LINEAR INVERSE PROBLEMSFOR MULTI-TERM EQUATION WITH RIEMANN - LIOUVILLE DERIVATIVE Текст научной статьи по специальности «Математика»

Серия «Математика»

2021. Т. 38. С. 36-53

Онлайн-доступ к журналу: http://mathizv.isu.ru

ИЗВЕСТИЯ

Иркутского

государственного

университета

УДК 518.517

MSC 35R30, 35R11, 34G10

DOI https://doi.org/10.26516/1997-7670.2021.38.36

Linear Inverse Problems

for Multi-term Equations with

Riemann — Liouville Derivatives *

M. M. Turov1, V. E. Fedorov1,2, B. T. Kien3

1 Chelyabinsk State University, Chelyabinsk, Russian Federation

2 South Ural State University (National Research University), Chelyabinsk, Russian Federation

3 Institute of Mathematics of Vietnam Academy of Science and Technology, Hanoi, Vietnam

Аннотация. The issues of well-posedness of linear inverse coefficient problems for

multi-term equations in Banach spaces with fractional Riemann - Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fractional part of the order of the highest fractional derivative. Abstract results are applied in the study of inverse problems for partial differential equations with polynomials from a self-adjoint elliptic differential operator with respect to spatial variables and with Riemann - Liouville derivatives in time.

Ключевые слова: inverse problem, Riemann - Liouville fractional derivative, degenerate evolution equation, initial-boundary value problem.

* The reported study was funded by RFBR and VAST (grants 21-51-54001 and QTRU 01-01/21-22).

LINEAR INVERSE PROBLEMS FOR EQUATIONS WITH MULTI-TERM... 37

1. Introduction

Problems for differential equations with unknown coefficients and overdetermination conditions, which are called inverse or inverse coefficient problems, play an important role in many applied research, when the nature of the process is known (the form of the equation of its dynamics is given), some of its parameters are not available for the direct measurement (unknown coefficients), but can be determined using additional measurements of the available parameters (setting overdetermination conditions) [21]. Among the studies of inverse problems for first order differential equations in Banach spaces in addition to the works of A. I. Prilepko, we note the works of I. V. Tikhonov, Yu. S. Eidelman [5], M. V. Falaleev [7], M. Al-Horani, A. Favini [11], S. G. Pyatkov [4]. In recent years, works on the study of inverse problems for equations with fractional derivatives [1; 8; 9; 12-14; 20] have appeared.

Multi-term fractional equations are of great interest to researchers [16; 18; 19]. For the multi-term equations with Riemann - Liouville derivatives unusual properties were revealed even in the scalar case [2; 15; 17]. In particular, the initial value problem of Cauchy type for such equations is solvable only if the initial conditions are given only for fractional derivatives of sufficiently large order [3; 10].

In this paper, for multi-term equations in Banach spaces with Riemann -Liouville derivatives and with bounded operators at them, we consider well-posedness issues for linear inverse problems with an unknown coefficient which is independent of time. In this case, the integral overdetermination condition with the Riemann - Stieltjes integral is used, which includes, as a partial case, the overdetermination condition at a fixed point of time. Both the equations resolved with respect to the highest derivative and equations with a degenerate operator at the highest derivative are considered. In the first case, conditions of the Cauchy type are specified, in the second, the initial conditions have a complex form and depend on the orders of the derivatives from the equation.

In the second section, a criterion for the well-posedness of the inverse problem for the equation in a Banach space, solved with respect to the highest Riemann - Liouville derivative, with bounded operators at the derivatives is obtained. In the third section, the inverse coefficient problem for the equation with a degenerate operator L at the highest derivative is reduced to a system of two inverse problems on subspaces for equations solved with respect to the highest derivative, under the condition that the operator at the second-highest derivative is (L, 0)-bounded. Two essentially different cases are considered: when the fractional part of the order of the second derivative coincides with the fractional part of the order of the highest derivative and when it differs (see [10]). For each case, a criterion for the correctness of the inverse problem is obtained. The fourth

38

M. M. TUROV, V. E. VEDOROV, B. T. KIEN

section contains applications of obtained abstract results to the study of the inverse problem for an equation with polynomials with respect to a self-adjoin elliptic differential operator in spatial variables and with Riemann -Liouville time derivatives in both the nondegenerate and degenerate cases.

2. Nondegenerate inverse problem

Let Z be a Banach space, by C(Z) denote the Banach space of linear bounded operators in Z, R+ := {a € R : a > 0}, h : R+ ^ Z. For fi > 0 denote the Riemann - Liouville integral

t

Jt h(t) := гщ J(t — sf-lh(s)ds.

0

By J° we denote the identity operator. Let a > 0, m := [a] be the smallest integer, which is greater or equal to a, D™ is the usual derivative of the order m € N, D? is the fractional Riemann - Liouville derivative,

i.e. D?h(t) := DnnJ]n-ah(t). At fi < 0 we will use the notation h(t) :=

h(t).

Consider the nondegenerate equation, i. e. the equation, resolved with respect to the highest derivative

m— 1

D?z(t) = ^2 AjD? m+1 z(t) + Y^ BiD?z(t) + X) CsJtS z(t) + V(t)u

j=1 1=1 S=1

(2.1)

with t € (0,T], where 0 < a1 < a2 < ■■■ < an < a, mi := \ai], m := [a], = « — m, l = 1,2,... ,n, fi1 > > ■ ■ ■ > fir > 0, operators Aj,

j = 1,2,... ,m — 1, Bi, l = 1,2,... ,n, Cs, s = 1,2,... ,r, are linear and bounded in the Banach space Z, and ip € C((0, T]; R) ПL1(0,T; R), и € Z.

Let a := max{o^ : l € {1, 2,..., n}, ai — mi < a — m}, m = \o], a := max{ai : l € {1,2,..., n}, ai — mi > a — m}, m = \o]. We denote

by m* := max{m — 1, m} the defect of the Cauchy type problem for the equation (2.1) [10]. Then the initial conditions of the Cauchy type have the form

D?~m+k2(0) = Zk, к = m*,m* + 1,...,m — 1. (2.2)

The overdetermination condition for the inverse problem take in the form

т

J z(t)dp(t) = zt,

(2.3)

where p has a bounded variation on the segment [0,T]. The integral is understood as a vector integral of Riemann - Stieltjes.

