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Vestnik KRAUNC. Fiz.-Mat. nauki. 2022. vol. 39. no. 2. P. 20-31. ISSN 2079-6641
MSC 35K05, 35K15 Research Article
On the control problem associated with the heating process in
the bounded domain
F. N. Dekhkonov1'2
1 National University of Uzbekistan named after Mirzo Ulugbek, 100174, University str., 4b, Tashkent, Uzbekistan
2 Institute of Mathematics named after V. I. Romanovskiy, Academy of Sciences of the Republic of Uzbekistan, 100174, University str., 4b, Tashkent, Uzbekistan
E-mail: [email protected]
The initial-boundary problem for the heat conduction equation inside a bounded domain is considered. It is supposed that on the boundary of this domain the heat exchange takes place according to Newton's law. The control parameter is equal to the magnitude of output of hot air and is defined on a givenmpart of the boundary. Then we determined the dependence T(0) on the parameters of the temperature process when 0 is close to critical value.
Key words: heat conduction equation, admissible control, initial-boundary value problem, integral equation.
d DOI: 10.26117/2079-6641-2022-39-2-20-31
Original article submitted: 01.07.2022 Revision submitted: 10.08.2022
For citation. Dekhkonov F. N. On the control problem associated with the heating process in the bounded domain. Vestnik KRAUNC. Fiz.-mat. nauki. 2022,39: 2, 20-31. d DOI: 10.26117/2079-6641-2022-39-2-20-31
The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)
© Dekhkonov F. N., 2022
1 Introduction
Consider in the bounded domain O c R3 with piecewise smooth boundary 9O the heat conduction equation
ut(x,t)= Au(x,t), x e O, t > 0, (1)
with boundary conditions
3u
— + h(x)u(x,t) = 0, x e 9O \ r, t>0, (2)
on
Funding. The work was done without financial support.
9u
— = a(x) M-(t), x E r, t>0, (3)
on
and initial condition
u(x,0)= 0. (4)
Here r is some subset of 3O (heater or air conditioner) with piecewise smooth boundary 3F and with mesF> 0 (we denote by mesF the surface measure of F, distinct from Lebesgue measure |F| ).
We suppose that h(x) (thermal conductivity of the walls) and a(x) (the density of the power of the heater or air conditioner) are given piecewise smooth non-negative functions, which are not identically zero. The condition (3) means that there is a blast of hot (or cold) air with magnitude of output given by a measurable real-valued function |x(t), and condition (2) means that on the surface 3O a heat exchange takes place according to Newton's law (see, e.g. [16], Sect. III.1.4).
We may extend both functions h(x) and a(x) to the whole boundary 3O by setting h(x) = 0 for x e r, and a(x) = 0 for x E F. In this case we may write the conditions (2) and (3) in the following form
9u|X,t) + h(x)u(x,t) = a(x) ^(t), x E 3O, t>0. (5)
3n
By the solution of the initial boundary value problem (1)-(5), we mean the generalized solution defined in [13] (see Chapter III, Sec. 5).
Let M > 0 be some given constant. We say that the function |x(t) is an admissible control if this function is measurable on the half line t > 0 and satisfies the following constraint
Wt)| < M, t > 0. (6)
Let the function p : O —> R satisfies conditions
p(x) dx = 1, p(x) > 0.
O
For any 0 > 0 consider the condition
u(x,t)p(x) dx = 0. (7)
O
Note that the weight function p(x) is not assumed to be strictly positive. In particular, the value (7) may be the average value over some subdomain of the main region O.
Denote by the symbol T(0) the minimal time required to reach the given value 0 by the average value of the temperature. This means that the equation (7) is fulfilled for t = T(0) and is not valid for t < T(0).
We present the critical value 0* such that for any 0 < 0* there exists the required admissible control |x(t) and corresponding value of T(0) < +oo, and for 0 > 0* the equality (7) is impossible.
The purpose of this work is to determine the dependence T(0) on the parameters of the temperature process when 0 is close to critical value.
We recall that the time-optimal control problem for partial differential equations of parabolic type was first investigated in [6] and [7]. More recent results concerned with this problem were established in [1], [2], [3], [4], [5], [10], [11]. Detailed information on the problems of optimal control for distributed parameter systems is given in [8] and in the monographs [9], [12] and [14].
