Finding the Optimal Place of Sensors for a 3-D Damped Wave Equation by using Measure Approach

In this paper‎, ‎we model and solve the problem of optimal shaping and placing to put sensors for a 3-D wave equation with constant damping in a bounded open connected subset‎ ‎of 3-dimensional space‎. ‎The place of sensor is modeled by a subdomain‎ ‎of this region of a given measure‎. ‎By using an approach based on the embedding process‎, ‎first‎, ‎the system is formulated in variational form; then‎, ‎by defining‎ ‎two positive Radon measures‎, ‎the problem is represented in a‎ ‎space of measures‎. ‎ In this way, the shape design problem is turned into an infinite linear problem whose solution is guaranteed. In this step, the optimal solution (optimal control, optimal region, and optimal energy) is identified by a 2-phase optimization search technique applying two subsequent approximation steps. Moreover, some numerical simulations are given to compare this new method with other methods.