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Математика
УДК 517.9 DOI: 10.14529/mmph160301
INVERSE PROBLEM FOR SOBOLEV TYPE EQUATION OF THE SECOND ORDER
A.A. Zamyshlyaeva, A.S. Muravyev
South Ural State University, Chelyabinsk, Russian Federation E-mail: zamyshliaevaaa@susu.ru
The paper deals with the inverse problem for the Sobolev type equation of the second order in Banach spaces. The introduction contains a problem statement and the historiography of Sobolev type equations. The second part includes preliminary information based on the results of the theory of higher-order Sobo-lev type equations. In the third part the initial problem is reduced to the inverse regular and singular problems. A theorem of unique solvability of regular problem is formulated and proved. Using the results of the third part, the solution for the singular problem is obtained in the fourth part. The sum of regular and singular solutions is a solution to the original problem, thus a theorem on the unique solvability of the inverse problem for Sobolev type equation of the second order is stated and proved.
Keywords: Sobolev type equation of the second order; inverse problem; a unique solvability theorem.
Introduction
Let U, F, Y be Banach spaces, operator X'-\0,T] ^L(Y;F), functions M :[0,T] ^F, ¥ :[0,T] ^ Y, operator C e L(U; F). Consider the following problem
AV(t) = + Bov(t) + %(t)q(t) + M(t), te [0,T], (1)
v(0) = vo, v(0) = vi, (2)
Cv(t) = ¥(t) (3)
The inverse is a problem of finding a pair of functions v e C2([0,T];U) and qe C2([0,T];Y) from relations (1)-(3). Note that (1) is a special case of equation
Av(n) (t) = Bn-iv( n-1) (t) +... + B0v(t) + f (t), (4)
which is called the higher order Sobolev type equation. Such operator-differential equations were firstly studied in [1, 2], and in more detail in [3]. Inverse Problems for Sobolev type equations, and other non-classical equations of mathematical physics were studied earlier in [4-6], and the inverse problem for the first-order equation was considered in [7].
Equations nonsolvable with respect to the highest derivative were firstly studied in 1885 in the works of Poincare. Often they were associated with studies of specific equations of hydrodynamics. Considerable interest in equations of this kind appeared in connection with the results C.W. Oseen, J. Leray, F.K.G. Odqvist and J. Schauder, E. Hopf and studies of S.L. Sobolev on the problem of small oscillations of a rotating fluid. This series of works initiated further research of equations nonsolvable with respect to the highest derivative. Thus, it was the basis for a new direction, which was originally developed by the disciples of S.L. Sobolev. After the first studies of S.L. Sobolev, I.G. Petrovsky spoke of need to investigate general differential equations and general differential systems that are not solvable with respect to highest derivative in time (such systems are called not of the Kovalevskaya type systems). In literature such equations and their specific interpretations are often called Sobolev type equations, saluting the founder.
The first attempt to study the phase space for the higher order equations was made in [8, 9]. In these works according to the ideology of M.V. Keldysh equation (4) was reduced to the equivalent first-order Sobolev-type equation, which has been studied by methods described in [10]. However, it should be
noted that the inverse reduction lead to a very complex algorithm of construction of the phase space. Moreover, it was not shown that all the initial values were lying in the same phase space.
Inverse problems often arise in different fields of science, particularly in the description of the internal characteristics of the medium in which the physical or chemical processes take place; when the results of observations of these processes are available for the measurement. The great interest to inverse problems appeared at the junction of the 19th and 20th centuries, particularly in geophysics. There was a pressing question: is it possible by the picture of movement of seismic wave fronts from the various earthquakes on the Earth's surface, to find the velocity of propagation of seismic waves in the Earth? There was formulated an inverse kinematic problem, firstly considered by the German geophysicists E. Wiechert and G. Herglotz. At the same time there arose another inverse problem in the potential theory. The studies of inverse problems in the potential theory in various productions were held by in V.N. Strakhov, A.N. Tikhonov, M.M. Lavrent'ev, V.K. Ivanov, A.I. Prilepko as well as their disciples. The inverse problems for dynamic reconstruction of the parameters of control systems and associated ill-posed problems were investigated in scientific schools of V.K. Ivanov, N.N. Krasovsky and others. In addition, there were studied inverse problems of electromagnetic intelligence, quantum scattering theory, and many others. Nowadays there appear new formulations of inverse problems and new results concerning their solvability.
Preliminary information
We use results of the theory of higher-order Sobolev type equations. Proofs of these results can be found in [2].
Definition 1. The sets
pA (B) = {i e C: (U"A - if~xBn_x -... - ^ - Bq)-1 e L( F, U)}
and cC (B) = C \ pA (B) are called an A-resolvent set of and an A-spectrum of pencil B.
