Stability of optimal shapes and convergence of thresholding algorithms in linear and spectral optimal control problems

We prove the convergence of the fixed-point (also called thresholding) algorithm in three optimal control problems under large volume constraints. This algorithm was introduced by Céa, Gioan and Michel, and is of constant use in the simulation of $L^{\mathrm{\infty }}-L^1$ optimal control problems. In this paper we consider the optimisation of the Dirichlet energy, of Dirichlet eigenvalues and of certain non-energetic problems. Our proofs rely on new diagonalisation procedure for shape hessians in optimal control problems, which leads to local stability estimates.