CC BY
15
MSC 35K70, 35Q80
DOI: 10.14529/mmp170204
PARAMETER IDENTIFICATION AND CONTROL IN HEAT TRANSFER PROCESSES
S. G. Pyatkov, O. V. Goncharenko
Yugra State University, Khanty-Mansyisk, Russian Federation E-mail: S_pyatkov@ugrasu.ru, ovg@ugrasu.ru
The article is devoted to the study of some mathematical models describing heat transfer processes. We examine an inverse problem of recovering a control parameter providing a prescribed temperature distribution at a given point of the spatial domain. The parameter is a lower order coefficient depending on time in a parabolic equation. This nonlinear problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. Existence and uniqueness theorems of solutions to this problem are stated and proved. Stability estimates are exposed. The main result is the global (in time) existence of solutions under some natural conditions of the data. The proofs rely on the maximum principle. The main functional spaces used are the Sobolev spaces.
Keywords: heat transfer; distributed control; mathematical model; parabolic equation; inverse problem; boundary value problem.
Introduction
We study the problem of recovering a lower order coefficient depending on time together with a solution in heat transfer mathematical models. This control parameter allows to ensure a given temperature distribution at a given point of spatial domain. Let G be a bounded domain in Rn with boundary r and Q = (0,T) x G. The mathematical model can be written as
ut — Lou + p(t)u = f (x,t), (t,x) e Q, (1)
n n
Lou = Aou + Bou,Aou = ^ 8Xi(aj(x)uXj), -Bou = ^ai(x)uXi + ao(x)u.
i,j=l i=l
Equation (1) is furnished with the initial and boundary conditions
u|t=o = uo, Bu\s = g(t,x), S =(0,T) x r, (2)
n
Bu = aj(x)niuXj + b(x)u) + (1 — a)u,
i,j=l
ni is the i-th coordinate of the unit outward normal to r, a(x) e C(r) is a continuous function taking two values 0, 1. Thus, on different connectedness components the boundary condition can be of different type (Dirichlet, Neumann, or Robin boundary condition). Let r = {x e r : a(x) = i} (i = 0,1) Si = (0,T) x r^ The unknowns in (1), (2) are the solution u and the function p(t). The overdetermination conditions are written as
u(x0 ,t) = ^(t). (3)
where ip(t) is some function specified below.
Determination of a single unknown time-dependent property such as the capacity, conductivity or diffusivity from additional local or non-local measurements of the main dependent variable at the boundary or inside the space domain represents a classical example of a coefficient identification problem (see, for instance, [1,2]). Problem (1) - (3) is classical and was studied by many authors. Numerical methods of solving the problem are developed in [3-5]. For local (in time) solvability results, see, for instance, [6,7]. The book [8, Ch. 6] contains some abstract theory of such problems and its applications. In particular, the conditions for a local (in time) solvability of (1) - (3) are presented in Corollary 9.4.2 of [8]. Moreover, similar result is also exposed in [9]. Some results on close inverse problems can be found in [10-13]. Inverse problems with integral overdetermination conditions are studied in [14-18]. The structure of the paper is as follows. In Section 1 we formulate our results. The main result is Theorem 4 which is a global (in time) existence and uniqueness theorem for solutions to the problem (1) - (3). Some stability estimates are given in Theorem 3. The solvability conditions are stated in terms of some inequalities and the proof relies on the maximum principle. Section 2 is devoted to the proofs of the main results.
