`a`
Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Optimal control problem for a viscoelastic beam and its galerkin approximation
Pages: 263 - 274, Issue 1, January 2018

doi:10.3934/dcdsb.2018018      Abstract        References        Full text (366.0K)           Related Articles

Andrzej Just - Centre of Mathematics and Physics, Lodz University of Technology, Al.Politechniki 11, 90-924 Lodz, Poland (email)
Zdzislaw Stempień - Institute of Mathematics, Lodz University of Technology, ul. Wolczanska 215, 90-924 Lodz, Poland (email)

1 A. S. Ackleh, H. T. Banks and G. A. Pinter, A Nonlinear Beam Equation, Appl. Math. Letters, 15 (2002), 381-387.       
2 J. Ahn and D. E. Stewart, A viscoelastic Timoshenko beam with dynamic frictionless impact, Discr. Cont. Dynam. Sys. B, 12 (2009), 1-22.       
3 K. T. Andrews, Y. Dumont, M. F. M'Bengue, J. Purcell and M. Shillor, Analysis and simulations of nonlinear elastic dynamic beam, ZAMP, 63 (2012), 1005-1019.       
4 N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comp. Optim. Applic., 23 (2002), 201-229.       
5 J. M. Ball, Stability theory for an extensible beam, J. Differ. Equat., 14 (1973), 399-418.       
6 J. M. Ball, Initial-boundary value problems for an extensible beam, J.Math.Anal. Appl., 42 (1973), 61-90.       
7 M. Barboteu, M. Sofonea and D. Tiba, The control variational method for beams in contact with deformable obstacles, ZAMM, 92 (2012), 25-40.       
8 A. Dębinska-Nagórska, A. Just and Z. Stempień, Approximation of an optimal control problem governed by a differential parabolic inclusion, Optim., 59 (2010), 707-715.       
9 A. Dębinska-Nagórska, A. Just and Z. Stempień, Galerkin method for optimal control of second-order evolution equations, Math. Meth. Appl. Sci., 27 (2004), 221-230.       
10 Z. Denkowski, S. Migórski and N. Papageorgiou, Nonlinear Analysis. Applications, Kluwer Academic Publishers, Boston/Dordrecht/London, 2002.
11 E. Feireisl and L. Herrman, Oscillations of a nonlinearly dumped extensible beam, Appl. Math., 37 (1992), 469-478.       
12 M. Galewski, On the optimal control problem governed by the nonlinear elastic beam equation, Appl. Math. Comput., 203 (2008), 916-920.       
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.       
14 J. Hwang, Optimal control problems for an extensible beam equation, J. Math. Anal. Appl., 353 (2009), 436-448.       
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.       
16 I. Lasiecka, Galerkin approximation of infinite-dimensional compensators for flexible structure with unbounded control action, Acta Appl. Math., 28 (1992), 101-133.       
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).       
18 M. L. Oliveira and O. A. Lima, Exponential decay of the solutions of the beam system, Nonlinear Anal., 42 (2000), 1271-1291.       
19 D. C. Pereira, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation, Nonlinear Anal., 14 (1990), 613-623.       
20 I. Sadek, M. Abukhaled and T. Abdulrub, Coupled Galerkin and parametrization methods for optimal control of discretely connected parallel beams, Appl. Math. Modelling, 34 (2010), 3949-3957.       
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.       
22 E. Zeidler, Nonlinear Functional Analysis and its Applications, II Springer-Verlag, Berlin, 1990.       

Go to top