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Серия «Математика». 2021. Т. 38. С. 36—53

LINEAR INVERSE PROBLEMS FOR EQUATIONS WITH MULTI-TERM... 39

By a solution of problem (2.1), (2.2), where the element и € Z is known, we will call a function z : (0,T] ^ Z, such that Тф—аz € Cm((0,T]; Z) П Cm—1([0, T ]; Z), Jfa—ai z € Cmt ((0, T ]; Z), l = 1,...,n, z € C ((0,Tj; Z),

s = 1, 2,..., r, and (2.2), (2.1) for t € (0, T] hold.

Theorem 1. [10]. Let a1 < a2 < ••• < an < a, m = \a], mi = \a{\, ai — mi = a — m, l = 1,2,..., n, fa > fa2 > • • • > fa > 0, Aj € C(Z), j = 1, 2,...,m — 1, Bi € C(Z), l = 1, 2,...,n, Cs € C(Z), s = 1, 2,...,r, Zk € Z, к = m*,m* + 1,... ,m — 1, € C((0,T]; R) П L1(0,T; R), и € Z.

Then there exists a unique solution to (2.1), (2.2), and it has the form

m—1 „

z(t) = ^2 Zp(t)zp + Zm-1 (t — s)(fi(s)uds,

p=m*

where at p = m*,m* + 1,... ,m — 1

(2.4)

z»(i) := ш / x-aR>

Г

m—1

xm-1-PI — ^ \j—1—pAj j=p+1

ext d\,

(m—1 n r

I — ^ Xj—mAj — ^ Xai—aBi — ^ \—l3°—aCs

j= 1 1=1 S=1

Г = Г+ U Г— U Го, Го = {X € C : |Л| = r0, arg Л € (—ж, ж)}, Г+ = [X € C : arg Л = -к,Х € [—г0, —то)}, Г— = [X € C : arg Л = —ж, X € (—то, — г0]}, г0 > 0 is a large enough number.

Now consider the inverse problem (2.1)-(2.3), assuming the element и to be unknown. By a solution of this problem we will understand a pair (z,u), where z : (0, T] ^ Z is a solution of problem (2.1), (2.2) with the corresponding и € Z, which satisfies condition (2.3). For brevity, we will also often call this element и € Z the solution of problem (2.1)-(2.3).

We call problem (2.1)-(2.3) well-posed if for any z^ €Z, к = m*,m* +

1,... ,m — 1, zt € Z it has a unique solution и € Z, satisfying the estimate

\W\\z < c(\\zm*\\z + ||2m*+1||z +-----+ \\zm—1\\z + \\zT\\z), where C > 0

does not depend on Zk, к = m*,m* + 1,... ,m — 1, and zt.

For given Zk € Z, к = m*,m* + 1,... ,m — 1, zt € Z we denote

T,. m—1 Tf }

Ф := zT — ^2 Zp(t)zpdp(t), x ■= dp(t) Zm—1(t — s)ip(s)ds.

0 P=m*

0 0

Theorem 2. Let a1 < a2 < ••• < an < a, m = \a], mi = \a{\,

ai — mi = a — m, l = 1,2,... ,n, fa > fa > • • • > fa > 0, Aj € C(Z),

40

M. M. TUROV, V. E. VEDOROV, B. T. KIEN

j = 1,2,..., m — 1, Bi € C(Z), l = 1,2 ,...,n, Cs € C(Z), s = 1,2 ,...,r, € C((0,T]; R) П Li(0,T; R), p : [0,T] ^ R have a bounded variation, for m* = 0 there exist e € (0,T] such that p € C 1([0,e]; R). Then problem

(2.1)-(2.3) is well-posed if and only if there exists the operator x-1 € C(Z). In this case the solution has the form и = х—1ф.

Proof. By Theorem 1, a solution of Cauchy type problem (2.1), (2.2) exists for all Zk € Z, к = m*,m* + 1,...,m — 1, и € Z and has form (2.4). Substituting solution (2.4) in condition (2.3), we obtain the equality

т

J dp(t) 0

m—1

t

^2 Zp(t)zp + I Zm-1(t — s)f (s)uds

p=m* 0

Zt .

It implies the equation xu = ф. Therefore, problem (2.1)-(2.3) is equivalent to the last equation. Its unique solvability for any ф € Z means exactly the existence of an inverse operator x-1 € C(Z). Then

(m—1

Ut ie + E

p=m*

In the work [10] the inequalities

Ker° *

ll^oWII c(z) < -px—XI, IIZP(t)llc(z) < P = 1, 2,...,m — 1, (T5)

were proved. Hence, for p = 1,2,... ,m — 1

т

Zp(t)zpdpi(t)

z>

T

Zp(t)zpdp1(t)

< KeroTHzvllzVoT(p),

o C(Z)

where H0T(Ф) is the variation of the function p on the segment [0, T]. If m* = 0, then

т /* в /* т /*

/ Zo(t)zodp(t) < / Zo(t)zop!(t)dt + / Zo(t)zodp(t)

J 0 C{Z) J 0 C{Z) J £

£

/Hi Kprot

Thus, the inverse problem is well-posed, if and only if x 1 € C(Z). □

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Серия «Математика». 2021. Т. 38. С. 36—53

LINEAR INVERSE PROBLEMS FOR EQUATIONS WITH MULTI-TERM... 41

3. Degenerate inverse problems

Let X, У be Banach spaces, L, Mi,M2,..., Mm-i ,Ni,N2, ..., Nn, Si,

S2,..., Sr € C(X; У) (linear continuous operators from A into У), ker L = {0}. All operators Ni, N2, ..., Nn, Si, S2,..., Sr are not zero, while some of (or all) Mj are zero operators. If Mj = 0 for all j = 1, 2,..., m — 1, then put j0 = 0; otherwise there exists j0 € {1,2,..., m — 1} such that Mj0 = 0, Mj = 0, for all j = jo + 1,jo + 2,...,m — 1.

