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An abstract version of the BC method is proposed as a chapter of the linear system theory dealing with dynamical systems with boundary control (DSBCs). A characterization of the response operator of DSBCs is given; a set of models (realizations) of DSBCs determined by the response operator is presented. As an application, a conditional existence theorem characterizing the dynamical Dirichlet-to-Neumann map of the Riemannian manifold is obtained. An abstract analogue of the Gelfand-Levitan-Krein-Marchenko equations is derived.

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... The inverse problem consists of determining the function ρ in Ω via the response operator R 2T provided T > T * . One of the natural ways to solve this problem is the Boundary Control method (BC-method, Belishev, 1986, see, e.g., [2][3][4] and works cited therein, and the version of the BC-method proposed in [12,13]). We do not give a BC-solution of the inverse problem in this paper. ...

... In contrast to the works cited above, we use measurements (waves) at the same part of the boundary as controls. This corresponds to the boundary triple technique in [2]. The boundary triple used in the present paper is associated with the Zaremba Laplacian with mixed boundary conditions studied in [9]. ...

... Different versions of dynamical systems with boundary controls are related (see [2]) to different choices of boundary triples for the operator in the space domain, see definitions in [6,8]. Here, we introduce the boundary triple corresponding to system (1.1)-(1.4). ...

We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω × (0; 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ Ω [0; T]; Σ ⊂ 𝜕Ω of the lateral surface of the cylinder Ω × (0; T). The domain of observation is Σ × [0; 2T]; and the pressure on another part (𝜕ΩnΣ) × [0; 2T]) is assumed to be zero for any control. We prove the approximate boundary controllability for functions from the subspace V ⊂ H¹(Ω) whose traces have vanished on Σ provided that the observation time is 2T more than two acoustic radii of the domain Ω. We give an explicit procedure for solving Boundary Control Problem (BCP) for smooth harmonic functions from V (i.e., we are looking for a boundary control f which generates a wave uf such that uf (., T) approximates any prescribed harmonic function from V ). Moreover, using the Friedrichs–Poincaré inequality, we obtain a conditional estimate for this BCP. Note that, for solving BCP for these harmonic functions, we do not need the knowledge of the speed of sound.

... вполне определяется частью многообразия Ω T , что и делает его атрибутом системы α T . 3 Он имеет абстрактный аналог, не предполагающий гиперболичности системы [19]. ...

... 3 Как нетрудно видеть, в силу конечности области влияния оператор R 2T * совпадает с (нерасширенным) оператором реакции R 2T системы α 2T . Однако последняя в наших рассмотрениях никак не участвует и методически правильно эти операторы различать [19]. Также отметим простой факт, следующий из определений и соотношений (1.13): оператор R 2T * определяет операторы R s при всех s 2T . ...

... Добавим в заключение, что оператор R 2T * выражается через операторы R T и C T явной формулой [19]. ...

The BC-method provides one of the approaches to inverse problems of mathematical physics. A characteristic feature of this method is the great variety of interdisciplinary relations involved: in addition to partial differential equations as a source of problems, use is made of control theory and systems theory, asymptotic methods, functional analysis, operator theory, Banach algebras, and so on. The purpose of this paper is to present the principal ideas and tools of the BC-method and to give a survey of some results. One of the main achievements of the method is chosen for presentation: The reconstruction of Riemannian manifolds from dynamical and spectral boundary data.

... вполне определяется частью многообразия Ω T , что и делает его атрибутом системы α T . 3 Он имеет абстрактный аналог, не предполагающий гиперболичности системы [19]. ...

... 3 Как нетрудно видеть, в силу конечности области влияния оператор R 2T * совпадает с (нерасширенным) оператором реакции R 2T системы α 2T . Однако последняя в наших рассмотрениях никак не участвует и методически правильно эти операторы различать [19]. Также отметим простой факт, следующий из определений и соотношений (1.13): оператор R 2T * определяет операторы R s при всех s 2T . ...

... Добавим в заключение, что оператор R 2T * выражается через операторы R T и C T явной формулой [19]. ...

BC-метод это один из подходов к обратным задачам математической физики. Его отличительная особенность - разнообразные междисциплинарные связи: помимо уравнений в частных производных, поставляющих задачи, используются теория управления и систем, асимптотические методы, функциональный анализ, теория операторов, банаховы алгебры и др. Цель работы - представить основные идеи и инструменты BC-метода и дать обзор некоторых результатов. Для презентации выбрано одно из главных его достижений - реконструкция римановых многообразий по динамическим и спектральным граничным данным. Библиография: 108 названий.

