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10Математические заметки СВФУ Октябрь—декабрь, 2022. Том 29, № 4
UDC 517.9
ON THE CONTROL PROBLEM ASSOCIATED WITH THE HEATING PROCESS F.N. Dekhkonov
Abstract: In the papaer, we consider the initial-boundary problem for the heat conduction equation inside a bounded domain. On the part of the border of the considered domain, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered domain gets a given value. 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 or cold air and is defined on a given part of the boundary, and the weight function is not assumed to be strictly positive in the given domain. Then, we found the dependence T(9) on the parameters of the temperature process when 9 is close to critical value.
DOI: 10.25587/SVFU.2023.82.41.005
Keywords: heat conduction equation, control function, initial-boundary value problem, admissible control, integral equation.
1. Introduction
Consider in the bounded domain Q c the heat conduction equation
with piecewise smooth boundary dO
ut(x, t) = Au(x, t), x G 0,, t > 0,
with boundary conditions du
dn
+ h(x)u(x,t) = 0, x G d0 \ Г, t > 0,
du dn
a(x) ^(t), x G Г, t > 0,
(1)
(2)
(3)
and initial condition
u(x, 0)=0. (4)
Here T is some subset of dQ (heater or air conditioner) with piecewise smooth boundary dT and with mes T > 0 (we denote by mes T the surface measure of T, distinct from Lebesgue measure |T| ).
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
© 2022 F.N. Dekhkonov
function ^(t), and condition (2) means that on the surface dO a heat exchange takes place according to Newton's law (see, e.g. [1, , Sect. III.1.4]).
We may extend both functions h(x) and a(x) to the whole boundary dO by setting h(x) = 0 for x € r, and a(x) = 0 for x € r. In this case we may write the conditions (2) and (3) in the following form
+ h(x)u(x, t) = a(x) n(t), x€dil, t > 0. (5)
dn
By the solution of the initial boundary value problem (1)-(5), we mean the generalized solution defined in [2] (see Chapter III, Sec. 5).
Let M > 0 be some given constant. We say that the function ^(t) is an admissible control if this function is measurable on the half line t > 0 and satisfies the following constraint
|M(t)| < m, t > 0. (6)
Let the function p : il —> R satisfies conditions
/P(X) dX =1, P(X) > ^ n
For any 0 > 0 consider the condition
/U(x't)p(x) dX = 0. (7) n
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 ^(t) and corresponding value of T(0) < 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.
The difference between the problem under consideration and paper [3] is that in this work the weight function p(x) is not required to be exactly greater than zero in the given domain. Therefore, we considered the three-dimensional domain in this work. A special case of this problem is studied in [4, 5].
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 [3-5,8-10]. Detailed information on the problems of optimal control for distributed parameter systems is given in [11] and in the monographs [12-14].
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) = Ak vk(x), x e O, (8)
with boundary condition
^^+h(x)vk(x) =0, X€dil. (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 ^ k ^ ro.
The corresponding eigenfunctions form a complete orthonormal system {vk}keN in ¿2(0, dx) and these functions belong to C(f2), where il = 0 U dil.
It is well-known that the asymptotic behavior of the solution of the heat conduction equation mainly depends on the first (minimal) eigenvalue of the corresponding selfadjoint extension of Laplace operator (see, e.g. [1]).
We obtain an estimate of the minimal time of heating by the characteristics of the first eigenfunction v1. According to (8), we get
Ak = -(Avk,vk) = / |Vvk(x)|2 dx + J |vk(x)|2h(x) da(x) > 0. n an
If h(x) > 0 and h(x) ^ 0 then A1 > 0. Indeed, assume that A1 = 0. Then the first eigenfunction is an harmonic function
Av1(x) = 0,
and, in accordance with the theorem of Giraud and Theorem I.5.II in the book [15], we may state that v1 = 0.
According to the non-negative of the first eigenfunction (see, e.g. [16]) and from the orthogonality of the eigenfunctions v1 and v2, we can write
A1 < A2.
