\prime (1, t)

\bigm| \bigm| 2 dt\biggr) 1/2

\leq | Eh(0) - E(0)| +\surd 2\alpha

\bigl( \| w\prime

N+1\| L2(0,\infty ) + \| w\prime (1, \cdot )\| L2(0,\infty )

\bigr) \| w\prime

N+1 - w\prime (1, \cdot )\| L2(0,\infty ).

(5.58)

This, together with Eh(0) \rightarrow E(0) and (5.56), shows that

limh\rightarrow 0

\| Eh - E\| C([0,\infty )) = 0.


6. Uniform observability and controllability. In this section, we discuss twoadditional preservation properties of the uniform observability and uniform controlla-bility of the finite difference semidiscrete scheme (5.1) to the following control systemin \scrH = H1

L(0, 1)\times L2(0, 1):\left\{ wtt(x, t) - wxx(x, t) = 0, 0 < x < 1, t > 0,w(0, t) = 0, wx(1, t) = U(t),w(x, 0) = w0(x), wt(x, 0) = w1(x).

(6.1)

Before we turn to the uniform approximation of exact controllability of system (6.1),we introduce some results and notation for the continuous problem.

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6.1. Continuous problem: Known results and notation. Lemmas 6.1 and6.2 are available in [26].

Lemma 6.1. System (6.1) is exactly controllable over [0, T ], T > 2. Precisely, forany T > 2 and any initial value (w0, w1) \in \scrH , there exists a control U(\cdot ) \in L2(0, T )such that the solution w(x, t) of system (6.1) satisfies

w(x, T ) = wt(x, T ) = 0, x \in (0, 1).(6.2)

Control U(t) having property (6.2) satisfies\int T

0

U(t)wt(1, t)dt = - \int 1

0

w1w1dx - \int 1

0

w0xw

0xdx,(6.3)

for any (w0, w1) \in \scrH , where (w,wt) is the solution of the following adjoint system:

(6.4)

\left\{ wtt(x, t) - wxx(x, t) = 0, x \in (0, 1), t > 0,

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

w(x, 0) = w0(x), wt(x, 0) = w1(x), x \in [0, 1].

It is known that among the admissible controls (i.e., all controls having prop-erty (6.2)), there is a unique control with minimal L2-norm. The optimal control,also referred to as HUM control, can be characterized as the minimizer of a suitablefunctional. Let us introduce the functional \scrJ : \scrH \rightarrow \BbbR by

\scrJ ((w0, w1)) =1

2

\int T

0

| wt(1, t)| 2dt+\int 1

0

w1(x)w1(x)dx(6.5)

+

\int 1

0

w0x(x)w

0x(x)dx \forall (w0, w1) \in \scrH .

It is well known that the observability inequality of system (6.4) holds true:

\=E(0) \leq 1

2(T - 2)

\int T

0

| wt(1, t)| 2dt, \=E(t) =1

2

\int 1

0

[w2x(x, t) + w2

t (x, t)]dx,(6.6)

which means that system (1.1) is exactly observable over [0, T ], T > 2, and ensuresthat \scrJ (\cdot ) is coercive and thus has a unique minimizer.

Lemma 6.2. For given initial state (w0, w1) \in \scrH , suppose that ( \widehat w0, \widehat w1) \in \scrH isthe minimizer of \scrJ (\cdot ). Then, U(t) = \widehat wt(1, t) is the HUM control of system (6.1) withproperty (6.2), where ( \widehat w(x, t), \widehat wt(x, t)) is the solution of adjoint system (6.4) with theinitial value ( \widehat w0(x), \widehat w1(x)).

6.2. Uniform observability inequality. Parallel to finite difference scheme(5.1), the semidiscretized finite difference scheme of control system (6.1) and adjointsystem (6.4) reads\left\{

1

4




h

4




2= Uh(t),


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and \left\{

1

4




h

4




2= 0,


(6.8)

respectively. The discrete energy \=Eh(t) is also defined for (6.8) paralleling as (5.12),which is conservative,

d \=Eh(t)

dt= 0,(6.9)

similar to that of Lemma 5.2 by setting \alpha = 0 in (5.13).Theorem 6.3 means that the observability inequality (6.6) is uniformly preserved

by the semidiscrete scheme (5.1), which is the discrete counterpart of the observationproblem of original system (1.1).

