Parameterization optimal control of an unsteady partial differential equations with convection term by an improved three-term spectrum conjugate gradient algorithm

Abstract

The unsteady partial differential equations (UPDE) with convection term gives a clear descriptions for the solidification process of a slab in dynamic production of continuous casting. To give a suitable setting value of secondary cooling water flow rate for the dynamic control system, this study investigates an optimal control problem (OCP) of UPDE with convection term. Firstly, control vector discretization of OCP and the solution of UPDE are given. Secondly, due to the rapidity for gradient, this paper analyzes the expression of the gradient calculation method based on Hamiltonian function costate system by approximate treatment, matrix calculation and composite trapezoidal integral method. Thirdly, an improved three-term spectrum conjugate gradient algorithm (ITSCGA) is proposed to solve the OCP of UPDE, and the global convergence of the ITSCGA is demonstrated. Lastly, the performance of ITSCGA is demonstrated by experimental simulations. The results demonstrate that the ITSCGA provides a smaller temperature fluctuations, and improves the quality of a slab.

Keywords

Control parameterization unsteady partial differential equations optimal control three-term spectrum conjugate gradient algorithm dynamic process

Introduction

Partial differential equations (PDEs) are widely applied in many fields, such as physics, chemistry, biology, medicine, and so on, and it is encouraged to solve the mathematical problems in economics, finance, image processing, and other fields. Especially in metallurgy, the solidification heat transfer model can be used to represent the cooling process of a slab. Most secondary cooling water control systems^1–4 of continuous casting explore this PDE model to adjust the water flow rate setting value. Therefore, the study of optimization problem based on the solidification heat transfer model of a slab is crucial for the secondary cooling systems of continuous casting process. Partial scholars^5–10 based on the unsteady heat transfer model of a slab study the optimization method for the water flow rate setting value in the static production of continuous casting. However, in actual production, some dynamic situations of continuous casting often occur due to the early or tardy arrival of a ladle and the change of superheat, which can lead to adjustments in casting speed. Therefore, an unsteady heat transfer model with convection term (UHTMCT) of a slab attracts the attention of some scholars to deal with the dynamic situations. Based on the UHTMCT of a slab, Yu et al.¹¹ optimizes the water flow rate setting value in dynamic environment. Hardin et al.¹² proposes a dynamic spray cooling control model, and applies them to determine the optimal amount of water volume at each cooling zone. Wang et al.¹³ introduces an open-loop dynamic spray cooling control strategy, which uses a control algorithm to track the casting speed and temperature of each slice.

In the view of above researches, our previous works¹⁴ study the nonlinear model predictive control (MPC) of the water flow rate setting value based on the UHTMCT of a slab. According to Canto and Banga¹⁵ and Ping et al.,¹⁶ optimal control can be an effective method to obtain the control trajectory for nonlinear MPC implementation. Due to the large computation in NMPC applications, it is very essential to develop more efficient solvers for the nonlinear MPC of UHTMCT. Therefore, optimal control problem (OCP) of UHTMCT and an efficient solution method are investigated in this paper. The optimization efficiency is an important work to solve the OCP, thus a highly efficient parameterization optimal method and an improved three-term spectrum conjugate gradient algorithm (ITSCGA) are presented as a prominent solver. Furthermore, for the parameterization optimal method, in order to reduce the computational expense of gradients, the gradient of OCP for UHTMCT is analyzed based on a Hamiltonian costate system by the approximate treatments of numerical integration strategy.

Optimization problem of boundary conditions of PDEs

OCP of UHTMCT^17,18 in continuous casting belongs to the optimization of boundary conditions of PDEs. At present, optimization problems of boundary conditions of PDEs are divided into two categories: (1) Boundary condition identification problem of PDEs. At present, boundary condition identification problems of PDEs have been applied in many fields.^19–22 Duda et al.^23,24 solve the transient heat conduction equation based on the control volume finite element method and the control volume method, and estimate an unknown boundary condition through the transient temperature distribution. Sun et al.²⁵ use the sequential quadratic programming algorithm as an optimization technique to determine the surface heat flux of the billet. Huang et al.²⁶ estimate the time-varying heat flux on the chip boundary, based on three-dimensional unsteady PDE by using the inverse method; Ruan et al.²⁷ develop a stochastic particle swarm optimization method to determine the heat transfer coefficient in the continuous casting process. Torrijos et al.²⁸ study the inverse heat problem for transient heat conduction equation of the heat flux on the unknown surface of the molten salt loop. (2) The optimal control problem of boundary conditions of PDEs. To solve this problem, scholars have carried out many research works and proposed various optimal control algorithms.^29–32 Gavrikov et al.³³ study the optimal boundary control problem with linear quadratic cost function in the heat transfer process of a cylinder; Mechelli and Volkwein³⁴ study an optimal boundary and bilateral control problem of heat transfer equation with convection term; Dekhkonov³⁵ recently studies the boundary control problem, based on the heat transfer model, for the heat exchange process; Bollo et al.³⁶ solve OCPs of Neumann boundary for a $n$ -dimensional heat equation; Abbasi and Malek³⁷ solve pointwise optimal control problem on and inside a tissue subject to thermal wave model with Dirichlet and Rubin boundary conditions.

The algorithms for solving the optimization problem of boundary conditions of PDEs are mainly divided into two categories. (1) Stochastic algorithms. The stochastic algorithms^38,39 have the random search’s global optimization characteristic, but the disadvantages of these methods are slow convergence speed and high computational cost. (2) Gradient algorithm. There are a number of efficient numerical methods based on the gradient, such as Newton method,⁴⁰ alternating direction multiplier method,⁴¹ and conjugate gradient algorithm (CGA). For the Newton method,⁴⁰ we have done the study on this method, which needs to calculate the second derivative of objective function, which is difficult to calculate when applied to our problem. Therefore, Newton method⁴⁰ is not suitable to this paper’s problem. For the alternating direction multiplier method, we only find one literature,⁴¹ which uses the alternating direction method of multipliers (ADMM) to solve the optimal control model constrained by one-dimensional partial differential equation. In Dawodu,⁴¹ the PDE does not include the convection term, and this PDE can be discretized into a derived discrete convex optimization form amenable to the ADMM with the help of Crank-Nicolson and Composite Simpson’s methods. However, the problem in our paper contains convection term, and has a higher dimension PDE, which leads to difficult to transform the original problem into the convex optimization form amenable to the ADMM. Furthermore, at present, much literature develop CGA to solve the OCP¹⁶ and optimization problem of PDE.^42–51 Zhou et al.⁴² estimate the temperature and heat flux on inaccessible surfaces by using the conjugate gradient method with temperature and heat flux measured on back surface. Yang et al.⁴³ propose a modified conjugate gradient method to identify the physical parameters of transient heat conduction problems in aerospace engineering. Wang et al.^44,45 propose an improved CGA to solve the OCP based on one-dimensional and two-dimensional parabolic PDEs; Razzaghi et al.⁴⁶ adopts an improved CGA to minimize cost function in the unsteady heat transfer model. They solve accurately an optimization problem by using CGA. Other notable surveys in this field can be found in Refs.^47–49. In view of the above research findings, our previous works^50,51 combine the regularization method with the improved conjugate gradient algorithm to identify the boundary condition of the heat transfer model. Based on the above reasons, we select the CGA to solve OCP of PDE, and develop an improved three-term spectrum conjugate gradient algorithm (ITSCGA).

Motivation and innovation

Based on the above analysis, few literature study the OCP of UHTMCT in continuous casting. Therefore, in the view of above valuable optimal control methods of unsteady PDE, this paper investigates the OCP of UHTMCT, and presents a parameterization optimal method that combines gradient analysis based on a Hamiltonian function costate system and an ITSCGA. Due to the rapidity for gradient, the control vectors of the OCP are discretized in time and space, and the expression of the gradient calculation method based on the Hamiltonian function costate system is analyzed. Then, an ITSCGA is proposed to solve the OCP of UHTMCT. Meanwhile, the global convergence of ITSCGA is demonstrated. The following are the main contributions of this study:

An OCP of the UHTMCT is investigated.

The gradient of OCP for UHTMCT is analyzed based on a Hamiltonian costate system by approximate treatment, matrix calculation, and composite trapezoidal integral method.

An ITSCGA is proposed in this paper, and the global convergence of ITSCGA is analyzed.

The structure of this paper is as follows: in Section “The solution of dynamic PDE and control vector discretization,” the parameterization of the control vector and the solution of dynamic PDE are described; in Section “The analysis of gradient calculation,” we give the strategy of fast gradient calculation; in Section “Improved three-term spectrum conjugate gradient algorithm and its global convergence,” the ITSCGA is proposed, and the global convergence of the ITSCGA is analyzed; Section “Experiment simulation” verifies the performance of the ITSCGA by experimental simulations.

