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

• Published in: Applied mathematics & optimization Vol. 91; no. 3; p. 63
• Main Authors: Bernardin, Frédéric, Lemoine, Jérôme, Münch, Arnaud
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2025; Springer Nature B.V
• Subjects: Advection-diffusion equation; Algorithms; Applied mathematics; Asphalt pavements; Boundary conditions; Calculus of Variations and Optimal Control; Optimization; Control; Deicing; Energy; Heat transfer; Heating; Hydraulics; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Nonlinear control; Numerical and Computational Physics; Optimal control; Optimization; Partial differential equations; Roads & highways; Simulation; Systems Theory; Temperature; Theoretical; 65C20; Energy; 49J20; Optimal control; Unilateral constraint; Road heating; Non linear advection–diffusion equation; De-icing
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-025-10261-7

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.