Research Article Control of Nonlinear Distributed...

Research ArticleControl of Nonlinear Distributed Parameter SystemsBased on Global Approximation

Chunyan Du1 and Guansheng Xing2

1 School of Electrical Engineering and Automation Tianjin University Tianjin China2 School of Electrical Engineering and Automation Hebei University of Technology Tianjin China

Correspondence should be addressed to Chunyan Du ducytjueducn

Received 6 June 2014 Accepted 24 June 2014 Published 14 August 2014

Academic Editor Giuseppe Marino

Copyright copy 2014 C Du and G Xing This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We extend an iterative approximation method to nonlinear distributed parameter systems given by partial differential andfunctional equations The nonlinear system is approached by a sequence of linear time-varying systems which globally convergesin the limit to the original nonlinear systems considered This allows many linear control techniques to be applied to nonlinearsystems Here we design a sliding mode controller for a nonlinear wave equation to demonstrate the effectiveness of this method

1 Introduction

The control of finite-dimensional nonlinear system of theform

(119905) = 119860 (119909119873 (119909 120579)) 119909 (119905) + 119861119906 (1)

has recently been studied via a sequence of linear time-varying (LTV) approximation of the form

[119894](119905) = 119860 (119909

[119894minus1] 119873 (119909

[119894minus1] 120579)) 119909

[119894](119905) + 119861 (119909

[119894minus1](119905)) 119906

119894 = 1 2 3 4

(2)

where 119873(119909 120579) is some nonlinear function defined over theinterval [119905 minus 120579 119905]

This iterative linear approximation method makes itconvenient to control nonlinear systems via linear feedbackcontrol technique It has been shown to be effective in slidingcontrol [1] and optimal control [2] The basic theory andconvergence is presented in [3 4]

However many real systems are distributed parametersystems described by partial differential and functional equa-tions Comparing with the lumped parameter systems it ismore difficult to control these systems especially for non-linear distributed parameter systems Different methods areapplied to different kinds of nonlinear distributed parameter

equations (PDE) The Galerkinrsquos method is used to transfernonlinear parabolic equations into nonlinear ordinary differ-ential equations (ODE) which allows many control methodsto be applied to nonlinear parabolic equations In [5] El-Farra et al develop feedback linearization method basedon Lyapunov techniques In [6] Wu and Li design a fuzzyobserver-based controller based on T-S model of the ODEwhile in [7 8] Wu and Li design linear model feedbackcontrollers based on neural network approximation of theODE In [9] Deng et al develop a spectral approximationmethod to distributed thermal processing and a hybridgeneral regression NN is trained to be a nonlinear model ofthe original PDE which also allowsmany control methods tobe applied to this kind of nonlinear PDEs

In this paper we will extend the above-mentioned iter-ative linear approximation theory to the nonlinear waveequation which is a typical infinite-dimensional nonlinearPDE of the form

120597

2120601

120597119905

2= 120596 (1 + 119903120601

2)

120597

2120601

120597119909

2+ 119906

119909 isin Ω 119905 isin (0 120591)

(3)

where 119903 and 120596 are constant and the wave frequency is anonlinear function of 120601 And we design a sliding modecontroller based on the approximation method

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 845370 6 pageshttpdxdoiorg1011552014845370

2 Journal of Applied Mathematics

The nonlinear wave equation is transformed into asequence of LTV approximations each of which has aninfinite-dimensional sliding surface which is time-varyingThe limit of these surfaces gives an effective nonlinear slidingsurface for the original system And the limit of these slidingmode controllers will exponentially stabilizes the nonlinearwave equation

Section 2 recalls the principles of the slidingmode controlfor finite-dimensional nonlinear systems based on LTVapproximation method Section 3 designs a sequence of LTVapproximations for nonlinear wave equation and proves theirconvergence Section 4 develops a sliding controller for thenonlinear wave equation based on LTV approximationsSections 5 and 6 present the simulation result and theconclusion

2 Sliding Control for Finite-DimensionalNonlinear Systems

21 LTV Approximation of Finite-Dimensional Nonlinear Sys-tems For a nonlinear finite-dimensional system given by afunctional differential equation

(119905) = 119860 (119909119873 (119909 120579)) 119909 (119905) (4)

where 119873(119909 120579) is some nonlinear function defined over theinterval [119905 minus 120579 119905] we can define a series LTV approximationsas follows

[0](119905) = 119860 (1199090 119873 (1199090 120579)) 119909

[0](119905)

[119894](119905) = 119860 (119909

[119894minus1](119905) 119873 (119909

[119894minus1](119905) 120579)) 119909

[119894](119905)

119909

[0](0) = 119909

[119894](0) = 1199090 isin 119877

(5)

The approximations of the form (5) are proved to be globalconvergent under a mild Lipschitz condition [4]

Theorem 1 Suppose that the nonlinear functional differentialequation (4) has a unique solution on the interval [119905minus120579 119905] andassume that the following condition holds

1003817

1003817

1003817

1003817

119860 (1199091 119873 (1199091 119902)) minus 119860 (1199092 119873 (1199092 119902))1003817

1003817

1003817

1003817

le 120572 (119870)

1003817

1003817

1003817

1003817

1199091 minus 11990921003817

1003817

1003817

1003817

119891119900119903 1199091 1199092 isin 119861 (119870 1199090)

