Бюллетени и Вестники - Библиотека аль-Фараби | Казахский национальный университет имени аль-Фараби

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ӘЛ-ФАРАБИ атындағы ҚАЗАҚ ҰЛТТЫҚ УНИВЕРСИТЕТI

ХАБАРШЫ

Математика, механика, информатика сериясы

КАЗАХСКИЙ НАЦИОНАЛЬНЫЙ УНИВЕРСИТЕТ имени АЛЬ-ФАРАБИ

ВЕСТНИК

Серия математика, механика, информатика

AL-FARABI KAZAKH NATIONAL UNIVERSITY

Journal of Mathematics, Mechanics and Computer Science

№2 (114)

Алматы

«Қазақ университетi»

2022

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Зарегистрирован в Министерстве информации и коммуникаций Республики Казахстан, свидетельство №16508-Ж от 04.05.2017 г. (Время и номер первичной постановки на учет

№766 от 22.04.1992 г.). Язык издания: казахский, русский, английский. Выходит 4 раза в год.

Тематическая направленность: теоретическая и прикладная математика, механика, информатика.

Редакционная коллегия

научный редактор–Б.Е. Кангужин, д.ф.-м.н., профессор, КазНУ им. аль-Фараби,

заместитель научного редактора–Д.И. Борисов, д.ф.-м.н., профессор, Институт математики с вычислительным центром Уфимского научного центра РАН, Башкирский государственный педагогический

университет им. М. Акмуллы, Россия

ответственный секретарь –С.Е. Айтжанов, к.ф.-м.н., доцент, КазНУ им. аль-Фараби Гиви Пэйман – профессор, Университет Питтсбурга, США

Моханрадж Муругесан – профессор, Инженерный и технологический колледж Хиндустана, Индия Эльбаз Ибрагим Мохаммед Абу Эльмагд – профессор, Национальный исследовательский

институт астрономии и геофизики, Египет

Жакебаев Д.Б. – PhD доктор, профессор, КазНУ им.аль-Фараби, Казахстан Кабанихин С.И. – д.ф.-м.н., профессор, чл.-корр. РАН, Институт вычислительной математики и математической геофизики СО РАН, Россия

Майнке М. – профессор, Департамент Вычислительной гидродинамики Института аэродинамики, Германия

Малышкин В.Э. – д.т.н., профессор, Новосибирский государственный технический университет, Россия

Ракишева З.Б. – к.ф.-м.н., доцент, КазНУ им.аль-Фараби, Казахстан

Ружанский М. – д.ф.-м.н., профессор, Имперский колледж Лондона, Великобритания Сагитов С.М. – д.ф.-м.н., профессор, Университет Гетеборга, Швеция

Сукочев Ф.А. – профессор, академик АН Австралии, Университет Нового Южного Уэльса, Австралия Тайманов И.А. – д.ф.-м.н., профессор, академик РАН, Институт математики

им. С.Л. Соболева СО РАН, Россия

Темляков В.Н. – д.ф.-м.н., профессор, Университет Южной Каролины, США Шиничи Накасука – PhD доктор, профессор, Университет Токио, Япония

Индексируется и участвует:

Научное издание

Вестник КазНУ. Серия “Математика, механика, информатика”, № 2 (114) 2022.

Редактор – С.Е. Айтжанов. Компьютерная верстка – С.Е. Айтжанов

Формат60×84 1/8. Бумага офсетная. Печать цифровая. Объем 9,87 п.л.

Заказ N 16302. Издательский дом “Қазақ университетi”

Казахского национального университета им. аль-Фараби. 050040, г. Алматы, пр.аль-Фараби, 71, КазНУ.

Отпечатано в типографии издательского дома “Қазақ университетi”.

c КазНУ им. аль-Фараби, 2022

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1-бөлiм Раздел 1 Section 1

Математика Математика Mathematics

IRSTI 27.29.17 DOI: https://doi.org/10.26577/JMMCS.2022.v114.i2.01

T.M. Aldibekov

Al-Farabi Kazakh National University, Kazakhstan, Almaty e-mail: [email protected]

ON BOUNDED SOLUTIONS OF DIFFERENTIAL SYSTEMS

The question of the existence of bounded solutions on an infinite interval of a linear inhomogeneous system of ordinary differential equations in a finite-dimensional space is considered. The study of bounded solutions of systems of ordinary differential equations is one of the most important problems in the qualitative theory of differential equations. In the study of the asymptotic behavior of solutions to differential systems, the works of A. Poincar´e and A.M. Lyapunov.

Various conditions for the existence of bounded solutions of a linear system of ordinary differential equations have been obtained by many authors. Note the works of O. Perron, A. Walter, H. Shpet, D. Caligo, N.I. Gavrilova, M. Hukukara, M. Nagumo, M. Caratheodori, U. Barbouti, N.Ya. Lyashchenko, B.P. Demidovich, A. Wintner, R. Bellman, Yu.S. Bogdanov, Z. Vazhevsky, N. Levinson, M. Markus, L. Cesari and others. In this paper, we establish sufficient conditions for the boundedness of all solutions of a linear inhomogeneous system of differential equations on an infinite interval. A coefficient criterion for the boundedness of all solutions on an infinite interval of a linear inhomogeneous system of differential equations in a certain class of differential systems is given. Applied methods of differential equations and function theory. The results obtained are used in applications of differential equations and are of practical value.

Key words: solution, boundedness, system, linear, differential equation.

Т.М. Алдибеков

Әл-Фараби атындағы Қазақ ұлттық университетi, Қазақстан, Алматы қ.

e-mail: [email protected]

Дифференциалдық жүйелердiң шектеулi шешiмдерi туралы

Ақырлы өлшемдi кеңiстiктегi қарапайым дифференциалдық теңдеулердiң сызықтық бiртектi емес жүйесiнiң шексiз аралықта шектелген шешiмдердiң болуы туралы мәселе қарастырыла- ды. Қарапайым дифференциалдық теңдеулер жүйесiнiң шектеулi шешiмдерiн зерттеу диф- ференциалдық теңдеулердiң сапалық теориясының маңызды мәселелерiнiң бiрi болып табы- лады. Дифференциалдық жүйелерге арналған шешiмдердiң асимптотикалық мiнез-құлқын зерттеу барысында А. Пуанкаре мен А.М. Ляпуновтың жұмыстары негiзiн қалаушы бо- лып табылады. Қарапайым дифференциалдық теңдеулердiң сызықтық жүйесiнiң шектел- ген шешiмдерiнiң болуының әр түрлi шарттары көптеген авторлармен алынған. О. Перрон, А. Вальтер, Х. Шпет, Д. Калиго, Н.И. Гаврилова, М. Хукукара, М. Нагумо, М. Каратеодори, У. Барбути, Н.Я. Лященко, Б.П. Демидович, А. Винтнер, Р. Беллман, Ю.С. Богданов, З. Ва- жевский, Н. Левинсон, М. Маркус, Л. Сезари және т.б. Бұл жұмыста бiз дифференциалдық теңдеулердiң сызықтық бiртектi емес жүйесiнiң барлық шешiмдерiнiң шексiз аралықта шек- телуiне жеткiлiктi шарттар орнатамыз. Дифференциалдық жүйелердiң белгiлi бiр класын- дағы дифференциалдық теңдеулердiң сызықтық бiртектi емес жүйесiнiң шексiз аралықтағы барлық шешiмдердiң шектелуiнiң коэффициент критерийi келтiрiлген. Дифференциалдық теңдеулер мен функциялар теориясының әдiстерi қолданылған. Алынған нәтижелер диффе- ренциалдық теңдеулердi қолдану кезiнде қолданылады және практикалық маңызы бар.