The solution of dynamic PDE and control vector discretization

The optimal control model for unsteady PDE with convective term is described as follows:

\begin{matrix} min_{h \in ℜ^{3}} J (h) = \int_{0}^{t_{f}} \int_{0}^{L} {‖ T_{c} (l, t, z; h) - T_{d} (l, t, z) ‖}^{2} d t d z \\ + α \int_{0}^{t_{f}} \int_{0}^{L} ϕ (h (t, z)) d t d z, \\ s . t . T_{t} + V_{cast} T_{z} = k_{0} T_{xx}, (x, t, z \in Ω_{T}), \\ T (x, 0, z) = T_{0}, \\ - k_{0} {T_{x} (x, t, z) |}_{x = 0} = 0, - k_{0} {T_{x} (x, t, z) |}_{x = l} = h (t, z), \\ - k_{0} {T_{z} (x, t, z) |}_{z = 0} = 0, - k_{0} {T_{z} (x, t, z) |}_{z = L} = 0, \end{matrix}

(1)

where $Ω_{T} : = {(x, t, z) \in ℜ^{3} : x \in [0, l], t \in [0, t_{f}], z \in [0, L], t_{f} > 0}$ , $t$ represent time; $x$ represent the width direction of the billet; $z$ represent length direction of the billet; $J (h)$ is a cost function; $h (t, z)$ is the heat flux $(W / m^{2})$ , which satisfies $h (\cdot) \in H (Ω_{T})$ ; $H (Ω_{T}) : = L_{2} (Ω_{T})$ is the set of admissible unknown heat fluxes of $h (\cdot)$ . ∥·∥ is norm; the function $ϕ (\cdot) = ∥ \cdot ∥^{2}$ . The $T$ subscripts $x$ , $z$ , and $t$ stand for the partial derivatives with respect to $x$ , $z$ , and $t$ , respectively; $V_{cast}$ is the casting speed $(m / \min)$ ; $T$ is the temperature of steel billets (K); $T_{c}$ is calculated temperature of steel billets (K); $T_{d}$ is target temperature of steel billets (K); $t_{f}$ is the final time (s); $k_{0}$ is a thermal diffusivity coefficient $(W / m \cdot K)$ that depends on the physical characteristics of the steel.¹⁷ $α$ is a positive number; $L$ and $l$ are the length and width of the steel billet (m), respectively. $T (x, 0, z) = T_{0}$ is initial condition, $T_{0}$ is the initial temperature (K), which is a constant; The boundary conditions at positions $x = 0$ and $x = l$ are as $- k_{0} T_{x} (x, t, z) |_{x = 0} = 0$ and $- k_{0} T_{x} (x, t, z) |_{x = l} = h (t, z)$ ; The boundary conditions at positions $z = 0$ and $z = L$ are as $- k_{0} T_{z} (x, t, z) |_{z = 0} = 0$ and $- k_{0} T_{z} (x, t, z) |_{z = L} = 0$ .

The solution of dynamic PDEs

Equation (1) contains an UHTMCT that can be used to describe the solidification process of continuous casting. As shown in Figure 1, to solve this UHTMCT, this paper considers a life cycle model method (LCMM).⁸ In the LCMM, the first slice is generated when continuous casting begins. The first slice moves $d$ distance along the direction of the casting speed, and the second one is generated. The slice is created at the surface of the mold, and dead at the outlet of the secondary cooling zone. Therefore, the whole billet is composed of many slicing elements, and the LCMM is used to solve the transient convective heat transfer model. The LCMM divides the whole billet into $Q$ slices along the direction of casting speed, and the distance between two adjacent slices is $d$ . The heat transfer equation of each slice is given as follows:

T_{t} = k_{0} T_{xx} .

(2)

Figure 1.

The solidification process and life cycle model method of the cast billet.

We solve the equation (2) by using the finite difference method, which is shown in the following:

\frac{T_{q}^{n + 1} - T_{q}^{n}}{Δ t} = k_{0} \frac{T_{q + 1}^{n} - 2 T_{q}^{n} + T_{q - 1}^{n}}{Δ x^{2}},

(3)

where $q$ is the indexes of space nodes in $z$ ; $Δ t$ is time step; $Δ z$ is spatial step size. $n + 1$ and $n$ are the backward and forward time intervals, respectively.

By organizing the equation (3), we can obtain

T_{q}^{n + 1} = T_{q}^{n} + r (T_{q + 1}^{n} - 2 T_{q}^{n} + T_{q - 1}^{n}),

(4)

where $r = k_{0} \frac{Δ t}{Δ x^{2}}$ .

The boundary condition can be written as:

\frac{T_{q}^{n + 1} - T_{q}^{n}}{Δ t} \frac{Δ x}{2} = k_{0} \frac{T_{q \pm 1}^{n} - T_{q}^{n}}{Δ x} + h (T_{w} - T_{q}^{n}),

(5)

where $T_{w}$ is cold water temperature.

So we can get the upper boundary:

T_{q}^{n + 1} = T_{q}^{n} + 2 r (T_{q - 1}^{n} - T_{q}^{n}) + \frac{2 h r Δ x}{k_{0}} (T_{w} - T_{q}^{n}),

(6)

and the lower boundary:

T_{q}^{n + 1} = T_{q}^{n} + 2 r (T_{q + 1}^{n} - T_{q}^{n}) + \frac{2 h r Δ x}{k_{0}} (T_{w} - T_{q}^{n}) .

(7)

The equations (2) to (7) are given to solve one slice model. Furthermore, the thermal convection term $\partial T / \partial z$ can be replaced by the movement of the slice based on the LCMM.

Control vector discretization

According to equation (1), we can obtain the cost function of the OCP as follow:

\begin{matrix} J (h) = \int_{0}^{t_{f}} \int_{0}^{L} {‖ T_{c} (l, t, z; h) - T_{d} (l, t, z) ‖}^{2} d t d z \\ + α \int_{0}^{t_{f}} \int_{0}^{L} ϕ (h (t, z)) d t d z . \end{matrix}

(8)

In the following, the specific discretized process of the control vector $h (t, z)$ is given to obtain an approximate problem. Firstly, the control vector is discretized into $N$ stages [ $t_{i - 1}, t_{i}$ ] within the time range of $[0, t_{f}]$ , and the space direction $z$ is divided into $M$ sections [ $z_{j - 1}, z_{j}$ ] respectively, which can be written as follows:

0 = 0 < t_{1} < t_{2} < \dots < t_{i - 1} < t_{i} \dots < t_{N} = t_{f},

(9)

0 = 0 < z_{1} < z_{2} < \dots < z_{j - 1} < z_{j} \dots < z_{M} = L .

(10)

Secondly, the piecewise constant $χ_{[t_{i - 1,} t_{i})}$ is used to discretize the time horizon of control vector,

\begin{matrix} χ_{[t_{i - 1,} t_{i})} = {\begin{matrix} 1, t \in [t_{i - 1,} t_{i}) \\ 0, t \notin [t_{i - 1,} t_{i}) \end{matrix}, \end{matrix}

(11)

where $i = 1, 2, 3, \dots, N$ and $j = 1, 2, 3, \dots, M$ .

Thirdly, by using the matrix calculation, the control vector $h (t, z)$ can be approximately expressed as,

h (t, z_{1}) = \sum_{i = 1}^{N} σ_{1}^{i} \cdot χ_{[t_{i - 1}, t_{i})},

(12)

h (t, z_{2}) = \sum_{i = 1}^{N} σ_{2}^{i} \cdot χ_{[t_{i - 1}, t_{i})},

(13)

h (t, z_{3}) = \sum_{i = 1}^{N} σ_{3}^{i} \cdot χ_{[t_{i - 1}, t_{i})},

(14)

⋮

h (t, z_{M}) = \sum_{i = 1}^{N} σ_{M}^{i} \cdot χ_{[t_{i - 1}, t_{i})} .

(15)

Generally, the control vector in a dynamic system satisfies the relation $h_{\min} (t, z_{j}) \leq h (t, z_{j}) \leq h_{\max} (t, z_{j})$ in the range of 0 to $t_{f}$ , where $h_{\min} (t, z_{j})$ and $h_{\max} (t, z_{j})$ are the given boundary limitations. Therefore, the limitation of $σ_{j}^{i}$ is $h_{\min} (t, z_{j}) \leq σ_{j}^{i} \leq h_{\max} (t, z_{j})$ , $i = 1, 2, 3, \dots, N$ , $j = 1, 2, 3, \dots, M$ .

