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of the control can be made arbitrarily large enough small time interval. For this purpose, we choose the initial data as follows: Zj0 = b, z°k = 0, k = 2,..., for every positive t < T
management u(t) in the interval [0,t] defined as a function e *
of the form u = uk (t) = 0,0 <t < t, k = 2,....
| e 2^ds
It is easy to notice so the selected control is valid. Then from the representation of the solution of the problem (l) -
(3) have z (t ) = Means
+ je((r)dr\fa, 0 < t < T.
)l
b+Pp
je 2vds
=dr
b + P\I e 2Vds
(8)
If we introduce the notation x(t ) = ||z (t )|| > from (8) we have x(0) = b and
c \
X(t )
-\b -XiPjj e 2Xsds +
je 2X'ds
Hence it is easy to see that for sufficiently small positive t, x'(t)> 0, t. e. x(t)> b at the same values t. Therefore ||z (t)|| e D (t) not for all t > 0. Theorem 3 is proved.
Note. As a note, you can show that the multi-valued mapping D(t), 0 < t < T always weakly invariant with respect to the problem (l) - (3).
References:
1. Egorov A. I. The optimal control of the heat and diffusion processes. Nauka Moscov, 1978 (in Russian).
2. Feuer A., Heymann M. <<Eqn118.eps>>invariance in control systems with bounded controls. J. Math.Anal.And Appl.53, no.2 (1976), 266-276.
3. Tukhtasinov M., Mustapokulov Kh.Ya.. Invariant sets in systems with distributed parameters. Theses of the reports of the International Russian-Balkarsky symposium. Nalchik city. 2010. 232-233. (in Russian).
4. Mustapokulov Kh.Ya.. On the invariance of the constant multi-valued mapping in the heat conductivity problem. J. European science review.№ 5-6, 2015, 27-29.
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Pirmatov Shamshod Turgunboevich, Head of Higher mathematics department, candidate of physical-mathematical sciences, an associate professor of Tashkent state technical university named after Abu Raikhon Beruni E-mail: shamshod@rambler.ru Kholkhodjaev Bokhodir Asatullaevich senior teacher of Tashkent state technical university named after Abu Raikhon Beruni
About the generalized continuity of functions in points of convergence of their spectral expansion connected with Schrodinger's operator
Abstract: It is paper 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, Riesz's means, generalized continuity, summability, operator Schrodinger.
Let flci3any limited area with smooth border ofdQe CR™ , and let q (x ) -non-negative function from a class L2 (Q).
We will consider, Schrodinger's operator of L (x, D) = -À + q (x) with range of definition of C"(Q). Let the operator H be one of self-conjugate expansions of the operator L with a discrete range. We will designate through own values, and through un (x) corresponding own functions of this operator, i. e. Hun = Xnun (x ).
Spectral expansion of any function f e L2 (Q) has an appearance
EJ ( * ) =X fnUn ( * ). (1)
Xn <x
and Riesz's mean of an order s are defined by equality
V (*)=f(1
o (
f u
J n n
-II1 -j\ dEJ.
(2)
Kf (x) =Sfl -T
k<I V l
Average value about a > 0 functions f (x) in this point * is defined as follows:
i i a-1 r(a + 3/2) u
saf (x )=
n3,2r(a)R3
I
\y <R
1 -1
R
f (x + y )dy. (3)
About the generalized continuity of functions in points of convergence of their spectral expansion connected with.
Теорема 1. Let f e L2(Q) . We assume that at some 5 > 0 in a point x ей spectral function expansion f e L2(Q) it is summarized by Riesz's means of order s. Then for any
a> max j 1,s| fairly following statement: jim^f(x) = f (x).
Before proving this theorem, we will establish justice of several lemmas. For any function g(t) having t > 0 locally limited variation on a half-line, we will enter its averages of Riesz following in a way:
g
(Я) = |(\ -jj dg(t).
Lemma 1. Let g(t) be the function of local
bounded variation on the half-line t > 0 and for some P > 0 the following inequality
J t- \dg(
t ) <œ
is valid. Set
gs(X) = J (1 -^dg(t).
If I > ß + s +1/2 then the following equality
»
J (Wi T'J, (Wf )dg (t ) =
= 2-
r(s+Do is valid.
Proofsee[4. 29-31].
Lemma 2. Let a > -1. Then
m 2
J (VI)-' -U'J,+1+, (R-JX)E IfdX
(4)
2 r(l +1) J (Ryfly' J, (R4l)dE, f (x) = S^ f (x) (5)
0
and integral converges absolutely and uniformly. Proof see [1. 255-258].
Comparing the relation (4) and (5), we receive equality
SaRf(x) = 2a-s-1/2 Г(а + 3/2) Rs-a-1/2 x
r(s +1)
xjl" ^ Ja+s+3/2 (RjlWJdX
œ a-s+3/2 2
(6)
Designating E[f = gs (A) and after, replacement of change of t = Ry/x we receive the following
SaRf (x) = 2'
. Г(а + 3/2)
J g(XR-2)i 2 1а++У2(Л )dt (7)
r(* +1) 0
Lemma 3. Let a > 0. We will designate
R f 2 Л"-1
Lnt) = Jj1 - R r cosЯ(r - t)dr,
(8)
0 < t < R. 2
Then the following assessment is fair Ln (R, t) = O(R ).