Theorem 6.3. Let T > 2 and \=Eh(t) be the discrete energy corresponding to (6.8).Then,

\=Eh(0) \leq 1

2(T - 2)

\int T

0

| w\prime N+1(t)| 2dt.(6.10)

Proof. Multiplying the first equation of (6.8) by jh(wj+1 - wj - 1)/2 and summingfor j from 1 to N give

0 =h

4

N\sum j=1

(w\prime \prime j - 1 + 2w\prime \prime

j + w\prime \prime j+1)j

wj+1 - wj - 1

2 - 1

h

N\sum j=1

(wj - 1 - 2wj + wj+1)jwj+1 - wj - 1

2

=d

dt

h

4

N\sum j=1

(w\prime j - 1 + 2w\prime

j + w\prime j+1)j

wj+1 - wj - 1

2+

h

8

N\sum j=0

(w\prime j + w\prime

j+1)2 - 1

8(w\prime

N + w\prime N+1)

2

+h

2

N\sum j=0

\biggl( wj+1 - wj

h

\biggr) 2

- 1

2

\biggl( wN+1 - wN

h

\biggr) 2

.

(6.11)

Multiply the second equation of system (6.8) by (N + 1)(wN+1 - wN ) to give

0 =d

dt

1

4(w\prime

N + w\prime N+1)(wN+1 - wN ) - 1

4(w\prime

N + w\prime N+1)(w

\prime N+1 - w\prime

N )(6.12)

+

\biggl( wN+1 - wN

h2

\biggr) 2

.

Adding (6.11) with (6.12), we obtain

dXh(t)

dt+ Eh(t) -

1

8(3| w\prime

N+1| 2 + 2w\prime Nw

\prime N+1 - | w\prime

N | 2) + 1

2

\biggl( wN+1 - wN

h

\biggr) 2

= 0,

(6.13)

where

Xh(t) =h

4

N\sum j=1

(w\prime j - 1 + 2w\prime

j + w\prime j+1)j

wj+1 - wj - 1

2+

1

4(w\prime

N + w\prime N+1)(wN+1 - wN ).

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Since

| Xh(t)| \leq h

N\sum j=1

\bigm| \bigm| \bigm| w\prime j - 1 + 2w\prime

j + w\prime j+1

4

\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| wj+1 - wj - 1

2h

\bigm| \bigm| \bigm| + h

2

\bigm| \bigm| \bigm| w\prime N + w\prime

N+1

2

\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| wN+1 - wN

h

\bigm| \bigm| \bigm| \leq h

2

N\sum j=1

\bigm| \bigm| \bigm| w\prime j - 1 + 2w\prime

j + w\prime j+1

4

\bigm| \bigm| \bigm| 2 + h

2

N\sum j=1

\bigm| \bigm| \bigm| wj+1 - wj - 1

2h


4


N+1

2

\bigm| \bigm| \bigm| 2+h

4

\bigm| \bigm| \bigm| wN+1 - wN

h

\bigm| \bigm| \bigm| 2\leq h

4

N\sum j=1

\bigm| \bigm| \bigm| w\prime j - 1 + w\prime

j

2


4

N\sum j=1

\bigm| \bigm| \bigm| w\prime j + w\prime

j+1

2


4


N+1

2

\bigm| \bigm| \bigm| 2+h

4

N\sum j=1

\bigm| \bigm| \bigm| wj+1 - wj

h


4

N\sum j=1

\bigm| \bigm| \bigm| wj - wj - 1

h


4

\bigm| \bigm| \bigm| wN+1 - wN

h

\bigm| \bigm| \bigm| 2 \leq Eh(t),

integrating both sides of (6.13) from 0 to T with respect to time produces

0 \geq Xh(t)\bigm| \bigm| \bigm| T0+

\int T

0

Eh(t)dt - 1

2

\int T

0

| w\prime N+1| 2dt \geq (T - 2)Eh(0) -

1

2

\int T

0

| w\prime N+1| 2dt,

where we used the fact (6.9). This gives the uniform observability inequality(6.10).

6.3. Uniform controllability. For notational simplicity, we introduce matricesP,B \in \scrM (N+1)\times (N+1) as

P =1

4

\left( 2 11 2 1

. . .. . .

. . .

1 2 11 1

\right) , B =

\left( 2 - 1 - 1 2 - 1

. . .. . .

. . .

