Optimal Control for Radiative Heat Transfer Model with Monotonic Cost Functionals

Abstract

A boundary optimal control problem for a nonlinear nonstationary heat transfer model is considered. The model describes coupled conduction and radiation within the P1 approximation. The control parameter is related to the emissivity of the boundary and varies with time. The optimal control problem is to minimize or maximize a cost functional which is assumed to be monotonic. Su cient conditions of optimality are derived and the convergence of a simple iterative method is shown.

Keywords

• optimal control
• radiative heat transfer
• conductive heat transfer
• su cient optimality conditions
• simple iterative method
• bang-bang

Introduction

Radiative heat transfer models can be used for describing various engineering processes. These models contain parameters related to some properties of a medium or a boundary surface. Optimal control problems for such models consist in determination of some parameters values in order to minimize (or maximize) a given cost functional. Papers [1{5] deal with problems of boundary temperature control for radiative heat transfer models including SPN approximations of the radiative transfer equation (RTE). Note that approximations of RTE are employed to simplify numerical solution of governing equations, and SP1 (P1) approximation is valid mainly for optically thick and highly scattering media at large optical distances from the boundary [6, 7].

In this paper, we consider a di usion model (P1 approximation of RTE) including a nonstationary heat equation combined with a stationary equation for the mean intensity of thermal radiation. The control parameter depends on the emissivity of the boundary. We will assume that the cost functional is monotonic. Optimality systems for such functionals become simpler and do not contain an adjoint equation. Moreover, the ? The research was supported by the Ministry of education and science of Russian Federation (project 14.Y26.31.0003) Copyright c by the paper's authors. Copying permitted for private and academic purposes. In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org monotonicity condition allows to obtain su cient conditions of optimality and prove the convergence of a simple iterative method.

Paper [8] deals with the analogous optimal control problem of obtaining a desired temperature distribution by controlling the emissivity of the boundary. Note that the cost functional, representing L2-deviation of the temperature from the desired eld, turns monotonic if the desired temperarure equals 0. A similar optimal control problem was investigated in [9, 10], where the emissivity does not vary with time, and the work [11] is devoted to an analogous problem for a steady-state P1 model. In the mentioned papers, an analog of the bang-bang principle for the optimal control was proven. Based on this principle, it is possible to construct e cient numerical algorithms for solving optimal control problems. In a general case, a simple iterative method fails to converge, that is why a generalized algorithm was applied in [9, 10]. However, if the cost functional is monotonic, the convergence of a simple iterative method can be proven. 2

Problem formulation

The nonstationary normalized P1 model of radiative-conductive heat transfer in a bounded domain R3 has the following form [9]: + b a(j j 3

') = 0; b)j = 0; ' + a(' b4)j j j 3) = 0; = 0; jt=0 = 0: ( 1 ) ( 2 ) ( 3 ) Here, is the normalized temperature, ' the normalized radiation intensity averaged over all directions, a the absorption coe cient, and b the boundary temperature taken in Newton's law of cooling. The parameters a, b, and are positive constants, and = (x), u = u(x; t), x 2 , t 2 (0; T ) are positive functions. The control parameter u depends on the emissivity " of the boundary surface as follows: u = "=2(2 "). The symbol @n denotes the derivative in the outward normal direction n on the boundary := @ .

De ne the set of admissible controls Uad of functions u(x; t) such that u1 u u2, where u1(x; t) and u2(x; t) are positive functions. The problem of optimal control consists in the determination of functions u 2 Uad, , and ' which satisfy the conditions ( 1 )-( 3 ) and minimize (or maximize) an objective functional J ( ; ') which is assumed to be monotonic. The precise de nition of monotonicity will be given in the next section. 3

Formalization of the optimal control problem

b (ii) 0

Denote H = L2( ), V = H1( ). Note that V H = H0 V 0. Let the value of a functional f 2 V 0 on an element v 2 V be denoted by (f; v), and (f; v) is the inner product in H if f and v are elements of H. De ne the space W = fy 2 L2(0; T ; V ) : y0 2 L2(0; T ; V 0)g, y0 = dy=dt, as well as the space of states Y = W L2(0; T ; V ) and the space of controls U = L2( ), Uad = fu 2 U : u1 u u2g.

De nition 1. A pair f ; 'g 2 Y is called weak solution of the problem ( 1 ){( 3 ), that corresponds to the control u 2 Uad, if the following equalities are ful lled for any v; w 2 V a.e. on (0; T ):

Z

Z u(' ( 0; v) + a(r ; rv) + ( b)vd and jt=0 = 0.

