- Research
- Open Access

# Optimal control of multiphase steel production

- Dietmar Hömberg
^{1, 2}, - Klaus Krumbiegel
^{3}and - Nataliya Togobytska
^{4}Email authorView ORCID ID profile

**9**:6

https://doi.org/10.1186/s13362-019-0063-x

© The Author(s) 2019

**Received:**13 January 2019**Accepted:**21 June 2019**Published:**3 July 2019

## Abstract

An optimal control problem for the production of multiphase steel is investigated that takes into account phase transformations in the steel slab. The state equations are a semilinear heat equation coupled with an ordinary differential equation, that describes the evolution of the steel microstructure. The time-dependent heat transfer coefficient serves as a control function. Necessary and sufficient optimality conditions for the control problem are derived. For the numerical solution of the control problem, a reduced sequential quadratic programming method with a primal-dual active set strategy is developed. The numerical results are presented for the optimal control of a cooling line in the production of hot-rolled Mo–Mn dual phase steel.

## Keywords

- Hot rolling
- Dual phase steels
- Optimal control

## 1 Introduction

We consider an optimal control problem that describes the hot rolling process of multiphase steel, in particular dual phase (DP) steel. Dual phase steels have shown high potential for automotive applications due to their remarkable property combination with high strength and good formability. The microstructure of DP steel typically consists of a soft ferrite phase with dispersed islands of a hard martensite as the secondary phase [3]. The essential industrial process route for the production of DP steel consists of the hot rolling and subsequent controlled cooling on the run out table (ROT) which is located behind the finishing mill.

The controlled cooling of stages (2)–(4) happens on the run out table. Here, the most important control parameters are the flow-rate of water and the feed velocity of the strip. Since the process window for the adjustment of the phase composition is very tight, the computation of optimal process parameters is an important task. The goal of this paper is the analysis of a mathematical optimal control problem to compute the desired ferrite fraction and temperature at the end of step 3 of the process.

The controlled cooling of steel is a well-studied topic in engineering science and mathematics. There are a variety of methods used for the control approaches. An algorithm for the computation of optimal strategies for the cooling of steel strips in hot strip mills was proposed by Landl et al. [17]. The authors considered the problem of determination of suitable cooling strategy as a discrete optimization problem and demonstrated the numerical results for the real hot rolling mill. While they considered an integer optimization problem for switching on and off cooling sections, the goal of this study is to optimize the amount of coolant in a single cooling section. Lezius and Tröltzsch [18] considered a simplified numerical approach for the controlled cooling of steel profiles. A method of model predictive control for the temperature evolution of the strip has been proposed by Hashimoto, Yoshioka and Ohtsuka [10]. In Zheng and Li [26] a control strategy based on Kalman filter and model predictive control is discussed for the hot-rolled strip laminar cooling process. Wang et al. [25] discussed the method to calculate the convective heat transfer coefficient by combining a mathematical model with a back propagation neural network. While previous optimal control approaches for run out tables solely focus on the evolution of temperature, the main novelty of this paper is that we put a special emphasis on the microstructure, i.e., the composition of steel phases produced upon cooling. As mentioned earlier, from application point of view this is of high relevance, especially for the production of modern multiphase steels such as dual phase or trip steels.

We formulate an optimal control problem which consists in obtaining the cooling strategy such that the desired dual phase microstructure in steel is reached most accurately. This problem is a nonlinear boundary control problem, in which the state system consists of a semilinear heat equation coupled with an ordinary differential equation. The latter describes the evolution of the ferrite phase fraction. The heat transfer coefficient in the Newton type cooling boundary condition acts as the control parameter. In a previous paper [4], we have shown how to relate this coefficient to the flow-rate of coolant in a real cooling process. The scope of this paper is to analyze the resulting boundary coefficient control problem subject to a semilinear heat equation and rate law to describe the evolution of ferrite phase. Due to the nonlinearity in the coupling term on the right-hand side of the heat equation, the state system requires a detailed analysis, especially concerning the regularity of the solutions, which is of crucial importance for the derivation of second-order sufficient optimality conditions.

