January  2018, 23(1): 263-274. doi: 10.3934/dcdsb.2018018

Optimal control problem for a viscoelastic beam and its galerkin approximation

1. 

Centre of Mathematics and Physics, Lodz University of Technology, Al.Politechniki 11, 90-924 Lodz, Poland

2. 

Institute of Mathematics, Lodz University of Technology, ul. Wolczanska 215, 90-924 Lodz, Poland

* Corresponding author: zdzislaw.stempien@p.lodz.pl

Received  November 2016 Revised  May 2017 Published  January 2018

This paper is concerned with the optimal control problem of the vibrations of a viscoelastic beam, which is governed by a nonlinear partial differential equation. We discuss the initial-boundary problem for the cases when the ends of the beam are clamped or hinged. We define the weak solution of this initial-boundary problem. Our control problem is formulated by minimization of a functional where the state of a system is the solution of viscoelastic beam equation. We use the Galerkin method to approximate the solution of our control problem with respect to a spatial variable. Based on the finite dimensional approximation we prove that as the discretization parameters tend to zero then the weak accumulation points of the optimal solutions of the discrete family control problems exist and each of these points is the solution of the original optimal control problem.

Citation: Andrzej Just, Zdzislaw Stempień. Optimal control problem for a viscoelastic beam and its galerkin approximation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 263-274. doi: 10.3934/dcdsb.2018018
References:
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A. S. AcklehH. T. Banks and G. A. Pinter, A Nonlinear Beam Equation, Appl. Math. Letters, 15 (2002), 381-387.  doi: 10.1016/S0893-9659(01)00147-1.  Google Scholar

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J. Ahn and D. E. Stewart, A viscoelastic Timoshenko beam with dynamic frictionless impact, Discr. Cont. Dynam. Sys. B, 12 (2009), 1-22.  doi: 10.3934/dcdsb.2009.12.1.  Google Scholar

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K. T. AndrewsY. DumontM. F. M'BengueJ. Purcell and M. Shillor, Analysis and simulations of~nonlinear elastic dynamic beam, ZAMP, 63 (2012), 1005-1019.  doi: 10.1007/s00033-012-0233-9.  Google Scholar

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N. AradaE. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comp. Optim. Applic., 23 (2002), 201-229.  doi: 10.1023/A:1020576801966.  Google Scholar

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J. M. Ball, Stability theory for an extensible beam, J. Differ. Equat., 14 (1973), 399-418.  doi: 10.1016/0022-0396(73)90056-9.  Google Scholar

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J. M. Ball, Initial-boundary value problems for an extensible beam, J.Math.Anal. Appl., 42 (1973), 61-90.  doi: 10.1016/0022-247X(73)90121-2.  Google Scholar

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M. BarboteuM. Sofonea and D. Tiba, The control variational method for beams in contact with deformable obstacles, ZAMM, 92 (2012), 25-40.  doi: 10.1002/zamm.201000161.  Google Scholar

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A. Dębinska-NagórskaA. Just and Z. Stempień, Approximation of an optimal control problem governed by a differential parabolic inclusion, Optim., 59 (2010), 707-715.  doi: 10.1080/02331930802434807.  Google Scholar

[9]

A. Dębinska-NagórskaA. Just and Z. Stempień, Galerkin method for optimal control of~second-order evolution equations, Math. Meth. Appl. Sci., 27 (2004), 221-230.  doi: 10.1002/mma.452.  Google Scholar

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Z. Denkowski, S. Migórski and N. Papageorgiou, Nonlinear Analysis. Applications Kluwer Academic Publishers, Boston/Dordrecht/London, 2002. Google Scholar

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E. Feireisl and L. Herrman, Oscillations of a nonlinearly dumped extensible beam, Appl. Math., 37 (1992), 469-478.   Google Scholar

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M. Galewski, On the optimal control problem governed by the nonlinear elastic beam equation, Appl. Math. Comput., 203 (2008), 916-920.  doi: 10.1016/j.amc.2008.04.022.  Google Scholar

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I. Hlaváček and J. Lovišek, Optimal control of semi-coercive variational inequalities with application to optimal design of beams and plates, ZAMM, 78 (1998), 405-417.  doi: 10.1002/(SICI)1521-4001(199806)78:6<405::AID-ZAMM405>3.0.CO;2-Z.  Google Scholar

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J. Hwang, Optimal control problems for an extensible beam equation, J. Math. Anal. Appl., 353 (2009), 436-448.  doi: 10.1016/j.jmaa.2008.12.020.  Google Scholar

[15]

A. Just and Z. Stempień, Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation, Opuscula Math., 36 (2016), 239-252.  doi: 10.7494/OpMath.2016.36.2.239.  Google Scholar

[16]

I. Lasiecka, Galerkin approximation of infinite-dimensional compensators for flexible structure with unbounded control action, Acta Appl. Math., 28 (1992), 101-133.  doi: 10.1007/BF00047552.  Google Scholar

[17]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations Springer-Verlag, Berlin/Heidelberg/New York, 1971, (Russian Edition, Mir, Moscow 1972).  Google Scholar

[18]

M. L. Oliveira and O. A. Lima, Exponential decay of the solutions of the beam system, Nonlinear Anal., 42 (2000), 1271-1291.  doi: 10.1016/S0362-546X(99)00155-8.  Google Scholar

[19]

D. C. Pereira, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation, Nonlinear Anal., 14 (1990), 613-623.  doi: 10.1016/0362-546X(90)90041-E.  Google Scholar

