Boundary Control Method

• Boundary Control Method is an inverse problem framework that uses boundary excitations, controllability, and operator factorization to reconstruct internal coefficients and geometry.
• It relies on operator-theoretic foundations like the connecting operator and Blagovestchenskii identity to convert interior inner products into boundary data evaluations.
• The method applies to hyperbolic, elliptic, and parabolic problems, offering explicit inversion formulas and numerical stability under strict controllability and regularization conditions.

Searching arXiv for recent and foundational papers on the Boundary Control method to support the article. Boundary Control Method (BC-method, BCM) is an approach to inverse problems based upon deep relations to control and system theory. In the standard hyperbolic setting, one excites a medium from the boundary, measures the boundary response, and uses controllability, duality, and operator factorization to reconstruct interior waves and, from them, the unknown coefficients or geometry. In more recent developments, the same architecture has been used for inverse spectral theory, generalized spectral estimation, inverse source problems, linearized inverse boundary value problems for acoustic equations, explicit time-to-frequency conversion of boundary data, and direct treatments of certain elliptic and parabolic inverse problems, including the Calderón problem (Belishev, 2017, Belishev et al., 13 May 2025, Avdonin et al., 29 May 2025, Kyriakopoulou, 18 Mar 2025, Avdonin et al., 14 May 2025, Yang, 14 Feb 2026).

1. Operator-theoretic core

A standard BCM formulation separates three ingredients: an outer space of boundary controls, an inner space of states at a fixed final time, and a measured boundary response. For the wave equation on a manifold or domain, one introduces a control operator

$W_T f = u^f(T,\cdot),$

mapping boundary data to the wave at time $T$, together with an observation or response operator $R_T$ or $O^T$ obtained from the measured boundary trace. The connecting operator is then

$C^T = (W^T)^*W^T,$

or equivalently $C^T = O^T W^T$ in formulations where the dual observation operator is emphasized. This operator is central because it encodes interior inner products using only boundary measurements (Ivanov et al., 2016, Belishev, 2017, Kyriakopoulou, 18 Mar 2025, Belishev et al., 13 May 2025).

The basic reciprocity statement is a Blagovestchenskii-type identity. In one formulation, for any two boundary controls $f,h$,

$$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$

and in operator form

$W_T^*W_T = J\,R_T - R_T^*\,J,$

with $J$ the time-reversal operator $T$0. In the acoustic setting with control restricted to a screen $T$1, the same principle appears as

$T$2

These identities convert inaccessible interior bilinear forms into boundary pairings and make the inverse problem operator-theoretic rather than purely PDE-local (Kyriakopoulou, 18 Mar 2025, Belishev, 2017, Ivanov et al., 2016).

The same pattern persists in abstract systems. In the generalized spectral-estimation framework, one studies

$T$3

with dual backward system

$T$4

and control operator $T$5. The adjoint relation $T$6 plays the role of boundary reciprocity. This abstract version shows that BCM is not tied to one particular PDE realization, although its most developed inverse results remain concentrated in wave-type problems (Avdonin et al., 14 May 2025).

2. Hyperbolic reconstruction, locality, and geometry

For hyperbolic inverse problems, BCM exploits finite propagation speed. If measurements are made for time $T$7, then one recovers only the region illuminated within travel time $T$8. In the geophysical formulation this region is

$T$9

and the method is explicitly described as time-optimal: measuring for time $R_T$0 allows recovery only to the depth determined by travel-time distance $R_T$1 into the domain (Belishev, 2017, Ivanov et al., 2016).

A key structural input is boundary controllability. In one formulation, the reachable set

$R_T$2

is dense in $R_T$3. In the compact Riemannian setting, exact controllability is tied to the Geometric Control Condition; if $R_T$4, then for any $R_T$5 there exists $R_T$6 such that $R_T$7. This makes it possible to sculpt interior states by boundary actuation and then recover them from boundary measurements through the connecting operator (Belishev, 2017, Yang, 14 Feb 2026).