(6)

where 119861(119870 1199090) isin 119877119873 is a ball of radius119870 centered at 1199090 and 119877is a constant related to 119870 Then the sequence of 119909[119894](119905) definedin (5) converges uniformly on [119905 minus 120579 119905]

This method makes it possible to control nonlinearsystem in form of (1) using linear feedback control techniquesuch as sliding control and optimal control

22 Sliding Control for Nonlinear Systems Based on LTVApproximation Apply the approximation technique for sys-tem (1) the following sequence of LTV can be obtained

[0](119905) = 119860 (1199090 119873 (1199090 120579)) 119909

[0](119905) + 119861119906

[0](119905)

[119894](119905) = 119860 (119909

[119894minus1](119905) 119873 (119909

[119894minus1](119905) 120579)) 119909

[119894](119905) + 119861119906

[119894](119905)

119909

[0](0) = 119909

[119894](0) = 1199090 isin 119877

(7)

For each of these LTV equations a slidingmode controller canbe designed as follows

120590

[119894](119905) = 0

[119894](119905) = minus sign (120590[119894] (119905))

(8)

where 120590[119894](119905) = 0 is a time-varying sliding surface Once thesystem hits the surface the reduced order system will remainstable The system is driven into the sliding surface by settingderivative of 120590[119894] equal to minus sign of 120590[119894] Details on how tochoose the sliding surface and get the control input for LTVsystems can be found in [10]

Thus we can get a series of sliding mode controller 119906[119894](119905)for the LTV systems (7) It is proved that under a mildcondition 119906[119894](119905) converge as 119894 rarr infin The limit of the slidingsurfaces gives an effective nonlinear sliding surface for thesystem and the limit of 119906[119894](119905) will exponentially stabilize theoriginal nonlinear equation (1) [3]

3 Linear Approximation ofNonlinear Wave Equation

In this section we extend the iterative approximationmethodto nonlinear distributed parameter systems given by partialdifferential and functional equations As an example wedesign a sequence of iterative LTV distributed parametersystems to approximate the nonlinear wave equation andprove their convergence

First we transfer (3) to a standard style Define a variableas follows

120595 (119909 119905) =

120597120601

120597119905

(9)

Thus (3) could be written as

120597120601

120597119905

= 120595

120597120595

120597119905

= 120596 (1 + 119903120601

2)

120597

2120601

120597119909

2

(10)

Journal of Applied Mathematics 3

From (5) we can get a sequence of LTV approximationsystem of (10) as follows

120597120601

[119894]

120597119905

= 120595

[119894]

120597120595

[119894]

120597119905

= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

119894 = 1 2 119873

(11)

where the initial values of each system are defined by theinitial value function 119891(119909) of (3) as

120601

[119894](119909 0) = 119891 (119909) 119909 isin (0 119897)

120595

[119894](119909 0) = 0 119909 isin (0 119897)

120601

[0](119909 119905) = 119891 (119909) 119905 isin (0 120591)

120595

[0](119909 119905) = 0 119905 isin (0 120591)

(12)

and the first LTV approximation system (when 119894 = 1) isdefined as

120597120601

[1]

120597119905

= 120595

[1]

120597120595

[1]

120597119905

= 120596 [1 + 119903(120601

[0])

2]

120597

2120601

[1]

120597119909

2

(13)

Obviously each LTV system of (11) is a distributed parametersystem and has a unique solution Here we will prove that thissequence of systems will converge to the original nonlinearsystem (10)

Proof Firstly we use a sequence of finite-dimensional differ-ence equations to approximate equation (11)

For 120601[119894](119909 119905) and 120595[119894](119909 119905) (119909 isin [0 119897]) define

120601

[119894]

119895 (119905) = 120601[119894](

119895119897

119872

119905)

120595

[119894]

119895 (119905) = 120595[119894](

119895119897

119872

119905)

119895 = 1 2 119872

(14)

with initial value(120601

[119894]

1 (0) 120601[119894]

2 (0) 120601[119894]

119872 (0))

= (119891(

119897

119872

) 119891(

2119897

119872

) 119891 (119897))

(120595

[119894]

1 (0) 120595[119894]

2 (0) 120595[119894]

119872 (0)) = (0 0 0)

(15)

Suppose that when119872 rarr infin (11) could be approximated bythe following finite-dimensional system [11]

[

Φ[119894](119905)

Ψ[119894](119905)

] = 119860

[119894minus1][

Φ[119894](119905)

Ψ[119894](119905)

] (16)

where

Φ[119894](119905) = [120601

[119894]

1 (119905) 120601[119894]

2 (119905) 120601[119894]

119872 (119905)]119879

Ψ[119894](119905) = [120595

[119894]

1 (119905) 120595[119894]

2 (119905) 120595[119894]

119872 (119905)]119879

(17)

and 119860[119894minus1] is defined as follows

119860

[119894minus1]=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0 0 sdot sdot sdot 0 1 0 sdot sdot sdot 0



minus2120596(1 + 120601

[119894minus1]

1 )2

(119897119872)

2

120596(1 + 120601

[119894minus1]

1 )2

(119897119872)

2sdot sdot sdot 0 0 0 sdot sdot sdot 0

120596(1 + 120601

[119894minus1]

