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February  2022, 11(1): 283-300. doi: 10.3934/eect.2021003

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

Mikhail I. Belishev 1,, and  Sergey A. Simonov 1,2,

1. 

St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023 Russia
2. 

St. Petersburg State University, 7–9 Universitetskaya nab., St. Petersburg, 199034 Russia

 

Received  July 2020 Published  February 2022 Early access  January 2021

Fund Project: The first author is supported by the RFBR grant 20-01 627A and Volkswagen Foundation. The second author is supported by the RFBR grant 19-01-00565A

The one-dimensional Dirac dynamical system
$ \Sigma $
 is
$ \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*} $
where
$ \sigma_{\!_3} = \begin{pmatrix}1&0 \\ 0&-1\end{pmatrix} $
 is the Pauli matrix;
$ V = \begin{pmatrix}0&p\\ \bar p&0\end{pmatrix} $
 with
$ p = p(x) $
 is a potential;
$ u = \begin{pmatrix}u_1^f(x, t) \\ u_2^f(x, t)\end{pmatrix} $
 is the trajectory in
$ \mathscr H = L_2(\mathbb R_+;\mathbb C^2) $
;
$ f\in\mathscr F = L_2([0, \infty);\mathbb C) $
 is a boundary control. System
$ \Sigma $
 is not controllable: the total reachable set
$ \mathscr U = {\rm span}_{t>0}\{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $
 is not dense in
$ \mathscr H $
, but contains a controllable part
$ \Sigma_u $
. We construct a dynamical system
$ \Sigma_a $
, which is controllable in
$ L_2(\mathbb R_+;\mathbb C) $
 and connected with
$ \Sigma_u $
 via a unitary transform. The construction is based on geometrical optics relations: trajectories of
$ \Sigma_a $
 are composed of jump amplitudes that arise as a result of projecting in
$ \overline{\mathscr U} $
 onto the reachable sets
$ \mathscr U^t = \{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $
. System
$ \Sigma_a $
, which we call the amplitude model of the original
$ \Sigma $
, has the same input/output correspondence as system
$ \Sigma $
. As such,
$ \Sigma_a $
 provides a canonical completely reachable realization of the Dirac system.
Keywords: 1D dynamical Dirac system, reachable sets, lack of controllability, propagation of singularities, canonical controllable model.
Mathematics Subject Classification: Primary: 35B30, 47N70, 46N20; Secondary: 93B28.
Citation: Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations and Control Theory, 2022, 11 (1) : 283-300. doi: 10.3934/eect.2021003
References:
[1]

M. I. Belishev, A unitary invariant of a semi-bounded operator in reconstruction of manifolds, Journal of Operator Theory, 69 (2013), 299-326.  doi: 10.7900/jot.2010oct22.1925.

[2]

M. I. Belishev, Boundary control method, in Encyclopedia of Applied and Computational Mathematics, (2015), 142–146. doi: 10.1007/978-3-540-70529-1_7.

[3]

M. I. Belishev, Boundary control and tomography of Riemannian manifolds, Russian Mathematical Surveys, 72 (2017), 581-644.  doi: 10.4213/rm9768.

[4]

M. I. Belishev and V. S. Mikhaylov, Inverse problem for one-dimensional dynamical Dirac system (BC-method), Inverse Problems, 30 (2014), 125013, 1–26. doi: 10.1088/0266-5611/30/12/125013.

[5]

M. I. Belishev and S. A. Simonov, Wave model of the Sturm-Liouville operator on the half-line, St. Petersburg Math. J., 29 (2018), 227-248.  doi: 10.1090/spmj/1491.

[6]

M. I. Belishev and S. A. Simonov, A wave model of metric spaces, Functional Analysis and Its Applications, 53 (2019), 79-85.  doi: 10.1134/S0016266319020011.

[7]

M. I. Belishev and S. A. Simonov, A wave model of metric space with measure, Sbornik: Mathematics, 211 (2020), 521-538. 

[8]

M. I. Belishev and S. A. Simonov, On evolutionary first-order dynamical system with boundary control, Zapiski Nauchnykh Seminarov POMI, in Russian, 483 (2019), 41–54.

[9]

I. C. Gohberg and M. G. Krein, Theory and Applications of Volterra Operators in Hilbert Space, Transl. of Monographs No. 24, Amer. Math. Soc, Providence. Rhode Island, 1970.

[10]

R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, New-York: McGraw-Hill, 1969.

show all references

References:
[1]

M. I. Belishev, A unitary invariant of a semi-bounded operator in reconstruction of manifolds, Journal of Operator Theory, 69 (2013), 299-326.  doi: 10.7900/jot.2010oct22.1925.

[2]

M. I. Belishev, Boundary control method, in Encyclopedia of Applied and Computational Mathematics, (2015), 142–146. doi: 10.1007/978-3-540-70529-1_7.

[3]

M. I. Belishev, Boundary control and tomography of Riemannian manifolds, Russian Mathematical Surveys, 72 (2017), 581-644.  doi: 10.4213/rm9768.

[4]

M. I. Belishev and V. S. Mikhaylov, Inverse problem for one-dimensional dynamical Dirac system (BC-method), Inverse Problems, 30 (2014), 125013, 1–26. doi: 10.1088/0266-5611/30/12/125013.

[5]

M. I. Belishev and S. A. Simonov, Wave model of the Sturm-Liouville operator on the half-line, St. Petersburg Math. J., 29 (2018), 227-248.  doi: 10.1090/spmj/1491.

[6]

M. I. Belishev and S. A. Simonov, A wave model of metric spaces, Functional Analysis and Its Applications, 53 (2019), 79-85.  doi: 10.1134/S0016266319020011.

[7]

M. I. Belishev and S. A. Simonov, A wave model of metric space with measure, Sbornik: Mathematics, 211 (2020), 521-538. 

[8]

M. I. Belishev and S. A. Simonov, On evolutionary first-order dynamical system with boundary control, Zapiski Nauchnykh Seminarov POMI, in Russian, 483 (2019), 41–54.

[9]

I. C. Gohberg and M. G. Krein, Theory and Applications of Volterra Operators in Hilbert Space, Transl. of Monographs No. 24, Amer. Math. Soc, Providence. Rhode Island, 1970.

[10]

R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, New-York: McGraw-Hill, 1969.

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