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

TL;DR

This paper proves the convergence of a thresholding algorithm in various linear and spectral optimal control problems with large volume constraints, introducing new stability analysis techniques.

Contribution

It introduces a new diagonalisation method for shape Hessians, enabling convergence proofs for the thresholding algorithm in multiple optimal control settings.

Findings

Convergence of the thresholding algorithm is established for Dirichlet energy optimization.

Convergence results extend to Dirichlet eigenvalues and certain non-energetic problems.

New stability estimates for shape Hessians are developed.

Abstract

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 C\'ea, Gioan and Michel, and is of constant use in the simulation of $L^\infty-L^1$ 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.