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

ýConsidering a damped wave system defined on a two-dimensional domainý, ýwith a dissipative term localized in an unknown subset with an unknown damping parameterý, ýwe 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 timeý. ýBy using a new approach based on the embedding processý, ýfirstý, ýthe system is formulated in variational form; thený, ýby transferring the problem into polar coordinates and defining two positive Radon measuresý, ýwe represent the problem in a space of measuresý. ýIn this wayý, ýthe shape design problem is changed into an infinite linear one whose solution is guaranteedý. ýIn this stageý, ýby applying two subsequent approximation stepsý, ýthe optimal solution (optimal controlý, ýoptimal regioný, ýoptimal damping parameter and optimal energy) is identified by a three-phase optimization search techniqueý. ýNumerical simulations are also given in order to compare this new method with another oneý.