Netuka, Horymír; Machalová, Jitka Optimal control of system governed by the Gao beam equation. (English) Zbl 1334.49006 Discrete Contin. Dyn. Syst. 2015, Suppl., 783-792 (2015). Summary: 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 the 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 an 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 the Euler-Bernoulli beam as well as the authors’ previous considerations. Cited in 2 Documents MSC: 49J15 Existence theories for optimal control problems involving ordinary differential equations 49K15 Optimality conditions for problems involving ordinary differential equations 49M25 Discrete approximations in optimal control 49N10 Linear-quadratic optimal control problems 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations Keywords:optimal control; Gao beam equation; Euler-Bernoulli beam; finite element method PDFBibTeX XMLCite \textit{H. Netuka} and \textit{J. Machalová}, Discrete Contin. Dyn. Syst. 2015, 783--792 (2015; Zbl 1334.49006) Full Text: DOI References: [1] A. Borzi, <em>Computational Optimization of Systems Governed by Partial Differential Equations</em>,, SIAM (2012) · Zbl 1240.90001 [2] I. Ekeland, <em>Convex Analysis and Variational Problems</em>,, SIAM (1999) · Zbl 0939.49002 [3] D. Y. Gao, Nonlinear elastic beam theory with application in contact problems and variational approaches., Mechanics Research Communications, 23, 11 (1996) · Zbl 0843.73042 [4] D. Y. Gao, Finite deformation beam models and triality theory in dynamical post-buckling analysis., Int. J. of Non-Linear Mechanics, 35, 103 (2000) · Zbl 1068.74569 [5] J.-L. Lions, <em>Optimal Control of Systems Governed by Partial Differential Equations</em>,, Springer-Verlag (1971) · Zbl 0203.09001 [6] J.-L. Lions, <em>Some Aspects of the Optimal Control of Distributed Parameter Systems</em>,, SIAM (1972) · Zbl 0275.49001 [7] Reddy, An Introduction to the Finite Element Method., Third edition. McGraw-Hill Book Co. (2006) [8] F. Tröltzsch, <em>Optimal Control of Partial Differential Equations. Theory, Methods and Applications</em>,, AMS (2010) · Zbl 1195.49001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.