1. Preliminaries
Given Banach spaces X, Y, the symbol L(X,Y) stands for the space of linear continuous operators defined on X with values in Y. Let E be a Banach space. By Lp(G; E) (G is a domain in Rn) we denote the space of strongly measurable functions defined on G with values in E endowed with the norm ||||u(x)||e||lp(g) [19]. We employ also the spaces Ck(G) comprising functions continuous in G with all their derivatives up to order k admitting continuous extensions on the closure G. The Sobolev space notations are conventional, i.e., Ws(G; E), Ws(Q; E), etc. (see the definitions in [19,20]). If E = C (E = R) or E = Cn (E = Rn) then the latter space is denoted by Ws(Q). Similarly, we use the notations W£(G) or Ck(G) rather than W£(G; E) or Ck (G; E). Thus, the membership u E Ws(G) (or u E Ck(G)) for a given vector-function u = (ul,u2,... ,uk) means that every of its component u belongs to Ws(G) (or Ck(G)). The norm of the vector is just the sum of the norms of the coordinates. Given an interval J = (0,T), put Ws'r(Q) = Ws(J; Lp(G)) n Lp(J; Wrp (G). Respectively, Ws'r(S) = Ws(J; Lp(r)) n Lp(J; Wrp (r)).
G
domain in Rn with boundary r E C2 (see the definition in [21, p. 17]). Expose the conditions on the data of the problem. All spaces below and the coefficients of equation (1) are assumed to be real. Fix p > n + 2 (this condition simplifies the arguments and it can be weakened). Let q = p/(p — 1) Denote (x0) = {y E Rn : \y — x01 <
The conditions on the coefficients of the operators L0, B are as follows:
a
E C\G), az,aa E Lp (G), b E C 1(r). (4)
The matrix {aj} is symmetric and the ellipticity condition
n
3Ô1 > 0 : J] al3tej > ¿lieI2 V£ E Rn, X E G, (5)
i,j=1
holds. The conditions on the data are of the form
Uo(x) E W2p-1/p(G), g(t,x) E Wl-1/2p,2-1/p(So) n wlp/2-1/2p>1-1/p(Si), (6)
f e Lp(Q), B(0,x)uo|r = g(0,x) Ух e Г, (7)
ио(х) > 0 (x e G), g(t,x) > 0, f > 0 ((t,x) e Q). (8) We also use some additional conditions
ф e W1(0,T), 382 > 0 : 1Ф(1)1 > 82 yt e [0,T], ф(0) = uo(xo), (9)
3^0 > 0: B&0(xo) С G, aij e W;(B&0(xo)), ahao e W1(B&0(xo)), (10)
Vuo(x) e W^-1/p(BSo(xo)). (11)
Assume that Qs = (0, T) x Bs(xo), QY = (0,y) x G. Present some auxiliary statements. Replace the equation (1) with the equation
Lu = ut — Lou = f (x,t), (t,x) e Q. (12)
Theorem 1. Assume that conditions (4) - (7) hold. Then there exists a unique solution to (2), (12) such that u e Wl'2(Q). Under the additional conditions (10), (11), a solution u possesses the property Vu e Wl'2(Qs) for a 11 8 < 8o. If condition (8) is valid then the uQ
Proof. If So = S or S1 = S then we can refer to the standard theorems on solvability (see, for instance, [21, Theorem 9.1, Ch. 4] in the case of the Dirichlet conditions or [21, Theorem 10.4, Ch. 8] in the case of more general boundary conditions). Examine our case. First we take homogeneous initial and boundary conditions in (2). Let D(A) = {u e W1,2(Q) : Buls = 0^. The claim in the сase of Lo = Ao results from Theorem 12.2 in [22]. In the general case the result is a consequence of Theorems 12.2 and 3.2 in [22]. To refer to Theorem 3.2 we need an additional estimate for the norm T;L(Ee сю,е0)) f°r some
в e (0,1) and p > max(l/(l — в),p), where Eo = Lp(G) and = (Eo,D(A))e^ is the space obtained by the real interpolation method (see the definitions [19]). Since the Bo t
\\Bou\\Lp(G) < c\\u\\ws(G), S < 2,
and use the embedding E^ С W£(G) for в > s/2 (which follows from the embedding (1.1) in [22] and Theorem 5.2 in [23]). This estimate results from the embedding theorems and conditions (4). We infer
\\Bou\\lp(G) < c\u\w^(G) < cl\\u\\ws(G), s e (1 + n/p, 2).