So we can obtain

h (t, z) \approx \bar{h} = σ \cdot χ_{M},

(16)

where $\bar{h} = [\begin{matrix} h (t, z_{1}) \\ h (t, z_{2}) \\ \begin{matrix} h (t, z_{3}) \\ ⋮ \\ h (t, z_{M}) \end{matrix} \end{matrix}], σ = [\begin{matrix} σ_{1}^{1} & σ_{1}^{2} & \begin{matrix} σ_{1}^{3} & \dots & σ_{1}^{N} \end{matrix} \\ σ_{2}^{1} & σ_{2}^{2} & \begin{matrix} σ_{2}^{3} & \dots & σ_{2}^{N} \end{matrix} \\ \begin{matrix} σ_{3}^{1} \\ ⋮ \\ σ_{M}^{1} \end{matrix} & \begin{matrix} σ_{3}^{2} \\ ⋮ \\ σ_{M}^{2} \end{matrix} & \begin{matrix} \begin{matrix} σ_{3}^{3} & \dots & σ_{3}^{N} \end{matrix} \\ \begin{matrix} ⋮ & ⋱ & ⋮ \end{matrix} \\ \begin{matrix} σ_{M}^{3} & \dots & σ_{M}^{N} \end{matrix} \end{matrix} \end{matrix}], χ_{M} = [\begin{matrix} χ_{[t_{0}, t_{1})} \\ χ_{[t_{1}, t_{2})} \\ \begin{matrix} χ_{[t_{2}, t_{3})} \\ ⋮ \\ χ_{[t_{N - 1}, t_{N})} \end{matrix} \end{matrix}] .$

According to Lin et al.,⁵² it has proved that the state trajectory can be calculated precisely. According to the equation (16), we can find that $h (t, z)$ is an approximate function of $σ$ . Figure 2 shows the discretization of the control vector in time at the $j$ space layer. The discrete stages $N$ of time layer can affect the degree of curve approximation. Obviously, the higher the discrete stage, the better the approximate accuracy.

Figure 2.

Control vector discretization approximation diagram.

Therefore, the equation (1) can be approximately converted into an OCP as follows:

\begin{matrix} J (σ) = \int_{0}^{t_{f}} \int_{0}^{L} {‖ T_{c} (l, t, z; σ) - T_{d} (l, t, z) ‖}^{2} d t d z \\ + α \int_{0}^{t_{f}} \int_{0}^{L} ϕ (σ) d t d z, \\ s . t . T_{t} + V_{cast} T_{z} = k_{0} T_{xx}, \\ T (x, 0, z) = T_{0}, \\ - k_{0} \frac{\partial T}{\partial x} |_{x = 0} = 0, - k_{0} \frac{\partial T}{\partial x} |_{x = l} = \bar{h}, \\ - k_{0} \frac{\partial T}{\partial z} |_{z = 0} = 0, - k_{0} \frac{\partial T}{\partial z} |_{z = L} = 0 . \end{matrix}

(17)

Based on the Loxton and Lin,⁵³ it is easy to conclude that the formula (17) converges to formula (1), so the proof process is not detailed description here.

The analysis of gradient calculation

By analyzing the equation (17), it can be seen that $σ$ is the decision variable of the OCP. Generally, gradient-based solvers are used to solve the OCP due to its rapidity. Therefore, the gradient calculation for the cost function $J$ in this paper is analyzed based on the Hamiltonian function costate system. The specific description is given as follows:

The Hamiltonian function of the cost function is defined as follow:

\begin{matrix} \bar{H} (T (x, t, z; σ), λ, σ, x, t, z) = ‖ T_{c} (l, t, z; σ) - T_{d} (l, t, z) ‖^{2} \\ + λ^{T} (k_{0} T_{xx} - V_{cast} T_{z}), \end{matrix}

(18)

where $\bar{H} (T (x, t, z; σ), λ, σ, x, t, z) = [\begin{matrix} H_{1} (T (x, t, z; σ_{1}), λ, σ_{1}, x, t, z) \\ H_{2} (T (x, t, z; σ_{2}), λ, σ_{2}, x, t, z) \\ ⋮ \\ H_{M} (T (x, t, z; σ_{M}), λ, σ_{M}, x, t, z) \end{matrix}],$ $σ = [\begin{matrix} σ_{1} \\ σ_{2} \\ ⋮ \\ σ_{M} \end{matrix}]$ . Meanwhile, $λ \in R^{n}$ is the adjoint vector or costate vector; The $T$ in the upper right corner of a vector represents the transposition of the vector. Correspondingly, we can obtain the costate system¹⁶ of the Hamiltonian function,

\begin{matrix} \overset{\cdot}{λ} (t) = - {[\frac{\partial \bar{H} (T (x, t, z; σ), λ, σ, x, t, z)}{\partial T}]}^{T}, \\ λ (t_{f}) = {[\frac{\partial ϕ (σ)}{\partial T}]}^{T} = 0, t \in [0, t_{f}] . \end{matrix}

(19)

Where $λ$ is the solution of costate system (19). Consequently, the following theorem can be used to determine the gradient of the cost function.

Theorem 3.1. If the Hamiltonian function is defined by equation (18), the costate system is generated by equation (19). The gradient formulate of $J$ can be expressed as below:

\begin{matrix} \frac{\partial J}{\partial σ} = α \int_{0}^{t_{f}} \int_{0}^{L} \frac{\partial ϕ (σ)}{\partial σ} d t d z \\ + \int_{0}^{t_{f}} \int_{0}^{L} \frac{\partial \bar{H} (T (x, t, z; σ), λ, σ, x, t, z)}{\partial σ} d t d z . \end{matrix}

(20)

Proof. Assuming $v (t)$ is an arbitrary continuous function. Then $J$ in equation (17) can be rewritten as:

\begin{matrix} J (σ) = \int_{0}^{t_{f}} \int_{0}^{L} {‖ T_{c} (l, t, z; σ) - T_{d} (l, t, z) ‖}^{2} d t d z \\ + α \int_{0}^{t_{f}} \int_{0}^{L} ϕ (σ) d t d z \\ = α \int_{0}^{t_{f}} \int_{0}^{L} ϕ (σ) d t d z + \int_{0}^{t_{f}} \int_{0}^{L} [\bar{H} (T (x, t, z; σ), λ, σ, x, t, z) \\ - v {(t)}^{T} \cdot T_{t} (x, t, z; σ)] d t d z \\ = α \int_{0}^{t_{f}} \int_{0}^{L} ϕ (σ) d t d z + \int_{0}^{t_{f}} \int_{0}^{L} \bar{H} (T (x, t, z; σ), λ, σ, x, t, z) d t d z \\ - \int_{0}^{t_{f}} \int_{0}^{L} v {(t)}^{T} \cdot T_{t} (x, t, z; σ) d t d z . \end{matrix}

(21)

Applied the integration by parts, $\int_{0}^{t_{f}} \int_{0}^{L} v {(t)}^{T} \cdot T_{t} (x, t, z; σ) d t d z$ can be transformed into

\begin{matrix} \int_{0}^{t_{f}} \int_{0}^{L} v {(t)}^{T} \cdot T_{t} (x, t, z; σ) d t d z \\ = \int_{0}^{L} v {(t_{f})}^{T} \cdot T (x, t_{f}, z; σ) d z - \int_{0}^{L} v {(0)}^{T} \cdot T (x, 0, z; σ) d z \\ - \int_{0}^{t_{f}} \int_{0}^{L} \overset{\cdot}{v} {(t)}^{T} \cdot T (x, t, z; σ) d t d z . \end{matrix}

(22)

So the following equation can be obtained:

\begin{matrix} J (σ) = α \int_{0}^{t_{f}} \int_{0}^{L} ϕ (σ) d t d z + \int_{0}^{t_{f}} \int_{0}^{L} \bar{H} (T (x, t, z; σ), λ, σ, x, t, z) d t d z \\ - \int_{0}^{L} v {(t_{f})}^{T} \cdot T (x, t_{f}, z; σ) d z + \int_{0}^{t_{f}} \int_{0}^{L} \overset{\cdot}{v} {(t)}^{T} \cdot T (x, t, z; σ) d t d z, \end{matrix}

(23)

where $T (x, 0, z) = 0$ .

Taking the derivative of the formula (23) with respect to $σ$ , we have

\begin{matrix} \frac{\partial J}{\partial σ} = α \int_{0}^{t_{f}} \int_{0}^{L} \frac{\partial ϕ (σ)}{\partial σ} d t d z + \int_{0}^{t_{f}} \int_{0}^{L} \frac{\partial \bar{H} (T (x, t, z; σ), λ, σ, x, t, z)}{\partial σ} d t d z \\ + \int_{0}^{t_{f}} \int_{0}^{L} \frac{\partial \bar{H} (T (x, t, z; σ), λ, σ, x, t, z)}{\partial T} \cdot \frac{\partial T (x, t, z; σ)}{\partial σ} d t d z \\ + \int_{0}^{t_{f}} \int_{0}^{L} \overset{\cdot}{v} {(t)}^{T} \cdot \frac{\partial T (x, t, z; σ)}{\partial σ} d t d z - \int_{0}^{L} v {(t_{f})}^{T} \cdot \frac{\partial T (x, t_{f}, z; σ)}{\partial σ} d z . \end{matrix}

(24)

Because $v (t)$ is a continuous function, we choose arbitrarily and let $v (t) = λ$ , we can obtain the gradient formula of Hamiltonian costate system,

\frac{\partial J}{\partial σ} = α \int_{0}^{t_{f}} \int_{0}^{L} \frac{\partial ϕ (σ)}{\partial σ} d t d z + \int_{0}^{t_{f}} \int_{0}^{L} \frac{\partial \bar{H} (T (x, t, z; σ), λ, σ, x, t, z)}{\partial σ} d t d z .

