Stabilization of a 1-D wave equation with anti-damping

GUO Chun-li,HU Rong

(School of Mathematics,Sichuan University of Arts and Science,Dazhou Sichuan 635000,China)

Abstract: This paper is to stabilize a 1-D wave equation with an anti-damping by boundary control.To use the backstepping method of boundary control,a new transformation of two kernel functions is introduced.The equations of kernel functions are more complicated mathematically.By some mathematical skill,solutions of the kernel equations are constructed.Finally,the inverse transformation is attained.Through boundedness of the transformation and its inverse,stability of the closed-loop system is established.

Key words: wave equation;boundary control;stability

1 Introduction

Wave equations describe a variety of natural phenomena,such as sound waves,water waves,etc.Therefore,wave equation has a rich engineering background.Stabilization of wave equations plays a role in practical applications.Some results can be found in[1-2].Stability of a wave equation with velocity recirculation is considered in[3].Also,applications in deep oil drilling can be found,for example,in[4].In recent years,boundary control of partial differential equations was developed(see,e.g.,[1-13]).In[5],a wave equation with Kelvin-Voigt damping through boundary control is considered.Stability of wave equations by output feedback boundary control are also concerned(see,e.g.,[8,12]).

In[10],stability of a wave equation with an antidamping at one boundary is addressed.Motivated by[10],we consider a 1-D wave equation with antidamping at an internal point,and stabilize it by boundary control.The control system can be written into

whereλ＞0is a constant andU(t)is the controller.The system(1)models a string vibration which is motived at the endx=1and is uncontrolled at the opposite end.The anti-damping on the internal point comes fromλu(x0,t).Motivated by[12-15],we will design a state feedback boundary controllerU(t)through backstepping to stabilize the closed-loop system.

2 Control design

The idea of control design is derived from the PDE backstepping method.It is that the control system(1)is converted into the stable target system by a bounded inverse transformation.

Firstly,to design the control input by backstepping method,consider the transformation

where the kernelsk(x,y)andr(x,y)will be calculated later.

Motivated by[13-15],we introduce the backstepping transformation(2)which maps the control system(1)into the following target system:

which is exponentially stable forc0＞0andc1＞0(see,e.g.,[8]).

Secondly,by computing the partial derivative of both sides of the transformation(2)with respect tox,we can attain that

Takingx=1in Eq.(4)and usingwx(1,t)=-c1wt(1,t)in Eq.(3),it gives that

By taking the partial derivative of the transformation(2)with respect tot,we get

Then,from Eqs.(5)-(6),the controllerU(t)can be obtained.

Thirdly,in order to construct the stabilization of the closed-loop system(1),it is necessary to prove the boundedness and reversibility of the transformation(2).In Section 4,we will find the inverse transformation.Then,choosing the suitable norm,the stabilization of the closed-loop system is constructed by using the boundedness of the transformation.

3 Calculation of kernels

Differentiating(4)with respect tox,we obtain that

wherek′(x,x)represents the derivative of functionk(x,x)as follows:

From the system(1),differentiating Eq.(6)with respect tot,we have

Through integration by parts andux(0,t)=0in Eq.(1),it gives that

To satisfy the equation in Eq.(3),the kernelsk(x,y)andr(x,y)need to satisfy the under equations

and the compatibility condition

From Eq.(4)and the conditionwx(0,t)=c0w(0,t)in Eq.(3),we get

Hence,we obtain two more conditionsrx(0,y)=c0r(0,y)andk(0,0)=-c0.Fromk(0,0)=-c0andk′(x,x)=0,we getk(x,x)=-c0.Therefore,the functionk(x,y)satis fies the following equations:

Solving Eq.(13),it can be obtained that(see[6]or verify directly)

Next,affected by the variable separation method of ODE,we suppose that the functionr(x,y)can be expressed as

From Eqs.(11)(15)andrx(0,y)=c0r(0,y),it is obtained tha

Now,to solve the equation(16),we search the possible solutions such thatq′′(x)/q(x)=p′′(x)/p(x)are constants.Let

wherea＞0is a constant to be determined.

Therefore,q(y)andp(x)satisfy the equations

Solving the problems(17)-(18),we have

wherebandcare constants to be determined.

Now,checking the compatibility condition(12),we can obtain the conditions which the constantsa,b,cneed to satisfy.First,from Eqs.(14)-(15)and Eqs.(19)-(20),we get

To satisfy the compatibility condition(12),we take

Then,by Eqs.(15)(19)-(20)and Eq.(24),it holds that

According to the above calculation and analysis,the following theorem can be obtained.