- 1 2 - 1 - 1 1

\right) .(6.14)

Let Ph = hP and Bh = 1hB. Then, the control systems (6.7) and adjoint system (6.8)

can be rewritten respectively into the following vectorial forms:

(6.15)

\Biggl\{ PhW

\prime \prime h (t) +BhWh(t) = Fh(t),

Wh(0) =W 0h , W

\prime h(0) =W 1

h ,

and

(6.16)

\Biggl\{ PhW

\prime \prime h(t) +BhWh(t) = 0,

Wh(0) =W0

h, W\prime h(0) =W

1

h,

where Wh(t) = (w1(t), w2(t), . . . , wN+1(t))T, W 0

h =\bigl( w0

1, w02, . . . , w

0N+1

\bigr) T, W 1

h =\bigl( w1

1, w12, . . . , w

1N+1

\bigr) T, Fh(t) = (0, 0, . . . , Uh(t))

Tand similarly for Wh(t), W

0

h, and

W1

h.

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The uniform observability (6.10) shows that for any h > 0, system (6.15) is exactlycontrollable over any [0, T ], T > 2. Precisely, for any initial state (W 0

h ,W1h ) \in \BbbR 2N+2

and T > 2, there exists a control Fh(t) \in L2(0, T ) such that

Wh(T ) =W \prime h(T ) = 0.(6.17)

Let \langle \cdot , \cdot \rangle be the canonical inner product in \BbbR N+1. Multiplying (6.15) by the solution

W\prime h of system (6.16) and integrating in time over [0, T ], we obtain

\int T

0

\langle Fh,W\prime h\rangle dt

=

\int T

0

[\langle PhW\prime \prime h ,W

\prime h\rangle + \langle BhWh,W

\prime h\rangle ]dt

= \langle PhW\prime h,W

\prime h\rangle \bigm| \bigm| \bigm| T0+ \langle BhWh,Wh\rangle

\bigm| \bigm| \bigm| T0 - \int T

0

[\langle PhW\prime h,W

\prime \prime h\rangle + \langle BhW

\prime h,Wh\rangle ]dt

= \langle PhW\prime h,W

\prime h\rangle \bigm| \bigm| \bigm| T0+ \langle BhWh,Wh\rangle

\bigm| \bigm| \bigm| T0 - \int T

0

[\langle W \prime h, PhW

\prime \prime h\rangle + \langle W \prime

h, BhWh\rangle ]dt

= \langle PhW\prime h(T ),W

\prime h(T )\rangle - \langle PhW

1h ,W

1

h\rangle + \langle BhWh(T ),Wh(T )\rangle - \langle BhW0h ,W

0

h\rangle .

(6.18)

Since (6.17), it follows from (6.18) that\int T

0

Uh(t)w\prime N+1(t)dt+ \langle PhW

1h ,W

1

h\rangle + \langle BhW0h ,W

0

h\rangle = 0,(6.19)

which is just the condition that the discrete control Uh(t) should satisfy to achieve(6.17).

Let us introduce the inner product \langle \cdot , \cdot \rangle 0 in \BbbR 2N+2

\langle (f0, f1), (g0, g1)\rangle 0 = \langle Phf1, g1\rangle + \langle Bhf

0, g0\rangle , f i, gi \in \BbbR N+1, i = 0, 1,(6.20)

and the corresponding inner product induced norm is \| \cdot \| 0. From the definition of\langle \cdot , \cdot \rangle 0, we can check that the energy of adjoint system (6.16) satisfies

\=Eh(0) =1

2

\bigm\| \bigm\| \bigm\| \Bigl( W 0

h,W1

h

\Bigr) \bigm\| \bigm\| \bigm\| 20.

Introduce functional \scrJ h : \BbbR 2N+2 \rightarrow \BbbR by

\scrJ h

\Bigl( \Bigl( W

0

h,W1

h

\Bigr) \Bigr) =

1

2

\int T

0

| w\prime N+1| 2dt+

\Bigl\langle PhW

1h ,W

1

h

\Bigr\rangle +\Bigl\langle BhW

0h ,W

0

h

\Bigr\rangle ,(6.21)

which is the discrete counterpart of continuous functional (6.5).

Lemma 6.4. The functional \scrJ h(\cdot ) defined by (6.21) has a unique minimizer

(\widehat W 0,\widehat W 1).

