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doi: 10.3934/eect.2021003

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

 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  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.
Citation: Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, doi: 10.3934/eect.2021003
##### References:
  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.  Google Scholar  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. Google Scholar  M. I. Belishev, Boundary control and tomography of Riemannian manifolds, Russian Mathematical Surveys, 72 (2017), 581-644.  doi: 10.4213/rm9768.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  M. I. Belishev and S. A. Simonov, A wave model of metric space with measure, Sbornik: Mathematics, 211 (2020), 521-538.   Google Scholar  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. Google Scholar  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. Google Scholar  R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, New-York: McGraw-Hill, 1969. Google Scholar

show all references

##### References:
  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.  Google Scholar  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. Google Scholar  M. I. Belishev, Boundary control and tomography of Riemannian manifolds, Russian Mathematical Surveys, 72 (2017), 581-644.  doi: 10.4213/rm9768.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  M. I. Belishev and S. A. Simonov, A wave model of metric space with measure, Sbornik: Mathematics, 211 (2020), 521-538.   Google Scholar  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. Google Scholar  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. Google Scholar  R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, New-York: McGraw-Hill, 1969. Google Scholar
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