Some optimal design problems in a dynamical context: Theory and Numerics

A Review on Optimal Shape Design problems in a Dynamical Context

Atypicalsituationinshapedesignproblemsconsistinoptimizingthe distributionof two materials along a given geometry in order to increase the performance of the resulting system. Mathematically, the problem is reduced to find the infimum - over a classofcharacteristicfunctions-ofa functional whichdependsonthesolutionofa given partial differential equation.

Formally, assuming that this infimum problem is well-posed, one may use level set or topological approach (based on domain derivation techniques) to approximate numerically and efficiently some (local) minima.
However, since the functional depends on the gradient of the solution of the PDE, this kind of problem is generally ill-posed and needs of relaxation. Such relaxed formulations may be obtain explicitly in some simple situation by using Homogeneisation technique (H-mesure, Young mesure) and provide information on the optimal distribution under the form of micro-structure. In this talk, I will detail some example in the context of stabilization and control for the wave and heat equation.