=Paper=
{{Paper
|id=Vol-1623/papermp8
|storemode=property
|title=Optimal Control for Radiative Heat Transfer Model with Monotonic Cost Functionals
|pdfUrl=https://ceur-ws.org/Vol-1623/papermp8.pdf
|volume=Vol-1623
|authors=Gleb Grenkin, Alexander Chebotarev
|dblpUrl=https://dblp.org/rec/conf/door/GrenkinC16
}}
==Optimal Control for Radiative Heat Transfer Model with Monotonic Cost Functionals==
https://ceur-ws.org/Vol-1623/papermp8.pdf
Optimal Control for Radiative Heat Transfer Model
with Monotonic Cost Functionals
Gleb Grenkin1,2 and Alexander Chebotarev1,2 ?
1
Far Eastern Federal University,
Sukhanova st. 8, 690950 Vladivostok, Russia,
2
Institute for Applied Mathematics,
Radio st. 7, 690041 Vladivostok, Russia
{glebgrenkin@gmail.com,cheb@iam.dvo.ru}
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.
Sufficient conditions of optimality are derived and the convergence of a simple
iterative method is shown.
Keywords: optimal control; radiative heat transfer; conductive heat transfer,
sufficient optimality conditions, simple iterative method, bang-bang
1 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 diffusion 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
Optimal Control for Radiative Heat Transfer Model 211
monotonicity condition allows to obtain sufficient 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 field,
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 efficient 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 Ω ⊂ R3 has the following form [9]:
∂θ/∂t − a∆θ + bκa (|θ|θ3 − ϕ) = 0, −α∆ϕ + κa (ϕ − |θ|θ3 ) = 0, (1)
a∂n θ + β(θ − θb )|Γ = 0, α∂n ϕ + u(ϕ − θb4 )|Γ = 0, (2)
θ|t=0 = θ0 . (3)
Here, θ is the normalized temperature, ϕ the normalized radiation intensity averaged
over all directions, κa the absorption coefficient, and θb the boundary temperature
taken in Newton’s law of cooling. The parameters a, b, and α are positive constants, and
β = β(x), u = u(x, t), x ∈ Γ , t ∈ (0, T ) are positive functions. The control parameter
u depends on the emissivity ε of the boundary surface as follows: u = ε/2(2 − ε). The
symbol ∂n denotes the derivative in the outward normal direction n on the boundary
Γ := ∂Ω.
Define 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 ∈ Uad , θ, and ϕ which satisfy the conditions
(1)-(3) and minimize (or maximize) an objective functional J(θ, ϕ) which is assumed to
be monotonic. The precise definition of monotonicity will be given in the next section.
3 Formalization of the optimal control problem
Suppose that Ω is a Lipschitz bounded domain, Γ = ∂Ω, Σ = Γ ×(0, T ), Q = Ω×(0, T ),
and the model data satisfy the following conditions:
(i) β ∈ L∞ (Γ ), u1 , u2 , θb ∈ L∞ (Σ), 0 < β0 ≤ β, 0 < u0 ≤ u1 ≤ u2 , β0 , u0 = const,
θb ≥ 0;
(ii) 0 ≤ θ0 , ϕ0 ∈ L∞ (Ω).
212 G. Grenkin, A. Chebotarev
Denote H = L2 (Ω), V = H 1 (Ω). Note that V ⊂ H = H 0 ⊂ V 0 . Let the value of
a functional f ∈ V 0 on an element v ∈ V be denoted by (f, v), and (f, v) is the inner
product in H if f and v are elements of H. Define the space W = {y ∈ L2 (0, T ; V ) : y 0 ∈
L2 (0, T ; V 0 )}, y 0 = dy/dt, as well as the space of states Y = W × L2 (0, T ; V ) and the
space of controls U = L2 (Σ), Uad = {u ∈ U : u1 ≤ u ≤ u2 }.
Definition 1. A pair {θ, ϕ} ∈ Y is called weak solution of the problem (1)–(3), that
corresponds to the control u ∈ Uad , if the following equalities are fulfilled for any
v, w ∈ V a.e. on (0, T ):
Z
(θ0 , v) + a(∇θ, ∇v) + β(θ − θb )vdΓ + bκa (|θ|θ3 − ϕ, v) = 0, (4)
Γ
Z
α(∇ϕ, ∇w) + u(ϕ − θb4 )wdΓ + κa (ϕ − |θ|θ3 , w) = 0, (5)
Γ
and θ|t=0 = θ0 .