... • We develop a general approach proposed in [2] and apply it to a concrete time-domain inverse problem for the wave equation with a potential. The approach elaborates the well-known and deep relations between inverse problems and triangular factorization of operators in the Hilbert space [13,1,2,9]. ...

... By the latter, this operator must be regarded as an intrinsic object of system α T (but not α 2T ). Note in addition that R 2T is meaningful at a very general level: see [2]. ...

... Namely, the selection works as follows. ⋆ Conditions 1, 2 appear at very general level of an abstract dynamical system with boundary control (DSBC) associated with a time-independent boundary triple [2]. Such a system necessarily satisfies (4.19) and (4.20). ...

We deal with a dynamical system \begin{align*} & u_{tt}-\Delta u+qu=0 && {\rm in}\,\,\,\Omega \times (0,T)\\ & u\big|_{t=0}=u_t\big|_{t=0}=0 && {\rm in}\,\,\,\overline \Omega\\ & \partial_\nu u = f && {\rm in}\,\,\,\partial\Omega \times [0,T]\,, \end{align*} where $\Omega \subset {\mathbb R}^n$ is a bounded domain, $q \in L_\infty(\Omega)$ a real-valued function, $\nu$ the outward normal to $\partial \Omega$, $u=u^f(x,t)$ a solution. The input/output correspondence is realized by a response operator $R^T: f \mapsto u^f\big|_{\partial\Omega \times [0,T]}$ and its relevant extension by hyperbolicity $R^{2T}$. Ope\-rator $R^{2T}$ is determined by $q\big|_{\Omega^T}$, where $\Omega^T:=\{x \in \Omega\,|\,\,{\rm dist\,}(x,\partial \Omega)<T\}$. The inverse problem is: Given $R^{2T}$ to recover $q$ in $\Omega^T$. We solve this problem by the boundary control method and describe the {\it ne\-ces\-sary and sufficient} conditions on $R^{2T}$, which provide its solvability.

... Proof. See in[20,32]. ...

... constitutes a model. Such a model possesses some interesting properties[20]; however, ˜ F T is not a distributional space. @BULLET Let T > 0 and an open σ ⊂ be fixed, the operator R 2T ...

... d T ) up to isometry. @BULLET Equation (2.39) is one of the BCm-versions of the classical Gelfand–Levitan–Krein– Marchenko equations (see[9,20]). Apropos of this, there is the conjecture that the Faddeev–Newton equations of the inverse scattering problem can also be interpreted in BCm-terms. ...

The review covers the period 1997–2007 of development of the boundary control method, which is an approach to inverse problems based on their relations to control theory (Belishev 1986). The method solves the problems on unknown manifolds: given inverse data of a dynamical system associated with a manifold it recovers the manifold, the operator governing the system and the states of the system defined on the manifold. The main subject of the review is the extension of the boundary control method to the inverse problems of electrodynamics, elasticity theory, impedance tomography, problems on graphs as well as some new relations of the method to functional analysis and topology.

... (6) In general, the passage from X to Ω A(X) can preserve or extend X. 2 In the last case, the set Ω A(X) \ X = ∅ is called a crown of A(X), whereas the G-transform extends functions of A(X) from X to the crown. ...

... All Hilbert spaces and algebras used in the rest of the paper are assumed to be real. The object introduced below can be specified as a version of an abstract dynamic system with boundary control (DSBC) introduced in [2]. Let Γ be a set with measure λ. ...

... Surely, such a description can be hardly regarded to as efficient and checkable, but we are rather sceptical of the possibility to obtain something better. One more version of the characteristic conditions proposed in [2] also uses P ξ mod σ . Then the following question arises: Whether it is possible to characterize the data in terms of C T in itself, not invoking these projections. ...

In the boundary value inverse problems on manifolds, it is required to recover a Riemannian manifold ʊ from its boundary inverse
data (the elliptic or hyperbolic Dirichlet-to-Neumann map, spectral data, etc). We show that for a class of elliptic and hyperbolic
problems the required manifold is identical with the spectrum of a certain algebra determined by the inverse data and, consequently,
to recover the manifold it suffices to represent the corresponding algebra in the relevant canonical form.

