Inverse problem for Sobolev type equation of the second order

Let U , F , Y be Banach spaces, operator X :[0, T ] ^ L(Y ; F ), functions M :[0, T ] ^ F , T :[0, T ] ^ Y , operator C e L(U ; F ). Consider the following problem

Av(t) = Biv(t) + Bоv(t) + x(t)q(t) + M(t), te [0,T],(1)

v (0) = vo, v(0) = Vi,(2)

Cv (t) = T( t)(3)

The inverse is a problem of finding a pair of functions v e C ²([0, T ]; U ) and q e C ²([0, T ]; Y ) from relations (1)–(3). Note that (1) is a special case of equation

Av(n) (t) = Bn-I v(n-1) (t) +... + Bov (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 Poincaré. 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

Математика 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) ={pe C :(pnA - p-1 Bn-1 -... - pB1 - B0)-1 e L (F, U)} and σA(B) = C\ρA(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

R _μ A ( B ) = ( μ n A - μ n ^- 1 B n _- 1 - ... - μ B 1 - B 0 ) ^- 1 with domain ρ A ( B ) is called an A-resolvent of pencil B.

Definition 3. The pencil B is called polynomially bounded with respect to the operator A (polynomially A-bounded), if

3 a e R + Vpe C (| A l > a ) ^ ( R p ( B ) ё L ( F ; U )).

For further considerations we require fulfillment of additional conditions

J P r P A ( B)d p = 0, к = 0,1,                                (A)

γ where the contour γ={μ∈C:|μ|=r>a}.

Lemma 1. Let the pencil Bbe polynomially A-bounded and condition (A) be fulfilled. Then the operators P ∈ L ( U ) and Q ∈ L ( F ) , defined by formulas

P = 1-R pA ( ( l?) p Ad p , Q =     ^ AR IA p (j?) d p ,

2 π i                  2 π i

γγ are projectors.

Theorem 1. Let the pencil Bbe polynomially A-bounded and condition (A) be fulfilled. Then the actions of operators split:

• (i)       A^k ∈ L ( U^k ; F^k ),         k = 0,1;

• (ii)      B_l^k ∈ L ( U^k ; F^k ),          k = 0,1, l = 0,1;

• (iii)    there exists an operator ( A ¹) ^{- 1} ∈ L ( F ¹; U ¹),

• (iv)     there exists an operator ( B ₀⁰) ^{- 1} ∈ L ( F ⁰; U ⁰).

Definition 4. Define a family of operators { K q 1 , K q 2 } in the following way:

K11=H0,       K12=-H1,     K1q+1=Kq2H0,          Kq2+1=K1q-Kq2H1, where operators H0 =(B00) A0, H1 =(B00) B10.

Definition 5. The point ~ is called

• (i)     a removable singular point of the A- resolvent ofpencil B, if K 1 = O , K 12 = O ;

• (ii)    a pole of order p e N of the A- resolvent of pencil B, if K 1_p ^ O , K 2 ^ O ,

but k p + i = O , K p + i = O .

• (iii)    an essential singularity of the A- resolvent ofpencil B, if K k ^ O , k e N .

Reduction of initial inverse problem

Let the pencil B = ( B 0 , B 1 ) be polynomially A-bounded, U 0 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 u e C ²([0, T ]; U 1 ), w e C ²([0, T ]; U 0 ), q e C ²([0, T ]; Y ) from

• u(t) = 51 u(t) + 50u(t) + A- 1Qx(t)q(t) + A-1 QM(t), te [0,T],(5)

u (0) = v0, ii(0) = vj.(6)

Cu(t) = Cv(t) = T(t), t e [0,T],(7)

H0w(t) = H1 w(t) + w(t) + (B00)-1 (I - Q)X(t)q(t) + (B0)-1 (I - Q)M(t),    t e [0,T],(8)

w (0) = v00, w (0) = v0,(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) = 51 u(t) + 50u(t) + Ф(t)q(t) + F(t), te [0,T],(10)

u (0) = u ₀, u (0) = u 1

Cu ( t ) = T ( t )

In order to obtain a solution of (8), (9), we need to require the smoothness of class C^{p + 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 q e C^{p + 2}([0, T ]; Y ).

Lemma 2. Let n e N, 5 e Cn ([0,T];L(X)), g e Cn ([0,T];X). Then n-1                             t

J 5 ( t - s ) g ( s ) ds     = £ 5 ⁽ i ) ( t ) g ^{( n - 1 -} i ) (0) + J 5 ( t - s ) g ^{( n} ) ( s ) ds .