The claim of the theorem in the case of nonhomogeneous initial and boundary conditions follows from the conventional theorems on extension of the boundary conditions inside the domain (see, for instance, Theorem 7.3 in [24]).
The proof of the fact that a solution u possesses the property Vu e Wl'2(Qs) for all 8 < 8o under conditions (10), (11) is realized with the finite difference method with the use of Lemma 4.6 of Ch. 2 in [25]; it is similar to that in the proof of Theorem 1.1 in [9] or in the proof of Theorem 3.1 in [26].
Different maximum principles for parabolic equations can be found in [27]. Unfortunately, they are not applicable in our case. Under stronger conditions on coefficients
we can refer to the generalized maximum principle in [22, Theorem 17.1]. We use conventional arguments those involved in the proof of the maximum principle for
generalized solutions. Let u- = l — 0; Note that u- G Wl(Q) (see [21,
I 0, \iu(t,x) > 0. p j \
Sect. 4 of Ch. 2]). Moreover, we have that u-\s0 = 0 and u-\t=0 = 0. Multiply (11) by u-
G
we obtain that
1 d I (u-)2 dx + 5i Wu-\2 dx + I b(u-)2 dr — I g(t,x)u-dr —
2 dt JG Jg Jr Jr
fu dx+\ G Jg i=i
— fu dx + \ I y aiuxiu + a0\u \ dx\.
Since the data are of constant sign, we derive that 1 d
G G G
--— f (u )2 dx + 5^' \Vu \2 dx —\f aiuxiu + a0\u \2 dx\ + \ i b(u )2 dr\. (13) 2 dt Jg Jg Jg i=l Jr
All summands on the right-hand side of (13) are estimated similarly. We use the conditions on the data, the Holder inequality, embedding theorems, and interpolation inequalities. Estimate the summands on the right-hand side under the integral sign. We have
aiuxiu dx
G
— mnvu - il (G) iiu- h2p/(p-2)(G) —
— CiJi(t)lVu h2(G)llu ^(G), Ji(t)= hihp(G), s = n/p.
Next, the inequalities (see [19])
(14)
llullws(G) — C2lulsW2l(G)lul1L-2s(G), \ab\ — e^ + (e> 0), r + r = -
+ -^r (e> 0), - + i imply that the right-hand side of (14) is estimated as
4Vu-ll2L2 (G) + c(e)Ji(t)2p/(p-n)llu-ll2L2 (G),
where 2p/(p — n) — p (since p > n + 2,)- All summands on the right-hand side of (13) except for the last of them are estimated similarly. The estimate for the last summand is simpler. We have
\ i b(u-)2 dr\ — cllulUr — cillu-^(g) — ellVu-llL2(G) + c(e)lu-lL2(G) (si E (1/p, 1)).
In this case inequality (13) can be rewritten as
1 d 2 dt
2dt I (u-)2 dx + 5i \Vu-\2 dx — c2ellVu-llL2(g) + c(e)J0(t)lu-lL2(g),
GG
where c2, c(e) are some positive constants and J0 E Li(0,T). Choosing c2e = we arrive at the inequality
y'(t) — Jt)y(t), y(0) = llu- IIL2(G)\t=0 = 0. Hence, we can conclude that u- = 0 almost everywhere in Q, i.e., u > 0 a.e. in Q. n
Corollary 1. There exists Ao > 0 such that —Lo + A for all A > Ao is an isomorphism of the space {u e W£(G) : Bulr = 0} onto Lp(G) (see [22, Remark 3.1 (b)]).
In view of Corollary 1 we assume below that the problem Lou = f e Lp(G), Bulr = 0 has a unique solution u e W2 (G), otherwise we make the change оf variables u = vext in (1) and reduce the arguments to this case.
Theorem 2. Let conditions (4) - (7) hold. Then, for y > 0, a solution u e Wl'2(Q) to (2), (12) with homogeneous initial and boundary conditions (i.e., uo = 0, g(t,x) = 0) satisfies the estimate
\\ue-Yt\\wip*(Q) + lYl\\ue-Ythp(Q) < c\\fe-Ythp(Q).