The constructive side of BCM is often organized through wave bases and amplitude formulas. One orthonormalizes a family of boundary controls with respect to the boundary Gram form $R_T$8, obtaining controls whose terminal waves form an orthonormal basis in the illuminated subdomain. The amplitude formula then identifies the “portrait”

$R_T$9

where $O^T$0 is the point reached along the inward geodesic ray from boundary point $O^T$1 after travel time $O^T$2, and $O^T$3 is the geometric amplitude factor. Taking $O^T$4 recovers ray coordinates, after which the sound speed follows from

$O^T$5

This is the mechanism behind the “see the waves” interpretation in geophysics (Belishev, 2017, Ivanov et al., 2016).

The same geometric logic underlies uniqueness theorems. A prototype BCM theorem states that if two compact Riemannian manifolds with common boundary have the same wave response operator $O^T$6 for some $O^T$7, then they are isometric by an isometry equal to the identity on the boundary. Recent work extends this line to polyhedral settings: “Polyhedral reconstruction via Boundary Control method” studies uniqueness of an elliptic Riemannian polyhedron using an elliptic version of BCM, and its abstract also states interface detection criteria for hyperbolic Riemannian manifolds through the waveguide notion and the four-wave mixing notion (Kyriakopoulou, 18 Mar 2025, Kyriakopoulou, 18 Mar 2025).

Numerically, the acoustic BCM has been tested on the Marmousi model. In that study, the recovered sound speed is an “averaged” profile; in most of the reconstructed tube the relative error stays between $O^T$8–$O^T$9, and the output is proposed as a starting approximation for higher-resolution iterative inversion (Ivanov et al., 2016).

3. Inverse spectral theory and classical integral equations

In one spatial dimension, BCM has a particularly explicit relation to inverse spectral theory. For the half-line Schrödinger operator

$C^T = (W^T)^*W^T,$0

with Dirichlet condition at $C^T = (W^T)^*W^T,$1, the dynamical inverse problem is to reconstruct $C^T = (W^T)^*W^T,$2 on $C^T = (W^T)^*W^T,$3 from the response function $C^T = (W^T)^*W^T,$4 in the boundary operator

$C^T = (W^T)^*W^T,$5

The control operator $C^T = (W^T)^*W^T,$6 is invertible, and the connecting operator

$C^T = (W^T)^*W^T,$7

is boundedly invertible and positive-definite. Conversely, any $C^T = (W^T)^*W^T,$8 for which the corresponding $C^T = (W^T)^*W^T,$9 is positive-definite arises from a unique $C^T = O^T W^T$0. In this formulation, positivity of $C^T = O^T W^T$1 is the necessary and sufficient condition for solvability of the BC inverse problem (Avdonin et al., 29 May 2025).

The local Gelfand–Levitan equation appears as an operator factorization identity. Writing

$C^T = O^T W^T$2

one obtains

$C^T = O^T W^T$3

which yields the kernel equation

$C^T = O^T W^T$4

This is identified explicitly as the local Gelfand–Levitan equation, and the potential is then recovered by

$C^T = O^T W^T$5

The same paper connects the BC framework to Krein’s method, Simon’s $C^T = O^T W^T$6-amplitude, and Remling’s de Branges approach, presenting BCM as a physically motivated route to classical inverse-spectral results (Avdonin et al., 29 May 2025).

A more general unification is given in the derivation of the Gelfand–Levitan, Krein, and Marchenko equations from special boundary control problems. In that viewpoint, each classical integral equation is the equation $C^T = O^T W^T$7 for a target stationary solution of $C^T = O^T W^T$8 or $C^T = O^T W^T$9, where $f,h$0 is constructed from the response data. The distinction between Gelfand–Levitan, Krein, and Marchenko then lies in the choice of boundary actuation, whether one prescribes $f,h$1 or $f,h$2 at final time, and whether the setting is finite-time or scattering (Belishev et al., 13 May 2025).

The spectral content can also be read directly from dynamical data. One result gives

$f,h$3

and

$f,h$4

This makes the passage between response functions and spectral measures explicit inside the BCM formalism (Avdonin et al., 29 May 2025).

4. Linearized BCM, explicit inversion formulas, and time-to-frequency conversion

A major recent direction is linearized BCM for inverse boundary value problems. In the acoustic wave equation with potential,

$f,h$5

one writes $f,h$6, $f,h$7, and linearizes the Neumann-to-Dirichlet map $f,h$8 at the background. The central object is the connecting operator

$f,h$9

with

$$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$0

Blagoveshchenskii’s identity becomes

$$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$1

and its linearization leads to explicit formulas for $$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$2 in terms of boundary integrals only. At $$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$3, the Fourier transform of $$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$4 is recovered by choosing controls that generate plane waves at time $$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$5; the reconstruction is Lipschitz-stable. At $$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$6, high-frequency total-wave solutions yield a limiting formula and Hölder-type stability in $$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$7 (Oksanen et al., 2021).

