First-Optimize-Then-Discretize Strategy for the Parabolic PDE Constrained Optimization Problem With Application to the Reheating Furnace

Abstract

Each slab entering to the reheating furnace has an optimal and unique reheating curve. The process of obtaining the optimal reheating curve is to solve the typical Partial differential equations (PDE) constrained optimization problem. Obviously, the solution of optimization problem is determined by both the precision of the mathematical PDE model and the numerical method. Firstly, the more accurate mathematical PDE model, in which some key parameters are reconsidered as temperature-dependent, is built for the reheating furnace. Secondly, the first-optimize-then-discretize approach is introduced to solve this PDE-constrained optimization problem. The analysis of the Fr <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">é</i> chet gradient of the cost functional is given and we can prove the gradient is Lipschitz continuous. Then, an improved conjugate gradient method is proposed to solve this problem. Finally, numerical simulations and experiment examples are given and analyzed. The results can prove the effectiveness of the proposed strategy.