Theorem 1. (cf. [9]) Let the conditions (i), (ii) be satisfied. For any u ∈ Uad the
problem (1)–(3) has a unique weak solution {θ, ϕ}, and the following inequalities are
fulfilled: 0 ≤ θ ≤ M , 0 ≤ ϕ ≤ M 4 , where M = max{kθb kL∞ (Σ) , kθ0 kL∞ (Ω) }.
Definition 2. The cost functional J : Y ∩ [L∞ (Q)]2 → R is called monotonic, if, given
any 0 ≤ θ1 ≤ θ2 , 0 ≤ ϕ1 ≤ ϕ2 a.e. in Q, we have J(θ1 , ϕ1 ) ≤ J(θ2 , ϕ2 ).
Next we state two optimization problems not depending on a specific monotonic
cost functional.
b ∈ Uad such that for any u ∈ Uad we have θb ≤ θ, ϕ
Problem 1. Find u b ≤ ϕ a.e. in Q.
b ∈ Uad such that for any u ∈ Uad we have θb ≥ θ, ϕ
Problem 2. Find u b ≥ ϕ a.e. in Q.
Here, θb = θ(bu), ϕ
b = ϕ(bu), θ = θ(u), ϕ = ϕ(u). A weak solution of the problem
(1)–(3), corresponding to the control u ∈ Uad , is denoted by {θ(u), ϕ(u)}.
Remark 1. It is readily seen that solutions of problems 1 and 2 are solutions of optimal
control problems J(θ, ϕ) → inf and J(θ, ϕ) → sup, respectively, where J is monotonic.
Definition 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 Γ1 ⊂ Γ is a
part of the boundary, on which u is given that is u = u1 = u2 on Γ1 . The functional
represents the energy outflow through Γ1 :
Z TZ
β(θ − θb ) + bu1 (ϕ − θb4 ) dΓ dt.
J(θ, ϕ) =
0 Γ1
Note that our goal is to minimize (or maximize) the temperature and radiative
intensity fields in the entire domain and time interval. Therefore, the answer will be
the same for any monotonic cost functional.
Optimal Control for Radiative Heat Transfer Model 213
4 Optimality conditions
Lemma 1. Let u, u e ∈ Uad , θ = θ(u), ϕ = ϕ(u), θe = θ(e
u), ϕe = ϕ(e
u), and one of the
following conditions
( is satisfied: (
u1 , if ϕ − θb4 < 0, e − θb4 < 0,
u1 , if ϕ
a) u = 4
b) u =
u2 , if ϕ − θb > 0; e − θb4 > 0.
u2 , if ϕ
Then ϕ ≤ ϕ,e θ ≤ θe a.e. in Q.
Proof. Set θ = θ− θ,
e ϕ = ϕ− ϕ e and define the functions η = max{θ, 0}, ψ = max{ϕ, 0}.
Set v = η, w = ψ in (4), (5) and integrate in t. We obtain
Z t Z
1
kη(t)k2 + ak∇ηk2 + βη 2 dΓ + bκa (θ + θ)(θ
e 2 + θe2 )η, η dτ =
2 0 Γ
Z t Z t
= bκa (ϕ, η)dτ ≤ bκa (ψ, η)dτ, (6)
0 0
Z t Z Z
2 2 4 2
αk∇ψk + eψ dΓ + (u − u
u e)(ϕ − θb )ψdΓ + κa kψk dτ =
0 Γ Γ
Z t Z t
2 2 e 2 + θe2 )η, ψ dτ.
= κa (θ + θ)(θ + θ )θ, ψ dτ ≤ κa
e e (θ + θ)(θ (7)
0 0
Condition a) implies that the third term in the left-hand side of (7) is nonnegative.
Z t Z t
2
Thus, we obtain the estimate kψ(τ )k dτ ≤ C1 kη(τ )k2 dτ . Then (6) yields the
0 0
estimate Z t
kη(t)k2 ≤ C2 kη(τ )k2 dτ.
0
It follows from Gronwall lemma that η = ψ = 0, and so θ ≤ θ,e ϕ≤ϕ
e a.e. in Q.
The statement for condition b) can be proven similarly. t
u
The following theorem follows from Lemma 1 and establishes a sufficient condition
of optimality.
Theorem 2. Let u ∈ Uad , ϕ = ϕ(u), and
(
u1 , if ϕ − θb4 < 0,
u=
u2 , if ϕ − θb4 > 0.
Then u is a solution of the problem 1.
Similar arguments lead to
Theorem 3. Let u ∈ Uad , ϕ = ϕ(u), and
(
u1 , if ϕ − θb4 > 0,
u=
u2 , if ϕ − θb4 < 0.
Then u is a solution of the problem 2.
214 G. Grenkin, A. Chebotarev
Next prove the uniqueness of the strong optimal control.