An analogous program has been carried out for density perturbations in acoustic wave equations. With

$$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$8

the linearized BC identity with free parameter $$\bigl\langle W_T f,\;W_T h\bigr\rangle_{L^2(M)} = \int_0^T\!\!\int_{\partial M} \bigl(f\,R_T h-h\,R_T f\bigr)\,dS\,dt,$$9 takes the form

$W_T^*W_T = J\,R_T - R_T^*\,J,$0

For constant background density, this yields explicit Fourier inversion formulas and a Lipschitz estimate for each Fourier coefficient. For variable background density, complex geometrical optics solutions lead to an increasing-stability estimate in $W_T^*W_T = J\,R_T - R_T^*\,J,$1 (Oksanen et al., 2024).

The damping reconstruction problem has been treated in the same spirit. For

$W_T^*W_T = J\,R_T - R_T^*\,J,$2

one linearizes $W_T^*W_T = J\,R_T - R_T^*\,J,$3 and derives a linearized Blagoveščenskiĭ identity with free complex parameter $W_T^*W_T = J\,R_T - R_T^*\,J,$4. For constant $W_T^*W_T = J\,R_T - R_T^*\,J,$5 and $W_T^*W_T = J\,R_T - R_T^*\,J,$6, choosing $W_T^*W_T = J\,R_T - R_T^*\,J,$7 so that $W_T^*W_T = J\,R_T - R_T^*\,J,$8 produces plane-wave terminal states and an explicit Fourier-inversion formula for $W_T^*W_T = J\,R_T - R_T^*\,J,$9, together with a Lipschitz-stable Fourier-space bound. For non-constant $J$0, CGO solutions yield an increasing-stability estimate in dimension $J$1 (Yang et al., 9 Mar 2026).

A related but distinct extension is the explicit conversion of time-domain wave boundary data to frequency-domain boundary data. On a compact Riemannian manifold, the frequency-domain Neumann-to-Dirichlet map $J$2 at any non-eigenfrequency can be represented in terms of the time-domain map $J$3. The final formula is

$J$4

with

$J$5

All operators in this formula act only on $J$6, and the derivation requires only boundary knowledge. For fixed Tikhonov parameter $J$7, the approximate map $J$8 is locally Lipschitz-stable with respect to perturbations of $J$9. In the reported one-dimensional experiments, the ND-map errors remain $T$00 at $T$01 Gaussian noise, $T$02 at $T$03, and $T$04 at $T$05, with breakdown only when $T$06 approaches a Neumann eigenvalue (Yang, 14 Feb 2026).

These linearized and time-frequency results make BCM explicitly computational. In one-dimensional experiments, the potential, density, and damping papers all implement non-iterative Fourier-type reconstructions from discrete linearized ND data and report numerical feasibility for smooth and discontinuous targets under low-level Gaussian noise (Oksanen et al., 2021, Oksanen et al., 2024, Yang et al., 9 Mar 2026).

5. Elliptic, parabolic, and abstract generalizations

Although BCM originated in hyperbolic inverse problems, recent work extends it directly to elliptic and parabolic boundary value problems without artificial hyperbolization. The proposed mechanism is to replace finite propagation by a spectral parameter or Laplace-transform parameter. For the elliptic family

$T$07

one considers the Dirichlet-to-Neumann map

$T$08

and generalized control/observation operators $T$09, $T$10. An analog of the Blagovestchenskii identity is then formed in each spectral slice, and the family $T$11 is used to encode boundary distance data. The same philosophy is applied to parabolic problems through the Laplace transform in time (Kyriakopoulou, 18 Mar 2025).

This extension is used to formulate a direct BCM approach to the Calderón problem. The conductivity equation is embedded into the elliptic spectral family

$T$12

and the paper states that full knowledge of $T$13 on an open set of $T$14 determines $T$15. It also states that the single map $T$16 suffices once one uses holomorphic continuation in $T$17 together with unique continuation for elliptic PDEs. A logarithmic stability estimate of the form

$T$18

is also stated under smoothness assumptions (Kyriakopoulou, 18 Mar 2025).