We investigate the existence of a solution and derive the first-order necessary and second-order sufficient optimality conditions, which form the basis for the convergence of the second-order optimization algorithms. Second-order optimality conditions for control problems governed by parabolic equations have been discussed, e.g., in Goldberg and Tröltzsch [7] and Raymond and Tröltzsch [20]. In comparison to the very general and abstract setting of the latter contribution, the main novelty of this paper is twofold, we consider a control in coefficient problem and we add an additional evolution equation to the state system to account for the evolution of steel microstructure.

To solve the control problem numerically, we use a reduced sequential quadratic programming (rSQP) method. This method has proven to be very effective in many areas of application, such as optimal control. A successful numerical application of the rSQP method to parabolic control problems has been reported by Hintermüller, Volkwein and Diwoky [12], Kupfer and Sachs [16].

In each iteration of rSQP method, the quadratic optimal control problem \((\mathit{QP}^{k})\) with control constraints has to be solved. To treat the \((\mathit{QP}^{k})\) problems, we apply a primal-dual active set strategy as, for instance, proposed by Bergonioux, Ito and Kunisch [2] for control constrained optimal control problems.

The paper is organized as follows: In Sect. 2, we analyze the optimal control problem and derive optimality conditions. In Sect. 3, we discuss the numerical optimization algorithms, i.e., the reduced SQP method with the active set strategy. The last section is devoted to numerical results.

## 2 The optimal control problem

### 2.1 Problem formulation and assumptions

*f*denotes the volume fraction of ferrite and

*θ*refers to the temperature. Typically, the function

*G*can be a nonlinear function in its arguments

*f*and

*θ*. For an example of concrete model for the austenite-ferrite phase transformation in the hot rolling process, we refer to [22]. The temperature distribution in the steel slab is described by the heat equation

*ρ*, the heat capacity \(c_{p}\), the heat conductivity

*κ*and the latent heat

*L*are assumed to be positive constants. The term \(\rho Lf_{t}\) describes the release of heat due to the phase transformation of ferrite. The boundary condition for the temperature imposed on the top and the bottom boundary of the domain

*Ω*is given as Newton’s law of cooling

*β*can describe, for instance, a profile of cooling medium distribution on the surface of the steel slab, see Fig. 2. The function

*u*can be expressed through a coolant flow-rate during the cooling and serves as the control variable in our problem.

*Ω*(see Fig. 2). The factors \(\alpha _{i}\), \(i=1,\ldots,3\), are positive constants. The third term in the cost functional represents a Tikhonov regularization term that can also be interpreted as a measure of the costs of the control. The control is bounded by two positive constants \(u_{a}\) and \(u_{b}\) since we consider only the cooling process and due to the restrictions on the maximal amount of coolant.

Further, we make some assumptions on the quantities of the optimal control problem that we need for the analysis.

### Assumptions

- (A1)
\(\varOmega \subset \mathbb{R}^{3}\) denotes a bounded domain with Lipschitz boundary

*∂Ω*. - (A2)The function \(G=G(\theta ,f)\) is twice continuously differentiable with respect to
*θ*and*f*. There is a constant \(M>0\), such thatThe second derivative of$$ \bigl\vert G(\theta ,f) \bigr\vert \le M,\quad \forall (\theta ,f)\in \mathbb{R}^{2}. $$*G*w.r.t. \((\theta ,f)\) is uniformly Lipschitz on bounded sets, i.e., for all \(M>0\) there exists \(L_{M}>0\) such that*G*satisfiesfor all \(\theta _{i}, f_{i}\in \mathbb{R}\) with \(\vert \theta _{i} \vert , \vert f_{i} \vert \le M\), \(i=1,2\).$$ \bigl\vert G''(\theta _{1},f_{1})-G''( \theta _{2},f_{2}) \bigr\vert \le L_{M}\bigl( \vert \theta _{1}- \theta _{2} \vert + \vert f_{1}-f_{2} \vert \bigr) $$ - (A3)
\(\beta \in L^{\infty }(\varSigma _{1})\), \(\theta _{w}\in L^{\infty }(\varSigma _{1})\), \(\theta _{0}\in C(\bar{\varOmega })\) and \(\theta _{d}\in L^{\infty }(Q)\).

- (A4)
\(f_{d}\in L^{\infty }(\varOmega )\), \(0\le f_{d}\le 1\) a.e. in

*Ω*.

### Remark 1

Assumption (A2) can be relaxed and has been chosen only to avoid technicalities when computing the derivatives. For more realistic phase transformation models we refer to [6].