Theorem 4. If u and u e a.e. in {(x, t) ∈
e are strong optimal controls, then u = u
Σ : ϕ(x, t) 6= θb4 (x, t)}.
Proof. By definition, ϕ(u) = ϕ(e u) = ϕ, θ(u) = θ(e
u) = θ a.e. in Q. It follows from (5)
that
Z Z
4 4
e)(ϕ − θb4 )vdΓ = 0 ∀v ∈ V a.e. on (0, T ).
u(ϕ − θb ) − u
e(ϕ − θb ) vdΓ = (u − u
Γ Γ
Hence (u − ue)(ϕ − θb4 ) = 0 a.e. on Σ, therefore, u = u
e a.e. in {(x, t) ∈ Σ : ϕ(x, t) 6=
4
θb (x, t)}. t
u
Remark 2. It follows from (4), (5) that an arbitrary modification of a strong optimal
control u in the set {(x, t) ∈ Σ : ϕ(x, t) = θb4 (x, t)} keeps the optimality of the control
u, because such modification does not influence 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.
Define the operator U : L∞ (Σ) → L∞ (Σ):
(
u1 , if ϕ − θb4 < 0,
U (ϕ) =
u2 , if ϕ − θb4 ≥ 0.
If u ∈ Uad and
U (ϕ(u)) = u, (8)
then, by Theorem 2, u is a strong optimal control.
Consider the simple iterative method for solving the equation (8). Choose an arbi-
trary initial guess u0 ∈ Uad . The iterative algorithm is as follows: uk+1 = U (ϕk ) where
ϕk = ϕ(uk ), k = 0, 1, . . ..
It follows from Lemma 1 that ϕk+1 ≤ ϕk (k = 0, 1, . . .) a.e. in Q. Thus, uk+1 ≤ uk
(k = 1, 2, . . .) a.e. on Σ. Taking into account that these sequences are bounded, we
obtain that uk → u∗ a.e. on Σ, ϕk → ϕ∗ a.e. in Q.
Lemma 2. ϕ∗ = ϕ(u∗ ) a.e. in Q.
Proof. Applying Lebesgue theorem, we obtain that uk → u∗ in L2 (Σ), ϕk → ϕ∗ in
L2 (Q).
It is easy to prove that the operator ϕ : L2 (Σ) → L2 (Q), defined on the set Uad , is
continuous. Therefore, ϕ∗ = ϕ(u∗ ) a.e. in Q. t
u
Lemma 3. u∗ = U (ϕ∗ ) a.e. on Σ.
Proof. Because ϕk+1 ≤ ϕk a.e. in Q, we have uk+1 = U (ϕk ) → U (ϕ∗ ) a.e. on Σ, and
so u∗ = U (ϕ∗ ) a.e. on Σ. t
u
Optimal Control for Radiative Heat Transfer Model 215
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 {θ, ϕ} of problem (1)–(3) be computed with absolute error
ε that is |ϕ − ϕ| e ≤ ε in Q, where ϕ
e is a component of the approximate solution. Then
the strong optimal control is determined ambiguously in the set {(x, t) ∈ Σ : |ϕ(x, t) −
θb4 (x, t)| ≤ ε}.
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 θ. Set θb =
0.4 at x = 0, and θb = 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], α = 3.3 . . . [cm],
κa = 0.01 [cm−1 ], and β = 10 [cm/s]. The initial function is θ0 (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 finite difference 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 {θ, ϕ} 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 in-
stants, and the minimum and maximum intensities of radiation are shown in Fig. 4.
Notice that the maximum and minimum fields 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.
216 G. Grenkin, A. Chebotarev
0.5
0.4
0.3
u(L, t)
0.2
0.1
0
0 5 10 15 20 25 30
t
Fig. 1. Strong optimal control in problem 1
0.5
0.4
0.3
u(L, t)
0.2
0.1
0
0 5 10 15 20 25 30
t
Fig. 2. Strong optimal control in problem 2
0.75
0.7
0.65
0.6
θ
0.55
0.5
min (t = 10)
max (t = 10)
min (t = 30)
0.45
max (t = 30)
min (t = 60)
max (t = 60)
0.4
0 10 20 30 40 50
x
Fig. 3. Minimum and maximum temperatures at t = 10, 30, and 60
Optimal Control for Radiative Heat Transfer Model 217
0.24
0.22
0.2
0.18
0.16
0.14
ϕ
0.12
0.1
0.08
min (t = 10)
max (t = 10)
0.06 min (t = 30)
max (t = 30)
0.04 min (t = 60)
max (t = 60)
0.02
0 10 20 30 40 50
x
Fig. 4. Minimum and maximum radiative intensities at t = 10, 30, and 60
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