2 )2

(119897119872)

2

minus2120596(1 + 120601

[119894minus1]

2 )2

(119897119872)


0 0 sdot sdot sdot

minus2120596(1 + 120601

[119894minus1]

119872)

2

(119897119872)

20 0 sdot sdot sdot 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(18)

Secondly we use a sequence of finite-dimensional differ-ence equations to approximate equation (10) define

120601119895 (119905) = 120601(119895119897

119872

119905)

120595119895 (119905) = 120595(119895119897

119872

119905)

119895 = 1 2 119872

(19)

with initial value

(1206011 (0) 1206012 (0) 120601119872 (0))=(119891(119897

119872

) 119891(

2119897

119872

) 119891 (119897))

(1205951 (0) 1205952 (0) 120595119872 (0)) = (0 0 0)

(20)


Suppose that when119872 rarr infin (10) could be approximated bythe following finite-dimensional system

[

Φ (119905)

Ψ (119905)] = 119860[

Φ (119905)

Ψ (119905)] (21)

where

Φ (119905) = [1206011 (119905) 1206012 (119905) 120601119872 (119905)]119879

Ψ (119905) = [1205951 (119905) 1205952 (119905) 120595119872 (119905)]119879

(22)

and 119860 is defined as follows

119860 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[




minus2120596(1 + 1206011)2

(119897119872)

2

120596(1 + 1206011)2

(119897119872)


120596(1 + 1206012)2

(119897119872)

2

minus2120596(1 + 1206012)2

(119897119872)


0 0 sdot sdot sdot

minus2120596(1 + 120601119872)2

(119897119872)


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(23)

Obviously matrix 119860 satisfies local Lipschitz condition FromTheorem 1 we could deduce that when 119894 rarr infin the linertime-varying systems (16) will converge uniformly to thenonlinear system (21)

Suppose at the same time that when 119872 rarr infin thedistributed parameter system (11) and (10) will be approxi-mated by difference approximation equations (16) and (21)separately Thus we could know that (11) will eventuallyconverge to (10)

Conclusion When 119894 rarr infin the linear time-varying dis-tributed parameter systems (11) will converge uniformly tothe nonlinear distributed parameter system (10)

Remark This approximation method could be extended toother nonlinear distributed parameter systems as far as thesystems could be approximated by difference equations andsatisfy local Lipschitz condition

4 Sliding Control of Nonlinear Wave Equation

In this section we design a sliding mode controller forthe nonlinear wave equation based on the approximationmethod

Consider the control problem

120597

2120601

120597119905

2= 120596 (1 + 119903120601

2)

120597

2120601

120597119909

2+ 119906 (119909 119905)

119909 isin [0 119897] 119905 isin (0 120591)

(24)

Here we will design a controller to stabilize this system

As in Section 3 system (24) could be approximated byLTV distributed parameter equation

120597120601

[119894]

120597119905

= 120595

[119894]

120597120595

[119894]

120597119905

= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2+ 119906

[119894](119909 119905)

(25)

Thus we could design a series of sliding mode surfaces andcontrollers for the above linear problem Under the localLipschitz condition when 119894 rarr infin the sliding mode surfacesand controllers are convergent The limit of the surfaces isan effective sliding mode surface of the original nonlinearsystem and the limit of the sliding mode controllers couldeventually stabilize the nonlinear system Here we noticethat both of the sliding mode surface and the controller aredistributed parameter system

We choose an infinite-dimensional time-varying slidingsurface for each system as

120590

[119894](119909 119905) = 120595

[119894]minus 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

(26)

When the system is on the surface that is 120590 = 0 we can getthe reduced order system

120595

[119894]= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

(27)

Notice that system (27) is a time-varying heating equationwhich is always stable [12] That means once the system hitsthe sliding surface it will remain stable

To drive the system to this surface set derivative of 120590[119894]

equal to minus sign of 120590[119894] as follows

[119894](119909 119905) = minus sign (120590[119894] (119909 119905)) (28)


By substituting (28) into (26) we get

119906

[119894]= minus sign (120590[119894]) + 2119903119908 120601[119894minus1]

120597

2120601

[119894]

120597119909

2

(29)

5 Simulation Result

Weapproximate each systemof (11) by a definite-dimensionalapproximation For

120601

[119894](119909 119905) 119909 isin [0 119897]

(30)

we could write

120601

[119894]

119895 (119909 119905) = 120601[119894](

119895119897

119873

119905) 119895 = 1 2 3 119873 (31)

Then the LTV approximation of nonlinear wave equation (11)can be written as

120601

[119894]

119895 (119905) = 120601[119894](

119895119897

119872

119905)

120595

[119894]

119895 (119905) = 120595[119894](

119895119897

119872

119905) 119895 = 1 2 119872

(32)

And the systems (26) can be written as

120601

[119894]

119895 = 120595119895[119894]

120595

[119894]

119895 =

120596 [1 + (119903120601

[119894minus1]

119895)

2] (minus2120601

[119894]

119895+ 120601

[119894]

119895minus1+ 120601

[119894]

119895+1)

(119897119872)

2+ 119906

[119894]

119895

(33)

with control input

119906

[119894]

119895 = minus sign (120590[119894]

119895 ) +

2119903120596

120601

[119894minus1]

119895(minus2120601

[119894]

