Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
권호 표지
DOI QR코드

DOI QR Code

FINITE ELEMENT METHOD FOR SOLVING BOUNDARY CONTROL PROBLEM GOVERNED BY ELLIPTIC VARIATIONAL INEQUALITIES WITH AN INFINITE NUMBER OF VARIABLES

  • Received : 2022.07.13
  • Accepted : 2023.06.01
  • Published : 2023.09.15
⟨ Previous Next ⟩

Abstract

In this paper, finite element method is applied to solve boundary control problem governed by elliptic variational inequality with an infinite number of variables. First, we introduce some important features of the finite element method, boundary control problem governed by elliptic variational inequalities with an infinite number of variables in the case of the control and observation are on the boundary is introduced. We prove the existence of the solution by using the augmented Lagrangian multipliers method. A triangular type finite element method is used.

Keywords

References

  1. M.A. Akinlar, Application of a finite element method for variational inequalities, J. Ineq. Appl., 44 (2013). 
  2. V. Barbu, Optimal Control of Variational Inequalities, Research, Notes in Mathematics 100, Boston-London-Melbourne: Pitman, 1984. 
  3. V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering; Boston-San Diego-New York, 189, 1993. 
  4. Ju.M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators "Naukova Dumka" Kiev, 1965. English Trans., Math. Monographs, Amer. Math. Soc., 17, 1968. 
  5. S.A. El-Zahaby, Necessary Conditions for Control Problems Governed by Elliptic Variational Inequalities With an Infinite Number of Variables, J. Advan. Mod. Anal., 44 (1994), 47-55. 
  6. S.A. El-Zahaby and Gh.H. Mostafa, Boundary Control Problem with Nonlinear State Equations with an Infinite Number of Variables, AMSE Press, France, 2007. 
  7. S.A. El-Zahaby and Gh.H. Mostafa, Necessary Conditions for Distributed Control Problems Governed by Variational Inequalities With an Infinite Number of Variables, The Mediterrenean J. Measurment and Control, 1(2005), 191-197. 
  8. I.M. Gali and H.A. El-Saify, Optimal Control of System Governed by Self-Adjoint Elliptic Operator With an Infinite Number of Variables, Proc. of the Second International Conference of Functional Differential Systems and Related Topics, II, Warsaw, Poland, (1981), 126-133. 
  9. I.M. Gali and H.A. El-Saify, Distributed Control of a System Governed by Dirichlet and Neumann Problems for a Self-Adjoint Elliptic Operator With an Infinite Number of Variables, J. Optim. Theory Appl., 39(2) (1983), 293-298.  https://doi.org/10.1007/BF00934534
  10. R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer, Berlin, 2008. 
  11. R. Glowinski and A. Marrocco, Numerical solution of two-dimensional magneto-static problems by augmented Lagrangian methods, Comput. Methods Appl. Mech. Eng., 12 (1977), 33-46.  https://doi.org/10.1016/0045-7825(77)90049-4
  12. J.L. Lions and E. Magnes, Non-homogeneous Boundary Value Problems and Applications, I, II, Springer-Verlag, New York, 1972. 
  13. A.H. Siddiqi, Functional Analysis and Applications, Springer Nature Singapore Pte Ltd, 2018.