To formulate the main result we describe some spectral properties of the corresponding self-adjoint extension of Laplace operator.
Consider the following eigenvalue problem for the Laplace operator
- Avk(x) = Akvk(x), x E O, (8)
with boundary condition
H(x)vk(x)= 0, x E 9O. (9)
on
Under assumptions made above this problem is self-adjoint in L2(O, dx) and there exists a sequence of eigenvalues {Ak} so that
Ai < A2 < ... < Ak —> +00, k^>oo.
The corresponding eigenfunctions form a complete orthonormal system {vk}kEN in L2(O,dx) and these functions belong to C(O), where O = OU 9O.
According to (8), we get
Ak = -(Avk,vk) =
|Vvk(x)|2dx +
|vk(x)|2h(x)da(x) > 0.
O 3O
If h(x) > 0 and h(x) = 0 then, Ai > 0. Indeed, assume that Ai = 0. Than the first eigenfunction is an harmonic function
Avi (x) = 0,
and, in accordance with the theorem of Giraud and Theorem I.5.II in the book [15], we may state that vi = 0.
According to the non-negative of the first eigenfunction (see, e.g. [17]) and from the orthogonality of the eigenfunctions vi and v2, we get
Ai < A2.
Recall that we consider the behavior of the function
U(t) =
u(x,t)p(x) dx, (10)
a
where the solution u(x,t) of the problem (1)-(4) depends on the control function |x(t).
Set
e* = m
and
[(-A)-1 p(x)]a(x) dff(x), (11)
i M b = v ■ (p,vi) Ai
vi(y)a(y)da(y). (12)
Theorem 1. Let 0* >0 be defined by equation (11). Then
1) for every 0 from the interval 0 < 0 < 0* there exist T(0) such that
U(t) < 0, 0<t<T(0),
and
U(T (0)) = 0.
2) for 0 —> 0* the following estimate is valid:
T(0) = to+ -^lnb + O(eA2-Al),
e(0) At
where
£ = |0* - 0|1/A1 .
3) for every 0 > 0* the T(0) does not exists.
2 The Main Integral Equation
We consider the following Green function:
oo
G(x,y,t) = Y_ e-AktVk(x)vk(y), x e O, y e O, t>0. k=1
This function is the solution of the initial-boundary value problem for the equation Gt(x,y,t) = AG(x,y,t), x e O, t>0, with boundary condition
dG(x,y,t) + h(x)G(x,y,t) = 0, x e 9O, t>0, on
and initial condition
G(x,y,0) = 5(x - y).
It follows from maximum principle that the Green function is non-negative (see, [1],
[3])
G(x,y,t) > 0, (x,y) e O x O, t > 0.
Set
H(x,t) =
p(y) G(x,y,t)dy, x g Q, t>0. (13)
a
It is clear that the function (13) is a solution of the following initial-boundary value problem:
Ht(x,t) — AH(x,t) = 0, x G O, t>0
9H(x,t) + h(x)u(x,t)= 0, x G dO, t>0, 9n
and
H(x,0) = p(x), x G O. In this using the spectral theorem in L2(O, dx) we may write
H(x,t) =
Obviously,
o
At
e-AtdEAp(x)
H(x,t) = (p,vi)e-Altvi(x) + Hi(x,t), t > 0, (14)
where
o
Pe-AtdEAp(x). (15)
Hi(x,t) =
Set
Ak =
vk(y)a(y)da(y).
Proposition 1. The following estimate is valid:
Ai =
vi (y)a(y)da(y ) > 0. (16)
Proof. Assume that this integral is equal to 0. Then on some surface Fi c F vi equals
0:
v(s)= 0, s e Fi.
It follows from (9) that
9v(s) 0 r = 0, S e ri.
Hence, vi (x) is a solution to homogeneous Cauchy problem and from the uniqueness of the solution vi (x) = 0, and this contradicts the assumption that vi (x) is an eigenfunction. □ Set
rr \ ^ vk(x)vk(y)
G2(x,y) = 2_--\2-. (17)
k=2 Ak
Proposition 2. The function Hi(x,t) satisfies the following estimate |Hi(x,t)| <|Ap|-^G2(x,x)e-A2t, t > 0,
uniformly in x E O.
