Inverse Problems for Kinetic and Other Evolution Equations [Reprint 2014 ed.] 9783110940909, 9783110363975

Table of contents :
Chapter 1. Formulas for solutions and coefficients of kinetic and other equations
1.1. Kinetic equations
1.2. Several formulas for solutions and coefficients of kinetic equations
1.3. Formulas in the inverse problems for kinetic equations with a potential
1.4. Formulas in tomography problems
1.5. Formulas of inverse problems for kinetic equation and integral geometry involving integration along geodesics
1.6. Differential and functional equations of inverse problems for nonlinear equations
Chapter 2. Theorems of uniqueness for inverse problems for kinetic equations
2.1. Inverse problem for a system of kinetic equations
2.2. Inverse problems for a system of quantum kinetic equations
2.3. On uniqueness of determination of a form by its integrals along geodesics
2.4. Dynamical model of the ethnic system. Formulas in direct and inverse problems
Chapter 3. Spherical harmonic method and inverse problem for kinetic equations
3.1. Spherical harmonics method
3.2. Steady-state transfer equation
3.3. Determining the dispersion index in the case of the P1-approximation
3.4. Definition of the dispersion index in the case of the P2-approximation
3.5. Reconstruction of the dispersion index and the source function
Chapter 4. Inverse problems for evolution equations of determining two coefficients
4.1. Nonlocal boundary-value problems for nonlinear equations and inverse problems of determining two coefficients
4.2. Recurrent formulas on derivatives of solutions
4.3. Integrodifferential equations in inverse problems of determining two coefficients for evolution equations
4.4. Inverse problem for a system of Maxwell equations
4.5. Determining two unknown coefficients of the parabolic-type equation
4.6. Inhomogeneous conditions of overdetermination
4.7. Representation of solutions and coefficients of partial differential equations of the second order
Chapter 5. Some results of multidimensional inverse problems theory
5.1. Formulas for coefficients in inverse problems for general evolutionary equations
5.2. Formulas in inverse problems for difference-differential equations
5.3. Inverse problem for evolutionary equations with degeneration and others
5.4. Group analysis and formulas in inverse problems of mathematical physics
5.5. Uniqueness of the solution of an integral equation of the first kind over real algebras with division of the finite dimension
5.6. Methods of geometry in the inverse seismic problem
5.7. Problems associated with projections of convex bodies onto planes
Bibliography

Citation preview

INVERSE A N D III-POSED PROBLEMS SERIES

Inverse Problems for Kinetic and other Evolution Equations

Also available in the Inverse and Ill-Posed Problems Series: Inverse Problems of Wave Processes AS. Blagoveshchenskii Uniqueness Problems for Degenerating Equations and Nonclassical Problems S.P. Shishatskii, A. Asanov and LR. Atamanov Uniqueness Questions in Reconstruction of Multidimensional Tomography-Type Projection Data V.P. Golubyatnikov Monte Carlo Method for Solving Inverse Problems of Radiation Transfer V.S.Antyufeev Introduction to the Theory of Inverse Problems A. L Bukhgeim Identification Problems of Wave Phenomena - Theory and Numerics S.I. Kabanikhin and A. Lorenzi Inverse Problems of Electromagnetic Geophysical Fields P.S. Martyshko Composite Type Equations and Inverse Problems A.I. Kozhanov Inverse Problems ofVibrational Spectroscopy A.G.Yagola, I.V. Kochikov, GM. Kuramshino andYuA Pentin Elements of the Theory of Inverse Problems A.M. Denisov Volterra Equations and Inverse Problems A.L Bughgeim Small Parameter Method in Multidimensional Inverse Problems AS. Barashkov Regularizaron, Uniqueness and Existence ofVolterra Equations of the First Kind A. Asanov Methods for Solution of Nonlinear Operator Equations V.P.Tanana Inverse and Ill-Posed Sources Problems Yu.LAnikonov, BA Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P.Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A.Asanov and LR. Atamanov Formulas in Inverse and Ill-Posed Problems Yu.LAnikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.LAnikonov Ill-Posed Problems with A Priori Information V.V.Vasin andA.LAgeev Integral Geometry ofTensor Fields VA Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

INVERSE AND ILL-POSED PROBLEMS SERIES

Inverse Problems for Kinetic and other Evolution Equations Yu.E.Anikonov

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Introduction Inverse problems for differential equations are problems of determining, besides the solution, the coefficients of equations, given additional information on the solution. Typical examples of inverse problems are inverse problems of scattering and potential theories, inverse kinematic and dynamic problems of seismic, inverse problems for heat- and radiation-transfer equations, inverse problems of integral geometry, and tomography problems. The difficulties that arise in the theory and applications of multidimensional inverse problems are mainly stipulated by the new mathematical modelling of natural and social phenomena for which there are no exact equations. The problems of identification, control, and image recovery are also directly linked with inverse problems for differential equations. Significant contribution to the theory and applications of inverse and ill-posed problems was made by Herglotz (1905), Radon (1917), Novikov (1938), Tikhonov (1943), Gelfand and Levitan (1951), Kostelyanec and Reshetnyak (1954), John (1955), Berezanskij (1958), Kirillov (1961), Marchuk (1964), Alekseev (1967), Elubaev (1969), Faddeev (1973), Prilepko (1973), Nizhnik (1973), Iskenderov (1975), Tikhonov and Arsenin (1977, 1979), Ivanov, Vasin, and Tanana (1978), Gelfand, Gindikin, and Graev (1980), Helgason (1980), Bukhgeim (1983), Prilepko and Orlovskii (1984), Yakhno (1985), Amirov (1985), Lavrent'ev, Romanov, and Shishatsky (1986), Natterer (1986), Tikhonov, Arsenin, and Timonov (1987), Gelfand and Goncharov (1987), Romanov (1987), Bubnov (1987a,b, 1988), Kabankhin (1988), Anger (1990), Sabatier (1990, 1996), Prilepko and Kostin (1993), Yamamoto and Nakagiri (1994), Savateev (1994), Anikonov, D. S. (1994, 1996, 1997), Puel and Yamomoto (1996), Belov and Kantor (1999), Denisov (1999), Sharafutdinov (2000),