[20]

I. SadekM. Abukhaled and T. Abdulrub, Coupled Galerkin and parametrization methods for optimal control of discretely connected parallel beams, Appl. Math. Modelling, 34 (2010), 3949-3957.  doi: 10.1016/j.apm.2010.03.031.  Google Scholar

[21]

F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems -strong convergence of optimal controls, Appl. Math. Optimz., 29 (1994), 309-329.  doi: 10.1007/BF01189480.  Google Scholar

[22]

E. Zeidler, Nonlinear Functional Analysis and its Applications Ⅱ Springer-Verlag, Berlin, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

show all references

References:
[1]

A. S. AcklehH. T. Banks and G. A. Pinter, A Nonlinear Beam Equation, Appl. Math. Letters, 15 (2002), 381-387.  doi: 10.1016/S0893-9659(01)00147-1.  Google Scholar

[2]

J. Ahn and D. E. Stewart, A viscoelastic Timoshenko beam with dynamic frictionless impact, Discr. Cont. Dynam. Sys. B, 12 (2009), 1-22.  doi: 10.3934/dcdsb.2009.12.1.  Google Scholar

[3]

K. T. AndrewsY. DumontM. F. M'BengueJ. Purcell and M. Shillor, Analysis and simulations of~nonlinear elastic dynamic beam, ZAMP, 63 (2012), 1005-1019.  doi: 10.1007/s00033-012-0233-9.  Google Scholar

[4]

N. AradaE. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comp. Optim. Applic., 23 (2002), 201-229.  doi: 10.1023/A:1020576801966.  Google Scholar

[5]

J. M. Ball, Stability theory for an extensible beam, J. Differ. Equat., 14 (1973), 399-418.  doi: 10.1016/0022-0396(73)90056-9.  Google Scholar

[6]

J. M. Ball, Initial-boundary value problems for an extensible beam, J.Math.Anal. Appl., 42 (1973), 61-90.  doi: 10.1016/0022-247X(73)90121-2.  Google Scholar

[7]

M. BarboteuM. Sofonea and D. Tiba, The control variational method for beams in contact with deformable obstacles, ZAMM, 92 (2012), 25-40.  doi: 10.1002/zamm.201000161.  Google Scholar

[8]

A. Dębinska-NagórskaA. Just and Z. Stempień, Approximation of an optimal control problem governed by a differential parabolic inclusion, Optim., 59 (2010), 707-715.  doi: 10.1080/02331930802434807.  Google Scholar

[9]

A. Dębinska-NagórskaA. Just and Z. Stempień, Galerkin method for optimal control of~second-order evolution equations, Math. Meth. Appl. Sci., 27 (2004), 221-230.  doi: 10.1002/mma.452.  Google Scholar

[10]

Z. Denkowski, S. Migórski and N. Papageorgiou, Nonlinear Analysis. Applications Kluwer Academic Publishers, Boston/Dordrecht/London, 2002. Google Scholar

[11]

E. Feireisl and L. Herrman, Oscillations of a nonlinearly dumped extensible beam, Appl. Math., 37 (1992), 469-478.   Google Scholar

[12]

M. Galewski, On the optimal control problem governed by the nonlinear elastic beam equation, Appl. Math. Comput., 203 (2008), 916-920.  doi: 10.1016/j.amc.2008.04.022.  Google Scholar

[13]

I. Hlaváček and J. Lovišek, Optimal control of semi-coercive variational inequalities with application to optimal design of beams and plates, ZAMM, 78 (1998), 405-417.  doi: 10.1002/(SICI)1521-4001(199806)78:6<405::AID-ZAMM405>3.0.CO;2-Z.  Google Scholar

[14]

J. Hwang, Optimal control problems for an extensible beam equation, J. Math. Anal. Appl., 353 (2009), 436-448.  doi: 10.1016/j.jmaa.2008.12.020.  Google Scholar

[15]

A. Just and Z. Stempień, Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation, Opuscula Math., 36 (2016), 239-252.  doi: 10.7494/OpMath.2016.36.2.239.  Google Scholar

[16]

I. Lasiecka, Galerkin approximation of infinite-dimensional compensators for flexible structure with unbounded control action, Acta Appl. Math., 28 (1992), 101-133.  doi: 10.1007/BF00047552.  Google Scholar

[17]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations Springer-Verlag, Berlin/Heidelberg/New York, 1971, (Russian Edition, Mir, Moscow 1972).  Google Scholar

[18]

M. L. Oliveira and O. A. Lima, Exponential decay of the solutions of the beam system, Nonlinear Anal., 42 (2000), 1271-1291.  doi: 10.1016/S0362-546X(99)00155-8.  Google Scholar

[19]

D. C. Pereira, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation, Nonlinear Anal., 14 (1990), 613-623.  doi: 10.1016/0362-546X(90)90041-E.  Google Scholar

[20]

I. SadekM. Abukhaled and T. Abdulrub, Coupled Galerkin and parametrization methods for optimal control of discretely connected parallel beams, Appl. Math. Modelling, 34 (2010), 3949-3957.  doi: 10.1016/j.apm.2010.03.031.  Google Scholar

[21]

F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems -strong convergence of optimal controls, Appl. Math. Optimz., 29 (1994), 309-329.  doi: 10.1007/BF01189480.  Google Scholar

[22]

E. Zeidler, Nonlinear Functional Analysis and its Applications Ⅱ Springer-Verlag, Berlin, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

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