Түйiн сөздер: шешiм, шектiлiк, жүйе, сызықтық, дифференциалдық теңдеу.

c 2022 Al-Farabi Kazakh National University

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4 On bounded solutions of differential systems

Т.М. Алдибеков

Казахский национальный университет имени аль-Фараби, Казахстан, г. Алматы e-mail: [email protected]

Об ограниченных решениях дифференциальных систем

Рассматривается вопрос существования ограниченных решений на бесконечном интер- вале линейной неоднородной системы обыкновенных дифференциальных уравнений в конечномерном пространстве. Изучение ограниченных решений систем обыкновенных дифференциальных уравнений является одной из важнейших задач качественной теории дифференциальных уравнений. В исследовании асимптотического поведения решений дифференциальных систем основополагающими являются работы А. Пуанкаре и А.М. Ля- пунова. Разнообразные условия существования ограниченных решений линейной системы обыкновенных дифференциальных уравнений получены многими авторами. Отметим работы О. Перрона, А. Вальтера, Х. Шпета, Д. Калиго, Н.И. Гаврилова, М. Хукукара, М. Нагумо, М. Каратеодори, У. Барбути, Н.Я. Лященко, Б.П. Демидовича, А. Винтнера, Р. Беллмана, Ю.С. Богданова, З. Важевского, Н. Левинсона, М. Маркуса, Л. Чезари и другие. В данной работе установлены достаточные условия ограниченности всех решений линейной неоднородной системы дифференциальных уравнений на бесконечном интервале.

Приведен коэффициентный признак ограниченности всех решений на бесконечном интерва- ле линейной неоднородной системы дифференциальных уравнений в определенном классе дифференциальных систем. Примененяюся методы дифференциальных уравнений и теории функций. Полученные результаты находят применения в приложениях дифференциальных уравнений и представляет собой, практическую ценность.

Ключевые слова: решение, ограниченность, система, линейная, дифференциальное урав- нение.

1 Introduction

The question of the existence of bounded solutions of differential systems on an infinite interval is considered. The study of bounded solutions of systems of ordinary differential equations is one of the most important problems in the qualitative theory of differential equations. In the study of the asymptotic behavior of solutions of differential systems, the works of A. Poincar´e [1] and A.M. Lyapunov [2]. Conditions for the existence of bounded solutions of a linear system of ordinary differential equations were obtained by the authors:

Dunkel O., Hukuhara M., Nagumo M., Caccioppoli R., Caratheodory M., Dini U., Spath H., Weyl H., Wiman A., Ascoli G ., Gavrilov N.I., Gusarova R.S., Conti R., Barbuti U., Lyashchenko N.Ya., Demidovich B.P., Faedo S., Wilkins I.E, Ghizzetti A., Sobol I.M. , Haupt O., Boas M., Boas R.P., Wintner A., Bellman R., Bogdanov Yu.S., Butlewski Z., Bylov B.F., Wazewski T., Walter A., Caligo D., Kitamura T., Landau E., Levinson N., Marcus M., Perron O., Cesari L., Spath H., Shtokalo I.Z., Sobol I.M., et al. For detailed references, see the book by Cesari L. [3]. General information is available in the books:

[4] V.V. Nemytsky and V.V. Stepanov, [5] Erugin N.P., [6] Sansone G., [7] Pliss V.A., [8] Bylov B.F., Vinograd R.E., Grobman D.M., Nemytskiy V.V., [9] Izobov N.A., [10]

Coddington E.A. and Levinson N., [11] Demidovich B.P., [12] Lefschetz S., [13] Massera H.L., Scheffer H.H., [14] Bellman R., [15] Coppel W.A., [16] Daletskiy Yu.L., Kerin M.G. We also note the works: [17–20] Wintner A., [21] Yoshizawa T., [22] Bihari I., [23] Hartman Ph., [24]

Hale J., Onuchic N.

In this paper, sufficient conditions are established boundedness of all solutions of a linear inhomogeneous system of differential equations on an infinite interval. A coefficient criterion

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for the boundedness on an infinite interval of all solutions of a linear inhomogeneous system of differential equations in a certain class of differential systems is given. Applied methods of differential equations and function theory. The results obtained are used in applications of differential equations and are of practical value.

2 Materials and research methods

A linear inhomogeneous system of differential equations is considered

˙

x=A(t)x+f(t) (1)

where

t∈I ≡(1,+∞), A(t)∈(I), f(t)∈(I);

and the corresponding linear homogeneous system of differential equations

˙

x=A(t)x (2)

Theorem 1 If the conditions

kA(t)k ≤Kγt^γ−1, 0< γ <1, K >0, kf(t)k ≤Kγt^γ−1, t∈[t₀,+∞); t₀ ∈I;

and the linear homogeneous system (2) is generalized correct, has negative upper generalized Lyapunov exponents with respect to t^γ,then any solution to the linear inhomogeneous system of differential equations (1) on the interval [t₀,+∞) limited.

Proof.From (1) multiplying scalarly by x(t) get

(x⁰, x) = (A(t)x, x) + (f(t), x). (3)

From (3) get

|(x⁰, x)≤ |(A(t)x, x)|+|(f(t), x)|. (4) From (4) get

|(x⁰, x)| ≤ kA(t)kkxk²+kf(t)kkxk. (5)

From (5) get

−kA(t)kv²− kf(t)kv ≤v⁰kA(t)kv² +kf(t)kv (6) where v(t) = kx(t)k.From (6) get

de^−Kt^γ ≤ kx(t)k ≤De^Kt^γ (7)

where d > 0, D > 0. From (7) we obtain that any nonzero solution to the linear inhomogeneous system (1) has a finite upper generalized Lyapunov exponent with respect to t^γ.In the linear homogeneous system (2) we take the largest upper generalized Lyapunov

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exponent λ₁ < 0. Let’s take α ∈ (0,|λ₁|) ) and in the linear inhomogeneous system of differential equation (1) we perform the transformation

x=ye^αt^γ, x(t0) = y(t0). (8)

Then from (1) we obtain

˙

y= [A(t) +αγt^γ−1E]y+e^αt^γf(t) (9)

where the corresponding linear homogeneous system of differential equations

˙

y= [A(t) +αγt^γ−1E]y (10)

is generalized correct and has negative upper generalized Lyapunov exponents.

