Numerical Computation for a Kind of Time Optimal Control Problem for the Tubular Reactor System

This paper is devoted to the study of numerical computation for a kind of time optimal control problem for the tubular reactor system.This kind of time optimal control problem is aimed at delaying the initiation time τ of the active control as late as possible, such that the state governed by this controlled system can reach the target set at a given ending timeT. To compute the time optimal control problem, we firstly approximate the original problem by finite element method and get a new approximation time optimal control problem governed by ordinary differential equations.Then, through the control parameterizationmethod and time-scaling transformation, the approximation problem becomes an optimal parameter selection problem. Finally, we use Sequential Quadratic Program algorithm to solve the optimal parameter selection problem. A numerical simulation is given for illustration.


Introduction
Tubular reactors have extensive applications in the industrial production.A lot of reaction processes in chemical and biochemical engineering can be described by tubular reactor models, such as chlorine dioxide bleaching model [1].The dynamic tubular reactors are typically described by nonlinear partial differential equations (PDEs), which include convection, reaction, and diffusion phenomena.For studies of such PDEs system, we would mention some works [2,3].Sometimes we regulate the reaction process artificially to improve the reactor rate or reduce the reaction time, by seeking some control strategies such as adding catalyst or placing a heating or cooling jacket.For this purpose, optimal control problems governed by such PDEs system arouse increasing attentions [4][5][6][7].As a kind of important optimal control problem, time optimal control problem plays an important role in many fields of applications.
Generally speaking, time optimal control problems can be divided into two types.The first one is to find a minimal time  and a control belonging to some constraint set which is acted upon from beginning time 0, such that the state governed by the controlled system can arrive at a given target set in the shortest time interval [0, ].This kind of problem is called the minimal time optimal control problem.The second one is to delay the initiation time  of the active control as late as possible, such that the state governed by this controlled system can reach the target set at the given ending time .This is a maximal time optimal control problem.There have been extensive researches on the first kind of time optimal control problem [8][9][10][11], but only a few works related to the second kind of time optimal control problem have been studied.For the second kind of time optimal control problem, we would mention the works [12,13].
The time optimal control problems governed by PDEs are infinite dimensional control problems and are difficult in finding out analytical solutions.Thus, the numerical approximations of time optimal control problems attract a lot of attentions.Here, we would like to mention some related works [16][17][18] on the computations of time optimal control problems.However, the works mentioned above focused on the first kind of time optimal control problem.To the best of our knowledge, it seems that no attempts have been made to develop the computation for the second kind of time optimal control problem governed by nonlinear PDEs systems.In this paper, we shall consider the computation for the second kind of time optimal control problem (TP).Firstly, we project the time optimal control problem (TP) by finite element method into an approximation problem (TP ℎ ).Although approximation problem (TP ℎ ) is governed by ordinary differential equations, it is difficult to solve directly due to the unknown time variables.Then, we translate problem (TP ℎ ) to an optimal parameter selection problem (TP ℎ  ) through the control parameterization method and timescaling transformation.Finally, we use Sequential Quadratic Program (SQP) algorithm to solve the optimal parameter selection problem.
The rest of this paper is organized as follows.In Section 2, we give the finite element approximation for the time optimal control problem (TP).In Section 3, we provide the control parameter method and time-scaling transformation to reduce the approximation problem to an optimal parameter selection problem.In Section 4, we give the procedure of solving the optimal parameter selection problem.In Section 5, we carry out some numerical experiments to illustrate the effectiveness of our approximation method.

The Finite Element Approximation of the Problem (TP)
In this section, we describe finite element approximation of the problem (TP).Firstly, we introduce a standard triangulation where  is a positive integer and   ,  = 0, 1, . . ., , are grid points which satisfy . ., } is called the mesh size of the triangulation T ℎ .Thus, we can write T ℎ as Corresponding to each triangulation T ℎ , we can define a finite dimensional space as follows: where  1 (  ) stands for the space of all linear polynomials defined on the subinterval   .Obviously, the basis of  ℎ can be taken as {  |  = 0, 1, . . ., }, where   ∈  ℎ ,  = 0, 1, . . ., , are continuous and piecewise linear polynomials that satisfy Thus, it is clear that  ℎ is a space of  + 1 dimensions and  ℎ ⊂  1 (0, ).