Consider the evolution equation

j о n r

D?Lx(t) = £ Mj D?~m++ x(t) + £ Nt Dat' x(t) + £ x(t) + <p(t)u,

j=i l=i S=1

(3.1)

which we call degenerate provided that ker L = {0}. Here 0 < a1 < a2 < ■ ■ ■ < an < a, m := [a], mi := \a{\, ai — mi = a — m, l = 1,2,... ,n, a := max{ai : l € {1, 2,..., n}, ai — mi < a — m}, m := \a], a := max{a^ : l € {1, 2,... ,n}, ai — mi > a — m}, m := \a], m* := max{m — 1,m}, Pi > p2 > ■■■ > l3r > 0, <p € C((0,T];У) П Li(0,T;У), и € Z. Consider the two possible cases: an < a — m + j0 and an > a — m + j0.

3.1. The first case

If an < a — m + j0, suppose that the operator Mj0 is (L, c)-bounded:

3a > 0 € C (|^| > a) ^ ((ц.Ь — Mj0)-i € С(У; A)).

In this case we define the operators R^(Mj0) := (ц,Ь — Mj0)-iL and L^(Mj0) := L(^,L — Mj0)-i, the projections

P :=2~j RL^M0)d^ € £(*), Q :=2tij LH(M*) d» € W),

i i

where 7 := {^ € C : 1^1 = r > a} (see [22, p. 89, 90]). Put A0 := ker P, Ai := imP, У0 := ker Q, Уi := imQ. Denote P0 := I — P, Q0 := I — Q for brevity, by Lq, Mj,q, Ni,q and Ss,q denote the restrictions of L, Mj, j = 1,2,..., jo, N1, l = 1,2,... ,n and Ss, s = 1,2,...,r, on Aq, q = 0,1. It is known (see [22, p. 90, 91]) that LP = QL, Mj0P = QMj0, Mj0,q € C{Xq; yq) and Lq € C{Xq; Уq), q = 0,1. Moreover, in the situation under consideration we have the operators M~oi0 € С(У0; X0) and L-1 €

£№; x1).

We also suppose that L0 = 0; in this case the (L, c)-bounded operator Mj0 is called (L, 0)-bounded [22]. Moreover, under the additional conditions

QMj = MjP, j = 1,2,...,jo — 1, QNi = N1P, l = 1,2,...,n,

QSs = SsP, s = 1, 2,...,r,

(3.2)

42

M. M. TUROV, V. E. VEDOROV, B. T. KIEN

equation (3.1) with the initial conditions

Da-m+kж(0) = Хк, k = m*,m* + 1,...,io - 1, ( )

Da-rn+k (px)(0) = Xk, fc = jo ,jo + 1, . . .,m - 1, .

and with the overdetermination condition

т

jxmm = *r <3-4>

o

can be reduced to the following problems for two equations on the mutually complemented subspaces X1 and X0:

30

Dfv(t) = £ L~-lMjD?~m++ v(t) + £ L-1^D?v(t) +

3 = 1

1=1 -1 1

+ £ L- ^s'tv(t) + v(t)L- 1u1, t € (0,T],

5=1

d:

.a-m+k t

v(0) = Vk, к = m* ,m* + 1,... ,m — 1,

T

v(t)d^(t) = vt

(3.5)

(3.6)

(3.7)

and

, . io-1 n

Drm+30w(t) = — £ M- 1oMjD?-m+3w(t) — £ M-oNiD?w(t) — 3=1 1=1

r

— £ Mj0,ossJtsw(t) — V(t)Min,oU°, t € (0,T],

5=1 ла-т+к

jo,ol

Df m+lcw(0) = wk, к = m*,m* + 1,...,jo — 1, T

/ w(t)d^(t) = wt,

(3.8)

(3.9)

(3.10)

o

where v(t) := Px(t), Vk := Pxu, к = m*,... ,m — 1, vx := Ржт, -u1 = Q«, ■w(t) := Pox(t), Wk := PoTfc, fc = m*,... ,jo — 1, wx := Pott and -uo = Qou.

Note that equations (3.5) and (3.8) are resolved with respect to the highest fractional derivative. Consequently, by theorem 2 problem (3.5)-(3.7) (problem (3.8)-(3.10)) is well-posed if and only if there exists the operator X\1 € C(X1; У1) (%£ € C(X°; £*)), wherein each of these problems has a unique solution и1 = Х\1'Ф1 Co = Xo 1 Vto). Here

T~ j. T~ m-1

X1 := / dp(t) Z^_1(t — s)L11^(s)ds, ^1 := vT — ^ Z^(t)vpd^(t)

p=m*

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LINEAR INVERSE PROBLEMS FOR EQUATIONS WITH MULTI-TERM... 43

ZpO := Ш I A-aR\\ Xm-1~PL\ - £ V-1~p мЛ ext dX,

Г1 \ j=p+1 )

( jo n r \

('

R\ := ( Li - £ Xi~mMj,i - £ Xai~aNl,i - £ A"л-“&,1 i=i 1=1 s=i

)

t ^ io-1

Xo := dp(t) Z^-i(t-s)M—0p(s)ds, ф0 := Z%(t)wpdp(t),

0 P=m*

^) := / Л-“Я°А ■ (xm-1-pMj0to + JE А^'-1-РМ, J eAtdA

Го V i=p+i )

)'

^ := (м,0,o + ~£ Xi-mMj)0 + £ A^-^ + £ X-^-aSsfl

30-1

£

3=1

Obvious transformations are used here and contours Г1 and Г0 are constructed as Г, taking into account the norms of the operators in the problem.

Now we introduce strict definitions and formulations.

A solution to problem (3.1), (3.3) is a function x : (0,T] ^ X such that J™-aLx € Cm((0,Tj; У) П Cm-1([0,T]; У), J™-ax € C*((0,Tj; A) П C>°-1 ([0, T]; A), Jp-агж € Cmi((0, T]; A), jfsж € C((0,T]; A), equality

(3.1) for all t € (0,T] and conditions (3.3) are satisfied.