From the linear system of differential equations (9) we obtain

y(t) =Y(t, t₀)y(t₀) +

t

Z

t0

Y(t, s)e^αs^γf(s)ds (11)

where Y(t, t₀) = Y(t)Y⁻¹(t₀) – the Cauchy matrix of a linear homogeneous system of differential equations (10). By virtue of the generalized correctness of the linear system (10), for any ε∈(0,|α|)exists Dε(t0)>0and the inequality

kY(t, τ)k ≤D_ε(t₀)e^ετ^γ (12)

att ≥τ ≥t₀. From (11), (12) get

ky(t)k ≤Dε(t0)ky(t0)k+

t

Z

t0

Dε(t0)e^εs^γKγs^γ−1ds. (13)

From (8), (13) get

kx(t)k ≤e^−αt^γD_ε(t₀)

kx(t₀)k+Ke^εt^γ ε

. (14)

From (14), using arbitrary smallness ε, directing ε→0get

kx(t)k ≤e^−αt^γD_ε(t₀)(kx(t₀)k+Kt^γ) (15) att ≥t₀.

It follows from (15) that any solution of the linear inhomogeneous system of differential equations (1), on the interval [t₀,+∞) limited. Theorem 1 is proved.

Consider a linear inhomogeneous system of differential equations dy_i

dt =

n

X

k=1

pik(t)yi+fi(t), i= 1, n; (16)

where p_ik(t), f_i(t), i = 1, n; k = 1, n; continuous real functions on the interval (1,+∞), t₀ ∈(1,+∞).

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Theorem 2 If for 1< t₀ ≤t conditions are met:

1) pi−1,i−1(t)≥p_ii(t) +βγt^γ−1, i= 2, n; β >0, γ >0;

2) limt→+∞|p_ik(t)|

γt^γ−1 = 0, i6≡k, i= 1, n; k = 1, n;

3) limt→+∞1 t^γ

t

Z

t0

p_ii(s)ds=β_i <0, i = 1, n;

4)|f_i(t)| ≤Kγt^γ−1, i = 1, n; K >0;

then any solution to the linear inhomogeneous system of differential equations (16) on the interval [t₀,+∞) limited.

Proof.The corresponding linear homogeneous system of differential equations dy_i

dt =

n

X

k=1

p_ik(t)y_i, i= 1, n; (17)

under conditions: 1), 2) and 3) is generalized correct and has negative generalized upper Lyapunov exponents with respect to t^γ. The largest generalized upper Lyapunov exponent of the linear homogeneous system of differential equations (17) is β1 <0.Using condition 4) and similarly to Theorem 1, we obtain that any solution to the linear inhomogeneous system of differential equations (16) bounded. Theorem 2 is proved.

Let’s look at an example. In systemx⁰₁ =− 1 4√

tx1+ 1 4√

t, x⁰₂ =− 1 2√

tx2− 1 4√

t; 1< t0 ≤t;

where γ = 1

2, 0 < β ≤ 1

2, β₁ = −1

2, β₂ = −1, 1

2 ≤ K, the conditions of Theorem 2 are satisfied; therefore, any solution to a linear inhomogeneous system of differential equations is bounded.

3 Result

In this work, sufficient conditions for the boundedness of solutions of a linear inhomogeneous system of differential equations are obtained.

References

[1] Poincar´e A. O krivyh, opredelyaemyh differencial’nymi uravneniyami [About Curves Defined by Differential Equations]

(M.-L.: Gostekhizdat, 1947).

[2] Lyapunov A.M. Obshchaya zadacha ob ustojchivosti dvizheniya [General problem of motion stability] (M.-L.: GITTL, 1950): 472.

[3] Cezari L. Asimptoticheskoe povedenie i ustojchivost’ reshenij obyknovennyh differencial’nyh uravnenij [Asymptotic behavior and stability of solutions of ordinary differential equations] (M.: Mir, 1964): 477.

[4] Nemytsky V.V., Stepanov V.V. Kachestvennaya teoriya differencial’nyh uravnenij [Qualitative theory of differential equations]Izd. 2-e, pererab. i dop. (M.-L.: GITTL, 1949): 545.

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[5] Erugin N.P.Linejnye sistemy obyknovennyh differencial’nyh uravnenij [Linear systems of ordinary differential equations]

(Minsk: Izd-vo Akademii nauk BSSR, 1963): 270.

[6] Sansone G.Obyknovennye differencial’nye uravneniya [Ordinary differential equations]T.1. (M.: In. lit., 1953): 346; T.

2. (M.: In. lit., 1954): 414.

[7] Pliss V.A.Nelokal’nye problemy teorii kolebanij [Nonlocal Problems of Oscillation Theory] (M.: Nauka, 1964): 367.

[8] Bylov B.F., Vinograd R.E., Grobman D.M., Nemyckij V.V.Teoriya pokazatelej Lyapunova i ee prilozheniya k voprosam ustojchivosti [Theory of Lyapunov exponents and its applications to stability issues]Monografiya (M.: Nauka, 1966): 576.

[9] Isobov N.A.Vvedenie v teoriyu pokazatelej Lyapunova [Introduction to the theory of Lyapunov exponents](Mn.: BGU, 2006): 319.

[10] Coddington E.A., Levinson N.Teoriya obyknovennyh differencial’nyh uravnenij [Theory of ordinary differential equations]

(M.: IL, 1958): 475.

[11] Demidovich B.P. Lekcii po matematicheskoj teorii ustojchivosti [Lectures on the mathematical theory of stability] (M.:

Nauka, 1967): 472.

[12] Lefschetz S.Geometricheskaya teoriya differencial’nyh uravnenij [Geometric theory of differential equations] (M.: Izd-vo IL, 1961): 388.

[13] Massera H.L., Schaeffer H.H. Linejnye differencial’nye uravneniya i funkcional’nye prostranstva [Linear differential equations and function spaces]Per. s angl. (M.: Mir, 1970): 458.

[14] Bellman R.Teoriya ustojchivosti reshenij differencial’nyh uravnenij [Stability theory of solutions of differential equations]

(M.: Izd-vo IL, 1954): 216.

[15] Coppel W.A.Stability and asymptotic behavior of differential equations(Boston, D.C. Heath, 1965): 166.