119895+ 120601

[119894]

119895minus1+ 120601

[119894]

119895+1)

(119897119872)

2

(34)

where the initial values

120601

[119894]

119895 (0) = 1206011198950 120595

[119894]

119895 (0) = 1205951198950(35)

and the boundary condition

120601

[119894]

0 (0) = 0 120601

[119894]

119872 (0) = 0(36)

The simulation is performed in MATLAB by using Eulernumerical integration technique The parameters are

119908 = 04 119903 = 05 119897 = 1

119873 = 100 119905 isin [0 1] 119904

1206011198950=

119895

50

if 1 le 119895 le 50

1 minus

(119895 minus 50)

50

if 50 lt 119895 le 100

1205951198950= 0

(37)

Figure 1 gives the approximation error of (11) where thered part denotes the error when 119894 = 2 and the blue partdenotes the error when 119894 = 4 As is shown in the figure theapproximation error decreases as iteration time increases

Figure 2 gives the control result when applying 119906[4] to theoriginal nonlinear wave system As is shown in the figure thesystem is stabilized

0

5

10

15

20

e

times10minus3

minus5

0 02 04 0608

1

001

0203

0405

x (m)t (s)

Figure 1 Approximation error

002

0406

081

001

0203

04050

02

04

06

08

1

x (m)t (s) 0 2

0406

08

0102

0304

)t (s)

120601

Figure 2 Control result

6 Conclusion

In this paper we extend a recently introduced approxima-tion method to nonlinear distributed parameter system anddesign a sliding mode controller for a nonlinear wave equa-tion The nonlinear wave equation is replaced by a sequenceof LTV systems which are proved to be globally convergentunder certain conditions An infinite-dimensional LTV slid-ing surface is designed for each of the LTV approximationThe limit of these surfaces gives an effective nonlinear slidingsurface for the original system And the limit of these slidingmode controllers exponentially stabilizes the nonlinear waveequation The control result shows the effectiveness of thismethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was partially supported by the National NatureScience Foundation of China (no 61271321) Natural ScienceFoundation of Hebei Province (no F2013202101) and Science


and Technology Support Program of Hebei Province (no13211827)

References

[1] M Tomas-Rodriguez S P Banks and M U Salamci ldquoSlidingmode control for nonlinear systems an iterative approachrdquoin Proceedings of the 45th IEEE Conference on Decision andControl pp 4963ndash4968 SanDiego Calif USADecember 2006

[2] T Cimen and S P Banks ldquoGlobal optimal feedback controlfor general nonlinear systems with nonquadratic performancecriteriardquo Systems and Control Letters vol 53 no 5 pp 327ndash3462004

[3] M Tomas-Rodriguez and S P Banks ldquoLinear approximationsto nonlinear dynamical systems with applications to stabilityand spectral theoryrdquo IMA Journal of Mathematical Control andInformation vol 20 no 1 pp 89ndash103 2003

[4] C Du X Xu S P Banks and A Wu ldquoControl of nonlinearfunctional differential equationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 71 no 12 pp e1850ndashe1857 2009

[5] N H El-Farra A Armaou and P D Christofides ldquoAnalysisand control of parabolic PDE systems with input constraintsrdquoAutomatica vol 39 no 4 pp 715ndash725 2003

[6] H-N Wu and H-X Li ldquo119867infin fuzzy observer-based control fora class of nonlinear distributed parameter systems with controlconstraintsrdquo IEEE Transactions on Fuzzy Systems vol 16 no 2pp 502ndash516 2008

[7] H-N Wu and H Li ldquoA Galerkinneural-network-based designof guaranteed cost control for nonlinear distributed parametersystemsrdquo IEEE Transactions on Neural Networks vol 19 no 5pp 795ndash807 2008

[8] H-NWuandH-X Li ldquoAdaptive neural control design for non-linear distributed parameter systems with persistent boundeddisturbancesrdquo IEEE Transactions on Neural Networks vol 20no 10 pp 1630ndash1644 2009

[9] H Deng H Li and G Chen ldquoSpectral-approximation-basedintelligent modeling for distributed thermal processesrdquo IEEETransactions on Control Systems Technology vol 13 no 5 pp686ndash700 2005

[10] H Dallali SlidingMode Control of Nonlinear System [MS thesisin Control Systems] University of Sheffield 2007

[11] Z Wensheng Finite Difference Methods for Partial DifferentialEquations in Science Computation China Higher EducationPress Beijing China 2006

[12] W Mingxin Equations of Mathematical Physics Tsinghua Uni-versity Press Beijing China 2009

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Stochastic AnalysisInternational Journal of


The nonlinear wave equation is transformed into asequence of LTV approximations each of which has aninfinite-dimensional sliding surface which is time-varyingThe limit of these surfaces gives an effective nonlinear slidingsurface for the original system And the limit of these slidingmode controllers will exponentially stabilizes the nonlinearwave equation

Section 2 recalls the principles of the slidingmode controlfor finite-dimensional nonlinear systems based on LTVapproximation method Section 3 designs a sequence of LTVapproximations for nonlinear wave equation and proves theirconvergence Section 4 develops a sliding controller for thenonlinear wave equation based on LTV approximationsSections 5 and 6 present the simulation result and theconclusion