Definition 2. The operator-valued function of a complex variable RA (B) = (in A - i"-1Bn-1 -... - iB1 - B0 )-1 with domain pA (B) is called an A-resolvent of pencil B.
Definition 3. The pencil B is called polynomially bounded with respect to the operator A (polyno-mially A-bounded), if
3a e R+Vi e C(| i |> a) ^ (R^(B) g L(F; U)). For further considerations we require fulfillment of additional conditions
{ v'rA (B)dl = 0, k = 0,1, (A)
Y
where the contour y= { Ie C :| i |= r > a}.
Lemma 1. Let the pencil Bbe polynomially A-bounded and condition (A) be fulfilled. Then the operators P e L(U) and Q e L(F), defined by formulas
P = n i Ri (B)iAdi, Q = 2- { lARA (B)di,
Y Y
are projectors.
Theorem 1. Let the pencil Bbe polynomially A-bounded and condition (A) be fulfilled. Then the actions of operators split:
(i) Ak e L(Uk; Fk), k = 0,1;
(ii) Bjk e L(Uk; Fk), k = 0,1, l = 0,1;
(iii) there exists an operator (A1)-1 e L(F 1;Ux),
(iv) there exists an operator (Bq)-1 e L(F0;U0).
Definition 4. Define a family of operators {Klq, K^} in the following way:
T/-1 _ tj js2 _ Tj T/~1 _ js2 Tj js2 __T^2 TT
K1 = H0, K1 =_H1, Kq+1 = KqH0, Kq+1 = Kq - KqH1,
Zamyshlyaeva A.A., Muravyev A.S.
Inverse problem for the second order Sobolev type equation
where operators Ho = (b00 ) 1 Ao, H1 = (b00 ) 1Bf.
Definition 5. The point ^ is called
(i) a removable singular point of the A- resolvent ofpencil B, if K\ = O, K^ = O;
(ii) a pole of order p e N of the A- resolvent ofpencil B, if K1p ^ O, K2 ^ O, but Kp+1 = O, Kp+i = O.
(iii) an essential singularity of the A- resolvent ofpencil B, if Kk ^ O, k e N.
Reduction of initial inverse problem
Let the pencil B = (B0, Bp) be polynomially A-bounded, U0 e ker C. Denote
v(t) = Pv(t) + (I -P)v(t). Then Pv(t) = u(t), (I - Q)v(t) = w(t). Then by Lemma 1 and Theorem 1,
problem (1)-(3) is equivalent to the problem of finding functions ue C2([0,T];U1), we C2([0,T];U0), qe C2([0,T];Y) from
u(t) = Su(t) + Sou(t) + A1-1Qj(t)q(t) + A^QM(t), te [0,T], (5)
u(0) = v1, ii(0) = v1. (6)
Cu(t) = Cv(t) = ¥(t), t e [0,T], (7)
How(t) = Hxw(t) + w(t) + (Bo0)-1(I - Q)x(t)q(t) + (Bo0)-1(I - Q)M(t), te [0,T], (8)
w(0) = vo0, w (0) = vO, (9)
Smooth solutions of regular problem
Inverse problem (5)-(7) is called regular. The direct problem (8), (9) with given function q is called singular. For convenience, we rewrite regular problem as follows:
u(t) = Sfi(t) + Sou(t) + 0(t)q(t) + ^(t), te [0,T], (10)
u (0) = uo, u(0) = u1, (11)
Cu(t) = ^(t) (12)
In order to obtain a solution of (8), (9), we need to require the smoothness of class Cp+2([0, T ];Y) of solution q of the regular inverse problem. Further we find a sufficient condition for the existence of a
smooth solution qe Cp+2([0,T];Y).
Lemma 2. Let ne N, Se Cn([0,T];L(X)), ge Cn([0,T];X). Then
f )("} n-1 t
{ S (t - s) g (s)ds = X S( }(t) g(n-1-i )(0) +{ S (t - s) g (n)(s )ds.
^ Y ) '=o 0
Proof. Let us prove formula (13) for n = 1: /
f * ^ * d * r '
j S (t - s) g (s )ds = S (0) g (t) + {-S (t - s) g (s )ds = S (0) g (t) + {
(13)
V 0
—S(t - s)
g (s)ds =
= S (t ) g (0) + j S (t - s) g '( s)ds.
