On the invariance of the constant multi-valued mapping in the heat conductivity problem Текст научной статьи по специальности «Математика»

On the invariance of the constant multi-valued mapping in the heat conductivity problem

Mustapokulov Khamdam Yangiboevich, teacher of Tashkent state technical university named after Abu Raikhon Beruni E-mail: mhy1506@yandex.ru

On the invariance of the constant multi-valued mapping in the heat conductivity problem

Abstract: In this work is considered the question about strong and weak invariance of the constant multi-valued mapping for equation heat conductivity with border controls. Gets the sufficient conditions for the strong and weak invariance given multi-valued mapping.

Keywords: the invariant set, the control, the multi-valued mapping, control of distributed systems.

1 Introduction

Formerly, interesting results on the invariance of given sets problem in respect to the system with fixed parameters had been received in the works of A. A. Feuer,

M. Heymann, V N. Ushakov, Kh. G. Guseynov, N. S. Rettiev,

A. Z. Fazilov and other authors. In works of Sh.Alimov are considered interesting applied problems of the control with convectors on the heat volume distribution [1, 23-39]. Let

We determine solution the problem (l)-(3) by Fourier method. If through fk (■) are designated Fourier function coefficients f (•) concerning the system {pk}, in this case solution the problem (l)-(3) is of the form

’(t,x ) = x

z0e^ + I

I Iu(T,s)<Pk (s)ds

e(t-T)dr fa (x )

Az=£ tt

i ,j=i dx.

^ Möz ^ dz , ч

a., (x I- , Pz =-+ h(x)z,x <

v dxj у

dn

Q,

where aij (x ) = aß (x )eC1 (Q), i, j = l,...,n, Q — is a

bounded domain in Rn with the piecewise-smooth boundary, A — the elliptic differential operator, i. e. positive constant

exists Y as ^aij (x)^i^j -Y^^,2, for all x eQ and real

i .j=1

(2)

(3)

number ft,...,ft, ^ft2 Ф 0, h (x)— the given positive

i=1

dz

continued function,------the derivative on the outer normal

dn

to the frontier of set ^ at x eöO. We consider the following problem of the heat exchange control [1, 30-35]

—( ’ ) = Az(t,x),0 < t < T, x eQ (l)

dt

with boundary and initial conditions

Pz(t,x) = u(t,x), 0 < t < T, x €dQ, z (0, x ) = z0 (x), x eQ, here z = z (t, x) — the unknown function, T — the arbitrary positive constant, z0 (■) e L2 (Q) — the initial function. The measurable functions are the control with u(■,■) e L2 (ST), where ST =|(t,x)| t e[0,T], x едП}. In [1, 23-39] is proved that for all u(v)e L2 (ST ) and z° (-)e L2 (Q) problem (l)-(3) has the only solution z = z(t,x) in Hilbert space W,1’0 (QT ), where QT = {(t,x)| t e(0,T), x eO}, consisting of elements space L2 (Qt ), having

quadratically integrable on QT the generalized derivatives z ,i = l,...,n. It is known that the elliptic operator A with the boundary condition Pz (t, x) = 0,0 < t < T, x e dQ has a discrete spectrum, i. e. eigenvalues i, such that 0 <A1 <A2 <... <Ak ^ +o, and the corresponding

eigenfunctions tyk (x), x eQ, composing the complete normalized orthogonally system L2 (Q).

0 < t < T, x eQ.

Further, we shall determine through U the plurality of the control, which is specified below by the certain positive number p.

Definition 1. The multi-valued mapping D: [0,T] 2R,

where R = (-да,да) named strongly invariant concerning problem (l)-(3), iffor all (z (')),(Q)^D (0) and u (v)eU the injection is carried out (z (t,■)) e D (t) for all 0 < t < T, where ( )— the standard norm, z (■,•)— the corresponding solution to the problem (l)- (3) [2,266-276; 3,232-233].

Definition 2. The multi-valued mapping D: [0,T] 2R,

where R = (—да, да), named weakly invariant concerningproblem (l)-(3), iffor all {z0 (■)) e D(0) the control exists u(■,■) eU namely (z(t,-))e D(t) for all 0 < t < T, where ( )— the standard norm, z (•,•)— the corresponding solution to theproblem

(l)-(3).

2 Statement of the problem

In this work the weak and strong invariance of the constant multi-valued mapping is studied in the form of D(t) = [0,b], 0 < t < T, where b — the positive constant.

Our further objective is to find the relations between parameters T, b, p so that to provide the strong and weak invariance of the multi-valued mapping D (t), t e[0,T ] concerning problem (l)-(3).