Fix an arbitrary 83 < 8o. If conditions (10), (11) are fulfilled then the following estimate holds:
\\Vxue-Yt\\wi,2(Qs3) + lYl\\Vxue-Yt\\l„(qSs) <
c(\\Vxfe-Ythp(QSo) + \\fe-Yt\\lpq), Y > 0. The constant c in this estimates is independent of the parameter y > 0.
Proof. Let u = veYt. Equation (12) is transformed to the equation
vt — Lov + yv = e-Ytf (x, t) with vlt=o = 0, Bvls = 0.
Next, we refer to the estimate in Theorem 3.1 of [26] and make the inverse change of variables.
□
Remark 1. Generally speaking our reference to Theorem 3.1 in [26] is not exact, since the case of different boundary conditions on different connectedness components of the boundary is not treated there. However the proof of Theorem 3.1 remains valid in this case as well, since it is based on a partition of unity and local considerations.
Lemma 1. Let u(t) e W^(0,T) and u(0) = 0. Then there exists a constant c > 0 independent of y > 0 such that
c
\\e-Ytu\\Lp(o,T) < y\\e-ltut\\Lp(o,T). The proof is elementary and we omit it.
Denote by Ф a solution to (12), (2) assuming that conditions (4) - (7) are fulfilled. We impose the following additional constraints on the data:
фt < &t(t,xo)&.e. on (0,T), Lo<^ < 0 a.e. on Q, З84 > 0: ф(t) > 84 Vt e (0,T). (15)
Assume that Я(Ф) = \^\\ltoQ) + \\lx(qSo), в(t) = &(t)№(t), ф'(т) = ф(т) —
$(r,xo), r(t) = —в(t)e-f°в(т)d\ Ro(Ф) = \\r(t)\\Lp(o,T)-
Now we can state our main results. The former half of Theorem 3 below (the existence theorem) is known (see, for instance, [9]). However, we present here this formulation for completeness of the exposition.
Theorem 3. Let conditions (4) - (7) and (9) - (11) hold. Then there exists a constant Y0 — T such that on the segment [0,y0] there exists a unique solution (u,p) to (1) -(3) with the property u E Wji'2(QY0), Vu E Wi2((0}Y0) x Bs(x0))) for a 11 5 < 50, and p(t) E Lp(0,j0). Assume that ui(x), gi(t,x), fi(t,x),fti(t) (i = 1, 2) are two different collections of the data satisfying the conditions of the theorem and the functions $i (i =
1, 2) are solutions to (1), (2), where p(t) = 0. Denote r() = —pi(t)e-&pi(T) dT (fc(t) = (^(t)-*i((t'x0))), fix a number R > 0, and assume that R($i) + R0($i) — R (i = 1, 2). Then there exist numbers j0 and c0(R) > 0 such that there exist unique solutions (ui,pi) (i = 1, 2) to (1) - (3) on the time segment [0,y0] satisfying the inequalities
lp — p2hp(0'Y0) — c-0 (R)(IIPi — p2hp(0'Y0) + Hi — ^2IW1(0'Y0) +
xBS( (X0)).
Theorem 4. Let conditions (4) - (11), (15) hold. Then there exists a unique solution to (1) - (3) such that u E Wi2(Q), p(t) E Lp(0, T) and Vu E Wi2(Qs) for a 11 5 < 50. The p(t)
p > c0($t(t,x°) — A(t)) ^ e [0,T],
J - m ' L' J'
c0
Remark 2. As it is easily seen, the statement of Theorem 4 remains valid if we change all signs in inequalities (8), (15), i.e., the functions —u0, —g(t,x), —f (t,x), —$(t,x), —ft meet conditions (8), (15).