Assume that for n = m equality (13) holds and prove that it is satisfied with n = m +1
\ (m+1)
j S (t - s) g (s)ds
t
+J
m—1
X s «(t ) g( m-1-i)(0) + J S (t - s) g (m)(s)ds i=0 0
= IS(i )(t ) g( m-1-i }(0) +
i=1
—S (t - s)
(m)
til 1 (s)ds = X S(i)(t)g(m-i}(0) + J S(t - s)g(m+1}(s)ds.
i=0
The lemma is proved.
Theorem 2. Suppose that the pencil S = (S0,S1) is polynomially bounded, Сe L(X;Y), pe N0, Cp+2([0,T];L(Y,X)), Fe Cp+2([0,T];X), k = 0,1,...,p, TeCp+2([0,T];Y), for all te [0,T] the operator СФ(t) is invertible, (СФ) -1 e Cp+2([0,T];L(Y)) and condition Cu0 =¥(0) is fulfilled. Then there exists a unique solution u e Cp+2([0,T ]; X), q e Cp+2([0, T];Y) of the inverse problem (10)-(12).
Proof. Before proceeding to the proof of the theorem denote S(t) = I" ( f21 -fS1 - S0)-1// 2e fdfi. According to theorem 6.2.3 [11], the searched function q 2niJ
Y
is a solution of equation
t
q(t) = q0 (t) + R(t) JCS(t - s)g(s)ds,
0
( ' } where q0(t) = R(t) ¥'(t) -CS(t)u0 - CJS(t - s)F'(s)ds -CF(t) , R(t) = (-CФ(t))-1.
V 0 J
Prove using the results of Lemma 2 that function qe Сp+2([0,T],Y).
t t q(t) = q0 (t) + R'(t) J CS(t - s)g(s)ds + R(t)CS(0)g(t) + R(t) J CS'(t - s)g(s)ds, 0 0
t
q' (t) = q№0 (t) + R" (t)J CS (t - s) g (s)ds + R' (t )CS (0) g (t) + R' (t )CS (0) g (t) +
0
t t +R(t)CS(0)g'(t) + R(t)CS'(0)g (t) + R(t) J CS'(t - s)g(s)ds + R'(t) J CS'(t - s)g(s)ds,
0 0
t
q"(t) = q0(t) + R"(t)J CS(t - s)g(s)ds + R"(t)CS(0)g (t) -
0
t
+R''(t) J CS'(t - s)g (s)ds + 2R''(t)CS(0)g(t) + 2R'(t)CS(0)g'(t) + R'(t)CS(0)g'(t) -
0
t
+R(t)CS(0)g*(t) + R'(t)CS'(0)g(t) + R(t)CS'(0)g'(t) + R'(t)JCS'(t - s)g(s)ds ■
0
t
+R' (t )CS '(0) g (t) + R' (t )J CS'(t - s) g (s)ds = q^(t) + R(t )C (S (t) g ' (0) + S '(t) g' (0) + S'(t) g (0)) +
0
t ( t +R(t)CJS(t - s)g"(s)ds + 3R'(t)C S(t)g'(0) + S'(t)g(0) + Js(t - s)g'(s)ds
l +
1 +
+3^' (t )C
S (t ) g (0) + j S (t - s) g ' ( s)ds + R" (t )C J S (t - s) g ( s) ds.
v
n—1 k
I Z
k=0 i=0 i=0
q( n)(t ) = q0 n)(t ) + £ £ CnCR (i)(t ) S( n—1—k >(t ) g( n—1—г)(0) +I CnCR (i)(t )j S (t — s ) g( n—' >( s )ds . Derivatives
q0n) (t ) = £ C'nR{i) (t ) i W( n+1) (t ) - CS( n) (t )u0 + C § S( k} (t ) F(i-1-k} (0) + C J S (t - s) F(i} ( s) ds - CF( n) (t ) ^
i=0 v k=0 0
exist because of the conditions of this theorem for n = 0,1,..., p +1.
Zamyshlyaeva A.A., Inverse problem for the second order
Muravyev A.S. Sobolev type equation
Denote ro = qo(0) and for n = 0,1,..., p +1 consistently define values
r„ = qOn)(0) +1 fJCnCR(\0)S(n-1-k\t)S(n-1-i\0)X O(l)(O)r .