(25)

Therefore, Theorem 3.1 is proved.

In order to deal with the integral term in equation (25), an approximate treatment and composite trapezoidal integral method is used. So equation (25) can be approximated by the following equation:

\begin{matrix} \frac{\partial J}{\partial σ} = α \int_{0}^{t_{f}} \int_{0}^{L} \frac{\partial ϕ (σ)}{\partial σ} d t d z + \int_{0}^{t_{f}} \int_{0}^{L} \frac{\partial \bar{H} (T (x, t, z; σ), λ, σ, x, t, z)}{\partial σ} d t d z \\ = α \sum_{j = 1}^{M} \int_{0}^{t_{f}} \frac{\partial ϕ (σ_{j})}{\partial σ_{j}} d t + \sum_{j = 1}^{M} \int_{0}^{t_{f}} \frac{\partial \bar{H} (T (x, t, z; σ_{j}), λ, σ_{j}, x, t, z_{j})}{\partial σ_{j}} d t . \end{matrix}

(26)

According to the compound gradient quadrature formula, we can obtain

\begin{matrix} \frac{\partial J}{\partial σ} = α \sum_{j = 1}^{M} \sum_{i = 1}^{N} \int_{t_{i - 1}}^{t_{i}} \frac{\partial ϕ (σ_{j})}{\partial σ_{j}^{i}} d t + \sum_{j = 1}^{M} \sum_{i = 1}^{N} \\ \int_{t_{i - 1}}^{t_{i}} \frac{\partial \bar{H} (T (x, t, z; σ_{j}), λ, σ_{j}, x, t, z_{j})}{\partial σ_{j}^{i}} d t + R_{n} (H (\cdot)) \\ \approx α \sum_{j = 1}^{M} \sum_{i = 1}^{N} \frac{\partial ϕ (σ_{j})}{\partial σ_{j}^{i}} + \sum_{j = 1}^{M} \sum_{i = 1}^{N} \\ (\begin{matrix} \frac{\partial \bar{H} (T (x, t_{i - 1}, z; σ_{j}), λ (t_{i - 1}), σ_{j}, x, t_{i - 1}, z_{j})}{\partial σ_{j}^{i}} \\ + \frac{\partial \bar{H} (T (x, t_{i}, z; σ_{j}), λ (t_{i}), σ_{j}, x, t_{i}, z_{j})}{\partial σ_{j}^{i}} \end{matrix}) \frac{t_{i} - t_{i - 1}}{2}, \end{matrix}

(27)

where $σ_{j} = {[σ_{j}^{1}, σ_{j}^{2}, σ_{j}^{3}, \dots, σ_{j}^{N}]}^{T}$ . In equation (27), $R_{n} (\bar{H} (\cdot))$ is a remainder term, which can be easily calculated by the compound gradient quadrature formula.¹⁶

R_{n} (\bar{H} (\cdot)) = \sum_{j = 1}^{M} \sum_{i = 1}^{N} [- \frac{{(t_{i} - t_{i - 1})}^{3}}{12} \bar{H} ″ (ξ_{i - 1}, z_{j})],

(28)

where $ξ$ is a constant that satisfies $ξ_{i} \in [t_{i - 1}, t_{i}]$ , $i = 1, 2, 3 \dots N$ , $j = 1, 2, 3 \dots M$ .

Meanwhile, the time is divided into $N$ equal intervals, we have

ϑ = t_{i} - t_{i - 1} = \frac{t_{f} - 0}{N} .

(29)

Due to the following inequality,

\begin{matrix} \min_{\binom{0 \leq j \leq M}{0 \leq i \leq N - 1}} \bar{H} ″ (ξ_{i - 1}, z_{j}) \leq \frac{1}{MN} \sum_{j = 1}^{M} \sum_{i = 1}^{N} \\ \bar{H} ″ (ξ_{i - 1}, z_{j}) \leq max_{\binom{0 \leq j \leq M}{0 \leq i \leq N - 1}} \bar{H} ″ (ξ_{i - 1}, z_{j}), \end{matrix}

(30)

we can obtain

H ″ (ξ, z_{j}) = \frac{1}{MN} \sum_{j = 1}^{M} \sum_{i = 1}^{N} \bar{H} ″ (ξ_{i - 1}, z_{j}) .

(31)

The remainder term $R_{n} (\bar{H} (\cdot))$ can be expressed as

R_{n} (\bar{H} (\cdot)) = - \frac{ϑ^{3}}{12} \bar{H} ″ (ξ, z_{j}) .

(32)

Proper selection of H can effectively reduce its influence on the accuracy of gradient calculation.¹⁶ On the basis of this, the impact of fast gradient calculation on optimization stability is also Insignificant.

Improved three-term spectrum conjugate gradient algorithm and its global convergence

In some famous CGAs, the choice of conjugate parameters has different influences on the optimal control problem, such as Polak-Ribiere-Polyak (PRP),⁵⁴ Fletcher-Reeves (FR) method,⁵⁵ and Dai-Yuan (DY).⁵⁶ According to Andrei,⁵⁷ the $β_{k}^{FR}$ and $β_{k}^{DY}$ have strong convergence properties, but they may have modest practical performance due to jamming. The $β_{k}^{PRP}$ may often have better computational performance, but they not generally be convergent. Therefore, many hybrid CGAs and improved CGAs have been proposed. Significantly, Liu and Li⁵⁸ proposed a spectral CGA to solve unconstrained optimization problems; Yin et al.⁵⁹ proposed a hybrid three-term CGA by incorporating the adaptive line search based on hybrid technique; Li et al.⁶⁰ proposed two three-term spectral CGAs using Quasi-Newton equations to solve unconstrained optimization problems; Waziri et al.⁶¹ proposed a double direction three-term spectral conjugate gradient method.

Therefore, CGA is a powerful technique for solving linear optimization problems, it is employed to deal with the OCP (17). However, the cost function in this paper contains control variable boundary constraints, which cannot be solved by restricted CGA directly. Therefore, this paper applies a trigonometric conversion function method to transform the original OCP into an unconstrained optimization problem. The temporary variable $ω (t, z)$ is defined and the transformed function is given by the following:

\bar{h} = 0.5 \times (h_{\max} (t, z_{j}) - h_{\min} (t, z_{j})) \times [\cos (ω (t, z)) + 1] + h_{\min} (t, z_{j}),

(33)

\begin{matrix} where \bar{h} = [\begin{matrix} h (t, z_{1}) \\ h (t, z_{2}) \\ \begin{matrix} h (t, z_{3}) \\ ⋮ \\ h (t, z_{M}) \end{matrix} \end{matrix}], ω = [\begin{matrix} ω (t, z_{1}) \\ ω (t, z_{2}) \\ \begin{matrix} ω (t, z_{3}) \\ ⋮ \\ ω (t, z_{M}) \end{matrix} \end{matrix}] . \end{matrix}

Because of $- 1 \leq \cos (ω (t, z)) \leq 1$ , the bounded variable $h (t, z)$ can substitute the unbounded variable $ω (t, z)$ , we have

\begin{matrix} min_{ω} J (ω) = \int_{0}^{t_{f}} \int_{0}^{L} {‖ T_{c} (l, t, z; ω) - T_{d} (l, t, z) ‖}^{2} d t d z \\ + α \int_{0}^{t_{f}} \int_{0}^{L} ϕ (ω) d t d z . \end{matrix}

(34)

Improved three-term spectrum conjugate gradient algorithm

In this section, we present an ITSCGA based on DY algorithm. An appropriate spectrum $μ_{k}$ is found to guarantee the descent direction and global convergence by an improved Quasi-Newton transversal equation.^60,62,63 Starting from an initial guess $ω_{o} \in ℜ^{3}$ , this method generates an approximation sequence ${ω_{k}}$ by the following iteration process:

ω_{k + 1} : = ω_{k} + α_{k} d_{k},

(35)

where $k$ is iterative number; $ω_{k}$ is the current iteration point, $α_{k}$ is the step-size in some line search and the search direction $d_{k}$ is expressed as:

d_{k} = {\begin{matrix} \begin{matrix} - \nabla J_{k}, \\ - μ_{k} \nabla J_{k} + β_{k} d_{k - 1} + θ_{k} y_{k}, \end{matrix} & \begin{matrix} k = 0 \\ k > 0 \end{matrix} \end{matrix},

(36)

where $\nabla J_{k}$ is gradient of cost function; $y_{k} = \nabla J_{k} - \nabla J_{k - 1}$ . The conjugate parameter $β_{k}$ and new parameter $θ_{k}$ can be determined as follows:

β_{k} = β_{k}^{DY} = \frac{{‖ \nabla J_{k} ‖}^{2}}{y_{k}^{T} d_{k - 1}},

(37)

θ_{k} = \frac{c \nabla J_{k}^{T} y_{k}}{\nabla J_{k}^{T} d_{k - 1}},

(38)

where $c$ is a constant, which satisfies $c > 0$ .