Yu. E. Anikonov. Inverse problems for kinetic equations This monograph is a continuation of the previous author's work (see, for example, Anikonov, 1968, 1978b, 1987, 1991; Anikonov, Bubnov, and Erokhin 1997) and deals with methods of studying multidimensional inverse problems lor kinetic and others evolution equations, and, in particular, for transfer equations. The methods used in the monograph were applied to concrete inverse problems. The author prefers constructive methods of studying multidimensional inverse problems applicable in linear and nonlinear statements. A significant part of the monograph contains formulas and relations for solving inverse problems. The reader can find in the monograph formulas for the solution and coefficients of kinetic equations, differential-difference equations, nonlinear evolution equations, second-order equations and others. The author hopes that the monograph will help mathematicians, engineers, and other specialists dealing with the inverse and ill-posed problems in their work. 'The publication of the monograph was supported by grant Ν 99-01-00607 of the Russian Foundation for Basic Research and by Integration grant Ν 43 of the Siberian Branch of the Russian Academy of Sciences.

Contents

Chapter 1. Formulas for solutions and coefficients of kinetic and other equations 1 1.1. Kinetic equations 1 1.2. Several formulas for solutions and coefficients of kinetic equations 5 1.3. Formulas in the inverse problems for kinetic equations with a potential 9 1.4. Formulas in tomography problems 11 1.5. Formulas of inverse problems for kinetic equation and integral geometry involving integration along geodesies 15 1.6. Differential and functional equations of inverse problems for nonlinear equations 21 Chapter 2. Theorems of uniqueness for inverse problems for kinetic equations 2.1. Inverse problem for a system of kinetic equations 2.2. Inverse problems for a system of quantum kinetic equations . . . 2.3. On uniqueness of determination of a form by its integrals along geodesies 2.4. Dynamical model of the ethnic system. Formulas in direct and inverse problems Chapter 3. Spherical harmonic method and inverse problem for kinetic equations 3.1. Spherical harmonics method 3.2. Steady-state transfer equation

35 35 40 55 59

73 73 79

Yu. E. Anikonov. Inverse problems for kinetic equations 3.3. Determining the dispersion index in the case of the Pi-approximation 3.4. Definition of the dispersion index in the case of the /^-approximation 3.5. Reconstruction of the dispersion index and the source function Chapter 4. Inverse problems for evolution equations of determining two coefficients 4.1. Nonlocal boundary-value problems for nonlinear equations and inverse problems of determining two coefficients . . . . . 4.2. Recurrent formulas on derivatives of solutions 4.3. Integrodifferential equations in inverse problems of determining two coefficients for evolution equations 4.4. Inverse problem for a system of Maxwell equations 4.5. Determining two unknown coefficients of the parabolic-type equation 4.6. Inhomogeneous conditions of overdetermination 4.7. Representation of solutions and coefficients of partial differential equations of the second order

93 101 121

127 127 139 144 152 160 172 184

Chapter 5. Some results of multidimensional inverse problems theory 199 5.1. Formulas for coefficients in inverse problems for general evolutionary equations 199 5.2. Formulas in inverse problems for difference-differential equations . 208 5.3. Inverse problem for evolutionary equations with degeneration and others 212 5.4. Group analysis and formulas in inverse problems of mathematical physics 216 5.5. Uniqueness of the solution of an integral equation of the first kind over real algebras with division of the finite dimension . . . . 237 5.6. Methods of geometry in the inverse seismic problem 244 5.7. Problems associated with projections of convex bodies onto planes 252 Bibliography

267

Chapter 1. Formulas for solutions and coefficients of kinetic and other equations

1.1.