[16] Daletskiy Yu.L., Kerin M.G. Ustojchivost’ reshenij differencial’nyh uravnenij v banahovom prostranstve [Stability of solutions of differential equations in a Banach space]Monografiya (Moskva: Nauka, 1970): 536.

[17] Wintner A., "Asymptotic equilibria" ,American Journal of Mathematics68(1) (1946): 125-132.

[18] Wintner A., "An Abelian lemma concerning asymptotic equilibria" , American Journal of Mathematics 68(3) (1946):

451-454.

[19] Wintner A., "Asymptotic integration constants" ,American Journal of Mathematics 68(4) (1946): 553-559.

[20] Wintner A., "On a theorem of Bocher in the theory of ordinary linear differential equations" , American Journal of Mathematics76(1) (1954): 183-190.

[21] Yoshizawa T., "Note on the boundedness of solutions of a system of differential equations" ,Mem. Coil. Sci. Univ. Kyoto A 28 (1954): 293-298.

[22] Bihari I., "A generalization of a lemma of bellman and its application to uniqueness problems of differential equations" , Acta Math. Acad. Sci. Hung.7 (1956): 81-94.

[23] Hartman Ph. "The existence of large or small solutions of linear differential equations" ,Duke Math. J.28 (1961): 421-429.

[24] Hale J., Onuchic N. "On the asymptotic behavior of solutions of a class of differential equations" , Contributions to Differential Equations1 (1963): 61-75.

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IRSTI 27.31.15 DOI: https://doi.org/10.26577/JMMCS.2022.v114.i2.02

K.B. Imanberdiyev^1,2^∗ , B.E. Kanguzhin^1,2 , A.M. Serik¹ , B. Uaissov³

1Al-Farabi Kazakh National University, Kazakhstan, Almaty

2Institute of Mathematics and Mathematical Modeling, Kazakhstan, Almaty

3Academy of logistics and transport, Kazakhstan, Almaty

∗e-mail: [email protected]

BOUNDARY CONTROL OF ROD TEMPERATURE FIELD WITH A SELECTED POINT

In this paper, we study the issue of boundary control of the temperature field of a rod with a selected point. The main purpose of the work is to clarify the conditions for the existence of a boundary control that ensures the transition of the temperature field from the initial state to the final state. Relations connecting the boundary controls with the initial and final states, as well as with the external temperature field are found. Such boundary controls, generally speaking, constitute an infinite set. For an unambiguous choice of the boundary control, a strictly convex objective functional is chosen. We are looking for a boundary control that minimizes the selected target functional. To do this, we first investigate the existence and uniqueness of solutions to the initial boundary value problem and the conjugate problem. And also, we present the derivation of a system of linear Fredholm integral equations of the second kind, which are satisfied by an optimal boundary control that minimizes a strictly convex target functional on a convex set.

Along the way, the linear part of the increment of the target functional is highlighted. Necessary and sufficient conditions for the minimum of a smooth convex functional on a convex set are established. The difference between the results of this work and the available ones is that in the proposed work, the temperature field is given by the heat conduction equation with a loaded term. As a result, the conjugate problem has a slightly different domain of definition than the domain of the conjugate problem in the case of no load.

Key words: initial-boundary value problem, heat equation, boundary control, Green’s function, Fredholm integral equation of the second kind, spectral properties, eigenfunction, eigenvalues.

Қ.Б. Иманбердиев^1,2^∗, Б.Е. Кангужин^1,2, А.М. Серiк¹, Б. Уаисов³

1Әл-Фараби атындағы Қазақ ұлттық университетi, Қазақстан, Алматы қ.

2Математика және математикалық модельдеу институты, Қазақстан, Алматы қ.

2Логистика және транспорт академиясы, Қазақстан, Алматы қ.

Таңдалған нүктесi бар өзекшенiң температуралық өрiсiн шекаралық басқару Бұл жұмыста таңдалған нүктесi бар өзекшенiң температуралық өрiсiн шекаралық басқару мәселесi зерттеледi. Жұмыстың негiзгi мақсаты – температуралық өрiстiң бастапқы күйден соңғы күйге өтуiн қамтамасыз ететiн шекаралық басқарудың бар болуы шарттарын аны- қтау. Шекаралық басқаруларды бастапқы және финалдық күйлермен, сондай-ақ сыртқы температура өрiсiмен байланыстыратын қатынастар табылды. Мұндай шекаралық басқару- лар, жалпы айтқанда, шексiз жиынды құрайды. Шекаралық басқарудың бiрегей таңдалуы үшiн қатаң дөңес мақсат функционал таңдалады. Таңдалған мақсат функционалды мини- мумдаушы шекаралық басқару iзделедi. Ол үшiн жұмыста алдымен бастапқы-шекаралық есеп пен түйiндес есептiң шешiмдерiнiң бар болуы мен жалғыздығын зерттеймiз. Сондай-ақ дөңес жиында қатаң дөңес мақсат функционалын минимумдаушы тиiмдi шекаралық басқа- румен қанағаттандырылатын Фредгольмның екiншi тектi сызықты интегралдық теңдеулер жүйесiнiң алынуы келтiрiлген. Осы орайда мақсат функционалдың өсiмшесiнiң сызықтық бөлiгi айқындалған. Дөңес жиында тегiс дөңес функционал минимумының қажеттi және жеткiлiктi шарттары анықталған. Жұмыстың нәтижесiнiң белгiлi жұмыстардан айырмашы- лығы температура өрiсi жүктелген мүшесi бар жылуөткiзгiштiк теңдеуi арқылы берiлгендiгi.

c 2022 Al-Farabi Kazakh National University

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10 Boundary control of rod temperature . . .

Нәтижесiнде, түйiндес есептiң жүктемесi болмаған жағдайда түйiндес есептiң облысынан бiршама өзгеше анықталу облысы болады.

Түйiн сөздер: бастапқы-шекаралық есеп, жылуөткiзгiштiк теңдеуi, шекаралық басқару, Грин функциясы, екiншi тектi Фредгольм интегралдық теңдеуi, спектрлiк қасиеттер, мен- шiктi функция, меншiктi мәндер.

К.Б. Иманбердиев^1,2^∗, Б.Е. Кангужин^1,2, А.М. Серiк¹, Б. Уаисов³

1Казахский национальный университет имени аль-Фараби, Казахстан, г. Алматы

2Институт математики и математического моделирования, Казахстан, г. Алматы

3Академия логистики и транспорта, Казахстан, г. Алматы

Граничное управление температурным полем стержня с выделенной точкой

В данной работе изучается вопрос о граничном управлении температурным полем стержня с выделенной точкой. Основная цель работы – выяснение условий существования граничного управления, обеспечивающего переход температурного поля из начального состояния в конечное состояние. Найдены соотношения, связывающие граничные управления с началь- ным и финальным состояниями, а также внешним температурным полем. Такие граничные управления, вообще говоря, составляют бесконечное множество. Для однозначного выбора граничного управления выбран строго выпуклый целевой функционал. Ищется граничное управление, которое минимизирует выбранный целевой функционал. Для этого в работе сначала исследуются существование и единственность решений начально-граничной задачи и сопряженной задачи. А также, дан вывод системы линейных интегральных уравнений Фредгольма второго рода, которым удовлетворяет оптимальное граничное управление, кото- рое минимизирует строго выпуклый целевой функционал на выпуклом множестве. По пути выделена линейная часть приращения целевого функционала. Установлены необходимые и достаточные условия минимума гладкого выпуклого функционала на выпуклом множестве.