3 Main results

A). Let {z (t,-)H\z (t,-)||,0 < t < T, and

U =

k=1 Van

(v):JZ( 1 u(t,x)(pk (x)dx \ <p,t e[0,T]

Here ||z (t, ■)) = j |z (t, x )| dx ^t^z] (t ),0 < t < T.

Proposition l. For any function u(■,■)efi is right the following inequality

2

27

Section 6. Mathematics

k=1 У 0

je h( Tj u(t,x)(pk (x)dxdz I <

(i - e^

У h J

■P

Theorem 1. The multi-valued mapping D(t), t e [0,T], is strongly invariant concerning problem (l)-(3), when is executed following inequality

P<\b. (4)

The proving of theorem 1. Let p < \b. We shall Show the strong invariance of the muilti-valued mapping D (< ),0S'* T concerning problem (l)-(3). For all z(0,-),||z(0,-)||< b and u(t,-),||u(t,-)||<p, we have

||z(t,-)||2 = I|z(t,x)) dx = Xzk (t) =

П k=1

= zf z°ke ~ht +| e ~h (t-t)| u (t, s )<pk (s )dsdr) =

dr +

<±\e-v + 2C" J e-‘4 ±\<| J и (t,s )</,, (s )ds

k=i о уk=i an

+zf 1e _Я' (t~-) 1 u (t, s )Pk (s )dsdz) .

k=l у о

Now applying twice Koshi-Bunyakovskiy inequality and proposition 1 we have

||z (t, -)|| < b2e 24 + 2bpe v j e k( -)dz +

0

Consequently, ||z(t,■) < be+ p

21 - e

4 +p f 1 - 1 Я1 e-ч л

1 Я1 J

P2-

Hence and resulting from inequality (4) z (t ,•) < b. This means that the multi-valued mapping D(t),0 < t < T is strongly invariant concerning problem (l)-(3). Theorem 1 is proved.

Note. We can demonstrate that the multi-valued mapping D (t ),0 < t < T is always weakly invariant concerning problem

(l)-(3).

Indeed for all z0 (■), |z0 (■)) ^ b, is possible to choose the control circuit u(■,■)eU so that the multi-valued mapping D(t) ,0 < t < T will be weakly invariant concerning problem

(l)-(3).

Really, for all z0 (■), ||z0 ()|| < b, with u(t,x) = 0, t > 0, we have

|Z(t,■)) ='E}z°ke~kt +|e~k{t~-) I u(t,s)(pk (s)dsdz ] =

k=1 V k an

= ±«e) <£|z;f <b

B).

k=1

Let

(z (v))=| Z У0Ц

U =

<•> •):

k=1 о \ en

Zjl J u (t, s )(Pk (s )ds| dt <p

and

Here

\\z (■>OK II\z (t >-f dt =JZ| zkdt.

k=l 0

Let z0 (■) any element space L2 (Q), satisfying the condition |Z° (')||eD(0), i. e. ||z° (')- b, but u (■,■) — any possible control, i. e. u(■,■) e U. We have

IZ(t’Of = 1 |z(t,x)) dx = ZZk (t) =

П k=1

= Zf z°ke~kk +1 e~h{(~T 1 u(t,s)pk (s)dsdz ] =

^Z

\zl\ e-2* + 2e-*\zl\ je~2A(t~r)drx

u(T,s)(Pk (s)dsj dT +

+|e~22i(t^dzji J u(t,s)<pk (s)ds I dv

2 Л

<

< b2e 24 + 2bpe ^. f e 2i‘(t r>dz + p21e 2h(t TdT.

V 0 0

Consequently,

IZ (t ,■)!< be +-j^ sjl—f2^.

11 v n л/2Л

(5)

We introduce the following function

Л = be~4 +-f^ Vl - e~22lt t > 0.

Proposition 2. For all positive parameters b, p is right the

following equality supf(t) = f(t0)= b2 +---------------, where

t>o У 2A,

1 ,

10 =-------In

0 2 A,

(

2 Л

1-

i V 2Aib J

The proving of Proposition 2. We study upon the function extremum f (t) on the half-line [0,да). We

Calculate derived

Г{( Щ1-,^ )•

Hence it is easy to demonstrate that if 0 < t < 10, that f'(f'о) = °> if to <t> that f '(t)< f '(f0) = 0.

Consequently, sup f (t ) = f (t0 ) =

t >0

b2 +P.