Proof. From (15), we can write
Hi(x,t) =
Ä2
e-AtdEAp(x) = ^(P,vk)e-Aktvk(x), t > 0.
k=2
Then, we have
|Hi(x,t)|2 =
^(P,vk)e-Aktvk(x)
k=2
oo
<
<
^|(p,Vk)|2Ak W^e-2Akt|vk(x)|2Ak^, t > 0.
vk=2 ) \k=2
□
Then, we get the following estimate
|Hi(x,t)| <||Ap||-VG2(x,x)e-A2t.
Now we introduce the kernel of a main integral operator:
K(t) =
According to (14), we may write
H(y,t)a(y)da(y).
K(t) = (p,vi)e-Alt
vi(y)a(y)da(y) +
Hi(y,t)a(y)da(y) =
(18)
= Ai ■ (p,vi)e-Alt + ß(t)e-A2t,
where
|ß(t)| < B = ||Ap||
VG2(y,y) a(y)da(y).
(19)
The proof of the following Proposition 3 can be seen [1].
Proposition 3. The derivative of the kernel (18) satisfies the following estimates:
K'(t) = , 0 < t < i, t
and
K'(t) =-AiAie-Al 1 + O(i)e-A2t, t > i.
where Ai is defined by the equality (16).
It is well-known (see, e.g. [13]) that the solution of the initial-boundary value problem (1) + (4) + (5) may be represented by the Green function:
u( x, t) =
|j.(s) ds
G(x,y,t- s) a(y) da(y).
2
According to condition (10) we can write
p(x)u(x,t)dx =
a
((s)ds
0 sa
a(y)da(y )
p(x)G(x,y,t — s)dx = U(t)
a
Then, from (13) and (18), we get the following integral equation
p(x)u(x,t)dx =
K(t — s) ((s) ds = U(t).
a
3 Proof of the Theorem 1
Set
L(x,t) =
H(x,s) ds.
Then we can write
(20)
(21)
L(x,t) = ^(p,vk)vk(x) k=i
e—Ak s ds =
L
k=i
I — e—AAkt
Ak
e—Alt
(p,vk)vk(x) = (—A) 'p(x)---—(p,vi)vi(x) — Li(x,t),
Ai
where
00 e—Akt
Li (x,t) = V_ —— (p,vk)vk(x). Ak
k=2
We have the following estimate
|Li(x,t)| < e—^f L|(p,vk^^ g ^ i/2
k=2
Hence, Further,
|Li (x,t)| < e—II PU.
L(x,t)a(x) da(x) =
— A (P,vi)e—Alt Ai
[(—A)—ip(x)]a(x) da(x) — Li (x,t)a(x) da(x).
We introduce a specific heating as
(22)
Q(t) =
K(t - s)ds =
K(s)ds.
(24)
0 0
The physical meaning of this function is evident: Q(t) equals the average temperature of O in case where the heater is acting unit load. It is clear that Q(0) = 0 and Q'(t) = K(t) > 0. According to (18), we have
L(x,t)a(x) da(x) =
ds
H(x,s) a(x)da(x) =
0 r
K(s)ds = Q(t).
Set
Q* = lim Q(t) =
t—>00
K(s)ds.
(25)
(26)
Obviously, the average temperature of O in the case where the heater is acting with unit load cannot exceed Q*. Set
0* = MQ*. (27)
Then, according to (22) and (23)
0(t) = MQ(t) = 9* - be-Alt + O(e-Â21),
(28)
where b defined by (12).
According to (26)-(28), for every 0 from the interval 0 < 0 < 0* there exist T(0) such
that and
U(t) < 9, 0<t<T(9),
U(T (9)) = 9.
Proposition 4. There exist T(0) > 0 and a real-valued measurable function |x(t) so that |p.(t)| < M and the following equality
K(T - s)^(s)ds = U(T ),
(29)
is valid.
