# American Institute of Mathematical Sciences

January  2018, 23(1): 263-274. doi: 10.3934/dcdsb.2018018

## Optimal control problem for a viscoelastic beam and its galerkin approximation

 1 Centre of Mathematics and Physics, Lodz University of Technology, Al.Politechniki 11, 90-924 Lodz, Poland 2 Institute of Mathematics, Lodz University of Technology, ul. Wolczanska 215, 90-924 Lodz, Poland

* Corresponding author: zdzislaw.stempien@p.lodz.pl

Received  November 2016 Revised  May 2017 Published  January 2018

This paper is concerned with the optimal control problem of the vibrations of a viscoelastic beam, which is governed by a nonlinear partial differential equation. We discuss the initial-boundary problem for the cases when the ends of the beam are clamped or hinged. We define the weak solution of this initial-boundary problem. Our control problem is formulated by minimization of a functional where the state of a system is the solution of viscoelastic beam equation. We use the Galerkin method to approximate the solution of our control problem with respect to a spatial variable. Based on the finite dimensional approximation we prove that as the discretization parameters tend to zero then the weak accumulation points of the optimal solutions of the discrete family control problems exist and each of these points is the solution of the original optimal control problem.

Citation: 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
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