### Remark 2

The choice of the cost functional in (1) is somewhat arbitrary. Mutatis mutandis, also a control of the temperature at end-time and/or a control of the distributed ferrite fraction is possible.

### 2.2 Analysis of the state system

Let us start with the discussion of the initial value problem (2a)–(2b) in the state system. In view of the assumptions, the following result can be proven by standard arguments. For a detailed proof, we refer to [13] or [14].

### Lemma 1

*Suppose that*(A2)

*holds true*.

*Then*,

*we have the following*:

- (a)
- (b)
*Let*\(\theta _{1}, \theta _{2}\in L^{p}(Q)\), \(1\le p <\infty \)*and let*\(f_{1}\), \(f_{2}\)*be the corresponding solutions of*(2a), (2b),*then there exists a constant*\(M_{2}>0\)*such that*$$ \Vert f_{1}-f_{2} \Vert _{W^{1,p}(0,T;L^{p}(\varOmega ))}\le M_{2} \Vert \theta _{1}- \theta _{2} \Vert _{L^{p}(Q)}. $$

*r*,

*u*, \(\theta _{w}\), \(\theta _{0}\), the following result can be obtained from Theorem 5.5 in Tröltzsch [24]:

### Lemma 2

*Suppose that*(A3)

*holds true*,

*and*\(r\in L^{s_{1}}(Q)\), \(u\in L^{ \infty }(0,T)\), \(u\ge 0\).

*Let*\(s_{1}>5/2\), \(s_{2}>4\),

*then the initial value problem*(3a)

*–*(3d)

*admits a unique solution*\(\theta \in W(0,T)\cap C(\bar{Q})\)

*satisfying the a priori estimate with a constant*\(C>0\)

It is a useful result for the proof of solvability of the state system (2a)–(2f), which is discussed below.

### Theorem 1

*Let*(A1)

*–*(A4)

*be satisfied*.

*Then*,

*the state system*(2a)

*–*(2f)

*admits for every control*\(u\in U_{\mathrm{ad}}\)

*a unique solution*

*satisfying*

### Proof

*c*denotes a generic constant, not to be confused with the heat capacity \(c_{p}\). To prove the existence of a local unique solution to (2c)–(2f), we apply the Banach’s fixed point theorem. For that purpose, we define an operator \(F: K\subset W(0,T)\rightarrow W(0,T)\) that maps \(\hat{\theta }\in W(0,T)\) to the solution

*θ*of

*f̂*solves (2a)–(2b) with

*θ̂*.

*F*is well-defined. Furthermore, the following a priori estimate with a constant \(C_{1}>0\) is valid

*M*is chosen big enough,

*F*is a self mapping on

*F*is a contraction. Let \(\hat{\theta } _{i}\in K\), \(i=1,2\), \(\theta _{i}=F(\hat{\theta }_{i})\) and \(\hat{\theta }=\hat{\theta }_{1}-\hat{\theta }_{2}\). Then, \(\theta = \theta _{1}-\theta _{2}\) solves

*G*in both variables (Assumption (A2)) and Lemma 1(b), we obtain

*F*is a contraction on \(W(0,T^{+})\). Since

*F*is also a self-mapping on

*K*, we can apply the Banach’s fixed point theorem to conclude that

*F*has a unique fixed point

*θ*, which is a local solution to (2c)–(2f). By a bootstrapping argument, the solution can be extended to the time interval \([0,T]\).

Moreover, in view of Lemma 1 we can apply Lemma 2 and obtain the additional regularity for *θ*. □

### Corollary 1

*Suppose that*(A1)

*–*(A4)

*hold true and let*\((\theta _{1}, f_{1})\), \(( \theta _{2}, f_{2})\)

*be the solutions of*(2a)

*–*(2f)

*corresponding to*\(u_{1}, u_{2}\in L^{\infty }(0,T)\).

*Then*,

*there exists a constant*\(C>0\),

*such that*

### Proof

*Ω*and over \((0,t)\) yields

Now, let us discuss the differentiability of the solution operator that we need for the derivation of first-order and second-order optimality conditions.

### Theorem 2

*Let Assumptions*(A1)

*–*(A4)

*be satisfied*.