Отличие результатов данной работы от имеющихся заключается в том, что в предлагаемой работе температурное поле задается уравнением теплопроводности с нагруженным членом.

Вследствие чего сопряженная задача имеет несколько отличительную область определения, чем область определения сопряженной задачи в случае отсутствия нагрузки.

Ключевые слова: начально-граничная задача, уравнение теплопроводности, граничное управление, функция Грина, интегральное уравнение Фредгольма второго рода, спектраль- ные свойства, собственная функция, собственные значения.

1 Introduction

In this paper, we study the issue of boundary control of rod temperature field with a selected point x₀.

ut(x, t)−uxx(x, t) +αu(x0, t) =f(x, t), (x, t)∈Q, (1) whereQ={(x, t) : 0< x < b, 0< t < T <+∞}.

It is assumed that at the initial moment t= 0 the temperature along the rod of lengthb is given by lawu(x,0) =u₀(x), 0< x < b,where u₀(x)is a twice continuously differentiable function. At the moment of timet =T the temperature of the rod is equal tou(x, T) =γ(x), 0< x < b,whereγ(x) is also a twice continuously differentiable function. The main purpose of the work is to clarify the conditions for the existence of the boundary controlu(0, t) =µ(t), u(b, t) = η(t),which ensures the transition of the temperature field from the state {u(x,0) = u0(x)} to the state {u(x, T) = γ(x)}. Similar problems were considered in [1, 2].

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According to the optimization method, we choose the following functional

J[µ, η] =ku(·, T;µ, η)−γ(·)k²_W1

2(0,b)+β1

Z T 0

|µ(t)|²dt+β2

Z T 0

|η(t)|²dt,

where β₁, β₂ are positive numbers,γ is a given function from class W₂¹(0, b).

The boundary control problem is as follows: it is required to find boundary controls (µ(t), η(t)) and the corresponding solution u(x, t), that satisfies equation (1) with initial boundary controls

u(0, t) =µ(t), u(b, t) =η(t), 0< t < T, (2)

u(x,0) =u₀(x), 0< x < b, (3)

and minimizes functional J[µ, η].

Many natural and fundamental physical phenomena can be modeled by partial differential equations (PDEs), such as heat conduction, sound, electrostatics, electrodynamics, fluid flow and quantum mechanics in which states depend on not only time but also space, for example, see [3–5]. In particular, heat diffusion phenomena are extended mainly in describing fluid, thermal, and chemical dynamics, including the wide applications of sea ice melting and freezing [6], lithium-ion batteries [7], etc. The work [8] is concerned with the problem of boundary observer-based finite-time output feedback control for a heat system with Neumann boundary condition and Dirichlet boundary actuator. Finite-time stabilization, which is a key feature in the sliding mode control theory, is investigated in the work [9]. More specifically, finite-time control for the heat equation with Dirichlet boundary condition and the Dirichlet control is investigated in [10]. In work [11] the heat equation with prescribed lateral and final data is studied in half-plane and the uniqueness of the bounded solution is proved. In work [12] the solvability problems of an nonhomogeneous boundary value problem in the first quadrant for a fractionally loaded heat equation are studied. For parabolic equations in a bounded domain, various aspects of inverse source problems has been studied in [13–16], etc.

The paper presents a derivation of a system of linear Fredholm integral equations of the second kind, which optimal boundary control is satisfied. In the proposed work, for the first time, the conjugate problem to a mixed boundary value problem for the heat conduction equation with a loaded term is explicitly written out. As a result, it was possible to obtain more precise information about the solutions of the conjugate problem. We note that in [1], the solution of the mixed boundary value problem for the heat conduction equation is decomposed by the eigenfunctions of a periodic problem with a specially selected period. In [2], the method of work [1] is extended to the heat conduction equation with a loaded term. In this paper, the expansion of the solution to the mixed problem for the heat equation with a loaded term is carried out in terms of the eigenfunctions of the corresponding spectral problem. At the same time, it is necessary to select a period and continue the solution in a wider area. Moreover, the solution of the conjugate problem is carried out similarly to the solution of the mixed problem for the heat conduction equation with a load.

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2 Existence and uniqueness of the solutions to the initial-boundary value problem and the conjugate problem

Before studying the boundary control problem, it is necessary to investigate the question of the existence and uniqueness of the solution to problem (1)–(3). To do this, select a function w(x, t)from class L₂((0, T);W₂¹(0, b))such that

w(x, t) = µ(t) + x

b(η(t)−µ(t)).

Then, instead of studying problem (1)–(3) it is enough to study the following problem:

v_t(x, t)−v_xx(x, t) +αv(x₀, t) =F(x, t), (x, t)∈Q, (4)

v(0, t) = 0, v(b, t) = 0, (5)

v(x,0) = v₀(x), 0< x < b, (6)

where

F(x, t) = f(x, t)−w_t(x, t)−αw(x₀, t), v₀(x) =u₀(x)−µ(0)− x

b(η(0)−µ(0)).

The solution to problem (4)–(6) is sought in the form v(x, t) =X

k>1

d_k(t)ϕ_k(t).

Here{ϕk} is the system of root functions of the following eigenvalue problem

−ϕ_xx(x) +αϕ(x₀) = λϕ(x), 0< x < b, (7)

ϕ(0) = 0, ϕ(b) = 0. (8)

In this case ϕ_k(x) ≡ ϕ_k(x, λ_k), where {λ_k} is a sequence of eigenvalues of (7)–

(8). The eigenfunctions ϕk(x) ≡ ϕ(x, λk) and the biorthogonal system of functions n

ψ_k(x) = _hϕ(·,λ^ψ(x,λ^k⁾

k),ψ(·,λ_k)i

o

are defined by formulas:

ϕ(x, λ) = sin√

√ λx

λ +α

sin√ λx₀

√ λ λ−α(1−cos√

λx₀)(1−cos√

λx), 0< x < b, ψ(x, λ) = sin√

λ(b−x)

√λ , x0 < x < b,

ψ(x, λ) = sin√

λ(b−x₀)

√λ cos√

λ(x₀ −x) +sin√

λ(x₀−x)

√λ cos√

α(b−x₀)−α1−cos√

λ(b−x₀) λ

!