2A,

Theorem 2. If p < b2At (l — T)/T, the multi-valued mapping D(t),0< t < T is strongly invariant concerning problem (l)-(3).

Theorem 3. If 2X1 > 1, the multi-valued mapping D(t),0< t < T is weakly invariant concerning the problem

(l)-(3).

The proving of theorem 2. Let p ^ &У2Aj (l — T)/T.

The following inequality results from (5) and proposition 2

IZ(v)|f =||Z(t>-)||2 dt ^ II f (t)|2 dt <|| f (t0) dt =

b 2

2A

T < b2.

Consequently, ||z (v)||e D(t) for all 0 < t < T, i. e. D(t) is strongly invariant concerning problem (l)-(3). Theorem 2 is proved.

2

2

ÖS2

k

9^2

k=l

2

0

9^2

ÖS2

9^2

2

2

2

2

0

28

Necessary conditions of summability of spectral expansion on eigenfuction of the operator laplace

The proving of theorem 3. Let 2A1 > 1. We Make sure that the multi-valued mapping D (t ),0< t < T is weakly invariant. For all ||z0 (■) eF(0) we choose the control u(t,x) = 0, t e[0,T], x е5П, i. e.

I u(t,s)(pk (s)ds = 0, k = 1,2,.... Then,

an

||z (v)||2 =|| |z (f,-)||2 dt =

0

« T 2 1 _ e-2*T b2

= t\«e-kt )2 dt < b2 —-------< — < b2.

’ 2\ 2\

Consequently, ||z ('>')|e D (t) for all 0 < t < T, i. e. D (t ),0< t < T is weakly invariant concerning the problem (l)-(3). Theorem 3 is proved.

References:

1. Egorov A. I. The optimal control of the heat and diffusion processes. Nauka Moscov, 1978 (in Russian).

2. Feuer A., Heymann M. П — invariance in control systems with bounded controls. J. Math.Anal.And Appl.53, no.2 (1976), 266-276.

3. Tukhtasinov M., Kh.Ya. Mustapokulov. Invariant sets in systems with distributed parameters. Theses of the reports of the International Russian-Balkarsky symposium. Nalchik city. 2010. 232-233. (in Russian).

Pirmatov Shamshod Turgunboevich, Head of Higher mathematics department, candidate of physico-mathematical sciences, an associate professor of Tashkent state technical university named after Abu Raikhon Beruni E-mail: shamshod@rambler.ru

Necessary conditions of summability of spectral expansion on eigenfuction of the operator laplace

Abstract: The spectral decomposition connected with self-conjugate expansion are considered the operator Laplace in N of dimensional area. It is proved that if spectral decomposition of any function in some point is summarized by Riesz’s means, its average value about a in the specified point possesses the generalized continuity.

Keywords: eigenfuction, eigenvalues, spectral expansion, local bounded variation, Riesz’s means, generalized continuity, summability, operator Laplace.

Introduction. The question on sufficient conditions at which performance it is possible to approximate function by its spectral decomposition connected with self-interfaced expansion of the elliptic operator, by present time is well studied and in detail shined in the mathematical literature as in our country, and abroad. Recently interest to these problems has noticeably increased, and the delicate questions connected with spectral decomposition of rough functions have undergone to research more.

From the mathematical literature well-known the examples showing, that spectral decomposition can converge even in those points where decomposed function has break so the usual requirement of smoothness is not necessary. However in all these examples functions in some the generalized sense nevertheless is continuous in a considered point. This fact for the spectral decomposition ad equating to self-interfaced expansion of operator Laplace in any multivariate area for the first time has been established in E. C. Titchmarsha’s work [1, 255-258]. Corresponding results have been received by them for Riesz means of spectral decomposition in case of when the order of averages is an integer, and also in the assumption, that dimension of considered area is not so great.

Conseder the following orthonormal system of eigenfuc-tions of Laplace operator:

-Au. (x) = A.u. (x),

x efic RN

Define the spectral expansions:

EA(x> f ) =X fkUk(x).

Af. <A

We introduce Riesz means of order s > 0:

E'J(x) = E(z-yj fu= J -j \ dEJ.

Denote

sa f (x)=

R -

|2 \

a>(a, N)

1

\y\<R

1 -1

R2

f (x + y )dy >

where

a>(a,N) = J (l - |yf j dy.

\y\^1

Theoreml. Let f e L2 (D) and s > 0. Suppose that for some x eQ the following ratio is carried out

lim E’J(x) = A.

A—\ /

Then for any a> s — (N — 2)/4 the following equality limS“ f (x) = A

R-^0

is valid.

a

29