*Then*,

*the solution operator*

*S*

*is twice Frechét*-

*differentiable from*\(L^{\infty }(0,T)\)

*to*\(Y\times {W^{1,p}(0,T;L^{p}(\varOmega ))}\), \(1\le p <\infty \).

*The directional derivative*\((\theta _{h}, f_{h})=S'(u)h=(S_{\theta }'(u)h,S _{f}'(u)h)\)

*at point*\(u\in L^{\infty }(0,T)\)

*in direction*\(h\in L^{ \infty }(0,T)\)

*is given by the solution of*

*with*\((\theta , f)=S(u)\).

*Furthermore*, \((\theta _{h_{1}h_{2}},f_{h _{1}h_{2}})=S''(u)[h_{1},h_{2}]\)

*is the solution of*

*with*\((\theta _{h_{i}},f_{h_{i}})=S'(u)h_{i}\), \(i=1,2\).

### Proof

*G*. Furthermore, one can analogously show Lipschitz continuity of the first derivative of the solution operator, i.e., for all \(u_{1}, u_{2}, h\in L^{\infty }(0,T)\), there exist a constant \(C>0\) such that

### 2.3 Existence and optimality conditions of optimal solutions

Since the state system is nonlinear, we cannot expect uniqueness of an optimal control and we have to deal with local optimal controls. We have the following result.

### Theorem 3

(Existence of optimal controls)

*Let Assumptions *(A1)*–*(A4) *be satisfied*. *Then*, *there exists at least one solution of the optimal control problem* (P).

To prove Theorem 3, we need the following auxiliary result:

### Lemma 3

*Assume*\(\{\theta _{k}\}\)

*is bounded in*\(L^{2} (0,T; H^{1} (\varOmega )) \cap L^{\infty }(Q)\)

*and*

*Then*,

*it also holds*

### Proof

*A*is linear and also continuous, since the application of the trace theorem yields

*φ*and

*χ*are smooth, using (18) and (19) we deduce that

With Lemma 3 at hand, we are now able to prove the existence of optimal solution of control problem (P).

### Proof of Theorem 3

*f*is the solution corresponding to

*θ*. We use test functions \(\varphi \in H^{1}(\varOmega )\) and \(\chi \in C^{1} [0,T]\) such that \(\chi (T) = 0\) and consider the weak formulation of (2c)–(2f) for \((\theta _{k'}, f_{n'}, u_{n'})\)

The optimality of \((\bar {\theta },\bar {f},\bar {u})\) follows by standard arguments using the lower semicontinuity of the cost functional w.r.t. *u*. □

In the following theorem first-order necessary optimality conditions are characterized by respective adjoint equations.

### Theorem 4

(Necessary optimality conditions)

*Let*\(\bar {u}\in U_{\mathrm{ad}}\)

*be an optimal control of problem*(P)

*and*\((\bar {\theta }, \bar {f})=S(\bar {u})\)

*the associated solution of the state system*(2a)

*–*(2f).

*Then there exists a unique solution*\((\bar{p}, \bar{q})\in Y\times W^{1,\infty }(0,T;L^{\infty }(\varOmega ))\)

*such that*

*Moreover*,

*the following variational inequality is valid*

### Proof

First observe that the system (24a)–(24f) is a linear backward-in-time system of the parabolic equation and ODE. After the time transformation \(t\mapsto T-t\) one can proceed as in the proof of Theorem 2 in order to prove the existence of the unique solution \((\bar{p},\bar{q}) \in W(0,T)\cap C(\bar{Q})\times W^{1, \infty }(0,T;L^{\infty }(\varOmega ))\) of the system (24a)–(24f).

*j*is differentiable and the set of admissible controls \(U_{\mathrm{ad}}\) bounded, closed and convex. Hence, the first-order necessary optimality conditions for a (local) optimal solution \(\bar {u}\in U_{\mathrm{ad}}\) is given by \(j'(\bar {u})(u-\bar {u}) \ge 0\), \(\forall u\in U_{\mathrm{ad}}\). For given direction \(h\in L^{\infty }(0,T)\) we have

*q̄*and integrate over

*Q*:

*q̄*, one can obtain for the first term in (26)

*Q*such that

*ū*are represented by the variational inequality (25). □

*ū*an admissible control of problem (P) with associated solution \((\bar {\theta },\bar {f})=S(\bar {u})\) of the state system (2a)–(2f). We suppose that the first-order optimality conditions given in Theorem 4 are satisfied with respective adjoint states \((\bar{p},\bar{q})\). Let us define the strongly active set associated to

*ū*. For fixed \(\tau >0\) we set

### Theorem 5

*Let*

*ū*

*be an admissible control of problem*(P)

*with associated state*\((\bar {\theta },\bar {f})=S(\bar {u})\)

*satisfying the first*-

*order necessary optimality conditions given in Theorem*4

*with associated adjoint states*\((\bar{p},\bar{q})\).