Theorem 1For anyλ＞0,Eq.(11)have classical solutions which are de fined by Eqs.(25)-(26).And the solutionsk(x,y)andr(x,y)are bounded on a triangle0≤y＜x≤1.

4 Stability

To establish the stabilization of the closed-loop system(1)under the controller(7),the inverse transformation of the transformation(2)is required.Then,the stabilization of the closed-loop system can be obtained by using the boundedness of the transformation.

4.1 Inverse transformation

The inverse transformation of the transformation(2)can be written as follows:

where the functionsh(x,y)andl(x,y)will be decided later.

From Eq.(28),to satisfy the equation in Eq.(1),the functionsl(x,y)andh(x,y)are determined by the following equations:

To satisfy the boundary conditionux(0,t)=0,takel(0,0)=-c0andhx(0,y)=0.Froml′(x,x)=0andl(0,0)=-c0,we getl(x,x)=-c0.The equations which the the functionsl(x,y)andh(x,y)satisfy are the following equations:

Now,to search a solution of Eq.(30),we consider the kernel functionl(x,y)is a constant.So we can obtain

Then the equations of the kernel functionh(x,y)is expressed as follows:

Using the similar method in Section 3,we consider that the problem ofh(x,y)has a solution of separation variables.h(x,y)is expressed as follows:

From Eqs.(32)-(33),we obtain

For simplicity,to solve Eq.(34),we suppose thatm(x)is a constant.Let

whereMis a nonzero constant to be determined.It is easy to verify thatm′(0)=0.By Eqs.(34)-(35),we obtain thatn(y)needs to satisfy the following equations:

Then the constantMand the solution of Eq.(36)need to satisfy the compatibility condition

Solving the second order ordinary differential Eq.(36),we can obtain

where the constantsd1andd2need to be determined.

To calculate the constantsd1,d2andM,we substitute Eq.(38)inton′(x0)=,n(x0)=0and the compatibility condition(37).So the constants satisfy the following equations.

Similarly,based on the above analysis and calculation,the following theorem can be constituted.

Theorem 2For anyλ＞0,Eq.(30)have classical solutions.And the solutionsh(x,y)andl(x,y)are bounded on a triangle0≤y＜x≤1.

4.2 Stability

From Theorems 1 and 2,we can establish the following the theorem.(see,e.g,.[6,10])

Theorem 3For anyλ＞0,the closed-loop system(1)with the controller(7)is exponentially stable in the sense of the norm

ProofFirstly,to obtain the stabilization of the closed-loop system(1),we will show that the invertible transformation of Eq.(2)is Eq.(27).From Theorem 1,the transformation(2)de fines a linear bounded operatorPin the sense of the norm(43),that is

By the same reason,from the transformation(27)and Theorem 2,a linear bounded operatorQis de fined in the sense of the norm(43),that is

Then,we will prove that the inverse operator ofPisQ.It is required thatPQ=IorQP=I,whereIis the identity operator of the norm(43).This means that(QP)u=u.Substituting Eq.(44)into Eq.(45)and exchanging the order of integration for the twice integrals,it gives that

Therefore,by Eqs.(52)-(54)(49)and(41),it gives that

By Eqs.(46)(51)and(55),we obtain that(QP)u=u.It is that the inverse operator ofPisQ.So,we get that the invertible transformation of Eq.(2)is Eq.(27).

By the boundedness of the operatorPandQ,there are two positive constantsC1andC2such that

where‖·‖is the norm of Eq.(43).

Then,we will show that the closed-loop system(1)is exponentially stable under the controller(7)for the initial stateu0(x),in the sense of the norm(43).

For the initial stateu0(x),u(x,t)is the solution of the closed-loop system(1).According to Theorem 1,the functionw(x,t)determined byw(x,t)=(Pu)(x,t)is the solution of the target system(3)under the initial statew0(x)=(Pu0)(x).By Eq.(56),it gives that

Meanwhile,the system(3)is exponentially stable in the sense of the norm(43)(see,e.g.,[6]).Hence,it means that the solutionw(x,t)of Eq.(3)satis fies the following inequality for the initial statew0(x)

whereNis a large positive number.

Finally,by Eqs.(44)-(45)(57)-(59)andQ=P-1,it holds that