KINETIC EQUATIONS

Kinetic equations describe the continuity of motion of substance and are the basic equations of mathematical physics and natural science. They can be used for quantitative and qualitative description of physical, chemical, biological, social and other processes. They are called "master equations", because mathematical simulation of the evolutionary processes based on the kinetic equations is fruitful and effective. The Liouville equation, the Boltzmann equation, the system of Vlasov's equations, the Fokker - Planck radiation transfer equations, the chain of Bogoljubov's equations, the Vegner kinetic equations for the matrix of density and diffraction are important kinetic equations. Let W = W(x,p, t) be a distribution function of a substance, depending on the point of space χ = ( χ ι , . . . , xn), impulse p= (pi,... ,pn), and time t. The classic kinetic equation describing continuous motion of the substance and conserving the phase volume is the Liouville equation, which allows the form dW

2

Yu. E. Anikonov. Inverse problems for kinetic equations

Here H = H(x,p,t) is the Hamilton function and {H,F} is the Poisson brackets. In the case when H is a matrix of order τη and W is a vectorcolumn, we obtain a system of the Liouville equations dWk , dt

dWs

[dp

kh i

dHskdWs\_n

dx

9x

>

>

The Hamilton function H(x,p,t) H(x,p,t) =

'••"m·

often has the representation ^\ρ\2-Φ(χ,ί),

where $(x,t) is a potential. In this case, the Liouville equation takes the form dW A/ dW + dx dt ^ i + dxi dPj ' ~ If there exist the sources J(x,p,t) and absorption σ(χ,ρ, t), then the Liouville equation transforms into the kinetic equation dW — + {H,W} + aW = J. Accounting of collision processes leads to appearance of the scattering term St F in the kinetic equation: dW —

+ {H, W} + aW = St F + J.

In particular, in the gas theory arises the Boltzmann equation dW A dW „ Trr 3=1

J

where St F = [ 3 1 •/M "

5(ρ,ρι;ρ',ρ!)[^(χ,ρ,ί)νΚ(α:,ρι,ί)

-ψ&ρ',νψ^,ρ'^άρ^ρ'

dp'v

Here the function g{p,pi',p' ,p[) is the probability of collision of two particles. The discrete Boltzmann equation has the form (Petrovskii, 1984) QT

rr

TTl = i Σ (A^F^-A^Fj), j,k,l=1 1

î = 1,2,... , m.

Chapter 1 . F o r m u l a s for k i n e t i c and other equations

3

Here the components of the tensor numbers A = { Α φ are positive and satisfy the conditions of symmetry

*kl ij —

ΛΜ

Alk — ^ij 1

Λ

kl * i j — ^kV

The linear kinetic equation of Fokker - Planck, which describes the motion of a heavy gas, has the form

dW + { H , F } = f [g(p + q , q , t ) W ( x , p + q , t ) - g { p , q , t ) W { x , p , t ) ] d q . dt J R" Here g ( p , q , t ) is the probability of changing of impulse of a heavy particle. Prom this integro-differential equation, using expansion of the function g ( p , q , t ) into the Tailor series, we can obtain the Fokker - Planck differential equation

A j ( p ) = J R Q j 9 Ì P , q , t ) d q , B k j ( p ) = J R " q k Q j 9 { p , q, t) dq. n

The kinetic equation may be used in modeling the ethnic process (Anikonov, 1995): v ^ γ dH y ι-— j ^ V dpj

dW dH dW -— * dxj dxj dPj J

* — —

+

σ

* W = 0.

Here W { x , p , t , y) is the distribution of persons in the phase space of physical ( t , y ) and social ( x , p ) variables; H ( x , p , y ) is the biochemical energy which determines the passionary field; χ is the potential opportunity of a person; ρ is the passionary impulse; t is the time variable; y is the space variable; the asterish denotes convolution with respect to the variables (i, y ) . For strong interaction, the following quantum kinetic equation is valid:

dW • { H , W } + aW = — V f [ $ \ W ( x , p ' , t ) e i y ( p ' ~ p i dp' dy+ St W + J . dt " ( 2 π Γ JR2" Here [Φ] = [ ^ ( x - ^ l 2 h y , t ) —

4

Yu. E. Anikonov. Inverse problems for kinetic equations

da/dpj = 0, then applying the Fourier transform with respect to the variable ρ, we can obtain the differential equation

Let ψ be a wave function, i. e., ih~ = -Αφ + Φ(χ, ί)ψ. dt Then the function W of the form w(x,p,t)

= —L.

J

ρ ( x - \ hy, ήψ(χ

+ Ì hy, t) e^ dy

satisfies the following quantum kinetic equation: aw dt

n

g »

57 =

L

W W ) * " - * * » « ·

Equations of the kinetic type arise in problems of integral geometry. Let Γ(χ,ί,ρ) be a half-line (x + p(r — t), τ), t < τ < oo beginning at the point (z, t) and having the direction p. Assume that oo

/

Χ(χ+ρ{τ

-t),r)

dr,

where X(x,t) is a finite continuously differentiate function. Then the function W(x,p, t) satisfies the kinetic equation dW

dW

.

„

J

If instead of the half-lines V(x,p,t) we take curves that satisfy the Hamilton system with the Hamiltonian H(x,p), then we obtain the Liouville equation for the integral of W(x,p,t) along the curves beginning from the point (x, t) and having the direction p: dW Here W and λ axe sought functions. As additional information, we take the trace of the function W on some submanifold of the variable t) (for example, W0 = W|x„=o or W0 = W|| x | = i).

Chapter

1. Formulas for kinetic and other

5

equations

Direct problems for kinetic equations can be stated as follows: find the distribution function of a quantity at an arbitrary moment of time for given initial and boundary conditions. An inverse problem for kinetic equations is a problem of simultaneous determining the distribution function of a quantity and some functions entering the equations for given additional information. As a rule, the additional information is the trace of the distribution function on some manifolds of variables. 1.2.