−αsin√

λ(x₀−x)

√λ

× cos√

λ(b−x₀)−cos√

λb−α(1−cos√

λ(b−x₀))(1−cos√ λx₀) λ

λ+α(1−cos√

λx0) , 0< x < x₀,

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The acceptable values of parameter λ are selected according to the conditions λ −α(1− cos√

λx₀) 6= 0, λ+α(1−cos√

λx₀) 6= 0. Each function f(x) from L₂(0, b) is decomposed into a Fourier series by the system {ϕ_k}:

f(x) = X

k>1

C_k(f)ϕ_k(x),

where Ck(f) =hf, ψki, k>1.

In this case, the Fourier coefficients {d_k(t), k > 1} in terms of system {ϕ_k(x)} of the solution v(x, t)satisfy equations

d_tk(t) +λ_kd_k(t) = D_k(t), t >0 (9)

and initial conditions

d_k(0) =d⁽⁰⁾_k . (10)

Here {D_k(t)} and {d⁽⁰⁾_k } are sequences of Fourier coefficients in terms of system {ϕ_k} of functions F(x, t)and v₀(x). Relations (9)–(10) imply the following representation

d_k(t) = d⁽⁰⁾_k e^−λ^k^t+ Z t

0

e^−λ^k^(t−τ)D_k(τ)dτ, t >0. (11)

Thus, problem (4)–(6) has a solutionv(x, t), representable in the form v(x, t) = X

k>1

dk(t)ϕk(x), (12)

and the coefficientsd_k(t)are calculated by formulas (11). Thus, we can formulate the following statement.

Theorem 1 Let v₀(x) be a twice continuously differentiable function on a finite segment [0, b], and the matching conditions v₀(0) = v₀(b) = 0 are satisfied. Suppose also that F(x, t) = L²((0, T);L₂(0, b)).Then there is a solutionv(x, t)of problem (4)–(6), which can be represented as a Fourier series (12), the coefficients {d_k(t)} of which are found by formulas (11).

Remark 1 Note that v0(x) is decomposed into a Fourier series by the system {ϕk} and the corresponding Fourier series on [0, b] converges uniformly. This follows from the fact that v₀(x) belongs to the domain of operator B. The monograph [17] contains theorems on the uniform convergence of spectral decompositions in such cases.

We denote by G(x, ξ, t) = P

k>1

e^−λ^k^tϕ_k(x)ψ_k(ξ), the function that represents the Green function of the Dirichlet problem for the heat equation with the selected point [18]. Then the statement follows.

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Corollary 1 Let the conditions of Theorem 1 be satisfied. Then there exists a solution u(x, t) of problem (1)–(3), which can be represented in the form

u(x, t) = Z b

0

u₀(ξ)G(x, ξ, t)dξ+ Z t

0

dτ Z b

0

f(ξ, τ)G(x, ξ, t−τ)dξ−u₀(0) Z b

0

G(x, ξ, t)dξ

− u₀(b)−u₀(0) b

Z b 0

ξG(x, ξ, t)dξ−α Z t

0

µ(τ)dτ Z b

0

G(x, ξ, t−τ)dξ

− α b

Z t 0

(η(τ)−µ(τ))dτ Z b

0

ξG(x, ξ, t−τ)dξ.

We now formulate and prove a uniqueness theorem for a solution.

Theorem 2 Let the conditions of Theorem 1 be satisfied. Then problem (4)–(6) has a unique solution.

Proof 1 The idea of this proof is borrowed from the work of V.A. Il’in [19]. Let r(x) be one of the eigenfunctions of operator B^∗. We denote by Φ(x, t) any of the functions of the form

Φ(x, t) =r(x)f(t),

wheref(t)is a function that is continuously differentiable on the entire numerical axis, which is equal to zero for all t > t₀, where t₀ is some number less than T.

Let v(x, t) be a solution to problem (4)–(6) for F ≡0, v₀ ≡0. Let consider integral 0 =

Z b 0

Z T 0

(v_t(x, t)−v_xx(x, t) +αv(x₀, t)) Φ(x, t)dtdx= Z b

0

r(x)dx Z T

0

v_t(x, t)f(t)dt +

Z T 0

f(t)dt Z b

0

Bv(x, t)r(x)dx = Z b

0

r(x)dx

ϕ(x, t)f(t)

T 0 −

Z T 0

v(x, t)ft(t)dt

+ Z T

0

f(t)dt Z b

0

v(x, t)B^∗r(x)dx=− Z b

0

Z T 0

v(x, t)r(x)ft(t)dtdx +λ

Z b 0

Z T 0

v(x, t)r(x)f(t)dtdx, (13) where λ is the eigenvalue of operator B^∗ corresponding to eigenfunction r(x).

Let us continue v(x, t) on domain t < 0, by setting it equal to zero there. Then, taking into account that f(t) = 0 for t > t₀, relation (13) can be rewritten in the form

Z b 0

Z ∞

−∞

v(x, t)r(x) −f_t(t) +λf(t)

dtdx= 0. (14)

We fix any ξ > 0. Then function f(ξ+t) is a priori equal to zero for t > t0. In equality (14) we substitute f(ξ+t) instead of f(t). Then for allξ >0 we have equality

Z b 0

Z ∞

−∞

v(x, t)r(x) −fξ(ξ+t) +λf(ξ+t)

dtdx= 0. (15)

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From (15) it follows Z b

0

Z ∞

−∞

v(x, t)r(x)f(ξ+t)dtdx=c·e^λξ, ξ >0. (16)

However, forξ >t₀ andt >0the functionf(ξ+t)≡0.Therefore, it follows from relation (16) that c= 0. Therefore, we have the equality

Z b 0

Z ∞

−∞

v(x, t)r(x)f(ξ+t)dtdx= 0, ξ >0. (17)

Since the system of eigenfunctions {ψ(x, λ_k), k >1} of operator B^∗ is complete in space L₂(0, b), equalities (17) imply

Z ∞ 0

v(x, t)f(ξ+t)dt≡0 in space L₂(0, b).

In particular, for ξ= 0 we find that Z T

0

v(x, t)f(t)dt= 0.

The latter equality holds for any function f(t), that has the properties described above.

Therefore v(x, t)≡0 for 0< x < b, 0< t < T. Theorem 2 is completely proved.

Therefore, the conjugate problem to problem (4)–(6) takes the form

−Ψ_t(x, t)−Ψ_xx(x, t) =E(x, t), (x, t)∈Q, (18)

Ψ(0, t) = 0, Ψ(b, t) = 0, t >0, (19)

Ψ(x₀+ 0, t) = Ψ(x₀−0, t), t >0, Ψx(x0 + 0, t) = Ψx(x0−0, t) +αRb

0 ψ(x, t)dx, t >0, (20)

Ψ(x, T) = ΨT(x), 0< x < b. (21)

Thus, we can formulate the following statement.