A solution to problem (3.1), (3.3), (3.4) is a pair (x,u) such that x is a solution of problem (3.1), (3.3) with the corresponding и € У and satisfies condition (3.4).

We call problem (3.1), (3.3), (3.4) well-posed if for any Xk € X, к = 0,1,.. .,j0 - 1, Xk € A1, к = j0,j0 + 1,.. .,m - 1 and хт € X it has a unique solution satisfying the estimate ||-u||y < C(\\xm*\\x + \\xm*+1\\x +

----+ ||жт-1||* + \\хт\\x), where C does not depend on Xk, к = m*,... ,m-1

and хт.

Theorem 1. Let 0 < a1 < a2 < ■■■ < an < a, m := \a], mi := \a{\, ai - mi = a - m, l = 1, 2,... ,n, fi1 > fi2 > ■ ■ ■ > fir > 0, L,Mj € C(Z), j = 1, 2,...,m - 1, Ni € C(Z), l = 1, 2,...,n, Ss € C(Z), s = 1, 2,...,r, conditions (3.2) hold, an < a - m + j0, operator Mj0 be (L, 0)-bounded, p € C((0,T]; R) П L1(0,T; R), function p : [0,T] ^ R have a bounded variation, for m* =0 there exist e € (0,T] such that p € C1 ([0,e]; R). Then problem (3.1), (3.3), (3.4) is well-posed if and only if there exist the operators x-1 € C(X1; У1), y-1 € C(X°; T°). In this case the solution has the form и = х-1ф1 + X- V0.

44

M. M. TUROV, V. E. VEDOROV, B. T. KIEN

3.2. The second case

Let an > a — m + j0, and let the operator Nn be (L, 0)-bounded. Then

P :=2- J RL,(Nn)d^ e £(*), Q :=2- JLi(N^ e £(У)

i i

and under the additional conditions

QMj = Mj P, j = 1, 2,...Jo, QNi = NiP, l = 1, 2,...,n — 1, ( )

QSs = SsP, s = 1,2,... ,r, (3.11)

by analogy with the first case consider two equations on the subspaces T1 and T0: equation

D?v(t)

30

L-1MjDt~m+j

П

v(t) + E L^NiDt1 v(t) +

3=1 1=1

+ E L-1 SsJtsv(t) + tp^L^u1,

endowed with conditions

(3.12)

Da-m+k v(0)

Vk, к = m*,m* + 1,... ,m — 1,

and equation

т

J v(t)d^(t) = vt , 0

(3.13)

(3.14)

30 П-1

D?nW(t) = — E Nr0oM3Drm+Jw(t) — E N-lN.D?w(t) 3=1 i=1

— E Nn,oSsJtsw(t) — <f(t)N~0u°, t e (0,TЬ

S=1

for which determine the parameters

Ш0 = \a0],mo = [aol,rn0 := max{m0 — 1,mo},

(3.15)

where

a0 = max{a — m + j0,ai : l = 1,2,... ,n — 1, ai — mi <an — mn, a — m < an — mn}, a0 = max{a — m + j0,ai : l = 1,2,... ,n — 1, ai — mi > an — mn, a — m > an — mn}.

Now we can formulate the initial conditions for equation (3.15):

D<xn-mn+k w(0) = Wk, ^ = m*0, m0 + 1,... ,mn — 1, (3.16)

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Серия «Математика». 2021. Т. 38. С. 36—53

LINEAR INVERSE PROBLEMS FOR EQUATIONS WITH MULTI-TERM... 45

and the overdetermination condition

т

j w(t)dp1(t) = wt . (3-17)

0

As before, v(t) = Px(t), и1 = Qu, w(t) = P0x(t), u° = Q0u.

In this case the conditions of the inverse problem for equation (3.1) can be given in the form

^a-m+k(px)(0) = Vk, fc = m*,m* + 1,..., m — 1, (3.18)

Dtn-mn+k(PoX)(0) = wk, к = m*, m* + 1,.. .,mn — 1, (3.19)

т

J x(t)dp,(t) = xt = vt + wt . (3.20)

0

Problem (3.1), (3.18)-(3.20) is called well-posed, if problems (3.12)-(3.14) and (3.15)-(3.17) are well-posed.

By Theorem 2, as in the first case, we obtain the following result.

Theorem 3. Let 0 < a1 < a2 < ••• < an < a, m := \a], mi := \a{\, ai — mi = a — m, l = 1,2,... ,n, > fi2 > ••• > fir > 0, L,Mj e C(Z),

j = 1,2,...,m — 1, Ni e C(Z), l = 1,2,...,n, Ss e C(Z), s = 1,2,...,r, an > a — m + j0, conditions (3.11) hold, operator Nn be (L, 0)-bounded, p e C((0,T]; R) П L1(0,T; R), function ц. : [0,T] ^ R have a bounded variation, for m*m0 = 0 there exist e e (0,T] such that p, e C1 ([0,e]; R). Then problem (3.1), (3.18)-(3.20) is well-posed if and only if there exist operators x-1 e C(X1; У1), x-1 e C(X°; T0). In this case the solution has the form и = x-V1 + X-V0.

Here

T~ j. T,. m-1

X1 ■= dp(t) Z^-1 (t — s)L-1p(s)ds, Ф1 := vT — ^ Z^(t)vpd^(t),

00

0 P=m*

z!(t) = 7^J a-aR\ j Xm-1-pL1 — ^ Xj-1-pMjA

p 2ni

-1-р*'з,1 I extdX,

('

Г1

30

j=p+1

R\ := [^1 — £ Х>-тМзЛ — £ Xai-aNi,1 — £ X-3^-aSs,1

j=1 1=1 S=1

)

1

•M )

X0 :

T t

J dp1(t) j Z°-1(t— s)N-0lф)ds, Ф0 : 00

т

/mn-1

E Z<0(t)wpdp,(t),

0 P=mo

46

M. M. TUROV, V. E. VEDOROV, B. T. KIEN

1 r mn-1 \

Z0(t) = — / X-anR\ I xmn-1~PNn,o + ^ \l-1~pTl I extd\

21 r0 V l=p+1 )

('

jo n—1 r

R°x := ( Nn,o + £ Ai-mMj + £ A";^ + £ X-^-aS.

j=1 1=1 s=1

Ti = Ni for ai - mi = an - mn, Tt = 0 for at - mb = an - mn.