Recall that we consider the behavior of the function
U (t) = / "(x'i)p(x) dx- (10)
n
where the solution u(x, t) of the problem (1)-(4) depends on the control function M(i).
Set
r = mJ[(-A)-1p(x)]a(x) da(x), (11)
r
and
M f
b = — -(p,v1)Jv1(y)a(y)dcT(y). (12)
1r
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(e)=ln^- + ^-lnb + 0(eX2-Xl),
e(0) Ai
where e = |0* - 0|1/Al;
(3) for every 0 > 0* the T(0) does not exists.
The proof of theorem we give step by step as propositions.
2. The main integral equation
We consider the following Green function:
G(x, y,t) = ^ e-Afctvk(x)vk(y), x € O, y € O, t> 0.
k=i
This function is the solution of the initial-boundary value problem for the equation
Gt(x, y, t) = AG(x, y, t), x € O, t > 0, with boundary condition
+ h(x)G(x, y,t) = 0, a; G i > 0,
dn
and initial condition
G(x, y, 0) = ¿(x — y).
Set
H(x,t)^y p(y)G(x,y,t) dy, x € O, t> 0. (13)
n
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 € O, t > 0,
+ h(x)H(x, t) = 0, x€dil, t> 0,
dn
and
H(x, 0) = p(x), x € O. In this using the spectral theorem in L2(O, dx) we may write
cc
Obviously,
H(x,t) = (p,vi)e-Altvi(x) + Hi(x,t), t > 0, (14)
where
to
Hi(x,t)^y e-At dEAp(x). (15)
A2
Set
Ak = J vk(y)a(y)
r
Proposition 1. The following estimate is true:
Ai = J vi(y)a(y)da(y) > 0. (16)
r
Proof. Assume that this integral is equal to 0. Then on some surface ri C T vi equals 0:
v(s) = 0, s e Ti.
It follows from (9) that
^ = 0, sGr,
dn
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.
Proposition 1 proved.
Set
G2(x,y) = 2_-72-• (17)
k=2 Ak
Proposition 2. The function H (x, t) satisfies the following estimate: I Hi (x, t)\ < ||Ap||v/G,2(x,x)e-A2t, t> 0,
uniformly in x € f2.
Proof. From (15) we can write
/TO
e-At dEAp(x) = ^(p,vk)e-Afctvk(x), t > 0.
7--O
k=2
A 2
Then we have
|Hi(x,t)|
2
¿(P, Vk)e-Ak*vk (x)
k=2
TO
< El(P'Vk)|2Ak Ee-2Afct|vk(x)|2A-M , t > 0.
k=2 k=2
2
Then we get the following estimate:
|ffi(M)l < ||Ap|| \JG2(a;, x)e~X2t. Proposition 2 proved.
Now we introduce the kernel of a main integral operator:
K (i) = y H (y,t)a(y)da(y). (18)
r
According to (14), we may write
K(t) = (p,vi)e-Xlt J vi (y)a(y) d^(y) + J Hi(y,t)a(y) da(y) rr
= Ai ■ (p,vi)e-Alt + ^(i)e-A2t, (19)
where
|/3(i)| < B = ||Ap|| J ^G^y)a(y)d<j(y).
r
The proof of the following Proposition 3 and Proposition 4 can be seen [8].
Proposition 3. The derivative of the kernel (18) satisfies the following estimates:
K'it) = °U, 0<id,
vi
and
K'(t) = -AiAie-Alt + O(1)e-A2t, t > 1.
where Ai is defined by the equality (16).
Proposition 4. Let u(x, t) be the solution of the initial-boundary value problem (1)-(4). Then the following equality
t
u(x, t) = J ^(s) ds J G(x,y,t — s)a(y) da(y), o r
is valid.
