Opuscula Math. 36, no. 2 (2016), 239-252
http://dx.doi.org/10.7494/OpMath.2016.36.2.239

 
Opuscula Mathematica

Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation

Andrzej Just
Zdzislaw Stempień

Abstract. Our goal is to study the Pareto optimal control system for a nonlinear one-dimensional extensible beam equation and its Galerkin approximation. First we consider a mathematical model of the beam equation which was obtained by S. Woinowsky-Krieger in 1950. Next we consider the Pareto optimal control problem based on this equation. Further, we describe the approximation of this system. We use the Galerkin method to approximate the solution of this control problem with respect to a spatial variable. Based on the standard finite dimensional approximation we prove that as the discretization parameters tend to zero then the weak accumulation point of the solutions of the discrete optimal control problems exist and each of these points is the solution of the original Pareto optimal control problem.

Keywords: nonlinear beam equation, Pareto optimal control, Galerkin approximation.

Mathematics Subject Classification: 49J20, 49M25, 58E17.

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  • Andrzej Just
  • Lodz University of Technology, Centre of Mathematics and Physics, al. Politechniki 11, 90-924 Lodz, Poland
  • Zdzislaw Stempień
  • Lodz University of Technology, Institute of Mathematics, ul. Wolczanska 215, 90-924 Lodz, Poland
  • Communicated by Marek Galewski.
  • Received: 2015-07-09.
  • Accepted: 2015-08-11.
  • Published online: 2015-12-18.
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Cite this article as:
Andrzej Just, Zdzislaw Stempień, Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation, Opuscula Math. 36, no. 2 (2016), 239-252, http://dx.doi.org/10.7494/OpMath.2016.36.2.239

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