A Linearization Technique for Optimal Design of the Damping Set with Internal Dissipation

Document Type : Research Article

Authors

1 Department of Mathematics, Faculty of Mathematics, Shiraz University of Technology, Shiraz, Iran.

2 Department of Mathematics, Jahrom University, Jahrom, Iran, P. O. Box: 74135-111 Dept. of Mathematics, Shiraz University of Technology, Shiraz , Iran.

Abstract

‎Considering a damped wave system defined on a two-dimensional domain‎, ‎with a dissipative term localized in an unknown subset with an unknown damping parameter‎, ‎we address the shape design ill-posed problem which consists of optimizing the shape of the unknown subset in order to minimize the energy of the system at a given time‎. ‎By using a new approach based on the embedding process‎, ‎first‎, ‎the system is formulated in variational form; then‎, ‎by transferring the problem into polar coordinates and defining two positive Radon measures‎, ‎we represent the problem in a space of measures‎. ‎In this way‎, ‎the shape design problem is changed into an infinite linear one whose solution is guaranteed‎. ‎In this stage‎, ‎by applying two subsequent approximation steps‎, ‎the optimal solution (optimal control‎, ‎optimal region‎, ‎optimal damping parameter and optimal energy) is identified by a three-phase optimization search technique‎. ‎Numerical simulations are also given in order to compare this new method with another one‎.

Keywords


1. Abbas, H. A. Marriage in Honey Bees Optimization (MBO): A Hap- lomelrosis Polyggnous Swarming Approach, The Congress on Evolution- ary Computation. (2001) 207-214.
2. Alexander, G. R. Spectral and scattering theory, United state of America, 1998.
3. Apostol, T. Mathematical Analysis, (2nd edition). Addison Wesley, 1974.
4. Cox, S. J. and Zuazua, E. The rate at which energy decays in a damped string, Partial Differential Equations. 19 (1994) 213243.
5. Debnath, L. and Mikusinski, P. Hilbert spaces with Applications, Elsevier academic Press, 3 rd Edition , 2005.
6. Fakharzadeh, J. A. and Rubio, J. E. Global Solution of Optimal Shape Design Problems, J. Anal. and Appl. 18 (1999) 143-155.
7. Fakharzadeh, J. A. and Rubio, J. E. Best Domain for an Elliptic Problem in Cartesian Coordinates by Means of Shape-measure, J of control. 11
(2009) 536-554.
8. Fakharzadeh, J. A., Alimorad, D. H. and Raei, Z. Using Linearization and Penalty Approach to Solve Optimal Shape Design Problem with an Obstacle, J. mathematics and computer Science. 7 (2013) 43-53.
9. Freitas, P. Optimizing the Rate of Decay of Solution of the Wave Equation Using Genetic Algorithms: A Counterex Ample to the Constant Damping Conjecture, SIAM J Control Optim. 37 (1999) 376-387.
10. George, B. T., Ross, L. F. and Maurice, D. W. Calculus and Analytic Geometry , 10 Edition, Wesley, 2001.
11. Hebrard, P. and Henrot, A. Optimal Shape and Position of the Actuators for the Stabilization of the String, J. syst. Control Letters 48 (2003) 199-209.
12. Lions, J. and Magenes, E. Problem aux Limites Non Homogenes A pplications, Dunod, Paris, 1968.
13. Lopez-Gomez, J. On the Linear Damped Wave Equation, J. Differ. Equations 134 (1997) 26-45.
14. Muller, S. Variational Model for Microstructure and Phase Transitions,
Lecture at the C.I.M.E summer school 'calculus of variations and geometric evolution problems', 1998.
15. Munch, A., Pedregal, P. and Periago, F. Optimal Design of the Damping set for the Stabilization of the Wave Equation, J. Differ. Equations 231(2006) 331-358.
16. Munch, A., Pedregal, P. and Periago, F. Optimal Internal Stabilization of the Linear System of Elasticity, Archive Rational Mechanical Analysis, 2009.
17. Munch, A. Optimal Internal Dissipation of Damped Wave Equation using a Topological Approach, Int. J. Appl. Math. Comp. Sci. 19 (2009) 15-37.
18. Privat, Y. and Trelat, E. Optimal Design of Sensors for a Damped Wave Equation, Madrid, Discrete Cont. Dynam. Syst. Proceedings of AIMS 2014 workshop.
19. Privat, Y., Trelat, E. and Zuazua, E. Optimal Shape and Location of Sensors for Parabolic Equations with Random Initial Data, to appear in Arch. Rat. Mech. Anal. Preprint (2014).
20. Rubio, J. E. Control and Optimization: the linear treatment of nonlinear problems, Manchester university Press, Manchester, 1986.
21. Rudin, W. Pronciples of Mathematical Analysis , third Edition, McGrawHill, new York, 1976.
22. Rudin, W. Real and Complex Analysis , second Edition, Tata McGrawHill Publishing Co Ltd., new Dehli, 1983.
23. Tcheugoue-Tebou, L. R. Stabilization of the Wave Equation with Localized Nonlinear Damping, J. Differ. Equations 145 (1998), 502-524.
24. Treves, F. Topology Vector Space, Distributions and Kernels , Academic Press, Now York and London, 1967.
CAPTCHA Image