According to Li et al.,⁶⁰ we can obtain

d_{k} \approx - B_{k}^{- 1} \nabla J_{k},

(39)

where $B_{k}$ is the approximation of the two-order Hessian matrix $\nabla^{2} J (ω_{k})$ . Because of the equation (36), the equation (39) can be written as

- B_{k}^{- 1} \nabla J \approx - μ_{k} \nabla J_{k} + β_{k} d_{k - 1} + θ_{k} y_{k} .

(40)

Letting $s_{k} = ω_{k + 1} - ω_{k}$ . Multiplying the equation (40) by $s_{k}^{T} B_{k}$ , we have

- s_{k}^{T} \nabla J_{k} \approx - μ_{k} s_{k}^{T} B_{k} \nabla J_{k} + β_{k} s_{k}^{T} B_{k} d_{k - 1} + θ_{k} s_{k}^{T} B_{k} y_{k} .

(41)

Making use of the Quasi-Newton equation $B_{k} s_{k} = y_{k}$ , the above expression can be rewritten as:

- s_{k}^{T} \nabla J_{k} \approx - μ_{k} y_{k}^{T} \nabla J_{k} + β_{k} y_{k}^{T} d_{k - 1} + θ_{k} ‖ y_{k} ‖^{2} .

(42)

Therefore, we can obtain spectral parameter

\begin{matrix} μ_{k} \approx \frac{y_{k}^{T} d_{k - 1}}{{‖ \nabla J_{k - 1} ‖}^{2}} + \frac{(s_{k}^{T} \nabla J_{k} {‖ \nabla J_{k - 1} ‖}^{2} + θ_{k} {‖ y_{k} ‖}^{2} {‖ \nabla J_{k - 1} ‖}^{2} + {‖ \nabla J_{k - 1} ‖}^{2} {‖ \nabla J_{k} ‖}^{2} - {‖ y_{k} ‖}^{2} \nabla J_{k}^{T} d_{k - 1})}{y_{k}^{T} \nabla J_{k} {‖ \nabla J_{k - 1} ‖}^{2}} . \end{matrix}

(43)

However, the optimization of OCP (17) by the ITSCGA is divergent. Therefore, we define $μ_{k}$ as

μ_{k} = \frac{y_{k}^{T} d_{k - 1}}{{‖ \nabla J_{k - 1} ‖}^{2}} + b .

(44)

A suitable value $b$ is needed to find to accelerate the convergence of the ITSCGA. By trial and error, we obtain $0 \leq b \leq 0.8$ , and $μ_{k} \in [τ_{1}, τ_{2}], τ_{1} \geq 1, τ_{2} \leq 20$ .

The solving process of the ITSCGA for OCP (17) are as follows:

Step 1: Set $k = 0$ , select $ω_{o} \in R$ , termination criteria $ε_{s}$ , maximum iteration steps $k_{\max}$ ;

Step 2: Use the LCMM to calculate the temperature $T$ of the billet;

Step 3: Use equations (27) and (36) to calculate the gradient $\nabla J (ω_{k})$ and the search direction $d_{k}$ respectively;

Step 4: Calculate spectrum $μ_{k}$ according to equation (44), if $μ_{k} < τ_{1}$ or $μ_{k} > τ_{2}$ , then $μ_{k} = 1$ ;

Step 5: Update the variable $ω_{k}$ according to equation (35);

Step 6: Use equation (34) to calculate the cost function $J (ω_{k})$ , if $J (ω_{k}) \leq ε_{s}$ , or the number of iterations reaches $k_{\max}$ ; then stop; else go to step 7;

Step 7: Set $k = k + 1$ , go to step 2.

The above steps are applicable to the optimization process when the convective term $\partial T / \partial z = 0$ . The ITSCGA based optimizer is difficult to apply to the case of $\partial T / \partial z \neq 0$ . Therefore, when the convective term $\partial T / \partial z \neq 0$ , dynamic optimization process is introduced. The whole dynamic process is divided into several static times, and heat flow at each time is obtained by the ITSCGA, which is taken as the initial heat flow at the next time. Therefore, dynamic optimization is obtained through iterative calculation.

The global convergence of improved three-term spectrum conjugate gradient algorithm

In this section, global convergence of the ITSCGA under the Wolfe search criteria¹⁶:

J (ω_{k} + α_{k} d_{k}) \leq J (ω_{k}) + δ α_{k} \nabla J_{k - 1}^{T} d_{k},

(45)

| \nabla J_{k}^{T} d_{k} | \leq - σ \nabla J_{k - 1}^{T} d_{k},

(46)

where $0 \leq δ \leq σ \leq 1$ .

Theorem 4.1. If sequence $ω_{k}$ and $d_{k}$ are generated by equations (35) and (36), we have the following equation

\nabla J_{k}^{T} d_{k} \leq - ‖ \nabla J_{k} ‖^{2} .

(47)

Proof. Induction can be used to demonstrate the conclusion. Since $\nabla J_{0}^{T} d_{0} = - \nabla {J_{0}}^{2}$ , the equation (47) holds for $k = 0$ . For $k \geq 1$ , we suppose that Theorem 4.1 is true for the last iteration $k - 1$ , and we can obtain $\nabla J_{k - 1}^{T} d_{k - 1} < 0$ . Multiplying the equation (36) by $\nabla J_{k}^{T}$ and we have

\begin{matrix} \nabla J_{k}^{T} d_{k} = \nabla J_{k}^{T} (- μ_{k} \nabla J_{k} + β_{k} d_{k - 1} + θ_{k} y_{k}) \\ = - μ_{k} \nabla J_{k}^{T} \nabla J_{k} + β_{k} \nabla J_{k}^{T} d_{k - 1} + θ_{k} \nabla J_{k}^{T} y_{k} \\ = - μ_{k} ‖ \nabla J_{k} ‖^{2} + \frac{\nabla J_{k}^{T} d_{k - 1}}{y_{k}^{T} d_{k - 1}} ‖ \nabla J_{k} ‖^{2} + c \frac{\nabla J_{k}^{T} y_{k}}{\nabla J_{k}^{T} d_{k - 1}} \nabla J_{k}^{T} y_{k} . \end{matrix}

(48)

Case 1: If $μ_{k} > 1,$ because of $0 \leq b \leq 0.8$ , we have

y_{k}^{T} d_{k - 1} > 0,

(49)

and

\nabla J_{k}^{T} d_{k - 1} > \nabla J_{k - 1}^{T} d_{k - 1} .

(50)

According to equations (48) and (50), we have

\begin{matrix} \nabla J_{k}^{T} d_{k} = - μ_{k} ‖ \nabla J_{k} ‖^{2} + \frac{\nabla J_{k}^{T} d_{k - 1}}{y_{k}^{T} d_{k - 1}} ‖ \nabla J_{k} ‖^{2} + c \frac{\nabla J_{k}^{T} y_{k}}{\nabla J_{k}^{T} d_{k - 1}} \nabla J_{k}^{T} y_{k} \\ \leq - μ_{k} ‖ \nabla J_{k} ‖^{2} + \frac{\nabla J_{k}^{T} d_{k - 1}}{\nabla J_{k}^{T} d_{k - 1} - \nabla J_{k - 1}^{T} d_{k - 1}} ‖ \nabla J_{k} ‖^{2} + c \frac{{(\nabla J_{k}^{T} y_{k})}^{2}}{\nabla J_{k - 1}^{T} d_{k - 1}} \\ \leq (1 - μ_{k}) ‖ \nabla J_{k} ‖^{2} + c \frac{{‖ y_{k} ‖}^{2}}{\nabla J_{k - 1}^{T} d_{k - 1}} ‖ \nabla J_{k} ‖^{2} . \end{matrix}

(51)

Due to $1 - μ_{k} < 0$ , $\nabla J_{k - 1}^{T} d_{k - 1} < 0$ , the sufficient descent property holds for $c > 0$ in this case.

Case 2: If $0.8 < μ_{k} < 1$ , we have $μ_{k} = 1$ . We also obtain the equation (50), so we have

\begin{matrix} \nabla J_{k}^{T} d_{k} = - ‖ \nabla J_{k} ‖^{2} + \frac{\nabla J_{k}^{T} d_{k - 1}}{y_{k}^{T} d_{k - 1}} ‖ \nabla J_{k} ‖^{2} + c \frac{\nabla J_{k}^{T} y_{k}}{\nabla J_{k}^{T} d_{k - 1}} \nabla J_{k}^{T} y_{k} \\ \leq c \frac{{‖ y_{k} ‖}^{2}}{\nabla J_{k - 1}^{T} d_{k - 1}} ‖ \nabla J_{k} ‖^{2} . \end{matrix}

(52)

Due to $\nabla J_{k - 1}^{T} d_{k - 1} < 0$ and $c > 0$ in this case.

