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<p>In order to design a road de-icing device by heating, we consider in a two dimensional setting the optimal control of an advection–diffusion equation with a nonlinear boundary condition of the Stefan-Boltzmann type. The problem models the heating of a road during a winter period to keep positive its surface temperature above a given threshold. The heating device is performed through the circulation of a coolant in a porous layer of the road. We prove the well-posedeness of the nonlinear optimal control problem, subject to unilateral constraints on the control and the state, set up a gradient based algorithm then discuss some numerical results associated with real data obtained from experimental measurements. The study, initially developed in a one dimensional simpler setting in [<CitationRef CitationID="CR1">1</CitationRef>], aims to quantify the minimal energy to be provided to keep the road surface without frost or snow.</p>

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A Non Linear Optimal Control Problem Related to a Road De-icing Device: Analysis and Numerical Experiments

  • 期刊论文

  • 发表日期：

    2025年4月27日

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  • Frédéric Bernardin,
  • Jérôme Lemoine,
  • Arnaud Münch

摘要

In order to design a road de-icing device by heating, we consider in a two dimensional setting the optimal control of an advection–diffusion equation with a nonlinear boundary condition of the Stefan-Boltzmann type. The problem models the heating of a road during a winter period to keep positive its surface temperature above a given threshold. The heating device is performed through the circulation of a coolant in a porous layer of the road. We prove the well-posedeness of the nonlinear optimal control problem, subject to unilateral constraints on the control and the state, set up a gradient based algorithm then discuss some numerical results associated with real data obtained from experimental measurements. The study, initially developed in a one dimensional simpler setting in [1], aims to quantify the minimal energy to be provided to keep the road surface without frost or snow.