... Note that such a local character of dependence of the response operator on the density holds true in the multidimensional case and representation (2.26) has a certain analog (see [7] ). However, the response operator of a multidimensional system is a much more complicated object. ...

... So, our approach provides the unified 'BC-interpretation' of the basic inverse problem theory equations. A main advantage of such an interpretation is that it leads to certain natural multidimensional analogs of GLK [4], [7]. ...

... Proof Recall, that 9 However, a certain conditional characterization is known: see [7] (see section 1.3). By Lemma 4, the wave u f ( · , T ) is continuous onΩ T . ...

This is the first paper of a conceived series under the common title “The boundary control method in inverse problems.” The
aim of the series is to expound systematically an approach to inverse problems based upon its relationship with control theory.
The 1d-variant of the method is shown with the example of the classical problem of recovering the density of an inhomogeneous
string, and both dynamical and spectral statements of the problem are considered. The paper is written in such a way as to
serve as an introduction to the multidimensional BC-method: the basic tools and constructions are amenable to further generalization
to multidimensional problems. Bibliography: 31 titles.

... Motivation comes from a program of constructing a functional model of such operators (the so-called wave model: see [3], [6]- [9]). The given paper develops the results [2,5] on the general properties of DSBC. Perhaps, most curious of new facts is that the finiteness principle of wave propagation speed (for short, FS principle), which is well known and holds in numerous applications, does have a relevant analog for abstract DSBC. ...

... where the summand u f L (t) ∈ Dom L corresponds to the element y ′ ∈ Dom L in decomposition (2). ...

Let $L_0$ be a positive definite operator in a Hilbert space $\mathscr H$ with the defect indexes $n_\pm\geqslant 1$ and let $\{{\rm Ker\,}L^*_0;\Gamma_1,\Gamma_2\}$ be its canonical (by M.I.Vishik) boundary triple. The paper deals with an evolutionary dynamical system of the form \begin{align*} & u_{tt}+{L_0^*} u=0 &&\text{in}\,\,{\mathscr H},\,\,\,t>0;\\ & u\big|_{t=0}=u_t\big|_{t=0}=0 && {\rm in}\,\,{\mathscr H};\\ & \Gamma_1 u=f(t), && t\geqslant 0, \end{align*} where $f$ is a boundary control (a ${\rm Ker\,}L^*_0$-valued function of time), $u=u^f(t)$ is a trajectory. Some of the general properties of such systems are considered. An abstract analog of the finiteness principle of wave propagation speed is revealed.

... The key fact of the BC-method is that the operator R 2T determines the operator C T := (W T ) * W T through an explicit formula [2], [3], [4]. ...

... Recall that the image and control operators I : H → G and W T : F T → H were introduced in items 16 and 22 respectively. The composition V T := IW T : F T → G is called a visualizing operator [2], [3], [4]. ...

With a densely defined symmetric semi-bounded operator of nonzero defect
indexes $L_0$ in a separable Hilbert space ${\cal H}$ we associate a
topological space $\Omega_{L_0}$ ({\it wave spectrum}) constructed from the
reachable sets of a dynamical system governed by the equation
$u_{tt}+(L_0)^*u=0$. Wave spectra of unitary equivalent operators are
homeomorphic.
In inverse problems, one needs to recover a Riemannian manifold $\Omega$ via
dynamical or spectral boundary data. We show that for a generic class of
manifolds, $\Omega$ is isometric to the wave spectrum $\Omega_{L_0}$ of the
minimal Laplacian $L_0=-\Delta|_{C^\infty_0(\Omega\backslash \partial \Omega)}$
acting in ${\cal H}=L_2(\Omega)$, whereas $L_0$ is determined by the inverse
data up to unitary equivalence. Hence, the manifold can be recovered (up to
isometry) by the scheme `data $\Rightarrow L_0 \Rightarrow \Omega_{L_0}
\overset{\rm isom}= \Omega$'.
The wave spectrum is relevant to a wide class of dynamical systems, which
describe the finite speed wave propagation processes. The paper elucidates the
operator background of the boundary control method (Belishev`1986), which is an
approach to inverse problems based on their relations to control theory.

... (0. 2) let u = u f (x) be the solution for a smoothf . With the problem (0.1),(0.2) one associates the DN (Dirichlet-to-Neumann)-map Λ g : f → ∂u f ∂ν | Γ (ν is the outward normal). ...