*Further*,

*it is assumed that*(

*SSC*)

*holds at ū*.

*Then there exist a*\(\tilde{\delta }>0\)

*and*\(\rho >0\)

*such that*

*holds for all*\(u\in U_{\mathrm{ad}}\)

*with*\(\Vert u-\bar{u} \Vert _{L^{\infty }(0,T)} \le \rho \)

*with associated states*\((\theta ,f)=S(u)\).

### Proof

The proof closely resembles that of Theorem 5.17 in [24], therefore we will not give here all details and refer to [24]. We only indicate some important arguments that need a bit more explanation.

*j*. We denote \(h=u-\bar{u}\). It follows from Taylor’s theorem with integral remainder (see, e.g., Theorem 8.14.3, p. 186 in [5]) that

Such kind of sufficient optimality conditions is an indispensable tool basis for carrying out numerical analysis of optimal control problems, e.g., convergence analysis of the sequential quadratic programming method in order to solve optimal control problems numerically.

## 3 Numerical implementation

In this section we introduce numerical algorithms for the solution of optimal control problem (P) analyzed in the previous section. This problem belongs to the class of the nonlinear boundary control problems with control constraints. The SQP (Sequential Quadratic Programming) method has turned out to be one of the most successful methods in nonlinear optimization (see, e.g., [1, 19]). The principal idea is to linearize the nonlinear equality constraints and to replace the cost functional by a quadratic approximation of the Lagrangian. It is well known that the SQP algorithm exhibits local quadratic convergence in finite-dimensional spaces. The convergence analysis for nonlinear parabolic boundary control problems was presented in the works of Tröltzsch [8, 23].

In this work we focus on the reduced SQP method (rSQP), where the reduction onto the control space takes place when solving the \((\mathit{QP}^{k})\)-subproblems. We also introduce the primal-dual active set (PDAS) strategy, used for the treatment of the quadratic \((\mathit{QP}^{k})\) problems in each iteration of rSQP method. The conjugate gradient (CG) method has been applied to solve the linear system of equations arising in the (PDAS) algorithm.

### 3.1 Reduced SQP method

The main idea of reduced SQP methods in contrast to usual SQP methods is to use only an approximation of the projected Hessian of the Lagrangian onto the kernel of the linearized constraint, instead of an approximation of the full Hessian of the Lagrangian.

*k*-th iterate. Introducing the notation \(\mathcal{L}_{(\theta ,f)}''\)—the second derivative of the Lagrangian \(\mathcal{L}\) with respect to the state pair variable \((\theta ,f)\), we can rewrite the KKT matrix as \(3\times 3\) block matrix. Since the linearized state system is uniquely solvable for every right hand side (it can be shown along the lines of Theorem 1), we can derive the following decomposition of the full KKT matrix in (36) by Gaussian block elimination

*H*is defined by

- (i)Solve the reduced Hessian system:$$ H\delta u=\underbrace{-J_{u}+e_{u}^{*}e_{(\theta ,f)}^{-*} \bigl(J_{(\theta ,f)}- L_{(\theta ,f)}''e_{(\theta ,f)}^{-1}e \bigr)+ L_{u(\theta ,f)}''e _{(\theta ,f)}^{-1}e}_{:=r}; $$(38)
- (ii)Solve the linearized state system, i.e.$$ e_{(\theta ,f)} \begin{pmatrix} \delta \theta \\ \delta f \end{pmatrix} =-e_{u}\delta u-e; $$
- (iii)Solve the adjoint state system, i.e.$$ e^{*}_{(\theta ,f)} \begin{pmatrix} p \\ q \end{pmatrix} =-J_{(\theta ,f)}-\mathcal{L}_{(\theta ,f)}'' \begin{pmatrix} \delta \theta \\ \delta f \end{pmatrix} -\mathcal{L}_{(\theta ,f)u}'' \delta u. $$