Proof. First, we observe that \scrJ h(\cdot ) is strictly convex in \BbbR 2N+2 and hence is con-tinuous. Next, we check that \scrJ h is coercive, i.e.,

lim\| (W 0

h,W1h)\| 0\rightarrow \infty

\scrJ h

\Bigl( \Bigl( W

0

h,W1

h

\Bigr) \Bigr) = +\infty .(6.22)

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Indeed, from observability inequality (6.10), we have

\scrJ h

\Bigl( \Bigl( W

0

h,W1

h

\Bigr) \Bigr) =

1

2

\int T

0

| w\prime N+1(t)| 2dt+ \langle

\bigl( W 0

h ,W1h

\bigr) ,\Bigl( W

0

h,W1

h

\Bigr) \rangle 0

\geq T - 2

2\| \Bigl( W

0

h,W1

h

\Bigr) \| 20 - \| (W 0

h ,W1h )\| 0

\bigm\| \bigm\| \bigm\| \Bigl( W 0

h,W1

h

\Bigr) \bigm\| \bigm\| \bigm\| 0,

which implies that \scrJ h((W0

h,W1

h)) \rightarrow \infty as \| (W 0h ,W

1h )\| 0 \rightarrow +\infty . Therefore, \scrJ h(\cdot )

has a unique minimizer (\widehat W 0h ,\widehat W 1

h ).

Theorem 6.5 means that the semidiscrete scheme (5.1) preserves uniformly theexact controllability of original system (1.1).

Theorem 6.5. Suppose that (\widehat W 0h ,\widehat W 1

h ) is the unique minimizer of \scrJ h(\cdot ). Then,Uh(t) = \widehat w\prime

N+1(t) is the control of system (6.15) satisfying property (6.17), where

(\widehat Wh(t),\widehat W \prime h(t)) is the solution of adjoint system (6.16) with the initial state (\widehat W 0

h ,\widehat W 1

h ).Furthermore,

\| Uh\| L2(0,T ) \leq \surd T - 2\| (W 0

h ,W1h )\| 0(6.23)

and

\| (\widehat W 0h ,\widehat W 1

h )\| 0 \leq 2

T - 2\| (W 0

h ,W1h )\| 0.(6.24)

Proof. The fact that \widehat w\prime N+1(t) is just the control that drives the state of (6.15)

from (W 0h ,W

1h ) to rest at time T is the counterpart of Lemma 6.2 for discrete system

(6.15), which comes from general theory of finite-dimensional linear systems; see, e.g.,[26, pp. 200--201]. We only need to show that the control Uh(t) is bounded in L2(0, T ).

Since (\widehat W 0h ,\widehat W 1

h ) is a minimizer of \scrJ h(\cdot ), it has

\scrJ h((\widehat W 0h ,\widehat W 1

h )) \leq \scrJ h((0, 0)) = 0.(6.25)

By observability inequality (6.10), we have

1

2

\int T

0

| \widehat w\prime N+1| 2dt \leq - \langle (W 0

h ,W1h ), (\widehat W 0

h ,\widehat W 1h )\rangle 0 \leq \| (W 0

h ,W1h )\| 0\| (\widehat W 0

h ,\widehat W 1h )\| 0

=\sqrt{}

2E \widehat wh (0)\| (W 0

h ,W1h )\| 0 \leq

\surd T - 2

\biggl( \int T

0

| \widehat w\prime N+1| 2dt

\biggr) 12

\| (W 0h ,W

1h )\| 0,

which implies that

\| Uh\| L2(0,T ) =

\Biggl( \int T

0

| \widehat w\prime N+1| 2dt

\Biggr) 12

\leq \sqrt{} 4(T - 2)\| (W 0

h ,W1h )\| 0.

Furthermore, from (6.25) and observability inequality (6.10), we obtain

T - 2

2\| (\widehat W 0

h ,\widehat W 1

h )\| 20 = (T - 2)E \widehat wh (0) \leq

1

2

\int T

0

| \widehat w\prime N+1| 2dt \leq \| (W 0

h ,W1h )\| 0\| (\widehat W 0

h ,\widehat W 1h )\| 0,

which implies that

\| (\widehat W 0h ,\widehat W 1

h )\| 0 \leq 4CT \| (W 0h ,W

1h )\| 0.

This ends the proof of the theorem.

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7. Numerical experiments. For a numerical experiment, we also need to dis-cretize the time derivative in our semidiscretized finite difference schemes. We carryout a numerical experiment for fully discretized finite difference schemes of both (2.5)and (5.1), separately.