Remark 1. In the considered cases, when the operator at the second highest derivative is (L, 0)-bounded, the conditions p®-m+k(Px)(0) = Xk are equivalent to р^-т+к (Lx)( 0) = yk with yk = Lx к, к = m*,m* + 1,..., m -1, since ker L = ker P and there exists L-1 € С(У1; X1).

4. Applications

Let Р1 (X) = £ ap\p, (A) = £ %Xp, P^(X) = £ clpXp, Pi(X) =

p=0 p=0 p=0

£ dspXp, ap, bp, clp, dp € C, p = 0,1,...,n € N, j = 1, 2,...,m - 1, l =

p=0

1,2,... ,n, s = 1,2,... ,r, au =0, where Q C Rd is a bounded domain with a smooth boundary dQ,

(Ah)(C) = £ aq(0 Ы<2р

(Bihm = £ blq(o dlq'm

Ы<Р1

dp1dp2 ...эр/ ’

aq € С“(Q),

д/1 dp2 ...d// ’

bi, € C“(dQ), l = 1,2,

P

q = (q1, Q2,..., Qd) € Nq, Iql = q1 + ■ ■ ■ + qd, and the operator pencil A, B1, B2, ..., Bp is regularly elliptic [6]. Let the operator A1 € Cl(L2(Q)) have the domain }(Q) := {h € H2p(Q) : Bih(£) =0,1 =

1,2,..., p, ( € 9Q}, A1h := Ah. Suppose that A1 is a self-adjoint operator; then its spectrum a (A/) is real and discrete [6]. Moreover, assume that the spectrum a(A{) is bounded from the right and does not contain zero, {<pk : к € N} is an orthonormal system of eigenfunctions of A1 in L2(Q) which is enumerated in nonincreasing order of the corresponding eigenvalues {Xk : к € N}, taking into account their multiplicities.

Let 0 < a1 < a2 < ■ ■ ■ < an < a, m - 1 < a < m € N, mi - 1 < щ < mi € N, ai - mi = a - m, l = 1,2,... ,n, /1 > /2 > ■ ■ ■ > /r > 0. Consider the equation in Q x (0, T]

D?P1(A)V(/t)

m—1

£ Pi(A)Drm+3v(£,t)+

3 = 1

П Г Q

+ £ H(A)D/v(i,t) + £ Pl(A)jfsv((,t) + <p(t)w(£),

1=1 S=1

(4.1)

Известия Иркутского государственного университета.

Серия «Математика». 2021. Т. 38. С. 36—53

LINEAR INVERSE PROBLEMS FOR EQUATIONS WITH MULTI-TERM... 47

with the boundary conditions at (£, t) € dQ x (0, T]

BlAkv(£,t) = 0, к = 0,1,...,u - 1, l = 1, 2,..., p,

and the overdetermination condition

т

j v(i,t)dp(t) = vT (£), £ € Q.

0

The form of initial conditions depends on situation. Put

^ = {h € H 2P" (Q) : Bi Ak h(£) =0,k = 0,1,...,u - 1, l = 1,2,...,p,£ € 9Q}, У = L2(Q),

(4.2)

(4.3)

L = Pi(^) € £(*; У), Mj = Pi>(Д) € £(*; ?), j = 1, 2,..., m - 1, Щ = P3 (Л) €£(*; У), l = 1,2,... ,n, Ss = Р|(Д) €£(*; ?), s = 1, 2,...,r.

If P1(\k) = 0 for all к € N, then there exists the inverse operator L-1 € С(У; X). Define the defect m* for the collection of numbers a1,a2, ..., an, a and consider the Cauchy type conditions

A“-m+fc v(Z, 0) = vk (£), к = m*,m* + 1,...,m - 1, £ € Q. (4.4)

Then problem (4.1)-(4.4) can be presented in the form of (2.1)-(2.3), where Z = X, Aj = L-1Mj € C(Z), j = 1, 2,...,m - 1, Bt = L-1Nt € C(Z),

l = 1,2,... ,n, Cs = L-1Ss € C(Z), s = 1,2,... ,r, zk = vk(■), к = m*,m* + 1,... ,m - 1, zt = vt(■), и = L-1w(-). By Theorem 2 problem

(4.1)-(4.4) is well-posed if and only if for all к € N

т t

J J J R\,kex(t s)d\p>(s)dsdp(t)

0 0 г

f m-1

R\k := ( \a?1 (Xk) - E Xa+2-mPi(Xk)

> c,

3 = 1

Г

J2xai н (xk) -£ x-l3s Pi(\k)

1=1

5=1

■)

1

It follows from the equation

т t

X = 2“ ^(-,<Pk)<Pk J J J R\,keX(t S)d\^(s)dsdp(t),

k=1 0 0 Г

(4.5)

where {■, ■) is the inner product in L2(Q).

48

M. M. TUROV, V. E. VEDOROV, B. T. KIEN

Example 1. Take a = 5/2, m = 3, n = 1, r = 1, a1 = 2/3, p1 = 1/2, Pi(A) = A2, P}( A) = 61A + 62A2, P22(A) = О, РЦХ) = cq + C2A2, Pi (A) =

d0 + d1A, d = 1, Q = (0, -r), p = 1, МЛ, = , ^1 = I, P is the single jump

at fQ € (0,T] function. Then m = 0, m = 1, m* = 1, A^ = —k2 for к € N, and (4.1)-(4.4) has the form

A5/2 Щ4 (06 = (bl Цт + Ь2 D^/2V(S,t)+ (cq + C2 A2/3U(S,f) +

+ (do + diJl/2v(i,t) + Ц£), (£,i) € (0,^) x (0,T], v(0, t) = u(^, t) = 0(0, i) = 0(ж, t) = 0, t € (0, T],

D1/2v(£, 0) = ui({), D3/2v(£, 0) = «2(0, C € (0, ж),

v((,to)= vT(0, £ € (0, ж).

