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<p>We consider a general optimal control problem in the setting of gradient flows. Two approximations of the problem are presented, both relying on the variational reformulation of gradient-flow dynamics via the Weighted-Energy-Dissipation variational approach. This consists in the minimization of global-in-time functionals over trajectories, combined with a limit passage. We show that the original nonpenalized problem and the two successive approximations admit solutions. Moreover, resorting to a <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="28_2025_1086_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-convergence analysis we show that penalized optimal controls converge to nonpenalized one as the approximation is removed.</p>

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Optimal control of gradient flows via the Weighted Energy-Dissipation method

  • 期刊论文

  • 发表日期：

    2025年5月24日

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  • Takeshi Fukao,
  • Ulisse Stefanelli,
  • Riccardo Voso

摘要

We consider a general optimal control problem in the setting of gradient flows. Two approximations of the problem are presented, both relying on the variational reformulation of gradient-flow dynamics via the Weighted-Energy-Dissipation variational approach. This consists in the minimization of global-in-time functionals over trajectories, combined with a limit passage. We show that the original nonpenalized problem and the two successive approximations admit solutions. Moreover, resorting to a \(\Gamma \) Γ -convergence analysis we show that penalized optimal controls converge to nonpenalized one as the approximation is removed.