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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3182_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^\infty -L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mo>-</mo> <msup> <mi>L</mi> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年5月24日

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  • Antonin Chambolle,
  • Idriss Mazari-Fouquer,
  • Yannick Privat

摘要

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 \(L^\infty -L^1\) L ∞ - 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.