According to condition (10) we can write t
' ) ' ^ a(
o o do o
Then, from (13) and (18), we get the following integral equation:
t
p(x)»(x, t) dx = J ^(s) ds J a(y) d<r(y)y p(x)G(x,y,t — s) dx = U(t)
t
/^»m =/ k(,—ds=u(i). ,20,
3. Proof of Theorem 1
Set
t
0
Then we can write
t
c f c 1 _ e-Afct L(x,t) = y2{p,vk)vk{x) / e~AfeSds = V^---(p,vk)vk(x)
k=i 0 k=i Ak
e-Ait
= --t—(p, vi)vi(x) - Li{x, t),
Ai
where
t„ (t. t) ='S^ -
Ak
Li{x,t) = —{p,vk)vk(x).
k=2
We have the following estimate:
( c \ i/2 ( » i r M2 x i/2
|Ll(M)l<e-A2t £l(/^)|2
A2
k=2 / \k=2 k
Hence,
Further,
|L,(i,()| < t-^Vftl'.J=)M- (22)
/ L{x,t)a{x) ^ /^ArVMW.)
- -J L1(x,t)a(x)da(x). (23)
rr
A
r
We introduce a specific heating as
t t Q(t) = J K(t - s)ds = J K(s) ds. (24)
00
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 (see, e.g. [3, 8]). It is clear that Q(0) = 0 and Q'(t) = K(t) > 0. According to (18), we have
t t J L(x,t)a(x) d<r(x) = J ds J H(x,s)a(x) d<r(x) = J K(s) ds = Q(t). (25) r 0 r 0
Set
cc
Q* = lim Q(t) = i K(s) ds. (26)
t^C J
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) = 0* — be-Alt + O(e-A2t), (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
U(t) <0, 0 < t < T(0), (29)
and we may write
U(T(0)) = 0. (30)
Proposition 5. There exist T(0) > 0 and a real-valued measurable function ^(t) so that |^(t)| < M and the following equality
T
J K(T — s)^(s) ds = U(T), (31)
0
is valid.
Proof. This follows from the properties of the function Q. Indeed, if we set ^(t) = M, then
t t J K(t — s)^(s) ds = M J K(t — s) ds = MQ(t), 00
and because of (31) there exists T(0) > 0 so that MQ(T) = U(T). Proposition 5 proved.
Remark. It is clear that the value T(0), which was found in Proposition 5, gives a solution to the problem. Namely, T(0) is the root of the equation
Proposition 6. Let f (r) be increasing on the interval (0,1] and for some ft > 0
f (r) = br + O(r1+^). (33)
Then for inverse function r = f-1(s) the following estimate is valid:
In- = In- + In b + Of/), (34)
rs
where b defined by (12).
Proof. According to (33),
s = br[1+ a(r)], (35)
where
a(r) = O(r^ ). (36)
Note that f (r) > 0 on the interval 0 < r < 1. Hence,
s > Cr, 0 < r < 1. (37)
Then
and Hence,
Then, according to (35),
r(s) = r\3) < ^ • S,
r(s) = O(s). (38)
a(r(s)) = O(s^ ).
In - = In + In -- = In -p— ln[l + a(r) 1
s or 1 + a(r) or
= In i + In i + 0(|a(r)|) =ln- -lnb + O^). (39) r o r
Proposition 6 proved.
Corollary. The following equality is true:
t = In |g._g(t)|1/Al + £ In 6 + 0(|r - WI^-AO/Ax ). (40)
Indeed, according to (28),
0* — 0(t) = be-Alt + O(e-A2t). (41)
Set
r = « = ft* - fi(t\ « =
Ai
r = e-Xlt, s = 6*-6{t), /3=^-1. (42)
Then
e-A2t = e-Ait(i+^) = ri+^.
We can apply Proposition 6 and get
Then, for 0 ^ 0*, we have the following estimate:
11
T(0)=\n— + — \nb + 0(eX2-Xl), e(0) Ai
where
e = |0* — 0|i/Al. The proof of Theorem 1 follows from Propositions 5 and 6.
Acknowledgements. The author is grateful to Professor S. A. Alimov for his valuable comments.
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Submitted August 12, 2022 Revised August 19, 2022 Accepted November 29, 2022
Farrukh N. Dekhkonov
National University of Uzbekistan
4, University street, Tashkent 100174, Uzbekistan
f.n.dehqonov@mail.ru
