ZHANG Xiao-ying,FENG Hong-yinping

(1.Department of Basic Courses,Shanxi Agricultural University,Taigu Shanxi 030801,China;2.School of Mathematical Sciences,Shanxi University,Taiyuan Shanxi 030006,China)

Abstract：This paper considers the output tracking for one-dimensional Schr?dinger equation with disturbance via noncollocated boundary control.Firstly,we design an infinite-dimensional disturbance estimator to estimate the disturbance by virtue of the output and infinite structure of the system.Secondly,the adaptive servomechanism is designed to achieve the performance output tracking and the tracking error ?u(1,t)∈L2(0,∞)and all the internal-loops are bounded.Finally,we present some numerical simulations to illustrate the effectiveness of the proposed scheme.

Key words：boundary control;disturbance;output tracking;Schr?dinger equation

1 Introduction

Output tracking problem is one of the important contents of control theory research.The main idea of the output tracking problem is to design a controller to make the output signal converges to a given reference signal and,at the same time,all the internal systems are required to be bounded.Compared with finitedimensional output tracking,the output tracking results about infinite-dimensional systems are relatively few and have attracted the attention of researchers in recent years.Some results can be found in [1-3]and [4-13],where performance output tracking for wave equation,heat equation are considered,respectively.In this paper,we are concerned with the performance output tracking for a one-dimensional Schr?dinger equation with a general boundary disturbance at one end and a boundary control at the other end.Different from[6-7,12,14-15],the disturbance is non-collocated to the controller and the performance output is non-collocated to the disturbance in the considered problem.This non-collocation configuration gives rise to new difficulty in the controller design because the control action must propagate through the entire spatial domain to compensate for the disturbance,which requires a deep understanding about the control plant.The considered problem is described by the following equation：

whereu(x,t)denotes the state of system,j is the imaginary unit,yout(t) is the output,U(t) is the control,c0is a positive constant,u0(x)is the initial value,andd ∈L∞(0,∞)is the external disturbance.System(1)is closed to the Euler-Bernoulli beam model[16],whose physical background is clear in [17].In this paper,we will design an output feedback controllerU(t)such that the performance outputu(1,t)and the reference signaluref(t)satisfyu(1,t)?uref(t)∈L2(0,∞).

2 Main results

In this section,we will give the process of the controller and the main results.

Step 1An infinite-dimensional disturbance estimator for system (1) is proposed by virtue of output.Inspired by[18],the estimator is designed as

wherec1is a positive tuning parameter.In fact,let

the errorε(x,t)is governed by

which is exponentially stable[19].

Define the operatorA0:D(A0)?H →Hby

When initial valueε(·,0)∈D(A0),

whereMis a positive constant independent oft(Proposition 2.2[14]).Again,combining(1)-(2)and(4),we get

Step 2Design a controller such thatu(1,t)?uref(t)∈L2(0,∞).By (2) and (8),we are able to construct a servo system by estimation and cancelation strategy.

Moreover,system(9)can be rewritten as

Let

then the error ?u(x,t) between (1) and (9) is governed by

Owing to our construction (9),=u(1,t)?uref(t) happens to be the tracking error.As a consequent,it is sufficient to stabilize the system (12) to achieve output tracking.So,the controller can be designed naturally

Under the controller (13),we get the following closed-loop system of(1)：

We will consider the closed-loop system (14) in state spaceX ∈H3with the norm given by,for any(fi,gi,hi)∈X,i=1,2,

Theorem 1Suppose initial value of the closedloop system (14)∈X,d ∈L∞(0,∞),uref∈ W1,∞(0,∞).Then the closedloop system (14) admits an unique boundedness solution(u(·,t),?u(·,t),?z(·,t))∈C(0,∞;X).Moreover,the tracking error satisfies that

More specially,ifd(t)≡0 anduref(t)≡0,then,the closed-loop system(14)is exponentially stable,i.e.there exist two constantsL ＞0 andω ＞0 such that

3 The proof of main result

By(4)(12)and(13),this section first considers the well-posedness and exponential stability of the following transformed system：

System(18)can be written abstractly as

where the operatorA:D(A)?H2→H2is defined b y

Lemma 1The operator defined by (20) generates aC0-semigroup of contractions eA tinH2.

