An iterative solver for time-periodic heat optimal control problems

， -， -,3

(1.School of Science, Southwest Petroleum University, Chengdu 610500, China;2.School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China;3.School of Science, Civil Aviation Flight University of China, Deyang 618300, China)

Abstract: In this paper, an iterative algorithm is introduced for solving a class of optimal control problems constrained by time-periodic heat equation, where the optimization is concerned by searching a best source term of the heat equation to minimize the objective function.By applying the optimality condition, the problem is firstly transformed into two coupled time-periodic heat equations.Then the iterative algorithm is applied to decouple the coupled PDE system.Finally, the equations are separately solved in the Gauss-Seidel pattern.Numerical examples are presented to illustrate the robustness of the convergence rate of algorithm with respect to the discretization parameters.

Keywords: Time-periodic heat equation; Optimal control; Iteration(2010 MSC 65M60)

1 Introduction

In this paper，we consider the optimal control problems constrained by time-periodic heat partial differential equation(PDE)[1-20].Applications of such problems include the design of reverse flow reactors[12], cyclically steered(bio-)reactors[5]and energy-producing kites[4],etc.Meanwhile, such optimal control problems also arise in a variety of chemical engineering applications[3, 8, 9, 15, 18, 19]such as the moving bed processes[14]which find widespread use in the pharmaceutical and food industry.

Different from the usual setting for linear parabolic control systems in the literature, a particular feature of the time-periodic parabolic control problems is the constraint that the solution of the underlying dynamical system is periodic in time.This kind of optimal control problems appear as the sub-problems in inexact Newton or inexact sequential quadratic programming methods for the solution of nonlinear optimization problems with time-periodic partial differential equation(PDE)constraints.

The computation of optimal controls is based on the optimal conditions and their approximate solutions using some numerical discretization.The applicability and the accuracy of this strategy depend on the availability of structure of the discretized optimality systems.If accurate solutions are required, the resulting discretized problems will inevitably be of large scale, because in this case we often need to use small discretization sizes.Thus, it is an important issue to design efficient solvers to treat the optimal control problems with time-periodic PDE constraint.Existing numerical methods for this kind of control problems include the relaxation techniques[6], the multi-grid method[2]and the interesting pre-conditioning technique which attract considerable attention in the past years(see,e.g., Refs.[1,7,10,11,13,16,20]).However, these existing approaches are more complicated than the one proposed in this paper.In a word, the new iterative algorithm studied here has essential difference with respect to mechanism, computational cost and complexity with the just mentioned algorithms.

In this paper, we propose a new approach to solve the time-periodic heat optimal control problems.We firstly reformulate the optimal control problem as two coupled time-periodic heat equations.Then we solve this coupled PDE system via an iteration process.By picking up an initial guess for the control variable(chosen randomly in practical computation), we solve the state equation and the solution plays a role of source term for the adjoint equation.Then we solve the adjoint equation and with the solution we can prepare for the next iteration.We show that the convergence rate of the proposed iterative algorithm is robust with respect to the space and time discretization parameters.

The rest of this paper is organized as follows.In Section 2, we present the optimal control problem studied in this paper.The optimality system is also derived in detail in this section.Section 3 presents the algorithm and the details concerning implementation in practice.In Section 4, we show numerical results which indicate that the convergence rate of the proposed algorithm is robust with respect to the change of discretization parameters.Section 5 concludes this paper.

2 The optimal control problem

The model that we are interested in is the following optimal control problem:

(1a)

wherey(the state variable)andu(the control variable)satisfy the following constraints

(1b)

In order to solve(1a～1b), we now derive the optimality system.Denote byy(u)the solution of the state equation in(1b)and byy′(u;δu)the first-order directional derivative ofyatualong the directionδu.Let

e(y,u)=?ty-μΔy-u.

Then, a routine calculation yields

ey(y,u)y′(u;δu)+eu(y,u)δu=0

(2)

It is easy to getey(y,u)=?t-μΔandeu(y,u)=I, whereIis the identity operator.Substituting these results in to(2)gives

?ty′(u;δu)-μΔy′(u;δu)-δu=0

(3)

(4)

(5)

This gives

(6)

In(3)and(6), by lettingδu=v-uwith some suitablevwe have

μΔy′(u;v-u)-(v-u)=0,

which is equivalent to

ey(y,u)y′(u;v-u)+eu(y,u)(v-u)=0

(7)

Letp(the so-called co-state variable)be the solution of the following equation

(8)

Then it follows by using(8)and the second equation in(7)that

〈ey(y,u)y′(u;v-u),p〉=

-〈eu(y,u)(v-u),p〉.

