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Optimal control of system governed by the Gao beam equation. (English) Zbl 1334.49006

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.

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
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References:

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