SEVERAL FORMULAS FOR SOLUTIONS A N D COEFFICIENTS OF KINETIC EQUATIONS

In this section we will give formulas, obtained by the author (Anikonov, 1995, 1997), for solutions and coefficients of kinetic equations. This formulas contain functions, determining of which in concrete inverse problems for kinetic equations is directly connected with inversion of the Hilbert, Fourier, and other integral transforms. This circumstance and generality of the formulas obtained allow one in some cases efficiently and constructively determine solutions of inverse problems. The results presented have main by a formal character. The sufficient conditions of correctness of the formulas and operations stated below depend on differentiability of determining functions and sufficiently rapid decrease of the functions and their derivatives at infinity. In a number of cases, integrals have to be interpreted in the sense of the main value, and subintegral functions can be generalized. Let us consider the multidimensional nonlinear kinetic equation dw

-Qt +

ττ-\ dw j=1

J

= λ

,

.

(Μ)/(Μζ,ρ,*Μ·

,

U· 2 ·!)

Here xeDcW^eDi C K n , D and D\ are domains in the real Euclidean space R n ; ÍQ < t < t\; f(a,p) ψ 0 is a given differentiate function, a G Κ 1 , ρ G £>i; w(x,p,t) and \(x,t) are sought functions. Our main task is to find new representations of the desired functions w(x,p,t and X(x,t) entering (1.2.1), which include arbitrary functions. Introduce a function Φ(α,ρ) such that Φα(Φ-1(α>Ρ)>Ρ) = / ( α , ρ), where Φ^ = dΦ/dQ;.

Φ(Φ _1 (α,ρ),ρ) = α,

6

Yu. E. Anikonov. Inverse problems for kinetic equations

The function Φ (α, ρ), according to its definition, is the inverse function with respect to the variable y to the primitive

ΙΤΕΪΥ That provides an obvious method of determining the function Φ(α,ρ). For example, for the functions / = 1, / = a, f — 1 + a2, the corresponding functions are Φ = α, Φ = e a , Φ = t a n a . In practice, the function f(w,p), as a rule, is nonlinear. For example, the evolution of a population considering age characteristics can be described by a kinetic equation of the form (1.2.1) including the quadrature nonlinearity. The nonlinearities of other types can also appear. In general, the desired function X(x,t) can depend on the solution w(x,p,t), for example, oo

/ -00 w2(x,p,t)

dp,

or X(x, t) —

M(x,t,w),

where M is a functional independent of p. The formulas on w(x,p,t) and X(x,t) stated below include functions A(a,y), B(z,y). The function B(z,y) defines the general solution of the uniform equation (1.2.1) taking into account nonlinearity, and the function A(a,y) defines λ(α:,ί). Such representations of the functions w(x,p,t) and \(x, t) are intended for solution of inverse problems for kinetic equations (in particular, for tomography and integral geometry problems). Besides, it is often assumed that the solution w(x,p,t) and information (for example, Hx=o — wo{p,t)) are defined by the function A(a,y) only, i.e., are defined in fact by the function λ(χ,ί). Therefore, in inverse problems, the function B(z,y) often equals zero. In general, determining of both the functions A and Β given the information ιυ| Ι= ο = wo{Pi t) o n ly is very difficult, especially when the variable ρ or t is fixed; so, we need additional information. Let us outline once again that, if the information io| x= o = WQ(jp,t) is given (B = 0), the domain of variation of the variables p,t defines, as it is well seen from the formulas stated below, the well-posedness of statements of integral geometry and tomography problems. We recall the definition of functions rapid decreasing at infinity. A function is rapidly decreasing at infinity, i. e. belongs to the class C°°(K m ), if it and all its derivatives decrease more rapidly than in an arbitrary order

Chapter 1. Formulas for kinetic and other equations

7

as |x| -» oo. Belonging of the function A(a,y) to such a class guarantees well-posedness of the formulas stated below. Hereafter we restrict ourselves by this case. Theorem 1.2.1. Let A{a,y), B(z,y), a G R1, y G R n , and ζ G R" be arbitrary differentiable functions and the function Α(α, y) rapidly decrease at infìnity. Then the functions

í d λ(Μ)=/ J Rn cn

^;A{t-{x,y),y)dy, η

xeD,

ρ E Di,

{χ,y) = j=1

satisfy the equation dw

dw j=ι

..

,. .

,,

J

Theorem 1.2.2. Let A(a,y), B(z,y), a G Κ1, y G R1, and ζ G M1 be arbitrary differentiable functions and the function Α(α, y) rapidly decrease at infìnity. Then the functions (X,p,t) = φ([[ K JJ R2

Α(α,ν)Σ k r r i k\p k=0 +

4- i (k-1)

A(x, i) = ^ j j 2 A{aiy)h{2yßcty)j# /R

dady,

where /o(z) = Y^=q{z/2)2k / {k\)2 is the modified Bessel function, satisfy the equation dw dw .. .