Case 3: If $μ_{k} < 0.8$ , we have $μ_{k} = 1$ . We have

y_{k}^{T} d_{k - 1} < 0,

(53)

and

\nabla J_{k}^{T} d_{k - 1} < \nabla J_{k - 1}^{T} d_{k - 1},

(54)

so we can obtain the following inequality:

\begin{matrix} \nabla J_{k}^{T} d_{k} = - ‖ \nabla J_{k} ‖^{2} + \frac{\nabla J_{k}^{T} d_{k - 1}}{y_{k}^{T} d_{k - 1}} ‖ \nabla J_{k} ‖^{2} + c \frac{\nabla J_{k}^{T} y_{k}}{\nabla J_{k}^{T} d_{k - 1}} \nabla J_{k}^{T} y_{k} \\ \leq c \frac{{‖ y_{k} ‖}^{2}}{\nabla J_{k}^{T} d_{k - 1}} ‖ \nabla J_{k} ‖^{2} . \end{matrix}

(55)

Due to $\nabla J_{k}^{T} d_{k - 1} < 0$ , and the sufficient descent property holds for $c > 0$ in this case.

According to equations (51), (52) and (55), we can obtain $\nabla J_{k}^{T} d_{k} \leq - \nabla {J_{k}}^{2}$ .

Theorem 4.1 is proved.

Remark: In our experiments, we conduct extensive testing and debugging on parameter $c$ and find that the optimal value range for parameter $c$ can be chosen as $1 \leq c \leq 7$ . The parameter in this range makes the algorithm have better performance.

Theorem 4.2. Based on the strong Wolfe line search,¹⁶ we have the following result,

lim_{k \to \infty} \inf ‖ \nabla J_{k} ‖ = 0 .

(56)

Proof. We suppose that equation (56) is not true, so there exist constants $γ_{1}, γ_{2}$ , which satisfy $γ_{1} < \nabla {J_{k}}^{2} < γ_{2}$ . According to equation (36), we have

{(d_{k} + μ_{k} \nabla J_{k})}^{2} = {(β_{k} d_{k - 1} + θ_{k} y_{k})}^{2} .

(57)

So we have

\begin{matrix} ‖ d_{k} ‖^{2} = - {(μ_{k} \nabla J_{k})}^{2} - 2 μ_{k} \nabla J_{k}^{T} d_{k} + {(β_{k} d_{k - 1} + θ_{k} y_{k})}^{2} \\ \leq - {(μ_{k} \nabla J_{k})}^{2} - 2 μ_{k} \nabla J_{k}^{T} d_{k} + {(β_{k} d_{k - 1})}^{2} + {(θ_{k} y_{k})}^{2} \\ \leq {β_{k}}^{2} ‖ d_{k - 1} ‖^{2} + {(μ_{k} - 1)}^{2} ‖ \nabla J_{k} ‖^{2} + {θ_{k}}^{2} ‖ y_{k} ‖^{2}, \end{matrix}

(58)

where $μ_{k} \in [τ_{1}, τ_{2}]$ . Multiplying the inequality (58) by $\frac{1}{{(\nabla J_{k} d_{k})}^{2}}$ , we have

\begin{matrix} \frac{{‖ d_{k} ‖}^{2}}{{(\nabla J_{k}^{T} d_{k})}^{2}} \leq \frac{{‖ \nabla J_{k} ‖}^{4}}{{(y_{k}^{T} d_{k})}^{2}} \frac{{‖ d_{k - 1} ‖}^{2}}{{(\nabla J_{k}^{T} d_{k})}^{2}} + {(μ_{k} - 1)}^{2} \frac{{‖ \nabla J_{k} ‖}^{2}}{{(\nabla J_{k}^{T} d_{k})}^{2}} + \frac{{θ_{k}}^{2} {‖ y_{k} ‖}^{2}}{{(\nabla J_{k}^{T} d_{k})}^{2}} \\ \leq \frac{{γ_{2}}^{2}}{{γ_{1}}^{2}} \frac{{‖ d_{k - 1} ‖}^{2}}{{(\nabla J_{k}^{T} d_{k})}^{2}} + \frac{1}{{‖ \nabla J_{k} ‖}^{2}} + \frac{1}{{‖ \nabla J_{k} ‖}^{2} - {‖ \nabla J_{k - 1} ‖}^{2}} . \end{matrix}

(59)

Using $\frac{{d_{1}}^{2}}{\nabla {J_{1}}^{T} d_{1}} = \frac{1}{\nabla {J_{1}}^{2}}$ , we have

\begin{matrix} \frac{{γ_{2}}^{2}}{{γ_{1}}^{2}} \frac{{‖ d_{k - 1} ‖}^{2}}{{(\nabla J_{k}^{T} d_{k})}^{2}} + \frac{1}{{‖ \nabla J_{k} ‖}^{2}} + \frac{1}{{‖ \nabla J_{k} ‖}^{2} - {‖ \nabla J_{k - 1} ‖}^{2}} \\ \leq \frac{{γ_{2}}^{2}}{{γ_{1}}^{2}} \sum_{k = 1}^{\infty} \frac{1}{{‖ \nabla J_{k} ‖}^{2}} + \frac{1}{{‖ \nabla J_{k} ‖}^{2}} + \frac{1}{{‖ \nabla J_{k} ‖}^{2} - {‖ \nabla J_{k - 1} ‖}^{2}} \\ \leq \frac{{γ_{2}}^{2}}{{γ_{1}}^{2}} \sum_{k = 1}^{\infty} \frac{1}{{‖ \nabla J_{k} ‖}^{2}} \leq \frac{{γ_{2}}^{2}}{{γ_{1}}^{3}} k . \end{matrix}

(60)

Furthermore, we can obtain

lim_{m \to \infty} \sum_{k = 1}^{m} \frac{{(\nabla J_{k}^{T} d_{k})}^{2}}{{‖ d_{k} ‖}^{2}} \geq \frac{{γ_{2}}^{2}}{{γ_{1}}^{3}} lim_{m \to \infty} \sum_{k = 1}^{n} \frac{1}{k} = \infty .

(61)

Based on Wolfe line search criteria and Lipschitz continuous,¹⁹ we have

‖ Δ J (x) - Δ J (y) ‖ \leq L_{τ} ‖ x - y ‖,

(62)

and

(σ - 1) \nabla J_{k}^{T} d_{k} \leq d_{k}^{T} (\nabla J_{k} - \nabla J_{k - 1}) \leq α_{k} L_{τ} ‖ d_{k} ‖^{2} .

(63)

Where $L_{τ}$ is the Lipschitz constant. According to equations (35) and (45), we can obtain the following inequality

J (ω_{k}) - J (ω_{k} + α_{k} d_{k}) \geq δ α_{k} \nabla J_{k - 1}^{T} d_{k} \geq C \frac{{(\nabla J_{k}^{T} d_{k})}^{2}}{{‖ d_{k} ‖}^{2}},

(64)

where $C = - δ \frac{σ - 1}{L}$ .

Hence we can obtain the following equation

\begin{matrix} \sum_{k = 1}^{\infty} \frac{{(\nabla J_{k}^{T} d_{k})}^{2}}{{‖ d_{k} ‖}^{2}} \leq \sum_{k = 1}^{\infty} [J (ω_{k}) - J (ω_{k} + α_{k} d_{k})] \\ \leq \sum_{k = 1}^{\infty} [J (ω_{k}) - J (ω_{k + 1})] . \end{matrix}

(65)

So we have

lim_{m \to \infty} \sum_{k = 1}^{m} \frac{{(\nabla J_{k}^{T} d_{k})}^{2}}{{‖ d_{k} ‖}^{2}} < \infty .

(66)

Obviously, equation (61) is contradict to equation (66), so we can obtain $lim_{k \to \infty} \inf \nabla J_{k} = 0$ .

Theorem 4.2 is proved.

Experiment simulation

We carry out the static and dynamic simulation experiments in this section. In the static situation, the simulation experiments are divided into three situations. In case 1, the parameters $c$ and $b$ are analyzed in the ITSCGA; In case 2, the comparison of the ITSCGA with other algorithms is given; In case 3, the control effects are tested and verified. In the dynamic situation, two continuous casting experiments with variable casting speeds are tested, and the optimized control method based on ITSCGA is compared with other algorithms.