7.1. Full-discretization scheme for (2.5). First, we rewrite the semidiscretizedsystem (2.5) into a vector form\left\{ W \prime (t) =

2

hM - 1NW (t),

W (0) =W 0,

(7.1)

whereW (t) =\bigl( U(t)T , V (t)T

\bigr) T, U(t) = (u0(t), u1(t), . . . , uN (t))

T, V (t) = (v1(t), v2(t),

. . . , vN+1(t))T ,

M =

\biggl( A CO AT

\biggr) , N =

\biggl( O B

- BT C

\biggr) .

The matrices A, B, C, O \in \scrM (N+1)\times (N+1), where O is a zero matrix and A, B, Cread

A =

\left(

1 1

1 1

. . .. . .

1 1

1

\right) , B =

\left(

1

- 1 1

. . .. . .

- 1 1

- 1 1

\right) , C =

\left(

0 0 . . . 0 0

0 0 . . . 0 0

......

. . ....

...

0 0 . . . 0 0

0 0 . . . 0 - \alpha

\right) .

Introducing a parameter \theta \in [0, 1], we can get the following fully discretized finitedifference scheme:

Wn+1 - Wn

\tau =

2

h(1 - \theta )M - 1NWn +

2

h\theta M - 1NWn+1,

where \tau is the time step. If we let the mesh grid ratio \lambda = \tau /h (the CFL conditionnumber), we can get the computational form of Wn+1

(I - 2\lambda \theta M - 1N)Wn+1 = (I + 2\lambda (1 - \theta )M - 1N)Wn,(7.2)

where I is the 2N + 2 order identity matrix. From (7.2) we see that if \theta = 0, it isan explicit difference scheme. If 0 < \theta \leq 1, it is an implicit difference scheme. In thesimulation process, we find that the proposed finite difference (7.2) is unconditionallystable when \theta \in [0.5, 1]. When \theta \in [0, 0.5), the finite difference (7.2) is conditionallystable, although we cannot obtain the sharp CFL condition number which dependson \theta and \alpha .

Now we set the initial conditions (u0(x), v0(x)) of system (2.5) to check the ex-ponential decay of difference scheme (7.2). Here we take

u0(x) = 1 - | 2x - 1| , v0(x) = 0, x \in (0, 1).(7.3)

The logarithmic energy decay curves with \theta = 0.4, \theta = 0.5 and \theta = 1 are depicted inFigures 1, 2, and 3, respectively.

It is seen in Figure 1 that the difference scheme (7.2) is conditionally stable when\theta = 0.4. A smaller CFL condition number gives better stability of the scheme. If the

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0 5 10 15 20 25

logF

h(t)

10-300

10-250

10-200

10-150

10-100

10-50

100

1050α=1, θ=0.4, λ=0.2

Time t

N=10N=20N=40N=80

0 10 20 30 40 50 60 70 80

logF

h(t)

10-300

10-200

10-100

100

10100

10200

10300α=1, θ=0.4, λ=1

Time t

N=10N=20N=40N=60

Fig. 1. Logarithmic energy decay with \alpha = 1, \theta = 0.4.

0 5 10 15 20 25

logF

h(t)

10-300

10-250

10-200

10-150

10-100

10-50

100

1050α=1, θ=0.5, λ=1

Time t

N=20N=40N=60N=80

0 10 20 30 40 50 60 70 80 90

logF

h(t)

10-300

10-250

10-200

10-150

10-100

10-50

100

1050α=1, θ=0.5, λ=2

Time t

N=20N=40N=60N=80

Fig. 2. Logarithmic energy decay with \alpha = 1, \theta = 0.5.

Time t0 10 20 30 40 50 60 70 80

logF

h(t)

10-40

10-35

10-30

10-25

10-20

10-15

10-10

10-5

100

105α=2, θ=1, λ=1

N=20N=40N=60N=80

Time t0 10 20 30 40 50 60 70 80

logF

h(t)

10-40

10-35

10-30

10-25

10-20

10-15

10-10

10-5

100

105α=2, θ=1, λ=2

N=20N=40N=60N=80

Fig. 3. Logarithmic energy decay with \alpha = 2, \theta = 1.

CFL condition number \lambda = 1, it diverges to infinity as N goes to infinity. Figures 2and 3 show that if \theta \in [0.5, 1], the difference scheme is unconditionally stable. In eachsubfigure of Figures 2 and 3, we can see that the discretized energy decays nearly atthe same exponential decay rate for N = 20, 40, 60, and 80.

