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

Document Type : Research Article


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.


‎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.


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