Proofsee[2. 238-239].
Lemma 4. We will put
f Î i
..2 Л
1 = "" 1 - F
rdr f sin цп (r -t)Çn (t)tdt. (9)
Я 0 V R ; 0
Then, for any a > 0 the following assessment is
" |2
XI in (R)| = O(RR).
(10)
Proof of the lemma 4. We enter the following function
rq(a+y)
h(y ) =
-, если y < r.
0, если |y| > r. Fourier's coefficients of function h(y) are equal
Pn (r ) = \ h( y )un ( y )dy.
a
Passing to spherical coordinates, we will receive
q(a + y )
ßn (r ) = J
-un (a + y )dy =
r 1 r = J -t 2dt J q(a + t0)un (a + tO)dQ = J^n(t )tdt.
o * e o
Having differentiated this equality, we will receive ßn(r) = r%n(r). Further we will square \ßn(r)\ and we will summarize on, using Parseval's equality:
»
Ж (r )|2 = J
q(a + y)
dy *!<«» J TJ = O(r) (11)
Then
1 R ( r2 I r
In = — JI 1 - R I rdr I sin — (r -1 )Çn (t )tdt =
= 1(1 -"~2T I rdrJsin —n(r-1)ßn(t)dt
R
Integrating in parts J sin jin (r -1(t)dt integral, we
0
will receive
J sin (r -1 )p'n(t )dt = sin nn (r -1 )pn (t )| t=0J cos (r -1 )pn (t )dt
0 0
= nn J cos nn(r -1)pn(t)dt
0
Hence,
R f 2 A"-1 r
In = i I 1 " R2 rdr | cos ( r -1 )pn (t )dt.
o V R ) o
We will change an integration order
I. =
R R f 2 \a
\ßn (t)dt\\\ - R2 r cos (r - t)dr. (12)
According to definition (8) equality (12) can be written
R
down in the following look In = JLn(R,t)pn(t)dt. On a lemma 3
0
R
\I„(R)| = O(R2)\\pn(t)|dt.
0
Further, follows from Parseval's equality:
XI In (R)|2 = O(R 5)\Z\pn(t )2 dt = O(R 5)J O(t )dt = O(R7).
0 n=1
The lemma 4 is proved.
1 №
Consequence. — ^ fnIn ( R) = O(4R )| f\\ . (13) R n=i
On Weber formula [1. Ch.II, p. 230.], average value of function F equals
0
a-1
0
0
a-1
r
n=1
SaRf (x ) =
r(a + 3/2) n3/2r(a)R3
y <R
|2
1 -1
R2
f (x + y )dy =
= c.i JR^ fun(x)+
r(a + 3/2)h f^ + n3'2r(a)R3 h
(14)
Z J|l-Rï I rdr\sinVn(r-1)|„(t)tdt.
where Ca= 2 2 r(a + —). We enter the following function
J i(0
a+—
Pa(t) = Ca <pa(0) = 1.
a+— (t) 2
Then for average value (14) in compliance (5) and (9) we will gain the following impression:
saf (x)=±t(R»n) f,un (x)+rsa+3:2) t Vn (R)
n=1 n r (a)R n=1
The first representation composed in integrated has an appearance (5). It agrees (10) and (13)
-p/ o /^n M a-s+3/2
saf (x)=2a-s-i/21a+3i/)2) J g (XR-2)t—ja+s+m(J )dt+
r(s +1) 0 (15)
O^R )|| flU
Proof of the theorem 1. In equality we will designate (15) subintegral function
Fr(t) = g(tR~2)(4~t)-'-1+Js+1(St). On a theorem condition 1 limit of function gs (t) at t ^^ exists therefore this function is limited. In that case follows from asymptotic estimates of function of Bessel that Fr (t) < F(t), where F(t) — the integrated function.
This assessment allows to apply Lebesgue's theorem of limit transition under a sign of integration, we receive
lim SaRf (x) = f (x).
R^O
References:
1. Titchmarsh E. C. Eigenfunction Expansions Associated with Second Order Differential Equations, part II, Oxford 1958.
2. Pirmatov Sh. About the generalized continuity of functions in points of convergence their spectral pa3Ao^eHHH.//the International conference. Modern problems of calculus mathematics and mathematical physics. Moscow, on June 16-18, 2009, P. 238-239.
3. Alimov Sh. A., V A. Ilyin. Condition exactes de convergence uniforme des developpements spectraux et de leurs moyennes de Riesz pour une extension autoadjoint arbitraire de l'operateur de Laplace.//Comptes Rendus Acad. Scien. Paris, Serie A, t. 237, (1970), P. 461-464.
4. Pirmatov Sh. Necessary conditions of summability of spectral expansion on eigenfuction of the operator Laplace//Euro-pean science review, «East West» Association for Advanced Studies and Higher Education GmbH. Vienna. 5-6. 2015, - P. 29-31.
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