7.2. Full-discretization scheme for (5.1). The semidiscretized finite differ-ence scheme (5.1) can be rewritten into a vector form

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Time t0 50 100 150 200 250 300

logE

h(t)

10-160

10-140

10-120

10-100

10-80

10-60

10-40

10-20

100

1020α=2, λ=1

N=20N=40N=60N=80

Time t0 50 100 150 200 250 300

logE

h(t)

10-40

10-35

10-30

10-25

10-20

10-15

10-10

10-5

100

105α=2, λ=2

N=20N=40N=60N=80

Fig. 4. Logarithmic energy decay with \alpha = 2, \lambda = 1, 2.

(7.4)

\left\{ PW \prime \prime (t) +1

h2BW (t) +

1

hAW \prime (t) = 0,

W (0) =W 0, W \prime (0) =W 1,

where W (t) = (w1(t), w2(t), . . . , wN+1(t))T, W 0 =

\bigl( w0

1, w02, . . . , w

0N+1

\bigr) T, W 1 =

(w11, w

12, . . . , w

1N+1)

T , P,B are as defined in (6.14), and the N + 1 order diagonalmatrix A = diag\{ 0, . . . , 0, \alpha \} . Let the time step be \tau , and discretize the time deriv-ative by the central difference operator and take an average operator for the spacedirection, to produce the following full-discretization finite difference scheme:

P\delta 2tWk +

1

4h2B\Bigl( W k - 1 + 2W k +W k+1

\Bigr) +

1

2\tau hA\Bigl( W k+1 - W k - 1

\Bigr) = 0, k = 1, 2, . . . .

(7.5)

Also let mesh grid ratio \lambda = \tau /h. Then, (7.5) implies that

\biggl( P +

\lambda 2

4B +

\lambda

2A

\biggr) W k+1 =

\biggl( 2P - \lambda 2

2B

\biggr) W k -

\biggl( P +

\lambda 2

4B - \lambda

2A

\biggr) W k - 1, k = 1, 2, . . . ,

(7.6)

which is the actual difference scheme for our numerical experiment. We set the initialconditions (w0(x), w1(x)) of system (5.1) as

w0(x) = 1 - | 2x - 1| , w1(x) = 0, x \in (0, 1),(7.7)

to check the exponential decay of difference scheme (7.6). The logarithmic energydecay curves with \alpha = 2 and \lambda = 1, 2 are displayed in Figure 4, which shows boththe unconditional stability and the uniformly exponential decay of difference scheme(7.6).

8. Concluding remarks. In this paper, a novel space semidiscretized finite dif-ference scheme is proposed for approximating the exponentially stable one-dimensionalboundary damped wave equation uniformly. The original system is first transformedinto an equivalent system by a universally used equivalent transformation for waveequations. An ordinary semidiscretized difference scheme is then constructed for theequivalent system. The convergence of the difference scheme to the equivalent sys-tem is shown to be of second-order convergence. The uniformly exponential decay isestablished for the semidiscretized finite difference scheme. Of equal significance, the

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semidiscretized difference scheme for the equivalent system is found to be a widelyused order reduction finite difference scheme to the original system. Both uniformlyexponential decay and the convergence of the solutions are established rigorously forthis new discrete scheme to the original system. A numerical experiment is conductedto verify the theoretical analysis under two different implicit time finite differenceschemes. In contrast to other fully discretized finite difference schemes of preservingthe uniformly exponential decay, the proposed implicit difference scheme possesses theunconditional stability of the finite difference scheme and the uniformly exponentialdecay of discretized energy, without using any numerical viscosity. In addition, this fi-nite difference semidiscrete scheme also preserves two additional important propertiesof uniformly exact observability and exact controllability.

This new finite difference scheme provides a basic principle for numerical compu-tation of PDE control systems such as wave equations and any other types of PDEs.There are at least four good features for this new scheme: (a) It preserves the uniformexponential stability and other control properties such as uniformly observability andcontrollability; (b) it can be used to deal with any other boundary conditions; (c) theproof for the semidiscretized finite difference system is completely analogous to thecontinuous counterpart; and (d) it naturally keeps the finite difference nature. Webelieve that this scheme can be used to study the convergence of the minimal energycontrol for driving a given state to the origin in finite time, which is our future work.It particularly tells what a proper finite difference scheme should be when one needsto approximate PDEs in numerical simulations or practical implementation.

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Documents

A New Semidiscretized Order Reduction Finite Difference