_v у                   J i = 0                         0

Proof. Let us prove formula (13) for n = 1: ‘

J 5 ( t - s ) g ( s ) ds к 0

t

= 5 (0) g ( t ) + J- 5 ( t - s ) g ( s ) ds = 5 (0) g ( t ) + ₀ dt

t

J ^d5 ( t - s ) g ( s ) ds = 0 L ^ds          -

= 5 ( t ) g (0) + J 5 ( t - s ) g ‘ ( s ) ds . 0

Assume that for n = m equality (13) holds and prove that it is satisfied with n = m + 1

( t

J 5 ( t - s ) g ( s ) ds

x ( m + 1)

y

d S 5 ^{( i )}( t ) g ^{( m} - 1 - i ) (0) + J 5 ( t - s ) g ^{( m} ) ( s ) ds = j m 5 ^{( i )}( t ) g ^{( m} -^H)(0) + ^dt L i = 0                         0                        - i = 1

t

+

tm j -5(t-s) g(m)(s)ds = S5(i)(t)g( 0 Lds J              i=0

m

m - i )

t

(0) + J 5 ( t - s ) g ^{( m + 1)}

( s ) ds .

Математика

The lemma is proved.

Theorem 2. Suppose that the pencil S = (S0, S1) is polynomially bounded, C e L(X; Y), p e N0, Фе Cp+2([0,T];L(Y,X)), Fe Cp+2([0,T];X), k = 0,1,...,p, Te Cp+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 =T(0) is fulfilled. Then there exists a unique solutionu 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) = —1— [(^21 -^S1 - S0)-1 ^2e,u du. According to theorem 6.2.3 [11], the searched function q 2nij Y is a solution of equation

t q (t) = q 0 (t) + R (t) j CS (t - s) g (s) ds,

R ( t ) = ( - C Ф ( t ) ) ^{- 1} .

where q0(t) = R(t) T'(t) - CS(t)u0 - C jS(t - s)F'(s)ds - CF(t) , к                     0

Prove using the results of Lemma 2 that function q e C^{p + 2}([0, T ], Y ).

q '(t) = q 0 (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, 00

q '' ( t ) = q " ( t ) + R '' ( t ) j CS ( t - s ) g ( s ) ds + R ' ( t ) CS (0) g ( t ) + R ' ( t ) CS (0) g ( t ) + 0

+R (t) CS (0) g '(t) + R (t) CS '(0) g (t) + R (t) j CS '(t - s) g (s) ds + R'tt) j CS'tt - s) g (s) ds, 00

q ''' ( t ) = q *( 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 + 2 R '' ( t ) CS (0) g ( t ) + 2 R ' ( t ) CS (0) g ' ( t ) + R ' ( t ) CS (0) g' ( t ) + 0

+ R ( t ) CS (0) g ' ( t ) + R ' ( t ) CS' (0) g ( t ) + R ( t ) CS' (0) g ' ( 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 = q 0( t) + R (t) C (S (t) g' (0) + S'(t) g' (0) + S '(t) g (0)) + t/ j S (t - s) g ' (s) ds 0

+

+R(t) Cj S(t - s)g''(s)ds + 3R'(t)C S(t)g'(0) + S'(t)g(0) + 0

+ 3 R ' ( t ) C S ( t ) g (0) + j S ( t - s ) g ' ( s ) ds

+ R ''' ( t ) C j S ( t - s ) g ( s ) ds .

q ( n ) ( t ) = q 0 ⁿ ) ( t ) + ^ - 1 ]T C n CR ( i ) ( t ) S ( ^{n - 1} - ^k ) ( t ) g ( ^{n - 1} - i ) (0) + j^ C n CR ( i ) ( t ) j S ( t - s ) g ( ⁿ - i ) ( s ) ds .

k=0i=0                                            i=00

Derivatives n                (                                        i-1                                  t^

q 0 n ) ( t ) = У C n R ( i ) ( t ) T ^{( n + 1)}( t ) - CS ( ⁿ ) ( t ) u ₀ + C ^ S ^{( k} ) ( t ) F ( i ^{- 1} - ^k ) (0) + C j S ( t - s ) F ( i ) ( s ) ds - CF ( ⁿ ) ( t )

i=0            к                                  k=0                             0

exist because of the conditions of this theorem for n = 0,1,..., p + 1.