2. Proofs of the Main Results
Proof of Theorem 3. Let $ be a solution to (2), (12). Make the change of variables u = v + $. We obtain that
Lv + p(t)(v + $) = 0, v\t=0 = 0, v\s = 0, v(t,x0)= ft(t) — $(t,x0) = ft(t). Next, we make the following change of variables: v = ue-$°p(T) dT. We infer
Lu + p(t)e&p(T)dT$ = 0, u\t=0 = 0, u\s = 0, u(t,x0) = ft)(t)e&p(t)dT. (16) Put x = x0 b (16). We arrive at the equation
ftefop(T) dT + ft(t)p(t)e&p(T) dT = L0u(t,x0). Denote a(t) = ef°p(T) dT. The equations can be rewritten as
ft a + ft(t)a' = L0u(t,x0). (17)
Expressing the function a, we arrive at the equality
a(t) = e-Uo № dT + f L0 u(T,x) e-UT № d€ dr. (18) _J0 ft(T)_
gg Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming
& Computer Software (Bulletin SUSU MMCS), 2017, vol. 10, no. 2, pp. 51-62
Thus, we have
a(t) = -me-ft ^)dT + - № J ^¡T^ e-ft mdS dT, (19)
where the function u is a solution to the problem
Lu = -a'(t)<£, u\t=o = 0, Bu\S = 0. (20)
Hence, we infer u = L-1(-a'Denote ao(t) = a'(t). In view of (19), (18), we derive that
a0(t) = r(t) + S(a0(t)), a(t) = a0(t) dT + 1, (21)
Jo
S(ao) = ^^trX1 - № it LMT\X0) e- ft md' dT, u = L-1(-ao$). (22)
¡(t) Jo ¡(t)
Demonstrate that equation (21) is uniquely solvable in the class a0(t) E Lp(0, T). Estimate S
\Lou(t,Xo)\ < c\\Lou\\w£(BS3(x0)), s E (n/p, 1), is < So- (23)
Lo
the class CP(G) is a pointwise multiplier in W^(G) with s < p (see, for instance, the item 3.3.2 in [281), we have
\K UXiXjWw^ (x0)) < c\\u\\w2+s(BH (xo)) < Cl\\U\\w3{BH (xo))\u\lp(B;3 (x0))- (24)
Lower order summands are estimated similarly. Note that the embedding theorems ensure that ai,ao E C 1-n/p(Bs0(x0)). The definition of the norm in yields
\\aiuxHWw^B^ (xo)) < Cl\\ai\\ws(BS3 (xo)) \\uxi\\c(Bs3 (xo)) +
(25)
+cl\\ai\\c (Bs3 (xo))\\^Xi\\ws(BS3 (xo)) < c2\W\\wl+s(BS3 (xo)) <
<C||, ,11 (1+s)/3 || ii (3-s)/3
< C3\Mw3(BS3(xo))\U\ip(BS3(xo))-
Estimates (23) - (25) imply that there exist constants 9 E (0,1) and c4 > 0 such that \Lou(t,Xo)\ < C4\u\dw3(BS3 (xo ))\U\1L-(BS3 (xo))- (26)
Now we estimate the norm \\e-YtS(ao)\\Lp(0,T)■ The above definition of the operator S yields
\\e-YtS (ao)hp (o,T) < C5\\e-YtLou(t,Xo)hp (o,T) + d\\P K (o,T) £ e-YT\Lou(t )\dT <
< C7\\e-YtLou(t,Xo)\\lp(o,t)- ^27>
Inequalities (26), (27) and Lemma 1 imply the estimate
\\e-YtS(ao< C8\\e-Ytu\\lp^^(B^e^-^-^Q^) <
< Y-\\e Ytu\\Lp (o,T;w$(Bs3 (xo)) \\e ltut\\Lp(QS3)
Next, applying Theorem 2 and the definition of u, we obtain the estimate
IIe-YtS (a0)hp(0,T) — Y— (IIe-Yta0$hp(Q) + He-Yta0V$HLp(QSo)) —
— Y— IIe-Yta0hp(0,T)R($), (28)
where the constant ci0 is independent of 7 and the norms of the data, and it depends on the norms of the coefficients of the equation, the constants in embedding theorems, interpolation inequalities and T. Choose a constant y0 such that
R($) = 1/2. (29)
Y0
In this case we have the estimate
He-ltS (a0)hp(0,T) — \ IIe-Yta0hp(0,T)
for all y > Thus, the operator S is contractive in some equivalent norm of the space Lp(0,T) and, thereby, (21) is solvable with respect to the function a0 = a'(t). Obviously, this solution satisfies the estimate
Ile-Yta'hp(0T) — 2Hr(t)e-Ythp(0t), 7 > Y0.