8

Yu. E. Anikonov. Inverse problems for kinetic equations

Theorem 1.2.3. Let the functions f{a,y) and A(a,y) be analytic in the neighbourhood of the point of origin, and the function B(a,y) be differentiable. Let also constants c > 0 and to > 0 exist such that, for an arbitrary integer s, the inequality ΡΦ-l(A(p,t),p) dp2

p—0

0 < t < to

< c,

holds true. Then the functions 00 OO / 00 00

( - 1 )mxmpk dm (m + k)\ dtT

m=0 fc=0 +

B{x

ΧΜ

(-l)V (m!)2

Σ

= H

771=0

¡dm+k$~l{A(p,t),p)^ QpTTi+k p=0

-pt,p),pj, dm+1 dtm+l

ΐ

dp"

p=0

satisfy the equation dw

dw

..

.

,

in the neighbourhood of the point of origin. Theorems 1.2.1-1.2.3 can be proved by a direct test. The main information in inverse problems for kinetic equations is often taken as a trace of the solution w(x,p,t) on a prescribed manifold that belongs to the space of the variables (x,t). Consider, for example, the case wo(p,t) = w\Xo. Let also

/ = 1,

B = 0,

which is usual for tomography problems. Then the function A(a, y) obeys one of the following equations: 1 . ^ , 0 = / JR»

2. w0(p,t)=

A{t,y)_ 1

^ dy; - (P, y)

if A(a,y)e^ap+^ J J R2

3. w0(p,t) = A(p,t).

dadt;

Chapter 1. Formulas for kinetic and other equations

9

Prom 1-3 it follows that the problem of determining the function A (a, y ) by given WQ can be reduced to the problems of inversion of integral transforms of the Hilbert or Fourier type, analytic continuation, determining prime integrals, and others. In particular, equality 2 leads to Theorem 1.2.4. Let ρ ζ Μ ' , ί Ε ® 1 , and the function WQ{p, t) — ω|Ι=ο rapidly decrease at infìnity. Then the following formula of inversion holds true:

Taking into account the hypothesis of Theorem 1.2.4 and the properties of the Fourier transform, the function A(a,y) rapidly decreases at infinity; therefore, determining the function X(x,t) by the information w|x=o = WQ {p,t) is correct in Theorem 1.2.2. We remark once again that the considered domain of variation of (p, t), p e t f . i e R 1 allows obtaining an inversion formula with respect to A(a, y), and, hence, with respect to λ(χ,ί). If m = 1, Β = 0, and p e l 1 (that is essential), then the inversion of equalities 1 and 2 leads to the Hilbert transform with respect to the parameter t (in equality 1 we have to set q = 1 /p). We omit the statement of theorems which are analogies to Theorem 1.2.4. In the analytic case and when all derivatives of the information WQ (p, t) are bounded in the neighbourhood of the point (0,0), it is reasonable to use equality 3 and Theorem 1.2.3, since the series are convergent.

1.3.

FORMULAS IN THE INVERSE PROBLEMS FOR KINETIC EQUATIONS WITH A POTENTIAL

In this chapter, we consider the kinetic equation with the nonlinear right side (1.3.1) Here the variables (x,t,p) belong to certain domains of the real axis; the functions μ(χ,ί), λ(χ, t) axe sufficiently smooth and are independent of the variable p; f(z) is a certain differentiable function, z e R 1 , and μ(χ, t) is a potential. The main goal of this section is to find formulas for the functions w(x,t,p), X(x,t) which enter the kinetic equation (1.3.1) and which contain arbitrary functions and functions determined by the data μ(χ, t) and f(z).

10

Yu. E. Anikonov. Inverse problems for kinetic equations

Here we also discuss the use of these formulas for solving direct and inverse problems for kinetic equations. Note that, as in to the other works of the author, we do not exclude the dependence of the functions μ(χ, t) and X(x, t) on the solution w(x,p, t), for example, oo

/ •oo R(x,p,t,w(x,p,t))

dp.

Here R is a certain function. Hence it follows that nonlinearity of Eq. (1.1) may be defined not only by the function f{z). We now introduce the functions Φ(ζ), ζ G M1, a(x,p,t), ß(x,p,t), (x,p,t) G R 3 , A(y,z), B(y,z), and (y, z) G R2 as follows. 1. The function Φ(ζ) is inverse to the function F(z), where F(z) is such that F'{z) = 1 /f{z) ( f ( z ) φ 0). For example, if f{z) = ζ then Φ(^) = e2. 2. The functions a(x,p,t) and ß(x,p,t) integrals) of the equation

are independent solutions (first

du du . . du ä i + ä ^ ^ ä r 0 · For example, if μ = 0, then a = pt — χ, β = p. 3. A(y,z) and B(y,z) are arbitrary differential, sufficiently smooth functions rapidly decreasing as ζ —> oo. The basic formal result is the following theorem (Anikonov, 1998). Theorem 1.3.1 (Anikonov, 1998a). The functions w(x,p,t) defìned by the formulas .0)=0

32

Yu. E. Anikonov. Inverse problems for kinetic equations

(for more details see Poluektov, Punakh, and Shvitov, 1980). These additional relations play am important role in our investigations, since they allow one either to obtain boundary conditions for differential equations or impose significant restrictions on the functions u(x), v(x), φ(ί), and F(y). For example, if for λ we assume that λ(χ,ί,Ο) = X(x,t, 1) = 0 in the formula of Theorem 1.6.7, then for the functions u(x,t) and v(x,t) we obtain a system of nonlinear evolution equations with coefficients and nonlinearities expressed by F and b. And when we determine the functions F and b, the information of concrete inverse problems will lead us to the Abel and Schröder functional equations. Therefore, this additional relations imply a constructive quantitative definition of the solutions w(x,t) and X(x,w). We now consider a one-dimensional case. Let w{x,t) = u(x)F(v{x)