Denote r0 = q0(0) and for n = 0,1,...,p +1 consistently define values rn = q0n)(0) + £ £CnCR(i)(0)S(n-1-k)(t)S(n-1-i)(0) £ Ф(1)(0)rl . k=0i=0

Consider the system of integral equations

С о ( t ) = q о ( t ) + R ( t ) j CS ( t - s ) Ф ( s ) q o ( s ) ds ,

q(n)(t) = q0n)(t) + £ £rCR(i)(t)S(n-1-k)(t) £ Ф(1)(0)rl + k=0i=0

n                  tn

+£CnCR(i)(t)jS(t-s)£CnФ(1)(0)qn-i-1 (s)ds, при n = 1,...,p +1. i=0               0

System (14) can be reduced to the Volterra equation of the second kind g (t) = g o( t) + jK (t, s) g (s) ds in the space (С([0,T];Y))p+2 with matrix-operator function K(t,s), defined in a triangle A = {(t, s) e R2:0 < t < T,0 < s < 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 qo, i (t) = qo (t) + R (t) j CS (t - s )Ф( s) qo, i - (s) ds, n-1 kn qn,i (t) = q0n)(t) + ££CnCR(1)(t)S(n-1-k)(t) £ Ф( 1 )(0)r + k=0i=0

n                  tn

+£CnCR(i)(t)jS(t-s)£CnФ( 1)(s)qn_i_i,i-1(s)ds, for n = 1,...,p +1. i=0               o which uniformly on [0,T] converges to qn, n = 1,...,p +1 while i ^ ^. For an initial approximation we take qn,0 = 0, then qn+1,0 = qn,0, n = 0,...,p. In view of (15)

qn,i(0) = rn,n = 0,...,p +1,ie N.(16)

Suppose that for all j = 1,...,i the equality cin+1j (t) = q'n j (t), n = 0,...,p is true. In this case, by Lemma 2 and (16), n n - it d dt

£ ^ CnCR(i) (t) j S (t - s) Cn _ i Ф( 1) (s) qn _ i _ 1, i (s) ds = i=01=0               0

n n - it

= £ £ Cn CR(i+1) (t) j S(t - s)Cn-1 ф( 1) (s)qn-i-1,i (s)ds + i=01=0o n n - i

+££CnCR(i)tt)S(t)Cn-1Ф( 1)(s)qn-1-1,i (0) + i=01=0

n n - it

+£ £ CnCR(i) (t) j S (t - s) Cn - i ф( 1+1) (s) qn - i -1, i (s) ds + i=01=00

n n - it

+£ £ CnCR(i) (t) j S (t - s) Cn - i ф( 1) (s) qn - i -1+1, i (s) ds = i=01=0o n n - i

= £ £ Cn CR(i) (t)S(t)Cn-1 Ф( 1+1) (s)qn-1-1,i (0) + i=01=0

Математика n+1n - i+1

+Z Z CiCR(i)(t)J5(t-s)c—iФ(1)(s)qn—i—i+1,i (s)ds + i=1 1=00

n n - i+1

+Z Z cncR(i)(t)J5(t-s)c-iФ(1)(s)qn-i-1+1,i(s)ds + i=0 1=10

n n - i

+Z Z Cn CR (i) (t) J 5 (t - s) c -i Ф (1) (s) qn -i -1+1,i (s) ds = i=01=00

n n - i

= ZZ CnCR(1)(t) 5 (t) СП - i Ф( 1+1)( s) qn - i -1, i (0) + i=0 1=0

n+1n - i+1

+Z Z СП+1CR(i)(t)J5(t-s)cn-1Ф( 1)(s)qn-1-1+1,(s)ds.(17)

i=0 1=00

Differentiate the second term:

d dt

Z Z n -E i c n CR ⁽ i ) ( t ) 5 ^{( n - 1 - k} ) ( t ) ф ^{( 1} ) (0) r

. k = 0 i = 0 I = 0

= Z Z n Z i c n cR ⁽ i ^{+ 1)}( t ) 5 ^{( n - 1 - k} ) ( t ) Ф ^{( 1} ) (0) r₁ + k = 0 i = 0 I = 0

+ Z Z n Z i c n cR ⁽ i ) ( t ) 5 ^{( n - k} ) ( t ) Ф ^{( 1 )}(0) r = z Z n Z i c n cR ⁽ i ^{+ 1)}( t ) 5 ^{( n - k} ) ( t ) Ф ^{( 1 )}(0) r_{1 +} k = 0 i = 0 1 = 0                                             k = 1 i = 1 1 = 0