Theorem 1. (cf. [9]) Let the conditions (i), (ii) be satis ed. For any u 2 Uad the problem ( 1 ){( 3 ) has a unique weak solution f ; 'g, and the following inequalities are ful lled: 0 M , 0 ' M 4, where M = maxfk bkL1( ); k 0kL1( )g. De nition 2. The cost functional J : Y \ [L1(Q)]2 ! R is called monotonic, if, given any 0 1 2, 0 '1 '2 a.e. in Q, we have J ( 1; '1) J ( 2; '2).

Next we state two optimization problems not depending on a speci c monotonic cost functional.

Problem 1. Find ub 2 Uad such that for any u 2 Uad we have b Problem 2. Find ub 2 Uad such that for any u 2 Uad we have b , '

b , ' b ' a.e. in Q. ' a.e. in Q.

Here, b = (ub), 'b = '(ub), = (u), ' = '(u). A weak solution of the problem ( 1 ){( 3 ), corresponding to the control u 2 Uad, is denoted by f (u); '(u)g. Remark 1. It is readily seen that solutions of problems 1 and 2 are solutions of optimal control problems J ( ; ') ! inf and J ( ; ') ! sup, respectively, where J is monotonic. De nition 3. Solutions of the problems 1 and 2 are called strong optimal controls.

Let us give an example of a monotonic cost functional. Suppose that 1 is a part of the boundary, on which u is given that is u = u1 = u2 on 1. The functional represents the energy out ow through 1:

J ( ; ') =

Z T Z 0 1 ( b) + bu1(' b4) d dt:

Note that our goal is to minimize (or maximize) the temperature and radiative intensity elds in the entire domain and time interval. Therefore, the answer will be the same for any monotonic cost functional.

Optimality conditions

a) u =

Then '

', e

Lemma 1. Let u; ue 2 Uad, = (u), ' = '(u), e = (u), 'e = '(ue), and one of the following conditions is satis ed: e (u1; if ' b4 < 0; ( b4 < 0; b) u = u1; if 'e u2; if 'e b4 > 0: Proof. Set = Set v = , w = e, ' = ' 'e and de ne the functions in ( 4 ), ( 5 ) and integrate in t. We obtain = maxf ; 0g, = maxf'; 0g. k (t)k2 + akr k2 + 2d + b a ( + e)( 2 + e2) ; d =

Then u is a solution of the problem 2.

Next prove the uniqueness of the strong optimal control.

Theorem 4. If u and ue are strong optimal controls, then u = ue a.e. in f(x; t) 2 : '(x; t) 6= b4(x; t)g.

Proof. By de nition, '(u) = '(ue) = ', (u) = (u) = that e a.e. in Q. It follows from ( 5 ) ue(' b4) vd

Z = (u , therefore, u = ue a.e. in f(x; t) 2 Remark 2. It follows from ( 4 ), ( 5 ) that an arbitrary modi cation of a strong optimal control u in the set f(x; t) 2 : '(x; t) = b4(x; t)g keeps the optimality of the control u, because such modi cation does not in uence on the second term in ( 5 ). 5

Iterative algorithm

Describe a simple iterative method converging to a strong optimal control. Discuss the problem 1, considerations for problem 2 are similar.

De ne the operator U : L1( ) ! L1( ):

U (') = 'k a.e. in Q, we have uk+1 = U ('k) ! U (' ) a.e. on . ( 8 ) tu tu , and

It follows from Lemmas 2, 3 that the simple iterative method converges to a solution of ( 8 ), therefore, u is a strong optimal control.

Theorem 5. Problem 1 (or 2) is solvable.

Remark 3. Let the solution f ; 'g of problem ( 1 ){( 3 ) be computed with absolute error " that is j' 'ej " in Q, where 'e is a component of the approximate solution. Then the strong optimal control is determined ambiguously in the set f(x; t) 2 : j'(x; t) b4(x; t)j "g. 6

Numerical example

As an example, consider a one-dimensional model describing the radiative heat transfer problem in a slab of thickness L = 50 [cm]. The physical parameters are taken from [12]. The maximum temperature is chosen as Tmax = 500 C. Notice that the absolute temperature is related to the normalized temperature as follows: T = Tmax . Set b = 0:4 at x = 0, and b = 0:7 at x = L. The thermodynamical characteristics of the medium inside the slab correspond to air at the normal atmospheric pressure and the temperature 400 C, namely a = 0:92 [cm2/s], b = 18:7 [cm/s], = 3:3 : : : [cm], a = 0:01 [cm 1], and = 10 [cm/s]. The initial function is 0(x) = 0:3 + 0:7x=L. The time interval length is chosen as T = 60 [s]. The bounds of the control are u1 = 0:01 and u2 = 0:5.