Static experiment simulation

This section considers the convective term $\partial T / \partial z = 0$ in formula (17), and the values of process parameters are given in Table 1. Table 2 shows specifications of the continuous caster. In addition, eight control points are select. The distance (m) between each control point and the meniscus are 0.1, 1.0, 1.5, 2.0, 2.6, 3.3, 4.0, and 4.5, respectively. We set the target temperature ( $° C$ ) values as 955, 644, 599, 567, 555, 547, 550, and 553, and select the maximum number of iterations $k_{\max} = 100$ .

Table 1.

Process parameters of steel billets in the static experiment.

Parameters	Values
Final time (s)	100
Liquid temperature ( $° C$ )	1514
Water temperature ( $° C$ )	30
The width of steel billet (m)	0.5
Latent heat (J/kg)	268,000
Density of steel $(kg / m^{3})$	$ρ_{s}$ = 7800, $ρ_{l}$ = 7200
Specific heat of steel $(J / (kg \cdot ° C))$	$C_{s}$ = 660, $C_{l}$ = 830
Thermal conductivity ( $W / m \cdot K$ )	$λ_{s} = 31, λ_{l} = 35$
Maximum iterative number	100

Table 2.

Continuous caster physical parameters.

Parameters	Values of parameter
Number of sections in the SCZ	8
Effective mould length (m)	0.9
Length of each section in the SCZ (m)	0.74, 1.93, 0.77, 1.52, 3.84, 3.84, 6.71, 9.69
Metallurgical length (m)	29.84

Case 1: In this case, we specifically analyze the values of parameters $b$ and $c$ in equations (38) and (44), where $0 \leq b \leq 0.8, 1 \leq c \leq 7$ . We take different values of the parameters $b$ and $c$ within this limit, and compare the cost function in the simulation experiments. We obtain the values of parameter $b$ and $c$ in Table 3. The value of parameter $h_{\min} (t, z_{j})$ and $h_{\max} (t, z_{j})$ are 20 and 2000 respectively. According to the equation (1), we can calculate cost function $J (h)$ , and Figure 3 gives the results of the ITSCGA for different parameters in Table 3. It can be seen from Figure 3 that the convergence speed of the ITSCGA gives a better result when the parameters are chosen as $c = 7$ $and$ $b = 0.0$ .

Table 3.

The values of parameters $b$ and $c$ .

Parameters	Values of parameter
$c$	1	2	3	4	5	6	7
$b$	0.0	0.8	0.7	0.6	0.5	0.2	0.0

Figure 3.

The convergence of the cost function with the number of iterations.

Case 2: In this case, we take the simulation for comparison, in which the ITSCGA is compared with the CGA,⁵⁵ the modified conjugate gradient algorithm (MCGA),⁵⁰ the new conjugate gradient algorithms (NCGA-JCJ− and NCGA-JCJ+),⁶⁰ and the hybrid conjugate gradient algorithm (HCGA).⁵⁸ The convergence of the cost function and the number of iterations are shown in Figure 4. In addition, the termination criteria are selected $ε_{s} = 1.0$ . It can be seen from Figure 4 that ITSCGA significantly accelerates the convergence speed. Table 4 provides the number of iterations and runtime of the six algorithms. By comparing the results in Table 4, it can be seen that ITSCGA only iterates 39 times and reaches the termination criterion. Compared with other methods, ITSCGA has fewer iterations and faster convergence speed.

Figure 4.

Comparison of computational performance of algorithms.

Table 4.

The comparison of algorithms.

Algorithm	CGA⁵⁵	HCGA⁵⁸	MCGA⁵⁰	NCGA-JCJ−⁶⁰	NCGA-JCJ+⁶⁰	ITSCGA
Iterative number	48	334	44	338	184	42
Running time (s)	84.28	2329.33	77.47	1714.07	392.26	75.43

Case 3: The control effects with six algorithms, including CGA,⁵⁵ MCGA,⁵⁰ NCG-JCJ+,⁶⁰ NCG-JCJ−,⁶⁰ HCGA,⁵⁸ and ITSCGA, are compared. We set the target temperatures as 955°C, 644°C, 599°C, 567°C, 555°C, 547°C, 550°C, and 553 $° C$ . Figure 5(a) and (b) show the control effects of the six methods. According to Figure 5(a) and (b), we can see that ITSCGA makes the temperature reach the target value within 38 s, which is smaller than other algorithms. So the control effect of ITSCGA is better than that of other five methods. Meanwhile, to compare the convergence effect more clearly, we calculate the control error for the six methods. The error curve is shown in Figure 6(a) and (b), and the averaged errors are given in Table 5 ( $T_{a}$ is the averaged error temperature between the control and target temperature). According to the calculation results, the error of the optimizer based on ITSCGA is smaller than that of other methods. We can see from Figures 5(a) and (b), 6(a) and (b), the optimized control based on ITSCGA gives a more stable control temperature.

Figure 5.

(a) Control effects comparison of control points 1 to 4 in static experimental simulation. (b) Control effects comparison of control points 5 to 8 in static experimental simulation.

Figure 6.

(a) Comparison of error variation curves of control points 1 to 4 in static experimental simulation. (b) Comparison of error variation curves of control points 5–8 in static experimental simulation.

Table 5.

Comparison of the averaged error of the six methods.

Control point	Algorithms
	CGA⁵⁵	MCGA⁵⁰	NCG-JCJ+⁶⁰	NCG-JCJ−⁶⁰	HCGA⁵⁸	ITSCGA
1	$T_{a} = 3.7$	$T_{a} = 2.6$	$T_{a} = 2.0$	$T_{a} = 12.1$	$T_{a} = 11.7$	$T_{a} = 2.3$
2	$T_{a} = 1.9$	$T_{a} = 1.0$	$T_{a} = 1.1$	$T_{a} = 4.0$	$T_{a} = 3.9$	$T_{a} = 1.2$
3	$T_{a} = 2.1$	$T_{a} = 1.2$	$T_{a} = 1.8$	$T_{a} = 3.6$	$T_{a} = 3.6$	$T_{a} = 1.3$
4	$T_{a} = 2.8$	$T_{a} = 1.9$	$T_{a} = 3.3$	$T_{a} = 5.5$	$T_{a} = 5.5$	$T_{a} = 1.8$
5	$T_{a} = 4.4$	$T_{a} = 3.0$	$T_{a} = 4.2$	$T_{a} = 13.0$	$T_{a} = 12.9$	$T_{a} = 2.7$
6	$T_{a} = 2.2$	$T_{a} = 1.5$	$T_{a} = 2.0$	$T_{a} = 5.2$	$T_{a} = 5.1$	$T_{a} = 1.5$
7	$T_{a} = 1.8$	$T_{a} = 1.5$	$T_{a} = 0.9$	$T_{a} = 4.1$	$T_{a} = 4.0$	$T_{a} = 1.9$
8	$T_{a} = 2.3$	$T_{a} = 1.4$	$T_{a} = 1.2$	$T_{a} = 6.4$	$T_{a} = 6.3$	$T_{a} = 1.3$

Unsteady experiment simulation

In this section, we carry out unsteady experimental simulation on the OCP. We choose $c = 1, b = 0.8$ , and $N = 60$ in formula (9) and $M = 40$ in formula (10). In this situation, the convective term $\partial T / \partial z \neq 0$ is considered in formula (17). $T_{c}$ is obtained by calculating the average value of the temperature of each cooling section. The initial water flow rates of sections 1–8 are set as 357, 331, 406, 286, 328, 290, 236, and 89 L/min, respectively. The target temperatures are 1055°C, 1035°C, 1027°C, 1017°C, 995°C, 975°C, 965°C, and 945 $° C$ . The specifications of the caster and the values of technical parameters are given in Tables 2 and 6 respectively.

Table 6.

Process parameters of steel billets in the dynamic experiment.

Parameters	Values
Ambient temperature ( $° C$ )	30
Spray water temperature ( $° C$ )	25
Final time (s)	100
Density of steel $(kg / m^{3})$	$ρ_{s}$ = 7800, $ρ_{l}$ = 7200
Thermal conductivity ( $W / m \cdot K$ )	$λ_{s} = 31, λ_{l} = 35$
Specific heat of steel $(J / (kg \cdot ° C))$	$C_{s}$ = 660, $C_{l}$ = 830
Running time (s)	100
The width of steel billet (m)	0.4
Liquid temperature ( $° C$ )	1514

Case 1: The casting speed is changed from 0.8 to 1.1 m/min. We compare the optimization performance of four algorithms, including the ITSCGA, CGA,⁵⁵ MCGA,⁵⁰ and HCGA.⁵⁸ Figure 7 shows the change curve of the cost function $J$ based on four optimizers. Obviously, the convergence speed of the cost function of ITSCGA is faster than that of other methods. Then, we give the temperature curves of eight cooling zones, as shown in Figure 8(a) and (b), and the errors between the control temperature and target temperature are given in Figure 9(a) and (b). In comparison to other algorithms, the ITSCGA gives smaller temperature fluctuations, so its control effect is better than that of the other three methods. Meanwhile, to compare the convergence effect more clearly, we calculate the average error and maximum error for the four methods, which are given in Tables 7 and 8. From these two tables, we can see that the average and maximum errors of ITSCGA are smaller than other methods.