k=Oi=O l=O
Consider the system of integral equations
t
qo (t) = qo (t) + R(t) j CS (t - s)0( s)qo (s)ds, o
q(n)(t) = qOn)(t) + £ ]XCnCR(i)(t) S(n-1-k >(t O(l)(0)rl + (14)
к=0 i=0 l=0
n-i
+XC'nCR(,)(t)\S(t-s)^C'^m^(s)ds, при n = 1,...,p +1. i=0 0 l=0 System (14) can be reduced to the Volterra equation of the second kind
g (t ) = g0(t ) + J K (t, s) g (s)ds
0
\ p+2
in the space (C([0,T];Y))p with matrix-operator function K(t,5), defined in a triangle A = {(t, 5) e R2 : 0 < t <T,0 <5 < t}. By the continuity of all elements of system (14), it has a unique solution. This solution is the limit of a sequence of approximations
t
(t) = qo (t) + R(t) j CS(t - 5)0(5)qo,i-1 (5)d5,
0
n-1 к n-1-i
n,i(t) = q0n)(t) + XXcncR{l}(t)s(n-1-к}(t) X ф(1 }(0)^i + (15)
к=0 i=0 l=0
n-i
+X C'nCR( }(t)j S (t - s)X C'n 0(l )(s)q„-,-l,,-1( s)ds, for n = 1,..., p +1.
i=O o l=o
which uniformly on [O,T] converges to qn, n = 1,...,p +1 while i For an initial approximation we take q„,o = O , then qn+lfi = q'nfi , n = O,...,p . In view of (15)
q„i(O) = r„,n = O,...,p +1,ie N. (16)
Suppose that for all j = 1,..., i the equality (t) = q'j (t), n = O,..., p is true. In this case, by Lemma 2 and (16),
d YLCnCR{,)(t)jS(t-s)Cln_l0{l)(s)qn.l.u(s)ds i=Ol=0 o
dt
= XXcnCR(i+1}(t ) J S (t - s )Cn-,. Ф(1 }(s )qn-i-i ,i ( s )ds +
i=0l=0 0
+X XXcnCR(i }(t ) S (t )cn-i ф(1 \s)qn_,-i ,i (0) +
i=0l=0
+XX X CnCR°) (t) J S(t - s)cn-iФ(l+1} (s)qn-i-l,i (s)ds +
i=0 l=0 0
+XX "Xc'nCR(i }(t ) Js (t - s )cn-i ф(l }(s )qn_l-l+1,i ( s )ds = i=0 l=0 0
= XX XcnCR(i)(t )S (t )cn-i Ф(1+l)(s)qn_l_l ,i (0) + i=0l=0
n+1 n-i+1 t
+I I C'nCR(i}(t)jS(t - s)C^(s)qn_1_l+li1 (s)ds + i=1 l=0 0 n n—i+1 t
+I I ClnCR(l\t)\S(t — s)Cn_1Ф"(s)qn—1—,+, ;(s)ds + i=0 l =1 0 n n—i t
i=0l=0
i=0l=0 n+1 n-i+1
i=0 l=0
Differentiate the second term:
I I C'nCR^')(t)jS(t — s)C"—iФ(l) (s)qn—i—l+1,i (s)ds -=0 l =1 0
II Ic"CR(i >(t )|S (t — s)C"—i Ф(l}(s)qn—i—l+u (s )ds =
0 l=0 0
= I Ic"CR('' >(t) S (t )Cn—i Ф(1+1}( s)qn—i—l ,i (0) +
i=0l=0 1 n—i+1 t
I I C'n+1CR(i)(t) jS(t — s)Cn—i^(l)(s)qn—i—l+1,i.(s)ds. (17)
d_
dt
I I Zi CnCR(i} (t )S(n—1—k} (t )Ф(1 } (0) rl
k=0i=0 l=0
n—1 k n—1—i
= II I CnCR(i+1}(t)S(n—1—k}(t)Ф(1 }(0)r +
k=0i=0 l=0
n—1 k n—1—i n k n—1—i
+II I CnCR(i}(t)S(n—k)(t)Ф(l)(0)rl = II I C'nCR(i+1)(t)S(n—k)(t)ф(l)(0)rl +
k=0 i=0 l=0 k=1 i=1 l=0 n+1 k n—1—i n k n—1—i
+III ClnCR(i+1)(t)S(n—kk)(t^(l}(0)r = II I ClnCR(i\t)S(n—k)(t)Ф(l}(0)r + k=0 i=1 l=0 k=0 i=1 l=0
+I "I' C'nCR() (t)S(t)Ф(l)(0)rl + I 2CR(t)S(t)Ф(1 )(0)rl =
i=1 l=0 k=0 l=0
= II InI—iCn+1CR(i)(t)S(n—k)(t^(l )(0)/i + InI—'c"CR (i)(t) S (t)Ф(l )(0)rl +
k=0 i=1 l=0 i=1 l=0 n+1n—1 n n—1—i
+IICR(t)S(t)Ф(l}(0)rl — I I Cn—1CR(i)(t)S(t)Ф(l}(0)rl =
k =1 l=0 i =1 l=0 n+1 k n-1-i n n-i
= ZEE cn+1CR (i)(o n-k)(t)°(l )(0)i - EEccr (0(t) (t)^(l )(0>/l. (18) k=0 i=0 l=0 i=0 l=0 Differentiating (15), from (17) and (18) we obtain the equality qn+u+1(t) = q'ni+1(t). Thus, the sequence q0 i converges to the function q0 uniformly on [0,T] when i ^ ^ , and moreover q0 t = qli. Similarly prove equality cj„+1(t) = qn(t), n = 1,...,p, consequently q0(t) = qe C p+2([0, T ];Y) and hence q(n) = qn, n = 1,...,p +1. So the function u is p + 2 times differentiable. This proves the theorem.