Proof. This follows from the properties of the function Q. Indeed, if we set |x(t) = M, then we have
K(t - s)^(s)ds = M
K(t - s)ds = MQ(t),
and because of (29) there exists T(0) >0 so that MQ(T) = U(T). □
Remark. It is clear that the value T(0), which was found in Proposition 4, gives a solution to the problem. Namely, T(0) is the root of the equation
Q ) = U(T) = 0 Q( ) M M
Proposition 5. Let f(r) be increasing on the interval (0, i] and for some b, (3 > 0
f(r) = br + O(ri+3). (30)
Then for inverse function r = f-i (s) the following estimate is valid:
In- = In - + ln b + O(s3). rs
Proof. According to (30),
s = br[i + a(r)], (31)
where
a(r) = O(r3). Note that f(r) > 0 on the interval 0 < r < i. Hence,
s > Cr, 0 < r < i.
Then
r(s) = f-i(s) < C ■ s,
and
r(s) = O(s).
Hence,
a(r(s)) = O(s3).
Then, according to (31),
ln^- = In + ln-—i—- = In -ln[i + a(r)] = s br i + a(r) br
= Ini + ln 1 + O(|a(r)|) = Ini -Inb + O(s3). r b r
□
Corollary. The following equality is true:
t = In--}t + -Unb + O (|0*-0(t)|(A2-Al)/Al
|0* - 0(t)|i/Ai Ai V
Indeed, according to (28),
0* - 0(t) = be-Al t + O(e-A2t). 28
Set
Then, we get
r = e-Alt, s = 0* - 0(t), ß = — - i.
Ai
e-A2t = e-A1t(i+ß) = ri+ß.
We can apply Proposition 5 and get
t = A-lne*--e(i)+ Ailnb + O (r- 9(t)|") •
Then, for 0 —> 0*, we have the following estimate
T(0) = to+ -^lnb + O(eA2),
e(0) Ai
where
£ = |0* - 0|i/Al .
The proof of Theorem 1 follows from Propositions 4 and 5.
Competing interests. The author declares no conflicts of interest with respect to authorship and publication.
Contribution and responsibility. The author has contributed to this article. The author is solely responsible for providing the final version of the article for publication.
Acknowledgements. The author is grateful to Academician Sh. A. Alimov for his valuable comments.
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Dekhkonov Farrukh Nuriddin ugliA - Ph.D. (Phys. & Math.), National University of Uzbekistan named after M.Ulugbek, Tashkent, Uzbekistan, ORCID 0000-0003-4747-8557.
Вестник КРАУНЦ. Физ.-мат. науки. 2022. Т. 39. №. 2. С. 20-31. ISSN 2079-6641
УДК 517.977.5 Научная статья
К задаче управления, связанной с процессом нагрева в
ограниченной области
Ф. Н. Дехконов1,2
1 Национальный университет Узбекистана имени Мирзо Улугбека, ул. Университетская, 4, Ташкент, 100174, Узбекистан
2 Институт математики имени И. В. Романовского, АН Уз, ул. Университетская, 4б, Ташкент, 100174, Узбекистан.
E-mail: [email protected]
Рассмотрена начально-краевая задача для уравнения теплопроводности внутри ограниченной области. Предполагается, что на границе этой области происходит теплообмен по закону Ньютона. Параметр управления равен величине выхода горячего воздуха и определяется на заданном участке границы. Затем определяли зависимость T (0) от параметров температурного процесса, когда 0 близко к критическому значению.
Ключевые слова: уравнение теплопроводности, допустимое управление, начально-краевая задача, интегральное уравнение.
d DOI: 10.26117/2079-6641-2022-39-2-20-31
Поступила в редакцию: 01.07.2022 Revision submitted: 10.08.2022
Для цитирования. Dekhkonov F. N. On the control problem associated with the heating process in the bounded domain // Вестник КРАУНЦ. Физ.-мат. науки. 2022. Т. 39. № 2. C. 20-31. DOI: 10.26117/2079-6641-2022-39-2-20-31
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Контент публикуется на условиях лицензии Creative Commons Attribution 4-0 International (https://creativecommons.org/licenses/by/4.0/deed.ru)
© Декхонов Ф. Н., 2022
Дехконов Фаррухжон Нуриддин углиА - кандидат физико-математических наук, Национальный университет имени Мирзо Улугбека, Ташкент, Узбекистан, СЖСГО 0000-0003-4747-8557.
Финансирование. Работа выполнена без финансовой поддержки.
