Optimal shape and position of the support for the internal exact control of a string

In this paper, we consider the problem of optimizing the shape and position of the support ω of the internal exact control of minimal L 2 ( 0 , T ; L 2 ( ω ) ) -norm for the 1-D wave equation. A relaxation for this problem is found and the minimizers of the relaxed problem are characterized through first-order optimality conditions.