The solvability of the initial problem
Denote x(t) = (I -Q)X(t) , M0(t) = (I -Q)M(t).
Lemma 3. Let the pencil B be polynomially A-bounded, condition (A) be fulfilled and ^ be a pole of order p e N of the A-resolvent of pencil B. Let x0 e Cp+2([0, T]; F0), M0 e Cp+2([0, T]; F0) and
p dq+k
initial conditions wk eU0 satisfy wk =~Z K^^)-1-—(((t)q(t) +M0(t) ) = , k = 0,1. Then there
q=0 dt =
exists a unique solution w e C2([0,T];U0)of problem (8), (9), which can be represented as
w(t) = -ZZ Kq (B0 )-1 ^(( (t)q(t) +M0 (t)).
q=0 dt
Zamyshlyaeva A.A., Muravyev A.S.
Inverse problem for the second order Sobolev type equation
p dq+k
Proof. Denote Xo(t)q(t) + ^(t) = fO(t), in this case wO = -£K2q(Bo0)-1 -—rfO(O),k = O,1.
q=O dt
Therefore all the conditions of Lemma 2.7.2 [2] hold. Thus there exists a unique solution w(t) = -]^Kq2(BoO)-1 (AO) of (8), (9).
q=O dt
Theorem 3. Let the pencil B = (BO, B1) be polynomially A-bounded, condition (A) be fulfilled and ^ be a pole of order p e N of the A-resolvent ofpencil B. Let the operator C e L(X ; Y), ker C be not empty, Xo e Cp+2([O,T];FO), Mo e Cp+2([O,T];FO), ¥eCp+2([O,T];Y), for all te [O,T] operator Cx(t) be invertible, (Cx)-1 e Cp+2([O,T];L(Y)) and condition CvO = ¥(O) hold. Then there exists a unique solution (v, q) of inverse problem (1)-(3), such that v = u + w, where u e Cp+2([O,T ];U1) and q e Cp+2([O,T ];Y) are defined in Theorem 2, w e C 2([O, T];UO) is defined in Lemma 3.
Proof. When reducing the original problem to the inverse regular and singular problems it has been shown that v = u + w. Thus all the conditions of Lemma 3 and Theorem 2 are fulfilled, therefore the
functions u e Cp+2([O,T];U1), qe Cp+2([O,T];Y) are a solution of the regular inverse problem and the
function we C2([O,T];UO) is a solution of the singular problem. Thus there exists a unique solution of problem (1)-(3).
References
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Received June 6, 2016
Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2016, vol. 8, no. 3, pp. 5-12
DOI: 10.14529/mmph160301
ОБРАТНАЯ ЗАДАЧА ДЛЯ УРАВНЕНИЯ СОБОЛЕВСКОГО ТИПА ВТОРОГО ПОРЯДКА
А.А. Замышляева, А.С. Муравьев
Южно-Уральский государственный университет, г. Челябинск, Российская Федерация E-mail: zamyshliaevaaa@susu.ru
Исследована обратная задача для уравнения соболевского типа второго порядка в банаховом пространстве. Введение содержит постановку задачи и историографию уравнений соболевского типа. Вторая часть включает в себя предварительные сведения, основанные на результатах теории уравнений соболевского типа высокого порядка. В третьей части исходная задача редуцирована к обратной регулярной и сингулярной задачам, сформулирована и доказана теорема об однозначной разрешимости регулярной задачи. Пользуясь результатами, полученными в третьей части, в четвертой части получено решение для сингулярной задачи. Сумма решений регулярной и сингулярной является решением исходной задачи, таким образом, в работе сформулирована и доказана теорема об однозначной разрешимости обратной задачи для уравнения соболевского типа второго порядка.
Ключевые слова: уравнение соболевского типа; уравнение второго порядка; обратная задача; теорема об однозначной разрешимости.
Литература
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Поступила в редакцию 6 июня 2016 г.