2 Sliding Control for Finite-DimensionalNonlinear Systems

21 LTV Approximation of Finite-Dimensional Nonlinear Sys-tems For a nonlinear finite-dimensional system given by afunctional differential equation

(119905) = 119860 (119909119873 (119909 120579)) 119909 (119905) (4)

where 119873(119909 120579) is some nonlinear function defined over theinterval [119905 minus 120579 119905] we can define a series LTV approximationsas follows

[0](119905) = 119860 (1199090 119873 (1199090 120579)) 119909

[0](119905)

[119894](119905) = 119860 (119909

[119894minus1](119905) 119873 (119909

[119894minus1](119905) 120579)) 119909

[119894](119905)

119909

[0](0) = 119909

[119894](0) = 1199090 isin 119877

(5)

The approximations of the form (5) are proved to be globalconvergent under a mild Lipschitz condition [4]

Theorem 1 Suppose that the nonlinear functional differentialequation (4) has a unique solution on the interval [119905minus120579 119905] andassume that the following condition holds

1003817

1003817

1003817

1003817

119860 (1199091 119873 (1199091 119902)) minus 119860 (1199092 119873 (1199092 119902))1003817

1003817

1003817

1003817

le 120572 (119870)

1003817

1003817

1003817

1003817

1199091 minus 11990921003817

1003817

1003817

1003817

119891119900119903 1199091 1199092 isin 119861 (119870 1199090)

(6)

where 119861(119870 1199090) isin 119877119873 is a ball of radius119870 centered at 1199090 and 119877is a constant related to 119870 Then the sequence of 119909[119894](119905) definedin (5) converges uniformly on [119905 minus 120579 119905]

This method makes it possible to control nonlinearsystem in form of (1) using linear feedback control techniquesuch as sliding control and optimal control

22 Sliding Control for Nonlinear Systems Based on LTVApproximation Apply the approximation technique for sys-tem (1) the following sequence of LTV can be obtained

[0](119905) = 119860 (1199090 119873 (1199090 120579)) 119909

[0](119905) + 119861119906

[0](119905)

[119894](119905) = 119860 (119909

[119894minus1](119905) 119873 (119909

[119894minus1](119905) 120579)) 119909

[119894](119905) + 119861119906

[119894](119905)

119909

[0](0) = 119909

[119894](0) = 1199090 isin 119877

(7)

For each of these LTV equations a slidingmode controller canbe designed as follows

120590

[119894](119905) = 0

[119894](119905) = minus sign (120590[119894] (119905))

(8)

where 120590[119894](119905) = 0 is a time-varying sliding surface Once thesystem hits the surface the reduced order system will remainstable The system is driven into the sliding surface by settingderivative of 120590[119894] equal to minus sign of 120590[119894] Details on how tochoose the sliding surface and get the control input for LTVsystems can be found in [10]

Thus we can get a series of sliding mode controller 119906[119894](119905)for the LTV systems (7) It is proved that under a mildcondition 119906[119894](119905) converge as 119894 rarr infin The limit of the slidingsurfaces gives an effective nonlinear sliding surface for thesystem and the limit of 119906[119894](119905) will exponentially stabilize theoriginal nonlinear equation (1) [3]

3 Linear Approximation ofNonlinear Wave Equation

In this section we extend the iterative approximationmethodto nonlinear distributed parameter systems given by partialdifferential and functional equations As an example wedesign a sequence of iterative LTV distributed parametersystems to approximate the nonlinear wave equation andprove their convergence

First we transfer (3) to a standard style Define a variableas follows

120595 (119909 119905) =

120597120601

120597119905

(9)

Thus (3) could be written as

120597120601

120597119905

= 120595

120597120595

120597119905

= 120596 (1 + 119903120601

2)

120597

2120601

120597119909

2

(10)



120597120601

[119894]

120597119905

= 120595

[119894]

120597120595

[119894]

120597119905

= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

119894 = 1 2 119873

(11)


120601

[119894](119909 0) = 119891 (119909) 119909 isin (0 119897)

120595

[119894](119909 0) = 0 119909 isin (0 119897)

120601

[0](119909 119905) = 119891 (119909) 119905 isin (0 120591)

120595

[0](119909 119905) = 0 119905 isin (0 120591)

(12)


120597120601

[1]

120597119905

= 120595

[1]

120597120595

[1]

120597119905

= 120596 [1 + 119903(120601

[0])

2]

120597

2120601

[1]

120597119909

2

(13)




120601

[119894]

119895 (119905) = 120601[119894](

119895119897

119872

119905)

120595

[119894]

119895 (119905) = 120595[119894](

119895119897

119872

119905)

119895 = 1 2 119872

(14)


[119894]

1 (0) 120601[119894]

2 (0) 120601[119894]

119872 (0))

= (119891(

119897

119872

) 119891(

2119897

119872

) 119891 (119897))

(120595

[119894]

1 (0) 120595[119894]

2 (0) 120595[119894]

119872 (0)) = (0 0 0)

(15)


[

Φ[119894](119905)

Ψ[119894](119905)

] = 119860

[119894minus1][

Φ[119894](119905)

Ψ[119894](119905)

] (16)

where

Φ[119894](119905) = [120601

[119894]

1 (119905) 120601[119894]

2 (119905) 120601[119894]

119872 (119905)]119879

Ψ[119894](119905) = [120595

[119894]