+ φ{ή)

and the boundary values H I = 0 = α ( ύ )> u0 = ula^o,

υ0

= υ|Ι=ο,

H*=i = « i = u|®=i,

V\ = υ| Ι= ι

(1.6.7)

be given. Substituting (1.6.2) into (1.6.1), we have a{t)

=

UqF{VQ

+

(l _

|

d ,dHkl dWt dpi V dxj dpj dx{ J

d /dHkl awldwk\ dpi V dxi dxj dpj J d sdHkldWtdwk\ dxi V dpi dxj dpj J

|

d /dHkl dwtdwk\ dxi V dxj dpj dpi / a ,dHkldwtdwkv] dxi V dpj dxj dpi J \

y d2Hkl dWl dWk y d2Hkl dWl dwk ' ¿ dxidxj f)'r :f)'r •dpi ftri-dpj Άη.· * ^dpidpj dxi dxj .

We integrate this expression over the domain Q. As in Anikonov and Amirov (1983), Anikonov (1992), Bardakov (1995), we can prove that the integrals from all the terms that have the divergent form will vanish. The following relation remains:

40

Yu. E. Anikonov. Inverse problems for kinetic equations f \ y y ( 9 2 d W k dWk JQ '-rrf ^dpidpj dxi dxj Κ—1 l)J — 1

9

2

d

W

dxidxj

dWl dWk + y y (a2Hkl ^ \dpidpi dxi dxj k>l 1,7=1 j j k,l=1

k

dWk

) dpi dpj '

d2Hkl

dxidxj

dw dwk

i )} dpi dpj ) j j

d f d~di =

o

The intergrand may be considered as the quadratic form of the variables fdWi \ dx\ ' dWi Ct dpi

dWi dWrn ' dxn ' ' dx\ ' ' dWi dWm CI ΐ'··5 dpn î · · · ) dpi

dWm dxη dW,m ' r\ JI dpn

with the matrix (Dp ® H ) ® (—D^ H). Transforming this matrix to the canonical form, and taking into account the conditions of the theorem, we obtain that either all the derivatives dWk/dxi, k = 1 , . . . , m, i = 1 , . . . , η, or all the derivatives dW^/dpi, k = 1 , . . . , m, i — 1 , . . . , η, vanish. Taking into account the boundary conditions and the form of the domain Q, we obtain W{x,p,t) = 0 in Q. Finally, from (2.1.1) it follows that B{x,p,t) = 0 in Q. •

2.2.

INVERSE PROBLEMS FOR A SYSTEM OF QUANTUM KINETIC EQUATIONS

Here we investigate an inverse problem of simultaneous determination of the solution and the right-hand side of a system of quantum kinetic equations. Under some restrictions on data and required functions, the uniqueness of the solution is proved. Moreover, the inverse problem of determination of the Hamiltonian matrix from the system of quantum kinetic equations is investigated under the assumption that a family of direct problems has a solution. It have been proved that, if some special matrices dependent on the coefficients of the system are positively definite, then the inverse problem has at most one solution. The results of this section develop the data obtained previously by the author (Anikonov, 1991, 1995a,b). We consider the system of quantum kinetic equations ™

+

- ^

I ,

[*(* - 5 * . β - · < * + s M y c'yy-r}

dp' dy + q(x,p, I)

K W . O (2.2.1)

Chapter 2. Inverse problems for kinetic

equations

41

defined in the domain of the variables (x,p,t), where χ G D C R n , n > 1, D is a domain with a smooth boundary Γ, ρ G R™,α < ί < 6. Here W{x,p, t) = (W\, W2, •••, Wm)* is the quantum distribution vector function and the asterisk * denotes transition to a conjugate matrix, i. e. we take a complex conjugate value in each place and then make transposition of the matrix obtained, in particular, in the case of a real matrix the asterisk * means transposition only; H(x,p) = ||i/¿j|| and Φ (χ,ρ) = ||$¿j|| are m χ m quadratic matrices, which are called the Hamiltonian matrix and the average potentials matrix, respectively; q(x,p, t) = (qi, οd (Οΰ ff\l dxi \dpj Vcto, dxj dxi dpi dp; ) J dxi (2.2.4)

Chapter 2. Inverse problems for kinetic equations

43

We consider system (2.2.1) in the case when the matrix Φ equals zero for (χ,ρ, t) in the domain Ω of the real Euclidean space R 2 n + 1 . Here the domain Ω is the Cartesian product of domains Ω — D χ Di x [a,b], where χ = (χι,..., xn) G D, ρ = ( p i , . . . ,pn) G Di, D and Di are domains in the space W1, t G [α, 6], a, b G R, a < b. By 9Ω we denote the boundary of the domain Ω. Then (2.2.1) takes the form dW — + {H,W}