The necessary and sufficient condition (4.5) of its well-posedness has the form

J [k4A3/2 + (b1k2 — Ы4)А-r

— (dQ — d1k2 )A-3/2) 1 (eXT

1/2 — (cq + C2k4)X-1/3 —

— 1)dX

> c

Vk € N.

Consider the degenerate case. Suppose that P1(Xk) = 0 for some к € N, while P™ = 0, Р/ = 0, j = jQ + 1,jQ + 2,...,m — 1, for some jQ € {0,1,... ,m — 1}, and an > a — m+jQ. Then provided that the polynomials P1 and РЦ do not have common roots on the set {Xk}, the operator Nn (L, 0)-bounded (see [22]), while the projections have the form

p = ^',^k, Q = ^,^k^k.

PlPk )=Q PlPk)=Q

By Remark 1, we take the initial conditions in the form Da-rn+k P1(J)U(^, 0) = yk (£), к = m*,m* + 1,...,m — 1, £ € Q, (4.6)

l^Da.n-mn+k u(^, 0),p.} = Ck,, p^.) = 0,k = mQ, mQ+1,..., mn—1. (4.7)

Here conditions (4.7) are given for j € N such that P1(Xj) = 0; this set is finite, since P1 is a polynomial. The finite set of n numbers c^j defines the projection Dln-mn+ku(^, 0) to the subspace XQ := ker P, к = mQ,m* +

1.. .., mn — 1. The defects m*, m* are determined as above. Now, (4.1)-(4.3), (4.6), (4.7) is representable in the form (3.1), (3.18)-(3.20) with the spaces X and У and the operators L, Mj ,N1, Ss, j = 1,2,...,m — 1, l =

1.2.. .. ,n, s = 1, 2,... ,r, which are chosen above.

Theorem 3 implies that problem (4.1)-(4.3), (4.6), (4.7) is well-posed if and only if for all к € N such that P1(Xk) = 0,

т t

J J J P\kcX(t dX^(s)dsd^(t)

Q Q Г1

> c,

(4.8)

Известия Иркутского государственного университета.

Серия «Математика». 2021. Т. 38. С. 36—53

LINEAR INVERSE PROBLEMS FOR EQUATIONS WITH MULTI-TERM... 49

and for all к € N such that P\(Xk) = 0,

т t

j J j R°x,kex(t—s^ dXp>(s)dsdp(t) = 0, (4.9)

о о Го

(m— 1 nr

£ A"t,—'fi?(A*) + £ A“‘(A*) + £ A-A-РЦХк)

3 = 1 1=1 «=1

Example 2. Take a = 5/2, m = 3, n = 1, r = 1, aq = 2/3, Д = 1/2, fi1(A) = A(A + 9), P1(A) = M + 62A2, P|(A) = 0, fiHA) = со + C2A2, Pi(A) = d0 + d1A, d = 1, Q = (0,^), p = 1, Ли = , Б1 = I, ^ is the

single jump at t0 € (0,T] function. Then m = 0, m = 1, m* = 1, j0 = 1, aq > a — m + j0, m1 = 1, m0 = 1, m0 = 0 and m* = 0, and problem

(4.1)-(4.3), (4.6), (4.7) has the form

a5/2 (|4 + 912) кен) = (b112 + &2jg4) a1/2^,*)+

+ (c0 + C2Ц2) D2triv(^,t) + (d0 + d1 Ц2) J^/2v(^,t) + ),

(£,*) € (0,эт) x (0,T],

c(0,f) = c(^,t) = (gf (0,i) = foi) = 0, t € (0,T], £ € (0,эт),

^1/2 (W + 9 W) u(^ 0) = ^ € (0, ^

D3/2 (/|4 + 9Jft) v(£ 0) = £ € (0, ^

(Jt1/3c(-, 0), sin3{) = c,

v(C,t0)= vt Ш, C € (0, ж).

Here c € C, {yk (■), sin3{) = 0, к = 1, 2, since А к = —к2, к € N, fij(A3) = 0. Conditions (4.8), (4.9) have the form

Го

J (Va3/2 + (Ы2 — Ы4)А—1/2 —

Г1

— (d0 — d1k2 )A—3/2) 1 (eXT — 1)dX (eXT — 1)dX

(C0 + C2^4)A 1/3—

> с, к € N \ {3},

(—9&1 +81&2)A 1/2 + (c0 + 81c2)A 1/3 + (^0 — 9^1)A 3/2

= 0.

References

1. Glushak A.V. On an Inverse Problem for an Abstract Differential Equation of Fractional Order. Mathematical Notes, 2010, vol. 87, no. 5, pp. 654-662. https://doi.org/10.1134/S0001434610050056

50

M. M. TUROV, V. E. VEDOROV, B. T. KIEN

2. Nakhushev A.M. Drobnoye ischisleniye i ego primeneniye [Fractional calculus and its applications]. Moscow, Fizmalit Publ., 2003, 272 p. (in Russian)

3. Pskhu A.V. Initial-value problem for a linear ordinary differential equation of noninteger order. Sbornik Mathematics, 2011, vol. 202, no. 4, pp. 571-582. http://dx.doi.org/10.1070/SM2011v202n04ABEH004156

4. Pyatkov S.G. Boundary Value and Inverse Problems for Some Classes of Nonclassical Operator-Differential Equations. Siberian Mathematical Journal, 2021, vol. 62, no. 3, pp. 489-502. https://doi.org/10.1134/S0037446621030125

5. Tikhonov I.V., Eidelman Yu.S. Problems on correctness of ordinary and inverse problems for evolutionary equations of a special form. Mathematical Notes, 1994, vol. 56, no. 2, pp. 830-839. https://doi.org/10.1007/BF02110743

6. Triebel H. Interpolation Theory. Function Spaces. Differential Operators. Berlin, VEB Deutscher Verlag der Wissenschaften, 1978, 531 p.