Figure 7.

Comparison of convergence of dynamic processes.

Figure 8.

(a) Control effect of cooling sections 1 to 4 in case 1 of dynamic experiment. (b) Control effect of cooling sections 5–8 in case 1 of dynamic experiment.

Figure 9.

(a) Errors comparison of cooling sections 1 to 4 in case 1 of dynamic experiment. (b) Errors comparison of cooling sections 5–8 in case 1 of dynamic experiment.

Table 7.

The average values of error in case 1 of dynamic experiment.

Cooling section	Algorithms
Cooling section	ITSCGA	MCGA⁵⁰	CGA⁵⁵	HCGA⁵⁸
1	$T_{a} = 0.034$	$T_{a} = 2.58$	$T_{a} = 3.19$	$T_{a} = 4.06$
2	$T_{a} = 0.33$	$T_{a} = 1.97$	$T_{a} = 1.86$	$T_{a} = 3.20$
3	$T_{a} = 0.38$	$T_{a} = 2.09$	$T_{a} = 2.37$	$T_{a} = 2.21$
4	$T_{a} = 0.46$	$T_{a} = 1.84$	$T_{a} = 2.16$	$T_{a} = 2.50$
5	$T_{a} = 0.30$	$T_{a} = 1.82$	$T_{a} = 2.40$	$T_{a} = 3.45$
6	$T_{a} = 0.38$	$T_{a} = 2.94$	$T_{a} = 3.64$	$T_{a} = 3.93$
7	$T_{a} = 0.37$	$T_{a} = 3.31$	$T_{a} = 3.68$	$T_{a} = 4.96$
8	$T_{a} = 0.38$	$T_{a} = 3.97$	$T_{a} = 3.44$	$T_{a} = 4.36$

Table 8.

The maximum values of error in case 1 of dynamic experiment.

Cooling section	Algorithms
Cooling section	CGA⁵⁵	HCGA⁵⁸	MCGA⁵⁰	ITSCGA
1	$T_{\max} = 8.15$	$T_{\max} = 10.63$	$T_{\max} = 3.90$	$T_{\max} = 0.20$
2	$T_{\max} = 5.46$	$T_{\max} = 6.53$	$T_{\max} = 5.56$	$T_{\max} = 1.92$
3	$T_{\max} = 5.86$	$T_{\max} = 3.75$	$T_{\max} = 4.50$	$T_{\max} = 3.62$
4	$T_{\max} = 3.99$	$T_{\max} = 5.37$	$T_{\max} = 3.55$	$T_{\max} = 3.53$
5	$T_{\max} = 6.01$	$T_{\max} = 5.60$	$T_{\max} = 3.62$	$T_{\max} = 0.87$
6	$T_{\max} = 5.94$	$T_{\max} = 4.50$	$T_{\max} = 2.60$	$T_{\max} = 5.66$
7	$T_{\max} = 5.50$	$T_{\max} = 5.16$	$T_{\max} = 2.51$	$T_{\max} = 0.83$
8	$T_{\max} = 3.27$	$T_{\max} = 3.11$	$T_{\max} = 2.63$	$T_{\max} = 0.34$

Case 2: The casting speed is changed from 1.0 to 1.6 m/min. In this case, we consider the impact of significant changes in casting speed on control performance of the four algorithms. The control effect of the simulation experiment is shown in Figure 10(a) and (b). It can be seen from Figure 10(a) and (b) that ITSCGA has faster convergence speed and more stable temperature fluctuation than other methods. Figure 11 gives the average error for the four algorithms. From Figure 11, the average error of the ITSCGA is smaller, and the average error values of all eight cooling zones are less than 4 $° C$ .

Figure 10.

(a) Control effect of cooling sections 1 to 4 in case 2 of dynamic experiment. (b) Control effect of cooling sections 5 to 8 in case 2 of dynamic experiment.

Figure 11.

Comparison of average error values in case 2 of dynamic experiment.

Obviously, especially in the case of variable casting speed, the dynamic optimization based on ITSCGA gives smaller temperature fluctuations and obtains a better control effect compared with other methods.

Conclusion

This paper investigates the OCP based on the UHTMCT, and presents a parameterization optimal method that combines gradient analysis based on a Hamiltonian function costate system and an ITSCGA. Firstly, we analyze the expression of the gradient calculation method based on the Hamiltonian function costate system to explore the rapidity of the gradient. Secondly, an ITSCGA is proposed to solve the OCP of the UHTMCT, and the global convergence of the ITSCGA is proved. Lastly, both static and dynamic continuous casting simulation comparisons are tested the optimization performance of ITSCGA, and the results show that the ITSCGA is better than other methods.

Specifically, in the static experiment, the ITSCGA has fewer iterative numbers and a faster convergence rate. Meanwhile, the optimizer based on the ITSCGA provides more stable temperature control, and the average error of the eight cooling zones is smaller than other methods.

In the dynamic experiment, we consider the simulation of continuous casting experiments under two different cases of variable casting speed to verify the stability of temperature control. When casting speed changes from 0.8 to 1.1 m/min, the ITSCGA has a better control effect, and its average error is less than 0.5 $° C$ , which is smaller than that of other methods. When casting speed changes from 1.0 to 1.6 m/min, the average error of the ITSCGA is less than 4 $° C$ . Therefore, we conclude that the optimizer based on the ITSCGA can converge faster and have more stable temperature fluctuations than other methods, which can help to improve the quality of the slab.

In this paper, the optimization of secondary cooling water control under the change of casting speed is studied. In future research, optimization of secondary cooling water flow rate based on the multi-physical coupling, as well as modeling and optimization of crack index model for a slab will be considered.

Footnotes

Appendix

Notations

$T$	Temperature of steel billet (K)
$z$	Length direction of steel billet
$t$	Time
$x$	Width direction of steel billet
$L$	Length of steel billet (m)
$t_{f}$	Final time (s)
$l$	Width of steel billet (m)
$h$	Heat flux $(W / m^{2})$
$J$	Cost function
$T_{d}$	Target temperature of steel billets (K)
$α$	A positive number
∥·∥	Norm
$ϕ (\cdot)$	Function ${∥ \cdot ∥}^{2}$
$T_{c}$	Calculated temperature of steel billets (K)
$V_{cast}$	Casting speed $(m / \min)$
$T_{0}$	Initial temperature of steel billets (K)
$k_{0}$	Thermal diffusivity coefficient
$Q$	Number of slices
$q$	The indexes of space nodes
$Δ t$	Time step
$Δ x$	Spatial step size
$n$	Time interval
$r$	$k_{0} Δ t / Δ x^{2}$
$T_{w}$	Cold water temperature ( $° C$ )
$i$	Time discrete position
$j$	Spatial discrete position
$N$	Time discrete quantity
$M$	Spatial discrete quantity
$χ$	Piecewise constant
$h_{\min}$	Minimum heat flow
$h_{\max}$	Maximum heat flow
$\bar{H}$	Hamiltonian function
$λ$	Costate vector
$v$	Arbitrary continuous function
$R_{n}$	Remainder term
$ω$	Temporary variable
$k$	Iterative number
$β_{k}$	Conjugate parameters at iteration number $k$
$\nabla J_{k}$	Gradient at iteration number $k$
$y_{k}$	Defined as $\nabla J_{k} - \nabla J_{k - 1}$
$d_{k}$	Search direction at iteration number $k$
$s_{k}$	Defined as $d_{k} - d_{k - 1}$
$μ_{k}$	Spectrum at iteration number $k$
$θ_{k}$	A parameter at iteration number $k$
$B$	The approximation of the two-order Hessian matrix
$c$	A constant and $1 \leq c \leq 7$
$b$	A constant and $0 \leq b \leq 0.8$
$τ_{1}$	Lower limit of the spectrum $μ_{k}$
$τ_{2}$	Upper limit of the spectrum $μ_{k}$
$α_{k}$	Step-length at iteration number $k$
$L_{τ}$	Lipschitz constant
$ε_{s}$	Termination criteria
$k_{\max}$	Maximum iteration steps
$λ_{l}$	Liquid thermal conductivity of steel ( $W / m \cdot K$ )
$ρ_{l}$	Liquid density of steel $(kg / m^{3})$
$C_{l}$	Liquid specific heat of steel $(J / (kg \cdot ° C))$
$λ_{s}$	Solid thermal conductivity of steel ( $W / m \cdot K$ )
$C_{s}$	Solid specific heat of steel $(J / (kg \cdot ° C))$
$ρ_{s}$	Solid density of steel $(kg / m^{3})$

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partly supported by the Natural Science Foundation of Liaoning Province of China (2021-BS-189), National Natural Science Foundation of China (61773269).

ORCID iDs

Yu Wang

Xinfu Pang

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