Optimal design of sensors via geometric criteria

I. Ftouhi, Zuazua E.. Optimal design of sensors via geometric criteria (2023) J. Geom. Anal., Vol. 33, No. 253, https://doi.org/10.1007/s12220-023-01301-1

Abstract. We consider a convex set Ω and look for the optimal convex sensor ω ⊂ Ω of a given measure that minimizes the maximal distance to the points of Ω. This problem can be written as follows

inf{dH(ω,Ω) | |ω|= c and ω ⊂ Ω},

where c ∈ (0, |Ω|), dH being the Hausdorff distance.

We show that the parametrization via the support functions allows us to formulate the geometric optimal shape design problem as an analytic one. By proving a judicious equivalence result, the shape optimization problem is approximated by a simpler minimization of a quadratic function under linear constraints. We then present some numerical results and qualitative properties of the optimal sensors and exhibit an unexpected symmetry breaking phenomenon.

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arxiv: 00.0000