7. Falaleev M.V. Degenerated abstract problem of prediction-control in Banach spaces. The Bulletin of Irkutsk State University. Series Mathematics, 2010, vol. 3, pp. 126-132. (in Russian)

8. Fedorov V.E., Nagumanova A.V. Inverse Problem for Evolutionary Equation with the Gerasimov - Caputo Fractional Derivative in the Sectorial Case. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 28, pp. 123-137. https://doi.org/10.26516/1997-7670.2019.28.123 (in Russian)

9. Fedorov V.E., Kostic M. Identification Problem for Strongly Degenerate Evolution Equations with the Gerasimov - Caputo Derivative. Differential Equations, 2020, vol. 56, pp. 1613-1627. https://doi.org/10.1134/S00122661200120101

10. Fedorov V.E., Turov M.M. The Defect of a Cauchy Type Problem for Linear Equations with Several Riemann - Liouville Derivatives. Siberian Mathematical Journal, 2021, vol. 62, no. 5, pp. 925-942. https://doi.org/10.1134/S0037446621050141

11. Al Horani M., Favini A. Degenerate first-order inverse problems in Banach spaces. Nonlinear Analysis, 2012, vol. 75, no. 1, pp. 68-77.

https://doi.org/10.1016/j.na.2011.08.001

12. Fedorov V.E., Ivanova N.D. Identification problem for degenerate evolution equations of fractional order. Fractional Calculus and Applied Analysis, 2017, vol. 20, no. 3, pp. 706-721. https://doi.org/10.1515/fca-2017-0037

13. Fedorov V.E., Nagumanova A.V., Avilovich A.S. A class of inverse problems for evolution equations with the Riemann - Liouville derivative in the sectorial case. Mathematical Methods in the Applied Sciences, 2021, vol. 44, no. 15, pp. 1196111969. https://doi.org/10.1002/mma.6794

14. Fedorov V.E., Nagumanova A.V., Kostic M. A class of inverse problems for fractional order degenerate evolution equations. Journal of Inverse and Ill-Posed Problems, 2021, vol. 29, no. 2, pp. 173-184. https://doi.org/10.1515/jiip-2017-0099

15. Hadid S.B., Luchko Yu.F. An operational method for solving fractional differential equations of an arbitrary real order. Panamerican Mathematical Journal, 1996, vol. 6, no. 1, pp. 57-73.

16. Jiang H., Liu F., Turner I., Burrage K. Analitical solutions for the multi-term time-space Caputo - Riesz fractional advection-diffussion equations on a finite domain. Journal of Mathematical Analysis and Applications, 2012, vol. 389, no. 2, pp. 1117-1127. https://doi.org/10.1016/j.jmaa.2011.12.055

17. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. Amsterdam, Boston, Heidelberg, Elsevier Science Publ., 2006, 540 p.

18. Li C.-G., Kostic M., Li M. Abstract multi-term fractional differential equations. Kragujevac Journal of Mathematics, 2014, vol. 38, no. 1, pp. 51-71. https://doi.org/10.5937/KgJMath1401051L

Известия Иркутского государственного университета.

Серия «Математика». 2021. Т. 38. С. 36—53

LINEAR INVERSE PROBLEMS FOR EQUATIONS WITH MULTI-TERM... 51

19. Fedorov V.E., Kostic M. On a class of abstract degenerate multi-term fractional differential equations in locally convex spaces. Eurasian Mathematical Journal, 2018, vol. 9, no. 3, pp. 33-57. https://doi.org/10.32523/2077-9879-2018-9-3-33-57

20. Orlovsky D.G. Parameter determination in a differential equation of fractional order with Riemann - Liouville fractional derivative in a Hilbert space. Journal of Siberian Federal University. Mathematics & Physics, 2015, vol. 8, no. 1, pp. 55-63.

21. Prilepko A.I., Orlovskii D.G., Vasin I.A. Methods for Solving Inverse Problems in Mathematical Physics. New York, Basel, Marcel Dekker Inc., 2000, 744 p.

22. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, VSP, 2003, 216 p.

Mikhail Turov, Postgraduate, Chelyabinsk State University, 129, Bratiev Kashirinykh st., Chelyabinsk, 454001, Russian Federation, tel.: (351)7997235, email: [email protected] ORCID iD https://orcid.org/0000-0003-4075-5099

Vladimir Fedorov, Doctor of Sciences (Physics and Mathematics), Professor, Chelyabinsk State University, 129, Bratiev Kashirinykh st., Chelyabinsk, 454001, Russian Federation, tel.: (351)7997235, email: [email protected] ORCID iD https://orcid.org/0000-0002-0787-3272 Bui Trong Kien, PhD, Doctor in Mathematics, Institute of Mathematics of Vietnam Academy of Science and Technology, 18, Hoang Quoc Viet road, Cau Giay district, Hanoi, Vietnam, tel.: 84 024 37563474, email: [email protected]

Received 29.10.2021

Линейные обратные задачи для уравнений с несколькими производными Римана — Лиувилля

М. М. Туров1, В. Е. Федоров1,2, Б. Т. Киен3

1 Челябинский государственный университет, Челябинск, Российская Федерация

2 Южно-Уральский государственный университет (национальный исследовательский университет), Челябинск, Российская Федерация

3 Институт математики Вьетнамской академии науки и технологий, Ханой, Вьетнам

Аннотация. Рассматриваются вопросы корректности линейных обратных коэффициентных задач для уравнений в банаховых пространствах с несколькими дробными производными Римана - Лиувилля и с ограниченными операторами при них. Получены критерии корректности как для уравнения, разрешенного относительно старшей дробной производной, так и в случае вырожденного оператора при старшей производной в уравнении. В вырожденной задаче исследованы два существенно различных случая: когда дробная часть порядка второй по старшинству производной равна дробной части порядка старшей дробной производной или отличается от нее. Абстрактные результаты использованы при исследовании обратных задач для уравнений в частных производных с многочленами от самосопряженного

52

M. M. TUROV, V. E. VEDOROV, B. T. KIEN

эллиптического дифференциального по пространственным переменным оператора и с производными Римана - Лиувилля по времени.

Ключевые слова: обратная задача, дробная производная Римана - Лиувилля, вырожденные эволюционные уравнения, начально-краевая задача.