... • proposes a mechanism (the amplitude formulas of Geometric Optics) recovering the initial system through the model (see [1], [2]). The same is done in this paper: the DN-map determines the trace algebra A(Γ) which is a model of the algebra A(Ω); the Gelfand transform recovers A(Ω) (together with manifold Ω). ...

This paper is a corrected and extended version of the preprint [3]. Our results are the following: ffl we show a relationship between the Calderon problem and Function Algebras and give a new proof of Theorem 1 exploiting this relationship; ffl a simple formula (see (1.6)) linking the DN-map to the Euler characteristic of the manifold is derived

... The details can be found in. 24,25 We denote T ∞ ∶= C ∞ 0 ((0, T), R m ), take , g ∈ T ∞ , and evaluate the quadratic form making use of the equivalent form of (23), ie, tt + = 0, ...

Avdonin and Kurasov proposed a leaf peeling method based on the boundary control to recover a potential for the wave equation on a tree. Avdonin and Nicaise considered a source identification problem for the wave equation on a tree. This paper extends the methodology to the wave equation with unknown potential and source distributed parameters defined on a general tree graph.

... Also, an appropriate analog solves the kinematic inverse problem for a class of two-dimensional manifolds (Pestov 2004). p0205 There exists an abstract version of the approach, embedding the BC method into the framework of linear system theory ( bib0020 Belishev 2001). The method is also related to the problem of triangular factorization of operators (Belishev and Pushnitski 1996). ...

... In the present paper, we address the same question without the assumption about selfajointness of A . The possibility of recovering the spectral data from the dynamical one is well-known for the dynamical system with a boundary control [11,12]. We extend these ideas to the case of the dual (observation) system. ...

We consider applications of the Boundary Control (BC) method to generalized spectral estimation problems and to inverse source problems. We derive the equations of the BC method for these problems and show that the solvability of these equations crucially depends on the controllability properties of the corresponding dynamical system and properties of the corresponding families of exponentials.

... The positivity of the connecting operator C T = (W T ) * W T holding in our case plays the role of a natural analog of the positivity by Krein-Bochner in the case of scalar functions. • The procedure of continuation (i-iii) in fact repeats the scheme of the paper [5] which, in its turn, follows the approach using the models of dynamical systems [3,6]. • The use of the boundary data continuation is a well known device in inverse problems (see, e.g. ...

The boundary control problem for the dynamical Lame system
(isotropic elasticity model) is considered. The continuity of
the “input → state" map in L
2-norms is established. A structure of the
reachable sets for arbitrary T>0 is studied.
In general case, only the first component
of the
complete state
may be controlled, an approximate controllability occurring in
the subdomain filled with the shear (slow) waves.
The controllability results are applied to the problem of the boundary
data continuation. If T
0 exceeds the time needed
for shear waves to fill the entire domain, then the response
operator (“input → output" map)
uniquely determines
RT
for any T>0. A procedure recovering R
∞
via
is also described.

... Besides offering yet another alternative to identification methods based on control and optimization [8,22,23], the BC formalism is entirely linear and appears to be essentially independent on dimensionality. A recent paper by Belishev [12] presents a derivation the Gelfand-Levitan equations for multi-dimensional inverse problems. See particular applications of the BC method in multidimensions for the wave equation [9,11,17,18] and the heat equation [10]. ...

We consider the inverse problem of determining the potential in the one-dimensional Schrödinger equation from dynamical boundary observations, which are the range values of the Neumann-to-Dirichlet map. Dynamical boundary data have not been used in the inverse problem for the Schrödinger equation, since the traditional Gelfand–Levitan–Marchenko approach reconstructs the potential from spectral or scattering data. Here we show that one can completely recover the spectral data from the dynamical boundary data. The construction of the spectral data uses new results on exact and spectral controllability for the Schrödinger equation, which we obtain by using the properties of exponential Riesz bases (nonharmonic Fourier series). From the spectral data, we solve the inverse problem using the boundary control method, which—unlike other identification methods based on control and optimization—is consistently linear and, in principle, independent of dimensionality.

... It is a topological space determined by L 0 and constructed from reachable sets of the DSBC. 0 Introduction 0.1 About the paper We develop ideas and results of the papers [2] and [5]. The future prospect and goal is a functional model of a symmetric semi-bounded operator outlined in [5]. ...