Theorem 3 Let Ψ_T(x) be a twice continuously differentiable function on a finite segment [0, b], moreover, forΨ_T(x) the matching conditions (19)–(20) are satisfied. Suppose also that E(x, t) ∈ L²((0, T);L₂(0, b)). Then there is a solution Ψ(x, t) to problem (18)–(21), which can be represented as a Fourier series dual to series (12).

Theorem 3 implies the following statement.

Corollary 2 Let the conditions of Theorem 3 be satisfied. Then there exists a solutionΨ(x, t) to problem (18)–(21), which can be represented in the form

Ψ(x, t) = Z b

0

Ψ_T(ξ)G(ξ, x, T −t)dξ+ Z T

t

dτ Z b

0

E(ξ, τ)G(ξ, x, T −τ)dξ.

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Now we formulate and prove the uniqueness theorem.

Theorem 4 Let the conditions of Theorem 3 be satisfied. Then problem (18)–(21) has a unique solution.

The proof of Theorem 4 repeats the proof of Theorem 2. Let r(x) be one of the eigenfunctions of operatorB. We denote by Φ(x, t) any function of the form

Φ(x, t) = r(f)f(t),

where f(t) is a continuously differentiable function on the entire numerical axis, which is equal to zero for all t < t₀, where t₀ is some positive number. Further, the reasoning from the proof of Theorem 2 is repeated.

3 Necessary conditions for maintaining the final temperature regime

In this section, we study the boundary control problem I:

W_t(x, t)−W_xx(x, t) +αW(x₀, t) =f(x, t), (x, t)∈Q. (22)

W(x, T) = u_T(x), 0< x < b, (23)

Statement of the boundary control Problem I:

LetW(x, t;f, u_T)be an arbitrary solution of problem (22)–(23). We denote the boundary controls corresponding toW(x, t;f, u_T),byµ(t) = W(0, t;f, u_T)andη(t) =W(b, t;f, u_T),as well as byu₀(x) = W(x,0;f, u_T) the initial temperature regime.

What necessary conditions doµ(t), η(t), u₀(x),satisfy ifW(x, t;f, u_T)satisfies (22)–(23)?

This boundary control Problem I corresponds to a given final temperature regimeu_T(x).

To answer the question posed, it is convenient to introduce solutionsΨ(x, t) = Ψ(x, t; Ψ_T)to conjugate equation

−Ψ_t(x, t)−Ψ_xx(x, t) = 0, (x, t)∈Q, x6=x₀, (24) with conditions

Ψ(0, t) = 0, Ψ(b, t) = 0, t >0, (25)

Ψ(x₀+ 0, t) = Ψ(x₀−0, t),

Ψ⁰(x₀+ 0, t) = Ψ⁰(x₀−0, t) +αRb

0 Ψ(ξ, t)dξ, t >0, (26)

and the final temperature distribution

Ψ(x, T) = Ψ_T(x), 0< x < b (27)

for an arbitrary functionΨ_T(x)from class W₂²[0, b].

Lemma 1 For an arbitrary solution u(x, t)≡u(x, t;f, µ, η, u₀) of equation

u_t(x, t)−u_xx(x, t) +αu(x₀, t) =f(x, t), (x, t)∈Q, (28)

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with boundary conditions

u(0, t) =µ(t), u(b, t) = η(t), T > t >0, (29)

and with the initial temperature distribution

u(x,0) =u₀(x),0< x < b, (30)

the following integral relation is valid Z T

0

Z b 0

f(x, t)Ψ(x, t)dxdt= Z b

0

u(x, T)Ψ_T(x)−u₀(x)Ψ(x,0) dx

− Z T

0

µ(t)Ψ_x(0, t)dt+ Z T

0

η(t)Ψ_x(b, t)dt,

where Ψ(x, t) ≡ Ψ(x, t; Ψ_T) is the solution to conjugate problem (24)–(27) for an arbitrary Ψ_T(x)∈W₂²[0, b].

Let us formulate another useful lemma.

Lemma 2 For an arbitrary solution v(x, t)≡v(x, t;f, µ₁, η₁, v_T) of equation

v_t(x, t)−v_xx(x, t) +αv(x₀, t) = f(x, t), (x, t)∈Q, (31) with boundary conditions

v(0, t) = µ₁(t), v(b, t) =η₁(t), T > t >0, (32)

and with the final temperature distribution

v(x, T) =v_T(x), 0< x < b, (33)

the following integral relation is valid Z T

0

Z b 0

f(x, t)Ψ(x, t)dxdt= Z b

0

v_T(x)Ψ_T(x)−v(x,0)Ψ(x,0) dx

− Z T

0

µ₁(t)Ψ_x(0, t)dt+ Z T

0

η₁(t)Ψ_x(b, t)dt,

where Ψ(x, t) ≡ Ψ(x, t; Ψ_T) is the solution to conjugate problem (24)–(27) for an arbitrary Ψ_T(x)∈W₂²[0, b].

We now formulate an important statement.

Theorem 5 For the solution u(x, t) ≡ u(x, t;f, µ, η, u₀) to problem (28)–(30) and for the solution v(x, t) ≡ v(x, t;f, µ₁, η₁, v_T) to problem (31)–(33) the following integral identity is valid

Z T 0

(η₁(t)−η(t))G_x(ξ, b, T −t)dt− Z T

0

(µ₁(t)−µ(t))G_x(ξ,0, T −t)dt

≡ Z b

0

(v(x,0)−u₀(x))G(ξ, x, T)dx, ∀ξ ∈(0, b), (34) where G(x, ξ, t) = P

k>1

e^−λ^k^tϕ_k(x)ψ_k(ξ) is a Green’s function.

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Proof 2 Lemmas 1 and 2 imply the integral identity 0 =−

Z b 0

(v(x,0)−u₀(x)) Ψ(x,0)dx

− Z T

0

(µ₁(t)−µ(t)) Ψ_x(0, t)dt+ Z T

0

(η₁(t)−η(t)) Ψ_x(b, t)dt, (35) for all Ψ(x, t) at any Ψ_T(x). Corollary 2 implies that

Ψ(x, t) = Z b

0

Ψ_T(ξ)G(ξ, x, T −t)dξ.

Therefore, relation (35) takes the form

− Z b

0

(v(x,0)−u₀(x))dx Z b

0

Ψ_T(ξ)G(ξ, x, T)dξ

− Z T

0

(µ₁(t)−µ(t))dt Z b

0

Ψ_T(ξ)G_x(ξ,0, T −t)dξ +

Z T 0

(η₁(t)−η(t))dt Z b

0

Ψ_T(ξ)G_x(ξ, b, T −t)dξ.