The boundary-value problem ( 1 ){( 3 ) was solved by the nite di erence method with Newton's linearization. Namely, we use the implicit time discretization (10001 grid points) that leads to a nonlinear algebraic system at each time step after the discretization in space (2501 grid points). After applying Newton's method to this system, one requires to solve a block-tridiagonal linear system with two blocks that is possible by using standard solvers.

It is worth noting that the simple iterative method for solving problem ( 8 ) does not require storing the solution f ; 'g for all time grid lines, because the optimality conditions do not contain the adjoint system. The simple iterative method is applied to each individual time step and needs approximately 3 iterations. It follows from the statement of the algorithm that the resulting control will always be bang-bang one.

The solution of the problem 1 at x = L is presented in Fig. 1. The strong optimal control in the problem 1 equals u2 = 0:5 at x = 0. The solution of the problem 2 at x = L is depicted in Fig. 2. The strong optimal control in the problem 2 equals u1 = 0:01 at x = 0.

Figure 3 indicates the minimum and maximum temperatures at several time instants, and the minimum and maximum intensities of radiation are shown in Fig. 4. Notice that the maximum and minimum elds at large t are close to the corresponding optimal states in the steady-state optimal control problem due to the stabilization of the radiative heat transfer process. The strong optimal controls at large t are equal to the respective steady-state strong optimal controls as well. u(L, t) u(L, t) 0

0 0.5 0.4 0.3 0.2 0.1 0.5 0.4 0.3 0.2 0.1

References

1. Pinnau , R. , Thommes, G.: Optimal boundary control of glass cooling processes , Math. Methods Appl. Sci. , 27 ( 11 ), 1261 { 1281 ( 2004 )

2. Pinnau , R. : Analysis of optimal boundary control for radiative heat transfer modeled by the SP1-system . Commun. Math. Sci. 5 ( 4 ), 951 { 969 ( 2007 )

3. Frank , M. , Klar , A. , Pinnau , R.: Optimal control of glass cooling using simpli ed PN theory, Transport Theory Statist . Phys . 39 ( 2 {4), 282 { 311 ( 2010 )

4. Tse , O. , Pinnau , R.: Optimal control of a simpli ed natural convection-radiation model . Commun. Math. Sci . 11 ( 3 ), 679 { 707 ( 2013 )

5. Kovtanyuk , A.E. , Chebotarev , A.Yu. , Botkin , N.D. , Ho mann, K.-H.: Optimal boundary control of a steady-state heat transfer model accounting for radiative e ects . J. Math. Anal. Appl . 439 ( 2 ), 678 { 689 ( 2016 )

6. Dombrovsky , L.A. : The use of transport approximation and di usion-based models in radiative transfer calculations . Computational Thermal Sciences 4 ( 4 ), 297 { 315 ( 2012 )

7. Andrienko , D.A. , Surzhikov , S.T.: P1 approximation applied to the radiative heating of descent spacecraft . J. Spacecraft Rockets 49 ( 6 ), 1088 { 1098 ( 2012 )

8. Grenkin , G.V. : Optimal control in a nonstationary complex heat transfer problem . Dal'nevost. Mat. Zh . 14 ( 2 ), 160 { 172 ( 2014 )

9. Grenkin , G.V. , Chebotarev , A.Yu. , Kovtanyuk , A.E. , Botkin , N.D. , Ho mann, K.-H.: Boundary optimal control problem of complex heat transfer model . J. Math. Anal. Appl . 433 ( 2 ), 1243 { 1260 ( 2016 )

10. Grenkin , G.V. : An algorithm for solving the problem of boundary optimal control in a complex heat transfer model . Dal'nevost. Mat. Zh . 16 ( 1 ), 24 { 38 ( 2016 )

11. Kovtanyuk , A.E. , Chebotarev , A.Yu. , Botkin , N.D. , Ho mann, K.-H.: Theoretical analysis of an optimal control problem of conductive-convective-radiative heat transfer . J. Math. Anal. Appl . 412 ( 1 ), 520 { 528 ( 2014 )

12. Kovtanyuk , A.E. , Chebotarev , A.Yu. , Botkin , N.D. , Ho mann, K.-H.: Unique solvability of a steady-state complex heat transfer model . Commun. Nonlinear Sci. Numer . Simul. 20 ( 3 ), 776 { 784 ( 2015 )