Список литературы

1. Глушак А. В. Об одной обратной задаче для абстрактного дифференциального уравнения дробного порядка // Математические заметки. 2010. Т. 87, вып. 5. С. 684-693. https://doi.org/10.4213/mzm4437

2. Нахушев А. М. Дробное исчисление и его применение. M. : Физматлит, 2003. 272 с.

3. Псху А. В. Начальная задача для линейного обыкновенного дифференциального уравнения дробного порядка // Математический сборник. 2011. Т. 202, №4. C. 111-122. https://doi.org/10.4213/sm7645

4. Пятков С. Г. Краевые и обратные задачи для некоторых классов

неклассических операторно-дифференциальных уравнений // Сибирский математический журнал. 2021. Т. 62, № 3. C. 603-618.

https://doi.org/10.33048/smzh.2021.62.312

5. Тихонов И. В., Эйдельман Ю. С. Вопросы корректности прямых и обратных задач для эволюционного уравнения специального вида // Математические заметки. 1994. Т. 56, вып. 2. С. 99-113.

6. Трибель Х. Теория интерполяции. Функциональные пространства. Дифференциальные операторы. М. : Мир, 1980. 664 с.

7. Фалалеев М. В. Абстрактная задача прогноз-управление с вырождением в банаховых пространствах // Известия Иркутского государственного университета. Серия Математика. 2010. Т. 3, № 1. C. 126-132.

8. Федоров В. Е., Нагуманова А. В. Обратная задача для эволюционного уравнения с дробной производной Герасимова - Капуто в секториальном случае // Известия Иркутского государственного университета. Серия. 2019. Т. 28. C. 123-137. https://doi.org/10.26516/1997-7670.2019.28.123

9. Федоров В. Е., Костич М. Задача идентификации для сильно вырожденных эволюционных уравнений с производной Герасимова - Капу-то // Дифференциальные уравнения. 2021. Т. 57, № 1. С. 100-113. https://doi.org/10.31857/S037406412101009X

10. Федоров В. Е., Туров М. М. Дефект задачи типа Коши для линейных уравнений с несколькими производными Римана - Лиувилля // Сибирский математический журнал. 2021. Т. 62, № 5. С. 1143-1162.

11. Al Horani M., Favini A. Degenerate first-order inverse problems in Banach spaces // Nonlinear Analysis. 2012. Vol. 75, N. 1. P. 68-77. https://doi.org/10.1016/j.na.2011.08.001

12. Fedorov V. E., Ivanova N. D. Identification problem for degenerate evolution equations of fractional order // Fractional Calculus and Applied Analysis. 2017. Vol. 20, N. 3. P. 706-721. https://doi.org/10.1515/fca-2017-0037

13. Fedorov V. E., Nagumanova A. V., Avilovich A. S. A class of inverse problems for evolution equations with the Riemann - Liouville derivative in the sectorial case // Mathematical Methods in the Applied Sciences. 2021. Vol. 44, N. 15. P. 1196111969. https://doi.org/10.1002/mma.6794

Известия Иркутского государственного университета.

Серия «Математика». 2021. Т. 38. С. 36—53

LINEAR INVERSE PROBLEMS FOR EQUATIONS WITH MULTI-TERM... 53

14. Fedorov V. E., Nagumanova A. V., Kostic M. A class of inverse problems for fractional order degenerate evolution equations // Journal of Inverse and Ill-Posed Problems. 2021. Vol. 29, N. 2. P. 173-184. https://doi.org/10.1515/jiip-2017-0099

15. Hadid S. B., Luchko Yu. F. An operational method for solving fractional differential equations of an arbitrary real order // Panamerican Mathematical Journal. 1996. Vol. 6, N. 1. P. 57-73.

16. Jiang H., Liu F., Turner I., Burrage K. Analitical solutions for the multi-term time-space Caputo - Riesz fractional advection-diffussion equations on a finite domain // Journal of Mathematical Analysis and Applications. 2012. Vol. 389, N. 2. P. 1117-1127. https://doi.org/10.1016/j.jmaa.2011.12.055

17. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. Amsterdam ; Boston ; Heidelberg : Elsevier Science Publishing, 2006. 540 p.

18. Li C.-G., Kostic M., Li M. Abstract multi-term fractional differential equations // Kragujevac Journal of Mathematics. 2014. Vol. 38, N. 1. P. 51-71. https://doi.org/10.5937/KgJMath1401051L

19. Fedorov V. E., Kostic M. On a class of abstract degenerate multi-term fractional differential equations in locally convex spaces // Eurasian Mathematical Journal. 2018. Vol. 9, N. 3. P. 33-57. https://doi.org/10.32523/2077-9879-2018-9-3-33-57

20. Orlovsky D. G. Parameter determination in a differential equation of fractional order with Riemann - Liouville fractional derivative in a Hilbert space // Journal of Siberian Federal University. Mathematics & Physics. 2015. Vol. 8, N. 1. P. 55-63.

21. Prilepko A. I., Orlovskii D. G., Vasin I. A. Methods for Solving Inverse Problems in Mathematical Physics. New York ; Basel : Marcel Dekker Inc., 2000. 744 p.

22. Sviridyuk G. A., Fedorov V. E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht ; Boston : VSP, 2003. 216 p.

Михаил Михайлович Туров, аспирант, кафедра математического анализа, Челябинский государственный университет, Российская Федерация, 454001,г. Челябинск, ул. Братьев Кашириных, 129, тел.: (351)7997235, email: [email protected] ORCID iD https://orcid.org/0000-0003-4075-5099

Владимир Евгеньевич Фёдоров, доктор физико-математических наук, профессор, кафедра математического анализа, Челябинский государственный университет, Российская Федерация, 454001, г. Челябинск, ул. Братьев Кашириных, 129, тел.: (351)7997235, email: [email protected] ORCID iD https://orcid.org/0000-0002-0787-3272

Буй Тронг Киен, доктор физико-математических наук, заведующий отделом теории управления и оптимизации, Институт математики Вьетнамской академии науки и технологий, Вьетнам, район Кау Гиай, ул. Хоанг Куок Вьет, 18, тел.: 84 024 37563474, email: [email protected]

Поступила в редакцию 29.10.2021