Advances in Mathematical Physics

Volume 2015, Article ID 585967, 10 pages

http://dx.doi.org/10.1155/2015/585967

## Optimal Control Method of Parabolic Partial Differential Equations and Its Application to Heat Transfer Model in Continuous Cast Secondary Cooling Zone

State Key Laboratory of Synthetical Automation for Process industries, Northeastern University, Shenyang 110004, China

Received 1 November 2014; Revised 16 December 2014; Accepted 8 January 2015

Academic Editor: Ricardo Weder

Copyright © 2015 Yuan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Our work is devoted to a class of optimal control problems of parabolic partial differential equations. Because of the partial differential equations constraints, it is rather difficult to solve the optimization problem. The gradient of the cost function can be found by the adjoint problem approach. Based on the adjoint problem approach, the gradient of cost function is proved to be Lipschitz continuous. An improved conjugate method is applied to solve this optimization problem and this algorithm is proved to be convergent. This method is applied to set-point values in continuous cast secondary cooling zone. Based on the real data in a plant, the simulation experiments show that the method can ensure the steel billet quality. From these experiment results, it is concluded that the improved conjugate gradient algorithm is convergent and the method is effective in optimal control problem of partial differential equations.

#### 1. Introduction

Continuous caster is a machine by which the molten steel is solidified to slabs by spraying on it the cooling water. Because the production situation of secondary cooling zone is very bad, the set point of water volume in secondary cooling plays an important role in the continuous casting [1]. Undercooling of the strand in continuous cast secondary cooling can result in a liquid pool that is too long. However, overcooling can lead to the formation of cracks. The quality of steel billet depends on the behaviour of the surface temperature [2]. Optimal control problem of partial differential equations is encountered [3–5] in many applications ranging from engineering to science. The mathematical model arises in many engineering and physical processes such as heat conduction [2], fluid mechanics [6], and material sciences [7]. Many researchers considered a similar optimal control problem of partial differential equations [8, 9]: where is temperature (K). is the surface temperature (K). is the objective temperature. is a coefficient. is a controlled variable. is a guess for the controlled variable by a priori knowledge. , .

Li [10] applied conjugate gradient algorithm to estimate boundary condition. Lee et al. [11] proposed a repulsive particle swarm optimization algorithm to solve this problem. Farag et al. [12] used a modified partial quadratic interpolation method to solve parabolic optimal control problem. Based on the adjoint problem approach, Kaya and Erdem [13] studied an inverse parabolic problem. Based on the gradient of cost function that is Lipschitz continuous, Hasanov [14] proved existence of a quasisolution of the inverse problem and proposed a monotone iteration scheme. Hasanov [15] used the gradient method and proved the convergence of this method. However, the step size in their gradient algorithm is a constant. Step size largely infects the convergent speed of this gradient algorithm. Hasanov and Pektaş [16] considered an inverse source problem. They used a conjugate gradient algorithm to solve this problem, and the good results were obtained.

In this paper, a class of optimal control problems of partial differential equations is abstracted from optimization problem for set-point values in secondary cooling zone process. In our optimal control of partial differential equations problem, the gradient of cost function is proved Lipschitz continuous. We present an improved conjugate gradient algorithm to solve this problem and prove convergence of this algorithm. The simulation experiment shows that this algorithm can effectively solve the optimization problem for set-point values in secondary cooling zone, and the quality of billet is ensured.

#### 2. Continuous Casting Operation Model of Secondary Cooling Zone

Continuous casting makes a liquid metal by a special set of cooling devices into a certain section in the shape of a casting process. As shown in Figure 1, , , and are the thickness direction of steel billet, the width direction of steel billet, and the length direction of steel billet.