In particular, we infer
la' HLp(0 T) — 2e'°T IIr(t)hp(0,T) = ci (R($),M$)). (30)
We can restore the function a = 1 + /0 a0(r) d^. Given a function a, find a solution u to
t0 — T
|| /' a'(t) —t||C([0M) — t0/qHa'hp(0,t0) — ci(R, R0). 0
Choose t0 so that tl0/qci(R, R0) = 3/4. In this case a > 1 — 3/4= 1/4 > 0 on [0,t0] and we can construct the function f0 p(r) dr = ln(1 + a(t)). Respectively, p(t) = a'(t)/(1 + a(t)). Obviously, p(t) E Lp(0, t0). Verify that the functions p(t),u(t) are a solution to the inverse problem (16). Integrating (19), we obtain (18) whose transformation validates equality (17) and, hence, ft a + ft(t)a' = L0u(t, x0) — a' $(t, x0). On the other hand, taking x = x0 in (20), we have u'Ut,x0) = L0u(t,x0) — a'$(t, x0). Hence, u'(t,x0) = (fta)' and thereby u(t,x0) = aft = 0p(T)dTft, i.e., equalities (16) hold. Proceed with stability estimates. Assume that we have two collections of the data ui(x),gi(t, x), fi(t, x),fti(t) (i = 1, 2) and functions $i (i = 1, 2) are solutions to (1), (2) with these data and p(t) = 0. The respective ai
ai(t) = n(t) + Si(ai(t)), i = 1, 2, Si u
problem
Thus, ui = L ^(—a'fei) and
SM) = LoU;{t\Xo) - ft(t) f LUT>Xo)e- ftm)d* dr.
Vi{t) Jo Vi(r)
Choose the parameter Yo as in (29), where we take the quantity R from the conditions of the theorem rather than R(§). Next we choose t0 as before inserting the quantity R rather than R0($^d R(§)- In this case a solution (ui,pi) to our problem satisfying the above initial and boundary data exists on the segment [0, y0] (y0 = t0). The corresponding functions a\,a2 satisfy the estimate
a%(t) > 1/4, \\at\\Wm,T) < c(R). (32)
We have
SM) - Si(a'2) = +L0U2(^ - -
-m /0 Lo(U1-$(T'X0) e- £ № dT- (33)
- 0 LO^2(T, xo)(^i^TT1 - dT-
Subtracting equalities (31) for i = 1, 2, we infer
Lux - Lu2 = L(ux - u2) = -(a'x(t) - a'2(t))&1 + d2($i - $2). Repeating the arguments those in the proof of (28), we obtain the inequality
\\e-Yt- U2)(t,xo)\\Lp(0Yo) < Y— (\\e-Yt(a[ - a>2)\bp(o^o)c(R) + +Cl(R)(\\$1 - ®2\\lUQ'*0) + \\У(Фх - ф2)\Ьто((0,7о)хБ5о(xo))))■
(34)
The claim of the theorem follows from (32) - (34), the equality — a'2 = r1 — r2 + S1 (al) — S1(a'2), and some simplest estimates.