ProofFor any(fi,gi)∈H2,i=1,2,the inner product of system(18)inH2is defined by

For any(f,g)∈D(A),a straightforward computation shows that

This shows thatA ?1∈L(H2)is compact onH2.By the Lumer-Phillips Theorem (Theorem 1.4.3[20]),the operatorAgenerates aC0-semigroup of contractions onH2.The proof is complete.

ProofThe existence of solution for (18) is obtained due to Lemma 1.We only prove the exponential stability of the system(18).Obviously,the system(18)is a cascade system fromε-subsystem and ?u-subsystem.The exponential stability ofε-subsystem is obvious due to(4).The exponential stability of ?u-subsystem can also be proved by the transformation：

wherez(x,t)is governed by

Define the operatorsA1:D(A1)?H →Hby

which shows that system(28)admits a unique solutionz(·,t)∈C(0,∞;H)such that

Remark 1Lemma 2 can also be proved by the method of Riesz base,whose proof is similar to Proposition 2.3 of paper[14].

Proof of Theorem 1We first introduce the following invertible transformation：

Then the closed-loop system(14)is converted into

which can be rewritten as an evolution equation inH3：

whereB=(0,0,δ(·)) with the Dirac distributionδ(·),the operatorAis defined by (20) and the operatorA:D(A)?H3→H3,is given by

By Lemma 2 and the method of Riesz base,the operatorAgenerates an exponentially stableC0-semigroup onH3(see e.g.Lemma 5.1[14]).Moreover,the operatorBis admissible to the semigroup eAt[6-7,9].Hence,the solution of system(33)can be written as

wherer1(t)=andr2(t)=d(t)+jc0uref(t).By the assumptionsd∈L∞(0,∞)anduref∈W1,∞(0,∞),bothr1andr2are bounded inL∞(0,∞).By the admissibility ofBfor eAtand the exponential stability of eAt,we conclude the uniform boundedness,there exist two constantsLA ＞0 andωA ＞0 such that

Then by (32) and (37),the closed-loop system (14) is well-posed and uniform bounded.

Now,we prove(16).In fact,if we define

Then,the time derivative ofE1(t)andE2(t)along the trajectory of systems(4)and(28)are,respectively,

which imply that

Combining(27)and(43),we obtain

Notice that

(16)can be obtained by(44)and(45)easily.

More specially,ifd(t)=0 anduref(t)=0,the overall system(14)can be transformed as

Repeating the above process,it yields that the system(46) is exponentially stable.Then (17) holds true by(32).The proof is complete.

4 Numerical simulation

In this section,some numerical simulations of control system (14) are presented to illustrate the effectiveness of our control scheme by us.The finite difference scheme is adopted to discretize the system directly.We take the time step is 2×10?5and the space step is 0.1.The external disturbance is chosen asd(t)=sin(2t)+j cost.The reference signal isuref(t)=sin(5t)+j cos(3t).Moreover,we take the parameters and the initial values：

Figs.1-3 display the displacements of the closed-loop system(14).Fig.4 shows the total disturbance estimation.Fig.5 shows that the performanceu(1,t)tracking reference signaluref(t).It can be seen from the simulations that all the states of the involved systems are bounded and the performance output tracks the reference signal.

Fig.1(a) Real part of displacements for u(x,t)in closed-loop(14)

Fig.1(b) Imaginary part of displacements for u(x,t)in closed-loop(14)

Fig.2(a) Real part of displacements for ?u(x,t)subsystem

Fig 2(b) Imaginary part of displacements for (x,t)subsystem

Fig.3(a) Real part of displacements for (x,t)subsystem

Fig.3(b) Imaginary part of displacements for(x,t)subsystem

Fig.4(a) Real part of displacements forx(0,t)?jc0(0,t)tracking d(t)

Fig.4(b) Imaginary part of displacements for(0,t)?jc0(0,t)tracking d(t)

Fig.5(a) Real part of displacements for performance output u(1,t)tracking uref(t)

Fig.5(b) Imaginary part of displacements for performance output u(1,t)tracking uref(t)

5 Concluding remarks

In this paper,we consider the output tracking for one-dimensional Schr?dinger equation with a general boundary disturbance,where the control and the disturbance are non-collocated.An infinite dimensional disturbance estimator based controller is designed to achieve output tracking.Meanwhile,all the states of the closed-loop system are uniformly bounded.Future work is to extend this result to the multi-dimensional Schr?dinger equation and other PDE-ODE system.