*H*is defined as in (37) and the residuum

*r*has to be evaluated by

### 3.2 Primal-dual active set (PDAS) strategy

*H*is not explicitly given after choosing a discretization strategy for the underlying partial differential equations. Hence, an iterative solver has to be established for tackling the reduced Hessian system, e.g. Conjugate gradient method (CG method) or Generalized minimal residual method (GMRES). In view of second-order sufficient optimality conditions for the original problem, we have applied the CG method for solving

## 4 Numerical results

In this section we discuss the numerical solution of the control problem (P). Firstly, we construct a test control problem in order to check the convergence of the reduced SQP method with a primal-dual active set strategy described above. Then we solve the optimal control problem for the hot rolling of DP steel. Here, for a globalization of the rSQP method, we use a projected gradient algorithm (see e.g. [15]) with a line search according to the Armijo rule to find suitable initial values for the rSQP method.

The numerical algorithms have been implemented in *WIAS-pdelib* software. For the solving the state and adjoint system the finite element toolbox *pdelib* was used.

### 4.1 A test problem

*Γ*denotes the boundary of

*Ω*and \(T> 0\). We apply the rSQP method discussed above to the semilinear parabolic boundary control problem

*θ̄*,

*p̄*is given by

*ū*.

Iterations history of the rSQP method with primal-dual active set strategy

Iter | \(J_{k}\) | \(e_{k}\) | \(\tau _{k}\) | #PDAS-loops |
---|---|---|---|---|

1 | 21.8504 | 0.94 | 0.22 | 3 |

2 | 20.3691 | 0.45 | 0.33 | 4 |

3 | 20.3517 | 0.0085 | 0.0938 | 3 |

4 | 20.3515 | 140.7 | 5⋅10 | 1 |

As reported in [8, 9], the quadratic convergence of the SQP methods is assured, if the quadratic subproblems \((\mathit{QP}^{k})\) are solved with a quite high precision. The time-space discretization has to match the current accuracy of the SQP step. In our test example, we observe that the speed of convergence of the rSQP method is limited after the third iteration by the time-space discretization error.

### 4.2 Optimal control problem for dual phase steel

In this subsection we present a numerical solution of the optimal control problem (P) formulated for the production of Mo–Mn dual phase (DP) steel.

*δ*to be \(\delta =10^{-2}\). The equilibrium volume fraction of ferrite \(f_{\mathrm{eq}}(\theta )\) and the temperature dependent factor \(g_{1}(\theta )\) are cubic spline functions interpolating the pointwise data as shown in Fig. 5. The factor representing the preconditioning of the initial phase austenite is given by \(g_{2}=10\). The model (42) for the austenite-ferrite phase transformation in the hot rolling process has been discussed in [22]. For further details about the modeling we refer to this article. We note, that Assumption (A2) is too strong for the function \(G(\theta ,f)\). Nevertheless, the existence and uniqueness of the solution to state system can be also shown for this function and all other theoretical and numerical considerations remain unchanged.

^{∘}C is chosen to be \(\rho =7.85\ \frac{\text{g}}{\text{cm}^{3}} \). The values for the heat conductivity

*κ*and specific heat \(c_{p}\) are set to

*L*of the austenite-ferrite phase transformation is specified according to [11] as \(L= 77.0\ \frac{\text{J}}{\text{g}}\).

*θ*is in the range of 20

^{∘}C–1200

^{∘}C. Therefore, in order to obtain useful results, an equilibrating of this two terms in cost functional is necessary. In the subsequent computations we set \(\alpha _{1}=1\), \(\alpha _{2}=5\cdot 10^{-6}\). The factor \(\alpha _{3}\) is a Tikhonov regularization parameter and is chosen as 0.1.

The nonlinear state system (2a)–(2f) as well the corresponding adjoint system in each iteration of projected gradient method can be solved numerically using semi-implicit Euler scheme. The rSQP method requires a solving of the linearized problems \((\mathit{QP}^{k})\). Here, the linear parabolic equation was discretized in a standard way using method of lines and ODE for the phase transition was treated numerically by explicit Euler scheme.