Rearranging the order of the integrals, we obtain the equality

Z b 0

Ψ_T(ξ) (Z T

0

(η₁(t)−η(t))G_x(ξ, b, T −t)dt

− Z T

0

(µ1(t)−µ(t))Gx(ξ,0, T −t)dt− Z b

0

(v(x,0)−u0(x))G(ξ, x, T)dx )

dξ= 0.

Since Ψ_T(ξ) is an arbitrary function from W₂²[0, b], then relation (34) follows from the last equality. Theorem 5 is completely proved.

This implies the following statement.

Corollary 3 Letu(x, t)≡u(x, t;f, µ, η, u0)andv(x, t)≡v(x, t;f, µ1, η1, vT)are solutions to problems (28)–(30) and (31)–(33), respectively. If u₀ =v(x,0), x∈(0, b), then the following identity is valid

Z T 0

(η₁(t)−η(t))G_x(ξ, b, T −t)dt− Z T

0

(µ₁(t)−µ(t))G_x(ξ,0, T −t)dt ≡0, ∀ξ ∈(0, b).

4 Optimality criteria

In this section, the target functional is investigated.

J[µ, η] = Z b

0

|u(x, T;µ, η)−γ(x)|²dx +

Z b 0

|u⁰(x, T;µ, η)−γ⁰(x)|²dx+β₁ Z T

0

µ²(t)dt+β₂ Z T

0

η²(t)dt.

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Let us take arbitrary controls (µ(t), η(t)) and (µ(t) +h(y), η(t) +q(t)), where h(0) = 0, q(0) = 0. We denote the corresponding solutions of problem (1)–(3) by u(x, t;µ, η) and u(x, t;µ+h, η+q). Let us introduce the notation

∆u(x, t) =u(x, t;µ+h, η+q)−u(x, t;µ, η).

Then from (1)–(3) it follows

∂

∂t∆u− ∂²

∂x²∆u+α∆u(x0, t) = 0, (x, t)∈Q, (36)

∆u

x=0 =h(t), ∆u

x=b =q(t), t >0, (37)

∆u

t=0 = 0, 0< x < b. (38)

Arguing as in the proof of Theorem 1, we obtain the representation

∆u(x, t) = 1 b

Z t 0

(q(τ)−h(τ)) ∂

∂τK₁(x, t−τ)dτ − αx₀ b

Z t 0

q(τ)K₀(x, t−τ)dτ

−α

1− x₀ b

Z t 0

h(τ)K0(x, t−τ)dτ+ Z t

0

h(τ) ∂

∂τK0(x, t−τ)dτ, where K₀(x, t) = P

k>1

β_k⁽⁰⁾e^−λ^k^t·ϕ_k(x), K₁(x, t) = P

k>1

β_k⁽¹⁾e^−λ^k^t·ϕ_k(x).

Consider the increment of the target functional J[µ, η].

∆J[µ, η] = 2 Z b

0

Re

u(x, T;µ, η)−γ(x)

∆u(x, T)

dx + 2

Z b 0

Re

u_x(x, T;µ, η)−γ_x(x) ∂

∂x∆u(x, T)

dx+ 2β₁ Z T

0

Re

µ(t)h(t) dt

+ 2β2

Z T 0

Re

η(t)q(t)

dt+o



 s

Z T 0

(|h(t)|²+|q(t)|²)dt





= 2 Z b

0

Re

u(x, T;µ, η)−γ(x)

∆u(x, T) dx + 2Re

u_x(x, T;µ, η)−γ_x(x)

∆u(x, T)

x=b x=0

−2 Z b

0

Re

u_xx(x, T;µ, η)−γ_xx(x) ∂

∂x∆u(x, T)

dx

+ 2β₁ Z T

0

Re

µ(t)h(t)

dt+ 2β₂ Z T

0

Re

η(t)q(t) dt+o



 s

Z T 0

(|h(t)|²+|q(t)|²)dt





= 2Re

u_x(b, T;µ, η)−γ_x(b) q(T)

−2Re

u_x(0, T;µ, η)−γ_x(0) h(T) + 2

Z b 0

Re

∆u(x, T)h

−

u_xx(x, T;µ, η)−γ_xx(x) +

u(x, T;µ, η)−γ(x)i dx

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+2β₁ Z T

0

Re

µ(t)h(t)

dt+2β₂ Z T

0

Re

η(t)q(t) dt+o



 s

Z T 0

(|h(t)|²+|q(t)|²)dt



, (39) where

q,h→0lim o

q RT

0 (|h(t)|²+|q(t)|²)dt

RT

0 (|h(t)|²+|q(t)|²)dt = 0.

Let us introduce the solution Ψ(x, t;µ, η) of the following conjugate problem (18)–(21) at E(x, t)≡ 0, Ψ_T(x) =

−_dx^d²2 +I

(u(x, T;µ, η)−γ(x)). For further purposes, we transform the integral

Z b 0

∆u(x, T) [−(u_xx(x, T;µ, η)−γ_xx(x)) + (u(x, T;µ, η)−γ(x))]dx

= Z T

0

∂

∂t Z b

0

∆u(x, t)Ψ(x, t;µ, η)dx

dt

= Z T

0

dt Z x0

0

∂²

∂x²∆u(x, t)−α∆u(x₀, t)

Ψ(x, t;µ, η)dx +

Z T 0

dt Z b

x0

∂²

∂x²∆u(x, t)−α∆u(x₀, t)

Ψ(x, t;µ, η)dx +

Z T 0

dt Z b

0

∆u(x, t)∂

∂tΨ(x, t;µ, η)dx

= Z T

0

dt (

∂

∂x∆u(x, t)Ψ(x, t;µ, η)−∆u(x, t) ∂

∂xΨ(x, t;µ, η)

x=x0−0 x=0

+ ∂

∂x∆u(x, t)Ψ(x, t;µ, η)−∆u(x, t) ∂

∂xΨ(x, t;µ, η) x=b

x=x0+0

)

+α Z T

0

∆u(x₀, t)Ψ(x, t;µ, η)dt +

Z T 0

dt Z b

x0

∆u(x, t) ∂²Ψ(x, t;µ, η)

∂x² + ∂Ψ(x, t;µ, η)

∂t

! dx

= Z T

0

h(t) ∂

∂xΨ(0, t;µ, η)dt− Z T

0

q(t) ∂

∂xΨ(b, t;µ, η)dt.

Thus, the following relation is true 2

Z b 0

Re

∆u(x, T)

− d² dx² +I

(u(x, T;µ, η)−γ(x))

dx

= 2 Z T

0

Re

h(t) ∂

∂xΨ(0, t;µ, η)

dt−2 Z T

0

Re

q(t) ∂

∂xΨ(b, t;µ, η)

dt.