1 (119905) 120595[119894]

2 (119905) 120595[119894]

119872 (119905)]119879

(17)


119860

[119894minus1]=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[




minus2120596(1 + 120601

[119894minus1]

1 )2

(119897119872)

2

120596(1 + 120601

[119894minus1]

1 )2

(119897119872)


120596(1 + 120601

[119894minus1]

2 )2

(119897119872)

2

minus2120596(1 + 120601

[119894minus1]

2 )2

(119897119872)


0 0 sdot sdot sdot

minus2120596(1 + 120601

[119894minus1]

119872)

2

(119897119872)


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(18)


120601119895 (119905) = 120601(119895119897

119872

119905)

120595119895 (119905) = 120595(119895119897

119872

119905)

119895 = 1 2 119872

(19)

with initial value

(1206011 (0) 1206012 (0) 120601119872 (0))=(119891(119897

119872

) 119891(

2119897

119872

) 119891 (119897))

(1205951 (0) 1205952 (0) 120595119872 (0)) = (0 0 0)

(20)



[

Φ (119905)

Ψ (119905)] = 119860[

Φ (119905)

Ψ (119905)] (21)

where

Φ (119905) = [1206011 (119905) 1206012 (119905) 120601119872 (119905)]119879

Ψ (119905) = [1205951 (119905) 1205952 (119905) 120595119872 (119905)]119879

(22)


119860 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[




minus2120596(1 + 1206011)2

(119897119872)

2

120596(1 + 1206011)2

(119897119872)


120596(1 + 1206012)2

(119897119872)

2

minus2120596(1 + 1206012)2

(119897119872)


0 0 sdot sdot sdot

minus2120596(1 + 120601119872)2

(119897119872)


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(23)








120597

2120601

120597119905

2= 120596 (1 + 119903120601

2)

120597

2120601

120597119909

2+ 119906 (119909 119905)

119909 isin [0 119897] 119905 isin (0 120591)

(24)



120597120601

[119894]

120597119905

= 120595

[119894]

120597120595

[119894]

120597119905

= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2+ 119906

[119894](119909 119905)

(25)



120590

[119894](119909 119905) = 120595

[119894]minus 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

(26)


120595

[119894]= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

(27)




[119894](119909 119905) = minus sign (120590[119894] (119909 119905)) (28)



119906

[119894]= minus sign (120590[119894]) + 2119903119908 120601[119894minus1]

120597

2120601

[119894]

120597119909

2

(29)

5 Simulation Result


120601

[119894](119909 119905) 119909 isin [0 119897]

(30)

we could write

120601

[119894]

119895 (119909 119905) = 120601[119894](

119895119897

119873

119905) 119895 = 1 2 3 119873 (31)


120601

[119894]

119895 (119905) = 120601[119894](

119895119897

119872

119905)

120595

[119894]

119895 (119905) = 120595[119894](

119895119897

119872

119905) 119895 = 1 2 119872

(32)


120601

[119894]

119895 = 120595119895[119894]

120595

[119894]

119895 =

120596 [1 + (119903120601

[119894minus1]

119895)

2] (minus2120601

[119894]

119895+ 120601

[119894]

119895minus1+ 120601

[119894]

119895+1)

(119897119872)

2+ 119906

[119894]

119895

(33)

with control input

119906

[119894]

119895 = minus sign (120590[119894]

119895 ) +

2119903120596

120601

[119894minus1]

119895(minus2120601

[119894]

119895+ 120601

[119894]

119895minus1+ 120601

[119894]

119895+1)

(119897119872)

2

(34)


120601

[119894]

119895 (0) = 1206011198950 120595

[119894]

119895 (0) = 1205951198950(35)


120601

[119894]

0 (0) = 0 120601

[119894]

119872 (0) = 0(36)


119908 = 04 119903 = 05 119897 = 1

119873 = 100 119905 isin [0 1] 119904

1206011198950=

119895

50

if 1 le 119895 le 50

1 minus

(119895 minus 50)

50

if 50 lt 119895 le 100

1205951198950= 0

(37)



0

5

10

15

20

e

times10minus3

minus5

0 02 04 0608

1

001

0203

0405

x (m)t (s)


002

0406

081

001

0203

04050

02

04

06

08

1

x (m)t (s) 0 2

0406

08

0102

0304

)t (s)

120601


6 Conclusion




Acknowledgments




References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597120601

[119894]

120597119905

= 120595

[119894]

120597120595

[119894]

120597119905

= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

119894 = 1 2 119873

(11)


120601

[119894](119909 0) = 119891 (119909) 119909 isin (0 119897)

120595

[119894](119909 0) = 0 119909 isin (0 119897)

120601

[0](119909 119905) = 119891 (119909) 119905 isin (0 120591)

120595

[0](119909 119905) = 0 119905 isin (0 120591)

(12)


120597120601

[1]

120597119905

= 120595

[1]

120597120595

[1]

120597119905

= 120596 [1 + 119903(120601

[0])

2]

120597

2120601

[1]

120597119909

2

(13)




120601

[119894]

119895 (119905) = 120601[119894](

119895119897

119872

119905)

120595

[119894]

119895 (119905) = 120595[119894](

119895119897

119872

119905)

119895 = 1 2 119872

(14)