+Z Zn-1- icncR(i+1)( t) 5(n-k)(t )Ф(1)(0) r = Z ZnZ icncR(i)(t) 5(n-k)(t )Ф (1)(0) r + k=0i=1 1=0                                               k=0i=1 1=0

n - 1 n - 1 - i                                      n n - 1

+ Z Z c n cR ⁽ i ) ( t ) 5 ( t ) Ф ^{( 1 )}(0) r + ZZ cR ( t ) 5 ( t ) Ф ^{( 1 )}(0) r = i = 1 1 = 0                                       k = 0 1 = 0

n k n - 1 - i                                               n n - 1 - i

= ZZ Z c n + 1 cR ⁽ i ) ( t ) 5 ^{( n - k} ) ( t ) Ф ^{( 1 )}(0) r 1 + Z Z c n cR ⁽ i ) ( t ) 5 ( t ) Ф ^{( 1 )}(0) г + k = 0 i = 1 1 = 0                                                i = 1 1 = 0

n + 1 n - 1                                n n - 1 - i

+ ZZ cr ( t ) 5 ( t ) Ф ^{( 1 )}(0) r - Z Z c n ^{- 1} cR ⁽ i ) ( t ) 5 ( t ) Ф ^{( 1 )}(0) r = k = 1 1 = 0                               i = 1 1 = 0

n + 1 k n - 1 - i                                              n n - i

= ZZZ c n _{+ 1} cr ⁽ i ) ( t ) 5 ^{( n - k} ) ( t ) Ф ⁽¹⁾ (0) r - ZZ c n cR ⁽ i ) ( t ) 5 ( t ) Ф ⁽¹⁾ (0) r .         (18)

k = 0 i = 0 1 = 0                                            i = 0 1 = 0

Differentiating (15), from (17) and (18) we obtain the equality <7n+1i+1(t) = <7"i+1(t). Thus, the sequence c[0i converges to the function 0 uniformly on [0,T] when i ^^, and moreover cj'Oi = <71 i. Si milarly prove equality <7 n+1( t) = <7" (t), n = 1,..., p, consequently <70( t) = q е Сp+2([0, T ]; Y) and hence q(n) = <7n, 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 o( t ) = ( I - Q ) x ( t ), M o( t ) = ( I - Q ) M ( t ).

Lemma 3. Let the penci1 B be po1ynomia11y A-bounded, condition (A) be fu1fi11ed and ^ be a po1e of order p е Nof the A-reso1vent of penci1 IB. Let x ₀ е c^{p + 2}([0, T ]; F 0 ), M ₀ е c^{p + 2}([0, T ]; F 0 ) and

p initia1 conditions wk е U0 satisfy wk =-Z Kq(.B0)

q = 0

d^q + k

¹ dt q +r ( X o^{( t} ) q ^{( t} ) ^{+ M} o^{( t} ) )| _t = ₀, ^k = ^0,1. Then^there

exists a unique so1ution wе c2 ([0, T]; U0) of prob1em (8), (9), which can be represented as pq w (t) = - Z Kq (B00 )-1 -- ( Xo (t) q (t) + M 0 (t)).

q=0 dt p            dq+k

Proof. Denote Xo(t)q(t)+ M0(t) = f0(t), in this case wk =-^Kq(B00)-1-—f 0(G),k = 0,1. q        q+k+k q =0            dt

Therefore all the conditions of Lemma 2.7.2 [2] hold. Thus there exists a unique solution pq

w ( t ) = - £ K q ²( B °F¹ —( f ⁰( t ) ) of (8), (9). q = 0 dt

Theorem 3. Let the pencil B = ( B ₀, B 1 ) 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, X 0 e C^{p + 2}([0, T ]; F 0 ), M 0 e C p ^{+ 2}([0, T ]; F 0 ), Te C p ^{+ 2}([0, T ]; Y ), for all t e [0, T ] operator C % (t ) be invertible, ( С х ) ^{- 1} e C p ^{+ 2}([0, T ]; L ( Y )) and condition Cv ₀ = T (0) hold. Then there exists a unique solution ( v , q ) of inverse problem (f)-(3), such that v = u + w, where u e C^{p + 2}([0, T ]; U 1 ) and q e C^{p + 2}([0, T ]; Y ) are defined in Theorem 2, w e C ²([0, T ]; U 0 ) 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 C^{p + 2}([0, T ]; U 1 ) , q e C^{p + 2}([0, T ]; Y ) are a solution of the regular inverse problem and the function w e C ²([0, T ]; U 0 ) is a solution of the singular problem. Thus there exists a unique solution of problem (1)–(3).