Abstract BCM has also been developed for generalized spectral estimation and inverse source problems in complex Hilbert spaces with possibly non-self-adjoint generator $T$19 having isolated eigenvalues of finite algebraic multiplicity. In that setting, BC moment equations are written in terms of shifted controls associated with dual Jordan chains. The top equation is a generalized eigenvalue problem whose eigenvalues reproduce $T$20, and the solvability of the BC equations depends crucially on controllability of the dual system and on minimality, or the Riesz basis property, of the exponential family

$T$21

The same framework reconstructs the initial source $T$22 from output data by recovering coefficients $T$23 through duality (Avdonin et al., 14 May 2025).

The published abstract of “Polyhedral reconstruction via Boundary Control method” adds another extension: uniqueness of an elliptic Riemannian polyhedron via an elliptic BCM, together with interface detection criteria for hyperbolic Riemannian manifolds through the waveguide notion and the four-wave mixing notion (Kyriakopoulou, 18 Mar 2025).

6. Assumptions, limitations, and terminological scope

BCM is constructive, but its strongest forms rest on stringent structural assumptions. Hyperbolic reconstruction typically requires sufficiently large observation time, and exact controllability is tied to a geometric condition. In the time-to-frequency conversion problem, exact controllability follows from the Geometric Control Condition and the requirement $T$24; in linearized acoustic potential recovery, the observation time must be large enough so that geometric control holds, meaning all rays of length $T$25 reach the boundary (Yang, 14 Feb 2026, Oksanen et al., 2021).

Classical geometric reconstructions also assume regular ray structure. In the geophysical presentation, smoothness of the boundary, sound speed, and potential is required to avoid caustics on the time interval $T$26 and to justify geometric-optics expansions. Partial-boundary variants exist, but the practical response operator is typically restricted to a screen and still depends on controlled illumination of the reachable tube (Belishev, 2017, Ivanov et al., 2016).

The linearized inverse problems have additional restrictions. The perturbation must be small, the background must be known, and high-frequency arguments may require CGO constructions or resolvent estimates. The density and damping papers explicitly distinguish constant-background cases, where explicit Fourier inversion is available, from variable-background cases, where one obtains increasing-stability estimates rather than equally direct formulas. The damping paper also notes that high-frequency reconstruction may amplify numerical noise, and the time-to-frequency paper excludes frequencies $T$27 at Neumann eigenvalues (Oksanen et al., 2024, Yang et al., 9 Mar 2026, Yang, 14 Feb 2026).

Inverse spectral versions impose different conditions. In the half-line Schrödinger setting, positivity of the connecting operator is necessary and sufficient for solvability. In the abstract spectral-estimation setting, solvability depends on exact spectral controllability of the dual system and on minimality of families of exponentials in $T$28 (Avdonin et al., 29 May 2025, Avdonin et al., 14 May 2025).

A persistent terminological issue is that “boundary control” is broader than the BC-method in inverse problems. The phrase also appears in control-constrained optimization with bilinear boundary action for semilinear elliptic equations, in exact boundary controllability for generalized wave equations with transmission, in approximate controllability of quantum systems by varying boundary conditions, and in Neumann-type feedback control for wildfire mitigation (Casas et al., 2024, Astudillo et al., 2019, Balmaseda et al., 2018, Belhadjoudja et al., 10 Jun 2025). This suggests a useful distinction: BCM is the inverse-problem framework built around control, observation, and connecting operators, whereas boundary control in the wider PDE-control literature refers more generally to actuation through the boundary.

Within that narrower meaning, BCM occupies a distinctive place because it turns inverse problems into control problems, then solves them by identities that expose interior geometry or coefficients through boundary data alone. The published record now shows this mechanism in hyperbolic, spectral, linearized, elliptic, parabolic, and abstract settings, while also making clear that controllability, positivity, spectral nondegeneracy, and regularization remain the decisive constraints (Belishev, 2017, Belishev et al., 13 May 2025, Avdonin et al., 29 May 2025, Kyriakopoulou, 18 Mar 2025, Avdonin et al., 14 May 2025).