[119894]

1 (0) 120601[119894]

2 (0) 120601[119894]

119872 (0))

= (119891(

119897

119872

) 119891(

2119897

119872

) 119891 (119897))

(120595

[119894]

1 (0) 120595[119894]

2 (0) 120595[119894]

119872 (0)) = (0 0 0)

(15)


[

Φ[119894](119905)

Ψ[119894](119905)

] = 119860

[119894minus1][

Φ[119894](119905)

Ψ[119894](119905)

] (16)

where

Φ[119894](119905) = [120601

[119894]

1 (119905) 120601[119894]

2 (119905) 120601[119894]

119872 (119905)]119879

Ψ[119894](119905) = [120595

[119894]

1 (119905) 120595[119894]

2 (119905) 120595[119894]

119872 (119905)]119879

(17)


119860

[119894minus1]=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[




minus2120596(1 + 120601

[119894minus1]

1 )2

(119897119872)

2

120596(1 + 120601

[119894minus1]

1 )2

(119897119872)


120596(1 + 120601

[119894minus1]

2 )2

(119897119872)

2

minus2120596(1 + 120601

[119894minus1]

2 )2

(119897119872)


0 0 sdot sdot sdot

minus2120596(1 + 120601

[119894minus1]

119872)

2

(119897119872)


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(18)


120601119895 (119905) = 120601(119895119897

119872

119905)

120595119895 (119905) = 120595(119895119897

119872

119905)

119895 = 1 2 119872

(19)

with initial value

(1206011 (0) 1206012 (0) 120601119872 (0))=(119891(119897

119872

) 119891(

2119897

119872

) 119891 (119897))

(1205951 (0) 1205952 (0) 120595119872 (0)) = (0 0 0)

(20)



[

Φ (119905)

Ψ (119905)] = 119860[

Φ (119905)

Ψ (119905)] (21)

where

Φ (119905) = [1206011 (119905) 1206012 (119905) 120601119872 (119905)]119879

Ψ (119905) = [1205951 (119905) 1205952 (119905) 120595119872 (119905)]119879

(22)


119860 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[




minus2120596(1 + 1206011)2

(119897119872)

2

120596(1 + 1206011)2

(119897119872)


120596(1 + 1206012)2

(119897119872)

2

minus2120596(1 + 1206012)2

(119897119872)


0 0 sdot sdot sdot

minus2120596(1 + 120601119872)2

(119897119872)


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(23)








120597

2120601

120597119905

2= 120596 (1 + 119903120601

2)

120597

2120601

120597119909

2+ 119906 (119909 119905)

119909 isin [0 119897] 119905 isin (0 120591)

(24)



120597120601

[119894]

120597119905

= 120595

[119894]

120597120595

[119894]

120597119905

= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2+ 119906

[119894](119909 119905)

(25)



120590

[119894](119909 119905) = 120595

[119894]minus 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

(26)


120595

[119894]= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

(27)




[119894](119909 119905) = minus sign (120590[119894] (119909 119905)) (28)



119906

[119894]= minus sign (120590[119894]) + 2119903119908 120601[119894minus1]

120597

2120601

[119894]

120597119909

2

(29)

5 Simulation Result


120601

[119894](119909 119905) 119909 isin [0 119897]

(30)

we could write

120601

[119894]

119895 (119909 119905) = 120601[119894](

119895119897

119873

119905) 119895 = 1 2 3 119873 (31)


120601

[119894]

119895 (119905) = 120601[119894](

119895119897

119872

119905)

120595

[119894]

119895 (119905) = 120595[119894](

119895119897

119872

119905) 119895 = 1 2 119872

(32)


120601

[119894]

119895 = 120595119895[119894]

120595

[119894]

119895 =

120596 [1 + (119903120601

[119894minus1]

119895)

2] (minus2120601

[119894]

119895+ 120601

[119894]

119895minus1+ 120601

[119894]

119895+1)

(119897119872)

2+ 119906

[119894]

119895

(33)

with control input

119906

[119894]

119895 = minus sign (120590[119894]

119895 ) +

2119903120596

120601

[119894minus1]

119895(minus2120601

[119894]

119895+ 120601

[119894]

119895minus1+ 120601

[119894]

119895+1)

(119897119872)

2

(34)


120601

[119894]

119895 (0) = 1206011198950 120595

[119894]

119895 (0) = 1205951198950(35)


120601

[119894]

0 (0) = 0 120601

[119894]

119872 (0) = 0(36)


119908 = 04 119903 = 05 119897 = 1

119873 = 100 119905 isin [0 1] 119904

1206011198950=

119895

50

if 1 le 119895 le 50

1 minus

(119895 minus 50)

50

if 50 lt 119895 le 100

1205951198950= 0

(37)



0

5

10

15

20

e

times10minus3

minus5

0 02 04 0608

1

001

0203

0405

x (m)t (s)


002

0406

081

001

0203

04050

02

04

06

08

1

x (m)t (s) 0 2

0406

08

0102

0304

)t (s)

120601


6 Conclusion




Acknowledgments




References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[

Φ (119905)

Ψ (119905)] = 119860[

Φ (119905)

Ψ (119905)] (21)

where

Φ (119905) = [1206011 (119905) 1206012 (119905) 120601119872 (119905)]119879

Ψ (119905) = [1205951 (119905) 1205952 (119905) 120595119872 (119905)]119879

(22)