The FE triangulation of the computational domain *Ω* is done by a uniform mesh with \(N=561\) degrees of freedom. For the time step, we take \(\Delta t=0.0125\). We approximate the control function \(u(t)\) with piecewise constant functions on the time grid such that the unknown control function is represented as \(u=(u_{1},\ldots,u_{n-1})^{T}\), \(u_{i}=u(t_{i})\), \(i=1,\ldots,n-1\).

As explained above, we use the gradient projection method for the globalization of the rSQP algorithm. As an initial guess for the gradient projection method we take \(u_{0}\equiv 0\). The algorithm was terminated after 7 iterations, provided the relative error \(\Vert u ^{k+1}-u^{k} \Vert _{L^{2}(0,T)}/ \Vert u^{k} \Vert _{L^{2}(0,T)}\) is smaller then 0.01. The obtained control function *û* with corresponding state variables *θ̂*, *f̂* and adjoint variables *p̂*, *q̂* serve as the initial iteration of the rSQP method.

Value of objective function \(J_{k}\), relative error \(\tau _{k}\) and number of PDAS loops in \(k^{\text{th}}\)-iteration of the rSQP method

Iter | \(J_{k}\) | \(\tau _{k}\) | # PDAS-loops |
---|---|---|---|

1 | 0.01669 | 0.1350 | 8 |

2 | 0.01438 | 0.01077 | 4 |

3 | 0.01434 | 6⋅10 | 2 |

^{∘}C and ferrite fraction of 85% are reached very accurately in the middle of the cross section.

## 5 Conclusions

We have studied the optimal control problem that describes the hot rolling process of multiphase steel. The nonlinear boundary control problem was analyzed and the first-order necessary and second-order sufficient optimality conditions were derived. The control problem was solved numerically by a reduced SQP method with active set strategy.

The approach has already been tested in an industrial setting. The results of the optimal control of the cooling line have been verified in hot rolling experiments at the pilot hot rolling mill at the Institute for Metal Forming (IMF), TU Bergakademie Freiberg. For more details we refer to a recent paper [4].

The challenging topic for the future research will be the real time control of the hot rolling process, which is an important task for the industrial employment of this approach. Here, recent developments in model reduction techniques seem to be a promising tool and will be subject of further work of the authors.

## Declarations

### Acknowledgements

The authors would like to thank Marcel Graf and Piyada Suwanpinij for carrying out the experiments on the hot rolling mill. Special thanks to Wolf Weiss for fruitful discussions about mathematical modeling and the interpretation of measurements. The third author is grateful to the financial support from the DFG.

### Availability of data and materials

Please contact author for data requests.

### Funding

The work on this paper has been partially supported by Deutsche Forschungsgemeinschaft (DFG) within the priority program 1204 “Algorithms for fast, material specific process-chain design and analysis in metal forming”.

### Authors’ contributions

The three authors are equally contributors to this paper. All authors read and approved the final manuscript.

