State feedback control for a class of variable coefficients fourth-order beam equations

CHEN Zhenjie，FU Qin，YU Pengfei，ZHANG Dan

（School of Mathematical Sciences，SUST，Suzhou 215009，China）

Abstract：This article studies the state feedback control problem for a class of fourth-order distributed parameter systems composed of a class of variable coefficient fourth-order beam equations.Based on the characteristics of the systems，the state feedback control laws were proposed for such systems by means of constructing an appropriate Lyapunov function.When the feedback control laws are applied to the systems，the state vectors of the closed-loop systems are globally asymptotically stable on L2（0，l）×L2（0，l）.Finally，a simulation example was given to show the effectiveness of this method.

Key words：state feedback；distributed parameter systems；fourth-order beamequations；globally asymptotic stability；Lyapunov function

1 Introduction

Since many practical engineering problems should be described by distributed parameter systems（DPSs）which are expressed by partial differential equations（PDEs），the applications of DPSs have covered various fields in the recent years，and a great deal of achievements have been obtained[1-8].Up to now，there are two methods often used to solve the control problem of DPSs：one of them is the boundary control[1-3]，and the other is the distributed control[4-8].This paper concerns with the distributed control of DPSs.

In the latest ten years，the study of DPSs has attracted a lot of attentions of researchers.As one of the important issues in the control fields of DPSs，the feedback controlling plays a crucial role.It considers how to design the feedback controllers so that the states of the closed-loop systems converge to zero on an appropriate Sobolev space as time increases.Some research results on the feedback control for parabolic and hyperbolic DPSs，as seen in[4-5]and[6-8]，respectively，have been achieved，and the corresponding convergence conclusions on the appropriate Sobolev spaces were obtained.However，it is noticed that all the DPSs considered in[4-8]are second-order，and it seems that so far no similar research on higher order DPSs are available in the literature.

Fourth-order wave equations，which arise commonly from the studies of vibration of beams and thin plates[9-10]，are of great research significance and have aroused widespread concern in the recent years[11-16].Until now，almost all the related researches on fourth-order wave equations have focused mainly on their well-posedness and numerical solutions[11-15].Paper[11]considered the exact solution problem for a class of variable coefficients fourth-order beam equations by using the Adomian method.A general approach based on the generalized Fourier series expansion was applied，and the analytic solution obtained there was simplified in terms of a given set of orthogonal basis functions.For DPSs，the control design problem can be discussed when the existence of solution to the corresponding PDEs has been proved.

Motivated by the above discussions，this article further investigates the feedback control problems for a class of fourth-order DPSs which are constructed by fourth-order beam equations in [11].More precisely，the state feedback controllers are obtained by constructing an appropriate Lyapunov functional.Moreover，when the feedback control laws are applied to the systems，the state vectors of the closed-loop systems are globally asymptotically stable onL2（0，l）×L2（0，l）space.

For functionw（x，t），x∈（0，l），t≥0，we callfor allt≥0.

2 Problem formulation

Firstly，we give a simple description of the work in[11].Consider the following variable coefficients fourthorder beam equation[11]

wherew（x，t），μ（x），EI（x）andq（x，t）are lateral beam displacement，mass per unit length，beam bending stiffness and load per unit length（source term），respectively.The homogeneous boundary conditions of equation（1）are a combination of the following[11]

or

with

or

The general solution of equation（1）expressed as a generalized Fourier series was given in[11]，see（46）in[11]for detailed informations.

Remark 1[6-8]，is the state vector of equation（1）.

Remark 2It is obvious that the solution of equation（1）exists whenq（x，t）≡0（for homogeneous cases，see 2.1 in[11]）.Denote

then we know thatμ1，μ2，EI1andEI2are positive by their physical meaningsμ（x）andEI（x）.

For the requirement of feedback control design，we replaceq（x，t）（load per unit length）given in（1）by the control variableu（x，t）.Then the fourth-order DPSgoverned by（1）is given as follows

3 Main results

Theorem 1Construct the following state feedback controller for system（2）

then the state vector of the closed-loop system（2）is globally asymptotically stable onL2（0，l）×L2（0，l），under the condition of

Proof.Construct the following Lyapunov functional

From（4），we knowλ2+2λ＞1，then

is a positive matrix.Therefore，V（t）is positive definite for

According to the definition of positive definite quadratic form，we know that there exists

such that

Taking the derivative ofV（t），it yields

By（2），（3），and integrating by parts，we obtain

By the homogeneous boundary conditions，it yields

From（4），we know

So，taking

we have

which together with（5）implies

Therefore

Inserting the above inequality into（5），we have

which means that the state vector of the closed-loop system（2）is globally asymptotically stable onL2（0，l）×L2（0，l）. □

Remark 3When the feedback control law（3）is applied to system（2），the closed-loop system can be rewritten as follows

Further，we have

The above equation can be transformed into

wherew（x，t）=eλtw（x，t）.Notice that the homogeneous boundary conditions are still satisfied forw（x，t）.Therefore，combining with Remark 2，it is actually shown that the general solution of the closed-loop system（2）under the action of the feedback control law（3）exists.

4 Simulation example

Construct the following system（μ（x）≡1，EI（x）≡1）

with the initial-boundary conditions

According to Theorem 1，takeλ=1 and construct the state feedback controller as follows

With the help of Mathematica，the simulation result of

can be obtained and is shown in Fig.1.

Fig.1 Trajectory of U（t）

5 Conclusion

In this article，we investigate the feedback control problem for a class of fourth-order DPSs.These systems are governed by a class of variable coefficients fourth-order beam equations and have well-posed boundary conditions.The state feedback control laws are proposed by constructing an appropriate Lyapunov functional，and under the action of the feedback control laws，the state vectors of the closed-loop systems are globally asymptotically stable onL2（0，l）×L2（0，l）.The simulation result shows the effectiveness of our theoretical method.Our next work is to study the problem of robust feedback control[17-18].