Let $L_0$ be a closed densely defined symmetric semi-bounded operator with
nonzero defect indexes in a separable Hilbert space $\cal H$. It determines a
{\it Green system} $\{{\cal H}, {\cal B}; L_0, \Gamma_1, \Gamma_2\}$, where
${\cal B}$ is a Hilbert space, and $\Gamma_i: {\cal H} \to \cal B$ are the
operators related through the Green formula $$(L_0^*u, v)_{\cal
H}-(u,L_0^*v)_{\cal H}=(\Gamma_1 u, \Gamma_2 v)_{\cal B} - (\Gamma_2 u,
\Gamma_1 v)_{\cal B}.$$ The {\it boundary operators} $\Gamma_i$ are chosen
canonically in the framework of the Vishik theory.
With the Green system one associates a {\it dynamical system with boundary
control} (DSBC) {align*} & u_{tt}+L_0^*u = 0 && {\rm in}\,\,\,{\cal H},
\,\,\,t>0 & u|_{t=0}=u_t|_{t=0}=0 && {\rm in}\,\,\,{\cal H} & \Gamma_1 u = f &&
{\rm in}\,\,\,{\cal B},\,\,\,t \geqslant 0. {align*} We show that this system
is {\it controllable} if and only if the operator $L_0$ is completely
non-self-adjoint.
A version of the notion of a {\it wave spectrum} of $L_0$ is introduced. It
is a topological space determined by $L_0$ and constructed from reachable sets
of the DSBC.

... The central idea of this approach is establishing explicit connection between the metrics of the input and state spaces of a dynamical system. Therefore, unlike other approaches, our method can be straightforwardly extended to infinite-dimensional identification problems (see [5,8]). ...

There exist many methods for solving the spectral estimation problem. This paper proposes a new approach to this problem based
on the Boundary Control method. We show that the problem of decomposition of a signal modeled by a sum of exponentials with
polynomial coefficients can be reduced to an identification problem for a discrete time linear dynamical system. It follows
that values of exponentials can be found solving a generalized eigenvalue problem as in the Matrix Pencil method. We also
give exact formulas for the polynomial amplitudes.
KeywordsSpectral estimation–Signal processing–Boundary Control method–Control theory–Matrix Pencil method

... i.e., C T connects the metrics of the outer and inner spaces. 5 See the remark at the end of sec 2.5 . ...

The paper deals with an approach to inverse problems (IPs) based on relations between the IP and the Boundary Control Theory (so-called BC-method). A vector variant of the method is elaborated for a class of dynamical systems with two different types of waves, which propagate with two different velocities interacting with one another. Controllability of these systems is investigated. The obtained results are applied to IP. In particular a procedure of solving the inverse scattering problem with finite potential is proposed.

... More about the extended (continued) response operator see in[5]. ...

A dynamical Maxwell system is et = curl h, ht = – curl e in Ω × (0, T), e|t=0 = 0, h|t=0 = 0 in Ω, eθ = ƒ in ∂Ω × [0, T], where Ω is a smooth compact oriented 3-dimensional Riemannian manifold with boundary, (·)θ is a tangent component of a vector at the boundary, e = e
ƒ (x, t) and h = h
ƒ (x, t) are the electric and magnetic components of the solution. One associates with this system a response operator RT : ƒ ↦ ν ∧ h
ƒ|∂Ω×(0,T), where ν is an outward normal to ∂Ω.The time-optimal setup of the inverse problem, which is relevant to the finiteness of the wave speed propagation, is as follows: given R
2T to recover the part ΩT ≔ {x ∈ Ω| dist(x, ∂Ω) < T} of the manifold. As was shown by Belishev, Isakov, Pestov, Sharafutdinov (Doklady Mathematics 61: 353–356, 2000), for T small enough the operator R
2T determines ΩT uniquely up to isometry.Here we prove that uniqueness holds for arbitrary T > 0 and provide a procedure that recovers ΩT from R
2T. Our approach is a version of the boundary control method (Belishev, 1986).

... The key fact of the BC-method is that the operator R 2t determines the operator (W t ) * W t through a simple and explicit relation: see [2], [3], [4]. Hence, R 2t determines the modulus |W t |. ...