□
Proof of Theorem 4■ Let the conditions of the theorem hold. As in Theorem 3 we reduce the problem to the study of equation (21) and justify its solvability. Since the norm S
method of successive approximations beginning with a0 = 0. Successive approximations are written as
ai(t) = r(t) + S (ai-1(t)). (35)
In view of (15), a1 = r(t) > 0. Assume that the function ai-1 is nonnegative. Demonstrate that a% is nonnegative too. The corresponding function w1 is a solution to (20), i.e., we have
LtJ = -ai-4, J\t=o = 0, Bui\s = 0.
Since condition (8) is fulfilled, by Theorem 1 Ф > 0 a.e. in Q. By condition L^ < 0 a.e. and ai-1 > 0 a.e. Hence, Lo(—ai-l$) > 0 a.e. Consider the problem
Lu* = -ai-1Lo§, w*|t=o = 0, Bw*\s = ai-1B<^\s = ai-1g(t,x) > 0.
By Theorem 1 a solution u* is nonnegative in Q. Define the function u = L0 lu*. We have uj|i=o = 0 Buj|s = 0 (by construction), uj G Wp, ' 2(Q), and L0u G Wp, ' 2(Q). Moreover, we can conclude that
u* — Lo(u* — ai~l<£) = 0.
Applying L-1 to this equality, we can claim that uj* — (L0u — a%~1^) = 0. Hence, ut — L0u = —In view of the uniqueness theorem, we derive that u% = u and thus L0u% = u* > 0 a.e. Consider equality (35) and recall the definition of S (see (22)):
S (ai~l) = L0^)0) — P (t) I * L0 ^r)0) e ft * T
Since P(t) < 0 every summand here is nonnegative and in view of (35), a1 > 0 a.e. Since the limit a0 is a strong limit of the sequence a1 in the space Lp(0,T), we conclude that a0 > 0 a.e. In this case the function /0 p(r) dr = ln(/J a0(r) dr + 1) is defined on the whole segment [0,T], respectively the function p(t) = is a solution to our problem.
Establish the desired estimate. We have (see (30)) that
ip(r) dT = f dT < Tl/q\\O\\lp(0,t) < Ci(R, R0)Tl/q, q = p/(p — 1).
J0 J0 1 + a(T)
From (21), it follows that a'(t) = p(t)e&p(t)dT > r(t) > 0. Thus,
p(t) > r(t)/Tl/qCi(R, R0) = r(t)c0, t G [0,T].
□
The authors were supported by the Russian Foundation for Basic Research and the Government of KhMAO-Yugra (Grant No. 15-4-1-00063, r_ural_a).
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Received March 4, 2017
УДК 517.95 Б01: 10.14529/ттр170204
ОПРЕДЕЛЕНИЕ ПАРАМЕТРА И УПРАВЛЕНИЕ В ПРОЦЕССАХ ТЕПЛОПЕРЕНОСА
С.Г. Пятков, О. В. Гавриленко
Югорский государственный университет, г. Ханты-Мансийск
Статья посвящена изучению некоторых математических моделей, описывающих процессы теплопереноса. Мы рассматриваем обратную задачу о восстановлении управляющего параметра, который обеспечивает заданное температурное распределение в данной точке пространственной области. Данный параметр - есть младший коэффициент в параболическом уравнении, зависящий от времени. Эта нелинейная задача сводится к операторному уравнения, разрешимость которого устанавливается при помощи априорных оценок и теоремы о неподвижной точке. Сформулированы и доказаны теоремы существования и единственности решений этой задачи. Установлены оценки устойчивости. Главный результат - глобальная по времени теорема существования решений при некоторых естественных условиях на данные задачи. Доказательство опирается на принцип максимума. Используемые функциональные пространства - пространства Соболева.
Ключевые слова: теплоперенос; распределенное управление; математическая модель; параболическое уравнение; обратная задача; краевая задача.
Сергей Григорьевич Пятков, доктор физико-математических наук, профессор, кафедра высшей математики, Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация), S_pyatkov@ugrasu.ru.
Олег Валериевич Гончаренко, аспирант, кафедра высшей математики, Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация), ovg@ugrasu.ru.
Поступила в редакцию 4 марта 2017 г.