=

q(x,p,t),

(2.2.5)

where H(x,p, t) = \\Hij\\ is the matrix of order m, q{x,p, t) — (qi,..., qm)*, Hij, Wi, qi, i, j = 1 , . . . , m are the real-valued functions dependent on the variables (x,p,t) in the domain Ω. The right-hand side of (2.2.5) satisfies the equations (2.2.6)

We formulate the inverse problem as follows: to find vector functions W(x,p, t) and q(x,p,t) defined in the domain Ω, which satisfy systems (2.2.5) and (2.2.6) under the assumption that the matrix H(x,p,t) G C2{Vt) and the trace of the solution W(x,p,t) on the boundary 9Ω of the domain Ω are given, i.e. W\dn = W°(x,p,t) for (x,p,t) G díl, where W° = (W^ 0 ,..., W^)* is a given vector function. The following theorem holds. Theorem 2.2.1 (Bardakov, 1999). If the matrix H is symmetric in the domain Ω, the matrices D^®H and —D^ ® H are nonnegative, and at least one of them is positively defìnite, then the inverse problem has at most one solution W(x,p, t) G C 3 (ÏÏ) and q{x,p,t) G (72(Ω). Proof. Let {Wl,ql)i (W2,q2) be two solutions of the inverse problem which satisfy all conditions of the theorem. We denote W = Wl — W2 and q = ql — q2. Then we have dW — + {H,W}

= q,

W\m

= 0.

We act on the fcth equation of system (2.2.7) by the operator η

(2.2.7)

44

Yu. E. Anikonov. Inverse problems for kinetic equations

for each k = 1 , . . . , m and obtain A dwk d2wk J

j=l

J

A - A dwk

d

J

1=1 ]=1

AdWkdqk

i u J

J

]=1

1

J

Combining the equalities, we come to the relation

k>l 3 = f

f

—

dxiJ

k=1j=l

^

dp/

To the expression in the first square brackets, we apply identities (2.2.2) and (2.2.3); and to the expression in the second square brackets, we apply identity (2.2.4). We transform the right-hand side using (2.2.6). As a result, we obtain

¿i

2

dx

¿t

j=1

i

J

dp

j

dx

i ^

J

+ l y Γ d ,dHkk dwkdwk\ 2 \-dxi V dpi dxj dpj J tyj — 1 +

dt

j=1

dp

dp

i

i

J

dt

dx

i

J

d sdHkk dwkawk^ dpi V dxi dxj dpj ) J

l y fd2Hkk dWk dWk _ d2Hkk dWkdWk\l 2 \dpidpj dxi dxj dxidxj dpi dpj J i l,J — ι

+ y Γ y f JLí^ We integrate the expression over the domain Ω. Due to the boundary conditions and the type of the domain Ω, the integrals from all terms that have the divergent form vanish. The expression reduces to dWk dWk f \y y (ef2Hkk JQ Lκ—1 fzri î)J—1 \dpidpj dxi dxj

+2

y

y

(d2Rkl

dwkdwk\

d2Rkk

dxidxj dpi dpj )

υΨι dWk

\dpidpj dxl dxj

kyl—1 îjj—1

d2Rkl

dx ¿ρ dí = o

dWl

dxidxj dp, dpj )\

k>l

(2.2.8)

The integrand may be considered as a quadratic form of the variables

dWi dXl ' " ' dWl dpi ' "

dWi dxn dWi ' 9pn ' "

dWm dxx dwm dpi

dWm dxn dWm dpn

with the matrix (D 2 ® Η) Θ {—D2 H). Bringing the quadratic form to the canonical form and taking into account the conditions of the theorem, we observe that either all derivatives dWk/dxi, k = 1 , . . . , m, i = 1 , . . . , η, or the derivatives dWk/dpi, k = 1,..., m, i = 1 , . . . , η, vanish. Taking into account the boundary conditions and the form of the domain Q, we obtain W(x,p,t) = 0 in Q. Finally, from (2.4) it follows that q(x,p,t) = 0 in Ω. The theorem is proved. • Now we consider system (2.2.1) under the assumptions that the variable ρ = ( p i , . . . ,p„) G R n varies in all the space M.n and the following data are given: W|r = W°(e,p,i),

seT

= dD,

pGRn,

α )dvdidí·

=

+|#

it follows from boundary conditions that the left-hand side of this equality is equal to zero. Therefore, we obtain

m m _

[*k,]

>n

OXj

(PkP.) dy dx di

(^-[^ksi)PkPsdydxdt.

;n \dXj

/

Chapter 2. Inverse problems for kinetic equations

51

Then (2.2.15) takes the form

ΣΣΐί

rb r

r PW*

ff

m

I ^ E w ^ d z d f

Now we will prove the following inequalities: η

m

Σ

Σ

r\

η

y j P k P s ^ k s ] > ο,

j=1

fc,s=l

J

m

]Γ Σ

v i Q k Q s ^ k s ] > o. J

j=1 k,s=1

We fix χ, y, and t and consider the function j=1

fc,í=l

v

Since 5(0) = 0, then S(z) = S'{9z)z, 0 < 0 < 1, and we have

* ) - -i Σ Σ i,j=1

^

. ( fc,s=l

^

^ ^

^

f

f

™J

)

·

Setting here ζ = h, we obtain the equality

¿j=l k,s=l The expression in the left-hand side is the quadratic form of η χ πι variables PkVi, k = 1 , . . . , m , i = 1 ,.,.,η. Due to condition 8), this form is nonnegative, i.e., S(h) > 0; and therefore, the following inequality holds: τι

τη

j=1 k,s=l

r, 3

The second inequality can be established similarly.

