A shape-measure method for solving free-boundary elliptic systems with boundary control function

Document Type : Research Article

Author

• A. Fakharzadeh Jahromi

Department of Mathematics, Shiraz University of Technology, P.O.Box:7155-313, Shiraz, Iran.

10.22067/ijnao.v3i2.24082

Abstract

This article deals with a computational algorithmic approach for obtaining the optimal solution of a general free boundary problem governed by an elliptic equation with boundary control and functional criterion. After determining the weak solution of the system, the problem is converted into a variational format. Then, by using some aspects of measure theory, the method characterize the nearly optimal pair of domain and its related optimal control function at the same time. This method has many advantages such as strong linearity, automatic existence theorem and the ability of obtaining global solution. Two sets of numerical examples is also given.

Keywords

• Elliptic controlled system
• Weak solution
• Atomic measure
• Linear programming
• Free boundary problem

20.1001.1.24236977.2013.3.2.5.8

---

References

1. Alt, W. and Dembo, M. Cytoplasm dynamics and cell motion: two-phase flow models, Int. J. Mathematical Biosciences, 156 (1999), 207–228.
2. Berger, M. P. F. and Wong, W. K. Apllied Optimal Designs, John Wiley & Sons Ltd, 2005.
3. Fakharzadeh Jahromi, A. and Rubio, J. E. Shapes and Measures, Journal of Mathematical Control and Information, 16 (1999), 207-220.
4. Fakharzadeh Jahromi, A. and Rubio, J. E. Shape-Measure Method for Solving Elliptic Optimal Shape Problems (Fixed Control Case), Bulletin of the Iranian Mathematical Society, 27 (2001), 41–63.

5. Fakharzadeh Jahromi, A. and Rubio, J. E. Best Domain for an Elliptic Problem in Cartesian Coordinates by Means of Shape-Measure, AJOP Asian J. of Control, 11, 5 (2009), 536–547.
6. Farahi, M. H. The Boundary Control of the Wave Equation, PhD thesis, Leeds University (1996).
7. Goberna, M. A. and Lopez, M. A. Linear Semi-infinite Optimization, Alicant Uiniversity, 1998.
8. Hadamard, J. Lessons on the Calculus of Variation. Gauthier-Villards, Paris, 1910. (in French).
9. Haslinger, J. and Neittaanamaki, P. Finite Element Approximation for Optimal Shape Design: Theory and Applications, Johan Wiley & Sons Ltd, 1988.
10. Haug, E. J. and Cea, J. Optimization of Disstributed Parameter Structures, Vols I and II, Sijthoff and Noordhoff, Alpen and Rijn, The Nether land, 1981.
11. Kamyad, A. V., Rubio, J. E. and Wilson, D. A. The optimal control of multidimensional diffusion equation, JOTA, 1, 70 (1991), 191–209.
12. Lancaster, K. E. Qualitative behavior of solution of elliptic free boundary problems, Pacific J. Maths., 154, 2 (1992), 297–317.
13. Mikhailov, V. P. Partial Differential Equation, MIR Publisher, Moscow, 1978.
14. Munch, A. Optimal design of the support of the control for the 2-d wave equation: Numerical investigation, Mathematical Modeling and Numerical Analysis, 5, 2 (2008), 331–351.
15. Munch, A. Optimal internal dissipation of a damped wave equation using a topological approach, Int. J. Appl. Math. Comput. Sci., 19, 1 (2009), 15–37.
16. Nelder, J. A. and Mead, R. A simplex method for function minimization, The Computer Journal, 7 (1964-65), 303-313.
17. Press, W. H.,Teukolsky, S. A., Vetterling, W. T. and Flannery, B. R. Neumerical Recipes in Fortran: The art of scientific computing, 2ed edition, Cambridge unversity press, 1992.
18. Pironneau, O. Optimal Shape Design for Elliptic System, Springer Verlag, New York - Berlin - Heidelberg - Tokyo, 1983.
19. Rubio, J. E. Control and Optimization: The Linear Treatment of Nonlinear Problems, Manchester University Press, Manchester, 1986.

20. Rubio, J. E. The global control of nonlinear elliptic equation. Journal of Franklin Institute, 330, 1 (1993), 29–35.
21. Rudin, W. Real and Complex Analysis, Tata McGraw-Hill Publishing Co.Ltd, New Delhi, second edition, 1983.
22. Sun, W. and Ya-Xiang, X. Optimization Theory and Methods: Nonlinear programming, Springer, 2006.
23. Vogel, T. A free boundary problem arising from galvanizing process, SIAM J. Math. Mech. Anal., 16 (1985) 970–979.
24. Young, L. C. Lectures on the Calculues of Variations and Optimal Control Theory, W.B. Sunders Company, 1969.