Boundary Control Method

Belishev, M. I.

doi:10.1007/978-3-540-70529-1_7

Cited by 7 publications

(6 citation statements)

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“…Thus, the A T -image of controls is formed by amplitudes of jumps that arise when projecting the reachable set U T onto 'shortened' sets U τ . The transforms of this type provide one of basic tools for solving inverse problems by the boundary control method [2,3,4]. One more view is as follows.…”

Section: By (39) We Easily Representmentioning

confidence: 99%

“…The Nature laws 1-4 constitute the 'ideological' background of the boundary control method [2,3] that is an approach to inverse problems of mathematical physics, which is based upon their relations to control and system theory.…”

Section: Such a Function Satisfiesmentioning

confidence: 99%

“…These relations describe propagation of singularities of waves u f initiated by singular controls f . Formulas of this type are the main tool for solving inverse problems by the boundary control method [2,3].…”

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confidence: 99%

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A canonical model of the one-dimensional dynamical Dirac system with boundary control

Belishev¹,

Simonov²

2022

EECT

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The one-dimensional Dirac dynamical system <inline-formula><tex-math id="M1">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> is<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} & iu_t+i\sigma_{\!_3}\, u_x+Vu = 0, \, \, \, \, x, t>0;\, \, \, u|_{t = 0} = 0, \, \, x>0;\, \, \, \, u_1|_{x = 0} = f, \, \, t>0, \end{align*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M2">\begin{document}$ \sigma_{\!_3} = \begin{pmatrix}1&0 \\ 0&-1\end{pmatrix} $\end{document}</tex-math></inline-formula> is the Pauli matrix; <inline-formula><tex-math id="M3">\begin{document}$ V = \begin{pmatrix}0&p\\ \bar p&0\end{pmatrix} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ p = p(x) $\end{document}</tex-math></inline-formula> is a potential; <inline-formula><tex-math id="M5">\begin{document}$ u = \begin{pmatrix}u_1^f(x, t) \\ u_2^f(x, t)\end{pmatrix} $\end{document}</tex-math></inline-formula> is the trajectory in <inline-formula><tex-math id="M6">\begin{document}$ \mathscr H = L_2(\mathbb R_+;\mathbb C^2) $\end{document}</tex-math></inline-formula>; <inline-formula><tex-math id="M7">\begin{document}$ f\in\mathscr F = L_2([0, \infty);\mathbb C) $\end{document}</tex-math></inline-formula> is a boundary control. System <inline-formula><tex-math id="M8">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> is not controllable: the total reachable set <inline-formula><tex-math id="M9">\begin{document}$ \mathscr U = {\rm span}_{t>0}\{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $\end{document}</tex-math></inline-formula> is not dense in <inline-formula><tex-math id="M10">\begin{document}$ \mathscr H $\end{document}</tex-math></inline-formula>, but contains a controllable part <inline-formula><tex-math id="M11">\begin{document}$ \Sigma_u $\end{document}</tex-math></inline-formula>. We construct a dynamical system <inline-formula><tex-math id="M12">\begin{document}$ \Sigma_a $\end{document}</tex-math></inline-formula>, which is controllable in <inline-formula><tex-math id="M13">\begin{document}$ L_2(\mathbb R_+;\mathbb C) $\end{document}</tex-math></inline-formula> and connected with <inline-formula><tex-math id="M14">\begin{document}$ \Sigma_u $\end{document}</tex-math></inline-formula> via a unitary transform. The construction is based on geometrical optics relations: trajectories of <inline-formula><tex-math id="M15">\begin{document}$ \Sigma_a $\end{document}</tex-math></inline-formula> are composed of jump amplitudes that arise as a result of projecting in <inline-formula><tex-math id="M16">\begin{document}$ \overline{\mathscr U} $\end{document}</tex-math></inline-formula> onto the reachable sets <inline-formula><tex-math id="M17">\begin{document}$ \mathscr U^t = \{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $\end{document}</tex-math></inline-formula>. System <inline-formula><tex-math id="M18">\begin{document}$ \Sigma_a $\end{document}</tex-math></inline-formula>, which we call the amplitude model of the original <inline-formula><tex-math id="M19">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula>, has the same input/output correspondence as system <inline-formula><tex-math id="M20">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula>. As such, <inline-formula><tex-math id="M21">\begin{document}$ \Sigma_a $\end{document}</tex-math></inline-formula> provides a canonical completely reachable realization of the Dirac system.

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Section: By (39) We Easily Representmentioning

confidence: 99%

Section: Such a Function Satisfiesmentioning

confidence: 99%

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A canonical model of the one-dimensional dynamical Dirac system with boundary control

Belishev¹,

Simonov²

2022

EECT

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“…This result may be interpreted as a point-wise boundary controllability of 2-jets by harmonic functions. Recall that w f is a solution to (7), (8). Proof.…”

Section: Controllability Of 2-jetsmentioning

confidence: 99%

“…Motivation There is an approach to inverse problems of mathematical physics (the socalled BC-method), which was originally based on the relations between inverse problems and the boundary control theory [4,7,9]. The BC-method recovers Riemannian manifolds via spectral and/or dynamical boundary data.…”

Section: Introductionmentioning

confidence: 99%

On algebraic and uniqueness properties of harmonic quaternion fields on 3d manifolds

Belishev

Vakulenko

2019

Cubo

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Let Ω be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair q = {α, u} of a function α and a vector field u on Ω. A field q is harmonic if α, u are continuous in Ω and ∇α = rot u, div u = 0 holds into Ω. The space Q(Ω) of harmonic fields is a subspace of the Banach algebra C (Ω) of continuous quaternion fields with the point-wise multiplica-We prove a Stone-Weierstrass type theorem: the subalgebra ∨Q(Ω) generated by harmonic fields is dense in C (Ω). Some results on 2-jets of harmonic functions and the uniqueness sets of harmonic fields are provided.Key words: 3d quaternion harmonic fields, real uniform Banach algebras, Stone-Weierstrass type theorem on density, uniqueness theorems. MSC: 30F15, 35Qxx, 46Jxx. * Saint-Petersburg Department of the Steklov Mathematical Institute, RAS, beli-shev@pdmi.ras.ru.

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Overview of Inverse Problems

McLaughlin¹

2015

Encyclopedia of Applied and Computational Mathematics

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Boundary Control Method

Cited by 7 publications

References 8 publications

A canonical model of the one-dimensional dynamical Dirac system with boundary control

A canonical model of the one-dimensional dynamical Dirac system with boundary control

On algebraic and uniqueness properties of harmonic quaternion fields on 3d manifolds

Overview of Inverse Problems

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