### Ethics approval and consent to participate

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

### Consent for publication

Not applicable.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Alt W. Nichtlineare Optimierung: Eine Einführung in Theorie, Verfahren und Anwendungen. Vieweg Studium: Aufbaukurs Mathematik. Wiesbaden: Vieweg+Teubner; 2002. View ArticleGoogle Scholar
- Bergounioux M, Ito K, Kunisch K. Primal-dual strategy for constrained optimal control problems. SIAM J Control Optim. 1999;37:1176–94. MathSciNetView ArticleGoogle Scholar
- Bhadeshia H, Honeycombe R. Steels: microstructure and properties. Amsterdam: Elsevier; 2011. Google Scholar
- Bleck W, Hömberg D, Prahl U, Suwanpinij P, Togobytska N. Optimal control of a cooling line for production of hot rolled dual phase steel. Steel Res Int. 2014;85:1328–33. View ArticleGoogle Scholar
- Dieudonne J. Foundations of modern analysis. Pure and applied mathematics. Read Books; 1960. MATHGoogle Scholar
- Fasano A, Hömberg D, Panizzi L. A mathematical model for case hardening of steel. Math Models Methods Appl Sci. 2009;19:2101–26. MathSciNetView ArticleGoogle Scholar
- Goldberg H, Tröltzsch F. Second order sufficient optimality conditions for a class of nonlinear parabolic boundary control problems. SIAM J Control Optim. 1993;31:1007–25. MathSciNetView ArticleGoogle Scholar
- Goldberg H, Tröltzsch F. On a Lagrange–Newton method for a nonlinear parabolic boundary control problem. Optim Methods Softw. 1998;8:225–47. MathSciNetView ArticleGoogle Scholar
- Goldberg H, Tröltzsch F. On a SQP-multigrid technique for nonlinear parabolic boundary control problems. Berlin: Springer; 1998. View ArticleGoogle Scholar
- Hashimoto T, Yoshioka Y, Ohtsuka T. Model predictive control for hot strip mill cooling system. In: Proceedings of the IEEE international conference on control applications. 2010. p. 646–51. Google Scholar
- Hengerer F, Strässle B, Bremi P. Berechnung der Abkühlungsvorgänge beim Öl- und Lufthärten zylinder- und plattenförmiger Werkstücke aus legiertem Vergütungsstahl mit Hilfe einer elektronischen Rechenanlage: Calcul, à l’aide d’une calculatrice électronique, des processus de refroidissement se déroulant lors de la trempe à l’huile et à l’air de cylindres et de plaques en acier allié. In: Bericht des Werkstoffausschusses des Vereins deutscher Eisenhüttenleute. Stahleisen; 1969. Google Scholar
- Hintermüller M, Volkwein S, Diwoky F. Fast solution techniques in constrained optimal boundary control of the semilinear heat equation. In: Control of coupled partial differential equations. Internat. series numer. math. vol. 155. Basel: Birkhäuser; 2007. p. 119–47. View ArticleGoogle Scholar
- Hömberg D, Sokolowski J. Optimal shape design of inductor coils for surface hardening. SIAM J Control Optim. 2003;42(3):1087–117. MathSciNetView ArticleGoogle Scholar
- Hömberg D, Volkwein S. Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition. Math Comput Model. 2003;37:1003–28. MathSciNetView ArticleGoogle Scholar
- Kelley CT. Iterative methods for optimization. Frontiers in applied mathematics. vol. 18. Philadelphia: SIAM; 1999. View ArticleGoogle Scholar
- Kupfer F-S, Sachs EW. Numerical solution of a nonlinear parabolic control problem by a reduced SQP method. Comput Optim Appl. 1992;1:113–35. MathSciNetView ArticleGoogle Scholar
- Landl G, Engl HW. Optimal strategies for the cooling of steel strips in hot strip mills. Inverse Probl Eng. 1995;2:103–18. View ArticleGoogle Scholar
- Lezius R, Tröltzsch F. Theoretical and numerical aspects of controlled cooling of steel profiles. In: Progress in industrial mathematics at ECMI 94. Berlin: Springer; 1996. p. 380–8. View ArticleGoogle Scholar
- Nocedal J, Wright S. Numerical optimization. 2nd ed. Springer series in operations research and financial engineering. New York: Springer; 2006. MATHGoogle Scholar
- Raymond JP, Tröltzsch F. Second order sufficient optimality conditions for nonlinear parabolic control problems with sate constaints. Discrete Contin Dyn Syst. 2000;6:431–50. View ArticleGoogle Scholar
- Spittel M, Spittel T. Metal forming data of ferrous alloys—deformation behaviour. Landolt–Börnstein—group VIII advanced materials and technologies. vol. 2C1. Berlin: Springer; 2009. Google Scholar
- Suwanpinij P, Togobytska N, Prahl U, Weiss W, Hömberg D, Bleck W. Numerical cooling strategy design for hot rolled dual phase steel. Steel Res Int. 2010;11:1001–9. View ArticleGoogle Scholar
- Tröltzsch F. An SQP method for the optimal control of a nonlinear heat equation. Control Cybern. 1994;23:268–88. MathSciNetMATHGoogle Scholar
- Tröltzsch F. Optimal control of partial differential equations: theory, methods, and applications. Graduate studies in mathematics. vol. 112. Providence: Am. Math. Soc.; 2010. MATHGoogle Scholar
- Wang B-X, Zhang D-H, Wang J, Yu M, Zhou N, Cao G-M. Application of neural network to prediction of plate finish cooling temperature. J Cent South Univ Technol. 2008;15:136–40. View ArticleGoogle Scholar
- Zheng Y, Li N, Li S. Hot-rolled strip laminar cooling process plant-wide temperature monitoring and control. Control Eng Pract. 2013;21(1):23–30. View ArticleGoogle Scholar