Let $L_0$ be a closed densely defined symmetric semi-bounded operator with nonzero defect indexes in a separable Hilbert space ${\cal H}$. With $L_0$ we associate a metric space $\Omega_{L_0}$ that is named a {\it wave spectrum} and constructed from trajectories $\{u(t)\}_{t \geq 0}$ of a dynamical system governed by the equation $u_{tt}+(L_0)^*u=0$. The wave spectrum is introduced through a relevant von Neumann operator algebra associated with the system. Wave spectra of unitary equivalent operators are isometric. In inverse problems on {\it unknown} manifolds, one needs to recover a Riemannian manifold $\Omega$ via dynamical or spectral boundary data. We show that for a generic class of manifolds, $\Omega$ is {\it isometric} to the wave spectrum $\Omega_{L_0}$ of the minimal Laplacian $L_0=-\Delta|_{C^\infty_0(\Omega\backslash \partial \Omega)}$ acting in ${\cal H}=L_2(\Omega)$, whereas $L_0$ is determined by the inverse data up to unitary equivalence. By this, one can recover the manifold by the scheme "the data $\Rightarrow L_0 \Rightarrow \Omega_{L_0} \overset{\rm isom}= \Omega$". The wave spectrum is relevant to a wide class of dynamical systems, which describe the finite speed wave propagation processes. The paper elucidates the operator background of the boundary control method (Belishev, 1986) based on relations of inverse problems to system and control theory. Comment: 32 pages

... to recover the metric g up to a function multiplier. 2 In dimension 2 the equalities g u = 0 and g u = 0 are equivalent. 13 ...

As was shown by Lassas and Uhlmann [Ann. Sci. Ecole Norm. Sup. (4), 34 (2001), pp. 771-787], the smooth two-dimensional compact orientable Riemann manifold with the boundary is uniquely determined by its Dirichlet-to-Neumann map (DN-map) up to conformal equivalence. We give a new proof of this fact based on relations between the Calderon problem and function algebras: the manifold is identified with the spectrum of the algebra of holomorphic functions determined by the DN-mapupto isometry; as such, the manifold is recovered from the DN-map by the use of the Gelfand transform. A simple formula linking the DN-map to the Euler characteristic of the manifold is derived.

We present an integral representation for unidirectional solutions of the Helmholtz equation which asymptotically correspond to solutions of the paraxial wave equation

We consider the dynamical inverse problem for two-velocity systems on finite trees in a time-optimal setting: i. e. we assume that the dynamical Dirichlet-to-Neumann map, which we use as inverse data, is known on some finite interval (the length of this interval depends on the optical diameter of a tree). Using the controllability of a dynamical system and ideas of the Boundary Control method, we can extract the spectral data from the dynamical one, and then extend the dynamical inverse data by an explicit formula, provided we understand it in a suitable (generalized) sense. Then we can construct the Titchmarsh-Weyl function and solve the inverse problem using the leaf-peeling method.

The dynamical inverse source problem is considered for an abstract, not self-adjoint operator in the Hilbert space, and an equation of the Boundary Control (BC) method for this problem is derived. It is shown that the solution of this equation crucially depends on the minimality property of a certain exponential family. Applications of this equation to the inverse source problem and to the problem of extension of inverse data are given.

As is known, the boundary spectral data of a compact Riemannian manifold with boundary are determined by its dynamical boundary data (the response operator of the wave equation) corresponding to any time interval: the response operator is represented in the form of a series over spectral data. The converse is true in the following sense: the response operator determines the manifold and, thus, its spectral data. To find these latter, one can reconstruct the manifold and then solve the (direct) boundary spectral problem. Obviously, such a way is not efficient and the question arises of whether one can extract the spectral data from the response operator without solving the inverse problem (without reconstructing the manifold). In the paper, a positive answer is given and a direct time-optimal procedure of extracting the spectral data from the response operator based on a variational principle is proposed. Bibliography: 9 titles.

One of the approaches to inverse problems based upon their relations to boundary control theory (the so-called BC method) is presented. The method gives an efficient way to reconstruct a Riemannian manifold via its response operator (dynamical Dirichlet-to-Neumann map) or spectral data (a spectrum of the Beltrami - Laplace operator and traces of normal derivatives of the eigenfunctions). The approach is applied to the problem of recovering a density, including the case of inverse data given on part of a boundary. The results of the numerical testing are demonstrated.

In this paper we consider realization problems (see [5]–[7]) for operator-valued R-functions acting on a Hilbert space E (dim E < ∞) as linear-fractional trans-formations of the transfer operator-valued functions (characteristic functions) of linear stationary conservative dynamic systems (Brodski ˘ i-Liv˘ sic rigged op-erator colligations). We specialize three subclasses of the class of all realizable operator-valued R-functions [7]. We give complete proofs of direct and inverse realization theorems for each subclass announced in [5], [6].

