American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 783-792. doi: 10.3934/proc.2015.0783

Optimal control of system governed by the Gao beam equation

Jitka Machalová 1, and  Horymír Netuka 1,

1. 

Palacký University, Faculty of Science, Department of Mathematical Analysis and Applications of Mathematics, 17. listopadu 1192/12, Olomouc, 771 46, Czech Republic, Czech Republic

Received  September 2014 Revised  April 2015 Published  November 2015

In this contribution several optimal control problems are mathematically formulated and analyzed for a nonlinear beam which was introduced in 1996 by David Y. Gao. The beam model is given by a static nonlinear fourth-order differential equation with some boundary conditions. The beam is here subjected to a vertical load and possibly to an axial tension load as well. A cost functional is constructed in such a way that the lower its value is, the better model we obtain. Both existence and uniqueness are studied for the solution to the proposed control problems along with optimality conditions. Due to the fact that analytical solution is not available for the nonlinear Gao beam, a finite element approximation is provided for the proposed problems. Numerical results are compared with Euler-Bernoulli beam as well as the authors' previous considerations.
Keywords: Optimal control, fi nite element method, nonlinear Gao beam, Euler-Bernoulli beam, numerical solution..
Mathematics Subject Classification: Primary: 49J15, 65L60; Secondary: 49N1.
Citation: Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783
References:
[1]

A. Borzi, V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations, SIAM, Philadelphia, 2012. Google Scholar

[2]

I. Ekeland, R. Témam, Convex Analysis and Variational Problems, SIAM, Philadelphia, 1999. Google Scholar

[3]

D.Y. Gao, Nonlinear elastic beam theory with application in contact problems and variational approaches. Mechanics Research Communications, 23, (1), pp. 11-17, 1996. Google Scholar

[4]

D.Y. Gao, Finite deformation beam models and triality theory in dynamical post-buckling analysis. Int. J. of Non-Linear Mechanics, 35, pp. 103-121, 2000. Google Scholar

[5]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971. Google Scholar

[6]

J.-L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems, SIAM, Philadelphia, 1972. Google Scholar

[7]

Reddy, J.N.:, An Introduction to the Finite Element Method. Third edition. McGraw-Hill Book Co., New York, 2006. Google Scholar

[8]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, AMS, Providence, Rhode Island, 2010. Google Scholar

show all references

References:
[1]

A. Borzi, V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations, SIAM, Philadelphia, 2012. Google Scholar

[2]

I. Ekeland, R. Témam, Convex Analysis and Variational Problems, SIAM, Philadelphia, 1999. Google Scholar

[3]

D.Y. Gao, Nonlinear elastic beam theory with application in contact problems and variational approaches. Mechanics Research Communications, 23, (1), pp. 11-17, 1996. Google Scholar

[4]

D.Y. Gao, Finite deformation beam models and triality theory in dynamical post-buckling analysis. Int. J. of Non-Linear Mechanics, 35, pp. 103-121, 2000. Google Scholar

[5]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971. Google Scholar

[6]

J.-L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems, SIAM, Philadelphia, 1972. Google Scholar

[7]

Reddy, J.N.:, An Introduction to the Finite Element Method. Third edition. McGraw-Hill Book Co., New York, 2006. Google Scholar

[8]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, AMS, Providence, Rhode Island, 2010. Google Scholar

[1]

Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029
[2]

Ammar Khemmoudj, Imane Djaidja. General decay for a viscoelastic rotating Euler-Bernoulli beam. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3531-3557. doi: 10.3934/cpaa.2020154
[3]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315
[4]

Arnaud Münch, Ademir Fernando Pazoto. Boundary stabilization of a nonlinear shallow beam: theory and numerical approximation. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 197-219. doi: 10.3934/dcdsb.2008.10.197
[5]

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
[6]

Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem. Networks & Heterogeneous Media, 2009, 4 (4) : 709-730. doi: 10.3934/nhm.2009.4.709
[7]

Jong Yeoul Park, Sun Hye Park. On uniform decay for the coupled Euler-Bernoulli viscoelastic system with boundary damping. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 425-436. doi: 10.3934/dcds.2005.12.425
[8]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183
[9]

Kamil Aida-Zade, Jamila Asadova. Numerical solution to optimal control problems of oscillatory processes. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021166
[10]

Maya Bassam, Denis Mercier, Ali Wehbe. Optimal energy decay rate of Rayleigh beam equation with only one boundary control force. Evolution Equations & Control Theory, 2015, 4 (1) : 21-38. doi: 10.3934/eect.2015.4.21
[11]

Dongsheng Yin, Min Tang, Shi Jin. The Gaussian beam method for the wigner equation with discontinuous potentials. Inverse Problems & Imaging, 2013, 7 (3) : 1051-1074. doi: 10.3934/ipi.2013.7.1051
[12]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721
[13]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021
[14]

Kaïs Ammari, Denis Mercier, Virginie Régnier, Julie Valein. Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Communications on Pure & Applied Analysis, 2012, 11 (2) : 785-807. doi: 10.3934/cpaa.2012.11.785
[15]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45
[16]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723
[17]

Marcelo Moreira Cavalcanti. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 675-695. doi: 10.3934/dcds.2002.8.675
[18]

Valentin Keyantuo, Louis Tebou, Mahamadi Warma. A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2875-2889. doi: 10.3934/dcds.2020152
[19]

Hamid Reza Marzban, Hamid Reza Tabrizidooz. Solution of nonlinear delay optimal control problems using a composite pseudospectral collocation method. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1379-1389. doi: 10.3934/cpaa.2010.9.1379
[20]

Pao-Liu Chow. Asymptotic solutions of a nonlinear stochastic beam equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 735-749. doi: 10.3934/dcdsb.2006.6.735

 Impact Factor: 

Tools

Metrics

  • PDF downloads (242)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy