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We will explain a new method for obtaining the nearly optimal
domain for optimal shape design problems associated with the
solution of a nonlinear wave equation. Taking into account the
boundary and terminal conditions of the system, a new approach is
applied to determine the optimal domain and its related optimal
control function with respect to the integral performance
criteria, by use of positive Radon measures. The approach, say
shape-measure, consists of two steps; first for a fixed domain,
the optimal control will be identified by the use of measures.
This function and the optimal value of the objective function
depend on the geometrical variables of the domain. In second step,
based on the results of the previous one and by applying some
convenient optimization techniques, the optimal domain and its
related optimal control function will be identified at the same
time. The existence of the optimal solution is considered and a
numerical example is also given.
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