119860 =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[




minus2120596(1 + 1206011)2

(119897119872)

2

120596(1 + 1206011)2

(119897119872)


120596(1 + 1206012)2

(119897119872)

2

minus2120596(1 + 1206012)2

(119897119872)


0 0 sdot sdot sdot

minus2120596(1 + 120601119872)2

(119897119872)


]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(23)








120597

2120601

120597119905

2= 120596 (1 + 119903120601

2)

120597

2120601

120597119909

2+ 119906 (119909 119905)

119909 isin [0 119897] 119905 isin (0 120591)

(24)



120597120601

[119894]

120597119905

= 120595

[119894]

120597120595

[119894]

120597119905

= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2+ 119906

[119894](119909 119905)

(25)



120590

[119894](119909 119905) = 120595

[119894]minus 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

(26)


120595

[119894]= 120596 [1 + 119903(120601

[119894minus1])

2]

120597

2120601

[119894]

120597119909

2

(27)




[119894](119909 119905) = minus sign (120590[119894] (119909 119905)) (28)



119906

[119894]= minus sign (120590[119894]) + 2119903119908 120601[119894minus1]

120597

2120601

[119894]

120597119909

2

(29)

5 Simulation Result


120601

[119894](119909 119905) 119909 isin [0 119897]

(30)

we could write

120601

[119894]

119895 (119909 119905) = 120601[119894](

119895119897

119873

119905) 119895 = 1 2 3 119873 (31)


120601

[119894]

119895 (119905) = 120601[119894](

119895119897

119872

119905)

120595

[119894]

119895 (119905) = 120595[119894](

119895119897

119872

119905) 119895 = 1 2 119872

(32)


120601

[119894]

119895 = 120595119895[119894]

120595

[119894]

119895 =

120596 [1 + (119903120601

[119894minus1]

119895)

2] (minus2120601

[119894]

119895+ 120601

[119894]

119895minus1+ 120601

[119894]

119895+1)

(119897119872)

2+ 119906

[119894]

119895

(33)

with control input

119906

[119894]

119895 = minus sign (120590[119894]

119895 ) +

2119903120596

120601

[119894minus1]

119895(minus2120601

[119894]

119895+ 120601

[119894]

119895minus1+ 120601

[119894]

119895+1)

(119897119872)

2

(34)


120601

[119894]

119895 (0) = 1206011198950 120595

[119894]

119895 (0) = 1205951198950(35)


120601

[119894]

0 (0) = 0 120601

[119894]

119872 (0) = 0(36)


119908 = 04 119903 = 05 119897 = 1

119873 = 100 119905 isin [0 1] 119904

1206011198950=

119895

50

if 1 le 119895 le 50

1 minus

(119895 minus 50)

50

if 50 lt 119895 le 100

1205951198950= 0

(37)



0

5

10

15

20

e

times10minus3

minus5

0 02 04 0608

1

001

0203

0405

x (m)t (s)


002

0406

081

001

0203

04050

02

04

06

08

1

x (m)t (s) 0 2

0406

08

0102

0304

)t (s)

120601


6 Conclusion




Acknowledgments




References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906

[119894]= minus sign (120590[119894]) + 2119903119908 120601[119894minus1]

120597

2120601

[119894]

120597119909

2

(29)

5 Simulation Result


120601

[119894](119909 119905) 119909 isin [0 119897]

(30)

we could write

120601

[119894]

119895 (119909 119905) = 120601[119894](

119895119897

119873

119905) 119895 = 1 2 3 119873 (31)


120601

[119894]

119895 (119905) = 120601[119894](

119895119897

119872

119905)

120595

[119894]

119895 (119905) = 120595[119894](

119895119897

119872

119905) 119895 = 1 2 119872

(32)


120601

[119894]

119895 = 120595119895[119894]

120595

[119894]

119895 =

120596 [1 + (119903120601

[119894minus1]

119895)

2] (minus2120601

[119894]

119895+ 120601

[119894]

119895minus1+ 120601

[119894]

119895+1)

(119897119872)

2+ 119906

[119894]

119895

(33)

with control input

119906

[119894]

119895 = minus sign (120590[119894]

119895 ) +

2119903120596

120601

[119894minus1]

119895(minus2120601

[119894]

119895+ 120601

[119894]

119895minus1+ 120601

[119894]

119895+1)

(119897119872)

2

(34)


120601

[119894]

119895 (0) = 1206011198950 120595

[119894]

119895 (0) = 1205951198950(35)


120601

[119894]

0 (0) = 0 120601

[119894]

119872 (0) = 0(36)


119908 = 04 119903 = 05 119897 = 1

119873 = 100 119905 isin [0 1] 119904

1206011198950=

119895

50

if 1 le 119895 le 50

1 minus

(119895 minus 50)

50

if 50 lt 119895 le 100

1205951198950= 0

(37)



0

5

10

15

20

e

times10minus3

minus5

0 02 04 0608

1

001

0203

0405

x (m)t (s)


002

0406

081

001

0203

04050

02

04

06

08

1

x (m)t (s) 0 2

0406

08

0102

0304

)t (s)

120601


6 Conclusion




Acknowledgments




References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Control of Nonlinear Distributed...

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