Control Parameterization Method and Time-Scaling Transformation
To solve the problem (TP ℎ ) numerically, which is a time optimal control problem in the finite dimensional space, we apply the classical control parametrization method [19], where the control function can be approximated by the piecewise constant functions.We subdivide the time horizon [0, ] into  subintervals [ −1 ,   ],  = 1, 2, . . ., , where  ≥ 2 is a given integer and   ,  = 0, 1, . . ., , satisfy Here each time point   ,  = 1, 2, . . .,  − 1, is called a switching time.We allow the approximate control to switch at each switching time.To find the optimal switching time points, we assume that these switching points are not prefixed and are the parameters, which need to be chosen optimally in the following set: Define We approximate the control function () by a constant vector on each subinterval: Then the control function can be approximated by the following piecewise constant control of the form (29) Since the switching times are unknown, (29) is not easy to solve numerically.It is also difficult to integrate (29) accurately if some of the subintervals [ −1 ,   ],  = 1, . . ., , are very short.To overcome these difficulties brought about by the unknown switching times, we make use of the timescaling transformation [20][21][22].Let  ∈ [0, ] be a new time variable, where  is the number of the time subintervals.The time-scaling transformation between  ∈ [0, ] and  ∈ [0, ] can be established through the following differential equation: where  is a piecewise constant function, which is defined by where   =   −  −1 ,  = 1, . . ., .Integrating (30) can yield that where ⌊⌋ denotes the largest integer which is less than or equal to .It follows from (32) that Noting that () =  in (30), we have t() =   .Thus Equality (34) shows that the time-scaling transformation maps  =  to the th switching time  =   ,  = 1, . . .,  − 1.
Following the procedure of Algorithm 1, we can solve the parameter selection problem (TP ℎ  ).Let ( * ,  * ) be the optimal parameter for the problem (TP ℎ  ).By making use of (32), the approximation of ( * ,  * ) and (, ) ⊤ for the problem (TP) can be obtained.

Numerical Simulation and Discussion
In this section, a numerical simulation for the problem (TP) is presented.We take in (2).The control bounds are taken as (42)  In this simulation, we divide the interval [0, 1] into  equal subintervals.Thus, we have ℎ = 1/.We choose the basis functions   (),  = 0, 1, . . ., , as follows: We carry out the numerical simulation experiments within the MATLAB environment (version R2016a).We use personal computer with the following configuration: Intel Core i5-7200 2.50 GHz CPU, 8.00 GB RAM, 64-bit Windows 10 Operating System.
The following simulations are conducted by Algorithm 1.Using the piecewise constant control parameterization method with  = 2, 5, 10, 20, we solve problem (TP ℎ  ) for  = 10, 12, 15, 20, 30, 40.The optimal time  * is given in Table 1.From Table 1, we can see that as  increases from 10, 12, 15 to 20, the approximation value of  * varies from 0.3680, 0.3682, 0.3685 to 0.3686, which means finer finite element triangulation (T ℎ ) can lead to shorter actively controlled interval [ * , ].But when  increases from 20, 30 to 40, the optimal time tends to be stable.Moreover, we see that the change of the value of  has little effect on the approximation results.The optimal controls for  = 40 when  = 2 and  = 20 are presented in Figure 2. It can be found in Figure 2 that the optimal control for the problem (TP ℎ  ) is bang-bang control, and the optimal control  * only switches one time.The corresponding states without control and with the optimal control acting on the interval ( * , ) for  = 40 and  = 20 are also presented in Figures 3 and 4, respectively.

Conclusions
In this paper, we propose a computation method for the second kind of time optimal control problem for the tubular reactors systems, which are widely used in chemical and Mathematical Problems in Engineering biochemical engineering.The aim of the second kind of time optimal control problem is to find a control such that the state governed by the tubular reactors system can arrive at a given target set at the ending time  with the shortest duration of the control action.Currently, the computation of the second kind of time optimal control problem did not attract enough attentions of the researchers.Our approach can be summarized in three steps: (1) A finite dimensional approximation problem can be obtained with finite element method; (2) through control parameterization method and time-scaling transformation, we translate the approximation problem into an optimal parameter selection problem; (3) we solve this optimal parameter selection problem by Sequential Quadratic Program algorithm.Moreover, we give an example for illustration to show the effectiveness of our approximation method.

Figure 3 :Figure 4 :
Figure 3: The state function without control.

Table 1 :
Optimal time  * for different  and different .