# American Institute of Mathematical Sciences

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

## Optimal control of system governed by the Gao beam equation

 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.
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, (2012). Google Scholar [2] I. Ekeland, R. Témam, Convex Analysis and Variational Problems,, SIAM, (1999). Google Scholar [3] D.Y. Gao, Nonlinear elastic beam theory with application in contact problems and variational approaches., Mechanics Research Communications, 23 (1996), 11. 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 (2000), 103. Google Scholar [5] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971). Google Scholar [6] J.-L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems,, SIAM, (1972). Google Scholar [7] Reddy, J.N.:, An Introduction to the Finite Element Method., Third edition. McGraw-Hill Book Co., (2006). Google Scholar [8] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications,, AMS, (2010). Google Scholar

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##### References:
 [1] A. Borzi, V. Schulz, Computational Optimization of Systems Governed by Partial Differential Equations,, SIAM, (2012). Google Scholar [2] I. Ekeland, R. Témam, Convex Analysis and Variational Problems,, SIAM, (1999). Google Scholar [3] D.Y. Gao, Nonlinear elastic beam theory with application in contact problems and variational approaches., Mechanics Research Communications, 23 (1996), 11. 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 (2000), 103. Google Scholar [5] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971). Google Scholar [6] J.-L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems,, SIAM, (1972). Google Scholar [7] Reddy, J.N.:, An Introduction to the Finite Element Method., Third edition. McGraw-Hill Book Co., (2006). Google Scholar [8] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications,, AMS, (2010). Google Scholar
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