Now, by using(5)and(8)we have

(9)

Since(9)holds for arbitrary directional variablev, it holds

(10a)

p(0,x)=p(1,x)forx∈Ω,p(t,x)=0

for(x,t)∈?Ω×(0,1)

(10b)

Substituting(8)and(9)into(7)gives

(11)

〈ηu-p,v-u〉=0,?v.

When the co-state variablepis ready, the control variableucan be chosen asu=ηp.

3 The algorithm

We now propose an iterative algorithm for solving(12)as follows.

(13)

wherek≥0 is the iteration index and fork=0 we need to pick an initial guessp0(x,t)for the co-state variable.In practical computation, such an initial guess is chosen randomly subject to the periodic condition and the boundary condition.In(13), withpk(x,t)known from the previous iteration, we can first solveyk+1(x,t)from the first PDE and then solvepk+1(x,t)from the second PDE.The algorithm is therefore of the Gauss-Seidel type.

Both the first and second PDEs in(13)are time-periodic heat equations and many existing numerical methods can be directly applied.As an illustration, we consider the case Ω=(0,1)dwithd=1,2,3 as follows.By a mesh withmnodes and denoting the value ofy(x,t)(resp.p(x,t))at thei-th nodexibyyi(t)(resp.pi(t)), the discrete solution

yk+1(t)≈(yk+1(x1,t),…,yk+1(xM,t))T

and

pk+1(t)≈(pk+1(x1,t),…,pk+1(xM,t))T

satisfy

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and

(15)

whereΔxdenotes the mesh size,mdenotes the number of spatial grids andIx∈RM×Mis the identity matrix withM=mdin thed-dimensional case(d=1,2,3).We can also consider other boundary conditions and spatial discretizations, such as finite element and finite volume, etc.

For temporal discretization, we consider the backward-Euler method for the state equation concerningyk+1(t)and the forward-Euler method for the co-state equation concerningpk+1(t).This numerical setting leads to the following full discrete formula:

(16a)

and

and

(17)

Then, we can represent(1b)as the following linear algebraic system:

(18)

Note that the matrixMgiven by(17)takes the form of block circulant and therefore ap-cyclic SOR(successive over-relaxation)iterative method[17]can be applied as an inner solver to handle each of the two linear systems in(18), which yields very efficient computation of the two systems in(18).

In this section, we present numerical results to validate the efficiency of the proposed iterative algorithm in Section 3.For all numerical results, the initial guessp0(x,t)for the proposed algorithm is chosen randomly under the periodic condition and the zero boundary condition.We consider the 1D case together with centered finite difference discretization for the Laplacian.We use the following data:

η=0.08,T=2.5

(19)

With this data, the solution of the optimality system(12)y(x,t)(left subfigure)andp(x,t)(right subfigure)is shown in Fig.1.

Fig.1 Numerical solution of the optimality system(12)with the data given by(19)

We now study whether or not the convergence rate of the full discrete version of the proposed iterative algorithm in Section 3 is robust with respect to the discretization parametersΔtandΔx.In Fig.2, we show the measured convergence rates of the algorithm in two situations: in the left subfigure we fixΔt=0.02 and choose forΔxthree values and in the right subfigure we fixΔx=0.025 and choose forΔtthree values.In both situations, we see clearly that the convergence rate is insensitive to the change ofΔxandΔt.

Fig.2 Convergence rate of the iterative algorithm(18)with different space mesh size(left)and time step size(right)

5 Conclusions

We have proposed an iterative algorithm for solving the optimal control problems with time-periodic heat equations as the constraint.We first derive the optimality system of such an optimal control problem, which consists of two coupled time-periodic heat equations.Then we apply the Gauss-Seidel iteration to such an optimality system, that is to say, we firstly solve the state equation and then solve the co-state equation in an iteration pattern.The full discrete version of the proposed iterative algorithm is also presented.Numerical results indicate that the proposed algorithm possesses robust convergence rate with respect to both the space and the time discretization parameters.