The present paper centers on second-order hyperbolic equations in the unknownw(t,x):$${w_{tt}} + A(x,\partial )w = f{\text{ in }}\Omega =(0,T]x\Omega $$ (1.1)augmented by initial conditions $$w(0, \cdot ) = {w_0};{\text{ }}{w_t}(0, \cdot ) = {w_1}{\text{ in }}\Omega $$ (1.2) and suitable boundary conditions either of Dirichlet type $$w{|_\Sigma } = u{\text{ in }}\Sigma = (0,T]x\Gamma ,$$ (1.3D) or else of Neumann type $$\frac{{\partial w}}{{\partial {\nu _A}}} = u{\text{ in }}\Sigma $$ (1.3N) where ∂/∂v
A
denotes the corresponding co-normal derivative. Here and throughout, Ω is a general open bounded domain inR
n
, n typically ≥ 2, with boundary ∂Ω = Γ assumed ‘smooth’ (the ‘degree’ of smoothness depending on the ‘degree’ of regularity of the solutions we wish to consider). Moreover,A(x, ∂) denotes a second-order elliptic operator on Ω: $$\left\{ {_{\sum\limits_{i,j}^n {aij} (x)\xi {}_i\xi {}_j \geqslant c\sum\limits_i^n {\xi {}_i^2\;cons\tan t\;c\; > \;0,} }^{A(x,\partial )=\; - \mathop \sum \limits_{i,j\;}^n \frac{n}{{\partial x{}_1}}(aij(x)\frac{\partial }{{\partial x{}_1}}),}} \right.$$ (1.4a)(1.4b) with suitably smooth coefficientsa
ij
(x) =a
ij
(x).

As is known, boundary spectral data of the compact Riemanman manifold Ω (spectrum of the Laplacian with zero Dirichlet boundary condition plus traces of normal derivatives of eigenfunctions at ∂Ω) determine its boundary dynamical data (dynamical Dirichlet-to-Neumann map) R 2T for all T>0. In the paper the procedures recovering spectral data of the submanifold Ω T ={x∈Ω∣dist(x,∂Ω)<T} via given R 2T with any prescribed T>0 and continuing R 2T from ∂Ω×(0,2T) onto ∂Ω×(0,∞) are proposed. The procedures do not invoke solving the inverse problems; main fragment is the constructing (via R 2T ) and the use of a model of a dynamical system associated with Ω T .

Various classes of extensions of symmetric operators with equal (finite or infinite) defect numbers are described in terms of abstract boundary conditions. The dual problem of the description of extensions of a symmetric binary relation is also considered.

The dynamical inverse problem for a class of dynamical systems with two types of waves propagating with different velocities and interacting with one another is considered. A characteristic description of the inverse data (the response function of a system) is obtained. The set of systems that have a given response function is described. Bibliography: 9 titles.

An operator integral, referred to as the amplitude integral (AI) and used in the BC-method (based on boundarycontrol theory) for solving inverse problems, is systematically studied. For a continuous operator and two families of increasing subspaces, the continual analog of the matrix diagonal in the form of an AI is introduced. The convergence of the AI is discussed. An example of an operator with no diagonal is provided. The role of the diagonal in the problem of triangular factorization is elucidated. The well-known result of matrix theory stating the uniqueness of triangular factorization with a prescribed diagonal is extended. It is shown that the corresponding factor can be represented in the AI form. The correspondence between the AI and the classical representation of the triangular factor of an operator that is a sum of the identity and a compact operator is established.

An approach to inverse problems based upon boundary control theory (the BC-method; M. Belishev, 1986) is developed. M. Brodskii's
operator integral is introduced, which works effectively in the inverse problems. It has dynamical nature connected with propagation
of discontinuities of wave fields. The integral is proved to converge for (large) times when the geodesic normal field starting
at the boundary loses its regularity. The operator integral is applied to solving the problem of recovering the potential
in the Schrdinger operator on a Riemannian manifold from its spectral data. Bibliography: 27 titles.

On the problem of extension of the Hermitian positive continuous functions

- M Krein

Krein M G 1940 On the problem of extension of the Hermitian positive continuous functions Dokl. Akad. Nauk
SSSR 26 17–21

- M Livshits

Livshits M S 1966 Operators, Oscillations, Waves (the Open Systems) (Moscow: Nauka)