52

Vu. E. Anikonov. Inverse problems for kinetic equations

We integrate (2.2.11) by x, p, and t taking into account (2.2.12)—(2.2.16) and obtain t í !

=

^-^(awt&w!,

Aawi

m 1 fb f f / / Vi Σ 2 f ^ h Ja JDJRn

1

n

1 rc ι

/*& /» r

1 f" f f y 2 Ja JD 7r" =

- ^ έ / 6 / j=1 Ja Jd

pi

we use identities (2.2.2)-(2.2.4) to transequality. Due to the boundary condithat have the divergent form vanish (see and we obtain the fundamental identity

Í d2Hkl dWt dWk ^ dpi dpj dxi dxj

y

f

Λ

ß PkPs^ksìdydxdt ra

As in the proof of Theorem 2.2.1, form the right-hand side of the tions, the integrals from all terms Anikonov, 1991; Anikonov, 1992),

a

yjf](PkPs k,s=l

d2Hkl dWt dWk χ ¿ dxl dxj dpi dpj J

+ QkQs)^-[^ks]dydxdt. i

According to condition 3), either the inequalities A

d2Hkl dWt dwk

"

>

Λ Λ

,dwk\2

y y d2Hkl dWt dWk ¿ u ^ d x i d x j dpi dPj

> Q

d2Hkl dWt dWk

> Q

-

or the inequalities y

y

^

(2.2.17)

Chapter 2. Inverse problems for kinetic equations A

d2Hkl

A

dWjdWk

53

>

are correct and, as was proved above, η

Σ

m

y j ( p k P s + Q k Q s ) ^ [ $ k s } > 0.

Σ

J

j=1 Jfc,s=l

Then from (2.2.17) it follows that either dWk/dxi = 0 or dWk/dpi = 0, 1 2 i — 1 , . . . , π, A; = 1 , . . . , τη. Therefore, we have W = W , and due to (2.2.1) it follows that q1 = q2· • Consider now a family of systems of quantum kinetic equations dW

dt

+

JR2n

L J

1 = 1,...,m

Φ

= φ

(

χ _

(2.2.18)

\ h y , t ) ~φ(χ

+

\Ην>ή

defined in the domain of the variables (x,p,t) where χ G D Ç R n , D is a domain with a smooth boundary Γ, ρ G R n , α < t < b. Let D\ Ç R n be a domain with a smooth boundary, Q = D χ D\ and the following data are given: W°(s,p,t)

= (W^Wl...,W^*

s e r ,

p e l " ,

= W |r α • · -,AmY B^(x,p)

= (B[l\...,B^y

)e^'dë (2.4.7)

H{x,p,y)=

f (R(P,Ô)e[iï yRn+i

άξ.

(2.4.8)

We obtain now the form of the functions /(ζ,ξ), g(z,ξ), from (2.4.6), (2.4.7) assuming that the energy function H(x,p,y) is

h(z,()

H(x,p, y) = ä(y)p2k + ß(y)x2t,

k,l Ε Ν.

T h e o r e m 2.4.2. Let W0(p,y) = W\x=Q,

Wl(x,y)

= W\p=0

W0(p, ξ) and W\(x, ξ) be the inverse Fourier transformations of the functions Wo(p,y) and Wi(x,£); and οι(ξ), β (ξ) be the inverse Fourier transformations of the functions ά(ξ) and β (ξ), respectively. Then, for the functions f , g,

66

Yu. E. Anikonov. Inverse problems for kinetic equations

and h, from (2.4.6)-(2.4.8), the following representation

r

ι

d,

holds:

,

- » ( s ) * © " "

Γ

:

l n

—

—Ζ ' .

dri•

M ' P r o o f . We have W = fWÔp2k

+ ζ

+ β(ξ)χ21,ξ)

h(ß(Ö(x21

e x p ( ζ 9 W Ö ( p 2 k - η2k) + β{ξ)χ2\ξ)

- η21) + α(ξ)ρ2\ ξ) άη)

Wo = / ( a ( ë ) p 2 f c , Ö exp ( J* g(a(Ô(p2k

-

V2kU)

άη).

Then Wo(p,ÖWo(-p,Ö Setting ζ = α(ξ)ρ2Ιί,

we obtain

=

f2(a(Öp2k,Ö.

άη

Chapter

2.

Inverse

problems

for kinetic

equations

Further, we have Wo(p,0 Wo ( - P , Ö

W 0 ( P

' °

or Jo

¿

Wo(-p,0

We set ζ - a(£)fa 2 * - rç2fc). Then άη =

dz

— ο,

fP 9(aiÖ(p2k Jo

~ i?2fc)iÖ

/ λ /

dr/ =

[ Jo

2k

z

λ ^ -

1

) / ^

a(t)p2k

g(z,Ödz 2k

z

So, we have Wo(p,Ö

=

(~P,0 W0(-P,f)

rr W > Jo Jo

g(z,j)dz

Ζ 2

λ

M

(2fc-l)/2* •

(I)-

/

Q

\

We set